Atomic phenomena in bichromatic laser fields

Physics Reports 345 (2001) 175}264

Atomic phenomena in bichromatic laser "elds Fritz Ehlotzky* Institute for Theoretical Physics, University of Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria Received April 2000; editor: J. Eichler

Contents 1. Introduction 1.1. Laser}atom interactions 1.2. Coherent phase control 1.3. Classical example of CPC 2. Electron}atom scattering in two-colour "elds 2.1. Potential scattering in a bichromatic "eld 2.2. E!ects of target dressing in bichromatic free}free transitions 2.3. Resonance scattering in a bichromatic "eld 3. Above threshold ionization and related topics in bichromatic "elds 3.1. CPC in ATI in the Keldysh}Faisal}Reiss model 3.2. More detailed calculations on CPC in ATI 3.3. Generalizations of the KFR model in bichromatic "elds 3.4. X-ray photoionization in presence of a laser "eld

178 178 178 180 182 182 186 195 197 198 200 207 212

4. Harmonic generation in a bichromatic "eld 4.1. HHG in incommensurate bichromatic "elds 4.2. HHG and CPC in commensurate bichromatic "elds 4.3. X-ray}atom scattering in a bichromatic "eld 5. Experiments on CPC and related phenomena 5.1. Ionization of atoms in two-colour "elds 5.2. HHG and CPC in bichromatic "elds 5.3. Other experiments in bichromatic "elds 6. Other investigations in two-colour "elds 6.1. A potpourri of atomic model calculations 6.2. CPC in solids in two-colour "elds 7. Concluding remarks Acknowledgements Note added in proof References

221 221 226 232 233 233 241 244 250 250 256 258 258 258 258

Abstract We present a review of work that has been done during the last 10 years on atomic scattering and reaction processes in bichromatic laser "elds. Of particular interest will be the case where the "eld is composed of two components of commensurate frequencies, usually consisting of a fundamental component
and one of its low harmonics 2
or 3
. These two components are in general out of phase by an angle
. The above

* Tel.: 0043-512-507-6214; fax: 0043-512-507-2964. E-mail address: [email protected] (F. Ehlotzky). 0370-1573/01/$ - see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 0 0 ) 0 0 1 0 0 - 9

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processes are then investigated as a function of the relative phase
. This procedure was termed the coherent phase control (CPC) of the atomic process considered. The idea was originally born in molecular physics as a possible means to manipulate molecular reactions.  2001 Elsevier Science B.V. All rights reserved. PACS: 32.80.Qk; 34.50.Rk; 32.80.Wr Keywords: Coherent control of atomic and molecular processes; Laser-modi"ed scattering and reactions; Other multiphoton processes

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1. Introduction 1.1. Laser}atom interactions The investigation of atomic processes in powerful laser "elds has become a domain of very active research thanks to the rapid development of laser sources. Reviews of this subject can be found in the books by Mittleman [1], Faisal [2], Gavrila [3] and Delone and Krainov [4]. Three processes turned out to be of particular interest in this area of investigations, namely (i) induced and inverse bremsstrahlung or free}free transitions in a laser "eld, (ii) above threshold ionization and (iii) higher harmonic generation. Below we shall mainly concentrate on these three topics. In the "rst case, the scattering of electrons by an atom in a laser "eld is treated and the laser-induced nonlinear bremsstrahlung processes are investigated. In the second case, laser-induced ionization of an atom is considered and it is found that more photons can be absorbed than the minimum number required for ionization yielding a series of electron energies E"N
#E !; where E is the    atomic binding energy and ; the quiver energy of the electron in the laser "eld. Therefore one  talks about above threshold ionization (ATI). Along with this process goes the generation of higher harmonics of the laser frequency
(HHG). There are, of course, several other processes related to these three main topics of our review and they will also be discussed besides. These investigations of laser}atom interactions revealed a number of hitherto unknown phenomena that are of interest for both, basic insight into atomic, molecular and solid state structures and for their practical applications in a large number of di!erent "elds of research like holography, "ber optics, telecommunication, material sciences, biology, plasma physics, thermonuclear fusion, and so on. The e$ciency of the above processes usually depends very much on the parameters of the laser "eld, like frequency, intensity, polarization, pulse duration, stochastic properties, etc. as well as on the atomic species considered. 1.2. Coherent phase control Controlling the yield of molecular reactions by means of lasers has been a longstanding goal in chemical dynamics. Early laser-based attempts at control relied either on the frequency resolution of lasers to locate a frequency which maximizes the yield, or on the use of high-power lasers to alter the dynamics. Both methods su!er from severe drawbacks; the former depends on the chance existence of a favorable branching ratio and the latter requires extremely high powers which make it impracticable. In addition, both of these methods are passive in the sense that the yield is primarily determined by molecular properties and cannot be controlled by experimental design. About 12 years ago, it was suggested in a series of pioneering papers by Shapiro, Brumer and co-workers [5}7] to actively manipulate chemical reactions by applying a bichromatic laser "eld of frequencies
and 3
and by varying the relative phase
and amplitudes E of the two "elds. The theoretical approach of these authors to actively control the yield of a chemical reaction is based on the following general idea. Suppose that we invoke two simultaneous coherent paths a and b to get to the reaction products. Under these circumstances the probability P of producing products is given by P"A #(A AH#AHA )#A  . ? ? @ ? @ @

(1.1)

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Here A is the probability amplitude of obtaining products through path i, so that A  and G ? A  are the probabilities for independently obtaining products from path a and path b. The second @ term, the crossed term, can be positive or negative and arises from the quantum interference between the two paths. If both paths a and b lead to more than one product channel then the branching ratio R Y for channel q and q is of the form OO R "P /P , OOY O OY

(1.2)

where now in (1.1) the amplitudes are AO and AOY (i"a, b), respectively. To have active control over G G the product distribution means that one can experimentally manipulate the magnitude of the numerator or denominator of (1.2). Thus, if one can design an experimental scenario such that by varying laboratory parameters one varies the sign and magnitude of the cross term, then one gets control over the product distributions and product yields. The above authors termed this overall approach, which relies upon coherence and the use of interference between a minimum of two path, coherent radiative control. Alternative methods, based upon time-dependent wave packet approaches, have been developed by others to which we shall return later. To be more speci"c, we consider a molecule, initially in a state E  of energy E , that is subjected G G to two electric "elds given by E(t)"E cos(
t#k ) R# )#E cos(
t#k ) R# ) .        

(1.3)

Here
"3
and both "elds are taken to have the same directions of linear polarization and   directions of propagation with k "3k . Then, as discussed above, the probability P(E, q; E ) of   G producing a molecular reaction product with energy E along the channel q from a state E  is G given by the probabilities P (E, q; E ) and P (E, q; E ) due to the
and
excitations, plus a cross  G  G   term P (E, q; E ) due to the interference between the two excitations, thus  G P(E, q; E )"P (E, q; E )#P (E, q; E )#P (E, q; E ) . G  G  G  G

(1.4)

Evaluating the corresponding probabilities in the weak radiation "eld limit by time-dependent perturbation theory [7], where in the case of excitation by the
component of (1.3) application of  third-order perturbation theory is required to get to the "nal state E, we only need to consider "rst-order perturbation to arrive at the same "nal state E using the
component of (1.3). Thus,  we get for the branching ratio R for channels q and q OOY FO!2x cos( !3 #O )FO #xE FO       , R "  OOY FOY!2x cos( !3 #OY)FOY#xE FOY       

(1.5)

where a normalized "eld amplitude E and "eld amplitude ratio x have been introduced [7]. The  experimental control over R is therefore obtained by varying the di!erence
" !3 and OOY   the parameter x. The former is the phase di!erence between the
and the
laser "elds and the   latter incorporates the ratio of the two laser amplitudes. Since `triplinga is a common method to produce
from
, of "xed relative phase, the subsequent variation of the phase of one of   these beams provides a straightforward method of altering
. A summary of the method and its application to photodissociation reactions can be found in reviews by Shapiro and Brumer [8,9].

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Early experiments are reported by Lu et al. [10] and Zhu et al. [11] and a more recent summary can be found in the report by Gordon and Rice [12]. A large theoretical and experimental body of work, using the above method of in#uencing molecular properties and reactions, has accumulated by now. In particular, the method has been extended to the use of powerful bichromatic laser "elds thus going beyond the application of perturbation theory. A su$cient amount of references on these works can be found in the papers by Charron et al. [13] on photodissociation, by Thompson et al. [14] and Bandrauk and Chelkowski [15] on dissociative ionization, by Bandrauk et al. [16] on harmonic generation, by Dion et al. [17] on molecular orientation, by McCullough et al. [18] on the coherent control of refractive indices and by Shapiro et al. [19] on the control of asymmetric synthesis with achiral light. Although we shall in our review mainly devote our attention to the application of coherent phase control (CPC), outlined above, to various atomic phenomena, we should also mention here other laser schemes that have been designed recently to tailor atomic and molecular processes. As suggested by Tannor and Rice [20] and improved by Koslo! et al. [21], the selectivity of molecular product formation can be induced by shaping and spacing pump and dump laser pulses. This pump}dump scheme for controlling the selectivity of product formation in a chemical reaction can be improved by developing a method for optimizing the "eld of a particular product with respect to the shapes of the pump and dump pulses. An even simpler scheme involves a series of identical laser pulses. By varying the delay time between two pump pulses with an accuracy better than the vibrational period, and probing the exited state population with a third pulse, Zewail and co-workers have shown that they could control selectively the wave packet motion in the B state of I [22]. More details of this method and its improvements can be found in the works of Blanchet  et al. [23], Kohler et al. [24] and Lozovoy et al. [25] and references cited therein. A comprehensive review on wave packet dynamics and its application to physics and chemistry in femto-time is presented in the papers by Garraway and Suominen [26,27]. Coherent control by single-phaseshaped femtosecond laser pulses is discussed, for example, by Assion et al. [28] and by Meshulach and Silberberg [29]. Finally, in the work of Bardeen et al. [30] feedback quantum control, where the sample `teachesa a computer-controlled arbitrary lightform generator to "nd the optimal light "eld, is experimentally demonstrated for a molecular system. Further details on the above control schemes can be obtained from an article by Manz [31] and an earlier review by Warren et al. [32]. 1.3. Classical example of CPC In order to demonstrate how a laser-induced nonlinear process can be e!ectively controlled by changing the relative phase of the stimulating bichromatic radiation "eld, we consider the following simple classical radiation problem. If a classical high-frequency plane wave E (t)"
cos
(t!n ) r/c) V V V

(1.6)

of unit "eld amplitude, expressed by the vector of linear polarization
, direction of propagation V n and frequency
is scattered by a free electron under the simultaneous action of a strong V bichromatic, low-frequency background "eld E(t)"E
[sin
t#sin(2
t#
)] 

(1.7)

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with
<
and equal "eld amplitude E and linear polarization
, then the phase of the absorbed V  and scattered X-ray "eld will be periodically modulated on account of the large-amplitude electron oscillations in the intense background "eld. This will induce an oscillating Doppler shift of the X-ray emission of the electron and the spectrum of scattered radiation will have the form
 "
#N
with N being a positive or negative integer or zero. Mathematically, this fact will be V V expressed by the retarded electron acceleration e V ( )"
cos
[ !(n!n) ) r( )/c] , V    m V

(1.8)

where r( ) is the amplitude of electron oscillations in the background "eld given by  r( )"!
[sin
 # sin(2
 #
)]     

(1.9)

taken at the average retarded time  .  " c/
is the amplitude of the classical electron   oscillations in the background "eld and is the corresponding velocity amplitude measured in units of the speed of light c. It is explicitly given by "eE /m
c"(I/I . This dimensionless  A parameter can also be used to measure the "eld intensity I of the low-frequency "eld as the ratio "I/I where I "
mc/2r  (r "e/mc"2.82;10\ cm and "/c) is the critical intenA A   sity at which  becomes equal to unity, thus turning the problem into a relativistic one. By inserting (1.9) into (1.8) and by then making a Fourier decomposition of the retarded acceleration into its in"nite number of harmonic components, we can evaluate the di!erential cross sections of X-ray scattering by a free electron and the concomitant nonlinear scattering processes induced by the bichromatic background "eld. We get d V d , " 2 B (a, b;
) . d , d

(1.10)

Here, d V /d is the Thomson cross section of X-ray scattering and B (a, b;
) is a generalized 2 , phase-dependent Bessel function of the integer order N de"ned by > B (a, b;
)"  J (a)J (b)e\ HP , , ,\H H H\

(1.11)

where the J are ordinary Bessel functions of the integer order . The generalized Bessel functions H (1.11) can be obtained by expanding its generating function into a Fourier series, viz. > exp i[a sin
t#b sin(2
t#
)]"  B (a, b;
) exp(iN
t) . , ,\

(1.12)

Here the various di!erent Fourier components of the expansions of the two exponentials on the left-hand side of (1.12) into ordinary Bessel functions J , yielding the same "nal harmonic H frequency N
, may be considered in this classical problem as the in"nite number of di!erent

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phase-dependent `reaction channelsa contributing to the cross sections (1.10). The arguments a and b are given in the present case by a" (K!K ) )
, b"a/4 

(1.13)

with K !K being the `wave vector transfera in X-ray scattering where K"
n and K "
 n, V V while n and n are the directions of propagation of the ingoing and scattered X-ray "elds. We shall see in the next chapter that formula (1.10) agrees very well with the corresponding low-frequency result of electron}atom scattering in a bichromatic low-frequency laser "eld of frequencies
and 2
and a relative phase
. From (1.11) it is quite clear that for a "xed value of a the cross sections (1.10) will strongly depend on the relative phase
of the two-"eld component in (1.7). Concerning notation and conventions used in our work, we should remark that by reproducing formulae and "gures of various authors we usually took over their notations and conventions. Therefore, the number of emitted or absorbed laser photons can be denoted by N, ¸, or n, respectively. Similarly, the phases
can be equivalently written as ,  or  and for the harmonic frequencies the authors may use
"i
(i"1, 2, 3). Finally, in most of the work atomic units (au) G are used but there will also be cases with "c"1.

2. Electron+atom scattering in two-colour 5elds The scattering of electrons by an atomic system in the presence of a laser "eld has been investigated by many authors since the pioneering work of Bunkin of Fedorov [33]. Summaries of these investigations can be found in the book by Mittleman [1], our recent review [34] and the survey of Joachain et al. [35]. In most of the theoretical work presented in this domain, the laser radiation has been treated as a classical radiation "eld with a single frequency
, or some narrow band multi-mode approximation has been employed, yielding better agreement with the experiments by Weingartshofer et al. [36]. Describing a laser beam by a monochromatic classical background "eld relies on the argument that in a laser beam the density of radiation quanta is so large that the depletion of this beam by emitting or absorbing quanta from it is negligible. If the laser frequency
and intensity I are su$ciently low so that the excitations of atomic transitions can be neglected, the atomic target can be described by a short range potential <(r) and the scattering can be treated in the "rst Born-approximation, as was done by Bunkin and Fedorov [33]. We shall therefore "rst consider free}free transitions in a bichromatic "eld for potential scattering and then discuss the in#uence of the laser-dressing of the atomic target. 2.1. Potential scattering in a bichromatic xeld We considered this problem several years ago in a series of papers [37}39]. Using the setup and the corresponding parameter values of the experiments by Weingartshofer et al. [36], our numerical investigations of the laser-induced scattering cross sections predict considerable asymmetries of the spectra of the scattered electrons as a function of the phase
if the frequency combination is
and 2
and we get particular symmetries for
and 3
, which should be easily observable e!ects.

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The scattering of electrons by a short range potential <(r) in a bichromatic laser "eld in dipole approximation is described in "rst order Born approximation (FBA) by the matrix element (using units "c"1)



¹ "!i dt p (t)<(r)p (t) , Y DG

(2.1)

where p (t) and p (t) are well-known Gordon}Volkov plane wave solutions [40,41] of the form Y R 1 p (t)"p exp !i d (p!eA()) , (2.2) 2m

 



where p is a plane wave of momentum p and energy E"p/2m. In (2.1) p and p refer to the initial and "nal electron momenta, respectively, corresponding to the kinetic energies E and E. For the two combinations of commensurate laser frequencies, considered above, we have the vector potentials A(t)"
(E /
) cos
t#
(E /2
) cos(2
t#
) , (2.3)    A(t)"
(E /
) cos
t#
(E /3
) cos(3
t#
) . (2.4)    Inserting these vector potentials into (2.2) and (2.1), respectively, we "nd after Fourier-decomposition, using the generating function exp(ia sin ) of ordinary Bessel functions, that (2.1) decomposes into an incoherent sum of multiphoton transition amplitudes, i.e. ¹ " ¹ , where for the , , DG bichromatic "eld of frequencies
and 2
we get ¹ "!2
ip<(r)pB (a, b ;
)(E!E!N
) , (2.5) , ,  where B (a, b ;
) are generalized Bessel functions of form (1.11) obtained in the foregoing chapter. ,  Here, however, the arguments are a" (p!p) )
, b " (p!p) )
, (2.6)     where the amplitudes  and  are given by  " /
and  " /2
with       "eE /mi
(i"1, 2). Correspondingly, we get for the frequency combination
and 3
G G ¹ "!2
ip<(r)pC (a, b ;
)(E!E!N
) , (2.7) , ,  where the generalized Bessel functions C (a, b ;
) have the form ,  > C (a, b ;
)"  J (a)J (b )e\ HP . (2.8) ,  ,\H H  H\ Here a is the same as in (2.6) but b " (p!p) )
with  " /3
and "eE /m3
. In the        low-frequency approximation p<(r)p is simply the amplitude of elastic electron scattering by the potential <(r) in the FBA. From (2.5) and (2.7) we can "nally evaluate the di!erential scattering cross sections (DSCS) of laser-induced bremsstrahlung processes of the nonlinear order N by absorbing or emitting N laser photons of the bichromatic "eld.

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Using the symmetry property J (z)"(!1)HJ (z) of the Bessel functions of the "rst kind, we \H H can easily verify that functions (1.11) and (2.8) ful"l the following symmetry relations: B (a, b ;
)"(!1),BH (a, b ;
!
) , \,  ,  C (a, b ;
)"(!1),CH (a, b ;
) . ,  \, 

(2.9) (2.10)

As will be shown in the numerical example, the "rst of these relations is responsible for pronounced asymmetries of the electron spectra for particular values of the phase
, whereas in the second case the spectra of scattered electrons for induced (N(0) and inverse (N'0) bremsstrahlung will always be symmetric, but here, for particular
, the central electron peak (N"0) may be strongly suppressed. For the numerical examples, the scattering geometry and parameter values of the experiment by Weingartshofer et al. [36] were used and a CO laser with
"0.17 eV and its second or  third harmonics were considered. The electron energy was taken E"10 eV so that the lowfrequency approximation is valid. Consequently, the relations d /d "B (a, b ;
) and , ,  d /d "C (a, b ;
) can be plotted as a function of
keeping the parameters a, b and , ,   b "xed while the Born cross section of elastic potential scattering, d , represents a normalization 

Fig. 1. Presents four-electron spectra evaluated from d /d "B (a, a/4;
) for (a)
"
/4, (b)
"
/2, (c)
"3
/2 , , and (d)
"
. Observe the asymmetry of the spectra (see Ref. [38]).

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185

factor. Moreover, the electric "eld amplitudes of the bichromatic "elds were considered to be equal such that E"E "E and therefore " /4 and " /9 and all polarizations were taken to be     equal
"
"
. This corresponds to two laser "elds of equal polarization and intensity in which   case we get b "a/4 and b "a/9, respectively. This shows that the use of higher harmonics for the   second "eld component becomes less e$cient for the observation of CPC e!ects, as we have explicitly demonstrated [39], since then the e!ect of the second Bessel function J (b) in (1.11) and H (2.8) has decreasing arguments and therefore becomes gradually unimportant. Finally, for the fundamental CO laser "eld an intensity of I"4;10 W cm\ was taken and the same angular  arrangements chosen as in the work of Weingartshofer et al. [36], yielding for a"3.52. In Figs. 1 and 2 we present d /d "B (a, a/4;
) and d /d "C (a, a/9;
), respectively for , , , , the phases
"
/2,
, 3
/2 and
showing in the "rst case the asymmetry of the spectra with respect to N'0 and N(0 and in the second case the symmetric behaviour of the cross sections with the dominance of certain scattering cross sections for particular values of N and
. As a generalization of the above potential scattering in a bichromatic laser "eld in FBA, we considered the corresponding Kroll}Watson low-frequency approximation [42] using the approach originally introduced by Mittleman [43]. Our considerations led to a rather complicated expression for the corresponding ¹-matrix elements of induced and inverse bremsstrahlung [44]. A more elegant and transparent approach to this problem was suggested by Milos\ evicH [45]. An

Fig. 2. Shows spectra calculated from d /d "C (a, a/9;
) for (a)
"0, (b)
"
/3, (c)
"
/2 and (d)
"
. , , Here the spectra are symmetric and particularly pronounced for certain N (see Ref. [38]).

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interesting relation between classical and quantum mechanical potential scattering in a mono- and bichromatic "eld was discussed by Rabadan et al. [46] and multichromatic Volkov waves were considered by Rosenberg [47]. 2.2. Ewects of target dressing in bichromatic free}free transitions In the foregoing section we considered scattering of electrons by a potential <(r) in a bichromatic "eld. In this case, the atom plays the role of a spectator not actually involved in the laser-induced process. For higher laser "eld intensities and frequencies it will become of importance to also consider the laser interaction with the atomic target. As long as the "eld strength E of the laser radiation is much less than the Coulomb "eld strength E on the Bohr orbit, roughly correspond ing to laser "eld intensities I(10 W cm\, we can use the method of time-dependent perturbation theory to treat the laser}atom interaction. If the laser}atom interaction is su$ciently far o!-resonance of any atomic transitions, the laser dressing will mainly lead to a laser-induced polarization of the target species which can be crudely described by a polarization potential. In this section we shall "rst describe work that has been done on elastic and inelastic electron}atom scattering for hydrogen and helium in a "eld of frequencies
and 2
and we shall then go on to consider electron}atom ionizing collisions, also called (e, 2e) collisions. Finally, we shall discuss some one-dimensional resonant model calculations. 2.2.1. Elastic electron}atom scattering A few years ago, we investigated the simple case of elastic electron}hydrogen scattering in a "eld of frequencies
and 2
[48]. The same problem was treated for electron}helium scattering with essentially the same conclusions by Ghalim and Mastour [49]. We shall therefore give a short outline of our work. We consider scattering of an electron by a hydrogen atom in a bichromatic "eld (2.3) in FBA, neglecting exchange e!ects. The laser-dressed target will be composed of the atomic potential in the absence of the "eld and a polarization term



 (r, t) e  dr . < (r, t)"! #e  r!r r

(2.11)

The dressed ground state wave function   of the hydrogen atom can be easily found in  "rst-order perturbation theory to be





(E !E ) sin
t#i
cos
t    (t)" 0!eE nn
) r0 L  (E !E )!
 L  L (E !E ) sin(2
t#
)#i2
cos(2
t#
)  # L (E !E )!(2
) L 



exp(!iE t) , 

(2.12)

where we assumed a bichromatic "eld of form (2.3) with equal polarizations
"
and "eld  strength E"E "E . The evaluation of (2.11) is then straightforward, in particular, if we use the  

F. Ehlotzky / Physics Reports 345 (2001) 175}264

187

closure approximation which is reasonably reliable for the ground state of hydrogen. This yields





e <(q) exp(iq ) r) dq 1#F(q) < (r, t)"!  (2
)



sin
t sin(2
t#
)  a E # !i   (
) q) 1!(
/EM ) 1!(2
/EM ) e



,

(2.13)

where <(q) is the Fourier transform of the atomic Coulomb potential and F(q) the atomic form factor.  is the static polarizability of hydrogen in its ground state in units of the Bohr radius  a and the energy di!erences E !E have been replaced by some adequate average value EM .  L  The scattering of electrons of about 100 eV kinetic energy and described by corresponding Gordon}Volkov states (2.2) for the "eld (2.3) will yield as in (2.1) and (2.5) the ¹-matrix elements





1  a k 4
e [1!F(Q)]B (a, a/4;
)#   a (B (a, a/4;
) ¹ "i2
, , 1!(
/EM ) ,> 2r 



!B (a, a/4;
) exp(!i
)) ,\

(E!E!N
) ,

(2.14)

where the explicit form of the Fourier transform for the Coulomb potential <(Q)"4
/Q was introduced with Q"p!p being the momentum transfer. B (a, a/4;
) are the generalized Bessel , functions de"ned in (1.11) with a" Q )
. Moreover, we introduced the classical electron radius  r "e/m and the wave number k. Finally, we can evaluate from (2.14) by standard methods the  nonlinear di!erential scattering cross sections with emission or absorbtion of N photons of the laser "eld. Using the work of Byron et al. [50], we have "xed the average transition energy to EM "(8/9)E  where E is the ground state energy of hydrogen.   For the presentation of numerical examples, it was necessary to consider a su$ciently high laser intensity of I"2;10 W cm\ for a Nd : YAG laser with frequency
"1.17 eV. The laser polarization was taken parallel to the ingoing electron momentum p and scattering angles were chosen in the range 044453. In Fig. 3 we show as a representative example the angular dependence of the nonlinear cross sections in units a (a being the Bohr radius) for (a) N"!1,   and (b) N"#1 both with
"0 and for (c) N"!2 and (d) N"#2 with
"
. In all four "gures, the continuous curves indicate the data obtained for a single frequency, the dashed lines refer to neglecting the polarization e!ects and the dotted lines belong to the cross section values evaluated including target dressing. For N"$2, in particular, there is a signi"cant di!erence between the data for scattering in a single "eld and in a bichromatic "eld, but the target dressing e!ects are considerable only for small scattering angles 453. The main conclusions drawn from the more general analysis of that paper were that the dressing e!ects have only a considerable in#uence on the nonlinear cross section data in a bichromatic, phase-dependent "eld under an extreme choice of parameters, namely at very small scattering angles :13 and for rather high laser

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Fig. 3. Presents a\d /d as a function of  for "xed
: (a) N"!1 (
"0); (b) N"#1 (
"0); (c) N"!2 (
"
);  , and (d) N"#2 (
"
). The values of
are those yielding the largest cross sections. Continuous curves refer to the data for a single frequency and target dressing, dashed curves to a bichromatic "eld and dotted curves to a bichromatic "eld and target dressing. Target dressing in a bichromatic "eld becomes prominent at very small , but at larger  than for a single "eld. Observe in the "gures N is denoted by n! (see Ref. [48]).

"eld intensities of about 10 W cm\ such that multiphoton ionization becomes a concomitant process. As was shown by Cionga and Zloh [51,52], for higher laser frequencies of about 4 eV higherorder dressing e!ects have to be taken into account going beyond the closure approximation in hydrogen. In general, the closure approximation can be used for low laser frequencies and small scattering angles. 2.2.2. Inelastic electron}atom scattering The inelastic electron}atom collisions in a bichromatic laser "eld of form (2.3) were investigated by Milos\ evicH et al. [53]. In this work, the excitation of the 2s and 2p states in the bichromatic "eld were considered. As in the elastic case, resonance and exchange e!ects were neglected and the problem was treated within the FBA. Similarly, the interaction of the scattered electron with the

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laser "eld was taken into account exactly, while the interaction of the laser "eld with the hydrogen atom was treated by "rst-order time-dependent perturbation theory (TDPT). The contribution of the target intermediate states to the scattering amplitude can no more be treated by the closure approximation but has to be taken into account using the Sturmian representation of the Coulomb Green's function, developed by Hostler [54] and Maquet [55]. The di!erential cross sections for the inelastic collision processes were then analyzed as a function of the relative phase
between the two laser "eld components and it was shown that the in#uence of the variation of
is more pronounced than for the elastic scattering discussed before. Furthermore, these CPC e!ects appear at lower laser "eld intensities and can also be important for larger scattering angles as we shall show for speci"c examples below. The symmetry properties of the DCS as a function of
were also analyzed and the minima of the DCS as a function of the scattering angle were explained as destructive interferences between the undressed part of the scattering amplitude and that part which corresponds to the dressing of the excited states. In the present work, the following perturbative solution of the dressed atomic states was found (expressed in atomic units):



i  (t)"E exp !i[A(t) ) r#E t] n! [G (E !
) exp(i
t) L L 2  L !G (E #
) exp(!i
t)#G (E !2
) exp(2i
t#i
)  L  L



!G (E #2
) exp(!2i
t!i
)]r )
n .  L

(2.15)

The time-independent Coulomb Green's function G (E), using the closure relation, can be ex pressed in the form G (E)" E!E \, where the summation is over the discrete and J J  continuum states of the hydrogen atom. For the calculation of the scattering amplitudes more useful is the Sturmian representation of the Coulomb Green's function [54,55], S S  LJK LJK G (E)"   1!n(!2E) LJK

(2.16)

since it is given as an expansion over the discrete complete set of the Coulomb Sturmian functions S (r) [56]. LJK Using the same Gordon}Volkov states (2.2), (2.3) as before, we get for the ¹-matrix elements of nonlinear inelastic electron}atom scattering in a bichromatic "eld in FBA i ¹ " f L (E!E !E!E !¸
), f L "f #f #f , L  * ' '' ''' * 2
*

(2.17)

where 2BH f "!n(exp(iQ ) r)!1)0 * , ' Q

(2.18)

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i f "! n exp(iQ ) r)[G (E #
)BH !G (E !
)BH #G (E #2
) exp(!i
)BH ''   *\   *>   *\ Q !G (E !2
) exp(i
)BH ]r )
0 *>  

(2.19)

i f "! nr )
[G (E !
)BH !G (E #
)BH #G (E !2
) exp(!i
)BH '''  L *\  L *>  L *\ Q !G (E #2
) exp(i
)BH ] exp(iQ ) r)0 *>  L

(2.20)

and the generalized Bessel functions B (a, a/4;
) are de"ned, as before, by (1.11). The amplitudes * f and f can be computed using the Sturmian representation (2.16). Approximate solutions can '' ''' be found by introducing average energies EM and EM and by employing the closure relation  L  "1. J As we see, the amplitudes f L have three contributions. The amplitude f is proportional to the * ' generalized Bessel function BH and this is essentially the same result as in potential scattering (2.5). * The amplitude f gives only small contributions to the DCS but the term f turns out to yield '' ''' a strong contribution to the CPC e!ects. If in (2.18)}(2.20) the excited state n is replaced by 0 the results for elastic scattering are recovered. As it turns out, in the present case of inelastic scattering no reasonable values for EM and EM can be found and therefore it is more adequate to use the  L Sturmian representation of the Coulomb Green's function (2.16). Furthermore, it can be shown that in the present case the symmetry relations for the amplitudes (2.18)}(2.20) read f (2
!
, l)"(!1)lf H(
, l) , H H

j"I, II, III ,

(2.21)

where l"0 for the 2s-state and l"1 for the 2p-state. These relations are closely related to the symmetry properties of the generalized Bessel functions B (2.9). Therefore it is su$cient to present * in the numerical examples data for 034
41803. In Fig. 4 we present a characteristic example for the CPC e!ects in the DCS data evaluated at "2/33 scattering angle. The initial electron energy is E"100 eV and its momentum p
. The laser intensity is I"3.5;10 W cm\ and the frequency
"1.17 eV. The phase-dependent data presented in (a) refer to the excitation of the 2s-state during scattering and the data of (b) belong to the excitation of the 2p-state. A few small values of ¸ were chosen (!3 full line, !2 broken line, !1 upper dotted curve, 0 lower dotted curve, #1 circles, #2 squares, #3 triangles). From these results we infer that the CPC, if compared with the case of elastic scattering, is much more pronounced for the inelastic process even for larger values of the emitted or absorbed laser quanta. These e!ects are still further increasing with increasing laser "eld intensity and, on the other hand, they become also visible at lower laser intensities than in the elastic case. 2.2.3. Electron}atom ionizing collisions The electron impact ionization of atoms in the absence of a laser "eld is well described in the textbooks of Mott and Massey [57] and of McDowell and Coleman [58]. These processes, also called (e, 2e)-reactions, are important for the electron momentum spectroscopy of atoms, molecules

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191

Fig. 4. Shows DCS as a function of the relative phase
for the inelastic process with the excitation of (a) the 2s state and (b) 2p state and for di!erent values of ¸ (!3 full curve, !2 broken curve, !1 upper dotted curve, 0 lower dotted curve, 1 circles, 2 squares, 3 triangles). The incident electron kinetic energy is E "100 eV, the scattering angle is "2/33,  the laser electric "eld stength is E"0.01 a.u. (I"3.51;10 W cm\), and the laser photon energy is
"1.17 eV (see Ref. [53]).

and solids. A pioneering experiment of this kind was performed by McCarthy in 1969 [59]. A "rst treatment of electron impact ionization in the presence of a laser "eld was published by Mohan and Chand [60]. This paper was followed by the work of many authors, who gave a more detailed evaluation of this process (see, for example, our review [34] and Mittleman's book [1]). Essentially,

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the expressions for the corresponding triple di!erential cross sections (TDCS) have a structure similar to the cross sections of other laser-assisted atomic collision processes derived in the FBA like in the work of Bunkin and Fedorov [33] or Kroll and Watson [42], i.e. the TDCS are expressed as the product of the square of an ordinary Bessel function and the TDCS for the (e,2e)-reaction without a laser "eld but with laser "eld-dependent momentum and energy shifts. Later on it was recognized by Joachain et al. [61] and others [62}65] that the in#uence of the dressing of the atomic states by a laser "eld cannot be neglected. The work by Milos\ evicH and the present author [66], presented here on the same process in a bichromatic "eld, follows along the same lines as in the work mentioned above. In our investigation of the CPC e!ects in the (e, 2e)-reactions we considered hydrogen only and, moreover, in Ehrhardt's asymmetric coplanar geometry only [67,68]. Ehrhardt's group has studied the situation in which a fast incident electron of momentum p and energy EK250 eV scatters at the target and the outgoing fast electron of momentum p is detected in coincidence with a slow ejected  electron of momentum p . All these momenta p, p and p are taken in the same plane and the  scattering angle  of the fast electron is "xed and small (K33) so that the momentum transfer  Q"p!p is relatively small. The angle  of the slow electron (E K5 eV) is varied. We also  chose, as before, a linearly polarized bichromatic "eld of form (2.3) with
"
and E "E . The    polarization
was taken parallel to p and we considered a laser "eld with
"1.17 eV. The theory of this process is a generalization of the calculations presented in the work of Joachain et al. [61] and Martin et al. [62]. With some caution the closure approximation was applied to perform the summation over the intermediate states which appear as in Section 2.2.2. The evaluation of the matrix elements of the various CPC processes for particular values of ¸ corresponding to the energy conservation relation E #E "E#¸
#E follows similar   lines as in Section 2.2.2 except that now the ionized electron of energy E has to be described by an appropriate continuum wave function which takes into account the Coulomb e!ects of the residual ion on the emitted slow electron. This continuum wave function evaluated in "rst-order timedependent perturbation theory is given by, using the method of Banerji and Mittleman [69], p (r, t)"exp !i[A(t) ) r#p ) (t)#E t] 

 

iE ; p (r)[1#ip ) (t)]#   (r)nr )
p  L 2 L ;

exp(i
t) exp(!i
t) exp(i2
t#i) exp(!i2
t!i) ! # ! E !E #
E !E !
E !E #2
E !E !2
L L L L



. (2.22)

Here (t)" [sin
t#(1/4) sin(2
t#)] and p (r) is the Coulomb wave function  p (r)"(2
)\ exp(
/2p )(1#i/p ) exp(ip ) r) ; F [!i/p , 1,!i(p r#p ) r)]  

(2.23)

with (z) being Euler's function and F a con#uent hypergeometric function. It is important to   remark that in the elastic and inelastic scattering processes, considered before, the A(t) term in the

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Gordon}Volkov states (2.2) can be eliminated. Here, however, it is impossible to drop this term from the wave function (2.22) of the ejected electron, except for su$ciently low laser "eld intensities such that ; /E ;1 where ; "E/4
 is the ponderomotive energy. If E K5 eV and  
"1.17 eV, then ; /E (10% if the laser "eld intensity is less than 4.8;10 W cm\. This puts  a limit on the laser "eld strength to be E(6;10 V m\. The corresponding ¹-matrix elements of the (e, 2e)-reaction can then be evaluated along the same lines as in Section 2.2.2 and we get i ¹ " f  (E #E !E!E !¸
) ,   * 2
*

(2.24)

where f  "f #f #f . The components f ( j"I, II, III) have very similar structure as in the * ' '' ''' H case of inelastic scattering in (2.18)}(2.20) and will not be written down here in their general form. They were approximately evaluated using the closure approximation putting E !E KEM "4/9 L  au and E !Ep "EM "0. In this approximation the amplitudes f read H L 1 f "! p  exp(iQ ) r)  2BH#p )  *   ' Q ;[BH !BH #(exp(!i)BH !exp(i)BH )] , *\ *>  *\ *>

(2.25)

R E f "
) p  exp(iQ ) r)   '' Q RQ



;



BH BH BH exp(!i) BH exp(i) *\ ! *> # *\ ! *> , 4/9!
4/9#
4/9!2
4/9#2


(2.26)

E R f "
) p  exp(iQ ) r)  '' Q RQ 



;



BH BH BH exp(!i) BH exp(i) *\ # *> # *\ # *> .

2
2


(2.27)

One can go beyond the closure approximation by using, as in Section 2.2.2, the Sturmian expansion of the Coulomb Green's functions that appear in f and f but no signi"cant changes are to be '' ''' expected for f since the laser frequency is much smaller than any excitation energy from the '' ground state of the hydrogen atom and the "eld strength E is not too strong. For the amplitude f the Sturmian expansion may not be adequate because for our values of the laser "eld and ''' scattering parameters the energy of the Green's function is positive and the Sturmian expansion may diverge. Finally, with the above made choice the evaluation of the TDCS is straightforward. For the numerical examples we show in the following, we chose E"250 eV, E "5 eV and
"1.17 eV. The laser polarization  was oriented parallel to the ingoing electron momentum p.

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Fig. 5. TDCS are shown as a function of the angle  of the slow ejected electron for di!erent values of the relative phase  between the bichromatic "eld components and for a di!enrent number of photons ¸, absorbed or emitted in the reaction. Continuous curve, "0; dotted curve, "
/2; and dashed curve, "
. (a) ¸"1, (b) ¸"!1, (c) ¸"2 and (d) ¸"!2. The incident electron kinetic energy was E "250 eV, the ejected electron energy was E "5 eV, and the   scattering angle was  "33. The laser electric "eld strength was E"5;10 V m\ and the laser photon energy 
"1.17 eV. The electric "eld polarization
was taken parallel to the incident momentum k (see Ref. [66]).

The electric "eld strength was taken E"5;10 V m\. Finally, the scattering angle was chosen  "33 and kept "xed. In Fig. 5 we present the TDCS as a function of the angle  of the ionized  electron for di!erent values of the relative phase  and for di!erent numbers ¸ of absorbed (emitted) photons. One can see that by changing the phase
the TDCS can be increased or

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195

decreased by a factor of 5. Hence, the CPC e!ects are signi"cant. We do not present results for ¸"0 because the phase e!ects are small in this case. We also note that the peak positions are not a!ected much by the change of the phase . Summarizing, the present analysis has shown that the introduction of the second "eld component can signi"cantly change the results for the TDCS. It is found that the CPC e!ects are quite important and that it is possible to increase or decrease the TDCS by almost one order of magnitude by changing the relative phase
. Also here the symmetry relations (2.21) were con"rmed and it was found that the TDCS have their maximum values for small scattering angles  . The general conclusion of this work was that the in#uence of the laser "eld on the atomic  processes becomes more pronounced if the atomic continuum states take part in the process. The in#uence of the laser "eld on the electrons in such states becomes comparable to the in#uence of the electric "eld of the atomic nucleus on these states, and therefore the laser "eld-induced e!ects, as in our case the CPC e!ects, become more pronounced. 2.3. Resonance scattering in a bichromatic xeld Here we consider electron resonance scattering in a phase-dependent bichromatic "eld of form (2.3) [70]. Since usually the main laser-induced phenomena are strongly aligned along the direction of linear polarization of the two "eld components, it is therefore su$cient to treat to a "rst order of approximation a one-dimensional model oriented along the z-axis. We took in our work a squarewell potential of depth !< and width 2a which supports two bound states of energy E and E    (depicted in Fig. 6). The frequency of the fundamental laser "eld was chosen such that
"E !E . This strongly laser-driven system yields two scattering resonances originating in the  

Fig. 6. Resonance scattering of electrons by a one-dimensional square-well potential in a powerful bichromatic laser "eld. The electrons impinge from the left in the positive z-direction with momentum p , and the laser "eld vector  potential A(t) in the dipole approximation is polarized in this direction. Electrons are re#ected from the well with momenta !p and are transmitted through the well with momenta p . The scattering potential of width 2a and depth L L !< supports two bound states E and E such that E !E is almost at resonance with the fundamental laser      frequency
(see Ref. [70]).

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Fig. 7. In (a) and (b) are shown the re#ection and transmission probabilities P (0) and P (0), respectively, of elastic 0 2 electron resonance scattering as a function of E /
for the four phases
"0.0, 0.25, 0.5, and 0.75 (in units of 2
) in the  bichromatic "eld. Observe the dramatic phase control of the resonance peaks of P (0) and P (0). Similarly, in (c) and (d) 0 2 are shown P (1) and P (1) of electron resonance scattering with stimulated absorption of a single quantum
from the 0 2 "eld now plotted as a function of E /
. All channels of stimulated emission are closed for the chosen initial electron  energy E (see Ref. [70]). 

two dressed bound states with complex quasi-energies (2.28) E!" E #E !
$[(E !E !
)#] ,      where  is the Rabi frequency of the two-level problem and the decay rates  are not written down explicitly [71]. The second "eld component has frequency 2
and phase-shift
. As we shall see, the electron scattering spectra, characterized by the nonlinear transmission and re#ection probabilities P (n) and P (n) (for the number n of emitted or absorbed photons
), respectively, show two 2 0 resonance peaks for particular electron scattering energies. These peaks are strongly modulated by

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197

changing the relative phase
of the two "eld components, thus yielding another example of considerable CPC e!ects. Moreover, we observe the phenomenon of resonance transfer, according to which for particular values of
either the one or the other resonance peak becomes dominant. For a single laser frequency, such a resonance process was discussed by Jung and Tayler [72] and more recently by Sacks and SzoK ke [73] and by KaminH ski [74]. In these references a Floquet method was used to solve consistently the scattering boundary value problem of the SchroK dinger equation for an electron in the laser "eld at the two potential boundaries $a of Fig. 6. The same method was used here and will not be presented explicitly. For the numerical examples in Fig. 7 the potential parameters < "1.0 and a"1.5 were chosen.  This potential supports two bound states having energies E "!0.745 and E "!0.145. As   fundamental laser frequency
"0.61 was taken and the two "eld components had about the same "eld strength such that the amplitudes " "0.3. In Fig. 7 we depict the evaluated re#ection  and transmission probabilities P (n) and P (n) as a function of the electron scattering energy E /
0 2 L (in units of the laser frequency) for the phase shifts
"0.0, 0.25, 0.5 and 0.75 (in units of 2
). The CPC of the relative height and general shape of the two scattering resonance peaks is particularly pronounced for low electron energies E 4
. The four diagrams in Fig. 7 represent (a) P (0),  0 (b) P (0), (c) P (1) and (d) P (1). The channel n"!1 is already closed. We observe symmetries in 2 0 2 the resonance spectra for
"0 and
in Fig. 7(a) and for
"
/2 and 3
/2 in Fig. 7(b). In a related work by KaminH ski et al. [75], it was shown that by changing
one can coherently control the shape and height of the resonance peaks, in particular, a pure Breit}Wigner distribution can be obtained, as discussed earlier by Jung and Taylor [72] for a single laser "eld. In another one-dimensional model calculation, phase-dependent on- and o!-shell e!ects in re#ection of electrons from an impenetrable potential wall in a bichromatic "eld of frequencies
and 2
was considered by VarroH and the present author [76]. The induced and inverse bremsstrahlung for this scattering process were evaluated and the CPC e!ects of the nonlinear currents j(N,
) of re#ection were discussed. If E is the initial electron energy and
the photon energy, two cases are of interest: (a) E<
and (b) EE
where in the latter case o!-shell e!ects are particularly relevant. The corresponding "ndings were related to the scattering of electrons by a hard sphere in a bichromatic "eld, in particular for close to backward scattering.

3. Above threshold ionization and related topics in bichromatic 5elds In addition to the books by Mittleman [1], Faisal [2], Gavrila [3] and Delone and Krainov [4], valuable information on ATI can be gained from the recent reviews by Protopapas et al. [77], DiMauro and Agostini [78] and by Delone and Krainov [79]. Two of the very e!ective methods to treat this highly nonlinear multiphoton process, are the R-matrix Floquet theory as described, for example, in the work by Joachain et al. [80] and for a bichromatic "eld by van der Hart [81] and the method of numerically integrating the time-dependent SchroK dinger equation as described by Kulander et al. in the book of Gavrila [3]. In the following, we shall "rst consider ATI in a bichromatic "eld, discussing the CPC e!ects for a laser "eld of two commensurate frequencies of low harmonic order and we shall besides consider the possibility of stabilization of the atomic target at high laser frequencies of the harmonic "eld. Of particular interest will be the investigation of CPC for high laser powers in which case the

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rescattering of the ionized electron at the target ion becomes of special importance to explain the extension of the ATI spectrum up to 10; (; "E/4
 being the poderomotive energy) in   the case of a single laser frequency, as discussed for example, in the review by Becker et al. [82]. We shall then go on and touch brie#y ATI in a "eld composed of two components, one of which is of low intensity but of su$ciently high frequency to ionize the atom by absorption of a single quantum, while the other component has high power but low frequency. If the ionization by the high-frequency component leads su$ciently far above the ionization threshold then this process has some similarity with our elementary classical model of X-ray scattering by a free electron in the presence of a powerful low-frequency "eld, discussed in Section 1.3, yielding a dominant ATI peak for N"0 and a number of sidebands for NO0 where N is the number of emitted or absorbed photons of the low frequency "eld. Finally, we shall discuss as a generalization X-ray photoionization in a bichromatic "eld and investigate the CPC e!ects in this con"guration. 3.1. CPC in ATI in the Keldysh}Faisal}Reiss model A simple treatment of CPC in ATI can be presented on the basis of the Keldysh}Faisal}Reiss model (KFR) [83}85], developed many years ago. A comprehensive presentation of this model can be found in the review by Reiss [86]. Although this model will only describe the basic features of the CPC e!ects in ATI, it is worthwhile to present a short discussion of the results of this model calculation, since it relates ATI to the scattering processes discussed in Section 2.1. Moreover, it was Keldysh [83] who pointed out for the "rst time that in the ionization of atoms by powerful laser beams two di!erent parameter regimes have to be distinguished which he characterized by his parameter "
, where
is the laser frequency and  the tunneling time, i.e. the time which an electron requires to tunnel through the potential barrier formed by the overlap of the atomic potential < (r) and the oscillating potential < (r, t)"!eE(t)
) r of the laser "eld. If the tunnel time  *  is very much larger than the laser period ¹"2
/
then the ionization of the atom by the laser "eld can be considered as a multiphoton process. On the other hand, for  very much shorter than the laser period ¹, the laser "eld is quasi-static and the ionization takes place as a tunneling process. Assuming that the tunnel velocity of an electron is equal to its velocity on the Bohr orbit, one can show by elementary considerations that the `Keldysh parameter a can be expressed by the ionization energy I "E  of the atom and the ponderomotive energy ; of the laser "eld in the    form "(I /2; ). Hence, with increasing laser "eld intensity ATI gradually becomes a tunnel  ing process. In one of the possible versions of the KFR model it is assumed that initially the electron to be ionized is in the ground state  (r, t)"u (r) exp(!iE t) of the atomic system where the Coulomb    "eld strength E is much larger than the laser "eld strength E so that the laser "eld can be  considered as a perturbation. Then, in lowest order of perturbation theory, the electron gets virtually scattered by the atomic potential <(r)"!Ze/r of nuclear charge Z and turns into a quasi-free particle embedded in the bichromatic laser "eld which leads to ionization. During ionization the laser "eld strength E is now much larger than the Coulomb "eld E so that the e!ect  of the latter can be neglected in a "rst order of approximation. In this way one may consider ATI as a `single-armed a potential scattering of electrons in a laser "eld. It is therefore not surprising that many of the theoretical methods developed for investigating free}free transitions are equally useful in the detailed treatment of ATI and vice-versa. In the case of a bichromatic "eld of frequencies

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199


and 2
of equal "eld strength E and polarization , the ionized electron will be described by a similar Gordon}Volkov plane wave state (2.2), (2.3) as we used for the discussion of the CPC e!ects in Section 2.1. Thus the ¹-matrix element of ATI will read in the KFR model



¹"!i dt dr exp i[(E#;)t!p ) r]  ;exp !i p )
[sin
t#sin(2
t#
)]<(r)u (r) exp(!iE t) ,    

(3.1)

where p is the momentum of the ionized electron of kinetic energy E"p/2m. The electron su!ers in the powerful bichromatic "eld a Stark shift which is equal to the total ponderomotive energy  ; in the two "elds. In the present case ;"(5/4); , where ; is the ponderomotive energy of     the fundamental "eld. For the ground state of the atom we use in the single active electron approximation the wave function of a hydrogen-like atom, i.e. u (r)"(4
)\ 2(Z/a ) exp(!Zr/a ) ,   

(3.2)

where a is the Bohr radius. The evaluation of the ¹-matrix element (3.1) is then straightforward  and decomposes into an in"nite set of matrix elements ¹ corresponding to ATI with the , absorption of N photons
of the bichromatic "eld




a Ze(Z/a )  B a, ;
(E#;#I !N
) ¹ "!8
i   , 4 (Z/a )#p , 

(3.3)

from which we can evaluate the di!erential cross sections of ATI in the KFR model to be given by




r 32Zp a d  ," B a, ;
, 
[1#(pa /Z)] , 4 d 

(3.4)

where r is the classical electron radius. The kinetic energy of the ionized electrons is determined  by the energy conservation relation found from (3.3) to be given by E"N
!I !; where   I "!E is equal to the ionization energy of the atom. As we can see, the threshold condition for   ionization in the bichromatic "eld is given by N
5I #;. Thus we may write N"N #S     where S50 counts the order of the above threshold ionization peaks. For the corresponding electron momentum we "nd p"[2m(N
!I !;)]. With increasing intensity of the laser   "eld, the number N of photons required to yield ionization will increase since the ponderomotive  energy ; is a linear function of the laser intensity I (see our discussion in Section 1.3). B (a, a/4;
)  , are the same generalized Bessel functions (1.11) which appeared in free}free transitions in Section 2.1, except that here a" p )
. Hence, we can expect that the CPC e!ects in ATI will yield  similar spectra as in Fig. 1 if we replace on the abscissa N by N #S where N corresponds to the   "rst above threshold peak and, of course, S50 will then only refer to the right hand parts of these "gures. Of course, this analogy is only true in a very crude sense. In particular, we have dropped in our derivation above the time-dependent contribution of the A-part of the electromagnetic interaction in the Gordon}Volkov state (2.2).

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3.2. More detailed calculations on CPC in ATI 3.2.1. Calculations at higher laser intensities The "rst more detailed calculation of CPC e!ects in ATI was performed by SzoK ke et al. [87] using a one-dimensional short-range potential and a simple model of a helium atom. As a result of their investigation they demonstrated two very di!erent behaviours that result from the variation of the relative phase
during simultaneous irradiation by a coherent laser beam and its harmonic. They showed that, for a simple, short range, one-dimensional potential, the presence of the third harmonic, 3
, of the laser can enhance or suppress the multiphoton ionization depending on its relative phase, even in realistic, time-dependent pulses. Also, there were strong changes observed in the details of the electron ATI spectrum because of the presence of the third harmonic. For the case of a laser and its third harmonic, they presented both, one-dimensional and three-dimensional calculations that show a strong dependence of the photo-electron angular distribution on the relative phase of the two colours, a strong enhancement of the ionization probability by the second harmonic, but relatively little dependence of it on the relative phase. These authors also showed that the asymmetry in the angular distribution must be taken into account in the interpretation of experiments which only probe one direction of emission. Their calculations thus suggested that the proposal of Shapiro et al. [6], discussed in the introduction, Section 1.2, to provide laser control over photo-dissociation processes, can be carried out using strong pulsed lasers. In a series of papers by Potvliege and Smith [88}90], CPC in ATI was considered for hydrogen for "eld combinations of frequencies r
and s
with r and s small integers. Ab initio nonperturbative Floquet calculations of the ionization rates were performed, considering also resonant ionization and the dependence of the "eld ionization rate on the relative phase
of the two "elds. They found, in particular, that the total ionization rates depend very little on the relative phase
. Moreover, in a paper by Pont et al. [91] the geometric phase was discussed which gets accumulated by the wave function of an atom ionizing in the presence of a bichromatic "eld as physical parameters are varied adiabatically around a closed circuit. As an illustration they calculated the geometric phase for a hydrogen atom in the presence of 355 nm light and its third harmonic when the phase and intensity of the two components are varied. It turned out, that the wave function need not be single valued after one complete circuit. Two circuits may be necessary to map the original eigenwave onto itself. Furthermore, the geometric phase may be complex, and may therefore modify the ionization yield calculated from the width of the instantaneous quasi-energy. Floquet theory was also applied to multiphoton detachment of H\ by two colour laser "elds by Telnov et al. [92], evaluating the angular distributions and partial rates at rather moderate laser "eld intensities. These authors presented a general non-perturbative formalism and an e$cient and accurate numerical technique for the study of angular distributions and partial widths for multiphoton above threshold detachment in two colour "elds. Their procedure was based on an extension of their earlier work [93] for one-colour detachment and the many mode Floquet theory of Ho et al. [94]. The procedure consisted of the following elements: (i) Determination of the resonance wave function and complex quasi-energy by means of the non-Hermitian Floquet Hamiltonian formalism. The Floquet Hamiltonian is discretized by the complex-scaling generalized pseudospectral technique developed by Wang et al. [95]. (ii) Calculation of the angular distribution and partial widths based on an exact di!erential formula and a procedure for the rotation of the resonance wave function back to the real axis. The method was applied to

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201

a non-perturbative study of multiphoton above threshold detachment of H\ by 10.6 m radiation and its third harmonic. The results showed strong dependence on the relative phase
between the fundamental frequency "eld and its harmonic. For the intensities used in these calculations (10 W cm\ for the fundamental frequency, 10 and 10 W cm\ for the harmonic) the total rate had its maximum at
"0 and minimum at
"
. However, this tendency, though valid for the "rst several above-threshold peaks in the energy spectrum, is reversed for the higher energy peaks. The energy spectrum for
"
was broader, and the peak heights decreased more slowly compared to the case of
"0. The strong phase-dependence was also manifested in the angular distributions of the ejected electrons. The CPC e!ects in ATI were also considered by numerically integrating the time-dependent SchroK dinger equation either for a one-dimensional model potential or, in the three-dimensional case, for hydrogen. In the work of Pont et al. [96] the time propagation was based on the split operator technique, with the full Hamiltonian split into two parts, the atomic Hamiltonian and the atom}"eld interaction. Both parts were represented on a complex Sturmian basis. The method was relatively e$cient, ionization yields and level populations of atomic hydrogen were easily computed on a work station for modest pulse durations (e.g., 50 cycles or so) and modest intensities (e.g., of order 10 W cm\ for a frequency of 0.2 au or 5.44 eV). The method was applied to atomic hydrogen and the results were compared with those obtained by Kulander [97]. They also illustrated the sensitivity of the ionization yield to the relative phase
in the case where the "eld is bichromatic with one "eld a harmonic of the other. Similar, more extensive calculations of the CPC e!ects in ATI were performed by Schafer and Kulander [98] by numerically integrating the time-dependent SchroK dinger equation (TDSE) for hydrogen. Their method of solution is described in detail in their review in the book of Gavrila [3]. They considered the ionization rates, angular distributions and ATI spectra in a strong, two-colour laser "eld. The two lasers were "rst and second harmonic "elds with the same intensity and a constant relative phase di!erence
between them. At longer wave length (1064 nm) and higher intensities ('10 W cm\), there was clear evidence in the phase-dependence of the ionization rates that ionization took place primarily through tunneling. Even though the total ionization rate in this regime depended on the peak value of the time-dependent electric "eld, the angular distributions showed additional phase-sensitive e!ects. In particular, there was a large forward}backward asymmetry in the emitted electron distributions that was not simply correlated with the maximum electric "eld. At shorter wavelength and/or lower intensities, there was a transition to multiphoton ionization with interference between ionization paths from the two lasers evident in the total ionization rates. In all cases, these authors found that the total ponderomotive shift of the ionization limit is the sum of the shifts for the two individual "elds (as predicted by the KFR model). As representative examples of their calculations we show in Fig. 8 the phase-dependence of the total ionization probability after 20 cycles, as well as the forward and backward components, for a Nd : YAG laser of
"1.17 eV and a constant intensity I"10 and 2;10 W cm\. The ionization is greater for "$
/2, which produces the largest electric "eld, as can be recognized from (2.3). The total ionization probability has a period of
, and depends only upon the maximum "eld strength. This is typical of ionization in the tunneling regime. The partial rates (i.e. the forward and backward distributions), however, have periods of 2
and show marked asymmetry, as can also be recognized from the approximate KFR results (3.4). This is true even for "0, when the peak forward and backward "eld strengths are equal. The

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Fig. 8. Presents for the "eld combination (1
, 2
) with
"1.17 eV the total (solid squares) and directional (solid circles, forward; open circles, backward) amount of ionization after 20 optical cycles as functions of the relative phase angle  for (a) I"1;10 W cm\, and (b) I"2;10 W cm\. Both "elds have the same "eldstrength (see Ref. [98]).

partial rates show a sensitivity to the detailed shape of the electric "eld that is not present in the total ionization rate. The emission in the forward direction is the same as in the backward direction for "$
, as expected. There is no connection between the forward and backward emission for # and !, other than the observation that the sum of the forward and backward emission is the same for both cases. Therefore, an experiment that measures the phase-dependence of the total ionization probability for this case must measure the emission in either the forward or backward directions separately over a range of 2
, or in both directions over
. In a paper by Protopapas et al. [99] the phase-dependence in two-colour excitation of a model atom by intense "elds was considered. They solved numerically the one-dimensional SchroK dinger

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203

Fig. 9. This shows the spatial and temporal evolution of an atomic wave packet formed under strong "eld excitation. The authors plotted in the Kramers}Henneberger frame  (x, t) in arbitrary units over a range of 400 atomic units for )& (
, 3
) two-colour excitation by a 50-cycle, sine-squared pulse with equal intensities (E "E "18.49 au) of funda  mental frequency (
"1 au) and third-harmonic frequency for the relative phases (a) "0 and (b) "
. Each pro"le is separated from its predecessor by a time step of 1/16 of the full pulse duration of 50 cycles (see Ref. [99]).

equation with a model potential for the case of excitation by two commensurate (the fundamental and third harmonic) intense laser "elds and found a strong dependence of the atomic evolution and high harmonic generation on the relative phase of the two "elds. They employed the Kramers}Henneberger frame [100,101], to exploit the insight gained from emphasizing the wave-packet dynamics of an electron oscillating under the in#uence of the strong "elds. As an example of their calculations we show in Fig. 9 the spatial and temporal evolution of an atomic wave packet formed under strong "eld excitation of a bichromatic "eld of components
and 3
for the relative phases "0 and
. The fundamental and third harmonic have equal intensities. One can clearly see that during the turn-on part of the pulse the initial ground state wave-packet is quickly ionized, producing outward going ripples of ionization. However, once this is over, the wave function quickly stabilizes, i.e. there is no further signi"cant ionization. In Fig. 9(a), where the relative phase "0, we observe a dichotomous wave packet being formed and destroyed in a way which follows the envelope function of the pulse. Fig. 9(b) shows the evolution for "
. There is a characteristic di!erence in the wave-packet shape; the dichotomy has now disappeared. These model calculations were extended to the near resonant case for a two-level atom and compared with more realistic numerical solutions of the one- and three-dimensional SchroK dinger equation by Protopapas and Knight [102]. In Fig. 10 we show their results on the CPC e!ects for the resonant ionization of the two-level atom in a (
, 3
) "eld con"guration. Using the density matrix formalism, Nakajima et al. [103] considered modulating ionization through phase control. They performed speci"c calculations for sodium atoms and found signi"cant modulation of the ion signal by chosing the appropriate combinations of laser intensities and frequencies in resonant as well as non-resonant processes.

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Fig. 10. Presents the two-colour (
, 3
) phase-dependent probabilities of ionization for square pulse excitation with equal coupling
3.2.2. CPC in ATI at moderate laser intensities Anderson et al. [104] and Baranova et al. [105] derived general phenomenological expressions for the angular distribution of electrons emitted when atoms or molecules are ionized by a light "eld containing the fundamental frequency
and the second harmonic 2
. They used timedependent perturbation theory and the quantum defect approximation to show that the polar asymmetry in the angular distribution appears because of interference of the one-photon process in the second harmonic 2
and the two-photon process in the "eld of frequency
. They evaluated the absolute phase of the interference term for alkali atoms. The same problem was investigated in the tunneling approximation by Baranova et al. [106] and Pazdzersky and Yurovsky [107]. The corresponding problem for the negative ion decay was considered by Pazdzersky and Usachenko [108] and by Kuchiev and Ostrovsky [109,110]. The problems considered by these authors at

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205

Fig. 11. Shows the di!erential ionization rates for the "rst above threshold peaks S"0 and 1 as a function of the relative phase
between the two-"elds of frequencies
and 3
for the two-"eld intensity combinations (A): I(
)"6.216;10\ au and I(3
)"3.108;10\ au and (B): I(
)"6.216;10\ au and I(3
)"1.554;10\ au. The spatial angles  are given in degrees. The strong intensity dependence of the CPC e!ects is remarkable (see Ref. [111]).

moderate laser "eld intensities will be discussed in more detail in Section 5.1.1 where we shall report on experiments which have been performed on atoms to observed CPC e!ects in bichromatic "elds. In a very instructive paper by Blank and Shapiro [111] it was pointed out that in the investigation of CPC e!ects in ATI one should not only consider the relative phase
of the two "elds but vary simultaneously their relative intensities. In particular, they considered ATI in hydrogen for a fundamental "eld
and its third harmonic 3
. Here they showed the importance of phase and intensity control of the ATI process. This is very nicely documented by their results reproduced in Fig. 11. They show in atomic units (1 au"e/a "27.2 eV) for
"  0.2567 au (K7 eV) and in (A) for I(
)"6.216;10\ au and I(3
)"3.108 au and in (B) for I(
)"6.126 au and I(3
)"1.554 au, respectively the phase-dependence of the angular distribution of the "rst two ATI peaks, i.e. for S"0 and 1. In these "gures the spatial angles  are given in degrees and the phase angles
in radians. For the conversion of the chosen intensities, we observe that 1 au of intensity "6.43;10 W cm\. As we can clearly see in these "gures, the phase- as well as intensity-dependence on the second "eld is remarkable. Such an importance of the intensity dependence of the second "eld was stressed in our analysis of the CPC e!ects in free}free transitions [39], discussed in Section 2.1.

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3.2.3. Inyuence of laser amplitude- and phase-yuctuations on CPC in ATI In the investigations of CPC in ATI presented up to now, it was assumed that the laser "eld can be described by idealized monochromatic plane waves. The lasers available for experiments do not, however, have those idealized properties. It is therefore of interest to investigate the e!ects of laser phase- and intensity #uctuations on CPC in bichromatic "elds. Such investigations were performed by Camparo and Lambropoulos [112,113]. They considered the e!ect of laser phase #uctuations on three-photon}one-photon, i.e. (
, 3
), phase control of resonance enhanced photoionization and investigated also the role of laser intensity noise in addition to laser phase noise. While their results indicate that relative phase and intensity #uctuations between the fundamental "eld and its third harmonic have a signi"cant e!ect on control, the contrast between constructive and destructive interferences is nontheless two orders of magnitude, even under worst case situations. Consequently, neither laser intensity nor phase #uctuations appear to pose a serious impediment to the e$cient phase control of atomic and molecular processes. Instead of changing the relative phase of the two harmonic "elds one can also e!ectively change the time-delay between the two components of the laser pulse. This was investigated for the control of two colour photoionization by Matulewski et al. [114]. 3.2.4. CPC in stabilization of atoms in high-frequency bichromatic xelds The stabilization of atomic systems in a powerful high-frequency laser "eld was predicted many years ago by Gavrila and co-workers (See the review in the book by Gavrila [3].) In this regime, high-intensity two-colour interference e!ects on atomic stabilization were investigated by Cheng et al. [115]. They considered the ionization dynamics of a one-dimensional model atom in ultrastrong dichromatic (1
, 2
) laser "elds in the stabilization regime. They focused on the phase control and the quantum interference e!ects. Their diagrams of total and partial yields versus relative phase show similar structure to that of the tunneling regime, although the underlying mechanism is quite di!erent. An enhancement or suppression of the total ionization rate by more

Fig. 12. Ionization rates are presented which were calculated from the high-frequency Floquet theory for a (
, 2
) bichromatic "eld of equal "eld amplitude E with classical amplitude  in the range 04 410 and for relative phases   044
/2. In the stabilization regime, the total ionization rate demonstrates a signi"cant phase dependence and reaches its maximum at "
/2 (see Ref. [115]).

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207

than two orders of magnitude were observed by adjusting the relative phase , which was attributed to the signi"cant e!ect of quantum interference in the stabilization regime. Asymmetry on the distribution of forward and backward ionization was also observed, i.e., for proper choice of the phase, the photoelectrons can be con"ned almost to one direction. All those "ndings were explained satisfactorily by the "rst-order high-frequency Floquet theory of Gavrila [3]. Fig. 12 shows the total ionization rate evaluated from this one-dimensional model atom as a function of  "E/
 (in au) and phases  in radians. The fundamental frequency is
"3 au (or K7.5 eV).  One easily recognizes the stabilization e!ect with increasing laser intensity ( &(I), except for  small oscillations, since the rate drops for increasing  . Moreover, we see that the phase-e!ect  becomes largest for "
/2. Their conclusions have recently been con"rmed by Potvliege [116] who performed similar one-dimensional model calculations. His main conclusion was that the CPC e!ects are signi"cant only at very high laser intensities. 3.3. Generalizations of the KFR model in bichromatic xelds 3.3.1. KFR model in a quantized two-colour xeld The KFR model [83}85], used in its simplest form at the beginning of this section to demonstrate CPC in ATI in an elementary way, has received various modi"cations and improvements, in particular, in the high-intensity tunneling ionization regime where the Keldysh parameter "
"(I/2; ) becomes smaller than unity [79], as we have discussed in Section 3.1. One of  the modi"cations of the KFR model has been suggested by Guo et al. [117], which is based on the formal scattering theory of Gellman and Goldberger [118]. In this modi"cation of the KFR model the classical radiation "eld, describing the laser, was replaced by a quantized monochromatic "eld and the corresponding Gordon}Volkov solution in this "eld was derived [119]. On the basis of this modi"ed KFR theory Gao et al. [120] considered phase-di!erence e!ects in two-colour above threshold ionization. The two beams of laser light were the fundamental "eld and its second harmonic with equal intensity and adjustable phase di!erence
. The photoelectron yield from krypton and xenon gases were studied as the phase-di!erence varies. The phase-di!erence e!ects in the ATI spectra were interpreted as a quantum interference between di!erent channels characterized by the numbers of transfered photons in the process of ionization in the "eld followed by the photoelectron exit process. The method turned out not to be speci"c for tunneling ionization and has no direct relation to the Keldysh parameter . 3.3.2. Rescattering in ATI in bichromatic xelds Another generalization of the KFR model is related to the recently observed new phenomena in ATI at very high laser powers. First it was found that the spectrum of the above threshold peaks in ATI of inert gases in a powerful linearly polarized laser light extends considerably further than the cuto! frequency of optical harmonic generation, given by
"I #3.2; as found by Krause

   et al. [121] performing a numerical integration of the TDSE. Soon after these experiments by Schafer et al. [122], new experiments by Yang et al. [123] on the angular distribution of the emitted electrons in the ATI peaks revealed unexpected sidelobes or rings at an emission angle of about 453 with respect to the direction of linear polarization of the laser "eld. These rings appeared at higher orders S of the ATI peaks near an electron energy of 9; . The above authors reproduced these  "ndings by a numerical integration of the TDSE and were able to interpret the appearance of these sidelobes as a result of rescattering of the emitted electrons by the residual ionic core. Subsequently,

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experiments by Paulus et al. [124,125] con"rmed the existence of these sidelobes at about 303 and they moreover detected the formation of a plateau of the ATI peaks up to about 10; , which they  con"rmed by one-dimensional model calculations. Several theoretical papers appeared to describe these new features of ATI, classically in the work by Paulus et al. [126], quasi-classically in a paper by Lewenstein et al. [127] and quantum mechanically in a report by Becker et al. [128]. In all these investigations a two- to three-step model calculation was performed in which it was assumed that in an initial step the atom gets ionized by tunneling and then the ripped o! electron moves freely in the laser "eld to be eventually rescattered by the parent ion core. The results of these papers were in qualitatively good agreement with the experiments and indicated that, at least for the present, the assumption of a single active electron in the atom is su$cient, even if the models are applied to inert gases. A nice survey of the present experimental and theoretical status on the results and interpretations of the new phenomena is given in a very recent review by Becker et al. [82], mentioned previously. In the KFR model, only the "rst-order term in the S-matrix expansion is considered, as is apparent from our elementary calculation in Section 3.1. In order to be able to describe the rescattering process, Becker et al. [82,128] considered the second-order term of the expansion describing the atomic potential by a short range -function approximation. Therefore, in this model the action of the residual ionic core on the ejected electron is not explicitly taken into account. Hence, we "rst considered a generalization of the KFR model by replacing the outgoing Gordon}Volkov wave (2.2) by an improved wave function which takes into account in an approximate way the Coulomb e!ects of the residual ion on the emitted electron. A derivation of this wave function can be found in our review on electron}atom scattering in a laser "eld [34]. This improved KFR model shows that close to ionization threshold the probabilities of ionization are considerably enhanced [129] and the angular distributions obtained are very di!erent and much richer in structure than those predicted by the KFR theory [130]. The rescattering process, however, cannot be described by this simple improvement of the KFR model, since it only describes that part of the ATI spectrum that extends to about 2; electron energies. In the case of  a bichromatic "eld of frequencies
and 2
and of equal linear polarization, the model also demonstrates the very di!erent phase e!ects obtained, if compared with the corresponding data of the KFR model [131]. Moreover, we showed in a one-dimensional model calculation of rescattering, the importance of the relative phases of the various components of the wave functions participating in the rescattering process [132]. In order to be able to describe the rescattering process on the basis of the S-matrix theory including the e!ects of the ionic Coulomb "eld, we generalized in a series of papers the work of Becker et al. [82,128] and applied it subsequently to the CPC e!ects in ATI [133,134]. In this model the corresponding S-matrix element reads in second order of perturbation theory [134]



S#S"!i DG DG

 

>

\

dt exp i[p ) (t)#U(t)#(E#I )t] 



; \ p A r ) E(t)u !i dq\ p A < q#A(t) > R  > R Q ;







dq#A(t!)r ) E(t!)u  exp[!iS(q; t, )] , 

(3.5)

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where the "rst (non-exponential) term in the curly brackets is equivalent to the matrix element of the improved KFR model described above [129}131] and the second term represents the rescattering process by a screened core-potential taken as < (r)"!exp(!r)/r. E and p are the kinetic  energy and momentum of the ionized electron and I "!E is the ionization energy.   For the bichromatic "eld (2.3) considered here (t)"
[ sin
t# sin(2
t#
)] and   U(t)"(1/2)RdtA(t) is the contribution of the A-part of the electromagnetic interaction which does not drop out in the case of ionization, as pointed out in Section 3.1. u  is chosen as the bare  ground state of the atom in the single-electron approximation neglecting target dressing by the laser "eld and \ p A  is the improved Coulomb}Volkov state found from the well-known > R Coulomb-wave (2.23) of Section 2.2.3 by replacing p by p#A(t) [34]. In the second term of the curly brackets of (3.5) the ionized electron propagates before rescattering by the potential < (r) in  the laser "eld alone, represented by the Gordon}Volkov propagator G (t, t), given by *



G (t, t)"!i(t!t) dqq (t)q (t) , *

(3.6)

where in the r ) E(t) gauge q (t)"q#A(t) exp !i[q ) (t)#U(t)#Eq t] .

(3.7)

Finally, the semiclassical action in the last integral of (3.5) is given by



S(q; t, )"

R

R\O

dt (1/2)[p#A(t)]#I  

"[Eq #I ]#p ) [(t)!(t!)]#U(t)!U(t!) . 

(3.8)

This quantity was introduced in connection with HHG [127,135]. The integral over the momenta q in (3.5) can be performed using the saddle point method, leading to a time-dependent WKB approximation [135]. This then yields



S "!i dt exp[i(E#I #; )]T ()   DG DG "!2
i (E#I #; !N
)¹ ,   , ,

(3.9)

where here "
t and "!
 such that r ) E()u  T ()"exp i[p ) ()#U ()] \ p A > (  DG  !i







< q #A() d(2
/i)\ p A > (  

;q #A()r ) E()u  exp[!iS(q ; ,
)]   

(3.10)

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and q "(1/)[(t!)!()], which is a solution of q S(q; t, )"0. Finally,  U ()"(1/4
) 2;  sin 2#;  sin(4#2
)    # 8(;  ;  )[sin(#
)#()sin(2#
)]   

(3.11)

in which ; ";  #;  is the total ponderomotive energy of the two "elds with ; H "     E/4j
, ( j"1, 2). The expression (3.10) is a 2
/
-periodic function of t and can be expanded in H a Fourier series, yielding the Fourier coe$cients ¹ of (3.9) from which we can evaluate the , di!erential ionization rates w(N, )"2
p¹ , where p"(2E is obtained from the energy , conservation ralation of (3.9) E"N
!I !; . The angle  which determines the angular   distribution of the ionized electrons is determined by p )
"p cos . The numerical examples presented were performed for a bichromatic "eld of equal intensity, such that ; "(5/4);  . As target a helium like atom was chosen, representing its ground state in   the single electron approximation by a hydrogen-like wave function u (r)"
\ exp(!r) and  E "0.5 au. The He>-like ion core potential was mentioned before. Although the actual core  potential is more complicated, our work showed [133] that our second order improved KFR model describes the basic features of ATI quite well, i.e., the enhancement of the rates near the threshold due to the Coulomb e!ects up to about 2; (mentioned before) and the corresponding  plateau features and cuto! at 10; (in the single frequency case) on account of rescattering,  basically by the core potential. For the following "gures, an intensity of I"10 W cm\ was chosen for which there are no experiments on CPC available yet. Hence, the extended plateau in the photoelectron energy spectrum which is obtained for rescattering in the above bichromatic "eld has not yet been observed. In a foregoing work on classical rescattering in a bichromatic "eld by Paulus et al. [136] it was shown that the rescattering plateau can extend to almost 21; . This classical model was  introduced by Corcum [137] and by Becker et al. [126,136]. As our quantum mechanical calculations showed, the cuto! for certain values of  and  can really be at 21; . As concerns the  sidelobes, we should notice that for a bichromatic laser "eld of frequencies
and 2
, and relative phase , the symmetry 1803!  , observed for a single frequency, gets broken and, therefore the sidelobes will not possess this symmetry any longer. Instead, in the bichromatic case, the di!erential ionization rates now obey the symmetry (#
, 1803!)  (, ). This can be shown explicitly by considering the ¹-matrix elements (3.9), (3.10) which satisfy the symmetry relation ¹ (#
; 1803!)"(!1),¹ (, ). According to this relation, the rates for backward direction , , ("1803) and "0 are the same as the rates for forward direction ("0) and "
. The same relation is also ful"lled by the generalized Bessel function B (a, a/4; ) in (3.4), as has to be expected. , In Fig. 13 we show the di!erential ionization rates as functions of E/;  for di!erent values of  the angle  of electron emission and for the relative phase "0. The direct ionization rates (computed from S in (3.5)) and the ionization rates which include rescattering and correspond to DG S#S are shown separately. For "1803 the direct rates are shown by dotted lines, while the DG DG rates which include the rescattering are represented by a continuous curve. For "903 only the direct rates are shown, while for "03 the direct rates are depicted as a continuous line. The rates which include rescattering for "03 are represented by dashed curves. It follows from this "gure that the direct ionization rates contribute to the total ionization rates near the threshold and for

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Fig. 13. Presents the di!erential ionization rates for a bichromatic linearly polarized laser "eld of frequencies
"1.58 eV and 2
, the relative phase
"0 and equal intensity I"10 W cm\ of both components, as a function of the kinetic energy E of the ionized electrons scaled to the ponderomotive potential ;  . The ground state wave function I  and energy are  "
\ exp(!r) and I "0.5 au, respectively. The short-range part of the core potential is   < (r)"!exp(!r)/r. The direct ionization rates and the ionization rates which include rescattering are shown  separately. For "1803 the direct rates are shown by a dotted curve, while the rates which include rescattering are represented by a continuous curve. For "903 only the rates of direct ionization are shown, while for "03 the direct rates are depicted as a continuous line. The rates which include rescattering at "03 are presented as a dashed curve (see Ref. [134]).

low electron energies E only, while the extended plateau starts at the cuto! of the direct rates and corresponds to the rescattering e!ects. The positions of the cuto!s strongly depend on the angle . The maximum of the outgoing electron energy, both for the direct and rescattering rates, is at "1803. The cuto! of the direct rates is around 5;  . The cuto! of the extended plateau is at  20.6;  , as predicted by the results of a classical consideration of rescattering [126,136,137]. The  maximum outgoing electron energies for smaller  are much lower than for "1803. The main conclusion drawn from Fig. 13 is that at "0 the backward emission of electrons is dominant and that in this direction one obtains many more high energy electrons. According to the symmetry relation, stated before, this situation will be reversed for "
. More details about the ATI angular distributions in the bichromatic "eld at di!erent energies of the ionized electrons at E(20.6;  can be found in the detailed analysis of our work [134]. They  can be interpreted as quantum mechanical interference e!ects by discussing in more detail the structure of the matrix elements (3.10). Also, a classical, more rigorous discussion of rescattering can be presented along the lines of our earlier work for a single frequency [138]. Summarizing, our generalized S-matrix theory of ATI in a "eld of frequencies
and 2
including rescattering predicted interesting e!ects which should be accessible to observation. We obtained an extended plateau for the emission of electrons in the backward direction for "0 or for
in the forward direction. The sidelobe structures were more complicated than in the

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monochromatic case, in particular for large angles  of electron emission. With increasing energy of the outgoing electrons, these angular intervals became narrower and at the cuto! at 21;  the  electrons are only emitted in the backward direction ("1803). The same rescattering process in ATI in a powerful bichromatic laser "eld of frequencies
and 2
has been investigated very recently in the tunneling regime by Chen et al. [139]. They considered the in#uence of the rescattering process on the two-colour laser ionization by using an improved two-step quasistatic model, in which the Coulomb focusing e!ect is taken into account. Special attention was paid to the phase-dependent rescattering process and the CPC e!ects. It was found that the rescattering leads to further forward/backward asymmetry, breaks the symmetry of rate-phase relation about "$
/2, and accounts for the departure of the predictions of a simple two-step model from the experimental data. Their results are in good agreement with the experimental observations. 3.4. X-ray photoionization in presence of a laser xeld Under this title we want to consider two types of processes which are closely related to each other. First, we shall report on those investigations in which an atomic system gets excited close to or above threshold by a high-frequency radiation "eld. This "eld is of moderate power and can be treated by time-dependent perturbation theory. A low-frequency laser "eld is present simultaneously and stimulates nonlinear ionization e!ects. This permits, for example, to discuss various laser-induced threshold phenomena. These processes are of course not directly related to the CPC e!ects to which this review is mainly devoted. Then we shall discuss processes where a soft X-ray "eld ionizes an atom in the presence of a bichromatic "eld and here we shall discuss the CPC as a function of the relative phase
of the two laser components. We should remark that by `X-raya "eld we denote here a high-frequency "eld,
, where this frequency can be very di!erent, & depending whether we consider photodetachment of a negative ion or ATI of an atom. 3.4.1. Two-colour ionization by incommensurate frequencies On this type of processes a nice short review was presented by VeH niard et al. [140]. Here it was shown, how present day coherent high-frequency radiation sources, obtained from higher harmonic generation, can be used to conveniently investigate ATI processes in which one harmonic photon
excites the atom into the continuum while a number of low-frequency laser photons &
are emitted or absorbed simultaneously yielding an ATI electron energy spectrum of the form * E"
#N
!I !; (NJ0) where ; is the ponderomotive energy of the high-power & *    low-frequency laser "eld. If the absorption of the photon
excites the ionized electron su$ciently & high into the continuum, one may expect that the spectrum of ATI peaks has certain similarities with the spectrum found for free}free transitions by Bunkin and Fedorov [33], as discussed in Section 2.1. We only have to replace the di!erential cross section for elastic electron}atom scattering by the corresponding cross section of X-ray ionization in the absence of the lowfrequency "eld. Thus, we get in this low-frequency approximation d V d V , "  J ( p )
) , d ,  d

(3.12)

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where the momentum of the ionized electron is determined by p"[2m(
#N
!I !; )]. & *   In (3.12) we neglected for simplicity the contribution of the time-dependent term of the A -part of * the low-frequency laser interaction which, as pointed out earlier, does not drop out in laser-induced or -assisted ionization processes. We simply assumed that ; ;2
. The spectrum that can be  * evaluated from (3.12) is similar to what was obtained by more explicit calculations, permitting also resonance e!ects and threshold phenomena if
KI , and so forth. &  In the work by Leone et al. [141] the above process was considered for hydrogen in a KFR-type of approximation, neglecting the laser-dressing of the atomic target, but including the Coulomb e!ects of the residual proton on the outgoing electron in an approximate way, using the Coulomb}Volkov wave function suggested by Jain and Tzoar [142] many years ago. They also addressed the gauge problem, considering the r ) E(t) and p ) A(t) interactions with the electromagnetic "eld. Since they were using for the high-frequency "eld
"50 and 100 eV their data are & su$ciently above ionization threshold and their ATI results are similar to those obtained for free}free transitions, as suggested by the approximate cross section formula (3.12). We show, for example, their total cross sections for
"100 eV,
"1.17 eV and I "10 W cm\ in Fig. 14. & * * In another paper by Bivona et al. [143] the same process was considered for photodetachment of H\. Here H\ was described by a short range potential with a single bound state and a Keldyshtype of approximation was su$cient to evaluate the ionization rates in the two-colour "eld. Since the energy required for photodetachment is very small, it was su$cient to take, for example,
"0.85 eV and
"0.0025 eV with I "10 W cm\, to evaluate the spectra. These authors & * * also considered the in#uence of a static electric "eld on the detachment process. A similar Keldysh-type of approximation as (3.12) was found by DoK rr and Shakeshaft [144], using TDPT. These authors also considered a one-dimensional model of two-colour photodetachment of a negative ion by numerically integrating the TDSE [145]. They considered, in particular, the threshold shift of detachment due to the change of the ponderomotive energy ; of the powerful  low-frequency "eld in the case where there are no shape resonances near threshold. A generalization of the work of Leone et al. [141] and DoK rr and Shakeshaft [144] was presented by Zhou and Rosenberg [146]. They introduced a variational approximation to derive a Kroll}Watson type [42] of formula for the two-colour ATI, yielding a generalization of (3.12). Also their numerical spectra of ionization resemble the data of Fig. 14. Solving the TDSE perturbatively in momentum space, it was shown by Han [147,148] that two-colour ATI of atoms can be easily generalized to the case where one or more photons
are absorbed to yield ionization with the simultaneous & absorption or emission of N
photons of the high-power low-frequency "eld. In this case, we may * expect in lowest order of approximation a cross section formula of the form r 32Zp d  L, " J( p )
)J ( p )
) & , * * 
# 
[1#(pa /Z)] L & d  & & * *

(3.13)

as a possible generalization of our KFR formula (3.4) for incommensurate frequencies
and
. & * Here the atom can get ionized by absorbing n photons from the high-frequency "eld
and can & absorb or emit in addition N photons from the low-frequency "eld
simultaneously. Thus we * expect a complicated spectrum of ATI peaks and the ionized electron would have the possible momenta p"[2m(n
#N
!I !; * )]. The oscillating character of the spectra anticip& *   ated from formula (3.13), are con"rmed by the numerical work of Han [147,148].

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Fig. 14. Shows the total cross sections of two-frequency multiphoton ionization of hydrogen atoms versus the number of exchanged low-frequency photons,
"1.17 eV, for di!erent values I ("1, 5, 10, 50) of the low-frequency laser *  intensity with I "10 W cm\. The energy of the high-frequency photon is
"100 eV. The spectra of this  & laser-assisted X-ray ionization process are similar to those of free}free transitions in electron}atom scattering according to Bunkin and Fedorov [33] (see Ref. [141]).

In a paper by DoK rr et al. [149] Floquet theory was employed to investigate two-colour ionization in a low-intensity "eld of high-frequency
and a high-intensity "eld of low-frequency &
assuming
On
, n being an integer. The ionization of hydrogen in its ground state was * & * considered. As low-frequency "eld they took
"1.17 eV. With the help of Floquet theory it was * possible to treat the following three processes: (i) resonance enhanced ionization, via levels that are ac-Stark mixed or/and very strongly shifted, (ii) laser-induced continuum structure, and (iii) nonresonant ionization in a regime where, despite the fact that the infrared "eld signi"cantly shifts the ground state relative to the continuum, it only redistributes the photoelectrons over the low-frequency channels. They also illustrated the sensitivity of the ionization yield to the spatiotemporal con"guration of the two "elds. The e!ect of the laser-induced continuum structure was

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also discussed in three papers by Rzaz ewski et al. [150], Wang et al. [151] and Bu!a et al. [152]. In & the "rst paper a model of atomic ionization by smooth pulses was solved with bound}continuum and energy-dependent continuum}continuum dipole coupling. The rotating wave approximation for the continuum}continuum transitions was avoided. In contrast with the #at continuum results, their ionization rate was independent of the continuum}continuum coupling. The authors also derived the photoelectron energy spectra and showed that counter rotating terms make important contributions. In the second paper dipole coupling between an in"nite number of continua, each having speci"c angular momentum, was considered. The continuum}continuum matrix elements depend on the energy of each continuum state and have a singularity, as expected for a hydrogenic atom. This model was analytically solvable without resource to the rotating wave approximation. The ionization rate was independent of the intensity of the low-frequency laser. The authors examined the angular distribution of the photoelectrons under di!erent circumstances of bound state parity and ionization laser polarization. Qualitative di!erences between the models were also illustrated using the photoelectron spectra. In the last paper similar calculations as by Wang et al. [151] were performed to obtain analytically solvable equations for the multiple continua model analyzed by Deng and Eberly [153]. In this model the atom is assumed to be described by the ground state  of energy !
(
'0) and n degenerate continua c (
) of free states of energy ? ? I I
(
'0). Weak UV laser radiation of frequency
couples the ground state  to the "rst I I & continuum c (
), while intense IR laser radiation of frequency
couples adjacent continua   * c (
) and c (
) (m"k$1). The results of this paper represented an extension of the work of I I K K the foregoing authors to two-colour photoionization with multiple continua, con"rming that no saturation of the total ionization rate is induced by saturating continuum}continuum transitions. The absence of any saturation of the ionization yield at increasing dressing laser intensity was also predicted by a numerical simulation of the two-colour photoionization, carried out for a threedimensional hydrogen atom by VeH niard et al. [154]. Target dressing e!ects in laser-assisted X-ray photoionization was investigated by Cionga et al. [155]. They considered X-ray photoionization of hydrogen in the presence of a low-frequency laser "eld using methods which partially take into account the radiative stimulated corrections to both bound and continuum states. Special attention was paid to the cases in which one low-frequency photon is exchanged (absorption or emission) between the atomic system and the laser "eld. It was demonstrated that the atomic dressing e!ects are important. While the ionization of hydrogen by the absorption of a single X-ray photon of energy

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Considering the presently available XUV femtosecond coherent radiation sources generated by high laser power harmonic generation on atoms, TaiG eb et al. [157] showed that, via a two-colour photoionization scheme involving a strong laser and such a short UV pulse, one can follow in time the population dynamics of the atomic states dressed by the time-varying laser "eld. The idea is to monitor the modi"cations of standard above threshold ionization spectra when a (weaker and much shorter) UV pulse is added at di!erent times during the laser pulse. Their predictions were supported by accurate numerical simulations involving the solution of the TDSE for a onedimensional model atom. In fact, they showed that short pulses of high harmonic UV radiation can be used to provide a time-dependent picture of the structure and population dynamics of atomic dressed states in a strong laser pulse. This could be achieved by monitoring the changes in the positions and magnitudes of related photoelectron lines originating from a two-colour process, while the short UV pulse scans the longer laser pulse. Their results indicated that the magnitudes of the changes should be large enough to be accessible to experiment. 3.4.2. CPC in ATI by two-colour commensurate frequencies In the above paper by VeH niard et al. [154] the TDSE was solved numerically for hydrogen in a bichromatic "eld, using methods developed by Schafer and Kulander [98,158]. They considered a powerful low-frequency pulse of a Ti : sapphire laser of frequency
"1.55 eV and chose as * high-frequency radiation of moderate power the 13th harmonic
"13
with intensity & * I "3;10 W cm\, which they kept "xed, while for the low-frequency laser pulse they took the & intensities (a) 5;10, (b) 3;10 and (c) 1.75;10 W cm\. In Fig. 15 we show the corresponding ATI spectra evaluated for the two "elds. While the Figs. 15(a) and (b) are similar to those shown in Fig. 14, Fig. 15(c) demonstrates that with increasing power of the laser "eld we get an interference with the direct ATI in the absence of the high-frequency "eld, while for lower laser intensities we observe the laser-assisted high-frequency ionization process, where the laser power is too low to yield direct ionization. Also the change of the ponderomotive energy shift ; , proportional to the  laser intensity I, gets visible in Figs. 15(a)}(c). As soon as the direct ATI process has comparable probabilities with the laser-assisted ATI, the spectrum becomes phase dependent. This is shown in Fig. 16. Here are shown the variations of the intensities of the two photoelectron peaks, which display a typical dependence on the relative phase  at t"0 between the laser "eld and its harmonic. As expected from our earlier discussions, one observes important changes in the magnitude of the peaks when the phase is changed from "0 to
/2 and to
. We note that the changes are quite important in spite of the fact that, in contrast to our other studies on phase e!ects in this report where the "elds had comparable intensities, the "eld strengths are very di!erent, since the ratio I /I K10\. We should also & * observe that, by studying the phase dependence of such two-colour ionization yields, one should be able to get an estimate of the relative values of ATI and laser-assisted transition amplitudes. In further investigations by VeH niard et al. [159] the above considerations were generalized. Here they solved the TDSE to calculate the main properties of atomic photoelectron spectra as they are obtained by using a radiation pulse containing N#1 frequencies associated with a `Dirac comba of N higher harmonics together with the laser which has been used to generate them. As before, they considered the physically relevant situation in which the harmonics have much weaker intensity than one of the laser. In such (N#1)-colour photoionization process, the atom can simultaneously absorb harmonic UV photons and exchange, i.e. absorb or emit, (via stimulated

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217

Fig. 15. Demonstrates the e!ect of the laser intensity on two-colour photoelectron spectra, for radiation pulses containing the fundamental frequency of a Ti : sapphire laser,
"1.55 eV and its 13th harmonic with a "xed intensity * I "3;10 W cm\. (a) I "5;10 W cm\; (b) I "3;10 W cm\; (c) I "1.75;10 W cm\. Note the & * * * changes in scale. With increasing power of the laser "eld one observes interference with direct ATI in the absence of
& (see Ref. [154]).

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Fig. 16. Indicates the dependence of two peaks of the two-colour photoelectron spectrum on the initial relative phase  of the Ti : sapphire laser "eld and its 13th harmonic. The dot-dashed line represents the ATI spectrum for the laser
alone. Long-dashed line, "0; full line, "
/2; dashed line, "
. Note that the changes are quite important in * spite of the fact that, in contrast with most previous studies on phase e!ects where the "elds had comparable intensities, the "eld strengths are very di!erent, since the ratio I /I "10\ (see Ref. [154]). & *

emission) laser IR photons. The results of their calculations show that, everything else kept "xed, the magnitude of the photoelectron peaks are strongly dependent on the di!erence of phases between successive harmonics. This strong dependence results from interference e!ects taking place between competing quantum paths leading to a given "nal state. An interesting feature is that these interferences involve transitions in the continuum states of the atom and do not depend on resonances in the discrete spectrum. Another interesting outcome of this study was to show that such e!ects should be observable with currently developed harmonic sources. In the case of a single harmonic frequency
"(2n#1)
one could understand these phenomena by generalizing & * our formula (3.13) to phase-dependent commensurate frequencies. This yields by means of the corresponding generating function of generalized Bessel functions, calling them D , namely , exp i[a sin
t#b sin((2n#1)
t#)], the approximate cross section formula * * d d , "  D [ (p )
),  (p )
); ] , & d , * d

(3.14)

where we called the appropriate prefactor of (3.13) d /d and D is found from the generating  , function to be given by > D (a, b; )"  J (a)J (b) exp(!i ) . , ,\L>H H H\

(3.15)

Without di$culties one could generalize these formulae to a whole set of harmonics to understand qualitatively, on account of the rapid variations of the Bessel functions, the above observed features.

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3.4.3. X-ray ionization and CPC in a two-colour xeld The other extreme where there is a high-frequency "eld,
, of low intensity and a low-frequency & "eld, not necessarily commensurate, of two components
and 2
which are out of phase by an * * angle
was considered by Bivona et al. [160] for the investigation of CPC e!ects in the photodetachment of H\ describing this atomic system by a short-range -potential having a single bound state [161]. Since the energy required for photodetachment is very low, E "!0.57 eV, it  is su$cient to take for the high-frequency "eld
"1.5 eV such that





d V  d V , "  B  (p )
),  (p )
);
. ,  d 4 d

(3.16)

In Fig. 17 we show the results of the above authors for the DCS of photodetachment and the CPC e!ects for the absorption of N photons from the bichromatic "eld as a function of the emission angle  with respect to
for
"
/2 (thin line) and
"0 (thick line). Evidently, the CPC e!ect is considerable. In the same spirit we investigated [162] X-ray photoionization of hydrogen in the presence of a bichromatic laser "eld of components
and 2
. The laser-dressing of the ground state of hydrogen was taken into account in "rst-order TDPT in the bichromatic "eld as in (2.15) of Section 2.2.2 and the laser-dressed outgoing electron wave was approximated by our improved Coulomb}Volkov wave (the "rst term in the curly brackets of (2.22), neglecting the contributions of the discrete states) of Section 2.2.3. The evaluation of the ¹ -matrix elements of laser-assisted X-ray L

Fig. 17. Presents DCS of the photoelectrons emitted into the channel characterized by n"5 for two di!erent values of the relative phase
: thin line,
"
/2; thick line,
"
. The radiation "eld parameters of two-colour (
, 2
) * * photodetachment of H\ are
"1.5 eV and
"0.01 eV and both components of the low-frequency "eld have the & * intensity I "2;10 W cm\ (see Ref. [160]). *

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ionization of hydrogen in a bichromatic "eld was then straightforward. Using the r ) E(t)-gauge, it turned out to be very simple to evaluate analytically the symmetry properties of the ¹ -matrix L elements. We showed that the symmetry (
#
,
!)  (
, ) is valid exactly, where
is the relative phase between the two laser components and  is the polar angle of the outgoing electron. In addition there is an approximate symmetry
 !
and

#
. All these symmetries, as well as the behaviour of the di!erential and total cross sections as a function of (
, ) and the number of exchanged photons n, were analyzed. The results presented showed the feasibility of CPC in laser-assisted X-ray photoionization. We also showed that in the monochromatic case our

Fig. 18. Shows the DCS for X-ray (
"50 eV) photoionization assisted by a bichromatic laser "eld of fundamental 6 frequency
"1.17 eV and its second harmonic with equal intensity I"3.51;10 W cm\ as a function of the relative phase
and the polar angle , for one absorbed photon (n"1). The polarization geometry considered is
"
6 (see Ref. [162]).

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model gives results which are in good agreement with the results of previous work by Cionga et al. [155], discussed earlier. In Fig. 18 we show the DCS for X-ray (
"50 eV) photoionization 6 assisted by a bichromatic "eld (
, 2
) of frequency
"1.17 eV and equal intensity I"3.51;10 W cm\ as a function of the relative phase
and the polar angle  for one absorbed laser photon and parallel geometry of the X-ray and laser polarizations,
"
. The strong 6 * phase-dependences and the various symmetry relations are well recognized.

4. Harmonic generation in a bichromatic 5eld By considering the unitarity of the S-matrix one can easily understand that higher harmonic generation (HHG) of the laser "eld frequency
is a concomitant process of above threshold ionization (ATI). Of course, in a more elementary way the connection between these two processes can be understood on the basis of the classical `simple man'sa theory as suggested by van Linden van den Heuvell and Muller [163], Gallagher [164] and Corkum [137]. According to this model, the ionized electron can be described by the classical equation of motion md *"eE sin
t. Its R time-integration yields m*"(eE/
)(cos
t!cos
t ) (assuming v "0 at time of ionization t ),    from which we get the maximum average kinetic energy which the electron can acquire in the "eld after ionization to be E "3; . If the electron gets kicked back to the nucleus by the    oscillating electric "eld vector E(t) of the laser radiation and gets reabsorbed by the residual ion, then the maximum harmonic photon energy will be
"I #3; as observed experimentally

   and con"rmed by detailed theoretical investigations. However, the electron may also rescatter from the ion and by induced bremsstrahlung absorb further photons from the laser "eld yielding the extended plateau of the ATI spectra having a cuto! at about 10; , as discussed in Section 3.3.2. In  this case, the average initial energy of the electron after ionization and before rescattering will be 2; . Solving the above Lorentz equation of motion with this initial condition, we "nd that during  rescattering the electron can acquire the maximum average kinetic energy 10; , as con"rmed  experimentally and theoretically. Translating this classical consideration into quantum mechanical language, a simple S-matrix model based on the KFR theory can be designed. We suggested such a theory which reproduced all basic features of HHG [165]. The same idea was taken up independently and worked out in greater detail by Lewenstein et al. [166]. In addition to those references, mentioned in the introduction in Section 1.1, in particular, the books by Gavrila [3] and Delone and Krainov [4] and the reviews by Protopapas et al. [77] and DiMauro and Agostini [78], we mention here the very nice recent reports on HHG by Platonenko and Stelkov [167] and by Salie`res et al. [168]. These authors consider HHG in a single and a bichromatic "eld and investigate in some detail various model calculations, phase-e!ects and so on. In the present section we shall "rst consider, as in the case of ATI in Section 3, harmonic generation in bichromatic "elds of incommensurate frequencies and then go on to a discussion of the CPC e!ects in HHG in bichromatic "elds of commensurate frequencies. 4.1. HHG in incommensurate bichromatic xelds In the present section we consider various combinations of bichromatic "elds of frequencies
and
and their e!ects on HHG. Such combinations may have di!erent polarizations,  

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di!erent intensities, large di!erences in frequencies and may be two "elds of equal frequency but having a time delay between the two laser pulses. All these combinations are designed to eventually control the HHG process. In the work by Antoine et al. [169] the generation of ultra-short pulses of harmonics was investigated. To this end they considered a single atom that is exposed to two short perpendicularly polarized laser pulsed. The two perpendicular electric "elds oscillate at two di!erent frequencies
and
. Therefore, the resultant "eld has a polarization that depends on time. Since harmonics   are emitted when the resultant oscillating "eld is linearly polarized, it is to be expected that a short pulse of harmonics may be emitted if the external "eld is linearly polarized during a short period of time. They have shown that indeed the atom may emit an ultra-short pulse of a given harmonic. This result was obtained by time}frequency analyzing the acceleration of the induced dipole moment with a "lter whose frequency bandwidth is smaller than twice the frequency of the external "eld. Their calculation of the dipole acceleration was based on the numerical solution of the TDSE. They then addressed the question how far it is possible to reduce the duration of the emitted pulse of one given harmonic by adjusting both
and
and keeping the amplitude of this pulse   signi"cant. In order to answer this question, they used the quantum version of the two-step model of Lewenstein et al. [166]. The authors showed by this method that it is actually possible for the atomic system to emit an ultra-short pulse of a given harmonic, whose time duration is of the order of the optical period. They stressed that this short pulse duration can be controlled externally. In a paper by Gaarde et al. [170] a theory of high-order sum and di!erence frequency mixing in a strong bichromatic "eld was developed. They presented a study of the interaction between free atoms and a two-colour "eld, consisting of a strong infrared laser "eld,
, and a much weaker * (by three orders of magnitude) optical "eld of variable frequency,
. Single-atom data as well as & calculations including propagation were shown. For the single-atom response, the authors generalized the quantum mechanical two-step approach of Lewenstein et al. [166] to the case of a two-colour "eld. Here they included the second "eld as a perturbation to the atom and the "rst "eld and went to second order in the sense that they allowed the second "eld to contribute one or two photons to the HHG process. This approximation was justi"ed because the second "eld was indeed much weaker then the "rst one and further motivated by the wish to do a full calculation including propagation. A detailed knowledge of the amplitude and phase of the atomic polarization as a function of the intensities of the driving "elds was necessary for propagation calculations. Then in their work the propagation of harmonics was generalized to the case of two-colour frequency mixing. The propagation equations were solved within the paraxial and slowly varying envelope approximation using the dipole-moments (dependent on the laser intensity of the intense "eld) expanded to second order in the weak "eld. They observed a di!erence between sum and di!erence frequency mixing. This di!erence was present already in the single-atom response and was enhanced due to phase matching. It came from the fact that the addition of even a weak second "eld in#uences the generation process. Di!erent interference e!ects result in di!erent phase response of the two processes. Comparison of their data with the corresponding results of the experiments of Gaarde et al. [171] showed very good agreement. Phase-matching e!ects in HHG have also been considered by Kan et al. [172]. In their work they solved the one-dimensional wave equation for high-order mixing of the type
"q
!l
&   (q
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mixed "eld can be increased by about an order of magnitude compared to non-phase-matched HHG but that fundamental limitations are introduced by axial electron density gradients. In a paper by Luca and Fiordilino [173] the Morlet wavelet spectrum of the radiation emitted by a two-level atom in presence of two laser pulses with very close frequency were obtained. The wavelet spectrum gives information on the time evolution of the full HHG spectrum and of a particular line. The beating condition stimulates the atom to emit pulses of harmonics with duration of the order of a few optical cycles of the pumping radiation. Pulse trains of three optical cycles were observed. Although the two-level atom does not permit ionization, its HHG spectrum is very similar to what has been observed experimentally. In particular, if the driving laser pulses are short, the probability of ionization tends to decrease. In the present work the HHG spectrum of the two-level atom was evaluated with the linearly polarized driving "eld given by E(t)"E P(t) cos[(1#)
t]#cos[(1!)
t] * *   "E P(t) cos[
t] cos[
t] ,  * *

(4.1)

where (1 and P(t) is the form of the pulse. If  is small enough, the superposition of the two "elds appears as a train of pulses oscillating with frequency
and modulated in amplitude by * P(t) cos(
t). The authors solved the TDSE with laser}atom interaction energy given by * <(t)"!eE(t) ) r by using an iterative analytical method proposed earlier [174]. The evaluation of the single harmonic lines, as shown in Fig. 19, display how the harmonics are emitted. One observes for every harmonic an emission as a train of pulses. The duration of a single spike strongly depends on the parameter . As  increases, the number of pulses increases and conversely the duration of a single spike decreases. Similarly, the generation of a harmonic quasi-continuum from beating laser "elds was addressed in a paper by Preston and Watson [175]. They presented simulations of the two-colour harmonic response of a one-dimensional model atom exposed to a superposition of two laser "elds of varying relative intensities and of non-commensurate frequencies. They showed that this interaction can be used to e$ciently generate a quasi-continuum of very closely spaced harmonics and they explained how such a scheme can be implemented experimentally. They showed how the generation process can be described in terms of beat frequencies between the two "elds. Another interesting control mechanism of harmonic generation was considered by Sanpera et al. [176]. Here they showed that by preparing the initial state as a coherent superposition of bound states one can obtain a harmonic spectrum with distinctive plateaus having di!erent conversion e$ciencies. They demonstrated how this scheme may provide a way to controlling the coherent output that is produced in an experiment. Their investigations were based on quantum interference e!ects in recombination via di!erent states. They prepared the initial state in a superposition of the ground state g and some excited state denoted by e with a "xed though arbitrary phase di!erence between both states, viz. (t)" g# exp(!i
)e

(4.2)

with #"1. To simplify the solution, they chose in particular
"0. The laser parameters (I and
) were taken such that only the excited state is depleted by ionization. They did this because it is su$cient, and requires much lower intensities, to promote the electrons into the continuum

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Fig. 19. Presents the temporal evolution of the 7th and 19th harmonic for "0.025. We observe for every harmonic an emission as a train of pulses. The duration of a single spike strongly depends upon the parameter . As  increases, the number of pulses increases and conversely the duration of a single spike decreases. The basic parameters of the calculation were < /
"2 and
"5
. Here < "eE ) r and
"3.27 eV (0 and 1 refering to ground and  *  *     excited state, respectively) while the laser intensity was I"8.8;10 W cm\ (see Ref. [173]).

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Fig. 20. Shows HHG spectra from the He> ion at I"1.15;10 W cm\ and
"0.042 au ( "1 m) starting from a coherent superposition of the ground and "rst excited state (2s) with equally weighted populations (1/(2)(g#2s) (solid line). Full dots correspond to the harmonic spectrum computed from the dipole projection onto the excited state and the squares correspond to the dipole projection onto the ground state. (The pulse lasted 16 cycles and was linearly ramped in the two "rst cycles.) (see Ref. [176]).

from the excited state. Since they aimed to describe a rather general way of possible control over the harmonic emission, they performed their calculations for a single hydrogenic He>. The results presented are not directly related to the structure of the atomic potential and therefore can be extended to any type of ion or atom. In a foregoing work [177] they demonstrated that the harmonic spectrum obtained from an initial state of form (4.2) consists of two distinct plateaus. One can see this by splitting the various contributions of the dipole acceleration for the coherent superposition (4.2). As an example, we present in Fig. 20 their results for harmonic generation from He> at I"1.15;10 W cm\ and
"0.042 au ( "1 m) starting from a coherent superposition of the ground state and the "rst excited state (2s) with equal weighted populations (1/(2)(g#e) (solid line). Full dots correspond to the harmonic spectra computed from the dipole projection onto the excited state and squares correspond to the dipole projection onto the ground state. It was stressed that the method does not allow to generate shorter harmonic wave length than the ones obtained from the ground state of the atom or ion alone (the cuto! will be always
"I #3; ). The method does, however, require much weaker intensities than in

   the latter case, and the signals corresponding to transitions back to the initial state are greatly enhanced compared to the case in which all the population is initially in the ground state. The initial coherent state can be prepared by a single laser "eld and one of its harmonics. In the work of Taranukhin and Shubin [178] HHG by atoms in strong bichromatic "elds was considered. They proposed to use bichromatic laser "elds to increase the e$ciency of generation of coherent short-wavelength radiation by atoms in the process of ATI in the tunneling regime. They showed in the framework of a semiclassical model that the use of a relatively weak static electric "eld, along with the basic pump radiation, results in the increase of the fraction of recombining photoelectrons and in the rise of their kinetic energy at the moment of recombination. They demonstrated that the experimental realization of this e!ect is possible with the use of a high-power

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CO laser radiation (instead of the static electric "eld) and ultra-short pulses of the basic pump  component with wavelength K1 m. Numerical calculations in the framework of a quantum mechanical model con"rmed the conclusions of the semiclassical model and showed that the addition of long-wavelength radiation enables tripling the maximum generation frequency. Under speci"c intensity of CO laser radiation, the so-called multi-plateau structure was discovered near  the high-frequency edge of the generated radiation spectrum. An interpretation of this e!ect was given in the framework of a semiclassical model. 4.2. HHG and CPC in commensurate bichromatic xelds Here we shall discuss work that has been done on HHG in bichromatic "elds mainly of commensurate frequencies in the combinations (
, 2
) and (
, 3
) and related processes of HHG depending on the CPC e!ects. Model calculations of polarization-dependent two-colour HHG were performed by Long et al. [179]. They evaluated emission rates for HHG by a zero-range potential model atom, proposed earlier [180}182], in the superposition of two monochromatic plane-wave "elds. Several polarizations of the driving "elds were considered: two linear polarizations enclosing an arbitrary angle, and two circularly polarized "elds that co- or counter-rotate in the same plane. Transition amplitudes were obtained in the form of sums of one-dimensional integrals that have to be computed numerically. For commensurate frequencies of the driving "elds the results depend critically on the relative phase between the two "elds. Parallel driving "elds are not always more e$cient in HHG than perpendicular "elds; also, two circular polarizations can be at least as e!ective. The odd harmonics of one "eld are usually weakened by the addition of the other "eld in favour of the mixed harmonics. If the ratio of the frequencies of the two incident "elds equals the ratio of two odd integers, then harmonics with elliptic polarization can be generated by two linearly polarized driving "elds. Harmonics with circular polarization can be readily produced with the help of two incident circularly polarized "elds whose "eld vectors co-rotate or counter-rotate in the same plane. It is important to point out that, for example, in the superposition of circularly polarized driving "elds small deviations from perfect circular polarization may have a dramatic e!ect on the observed spectra. Similarly, for linearly polarized driving "elds small deviations from the perfect perpendicularity may completely alter the spectra. Equally important, for commensurate frequencies variations of the relative phase can easily a!ect the intensity and ratio of nearby harmonics by two orders of magnitude. The e!ects are likely to make the interpretation of experimentally observed spectra very tricky, in addition to the collective e!ects which are understood in principle, but nontheless not easy to identify in practice. On the other hand, the single-atom polarization and phase e!ects o!er great potential for coherent control of high harmonic emission. Comparison of the results of this paper with experiments will be discussed in the next section (see Section 5.2.1). A particular example of the above mixing scheme, namely of generation of circularly polarized high-order harmonics by two-colour coplanar "eld superposition was considered very recently by Milos\ evicH et al. [183]. They investigated an e$cient method for the generation of circularly polarized high-order harmonics by a bichromatic laser "eld whose two components with frequencies
and 2
are circularly polarized in the same plane, but rotate in opposite directions. The generation of intense harmonics by such a driving-"eld con"guration was already con"rmed by

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227

a previous experiment (see Section 5.2.1). With the help of both a semiclassical three-step model as well as a saddle-point analysis, the mechanism of harmonic generation in this case was elucidated and the plateau structure of the harmonic response and their cuto!s were established. The sensitivity of the harmonic yield and the polarization of the harmonics to imperfect circular polarization of the driving "elds were investigated. Optimization of both the cuto! frequency and the harmonic e$ciency with respect to the intensity ratio of the two components of the driving "eld were discussed. The electron trajectories responsible for the emission of particular harmonics were identi"ed. Unlike the case of linearly polarized driving "eld, they have a nonzero start velocity. By comparison with the driving-"eld con"guration where the two components rotate in the same direction, the mechanism of intense harmonic emission was further clari"ed. Depending on the (unknown) saturation intensity for the bichromatic "eld with counter-rotating polarizations, this scheme might be of practical interest not only because of the circular polarization of the produced harmonics, but also because of their production e$ciency. The CPC e!ects in HHG were also investigated for high-intensity two-colour interactions in the tunneling and stabilization regime by Protopapas et al. [184]. Here, the two-colour excitation of a hydrogen atom was studied by solving numerically the TDSE. They considered the two laser frequencies to be the fundamental
and its third harmonic 3
with an adjustable phase
between the two laser "elds such that the interaction term of the atom with the "eld is given by <(t)"!er )
[E sin
t#E sin (3
t#
)] f (t) ,  

(4.3)

where E and E are the "eld amplitudes of the two-frequency components of equal polarization
.   The pulse-shaped function was chosen in the form f (t)"sin(
t/), with  being the pulse length, taken to be 32 optical cycles of the fundamental "eld. To produce maximum enhancement of the harmonic conversion e$ciency, they chose E "E . With these conditions, the time-varying   electric "eld has a maximum for a relative phase-di!erence between the two "elds of
"
, when the maximum "eld strength of the combined "elds reaches the value E"2E twice each cycle. For  a phase-di!erence of
"
/2 the maximum amplitude achieved is E"1.8E and "nally for
"0  the maximum "eld amplitude is E"1.6E . Their calculations have been extended into the  high-intensity domain and well into the tunneling and stabilization regimes. They showed that in the tunneling regime the two-colour excitation can enhance the intensity of the harmonics by more than two orders of magnitude compared to a single colour at the same e!ective intensity. In the stabilization regime they con"rmed that two-colour excitation can be used to generate harmonics. As an example of their results we present in Fig. 21 data for a hydrogen atom irradiated by the combination of the Nd : YAG laser frequency
"1.17 eV ( "1064 nm) and its third harmonic ( "345 nm) at E "E "0.05 au (I"8.3;10 W cm\). For the sake of comparison, they    also included the ionization produced by the fundamental laser alone at E and at twice this  amplitude E"2E . The "gure shows the harmonic spectra in the tunneling regime for one- and  two-colour cases (the corresponding ionization yields are shown in the inset). All these spectra were normalized to the "rst harmonic so that a better comparison of conversion e$ciencies can be made. For clarity, only the harmonic peak intensities at the frequencies encountered are plotted. A dotted line is used to join the "rst harmonics with the cuto! region where the harmonics are clearly produced again. The most striking di!erence in the data is the di!erent conversion e$ciency in both cases, regardless of the relative phase. Similar calculations for hydrogen were performed and

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Fig. 21. Presents comparison between one- ("lled symbols) and two-colour (open symbols) harmonic spectra in hydrogen in the tunneling regime. Filled circles correspond to E"0.05 au; "lled squares to E"0.10 au. Two colours E"E "0.05 au and
"0 (open circles) and
"
/2 (open squares). The inset window shows the corresponding  ionization yields (see Ref. [184]).

conclusions drawn in the work of Telnov et al. [185], using a time-independent Floquet approach, developed by Wang et al. [95]. In the work of Milos\ evicH and Piraux [135], HHG in a bichromatic elliptically polarized laser "eld was considered. The strong "eld approximation of HHG, developed by Lewenstein et al. [166], was generalized to the case of a bichromatic elliptically polarized radiation "eld. The quasi-classical cuto! law of Krause et al. [121] and Corkum [137] was analyzed for this bichromatic case. Numerical results for a linearly polarized bichromatic laser "eld were presented and analyzed for di!erent laser "eld frequencies, intensities and relative phases. Harmonic intensities are many orders of magnitude higher in the bichromatic case than in the monochromatic one, but the cuto! is shifted towards the lower harmonics. The plateau height can be controlled by changing the relative phase of the "elds. A qualitative agreement with recent experiments was shown. As a representative example for the CPC e!ects we present in Fig. 22 their results for a linearly polarized bichromatic "eld of components
and
"3
of equal normalized    amplitude a"a "a "0.5. Here the "eld strengths were de"ned by E "A a
( j"1, 3) and   H  H A can be found from ; "A /4 with the choice ; "20
while the ground state binding energy     E "!13.6
, choosing
"1 eV. In the "gure di!erent values of the phase
were considered.  A time}frequency analysis of two-colour HHG was considered by Figueira de Morrison Faria et al. [186]. They investigated the validity of the classical rescattering models for describing high harmonic generation with coherent bichromatic driving "elds. They compared the fully timedependent solution of the one-dimensional SchroK dinger equation to the predictions of the semiclassical three-step model, with particular emphasis on the return times and cuto! energies. The classical return times can be compared to the time of harmonic emission, obtained from the solution of the TDSE with the help of a time}frequency analysis. Their strikingly coincident results indicated that also in the bichromatic case, provided the "eld is su$ciently intense and the two

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229

Fig. 22. Shows the harmonic intensity as a function of the harmonic order for the Gaussian model [166]. The parameters are
"3
, a "a "0.5, ; "20
, I "13.6
,
"1 eV, and the data presented are for di!erent values of the          relative phase
: (a)
" 0, 3
/2,
/2 and (b)
"
,
/3, 2
/3. From these "gures one can see that both the height of the plateau and the position of the cuto! change with the change of
(see Ref. [135]).

driving harmonic "elds are of comparable intensity, the three-step model describes the essential physical process and has a very good predictive power. Data were presented for a two-colour "eld of frequencies
and 2
of equal linear polarization
and equal "eld amplitudes E. In a further investigation by Figueira de Morrison Faria et al. [187] the phase-dependent e!ects in bichromatic high-order harmonic generation were considered. They addressed high-order harmonic generation

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with linearly polarized bichromatic "elds, concentrating on a modulation in the harmonic yield as a function of the relative phase between the two "eld components, and on an o!set phase shift of this modulation for neighboring cuto! harmonics. These e!ects were recently observed in experiments where the relative phase between the two driving "elds was controlled. Using the three-step model and the fully numerical solution of the TDSE, they discussed the phase-dependent modulation and showed that the o!set phase is inherent to a particular set of semiclassical trajectories for the returning electron. These trajectories were identi"ed using classical arguments and isolated by means of the saddle-point method, which allows a detailed investigation of their interference. Thus, by adding a second driving "eld whose amplitude lies within an adequate parameter range, they were able to single out a set of trajectories according to its behaviour with respect to the relative phase. This e!ect is already present at the single-atom-response level. The CPC of HHG in a two-level atom driven by intense two-colour laser "elds was investigated by Gong et al. [188]. This model is known to generate high harmonics as well as hyper-Raman lines with a plateau and a cuto!. Furthermore, for one laser "eld case, they found, when the hyper-Raman lines are located at the position of the high harmonics, that the high harmonics are strongly enhanced due to interference e!ects between the high harmonics and the hyper-Raman lines. Within these conditions, by combining the laser "eld with another much weaker laser "eld, they showed the appearance of plentiful and strong high harmonics in the light spectrum of the driven atomic system. For their numerical examples they chose similar parameter conditions as in the foregoing work. In the model of Lewenstein et al. [166], mentioned previously, the main e!ect of harmonic generation comes from the quasi-classical motion of the ionized electron in the laser "eld, described by the semi-classical action, introduced in formula (3.8) of Section 3.3.2. Cormier and Lambropoulos [189] discussed instead in their paper the e!ect of a strong dressing "eld on the polarizability of atomic helium at harmonic frequencies. This study was carried out by solving numerically the TDSE describing a one-electron atom interacting with the combination of two "elds: a strong pump "eld at the fundamental frequency
and a weak probe "eld at some harmonic frequency N
. It was then examined how the dressed atom scatters the probe radiation as compared to the bare atom. This was readily achieved by comparing calculations with the pump "eld on and o!. In all cases, the probe high-frequency "eld was kept at low intensity, so as to not in itself induce a nonlinear behaviour. The calculation for both "elds was nontheless performed through the non-perturbative solution of the TDSE using techniques that were developed and applied to strong "eld problems earlier [190,191], and, in order to obtain quantitative answers corresponding to realistic laboratory situations, they choose to study this problem for helium atoms driven by a strong pump laser of
"2 eV photon energy. The "eld peak intensity of the pump "eld varied from I"10 to 2;10 W cm\. The harmonic distribution was obtained by Fourier transforming the time-dependent electric dipole



S(
)"d(
)"

1 ¹

2



(r, t)r )
(r, t)exp(i
t) dt



,

(4.4)

where (r, t) is the numerical solution of the TDSE. Calculations of the polarization were performed for a particular odd harmonic frequency, say N
, in the presence or absence of the strong dressing "eld of frequency
. Therefore, the polarization of the atom will result from its

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231

Fig. 23. Demonstrates the e!ect of the mixing of a powerful fundamental "eld
and its weak 13th harmonic on the complete spectrum of HHG. In the present case
"2 eV, and I "2;10 W cm\. Observe the enhancement of S the HHG probabilities due to the presence of the 13th harmonic "eld of low intensity, in particular, near the 13th order (see Ref. [189]).

interaction with the following bichromatic "eld: E(t)"
[E sin
t#E sin(N
t#
)] f (t) , (4.5) , where E is the Nth harmonic probe "eld peak "eld strength and E refers to the pump (dressing) , laser "eld. Both were taken of equal polarization
and f (t) is the pulse envelope. As an example we show in Fig. 23 the e!ect on harmonic generation of mixing of the fundamental "eld with

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I"2;10 W cm\ and the 13th harmonic with I"0, 10 and 10 W cm\. The corresponding phase-dependence of the spectra, i.e. the CPC e!ects, turn out in this case to be rather marginal. As our "gure shows, the modi"cation of the polarizability of an atom due to the non-perturbative dressing by a strong pump "eld has a signi"cant e!ect on the corresponding harmonics, in particular, for frequencies close to the weak harmonic "eld.

4.3. X-ray}atom scattering in a bichromatic xeld Closely related to the foregoing problem is the investigation of CPC e!ects in X-ray}atom scattering in the presence of a bichromatic "eld considered by MilosH evicH and Starace [192]. These authors considered the (elastic) scattering of 50 eV X-rays by hydrogen atoms in the presence of a bichromatic, linearly polarized laser "eld with frequencies
and r
, where r"2, 3. They chose
"1.17 eV and varied the relative phase  between the bichromatic laser "eld components. Numerical results for the DCS as a function of the number n, where n
is the energy exchanged with the laser "eld, were presented. For either a monochromatic laser "eld or a bichromatic laser "eld with the frequencies
and 3
, the integer n can only be even, while for a bichromatic "eld with frequencies
and 2
, the integer n can have any value. Both conditions follow from elementary parity considerations. For small values of n they found pronounced maxima in the DCS. For large n they obtained plateaus which are di!erent for negative and positive values of n. For lower values of the laser "eld intensity, the plateau for negative values of n is much more extended and greater

Fig. 24. Presents the DCS of X-ray-hydrogen atom scattering in units of r as a function of the number n of emitted or  absorbed laser photons. n
is the energy exchanged with either the monochromatic laser "eld of frequency
"1.17 eV (curve denoted by `monoa) or with the bichromatic laser "eld with frequencies (
, 3
) and the relative phase "
. The intensities of both laser "eld components are I"10 W cm\. The energy of the incident X-ray photon was chosen
"50 eV. The three radiation "elds were parallel linearly polarized. The "gure shows that, as in the monochromatic 6 case, the number n can only be even but that now the DCS are increased by many orders of magnitude, depending on the value of n and on the relative phase  (see Ref. [192]).

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233

in magnitude than that for positive values of n. The height of either plateau is also higher for the bichromatic laser "eld than for a monochromatic "eld. In the (
, 2
) case they derived the symmetry relation of the cross sections DCS(#
)"DCS(). The authors showed that the relative phase  in#uences the DCS so that CPC of the X-ray}atom scattering process is possible. For some values of , the energy of the outgoing X-ray photons can be increased. Finally, a quasiclassical explanation of the results was presented. The method of calculation used for that paper was a generalization of our earlier work [193]. A typical example of their results is presented in Fig. 24. It shows the DCS of X-ray hydrogen atom scattering at
"50 eV in units of r 6  (r being the classical electron radius) as a function of n of absorbed or emitted laser photons 
"1.17 eV for a bichromatic "eld of components (
, 3
) as a function of
at a laser "eld intensity for both components of I"10 W cm\.

5. Experiments on CPC and related phenomena In our discussion of the experiments performed on CPC of laser-induced processes in the atomic domain we shall "rst consider what has been done on the ionization of atoms and then proceed to a discussion of HHG experiments in bichromatic "elds. There were, of course, also experiments done on other processes which we shall consider later on. Not all of these experiments were devoted to the CPC e!ects but other control mechanisms were treated instead which we shall discuss besides. 5.1. Ionization of atoms in two-colour xelds 5.1.1. Experiments in the perturbative regime The "rst experiments on ionization of atoms in a phase-dependent bichromatic laser "eld were performed in the perturbative regime. In particular, there is the set of experiments performed by Chen et al. [194], Chen and Elliott [195], Yin et al. [196] and a comment by Chan et al. [197]. In the "rst of these experiments the excitation and ionization of mercury in a bichromatic "eld of frequencies
and 3
via the transition 6sS P6pP was considered as depicted in Fig. 25. The   fundamental "eld
had phase
and the harmonic "eld phase
. The average ionization rate can   then found to be wK[1#(5/8)M](M/2) cos (3
!
) , (5.1)   where M describes the relative contribution of the two processes, depicted in Fig. 25. By sending the two laser beams through a chamber with argon gas and changing the gas pressure the relative phase of the two radiation "elds can be changed on account of the di!erent refractive indices of argon for the
and 3
"elds. The corresponding variation of the ionization yield as a function of the gas pressure (hence of the relative phase) is shown in Fig. 26. In the further experiments their investigations were performed with improved sophistication and, in particular, the angular distribution of the ionized electrons of rubidium via the transition 5sS P
P in a "eld of frequencies 
and 2
was measured and the corresponding phase-dependence of the ionization yields. A sketch of the setup of this experiment is shown in Fig. 27 and the phase-dependence of the ionization yields for di!erent polar angles  is depicted in Fig. 28. In this case an asymmetric distribution of the

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Fig. 25. Shows the two processes which coherently interfere in the present experiment of Chen et al. The transition gP f  is three- and one-photon allowed, as shown in (a) and (b) (see Ref. [194]).

Fig. 26. Presents the ionization signal measured in the experiment of Chen et al. as a function of the argon pressure in the phase-changing gas chamber. Solid line indicates a best "t to the data. Error bars showing 1 standard deviation of the mean are shown for a few data points (see Ref. [194]).

ejected photoelectrons was observed from a symmetric atom and this asymmetry can be reversed through variation of the relative phase of the two "eld components. Another experimental determination of the cross sections and continuum phases of rubidium through complete measurements of atomic multiphoton ionization was performed recently by Wang and Elliott [198] using a single elliptically polarized laser "eld.

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Fig. 27. Shows the experimental setup of the experiment of Yin et al. Elements in the diagram include the dye laser (DL), mirrors (M), beam splitter and combiner (S), BBO doubling crystal (B), UV transmitting "lter (F), /2 retardation plate (P), and phase delay cell (DC). The polarization of the "elds were as indicated. The four-channel electron multiplier (CEM) detectors were positioned as shown (see Ref. [196]).

Fig. 28. Presents the experimental data of the work of Yin et al. The total electron count as a function of the pressure of the N gas in the phase delay cell for the four detectors positioned at (a) 03, (b) 453, (c) 903 and (d) 1803. The solid line is the  result of a least-squares "t of a sinusoidally varying curve to the data (see Ref. [196]).

Such an asymmetry in the angular distributions, as mentioned before, was predicted on general grounds and observed experimentally for the "rst time by Baranova et al. [199}201]. (See also the work by Anderson et al. [104] and Baranova et al. [105] mentioned in Section 3.3.2.) Performing

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Fig. 29. Indicates ionization of an atom produced simultaneously by single-photon (2
) and by two-photon (2;
) transitions.
is the ionization potential (see Ref. [199]).  Fig. 30. Presents the atomic levels and ionization paths involved in the interference process of the experiment of Cavalieri et al. The Nd : YAG emission (
) and its third harmonic (
) ionize the sodium atom, initially prepared in the 3P    state by the dye laser radiation (
) (see Ref. [202]). 

a simple model calculation for the perturbative ionization of an atom in a bichromatic "eld (
, 2
), described by a -potential having a single-bound state, one can easily show by inspecting Fig. 29 that the photoionization probability has the form dw(n) "B f (E ) n)#f (E ) E)#f [(E ) n)!(E ) E)/3] ,     d

(5.2)

where n is the direction of electron emission and E and E are the "eld amplitude vectors of the two  laser "elds of frequency
and 2
, respectively. B, f , f and f are frequency-dependent amplitudes.    Evaluating the interference term between the single-photon and two-photon ionization, we get the polar asymmetry, which in the general case will be phase-dependent. The relative phase of the corresponding ionization experiment of sodium, 
"
!
, was in the present experiment   changed by using the di!erent refractive index of a glass plate for
and 2
. The general results of these experiments were essentially the same as those reported before. Experiments similar to the one by Chen et al. [194] were performed on the CPC e!ects in two-colour photoionization of sodium by Cavalieri et al. [202]. They considered the ionization scheme of two-colour ionization of sodium depicted in Fig. 30 in which the electron gets "rst excited by a dye laser from its 3S ground state into the 3P excited state from which it gets   ionized via two competing pathways into the continuum, absorbing either three photons from the fundamental Nd : YAG laser
"1.17 eV (1064 nm), or one photon from its third harmonic (355 nm). The CPC e!ects in the ionization process were observed by passing the bichromatic "eld through a cell "lled with argon gas, the pressure of which could be changed. This experiment has provided fringe visibility as high as 42%. If the path length through the cell is ¸, the phase di!erence between the two "elds is given by 
"3
!
"¸
(n !n )/c, where n is      I

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237

Fig. 31. Shows the partial energy level diagrams of krypton and xenon indicating the excitation/ionization channels employed in the experiment of Karapanagioti et al. for investigating CPC in a (
, 3
) "eld con"guration (see Ref. [204]).

the index of refraction at frequency
"k
(k"1, 3). The di!erence n !n is proportional to I   the pressure of the tuning gas. The same type of problems to observe the CPC e!ects in atomic photoionization in bichromatic "elds were considered by Karapanagiotis et al. [203,204] and by Papastathopoulos et al. [205]. In the "rst set of papers a four-photon resonant, "ve-photon ionization scheme, shown in Fig. 31, has been employed to achieve coherent control of the ionization of Xe and Kr. The control occurred through the simultaneous irradiation of atoms by a laser "eld and its third harmonic. Single- and three-photon absorption interfere at a virtual level, resulting in large modulation of ionization when the relative phase of the two "elds is varied, while in Xe, third harmonic generation is also used to probe the interference. The non-linearity of the two interfering ionization pathways ensured large modulation depth as both processes are naturally con"ned to the focus volume of the detecting region, while the fact that a high number of photons was used, allowed reaching high level of excitation. In the third paper ionization of Xe occurred via two phase-correlated electromagnetic "elds, namely, the fundamental and the second harmonic of the laser. Since the "elds were in the visible and ultraviolet regime, they had readily controllable amplitudes. Thus excitation interference conditions and, consequently, phase-sensitive ionization were readily and e$ciently established. Another control scheme has been considered by Bowe et al. [206] in order to investigate two-colour intensity suppression of multiphoton ionization of atomic deuterium. These authors reported a strong suppression of the three-photon ionization resonant with the 2s state of atomic deuterium at 243 nm by the introduction of a second colour (486 nm) coupling the 2s and 4p states. This coupling reduces the ionization probability by several orders of magnitude. The experiment was performed using fundamental and second harmonic frequencies of the same laser, with laser intensity up to a maximum value of 88 MW cm\ and an atomic target density of the order of 5;10 cm\. The production of both D> ions and D(2s) atoms was measured as a function of the

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intensity of the fundamental beam (486 nm). The experimental data were compared with a theoretical calculation using essential atom#,eld states and taking into account the experimental laser bandwidth. 5.1.2. High-power ATI experiments in two-colour xelds Next, we turn to those experiments on atomic ionization which were performed at much higher laser power of the bichromatic "eld such that ATI becomes possible. In a series of investigations by Muller et al. [207], Schumacher et al. [208] and Schumacher and Bucksbaum [209] ATI in a two-colour "eld of frequencies
and 2
was considered for krypton and xenon and the CPC e!ects were investigated with increasing precision and insight into the underlying mechanism. The fundamental laser was a Nd : YAG source with
"1.17 eV and intensity of about 10}10 W cm\ such that in the later experiments the Keldysh parameter "[I /2; ] was   of the order unity. Hence, the ionization process took place in the transition region between multiphoton and tunneling ionization. This permitted to interpret the experimental results of ionization in a "eld of the form E(t)"E cos
t#E cos(2
t#)  

(5.3)

by means of the `simple man'sa theory [163] in which ionization occurs in two steps. In the "rst the electron ionizes with a rate (t) that depends on the instantaneous "eld amplitude E(t) in the tunneling regime [210]



1 1 3 exp ! (I (t)"4(I E(t) E(t) 2



(5.4)

and then the electron propagates in the "eld as a classical particle, starting from rest at the moment of ionization t . Neglecting the ion "eld, the electron acquires in the bichromatic "eld (5.3)  (following the same line of calculation as at the beginning of chapter 4) the average kinetic energy E ";  #2;  sin
t #;        # 2;  sin(2
t #)#4(;  ;  sin
t sin(2
t #)      

(5.5)

with ; G "eE/4m
 (i"1, 2) being the ponderomotive potentials of the two "eld components. G G  According to (5.5) the energy is composed of the total ponderomotive energy ;  #;  and the   net drift energy of the electrons, which depends on the moment t of ionization and on the relative  phase . The energy spectrum is therefore phase-dependent. The laser pulse duration is long compared to the electron travel time out of the laser focus, so that the electron quiver energy component is converted into drift energy, and (5.5) is the energy seen at the detector. Since the tunneling potential is oscillating back and forth with the laser "eld, electrons get emitted in both directions along the laser polarization. The phase-dependent emission direction could be measured by a detector placed so that it only sees the current in one direction. The minimum energy electron, one that is at rest in the laser "eld, leaves the focus with the ponderomotive energy ;  #;  .   There is also a maximum energy. For a single colour, this is 3; , but for two colours it depends on  the relative phase . But the lowest and highest energy limits are sharp. The above tunneling

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Fig. 32. Presents the two-colour photoelectron spectra of the (
, 2
) experiment of Schumacher and Bucksbaum: krypton (top) and xenon (bottom). The krypton spectrum was taken with I "I "2;10 W cm\; the xenon   spectrum with I "I "1;10 W cm\. For both spectra the phase di!erence  beween the two "elds was zero. The   large and small peaks correspond to ATI transitions to the P and P ion ground state, respectively. The intensities   were chosen so that the fundamental light alone produces almost no signal, and the second harmonic alone produces only a few ATI peaks. The combined spectrum clearly indicates nonlinear combinations of the two "elds (see Ref. [209]). Fig. 33. Shows for the same experiment as in Fig. 32 the phase dependence of ATI for krypton at low intensity. Presented are the "rst through sixth P ATI peaks (bottom to top). Left side: data taken with I "I "2;10 W cm\. Right    side: results of the two-step model using I "I "4;10 W cm\. The error bars are statistical. ATI peak energy listed   in the corner of each data graph is in eV. Vertical axes are labeled to indicate the relative heights between the various ATI peaks. The lower energy electrons peak at "0 and "
, where
and 2
constructively interfere to produce the largest "eld (see Ref. [209]).

picture says nothing about photons or wave mechanical interferences, so the spectrum is not broken up into integer photon peaks. As representative examples of the experimental results of this work, we present in Fig. 32 the two photoelectron spectra for krypton (top) and xenon (bottom) for I "I "2;10 and 1;10 W cm\, respectively. The phase was taken "0. In Fig. 33 we   show the phase dependence of ATI for krypton at low intensity I "I "2;10 W cm\. On the   left-hand side are the experimental data, and on the right-hand side are the calculations based on the above model (using I "I "4;10 W cm\). The overall agreement is reasonable. At   higher intensities it turned out that the agreement between the experimental data and the model calculations was quite satisfactory for krypton, not, however, for xenon. This indicated the

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Fig. 34. Presents in (a) the Michelson interferogram of the fundamental beam (745 nm) from the Ti : sapphire laser and in (b) the ion yield of Ne> versus relative phase between
and 3
for the experiment of Watanabe et al. The relative phase () is de"ned by E cos(
t#)#E cos 3
t. The intensity of
and the "eld ratio (E /E ) are, respectively, S S S S 5.7;10 W cm\ and 0.4 (see Ref. [211]).

involvement of the rescattering process at higher laser "elds. This point was discussed and clari"ed very recently in the work by Chen et al. [139]. Another experiment along the same lines, but at higher laser intensities at which the tunneling approximation (5.4) is even more justi"ed, was performed by Watanabe et al. [211]. These authors considered two-colour CPC in tunneling ionization and HHG by a strong "eld and its third harmonic. Here they measured the phase dependence of the total ionization yield of neon in the tunneling regime using a Ti : sapphire laser and its third harmonic. Adding the third harmonic with an intensity of only 10% enhanced the ion yield by a factor of 7. In the photoelectron spectra, above threshold ionization peaks due to the third harmonic disappeared when two colours were superimposed, resulting in a continuum spectrum. This clearly showed two-colour interference. The intensity of HHG in the plateau region was enhanced by a magnitude of order one. Fig. 34 shows in (a) the Michelson interferogram of the fundamental beam (745 nm) from the Ti : sapphire laser and in (b) the ion yield of Ne> versus relative phase
between
and 3
. The relative phase
was de"ned by the following choice of the two-colour "eld: E(t)"E cos (
t#
)#E cos 3
t . (5.6) S S The intensity of
and "eld ratio E /E were 5.7;10 W cm\ and 0.4, respectively. The S S presented calculations were performed using the two-step model discussed above. Another scheme of phase control than the one considered before was investigated by Jones [212]. Here the modulations in the ATI yield of sodium were observed as a function of the relative delay between two identical 150 fs laser pulses. The "rst pulse populated a coherent superposition state composed of ground state and excited levels. The ionization yield during the second pulse depended on its phase relative to that of the nonstationary superposition state. Ionization enhancements of more than one order of magnitude were achieved by exploiting coherence in the atom}laser system.

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5.2. HHG and CPC in bichromatic xelds Although we want to mainly concentrate on the CPC e!ects in HHG in a bichromatic "eld, we also mention a few experiments in which HHG has been controlled by other means. 5.2.1. Control of HHG by polarization and propagation manipulation In the work by Perry and Crane [213] an experiment was performed to measure the relative conversion e$ciency of 1
(1053 nm) and 2
(526 nm) to HHG in helium. The 2
"eld was over one order of magnitude more e$cient in producing harmonics in the plateau region but exhibited a plateau cuto! much earlier than the 1
"eld, consistent with the I #3; scaling predicted by   Krause et al. [121]. By mixing a weak 1
"eld with a strong 2
"eld they could produce sum- and di!erence-frequency radiation including `evena harmonics. Although the strength of the 1
"eld was too weak to cause the plateau to extend beyond the 2
cuto!, the e$ciency of the mixing products equaled the e$ciency of the 2
harmonics and was one order of magnitude stronger than the 1
harmonics. The strength of the mixed "eld emission exhibited a strong dependence on the relative polarization of the two "elds. The use of orthogonally polarized "elds produced harmonics with varying linear polarization depending on the order, while the use of circular polarization in the weak "eld should produce circularly polarized harmonics. By controlling the relative polarization of the two "elds, they could control the mixing e$ciency and produce coherent extreme ultraviolet (X;<) radiation polarized orthogonally to the strong driving "eld. An example of the results of "eld mixing is shown in Fig. 35. The polarization dependence of high-order two-colour mixing was also discussed in the work by Eichmann et al. [214]. They used the radiation of a high-power Ti : sapphire laser and its second harmonic (2
) and considered linearly and circularly polarized light "elds with comparable intensities. For the theoretical description a three-dimensional quantum mechanical calculation with a -potential has been applied, as developed by Becker et al. [180], showing quite good agreement with the experiments. The main conclusions of their investigations were: (i) Two parallel linearly polarized "elds produce more intense signals than two perpendiculary polarized "elds through the entire frequency regime. (ii) The harmonics produced by the perpendicularly polarized "elds exhibit a characteristic even/odd behaviour, the odd harmonics being more intense than the even ones. Notice, that according to the theoretical model the odd harmonics for the perpendicular polarization can approach the one-colour (1
) harmonics in the upper part of the plateau. (iii) In a certain frequency region above I the mixing signals are stronger than the one-colour harmonics  by one or two orders of magnitude. (iv) As opposed to naiv expectations derived from the classical picture, non-collinear polarization con"gurations produce mixing signals that are equally or even more intense than the harmonics of linearly polarized monochromatic "elds. Examples of the results of this work are shown in Fig. 36. In the work of Tong and Chu [215] the generation of circularly polarized multiple high-order harmonic emission from two-colour crossed laser beams was proposed. The experimental setup involved the use of two-colour laser "elds, consisting of a circularly polarized fundamental laser "eld and a linearly polarized second-harmonic laser "eld, in crossed-beam con"guration. The #exibility of such a scheme was demonstrated by an ab initio quantal study of the HHG power spectrum of the atoms by means of the time-dependent density-functional theory, developed earlier [216,217], with optimized e!ective potential and self-interaction correction. Although in this

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Fig. 35. Shows partial harmonic spectra obtained in the experiment of Perry and Crane from (a) strong 526 nm (2
) harmonic "eld only, (b) from strong 256 nm "eld and weak orthogonally polarized 1053 nm (1
) "eld with I(
)42;10 W cm\ and (c) from strong 526 nm "eld and a weak parallel polarized 1053 nm "eld with I(
)42;10 W cm\. We observe that by mixing a weak 1
"eld with a strong 2
"eld one can produce sum- and di!erence-frequency radiation including `evena harmonics. The strength of the mixed-"eld emission exhibits a strong dependence on the relative polarization of the two "elds (see Ref. [213]).

scheme both, linearly polarized and circularly polarized HHG were produced simultaneously, they propagate in perpendicular directions and are completely separable. The suppression of the background intensity, leading to cleaner HHG, is another potential advantage. For a given incident linearly polarized "eld intensity, they could also tune the incident circularly polarized "eld intensity to "nd the optimal range for the production of circularly polarized HHG. The selection rules for the harmonic generation within the two-colour crossed-beam scheme proposed by these authors were shown by Averbukh et al. [218] to be readily understood on the basis of the dynamical symmetry of the Floquet Hamiltonian by Alon et al. [219], represented within the dipole approximation. The scheme of Tong and Chu [215] was also reanalized in recent work by Becker et al. [220] and compared to the alternative scheme, discussed above [214], and some characteristic di!erences between the two were elaborated. Another mechanism for controlling HHG was considered by Mercer et al. [221]. They demonstrated in their experiments that by modulating the ellipticity of a laser "eld in space one can

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Fig. 36. Presents data of the experiments of Eichmann et al. on 1
#2
frequency mixing signals in argon for di!erent relative polarizations of the "elds in units of the photon energy of the pump laser ( 1.16 eV). The relative intensity scales are identical for (a)}(c) and for (d)}(i), respectively. In (a)}(c) are the experimental data. I corresponds to the ionization  potential of argon (15.76 eV). (d)}(i) are calculated intensities. For (d)}(f), I(
)"1.33;10 W cm\, I(2
)"0.58;10 W cm\; (e) is averaged over the relative phase. (f) The results for corotating circular polarizations have been multiplied by 10. For (g)}(i) I(
)"I(2
)"2;10 W cm\. In (h) relative phase is 1803. The experimental data points (a)}(c) are proportional to the photon numbers; the theoretical data points in (d)}(i) give a quantity which is proportional to the square of the dipolar moment (see Ref. [214]).

control the spatial mode of the harmonics and obtain beam pro"les ranging from Gaussian to annular or even to several rings. This was achieved by using bifringent focusing optics, with axes at 453 to the laser polarization, and by introducing a variable phase shift between the two components of the laser "eld with a Babinet compensator (e!ectively a bifringent material of variable thickness). The nature of the control was twofold: (i) From a microscopic point of view, one imposes, for each atom at a given point in space, a particular laser ellipticity, and, therefore, a particular intensity and polarization of the harmonic light. (ii) More interesting, from a macroscopic point of view, they controlled the volume where harmonics were generated, which, under certain conditions, allowed them to modify and choose at will the spatial mode. Bichromatic frequency conversion in potassium niobate was considered by Hohla et al. [222]. Here two frequency components of an IR laser beam near 980 nm were simultaneously coupled into two adjacent longitudinal modes of a passive ring resonator. A potassium niobate cristal inside

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Fig. 37. Schematic sketch of the experimental setup in the work of Andiel et al. (see Ref. [223]).

the resonator converted the circulating IR light into coherent blue radiation. The total conversion e$ciency was enhanced by a factor of 1.4 compared with that of conventional single-mode intracavity second-harmonic generation with the same circulating total power, and they obtained a total output power of 560 mW from 78 mW IR light incident upon the cavity. The spectra of the generated blue radiation and the circulating IR light contained a number of equidistant frequency components that were due to consecutive sum- and di!erence-frequency mixing. 5.2.2. CPC in HHG in a two-colour xeld Finally, and very recently, the high-order harmonic amplitude modulation in two-colour phase-controlled frequency mixing was observed by Andiel et al. [223]. Using the experimental setup shown in Fig. 37, they measured in helium and argon the phase-dependence of HHG in a bichromatic "eld of fundamental frequency
of a 200 fs Ti : sapphire laser and its second harmonic (2
) at an intensity of I "2;10 W cm\ of the fundamental "eld and "eld S ratio E /E "0.8. In the detection range of harmonics, covering the 7th to the 20th harmonic S S order, a phase-dependent modulation of the HHG signal of up to 27% was observed by precisely varying the relative phase between the two colours, where the total laser "eld was chosen in the form E(t, )"E cos
t#E cos(2
t#) . S S

(5.7)

Simulations based on a single-particle response, according to the work of Cormier and Lambropoulos [189,191], predicted enhancement of the high harmonic e$ciency by orders of magnitude for a given relative phase between the two colours. The above e!ect was shown to be particularly pronounced for harmonics near the cuto! region where proper choice of the relative phase results in the increase of the high harmonic yield by orders of magnitude. In Fig. 38 we show the CPC e!ects observed in this experiment. 5.3. Other experiments in bichromatic xelds In the work of Xenakis et al. [224], CPC was observed in autoionization in a bichromatic "eld of frequencies
and 3
. They reported on the "eld phase-dependent autoionization rate of calcium in the region of the 4p7s[] doubly excited state. Excitation of the autoionizing state occurred from  

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Fig. 38. Shows in (a) the laser energy #uctuations over 3000 laser shots and in (b) the harmonic emission signal demonstrating the modulation resulting by continuously varying the phase  between the two frequencies
and 2
. The open circles are experimental points. The dashed line is the running average over 100 data points. The solid line represents the indicated sinusoidal "t to the experimental points with "t parameters m +0.27 and +1.38. The O O intensity of the fundamental "eld was I "2;10 W cm\ and the ratio E /E "0.8 (see Ref. [223]). S S S

the atomic ground state through two phase-related and hence interfering channels, namely a three-photon channel (3
) and a single-photon channel (
),
being the third harmonic of  
(see Fig. 39). The autoionizing rate exhibits a sinusoidal modulation as a function of the relative phase
of the two excitation "elds. Both ionization "elds were not focused in the interaction region, thus demonstrating the possibility of phase control in a large interaction volume and free of phase shift e!ects associated with the focused geometries. De"ning the modulation depth by (I !I )/(1/2)(I #I ) we show in Fig. 40 their CPC results as a function of the cell







 pressure of argon used to change the phase
, as explained previously. The CPC of the energy and angular distribution of autoionized electrons was considered by Leeuwen et al. [225]. Here nonstationary, autoionizing wave packets were produced by exposing calcium Rydberg atoms to a pair of identical phase-coherent subpicosecond laser pulses. The energy and angular distribution and time of ejection of electrons was altered by changing the relative phase and delay between the two laser pulses. It was demonstrated in this experiment that con"guration interaction between quasibound dielectronic autoionizing states can be exploited to control the energy and angular distribution of ejected electrons in the time domain. The method, which used two identical phase-coherent pulses, can also be used to alter the temporal structure of the di!erential yield, and is applicable to any multicon"guration system with structured continua. Frequency metrology by use of quantum interference was investigated by Georgiades et al. [226]. They found that, by exploiting quantum interference of multiple excitation pathways like the

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Fig. 39. Presents the partial energy level diagram of the calcium atom indicating the (
, 3
) excitation scheme used in the experiments by Xenakis et al. (see Ref. [224]).

Fig. 40. Shows the experimental results displaying the modulation of the ionization signal in the region of the 4p7s[1/2]  autoionizing state of Ca (full squares) as the phase cell pressure is varied. The "tted curve is a linearly decreasing sinusoid. For comparison ionization data of Mg (open squares) are also shown, which due to the uneven amplitudes of the two excitation channels do not undergo interference modulation (see Ref. [224]).

parallel excitation paths for two-photon transitions, they could observe the beat note arising from a small frequency o!set (+10 Hz) between three "elds at wavelengths 852, 884 and 917 nm, demonstrating in this way demodulation of optical e!ects of $12.5 THz. By extending the idea of using atoms as nonlinear mixing elements to more general two photon transitions, they proposed

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a scheme for absolute-frequency comparison between an arbitrary target frequency and a small set of reference frequencies. We now consider some problems on coherent control in Rydberg atoms. In the work of Sirko et al. [227] the microwave excitation of helium 28S Rydberg atoms was investigated in a two-frequency "eld. They found that a second microwave "eld lifts the degeneracy of certain Floquet states describing the 28S He atom plus single microwave "eld system. As a result a microwave pulse produces two patterns of StuK ckelberg oscillations. The new, two-frequency oscillation patterns depend sensitively on the power and frequencies of both microwave "elds. Two frequency Floquet calculations gave insight into the interactions, but were directly applicable only when the frequency ratio was well approximated by low-order rational fractions. It was shown that the interference patterns can be destroyed by additive noise, but di!erent parts of the interference structure are a!ected in di!erent ways by the same noise "eld. Phase-sensitive dynamics of bichromatically driven two-level atoms was considered by Wu et al. [228]. They showed that qualitatively di!erent coherent transient phenomena associated with bichromatic optical excitation can be observed. The e!ects demonstrated included the control of atomic dynamics through a variation of the initial relative phase of the driving "elds and the polarization of population within atom-"eld dressed states. The dynamics observed were fundamentally more complex then those characteristic of monochromatically driven atoms. It was found that such di!erences can persist even when one of the bichromatic "eld components is arbitrarily weak. It was shown that the perturbers impact can be understood in the present case because of its resonance with transitions between atom-pump dressed states. It turned out that the initial relative phase of the bichromatic "eld components play a vital role in determining the atomic response such that certain initial relative phases lead to complete atom-pump-"eld dressed-state polarization. Numerical examples of their experimental results are shown together with data of the integration of the equation of motion in Fig. 41. Another control mechanism was considered in the work of Noel et al. [229]. Here the frequency-modulated excitation of a two-level atom was considered. They presented a detailed experimental study of the frequency-modulated excitation of a two-level atom, using a microwave "eld to drive transitions between two Rydberg Stark states of Potassium. In the absence of a modulation, the interaction is the standard model of the Rabi problem, producing sinusoidal oscillations of the population between the two states. In the presence of a frequency modulation of the interacting "eld, however, the time evolution of the system is signi"cantly modi"ed, producing square wave oscillations of the population, sinusoidal oscillations at a di!erent frequency, or even sinusoidal oscillations built up in a series of stair steps. The three responses described above were each found in a di!erent regime for the frequency of the modulation with respect to the unmodulated Rabi frequency: the low-, high- and intermediate frequency regime, respectively. In another experiment on Rydberg states, the CPC e!ects were considered in two-mode multiphoton transitions in a two-level atom by Ko et al. [230]. They investigated multiphoton transitions driven by two harmonically related microwave "elds in a two-level system. They observed pronounced dependence of the Rabi frequency on the amplitude and relative phase of the "elds in the low "eld limit and found that the Rabi frequency of the combined "elds was given by the phasor sum of the Rabi frequencies of the individual "elds. For strong "elds, each individual Rabi frequency is modi"ed by the presence of the other "eld and this intuitive phasor addition is no

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Fig. 41. In (a)}(e) is presented the experimentally measured atomic #uorescence I (t), following the onset of a step turn-on  bichromatic "eld in the work of Wu et al. Changes in the relative phase of the bichromatic "eld components are indicated at the lower right of each trace. (f )}( j) shows corresponding calculations of I (t) by numerical integration of the equations  of motion accounting for spontaneous radiative decay and the "nite rise time of the step excitation "eld. (k)}(o) shows the 1 component of the atomic state vector R, derived from the same calculations as parts (f )}( j). R represents the inversion  of the atom-pump dressed states. Here the temporal evolution of the atom is displayed by the vector model. In this model, the complete atomic state is represented by a vector R"(R , R , R ) where R (t) represents the atomic inversion and is     directly related to the experimental signal. The experiments were performed in an atomic beam of nuclear-spin-zero Yb atoms utilizing the S }P transition at 556 nm (see Ref. [228]).  

longer valid. In Fig. 42 we show their results on the phase-dependence of the Rabi frequencies of the combined excitation by a "eld of the form E (t)"E cos
t#E, cos (N
t#) KT

(5.8)

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Fig. 42. Shows for the experiments of Ko et al. the multidimensional representation of the phase dependence for interference between (a) one- and two-photon transitions, (b) one- and three-photon transitions and (c) one- and eight-photon transitions. For two given "eld values E and E, the surface height in these plots is de"ned as the maximum of the combined Rabi frequency as a function of , max[ ()]. The grey scale overlaid on this surface is the modulation , depth of the phase dependence. The black curve is where the individual Rabi frequencies are equal,  " , and the  , white dots are at the "eld values for the experimental data (see Ref. [230]).

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with N chosen 2, 3 and 8. The combined Rabi frequency was shown to be given by  ()"[M  #M  #2M M cos ], ,  ,  ,

M " J (kEG/N
); (i"1, N) , G G 

(5.9)

where J is an ordinary Bessel function of the order zero and k"549 MHZ/Vcm\ being the  electric dipole moment. We also mention a paper by Lange et al. [231] in which the decay of bichromatically driven atoms was considered. Here the decay characteristics of two-state Rydberg atoms was investigated in a microwave cavity driven simultaneously by two strong rf "elds. Tuning one of the driving "elds leads to sharp resonances in the decay rate of the upper atomic level. This width may become smaller than the cavity line width and transit time broadening. A theoretical model was presented, interpreting the resonances as cavity-induced decay of Floquet states of the composite atom-"eld system. The observed phenomena arise from the interplay of cavity-modi"ed BlochSiegert shifts and dynamic enhancement of spontaneous emission at single-photon and multiphoton resonances. These experiments showed that there is a considerable di!erence between driving an atom in a bichromatic "eld in free space or by placing the atom in a cavity simultaneously. Such systems exhibit new phenomena due to the interplay of cavity enhancement of radiative transitions and intensity induced modi"cation of the atomic levels close to the cavity resonance. Finally, we quote work by Williams et al. [232] on the measurement of the bichromatic optical force on Rb atoms. Here it was shown that the limit k/2 imposed on the magnitude of radiative forces by the spontaneous decay rate  of the excited state can be overcome by coherent control of the momentum exchange between atoms and the light "eld. This can be implemented with two light beams, each containing two frequencies whose phases, amplitudes and frequency di!erence are carefully chosen. The authors made precise measurements of the extremely large magnitude and velocity range of this bichromatic force, and have shown that its velocity dependence near the edge of its range is suitable for atomic beam slowing and laser cooling. Their measurements have corroborated various models and calculations of this bichromatic force.

6. Other investigations in two-colour 5elds 6.1. A potpourri of atomic model calculations In the work by Kosachiev et al. [233] the in#uence of the phase of electromagnetic "elds interacting with a multilevel system on the character of excitation of this system was considered. On the basis of analyzing the equation for an N-level quantum system density matrix, it was established that in a multilevel system containing a closed contour of transitions, resonantly interacting with the "eld, stationary populations always depend on the total interaction contour phase . For the simplest multilevel system, i.e. three-level, analytical expressions were obtained, connecting populations with the relative "eld phase in such systems. Thus for a  system, interacting with three resonant electromagnetic "elds, one "nds that, depending on the total phase value of the transition contour, both destruction and successive restoration of a coherent trapping state can take place.

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Fig. 43. Presents the schemes studied in the present work on autoionization. The shaded area is the atomic continuum. One bound state 1 and one (two) autoionizing state(s) 2 (and 3) are coupled by two "elds with frequencies
 and
 "3
 and the relative phase . (a) Single autoionizing state. (b) two autoionizing states. The    dashed line in (a), denoted as D, is the two-photon coupling between 2 and continuum states (shaded area). As the "eld intensity with frequency
 increases, the amplitude of this radiative coupling D increases and may become  comparable to the intra-atomic con"guration interaction <, and the autoionizing spectra may be signi"cantly modi"ed (see Ref. [235]).

The manipulation of the line shape and "nal products of autoionization through the phase of the electric "elds was considered in two papers by Nakajima and Lambropoulos [234,235]. Their choice of studying autoionizing states stems from the fact that they represent prototypes of channel coupling which also involve continua (see Fig. 43). Most of the theoretical discussion of phase control above was centered on transitions between bound states. An autoionizing state, however, represents a more complex situation in that it involves a continuum which acquires structure due to intra-atomic interactions. In the present work a formal theory was developed and detailed calculations were performed for phase-dependent laser}atom interactions involving autoionizing states. First, through simple models, they demonstrated that the simultaneous one- and threephoton excitation of one or two neighbouring autoionizing states can exhibit profound changes of the line shape, as the relative phase of the two "elds is varied from 0 to
. Through the proper choice of the "eld intensities and the phase, they obtained analytical results showing that one can cancel the transition to the discrete or the continuum part of the wave function, thereby leading to a #at or a completely symmetric line shape, respectively. At higher intensities, additional e!ects come into play, providing additional coupling between the discrete and continuum parts, which also exhibit a phase dependence. Finally, their theory was applied to a much more complex situation in Xe, involving many channels, not amenable to simple analytic expressions, but exhibiting nevertheless equally profound e!ects, including a modi"cation of the branching ratio of two di!erent products. The general formalism of the present calculations can be found in the work of Lambropoulos and Zoller [236] and Alber and Zoller [237]. As examples of the results of the above calculations we show in Figs. 44 and 45 the photoelectron angular distribution and change of the autoionizing line shape of Xe from the "nal energy 133.800 cm\ under the presence of two laser "elds of frequency
and
"3
as a function of the relative phase  and the intensities  I and I .  

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Fig. 44. Shows the photoelectron angular distribution (PAD) for xenon from the "nal energy 133.800 cm\ under the presence of two "elds at di!erent relative phase "0 (solid line) and "
(dotted line). After the emission of a photoelectron, a Xe core is left in either P or P ionic states of Xe>. PAD's corresponding to each P or P ionic     state are plotted separately and indicated in each graph. The laser intensities are I "10 W cm\ ["xed through (a)}(d)]  and (a) I "10 W cm\, (b) I "10 W cm\, (c) I "5;10 W cm\, (d) I "10 W cm\ (see Ref. [235]).     Fig. 45. Indicates the change of the autoionizing line shape of Xe through the relative phase  of the two laser "elds. The intensities are I "10 W cm\ ["xed through (a)}(c)] as 5 ns square pulse and (a) I "10 W cm\,   (b) I "5;10 W cm\, (c) I "10 W cm\. Dashed line corresponds to the incoherent ionization by the two "elds.   8s, 6d and 6d in the "gures correspond to the autoionizing states [P ]8s (J"1), [P ]6d (J"1) and       [P ]6d (J"3), respectively (see Ref. [235]).  

The same theoretical framework was applied in two other papers by Nakajima et al. [238,239] to control the branching ratio of photoionization products under two-colour excitation, considering the competition between AC Stark splitting and two-path interference. In the "rst of these papers they considered the phase control which governs the time-dependent behaviour of a system involving one bound and one decaying discrete state embedded in several continua belonging to several ionization thresholds. Representative numerical results were shown, focusing on the behaviour of the branching ratio as a function of the relative phase of two lasers, detuning and Rabi frequencies (or intensities). Speci"c quantitative results on Ca atoms were also reported. In the second paper they investigated the variation of the photoionization yield into di!erent ionic channels by means of the one-photon-near resonant two-photon ionization scheme under

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Fig. 46. Shows the level scheme considered in this paper on photoionization. Atoms in an initially unoccupied state 0 are excited into the continua through two coherent pathways (0P1PA and 0P2PA) by absorbing
#
? @ and
#
, respectively. Ionization into the incoherent channels (B, C, D, and E) are denoted by dashed arrows (see @ ? Ref. [239]). Fig. 47. Presents the ionization pro"les into the di!erent ionic core states of Ca>: 4s, 3d, and 4p as a function of detuning  . The laser intensities were I "I "10 W cm\ with 10 ns Gaussian pulse for both lasers. The Lorentzian laser    bandwidth were   "  "0.1 cm\ and the cuto! parameter "0.2 cm\. The solid, dashed and dot-dashed lines in * * each graph represent the core states 4s, 3d and 4p, respectively (see Ref. [239]).

two-colour excitation shown in Fig. 46. They presented a general formulation and the results of speci"c calculations pertaining to the Ca atom with realistic parameters. A signi"cant change of the ionization into Ca> 4s, 3d and 4d channels was observed as detuning was varied, which agrees quantitatively well with the observation of Wang et al. [240] for the Ba atom. The importance of the laser intensity e!ects was also addressed. In Fig. 47 we show their results for the ionization pro"les into the di!erent ionic core states of Ca>: 4s, 3d and 4p as a function of detuning  shown in Fig. 46. The possibility of direct determination of the quantum phase of continua utilizing the phase of lasers was recently addressed by Nakajima [241]. He showed that the quantum phase of continua can be directly determined utilizing the phase of linearly/circularly polarized laser light. With this method the phase di!erence of continua with opposite as well as same parities can be obtained from the phase lag of the angle-resolved photoelectron signal with respect to the relative phase of lasers. For illustration, the proposed method is speci"cally applied to the Na atom using the phase of lasers based on the
}2
scenario. Two-colour photoionization of helium above the N"2 ionization threshold was considerd by Grosges et al. [242]. They studied this process in the region of the 3s3dD autoionizing state with

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Fig. 48. The schematic diagram showing Autler}Townes (AT) splitting of a pair of resonantly coupled bound states i and  j and the associated AT doublet structure (A and B) in the ionization continuum (see Ref. [244]).

resonant excitation of the intermediate 2s2pP autoionizing state. They showed that the corresponding TDSE may be solved by expanding the total wave function on a few complex energy states. Perimetric coordinates were used to describe correlation e!ects accurately. Population at the end of interaction as well as photoionization probabilities were studied as a function of the laser pulse duration and both frequencies. Their calculations showed that many continua play a crucial role at high intensities. Multiple phase control in Mg through the continuum was studied by Lyras and Bachau [243]. They presented results for perturbative calculations on a scheme of CPC in Mg. The scheme involved the excitation of Mg in the vicinity of an autoionizing resonance lying above the "rst two ionization thresholds by a two-colour "eld composed of a fundamental frequency and its third harmonic, whose relative phase can be continuously controlled. Four- as well as two-photon coherent excitation pathways, involving appropriate combinations of the two frequencies, proceed through the atomic continuum simultaneously contributing to the excitation, while three-photon ionization by the fundamental is also energetically possible. As a result, some of the excitation processes involve ATI. The calculated ionization yields for the various groups of photoelectrons reveal signi"cant modulation as the relative phase of the "eld component frequencies is varied. The modulation patterns for the di!erent photoelectron groups are mutually shifted due to the multiphoton matrix element phases associated both with the ATI process and the multichannel nature of the "nal state. The e!ect is particularly pronounced when the autoionizing state is in near four-photon resonance with the fundamental frequency. The role of an autoionizing state as a "nal state as well as of the continuum as an intermediate state in coherent control processes was discussed. The possibility of modulation of an Autler}Towns (AT) doublet structure in an atomic continuum (see Fig. 48) by changing the phase di!erence between two incident femtosecond laser pulses was investigated theoretically by Lambrecht et al. [244]. Numerical simulations were carried out by solving the three-dimensional TDSE for a Na atom interacting with a two-colour

F. Ehlotzky / Physics Reports 345 (2001) 175}264

255

Fig. 49. Shows the phase modulation of the AT doublet A and B as a function of the relative phase  between 03 and 3603 for three di!erent intensities I "I "2;10 W cm\, peak A is shown in (a) and peak B in (b); I "I "   5;10 W cm\, A is shown in (c) and B in (d); and I "I "1;10 W cm\, A is shown in (e) and B in (f ). The  two-colour laser "eld has the frequencies
"2.0325 eV and
"4.065 eV (see Ref. [244]). 

"eld, a fundamental and its second harmonic. It was found that a sinusiodal modulation of the individual members of the doublet, as a function of the phase di!erence, can be achieved at low intensities of the order of 10 W cm\. Such modulations of the sharply de"ned electron currents (where the width is of the order of the "nestructure of the AT levels) may be detected by high-resolution photoelectron spectroscopy. It was also shown that at higher intensity of the order of 10 W cm\ the amplitude of modulation diminishes, indicating that lower intensities are to be preferred for the present purpose. Fig. 49 shows the phase modulation of the AT doublet A and B of Fig. 48 as a function of the relative phase . Resonance #uorescence in a bichromatic "eld as a source of nonclassical light was considered by Kryuchkian et al. [245]. They investigated the photon correlation phenomenon and photon count

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statistics in the resonance #uorescence of a two-level atom interacting with a strong bichromatic "eld. They obtained an analytical result for the second-order correlation function. Unusual quantum statistical properties of the #uorescent radiation were obtained, and e!ects on the photon correlation in#uenced by the electromagnetic reservoir were shown. The interaction with a bichromatic driving "eld showed a number of new features which are not present in the monochromatic driving "eld interaction. Among them were unusually strong superbuching e!ects in the secondorder correlation function as a result of strongly correlated two-photon emissions, and the alternating dependence of the photon statistics on the interaction strength parameter. In another, related paper by Jakob and Kryuchkyan [246] the interaction of a bichromatically driven two-level atom with a squeezed vacuum was considered recently. They investigated e!ects of a broadband squeezed reservoir on the second-order intensity correlation, and on squeezing properties in the resonance #uorescence of a bichromatically driven two-level atom in a cavity of moderate Q. Phase-dependent squeezed reservoir e!ects change the photon statistics, and lead to an ampli"cation of the degree of squeezing. Squeezed reservoir e!ects in the second-order correlation function g() are determined by two-photon emission processes which can be enhanced or suppressed in dependence of the squeezing phase. Coherent control of the polarization of an optical "eld was studied by Wielandy and Gaeta [247]. Through use of quantum coherence in an atomic system, they demonstrated that an optical "eld can be used to completely control the polarization state of another "eld. Theoretical and experimental results show that by applying a "eld with a frequency tuned near one transition, an initially isotropic atomic vapour can be made to behave as a linearly or circularly bifringent material for "elds tuned near an adjoining transition. Phase control of spontaneous emission was investigated by Paspalakis and Knight [248,249]. They used the phase di!erence of two lasers with equal frequency for the control of spontaneous emission in a four-level system. E!ects such as extreme spectral narrowing and selective and total cancelation of #uorescence decay were shown as the relative phase was varied. The above authors also considered in detail the trapping conditions, the population dynamics and the behaviour of the (long time) spontaneous emission spectrum, which are now phase dependent. Inhibition of spontaneous emission and extreme spectral narrowing were shown as the relative phase is varied. Their results were interpreted using a dressed state analysis of the dynamics. Finally, two-photon coherent control of atomic collisions by light with entangled polarization was recently considered by Havey et al. [250]. They described a new method of coherent optical control of internal dynamics of atomic collisions by means of two correlated light beams having entangled polarizations. They showed that, if excitation of a colliding pair of atoms is by two photons having entangled polarizations, it is possible to redirect the output fragments of the collision into certain channels with a selected type of internal transition symmetry. The transition symmetry is de"ned in the body-"xed coordinate frame which has random and originally unknown orientation in space. 6.2. CPC in solids in two-colour xelds To conclude, we mention a few papers in which the idea of CPC of a physical process in a bichromatic "eld of commensurate frequencies was applied to solid state phenomena.

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257

In the work of Kurizki et al. [251] it was suggested to coherently control the photocurrent directionality in semiconductors. They showed that one can generate and control photocurrents in semiconductors, with and without bias voltage, through multiple frequency phase-coherent laser excitation of donors. The proposed scheme could have potential applications. It could serve to measure the relative phase of two beams originating from a common laser source, which is accumulated in their passage through di!erent nonlinear optical elements. By modulating this phase, information could be transfered via corresponding modulations in the current through the crystal. Laser phase #uctuations, on a longer time scale than optical excitations, could cause current reversals and thus be detectable using these schemes. An interesting application of these schemes would be the monitoring of elastic (direction changing) collisions which ballistic electrons undergo in semiconductors. Experiments on this topic were performed by Dupont et al. [252]. In this work the direction of emission of photoexcited electrons in semiconductors were controlled by adjusting the relative phase di!erence between a midinfrared radiation and its second harmonic. This was achieved by using quantum interference of electrons produced with one- or two-photon bound-to-free interband transitions in AlGaAs/GaAs quantum-well superlattices. Two-colour control of localization: from lattices to spin systems was considered by Karczmarek et al. [253]. They demonstrated the control of quantum dynamics in a "nite model system described by a tight-binding Hamiltonian, through interaction with a multifrequency external "eld. E!ective defects can be introduced into the lattice by a two-frequency "eld, and the character of the defects can be controlled by the relative phase between the two "eld components. These "eldinduced defects imply robust localization of dressed (Floquet) states on lattice sites. Implications for a spin system in crossed magnetic "elds were discussed. A qubit for quantum computation was suggested by Woldeyohannes and John [254] through coherent control of spontaneous emission near a photonic band edge. They demonstrated the coherent control of spontaneous emission for a three-level atom located within a photonic band-gap structure with one resonant frequency near the edge of the photonic band gap. Spontaneous emission from the three-level atom can be totally suppressed or strongly enhanced depending on the relative phase between the steady-state control laser coupling the two upper levels and the pump laser pulse used to create an excited state of the atom in the form of a coherent superposition of the two upper levels. Unlike the free-space case, the steady-state inversion of the atomic system is strongly dependent on the externally prescribed initial conditions. This no-zero steady-state population is achieved by virtue of the localization of light in the vicinity of the emitting atom. It is robust to decoherence e!ects provided that the Rabi frequency of the control laser "eld}atom interaction exceeds the rate of dephasing interaction. As a result, such a system may be relevant for a single-atom, phase-sensitive, optical memory device on the atomic scale. The protected electric dipole within the photonic band gap provides a basis for a qubit to encode information for quantum computation. Some more information on these kind of topics can be found in the review by Grifoni and HaK nggi [255]. Finally, we investigated HHG at metal surfaces [256] by a powerful bichromatic laser "eld by generalizing model calculations on HHG at metal surfaces in a monochromatic femtosecond laser pulse [257]. The "eld had frequencies
and 2
and its two components were out of phase by
. We discussed CPC of HHG at metal surfaces and related our results to the corresponding e!ects in atoms.

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7. Concluding remarks In the present work we have given an overview on what has up to now been done on the investigation of CPC e!ects in atomic processes in bichromatic laser "elds, mainly of frequencies
and 2
or
and 3
. The considered e!ects can strongly depend on the relative intensities and polarization of the two "elds and, in particular, on the relative phase
. While on the theoretical side quite a number of investigations of various atomic processes have been performed and new e!ects predicted, till now comparatively little was achieved to demonstrate experimentally the CPC e!ects and their relevance for gaining further insight into laser-induced and -assisted atomic phenomena. The largest number of experiments were done in the perturbative regime of laser "elds of low intensity. For the theoretical treatment of these processes, those methods were used to describe the CPC e!ects which were also employed in the case of a single laser "eld, in particular, time-dependent perturbative methods, essential state approximations, two-level atoms, Floquet methods close to resonance, S-matrix approximations, one-dimensional atomic models and numerical integrations of the TDSE. Though in the molecular domain the method of coherently controlling a molecular reaction in a phase-dependent two-colour laser "eld seems to be quite promising, in the "eld of atomic processes the results achieved appear to be not so convincing to actually have a considerable impact on future developments in the "eld of laser phenomena, in particular, at larger laser powers and shorter laser pulses. There appear to be other control mechanisms available, of which some were discussed here, that yield equally interesting e!ects and could be more e$cient.

Acknowledgements The "rst draft of this work was prepared while the author was visitor at the Theoretical Physics Institute of the University of Alberta, Edmonton, Canada. The author wishes to acknowledge the kind hospitality extended to him there. In particular, he wishes to thank Professor A. Z. Capri for his interest in this work and for his support. The author also wishes to thank his co-authors Drs. A. Cionga, J.Z. KaminH ski, D.B. Milos\ evicH and S. VarroH for their fruitful collaboration.

Note added in proof After submission of this review the papers [258}273] on various aspects of controlled laserinduced atomic phenomena were published.

References [1] [2] [3] [4]

M.H. Mittleman, Introduction to the Theory of Laser}Atom Interactions, Plenum, New York, 1993. F.H.M. Faisal, Theory of Multiphoton Processes, Plenum, New York, 1987. M. Gavrila (Ed.), Atoms in Intense Laser Fields, Academic Press, New York, 1992. N.D. Delone, V.P. Krainov, Multiphoton Processes in Atom, Springer, Berlin, 1994.

F. Ehlotzky / Physics Reports 345 (2001) 175}264 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49]

259

M. Shapiro, P. Brumer, J. Chem. Phys. 84 (1986) 4103. M. Shapiro, J.W. Hepburn, P. Brumer, Chem. Phys. Lett. 149 (1988) 451. C.K. Chan, P. Brumer, M. Shapiro, J. Chem. Phys. 94 (1991) 2688. M. Shapiro, P. Brumer, in: P. Lambropoulos, H. Walther (Eds.), Multiphoton Processes 1996, Institute of Physics Conference Series, Vol. 154, Institute of Physics, Bristol, 1997, p. 231. M. Shapiro, P. Brumer, Adv. At. Mol. Opt. Phys. 42 (2000) 287. S.-P. Lu, S.M. Park, Y. Xie, R.J. Gordon, J. Chem. Phys. 96 (1992) 6613. L. Zhu, V. Kleiman, X. Li, S.-P. Lu, K. Trentelman, R.J. Gordon, Science 270 (1995) 77. R.G. Gordon, S.A. Rice, Annu. Rev. Phys. Chem. 48 (1997) 595. E. Charron, A. Giusti-Suzor, F.H. Mies, J. Chem. Phys. 103 (1995) 7359. M.R. Thompson, M.K. Thomas, P.F. Taday, J.H. Posthumus, A.J. Langley, L.J. Frasinski, K. Kodling, J. Phys. B 30 (1997) 5755. A.D. Bandrauk, S. Chelkowski, Phys. Rev. Lett. 84 (2000) 3562. A.D. Bandrauk, S. Chelkowski, H.Yu, E. Constant, Phys. Rev. A 56 (1997) R2537. C.M. Dion, A.D. Bandrauk, O. Atabek, A. Keller, H. Umeda, Y. Fujimura, Chem. Phys. Lett. 302 (1999) 215. E. McCullough, M. Shapiro, P. Brumer, Phys. Rev. A 61 (2000) 041801(R). M. Shapiro, E. Frishman, P. Brumer, Phys. Rev. Lett. 84 (2000) 1669. D.J. Tannor, S.A. Rice, J. Chem. Phys. 83 (1985) 5013. R. Koslo!, S.A. Rice, P. Gaspard, S. Tersigni, D.J. Tannor, Chem. Phys. 139 (1989) 201. J.J. Gerdy, M. Dantus, R.M. Bowman, H.A. Zewail, Chem. Phys. Lett. 171 (1990) 1. V. Blanchet, M.A. Bouchene, O. Cabrol, B. Girard, Chem. Phys. Lett. 233 (1995) 491. B. Kohler, V.V. Yakovlev, J. Che, J.L. Krause, M. Messina, K.R. Wilson, N. Schwentner, R.M. Whitnell, Y.-J. Yan, Phys. Rev. Lett. 74 (1995) 3360. V.V. Lozovoy, S.A. Antipin, F.E. Gostev, A.A. Titov, D.G. Tovbin, O.M. Sarkisov, A.S. Vetchinkin, S.Ya. Umanskii, Chem. Phys. Lett. 284 (1998) 221. B.M. Garraway, K.-A. Suominen, Rep. Prog. Phys. 58 (1995) 365. B.M. Garraway, K.-A. Suominen, Phys. Rev. A 58 (1998) 440. A. Assion, T. Baumert, J. Helbing, V. Seyfried, G. Gerber, Chem. Phys. Lett. 259 (1996) 259. D. Meshulach, Y. Silberberg, Phys. Rev. A 60 (1999) 1287. C.J. Bardeen, V.V. Yakovlev, K.R. Wilson, S.D. Carpenter, P.M. Weber, W.S. Warren, Chem. Phys. Lett. 280 (1997) 151. J. Manz, in: V. SundstroK m (Ed.), Femtochemistry and Femtobiology: Ultrafast Reaction Dynamics at Atomic Scale Resolution, Imperial College Press, London, 1997, p. 80. W.S. Warren, H. Rabitz, M. Dahleh, Science 259 (1993) 1581. F.V. Bunkin, M.V. Fedorov, Sov. Phys. JETP 22 (1966) 844. F. Ehlotzky, A. JaronH , J.Z. KaminH ski, Phys. Rep. 297 (1998) 63. C.J. Joachain, M.DoK rr, N.J. Kylstra, Adv. At. Mol. Opt. Phys. 42 (1998) 225. A. Weingartshofer, J.K. Holmes, J. Sabbagh, S.L. Chin, J. Phys. B 16 (1983) 1805. S. VarroH , F. Ehlotzky, Phys. Rev. A 47 (1993) 715. S. VarroH , F. Ehlotzky, Opt. Commun. 99 (1993) 177. S. VarroH , F. Ehlotzky, J. Phys. B 28 (1995) 1613. W. Gordon, Z. Phys. 40 (1926) 117. D.V. Volkov, Z. Phys. 94 (1935) 250. N.M. Kroll, K.M. Watson, Phys. Rev. A 8 (1973) 804. M.H. Mittleman, Phys. Rev. A 19 (1979) 134. F. Ehlotzky, Nuovo Cimento D 16 (1994) 453. D.B. Milos\ evicH , J. Phys. B 29 (1996) 875. I. RabadaH n, L. MeH ndez, A.S. Dickinson, J. Phys. B 29 (1996) 163. L. Rosenberg, F. Zhou, Phys. Rev. A 47 (1993) 2146. S. VarroH , F. Ehlotzky, J. Phys. B 30 (1997) 1061. M. Ghalim, F. Mastour, J. Phys. B 32 (1999) 3794.

260 [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94] [95] [96] [97] [98] [99]

F. Ehlotzky / Physics Reports 345 (2001) 175}264 F.W. Byron Jr., P. Francken, C.J. Joachain, J. Phys. B 20 (1987) 5487. A. Cionga, G. Zloh, Laser Phys. 9 (1999) 69. A. Cionga, G. Zloh, Laser Particle Beams 18 (2000), in press. D.B. Milos\ evicH , F. Ehlotzky, B. Piraux, J. Phys. B 30 (1997) 4347. L. Hostler, J. Math. Phys. 11 (1970) 2966. A. Maquet, Phys. Rev. A 15 (1977) 1088. M. Rotenberg, Ann. Phys. NY 19 (1962) 262. N.F. Mott, H.S.W. Massey, The Theory of Atomic Collisions, Clarendon, Oxford, 1971. M.R.C. McDowell, J.P. Coleman, Introduction to the Theory of Ion}Atom Collisions, North-Holland, Amsterdam, 1970. I.E. McCarthy, Can. J. Phys. 74 (1996) 703. M. Mohan, P. Chand, Phys. Lett. 65 (1978) 399. C.J. Joachain, P. Francken, A. Maquet, P. Martin, V. VeH niard, Phys. Rev. Lett. 61 (1988) 165. P. Martin, V. VeH niard, A. Maquet, P. Francken, C.J. Joachain, Phys. Rev. A 39 (1989) 6178. X.J. Chen, Phys. Rev. A 40 (1989) 1795. P. Francken, C.J. Joachain, J. Opt. Soc. Am. B 7 (1990) 554. R. TaiG eb, V. VeH niard, A. Maquet, S . Vuc\ icH , R.M. Potvliege, J. Phys. B 24 (1991) 3229. D.B. Milos\ evicH , F. Ehlotzky, Phys. Rev. A 56 (1997) 3879. H. Ehrhardt, K.H. Hesselbacher, K. Jung, M. Schulz, K. Willmann, J. Phys. B 5 (1972) 2107. H. Ehrhardt, K. Jung, G. Knoth, P. Schlemmer, Z. Phys. D 1 (1986) 3. J. Banerji, M.H. Mittleman, J. Phys. B 14 (1981) 3717. J.Z. KaminH ski, F. Ehlotzky, Phys. Rev. A 50 (1994) 4404. J.N. Bardsley, A SzoK ke, M.J. Comella, J. Phys. B 21 (1988) 3899. C. Jung, H.S. Taylor, Phys. Rev. A 23 (1987) 1115. R.A. Sacks, A. SzoK ke, Phys. Rev. A 40 (1989) 5614. J.Z. KaminH ski, Z. Phys. D 16 (1990) 153. J.Z. KaminH ski, A. JaronH , F. Ehlotzky, J. Phys. B 28 (1995) 4895. S. VarroH , F. Ehlotzky, J. Phys. B 28 (1995) 2729. M. Protopapas, C.H. Keitel, P.L. Knight, Rep. Prog. Phys. 60 (1997) 389. L.F. DiMauro, P. Agostini, Adv. At. Mol. Opt. Phys. 35 (1995) 79. N.B. Delone, V.P. Krainov, Phys. Usp. 41 (1998) 469. C.J. Joachain, M. DoK rr, N.J. Kylstra, Comments At. Mol. Phys. 33 (1997) 247. H.W. van der Hart, J. Phys. B 29 (1996) 2217. W. Becker, A. Lohr, M. Kleber, Quantum Semiclass. Opt. 7 (1995) 423. L.V. Keldysh, Sov. Phys. JETP 20 (1965) 1307. F.H.M. Faisal, J. Phys. B 6 (1973). H.R. Reiss, Phys. Rev. A 22 (1980) 1786. H.R. Reiss, Prog. Quantum Electron. 16 (1992) 1. A. SzoK ke, K.C. Kulander, J.N. Bardsley, J. Phys. B 24 (1991) 3165. R.M. Potvliege, P.H.G. Smith, J. Phys. B 24 (1991) L641. R.M. Potvliege, P.H.G. Smith, J. Phys. B 25 (1992) 2501. R.M. Potvliege, P.H.G. Smith, Phys. Rev. A 49 (1994) 3110. M. Pont, R.M. Potvliege, R. Shakeshaft, P.H.G. Smith, Phys. Rev. A 46 (1992) 555. D.A. Telnov, J. Wang, S.-I. Chu, Phys. Rev. A 51 (1995) 4797. D.A. Telnov, S.-I. Chu, Phys. Rev. A 50 (1994) 4099. T.S. Ho, S.-I. Chu, J.V. Tietz, Chem. Phys. Lett. 96 (1983) 464. J. Wang, S.-I. Chu, C. Laughlin, Phys. Rev. A 50 (1994) 3208. M. Pont, D. Proulx, R. Shakeshaft, Phys. Rev. A 44 (1991) 4486. K.C. Kulander, Phys. Rev. A 35 (1987) 445. K.J. Schafer, K.C. Kulander, Phys. Rev. A 45 (1992) 8026. M. Protopapas, P.L. Knight, K. Burnett, Phys. Rev. A 49 (1994) 1945.

F. Ehlotzky / Physics Reports 345 (2001) 175}264 [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149]

261

H.A. Kramers, Collected Scienti"c Papers, North-Holland, Amsterdam, 1956. W.C. Henneberger, Phys. Rev. Lett. 21 (1968) 838. M. Protopapas, P.L. Knight, J. Phys. B 28 (1995) 4459. T. Nakajima, P. Lambropoulos, S. Cavalieri, M. Matera, Phys. Rev. A 46 (1992) 7315. D.Z. Anderson, N.B. Baranova, K. Green, B.Ya. Zel'dovich, Sov. Phys. JETP 75 (1992) 210. N.B. Baranova, B.Ya. Zel'dovich, J. Opt. Soc. Am. B 8 (1991) 27. N.B. Baranova, H.R. Reiss, B.Ya. Zel'dovich, Phys. Rev. A 48 (1993) 1497. V.A. Pazdzersky, V.A. Yurovsky, Phys. Rev. A 51 (1995) 632. V.A. Pazdzersky, V.I. Usachenko, J. Phys. B 30 (1997) 3387. M.Yu. Kuchiev, V.N. Ostrovsky, J. Phys. B 31 (1998) 2525. M.Yu. Kuchiev, V.N. Ostrovsky, Phys. Rev. A 59 (1999) 2844. R.A. Blank, M. Shapiro, Phys. Rev. A 52 (1995) 4278. J.C. Camparo, P. Lambropoulos, Phys. Rev. A 55 (1997) 552. J.C. Camparo, P. Lambropoulos, Phys. Rev. A 59 (1999) 2515. J. Matulewski, A. RaczynH ski, J. Zaremba, Phys. Lett. A 251 (1999) 805. T. Cheng, J. Liu, S. Chen, Phys. Rev. A 59 (1999) 1451. R.M. Potvliege, Phys. Rev. A 60 (1999) 1311. D.-S. Guo, T. Aaberg, B. Craseman, Phys. Rev. A 40 (1988) 4997. M. Gell-Mann, M. L. Goldberger, Phys. Rev. 91 (1953) 393. D.-S. Guo, G.W.F. Drake, J. Phys. A 25 (1992) 5377. L. Gao, X. Li, P. Fu, D.-S. Gao, Phys. Rev. A 58 (1998) 3807. J.L. Krause, K.J. Schafer, K.C. Kulander, Phys. Rev Lett. 68 (1992) 3535. K.J. Schafer, B. Yang, L.F. DiMauro, K.C. Kulander, Phys. Rev. Lett. 70 (1993) 1599. B. Yang, K.J. Schafer, B. Walker, K.C. Kulander, P. Agostini, L.F. DiMauro, Phys. Rev. Lett. 71 (1993) 3770. G.G. Paulus, W. Nicklich, H. Xu, P. Lambropoulos, H. Walther, Phys. Rev. Lett. 72 (1994) 2851. G.G. Paulus, W. Nicklich, H. Walther, Europhys. Lett. 27 (1994) 267. G.G. Paulus, W. Becker, W. Nicklich, H. Walther, J. Phys. B 27 (1994) L703. M. Lewenstein, K.C. Kulander, K.J. Schafer, P.H. Bucksbaum, Phys. Rev. A 51 (1995) 1495. W. Becker, A. Lohr, M. Kleber, J. Phys. B 27 (1994) L325. J.Z. KaminH ski, A. JaronH , F. Ehlotzky, Phys. Rev. A 53 (1996) 1756. J.Z. KaminH ski, F. Ehlotzky, Phys. Rev. A 55 (1997) 4625. J.Z. KaminH ski, F. Ehlotzky, J. Phys. B 30 (1997) 5729. J.Z. KaminH ski, A. JaronH , F. Ehlotzky, Nuovo Cimento D 20 (1998) 19. D.B. Milos\ evicH , F. Ehlotzky, Phys. Rev. A 58 (1998) 3124. D.B. Milos\ evicH , F. Ehlotzky, J. Phys. B 32 (1999) 1585. D.B. Milos\ evicH , B. Piraux, Phys. Rev. A 54 (1996) 1522. G.G. Paulus, W. Becker, H. Walther, Phys. Rev. A 52 (1995) 4043. P.B. Corkum, Phys. Rev. Lett. 71 (1993) 1994. D.B. MilosH evicH , F. Ehlotzky, J. Phys. B 31 (1998) 4149. J. Chen, J. Liu, S.G. Chen, Phys. Rev. A 61 (2000) 033402. V. VeH niard, R. TaiG eb, A. Maquet, Comments At. Mol. Phys. 33 (1996) 53. C. Leone, S. Bivona, R. Burlon, G. Ferrante, Phys. Rev. A 38 (1988) 5642. M. Jain, N. Tzoar, Phys. Rev. A 18 (1978) 538. S. Bivona, R. Burlon, C. Leone, Phys. Rev. A 48 (1993) R3441. M. DoK rr, R. Shakeshaft, Phys. Rev. A 36 (1987) 421. M. DoK rr, R. Shakeshaft, Phys. Rev. A 40 (1989) 459. F. Zhou, L. Rosenberg, Phys. Rev. A 44 (1991) 3270. C.S. Han, Phys. Rev. A 53 (1996) 4268. C.S. Han, Phys. Rev. A 55 (1997) 3957. M. DoK rr, R.M. Potvliege, D. Proulx, R. Shakeshaft, Phys. Rev. A 44 (1991) 574.

262 [150] [151] [152] [153] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179] [180] [181] [182] [183] [184] [185] [186] [187] [188] [189] [190] [191] [192] [193] [194] [195] [196] [197] [198] [199]

F. Ehlotzky / Physics Reports 345 (2001) 175}264 K. Rza1 z ewski, L. Wang, J.W. Haus, Phys. Rev. A 40 (1989) 3453. L. Wang, J. W. Haus, K. Rza1 z ewski, Phys. Rev. A 42 (1990) 6784. R. Bu!a, S. Cavalieri, R. Eramo, M. Matera, J. Opt. Soc. Am. B 13 (1996) 492. Z. Deng, J.H. Eberly, J. Opt. Soc. Am. B 2 (1985) 486. V. VeH niard, R. TaiG eb, A. Maquet, Phys. Rev. Lett. 74 (1995) 4161. A. Cionga, V. Florescu, A. Maquet, R. TaiG eb, Phys. Rev. A 47 (1993) 1830. M. Fi"rig, A. Cionga, V. Florescu, J. Phys. B 30 (1997) 2599. R. TaiG eb, V. VeH niard, A. Maquet, Phys. Rev. A 57 (1998) R4098. K.J. Schafer, Comput. Phys. Commun. 63 (1992) 427. V. VeH niard, R. TaiG eb, A. Maquet, Phys. Rev. A 54 (1996) 721. S. Bivona, R. Burlon, C. Leone, Phys. Rev. A 57 (1998) R16. M.L. Du, J.B. Delos, Phys. Rev. A 38 (1988) 5609. D.B. Milos\ evicH , F. Ehlotzky, Phys. Rev. A 57 (1998) 2859. H.B. van Linden van den Heuvell, H.G. Muller, in: S.J Smith, P.L. Knight (Eds.), Multiphoton Processes, Cambridge Studies in Modern Optics, Vol. 8, Cambridge University Press, Cambridge, 1988. T.F. Gallagher, Phys. Rev. Lett. 61 (1988) 2304. F. Ehlotzky, Nuovo Cimento D 14 (1992) 517. M. Lewenstein, Ph. Balcou, M.Yu. Ivanov, A. L'Huillier, P.B. Corkum, Phys. Rev. A 49 (1994) 2117. V.T. Platonenko, V.V. Strelkov, Quantum Electron. 28 (1998) 564. P. Salie`res, A. L'Huillier, P. Antoine, M. Lewenstein, Adv. At. Mol. Opt. Phys. 41 (1999) 83. Ph. Antoine, B. Piraux, D.B. MilosH evicH , M. Gajda, Phys. Rev. A 54 (1996) R1761. M.B. Gaarde, A. L'Huillier, M. Lewenstein, Phys. Rev. A 54 (1996) 4236. M.B. Gaarde, P. Antoine, A. Persson, B. CarreH , A. L'Huillier, C.-G. WahlstroK m, J. Phys. B 29 (1996) L163. C. Kan, C.E. Capjack, R. Rankin, T. Brabec, N.H. Burnett, Phys. Rev. A 54 (1996) R1026. S. deLuca, E. Fiordilino, J. Mod. Opt. 45 (1998) 1775. E. Fiordilino, F. Trombetta, J. Mod. Opt. 35 (1988) 1635. S.G. Preston, J.B. Watson, J. Phys. B 31 (1998) 2247. A. Sanpera, J.B. Watson, M. Lewenstein, K. Burnett, Phys. Rev. A 54 (1996) 4320. J.B. Watson, A. Sanpera, X. Chen, K. Burnett, Phys. Rev. A 53 (1996) R1962. V.D. Taranukhin, N. Yu. Shubin, Quantum Electron. 29 (1999) 638. S. Long, W. Becker, J.K. McIver, Phys. Rev. A 52 (1995) 2262. W. Becker, S. Long, J.K. McIver, Phys. Rev. A 41 (1990) 4112. W. Becker, S. Long, J.K. McIver, Phys. Rev. A 46 (1992) R5334. W. Becker, S. Long, J.K. McIver, Phys. Rev. A 50 (1994) 1540. D.B. Milos\ evicH , W. Becker, R. Kopold, Phys. Rev. A 61 (2000) 063403. M. Protopapas, A. Sanpera, P.L. Knight, K. Burnett, Phys. Rev. A 52 (1995) R2527. D.A. Telnov, J. Wang, S.-I. Chu, Phys. Rev. A 52 (1995) 3988. C. Figueira de Morrison Faria, M. DoK rr, W. Becker, W. Sandner, Phys. Rev. A 60 (1999) 1377. C. Figueira de Morrison Faria, D.B. Milos\ evicH , G.G. Paulus, Phys. Rev. A 61 (2000) 063415. S.-Q. Gong, Z.-Y. Wang, S.-D. Du, Z.-Z. Xu, J. Mod. Opt. 46 (1999) 1669. E. Cormier, P. Lambropoulos, J. Phys. B 30 (1997) 3095. E. Cormier, P. Lambropoulos, J. Phys. B 29 (1996) 1667. E. Cormier, P. Lambropoulos, J. Phys. B 30 (1997) 77. D.B. Milos\ evicH , A.F. Starace, J. Phys. B 32 (1999) 1831. D.B. Milos\ evicH , F. Ehlotzky, Phys. Rev. A 58 (1998) 2319. C. Chen, Y.-Y. Yin, D.S. Elliot, Phys. Rev. Lett. 64 (1990) 507. C. Chen, D.S. Elliot, Phys. Rev. Lett. 65 (1990) 1737. Y.-Y. Yin, C. Chen, D.S. Elliot, A.V. Smith, Phys. Rev. Lett. 69 (1992) 2353. C.K. Chan, P. Brumer, M. Shapiro, Phys. Rev. Lett. 64 (1990) 3199. Z.-M. Wang, D.S. Elliott, Phys. Rev. Lett. 84 (2000) 3795. N.B. Baranova, A.N. Chudinov, B.Ya. Zel'dovich, Opt. Commun. 79 (1990) 116.

F. Ehlotzky / Physics Reports 345 (2001) 175}264

263

[200] N.B. Baranova, B.Ya. Zel'dovich, A.N. Chudinov, A.A. Shulginov, Sov. Phys. JETP 71 (1990) 1043. [201] N.B. Baranova, I.M. Peterov, B.Ya. Zel'dovich, I.I. Ryabtsev, A.A. Chudinov, A.A. Shulginov, JETP Lett. 55 (1992) 439. [202] S. Cavalieri, R. Eramo, L. Fini, Phys. Rev. A 55 (1997) 2941. [203] N.E. Karapanagioti, D. Xenakis, D. Charalambidis, C. Fotakis, J. Phys. B 29 (1996) 3599. [204] N.E. Karapanagioti, D. Xenakis, D. Charalambidis, C. Fotakis, in: P. Lambropoulos, H. Walther (Eds.), Multiphoton Processes 1996, Institute of Physics Conference Series, Vol. 154, Institute of Physics, Bristol, 1997, p. 248. [205] E. Papastathopoulos, D. Xenakis, D. Charalambidis, Phys. Rev. A 59 (1999) 4840. [206] P. Bowe, R.W. O'Neill, P.J.M. van der Burgt, J.A. Slevin, A. RaczynH ski, J. Zaremba, S. Chwirot, J. Phys. B 31 (1998) 1003. [207] H.G. Muller, P.H. Bucksbaum, D.W. Schumacher, A. Zavriyev, J. Phys. B 23 (1990) 2761. [208] D.W. Schumacher, F. Weihe, H.G. Muller, P.H. Bucksbaum, Phys. Rev. Lett. 73 (1994) 1344. [209] D.W. Schumacher, P.H. Bucksbaum, Phys. Rev. A 54 (1996) 4271. [210] P.B. Corkum, N.H. Burnett, F.B. Brunel, Phys. Rev. Lett. 62 (1989) 1259. [211] S. Watanabe, K. Kondo, Y. Nabekawa, A. Sagisaka, Y. Kobayashi, Phys. Rev. Lett. 73 (1994) 2692. [212] R.R. Jones, Phys. Rev. Lett. 75 (1995) 1491. [213] M.D. Perry, J.K. Crane, Phys. Rev. A 48 (1993) R4051. [214] H. Eichmann, A. Egbert, S. Nolte, C. Momma, B. Wellegehausen, W. Becker, S. Long, J.K. McIver, Phys. Rev. A 51 (1995) R3414. [215] X.-M. Tong, S.-I. Chu, Phys. Rev. A 58 (1998) R2656. [216] X.-M. Tong, S.-I. Chu, Phys. Rev. A 57 (1998) 452. [217] X.-M. Tong, S.-I. Chu, Chem. Phys. 217 (1997) 119. [218] V. Averbukh, O.E. Alon, N. Moiseyev, Phys. Rev. A 60 (1999) 2585. [219] O.E. Alon, V. Averbukh, N. Moiseyev, Phys. Rev. Lett. 80 (1998) 3743. [220] W. Becker, B.N. Chichkov, B. Wellegehausen, Phys. Rev. A 60 (1999) 1721. [221] I. Mercer, E. Mevel, R. Zerne, A. L'Huillier, P. Antoine, C.-G. WalstroK m, Phys. Rev. Lett. 77 (1996) 1731. [222] A. Hohla, V. Vuletic, T.W. HaK nsch, C. Zimmermann, Opt. Lett. 23 (1998) 436. [223] U. Andiel, G.D. Tsakiris, E. Cormier, K. Witte, Europhys. Lett. 47 (1999) 42. [224] D. Xenakis, N.E. Parapanagioti, O. Faucher, E. Hertz, D. Charalambidis, J. Phys. B 32 (1999) 341. [225] R. van Leeuwen, M.L. Bajama, R.R. Jones, Phys. Rev. Lett. 82 (1999) 2852. [226] N. Ph. Georgiades, E.S. Polzik, H.J. Kimble, Opt. Lett. 21 (1996) 1688. [227] L. Sirko, S. Yoakum, A. Ha!mann, P.M. Koch, Phys. Rev. A 47 (1993) R782. [228] Q. Wu, D.J. Gauthier, T.W. Mossberg, Phys. Rev. A 49 (1994) 1519. [229] M.W. Noel, W.M. Gri$th, T.F. Gallagher, Phys. Rev. A 58 (1998) 2265. [230] L. Ko, M.W. Noel, J. Lambert, T.F. Gallagher, J. Phys. B 32 (1999) 3469. [231] W. Lange, H. Walther, G.S. Agarwal, Phys. Rev. A 50 (1994) R3593. [232] M.R. Williams, F. Chi, M.T. Cashen, H. Metcalf, Phys. Rev. A 60 (1999) R1763. [233] D.V. Kosachiov, B.G. Matisov, Yu.V. Rezhdestvensky, J. Phys. B 25 (1992) 2473. [234] T. Nakajima, P. Lambropoulos, Phys. Rev. Lett. 70 (1993) 1081. [235] T. Nakajima, P. Lambropoulos, Phys. Rev. A 50 (1994) 595. [236] P. Lambropoulos, P. Zoller, Phys. Rev. A 24 (1981) 379. [237] G. Alber, P. Zoller, Phys. Rev. A 26 (1982) 1373. [238] T. Nakajima, J. Zhang, P. Lambropoulos, J. Phys. B 30 (1997) 1077. [239] T. Nakajima, J. Zhang, P. Lambropoulos, Phys. Rev. Lett. 79 (1997) 3367. [240] F. Wang, C. Chen, D.S. Elliot, Phys. Rev. Lett. 77 (1996) 2416. [241] T. Nakajima, Phys. Rev. A 61 (2000) 041403(R). [242] T. Grosges, B. Piraux, H. Bachau, Phys. Rev. A 60 (1999) 1371. [243] A. Lyras, H. Bachau, Phys. Rev. A 60 (1999) 4781. [244] U. Lambrecht, M. Nurhuda, F.H.M. Faisal, Phys. Rev. A 57 (1998) R3176. [245] G.Yu. Kryuchkian, M. Jakob, A.S. Sargsian, Phys. Rev. A 57 (1998) 2091. [246] M. Jakob, G.Yu. Kryuchkyan, Phys. Rev. A 61 (2000) 053823.

264 [247] [248] [249] [250] [251] [252] [253] [254] [255] [256] [257] [258] [259] [260] [261] [262] [263] [264] [265] [266] [267] [268] [269] [270] [271]

[272] [273]

F. Ehlotzky / Physics Reports 345 (2001) 175}264 S. Wielandy, A.L. Gaeta, Phys. Rev. Lett. 81 (1998) 3359. E. Paspalakis, P.L. Knight, Phys. Rev. Lett. 81 (1998) 293. E. Paspalakis, P.L. Knight, J. Mod. Opt. 47 (2000) 1025. M.D. Havey, D.V. Kupriyanov, I.M. Sokolov, Phys. Rev. Lett. 84 (2000) 3823. G. Kurizki, M. Shapiro, P. Brumer, Phys. Rev. B 39 (1989) 3435. E. Dupont, P.B. Corkum, H.C. Liu, M. Buchanan, Z.R. Wasilewski, Phys. Rev. Lett. 74 (1995) 3596. J. Karczmarek, M. Stott, M. Ivanov, Phys. Rev. A 60 (1999) R4225. M. Woldeyohannes, S. John, Phys. Rev. A 60 (1999) 5046. M. Grifoni, P. Haenggi, Phys. Rep. 304 (1998) 229. S. VarroH , F. Ehlotzky, Phys. Rev. A 57 (1997) 2439. S. VarroH , F. Ehlotzky, Phys. Rev. A 54 (1996) 3245. M.D. Havey, D.V. Kuprianov, I.M. Sokolov, Two-photon coherent control of atomic collisions by light with entangled polarization, Phys. Rev. Lett. 84 (2000) 3823. T. Cheng, J. Liu, S. Chen, Iterative treatment for atoms in high-frequency laser "elds: Two-color interference e!ect, Phys. Rev. A 62 (2000) 033402. A. JaronH , E. Mese, R.M. Potvliege, Floquet analysis of laser-induced continuum structures, J. Phys. B 33 (2000)  1487. S. Bivona, R. Burlon, C. Leone, Ponderomotive threshold shift and phase control in multiphoton detachment of negative ions, J. Phys. B 33 (2000) L591. Z. Ficek, J. Seke, A. Soldatov, G. Adam, Phase control of subharmonic resonances, Opt. Commun. 182 (2000) 143. R.G. Unanyan, S. GueH rin, H.R. Jauslin, Coherent population trapping under bichromatic "elds, Phys. Rev. A 62 (2000) 043407. D.B. Milos\ evicH , W. Becker, R. Kobold, Generation of circularly polarized high-order harmonics by two-color coplanar "eld mixing, Phys. Rev. A 61 (2000) 063403. C. Figueira de Morisson Faria, D.B. Milos\ evicH , G.G. Paulus, Phase-dependent e!ects in bichromatic high-order harmonic generation, Phys. Rev. A 61 (2000) 063415. D.B. Milos\ evicH , W. Becker, Attosecond pulse trains with unusual nonlinear polarization, Phys. Rev. A 62 (2000) 011403. J. Kuba, A. Klisnick, D. Ros, P. Fourcade, G. Jamelot, J.-L. Miquel, N. Blanchot, J.-F. Wyart, Two-color transient pumping in Ni-like Silver at 13.9 and 16.1 nm, Phys. Rev. A 62 (2000) 043808. G. Ferrante, M. Zarcone, S.A. Uryupin, Harmonic generation and wave mixing in a plasma in the presence of two linearly polarized laser "elds, J. Opt. Soc. A 17 (2000) 1383. S.K. Dutta, N.V. Morrow, G. Raithel, Enhancement of Sisyphus cooling using a bichromatic standing wave, Phys. Rev. A 62 (2000) 035401. K. Sundermann, H. Rabitz, R. de Vivie-Riedle, Compensating for spatial laser pro"le e!ects on the control of quantum systems, Phys. Rev. A 62 (2000) 013409. E. Springate, N. Hay, J.W.G. Tisch, M.B. Mason, T. Ditmire, J.P. Marangos, M.H.R. Hutchinson, Enhanced explosion of atomic clusters irradiated by a sequence of two high-intensity laser pulses, Phys. Rev. A 61 (2000) 044101. F. Ghafoor, S.-Y. Zhu, M.S. Zubairy, Amplitude and phase control of spontaneous emission, Phys. Rev. A 62 (2000) 013811. D. Petrosian, P. Lambropoulos, Phase control of photoabsorption in optically dense media, Phys. Rev. Lett. 85 (2000) 1843.