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Creation of population inversion by Fano interference in three-level cascade-type system
1 May 1996
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
126 (1996) 38-44
Creation of population inversion by Fano interference in three-level cascade-type system Yu.P. Malakyan, R.G. Unanyan Institute for Physical Research of Armenian Nat Academy of Sciences, Ashrarak-2,37&110, Armenia
Received 10 July 1995, accepted 9 November 1995
Abstract We demonstrate how Fano interference produces a large population inversion in a three-level cascade-type system involving an autoionizing state. We show that when an intense monochromatic field driving the lower transition of the atom by a two-photon resonant interaction creates a pair of dressed states which then are coupled by a second coherent field to ionization continuum and an autoionizing state, the Fano interference prevents the photoionization from the upper dressed state whereas the second state is quickly exhausted, thus making possible the coherent generation of short-wavelength light free of phase-matching condition. We discuss the incoherent processes which can destroy the Fano interference.
1. Introduction The attempts for many years to avoid the problem of population inversion (PI) in high-excited atomic levels, which stood for a long time in the way of short-wavelength laser development, have recently led to the prediction of lasing without inversion (LWI) [l-3] based on atomic coherence and quantum interference effects. Several experiments in the visible region [4-‘71 confirm the initial ideas of noninversion lasing. However, the ultimate goal of LWI to generate high-frequency coherent light is still far from being reached and the ways to produce the PI in high-lying atomic levels remain a subject of urgent study. At the same time some realistic considerations show that the solution of the problem can be found by utilizing coherent population trapping mechanisms. The idea we develop in this paper is the following. Intense monochromatic field strongly coupling 0030418/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0030-4018(95)00725-3
the ground and excited states of an atom by multiphoton resonant interaction produces the pair of dressed states which in the absence of the field turn into the bare ground and excited atomic states, respectively. The conditions are then constructed such that the atoms in the first dressed state rapidly decay to far removed states or are ionized whereas the population of the second one remains unaffected or is only slightly reduced using the population trapping mechanisms employed in many models of LWI (for a review see Refs. [8,9]) and in laser cooling of the atoms [lo], thus, a large population inversion is produced between the two states. The applicability of these mechanisms in the case of excited atomic levels unstable against radiative decay was not yet studied, however we will not discuss this question here since the relevant time-scale considered below is much shorter than the inverse radiative decay time. In this paper we will apply this mechanism to the three-level ladder system involving a single autoion-
Yu.P. Malakyun. R.G. Unnnyan/Optic.s
izing state (AS), as depicted in Fig. la and will show how the Fano interference leads to large PI between AS and the ground state of the atom. Remind that Fano-type interference has been exploited in Harris proposal of LWI [l]. In our case the driving field at a frequency o, couples the ground and excited levels by two-photon resonant interaction producing the dressed states I I), > (Fig. lb) separated by the Rabi frequency 0,. The second field at w2 ionizes the atoms from the I t,!_ > state, while being tuned resonantly to the frequency of “ I I/J+) + AS” transition it creates under the right conditions for the applied field the Fano resonance [l 11, thus trapping the atoms in the I J./J+)and autoionizing states. The same can be true for a one-electron atom if one replaces the AS by a laser-induced continuumstructure (LICS) [ 121. For the recent experiments on LICS see Refs. [ 13,141, where a convincing demonstration of LICS has been reported. Although in this case the situation is more complicated, there are
Communications
126 (1996) 38-44
39
much more possibilities to provide the trapping conditions due to the fact that the LICS position and width are easily varied by tuning the frequency of a structure-inducing laser. In this paper, however, we restrict our consideration to the AS case. In the next section we derive the basic equations describing the temporal evolution of the dressed states amplitudes. The Fano interference and population trapping are discussed in Section 3. A brief analysis of the limiting processes which can destroy the trapping condition are also included. Finally, Section 4 contains a summary of the results reported in this paper.
2. Basic equations
of the system
The equations for the amplitudes of two bound 1 and 2 and autoionizing 3 states in a three-level ladder system shown in Fig. la are found using the
b Fig. 1. (a) Schematic diagram of three-level systems studied. The driving field at w, couples the ground I and excited 2 levels and the second laser wr couples the excited state to AS. The Rabi frequencies R, and fla are taken as real. (b) The dressed states 1t/j* > separated > and AS due to this interaction by R are produced. The second laser is tuned to the I@+) + AS transition frequency. The splitting of 1t+h+ is not shown. The generation of light at the frequency wj is expected.
40
Yu.P. Malakyan, R.G. CJnanyan/ Optics Communications
standard technique [ 151 when the continuum is eliminated by substituting the solution for its amplitude to the three other equations.
126 (1996) 38-44
The new basis vector is connected to the bare atomic states by
tcI-
J=
(1) where the total Hamiltonian is non-Hermitian and has the form:
+h&(l
- i/q)(o,,
+ 5&
(2)
In these equations A, = w2, - 2w,, A, = oj2 - w2, n, = matrices, are the Pauli aij wi) and 0, = ~~~E*/fi are c CLzn/-%IEVfi(%, real-valued Rabi frequencies of driving E, and coupling E, fields respectively, I-i = 2~ 1V,, I* is the ionization rate from 12) due to laser 2 and r, = 27r I U I * is the autoionization width of 13) due to configuration Fano interaction U. q is the Fano parameter which represents the relative magnitude of two quantum mechanical pathways coupling the bound state to the AS:
(3)
q=2n2(rirJ-“‘.
We will now assume that Ti and r, are larger than all relaxation rates of the medium but they are sufficiently small compared to 0, allowing to consider the I I,!I+ > as the isolated states. The Stark-shift 0, due to the coupling E, field is also assumed to be small by the reason explained below. Thus our model requirement is: R,, fzi, r, en,.
(4)
The interaction of the atom with the driving field can then be treated exactly if one transforms Eqs. (1) to the corresponding dressed states basis which are found from the eigenvalue problem: ~~,~**+~~",*+~*~~lI~*~=~*l19*~~ &I
I 43) = 0,
where A,= (A, + 0)/2,
(5) L?= (A: + 4L!f)‘/*.
0 (1 =&St:.
*+
3
*3
St =
cos
e
sin
e
-sin 13 0 ~0~8
0
0
,
(6)
1I
0
where tan 8 = [(L? - A)/( ii2 + A)]“‘. In the new states basis the Hamiltonian fi = S’HS governs the temporal evolution of dressed states amplitudes A^= St& Upon transforming fi to the interaction picture by @ = exp( - i fit, r/h) g X exp(i fiOt/h) where
go=
(;
;
(7)
.%,,)
one finds the following equations for A: A_(t) = -(iA_+T_/2)A_, A,(f)
= -(ih++r+/2)A+ -ifi,cose(l
-i/q)A,,
(8)
A,(f) = - [i( A, + A*) + KJ2)A, -ifl,cose(l
-i/q)A+,
where r* = Ti cos*e(sin*e). In deriving Eq. (8) the pronounced oscillating terms exp( A+ - A_ )t have been neglected considering two-photon detuning A, + A, to be close to A+ (see Eq. (12)) and t ak’mg into account that the time scale of interest is r- ‘. This allows to decouple the equation for A_ from two others. As a result the / t,!_ > state independently decays into the continuum with the ionization rate r_. Eqs. (8) are solved subject to the conditions that the atom is initially prepared in the dressed states I +!r, > with the populations R, (0) = p0 and R_ (0) = 1 - p0 where the maximum value of p0 is l/2. This means that the coupling field is turned on with some delay until the coherently driven system settles
Yu.P. Mulakyan, R.G. Unanyan / Optics Communications
41
126 (1996) 38-44
down into steady-state due to the radiative decay or collisions which are caused by the own pressure of the active atoms or take place with a buffer gas. Besides, since the coupling field would not mix the dresses states I $,, > ( f12 ez a,>, its rise time T must satisfy the adiabatic condition ~0 Z+ 1. On the other hand r should be small enough to neglect the ionization from 1++) in the course: ~~~* 1. Both requirements are compatible with (4). Then, from (8) the populations of I J/, > and autoionizing states are
(9)
R(t)=(l-p,)exp(-r-t),
0
R+(r)=~~l(h+-iS2-ir+/2)exp(S,t)
4
2
6
6,
10
-(A+-iS,-ir+/2)
R,(r)
=p,~n,2cos28(1
.I d
(10)
Xexp(S,t)l*/lS,-S212,
AYdW,
,
1
lb)
+ l/q2)lexp(S,t)
-exp(S2r)12/lS,-~212,
(11)
_..__..________
..__......_....______ _
where S,,,=i
1
[A++A,+A,-i(c+r+)/2] 2 -.l -
+ (A+-A,-A2+i(<-1;)/2)2
If
[
I
+ 4L?,2cos28( 1 - i/q)*] “*)/2.
Thus, in general the system decays into the continuum during the time t = ( < i)- ’ . In the next section we discuss the trapping condition under which the transition from I $+) and AS to the continuum closes down.
3. Trapping
condition.
Influence
of incoherent
0
2
4
6 ri
3
IO
( Ill unitsof r* )
Fig. 2. (a) Population differences between the AS and ground level (curve 1) and between AS and excited 2 level (curve 2) as a function of T, fin units of f,) for A, /0, = 1.15, y= I and p0 = 0.5. Note that always AN, > AN,. (b) The same except that A, = 0. In this case AN, = AN,. In both cases AN,,, reach their maximum near r, cos% - 4.
processes
It is well known [12] that for trapping the atom in the bound and autoionizing states the Fano resonance condition must be hold which in our case is written as:
so that for the times I > (r,,, 1-l the nonvanishing populations in these states are found: R+=P,[q*(r,+~+)2+4r,2]
/+I*+ l)(r,+r+>‘~
(15)
In this case Ret&,) = 0 and S,,, become
R3=PJ+Ta/K+r+)*.
(16)
S,= -i[A++q(Ta+r+)/4]
From Eqs. (15), (16), R+,, depend on the intensity of the coupling field through r+ = ri cos20 _ \ E, I*. In Figs. 2 and 3 we show the population difference between the atomic states AN, = R, - R, and A N2
A+- A, - A, =
q( r+-
r,)/2.
( 12)
-(C+r+)/2, ( 13)
S,=
-i[A+-q(&+r+)/4],
(14)
42
Yu.P. Malakyan, R.G. Unanyan/ Oprics Communicarions 126 11996) 38-44
.6
0
5 time
IO ( in unitsof r,
15 )
Fig. 3. Temporal behavior of ground (curve I), excited (2) and autoionizing (3) states for q = 1, A, /r, = 10, ri cos*fl= r, and cos% = 0.75.
= R, - R, as a function of ri calculated for two values of A,/0, using Eqs. (91, (15), (16) and the reverse transformation B = Si. First, in the case of exact two-photon resonance we have AN, = A N2 independent of ri. However, if cos20 > l/2, AN, > AN2 in all ri region. Secondly, the populations reach their maximum at the point r+ = ri cos2e = r,.
(17)
The physical meaning of this condition is clear. For values of ri smaller than (17) the AS is basically damped by the autoionization long before the steady-state sets in. For larger values the same takes place but now by the pathway AS + 1+!J+> + continuum. The dependence of ANi on the coupling field detuning A, has a similar form and ANi has a maximum at the value A, = h, - A, following from (12). It is interesting to compare our results with that obtained in Ref. [I61 where the doubly cascade three-level system has been discussed and the inversion between the upper and intermediate or intermediate and ground states has been found. In contrast to these results we have obtained the simultaneous inversion between cascade states AS + 2 + 1, among which the inversion between AS and the ground state is the larger one. It must be noted that the effect predicted in this paper can be also observed in a cascade system of bound states if the mechanism of
1I,!_) depletion and coherent population trapping in high-lying bound states is employed. In Fig. 3 the temporal behavior of the bare atomic level populations is shown for the case of (17). R, and R, show the strong oscillations with two-photon Rabi frequency 0. Thus the quantum interference (12) leads to population trapping in the AS and, hence, to the large PI between AS and ground level making possible the generation of a coherent light of frequency w3 u wj, (Fig. lb) without phase-matching requirement. At this point one needs to prove the existence of the gain at this frequency since in the presence of both driving and coupling fields the atomic coherence can lead to an inversion without amplification and vice versa. This question, in general form, has been discussed in Ref. [17]. However, neglecting the alteration of the atomic population by the w,-field one can derive a simple equation for its amplitude, d*E,/dzdt
= 2rrNw,
1 /_L,~1‘(R,
+ phase-matching
- R,) E,/hc term,
(18)
showing the exponential gain proportional to the inversion R, - R, > 0. In the rhs of Eq. (18) the second term is responsible for the phase-matching generation leading to linear amplification which is not essential in our case. Here N is the atomic number density. Eq. (18) describes the transient regime of generation since the I $+> and AS are unstable against spontaneous emission and we have gain only for the times t I y- ’ , where y is a relevant decay rate. So far we have considered the simplest case of coherent fields and have ignored their fluctuations as well as the incoherent processes involved, e.g. the free-free transitions in the same continuum (above threshold ionization (ATI)), ionization from AS, etc, which can destroy the trapping condition (12). Obviously, our results remain valid if r, and, hence ri, as is required from (121, exceed all damping rates of the medium as well as spectral width of the applied fields. For a vapor pressure of a few Torr and a temperature up to 1000°C the collisional rate and Doppler broadening are only _ 0.01 - 0.1 cm-’ while usually r,/2rrc > 1 cm-‘. The weak restrictions follow from incoherent processes, as AT1 becomes significant for laser intensities beyond lOI
Yu.P. Malakyan, R.G. Unanyun/
Optics Communications
W/cm*, and the photoionization from AS is due to a two-electron transition and thereby is suppressed or occurs beginning at a specific frequency which restricts from above the laser frequency. On the other hand T, should not be too large, since otherwise it is very difficult to satisfy (4) taking into account that the multiphoton Rabi frequency is usually small, if intermediate resonances are absent. For a resonant enhancement of fi, the nondegenerate driving field can be used provided, however, that the ionization from I 9, > by laser w, is again forbidden or neglectly small. It is useful to demonstrate how these special conditions, especially the requirements (41, are satisfied in a real atomic system. The estimations show that the most preferable value of r, is - 1 - 2 cm-’ or less. Note that in heavy metals r, decreases with increasing nuclear charge Z as r, l/Z2 [18]. Recently such AS have been found in rare earth metals, in particular in ytterbium (r, - 1 cm-‘> and gadolinium (- 0.07 cm-‘> [l&19]. In Yt the nondegenerate laser pulses driving the two-photon resonant transition 6s2 ‘S, --) 6s5d ‘D, can be used with frequencies smaller than is needed for one-photon ionization from the 6s5d ‘D, state. Due to the resonant enhancement the Rabi frequency 0, can be of the order of several cm-’ at least for moderate values of driving field intensity. Further, the oscillator strength of two-electron transition from 6s5d to the 7s6p ‘P, AS is expected to be f- 10-j [20] making it possible to get J& < fl, even for an intense w,-field but having at the same time a large ionization ( ri - 1 cm-‘) from the 6s5d ‘D, level. In this configuration a coherent light generation on the transition AS -+ ground state with wavelength A160 nm is expected.
4. Conclusions In summary, we have discussed a general mechanism for creation of PI in a high-lying atomic level which is realized in two stages. At the first step an intense coherent light coupling this level with the ground state of an atom produces a pair of dressed states which in the zero-field limit are transformed into bare ground and excited atomic levels. In the second stage the first dressed state is quickly de-
126 (1996) 38-44
43
pleted, while the population in the other is coherently trapped. In this paper we have applied this mechanism to the particular case of a three-level cascadetype system involving AS where the population trapping is attained by means of Fano interference. We have pointed out that the considered scheme is suitable for sum-frequency generation from AS due to PI and, hence, free of phase-matching requirement. The showed possibility of PI creation by Fano interference encourages the research of the problem in the LICS model where the population trapping can be certainly realized much easier than in the static AS case.
Acknowledgements Yu.M. would like to express his gratitude to Prof. H. Wahher for the kind hospitality at the MaxPlanck-Institut fur Quantenoptik in Garching (Germany), where this work was initiated. This work is partly supported by the International Science Foundation under Grant No RY-8000.
References [II S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033. 121O.A. Kocharovskaya and Ya.1. Khanin, Zh. Eksp. Teor. Fiz. Pis. Red. 48 (1988) 58 I. [Sov.JETP Lett. 48 ( 1988) 6301. I31 M.O. Scully, S-Y Zhu and A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813. [41 J.Y. Gao, C. Gao, X.2. Guo, G.X. Jin, Q.W. Wang, J. Zhao, H.Z. Zhang, Y. Jiang, D.Z. Wang and D.M. Jiang, Optics Comm. 93 (1992) 323. [51 A. Nottelman, C. Peters and W. Lange, Phys. Rev. Lett. 70 (1993) 1783. 161ES. Fry, X. Li, D. Nikonov, G.G. Padmabandu, M.O. Scully, A.V. Smith, F.K. Tittel, C. Wang, S.R. Wilkinson and S.-Y. Zhu, Phys. Rev. Lett. 70 (1993) 3235. and H.B. Linden van [71 W.E. Van der Veer, A. Donszelmann den Heuvell, Phys. Rev. Lett. 70 (1993) 3243. Phys. Rep. 219 ( 1992) 175. I81 0. Kocharovskaya, [91 M.O. Scully, Phys. Rep. 219 (1992) 191. II01 A. Aspect, E. Arimondo, R. Kaiser, N. Vansteekiste and C. Cohen-Tanoudji, Phys. Rev. Lett. 61 (1988) 826. [I 11 U. Fano, Phys. Rev. 124 (1961) 1866. 1121P.L. Knight, M.A. Lauder and B.J. Dalton, Phys. Rep. 190 (1990) 1. 1131Y.L. Shao, D. Charalambidis, C. Fotakis, J. Zhang and P. Lambropoulos, Phys. Rev. Lett. 67 (1991) 3669.
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Yu.P. Malakyan, R.G. Unanyan/Optics
[ 141 S. Cavalieri, F.S. Pavone and M. Matera, Phys. Rev. Lett. 67
(1991) 3673. [15] Bo-nian Dai and P. Lambropoulos, Phys. Rev. A36 (1987) 5205. [16] R.M. Whitley and C.R. Stroud, Jr., Phys. Rev. Al4 (1976) 1498. [17] 0. Kocharovskaya and P. Mandel, Quantum Optics 6 (1994) 217.
Communications 126 (1996) 38-44
[18] G.I. Bekov, E.P. Vidolova-Angelova, L.N. Ivanov, V.S. Letokhov and V.I. Mishin, Optics Comm. 35 (1980) 194. [ 191 V.S. Letokhov, Laser photoionization spectroscopy, (Moscow, Nauka, 1987) [in Russian]. [20] II. Fano and J.W. Cooper, Rev. Mod. Phys. 40 (1968) 441.
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
126 (1996) 38-44
Creation of population inversion by Fano interference in three-level cascade-type system Yu.P. Malakyan, R.G. Unanyan Institute for Physical Research of Armenian Nat Academy of Sciences, Ashrarak-2,37&110, Armenia
Received 10 July 1995, accepted 9 November 1995
Abstract We demonstrate how Fano interference produces a large population inversion in a three-level cascade-type system involving an autoionizing state. We show that when an intense monochromatic field driving the lower transition of the atom by a two-photon resonant interaction creates a pair of dressed states which then are coupled by a second coherent field to ionization continuum and an autoionizing state, the Fano interference prevents the photoionization from the upper dressed state whereas the second state is quickly exhausted, thus making possible the coherent generation of short-wavelength light free of phase-matching condition. We discuss the incoherent processes which can destroy the Fano interference.
1. Introduction The attempts for many years to avoid the problem of population inversion (PI) in high-excited atomic levels, which stood for a long time in the way of short-wavelength laser development, have recently led to the prediction of lasing without inversion (LWI) [l-3] based on atomic coherence and quantum interference effects. Several experiments in the visible region [4-‘71 confirm the initial ideas of noninversion lasing. However, the ultimate goal of LWI to generate high-frequency coherent light is still far from being reached and the ways to produce the PI in high-lying atomic levels remain a subject of urgent study. At the same time some realistic considerations show that the solution of the problem can be found by utilizing coherent population trapping mechanisms. The idea we develop in this paper is the following. Intense monochromatic field strongly coupling 0030418/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0030-4018(95)00725-3
the ground and excited states of an atom by multiphoton resonant interaction produces the pair of dressed states which in the absence of the field turn into the bare ground and excited atomic states, respectively. The conditions are then constructed such that the atoms in the first dressed state rapidly decay to far removed states or are ionized whereas the population of the second one remains unaffected or is only slightly reduced using the population trapping mechanisms employed in many models of LWI (for a review see Refs. [8,9]) and in laser cooling of the atoms [lo], thus, a large population inversion is produced between the two states. The applicability of these mechanisms in the case of excited atomic levels unstable against radiative decay was not yet studied, however we will not discuss this question here since the relevant time-scale considered below is much shorter than the inverse radiative decay time. In this paper we will apply this mechanism to the three-level ladder system involving a single autoion-
Yu.P. Malakyun. R.G. Unnnyan/Optic.s
izing state (AS), as depicted in Fig. la and will show how the Fano interference leads to large PI between AS and the ground state of the atom. Remind that Fano-type interference has been exploited in Harris proposal of LWI [l]. In our case the driving field at a frequency o, couples the ground and excited levels by two-photon resonant interaction producing the dressed states I I), > (Fig. lb) separated by the Rabi frequency 0,. The second field at w2 ionizes the atoms from the I t,!_ > state, while being tuned resonantly to the frequency of “ I I/J+) + AS” transition it creates under the right conditions for the applied field the Fano resonance [l 11, thus trapping the atoms in the I J./J+)and autoionizing states. The same can be true for a one-electron atom if one replaces the AS by a laser-induced continuumstructure (LICS) [ 121. For the recent experiments on LICS see Refs. [ 13,141, where a convincing demonstration of LICS has been reported. Although in this case the situation is more complicated, there are
Communications
126 (1996) 38-44
39
much more possibilities to provide the trapping conditions due to the fact that the LICS position and width are easily varied by tuning the frequency of a structure-inducing laser. In this paper, however, we restrict our consideration to the AS case. In the next section we derive the basic equations describing the temporal evolution of the dressed states amplitudes. The Fano interference and population trapping are discussed in Section 3. A brief analysis of the limiting processes which can destroy the trapping condition are also included. Finally, Section 4 contains a summary of the results reported in this paper.
2. Basic equations
of the system
The equations for the amplitudes of two bound 1 and 2 and autoionizing 3 states in a three-level ladder system shown in Fig. la are found using the
b Fig. 1. (a) Schematic diagram of three-level systems studied. The driving field at w, couples the ground I and excited 2 levels and the second laser wr couples the excited state to AS. The Rabi frequencies R, and fla are taken as real. (b) The dressed states 1t/j* > separated > and AS due to this interaction by R are produced. The second laser is tuned to the I@+) + AS transition frequency. The splitting of 1t+h+ is not shown. The generation of light at the frequency wj is expected.
40
Yu.P. Malakyan, R.G. CJnanyan/ Optics Communications
standard technique [ 151 when the continuum is eliminated by substituting the solution for its amplitude to the three other equations.
126 (1996) 38-44
The new basis vector is connected to the bare atomic states by
tcI-
J=
(1) where the total Hamiltonian is non-Hermitian and has the form:
+h&(l
- i/q)(o,,
+ 5&
(2)
In these equations A, = w2, - 2w,, A, = oj2 - w2, n, = matrices, are the Pauli aij wi) and 0, = ~~~E*/fi are c CLzn/-%IEVfi(%, real-valued Rabi frequencies of driving E, and coupling E, fields respectively, I-i = 2~ 1V,, I* is the ionization rate from 12) due to laser 2 and r, = 27r I U I * is the autoionization width of 13) due to configuration Fano interaction U. q is the Fano parameter which represents the relative magnitude of two quantum mechanical pathways coupling the bound state to the AS:
(3)
q=2n2(rirJ-“‘.
We will now assume that Ti and r, are larger than all relaxation rates of the medium but they are sufficiently small compared to 0, allowing to consider the I I,!I+ > as the isolated states. The Stark-shift 0, due to the coupling E, field is also assumed to be small by the reason explained below. Thus our model requirement is: R,, fzi, r, en,.
(4)
The interaction of the atom with the driving field can then be treated exactly if one transforms Eqs. (1) to the corresponding dressed states basis which are found from the eigenvalue problem: ~~,~**+~~",*+~*~~lI~*~=~*l19*~~ &I
I 43) = 0,
where A,= (A, + 0)/2,
(5) L?= (A: + 4L!f)‘/*.
0 (1 =&St:.
*+
3
*3
St =
cos
e
sin
e
-sin 13 0 ~0~8
0
0
,
(6)
1I
0
where tan 8 = [(L? - A)/( ii2 + A)]“‘. In the new states basis the Hamiltonian fi = S’HS governs the temporal evolution of dressed states amplitudes A^= St& Upon transforming fi to the interaction picture by @ = exp( - i fit, r/h) g X exp(i fiOt/h) where
go=
(;
;
(7)
.%,,)
one finds the following equations for A: A_(t) = -(iA_+T_/2)A_, A,(f)
= -(ih++r+/2)A+ -ifi,cose(l
-i/q)A,,
(8)
A,(f) = - [i( A, + A*) + KJ2)A, -ifl,cose(l
-i/q)A+,
where r* = Ti cos*e(sin*e). In deriving Eq. (8) the pronounced oscillating terms exp( A+ - A_ )t have been neglected considering two-photon detuning A, + A, to be close to A+ (see Eq. (12)) and t ak’mg into account that the time scale of interest is r- ‘. This allows to decouple the equation for A_ from two others. As a result the / t,!_ > state independently decays into the continuum with the ionization rate r_. Eqs. (8) are solved subject to the conditions that the atom is initially prepared in the dressed states I +!r, > with the populations R, (0) = p0 and R_ (0) = 1 - p0 where the maximum value of p0 is l/2. This means that the coupling field is turned on with some delay until the coherently driven system settles
Yu.P. Mulakyan, R.G. Unanyan / Optics Communications
41
126 (1996) 38-44
down into steady-state due to the radiative decay or collisions which are caused by the own pressure of the active atoms or take place with a buffer gas. Besides, since the coupling field would not mix the dresses states I $,, > ( f12 ez a,>, its rise time T must satisfy the adiabatic condition ~0 Z+ 1. On the other hand r should be small enough to neglect the ionization from 1++) in the course: ~~~* 1. Both requirements are compatible with (4). Then, from (8) the populations of I J/, > and autoionizing states are
(9)
R(t)=(l-p,)exp(-r-t),
0
R+(r)=~~l(h+-iS2-ir+/2)exp(S,t)
4
2
6
6,
10
-(A+-iS,-ir+/2)
R,(r)
=p,~n,2cos28(1
.I d
(10)
Xexp(S,t)l*/lS,-S212,
AYdW,
,
1
lb)
+ l/q2)lexp(S,t)
-exp(S2r)12/lS,-~212,
(11)
_..__..________
..__......_....______ _
where S,,,=i
1
[A++A,+A,-i(c+r+)/2] 2 -.l -
+ (A+-A,-A2+i(<-1;)/2)2
If
[
I
+ 4L?,2cos28( 1 - i/q)*] “*)/2.
Thus, in general the system decays into the continuum during the time t = ( < i)- ’ . In the next section we discuss the trapping condition under which the transition from I $+) and AS to the continuum closes down.
3. Trapping
condition.
Influence
of incoherent
0
2
4
6 ri
3
IO
( Ill unitsof r* )
Fig. 2. (a) Population differences between the AS and ground level (curve 1) and between AS and excited 2 level (curve 2) as a function of T, fin units of f,) for A, /0, = 1.15, y= I and p0 = 0.5. Note that always AN, > AN,. (b) The same except that A, = 0. In this case AN, = AN,. In both cases AN,,, reach their maximum near r, cos% - 4.
processes
It is well known [12] that for trapping the atom in the bound and autoionizing states the Fano resonance condition must be hold which in our case is written as:
so that for the times I > (r,,, 1-l the nonvanishing populations in these states are found: R+=P,[q*(r,+~+)2+4r,2]
/+I*+ l)(r,+r+>‘~
(15)
In this case Ret&,) = 0 and S,,, become
R3=PJ+Ta/K+r+)*.
(16)
S,= -i[A++q(Ta+r+)/4]
From Eqs. (15), (16), R+,, depend on the intensity of the coupling field through r+ = ri cos20 _ \ E, I*. In Figs. 2 and 3 we show the population difference between the atomic states AN, = R, - R, and A N2
A+- A, - A, =
q( r+-
r,)/2.
( 12)
-(C+r+)/2, ( 13)
S,=
-i[A+-q(&+r+)/4],
(14)
42
Yu.P. Malakyan, R.G. Unanyan/ Oprics Communicarions 126 11996) 38-44
.6
0
5 time
IO ( in unitsof r,
15 )
Fig. 3. Temporal behavior of ground (curve I), excited (2) and autoionizing (3) states for q = 1, A, /r, = 10, ri cos*fl= r, and cos% = 0.75.
= R, - R, as a function of ri calculated for two values of A,/0, using Eqs. (91, (15), (16) and the reverse transformation B = Si. First, in the case of exact two-photon resonance we have AN, = A N2 independent of ri. However, if cos20 > l/2, AN, > AN2 in all ri region. Secondly, the populations reach their maximum at the point r+ = ri cos2e = r,.
(17)
The physical meaning of this condition is clear. For values of ri smaller than (17) the AS is basically damped by the autoionization long before the steady-state sets in. For larger values the same takes place but now by the pathway AS + 1+!J+> + continuum. The dependence of ANi on the coupling field detuning A, has a similar form and ANi has a maximum at the value A, = h, - A, following from (12). It is interesting to compare our results with that obtained in Ref. [I61 where the doubly cascade three-level system has been discussed and the inversion between the upper and intermediate or intermediate and ground states has been found. In contrast to these results we have obtained the simultaneous inversion between cascade states AS + 2 + 1, among which the inversion between AS and the ground state is the larger one. It must be noted that the effect predicted in this paper can be also observed in a cascade system of bound states if the mechanism of
1I,!_) depletion and coherent population trapping in high-lying bound states is employed. In Fig. 3 the temporal behavior of the bare atomic level populations is shown for the case of (17). R, and R, show the strong oscillations with two-photon Rabi frequency 0. Thus the quantum interference (12) leads to population trapping in the AS and, hence, to the large PI between AS and ground level making possible the generation of a coherent light of frequency w3 u wj, (Fig. lb) without phase-matching requirement. At this point one needs to prove the existence of the gain at this frequency since in the presence of both driving and coupling fields the atomic coherence can lead to an inversion without amplification and vice versa. This question, in general form, has been discussed in Ref. [17]. However, neglecting the alteration of the atomic population by the w,-field one can derive a simple equation for its amplitude, d*E,/dzdt
= 2rrNw,
1 /_L,~1‘(R,
+ phase-matching
- R,) E,/hc term,
(18)
showing the exponential gain proportional to the inversion R, - R, > 0. In the rhs of Eq. (18) the second term is responsible for the phase-matching generation leading to linear amplification which is not essential in our case. Here N is the atomic number density. Eq. (18) describes the transient regime of generation since the I $+> and AS are unstable against spontaneous emission and we have gain only for the times t I y- ’ , where y is a relevant decay rate. So far we have considered the simplest case of coherent fields and have ignored their fluctuations as well as the incoherent processes involved, e.g. the free-free transitions in the same continuum (above threshold ionization (ATI)), ionization from AS, etc, which can destroy the trapping condition (12). Obviously, our results remain valid if r, and, hence ri, as is required from (121, exceed all damping rates of the medium as well as spectral width of the applied fields. For a vapor pressure of a few Torr and a temperature up to 1000°C the collisional rate and Doppler broadening are only _ 0.01 - 0.1 cm-’ while usually r,/2rrc > 1 cm-‘. The weak restrictions follow from incoherent processes, as AT1 becomes significant for laser intensities beyond lOI
Yu.P. Malakyan, R.G. Unanyun/
Optics Communications
W/cm*, and the photoionization from AS is due to a two-electron transition and thereby is suppressed or occurs beginning at a specific frequency which restricts from above the laser frequency. On the other hand T, should not be too large, since otherwise it is very difficult to satisfy (4) taking into account that the multiphoton Rabi frequency is usually small, if intermediate resonances are absent. For a resonant enhancement of fi, the nondegenerate driving field can be used provided, however, that the ionization from I 9, > by laser w, is again forbidden or neglectly small. It is useful to demonstrate how these special conditions, especially the requirements (41, are satisfied in a real atomic system. The estimations show that the most preferable value of r, is - 1 - 2 cm-’ or less. Note that in heavy metals r, decreases with increasing nuclear charge Z as r, l/Z2 [18]. Recently such AS have been found in rare earth metals, in particular in ytterbium (r, - 1 cm-‘> and gadolinium (- 0.07 cm-‘> [l&19]. In Yt the nondegenerate laser pulses driving the two-photon resonant transition 6s2 ‘S, --) 6s5d ‘D, can be used with frequencies smaller than is needed for one-photon ionization from the 6s5d ‘D, state. Due to the resonant enhancement the Rabi frequency 0, can be of the order of several cm-’ at least for moderate values of driving field intensity. Further, the oscillator strength of two-electron transition from 6s5d to the 7s6p ‘P, AS is expected to be f- 10-j [20] making it possible to get J& < fl, even for an intense w,-field but having at the same time a large ionization ( ri - 1 cm-‘) from the 6s5d ‘D, level. In this configuration a coherent light generation on the transition AS -+ ground state with wavelength A160 nm is expected.
4. Conclusions In summary, we have discussed a general mechanism for creation of PI in a high-lying atomic level which is realized in two stages. At the first step an intense coherent light coupling this level with the ground state of an atom produces a pair of dressed states which in the zero-field limit are transformed into bare ground and excited atomic levels. In the second stage the first dressed state is quickly de-
126 (1996) 38-44
43
pleted, while the population in the other is coherently trapped. In this paper we have applied this mechanism to the particular case of a three-level cascadetype system involving AS where the population trapping is attained by means of Fano interference. We have pointed out that the considered scheme is suitable for sum-frequency generation from AS due to PI and, hence, free of phase-matching requirement. The showed possibility of PI creation by Fano interference encourages the research of the problem in the LICS model where the population trapping can be certainly realized much easier than in the static AS case.
Acknowledgements Yu.M. would like to express his gratitude to Prof. H. Wahher for the kind hospitality at the MaxPlanck-Institut fur Quantenoptik in Garching (Germany), where this work was initiated. This work is partly supported by the International Science Foundation under Grant No RY-8000.
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[18] G.I. Bekov, E.P. Vidolova-Angelova, L.N. Ivanov, V.S. Letokhov and V.I. Mishin, Optics Comm. 35 (1980) 194. [ 191 V.S. Letokhov, Laser photoionization spectroscopy, (Moscow, Nauka, 1987) [in Russian]. [20] II. Fano and J.W. Cooper, Rev. Mod. Phys. 40 (1968) 441.