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Electron spin polarisation in laser induced autoionisation
Volume 53, number 2
OPTICS COMMUNICATIONS
15 February 1985
ELECTRON SPIN POLARISATION IN LASER INDUCED AUTOIONISATION Jakub ZAKRZEWSKI * Institute for Theoretical Physics, Polish Academy of Sciences, Warsaw,Poland
Received 18 July 1984 Revised manuscript received 5 December 1984
The electron spin polarisation in the strong laser light induced autoionisation is discussed in the simple model. It is shown that confluence of coherences manifests itself in the dependence of polarisation on the laser detuning as a field strength dependent resonance, which enables verification of the phenomenon by means of the spin polarisation measurement. The separation of the confluence of coherences point and the point at which the maximal slowing down of the ionisation process (population trapping) occurs is showed.
The investigations of the spin polarisation (SP) and angular distribution of electrons obtained in photoionisation are of considerable recent interest [ 1 - 3 ] . It stems from the fact, that measurements of these quantities provide a lot of information about the atomic structure as well as about the excitation process itself. Traditionally, the ionisation step of the excitation is included in the theoretical studies perturbatively i.e. continuum of electron states is treated as a sink for the loss of the bound states populations [3]. Recently attention has been paid to strong field effects arising due to saturation of bound-free transition, possible in the vicinity o f the narrow, autoionisation (AS) (or externally induced autoionising-like [4,5]) resonance [5,11]. Particularly interesting was the discovery of drastic narrowing of the photoelectron energy spectra for strong excitation of AS resonance [7]. It appeared when one of the AutlerTownes [12] doublet was tuned to the Fano zero [ 13 ] o f the dipole matrix element o f bound-free transition and therefore the phenomenon was termed [7] the confluence o f coherences (CC). Further research incorporated incoherence effects such as finite laser bandwidth [8], radiative damping [9], inhomogeneous broadening [10] or the existence of other atomic continua [ 11 ]. * Present address: Institute of Physics, Jagellonian University Reymonta 4, Krakow, Poland. 0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
However, the phenomenon has not been yet verified experimentally and its direct observation by electron spectra measurement will be impossible for some time due to insufficient electron energy resolution. Therefore the interest arose for finding other experimental arrangement which will make observation of CC possible. Up till now it was shown that CC manifests itself in the spectra of photons emitted during recombination accompanying autoionisation [9]. The aim of this communication is to propose another experiment capable of verifying the CC phenomenon, namely SP experiment. To this end we discuss the simple atomic model introduced before [11 ]. It consists (fig. 1) of the initial state 10 ) coupled by a laser light of frequency o3 L to two atomic continua I~o; A) and I~; B). In ref. [11] we have shown that the existence o f additional (as compared with the standard [7 ] model) ionisation channel may diminish the CC by irreversible decay. On the other hand the very existence of the additional channel forms the basis for SP experiment (provided the resonant, AS channel and the non-resonant one have different, appropriate quantum numbers). As a simple example o f the atomic structure we have in mind may serve excitation of 6 2D3/2 ( m / = 1/2) state of T1 autoionising to e 2D3/2 ( m / = 1/2) continuum [1 ]. Assuming n 2P1/2(m/ = -1/2)initial atomic state o + excitation couples it to 6 2S1/2(m/ = 1/2) continuum also. 99
Volume 53, number 2
OPTICS COMMUNICATIONS
)
I0)
Fig. 1. Scheme of the discussed system.
The hamiltonian of our system (in ~ = c = 1 units) reads H=co010X01 + f dcoco IcO;A)(cO;AI
i=A,B
of the AS resonance, COL its position, q is the Fano asymmetry parameter [ 13], f20 is a bound-free equivalence of the Rabi frequency and 77 denotes the relative strength of the 10) ~ IcO; B) coupling. We denote the SP of electrons that belong to rco; A)/Iw; B) continuum (A/B channel) by PA(co)/PB(co). Their w-dependence is determined in general by the detailed atomic structure. Here we assume the simplest possible choice making Pi(w), i = A, B independent of co (in the vicinity of 6 2D3/2 resonance this condition is satisfied in Tt example). In ref. [11] we discussed the long time (t -+ oo) photoelectron spectra resulting from the model defined by refs. [1] and [2]. Now we are interested in full time evolution of the system. We expand the Schrodinger wave vector in the basis of states { 10), Iw; A), Ico; B) } I~(t)) = ~(t) 10) + exp(--icoLt )
+ fdco co IcO;B)(w;BI f d~o[s-zAco) exp(icoLt) 10)(cO; il
+ h.c.].
(1)
Further the arbitrary zero on the energy scale is chosen to coincide with the initial state energy, thus cOo = 0. We assume the following form of the couplings ~2A(cO), ~2B(cO)
X f dco[t3A(co,t)lcO; A) +/~B(cO't)lco; B ) ] .
Because I/3i(co, T) I2 is the probability density for the electron with energy 6o to be ejected via i-channel between t = 0 and t = T, the total SP of electrons emitted during this time interval reads
P(T)=
fd6o [ [/3A(co, 7")12 + [/3B(cO,T)I 2]
~A(C°) - (47r70)1/2 70 co-Col-iT0 g2o S2B(~O ) = n -
1 i71 ). q +i cO-Wl-iT1
(2)
which as Pi(co) are assumed to be independent of co can be cast into the form
-i~' 1
-
(47r70) 1/2 co - co1 - i71"
Note that ~A(co) has the frequently used [7-11 ] two-lorentzian shape which takes into account the autoionising character of the 10) ~ Ico; A) transition and ~2B(cO) is essentially "flat in energy". (In our T1 example e 2S1/2(m / = 1/2) continuum serves as a flat channel. It couples to different AS state 6 2P1/2(m/ = 1/2) which however is well off resonance, thus "flat in energy" approximation is well justified.) In ref. [2] 71 is the width of the fiat background (taken to be infinite at the end of calculation), 70 is the width I00
(3)
fdco PA(co) I/~A(co, T) I 2 + fdcO eB(cO) lflB(W, T) 12
~20
X(
15 February 1985
P(T, A) =
PANA(T,A) +PBNB(T, A ) NA(T,A) +NB(T , A) '
(5)
where by Ni(T, &) --- fd~ol/3i(co, T) I 2 we denoted the number of electrons ejected via A/B channel and by A = cOL -- cO1 the detuning of the laser light from AS resonance. In the usual SP experiments [1 ] the quantity P(T, A) is measured. We shall discuss below how the confluence of coherences affects the shape of the polarisation P(T, A). From the set of coupled differential equations for the Schrodinger amplitudes a(t),/3i(co, t), i = A, B follows that due to the flat character of the B channel a
Volume 5 3, number 2
OPTICS COMMUNICATIONS
simple rate equation can be found for NB(t, A)
(d/dt~B(t, A) = P la(t, A) I2,
(5)
,I
T = .1
P
where F = ~22 [r/I 2/2. Eq. (5) combined with the requirement of the probability conservation Ic~(T, A) I2 + NA(T, A) + NB(T, A) = 1 allows expressing (4) in the form
~
FofTdt la(t, A)] 2 P(T,
= PA + (PB - PA)
A)
-
I
I
I
I
[
I
-6
-4
-2
0
2
4 a l ~o
(7)
where z 1 and z 2 are the roots of the complex quadratic equation
(z + Cf)(z - iA + 3'0) -- C(1 + iq) 2 = 0, and f = 1 + It/12(1 + q2)
-I
Fig. 3. SP for different observation times (denoted in the 2 figure in the 3,01 units);f= 1.1, S2o/3` 02 = 64.
z2 Zl - exp(z 1 t) + exp(z2t), z 2 -- z 1 z 1 -- z 2
C = ~22/(43'0(1 + q2)),
T:IO0.
(6)
1 - I o ~ ( T , A) 12
By means o f the Laplace transform of the set of equations for a(t),/3i(6o, t) we obtained in ref. [11 ] the resolvent of the model defined by (1) and (2) ([11] eq. 11). Making use of it we find that
a(t, A)
15 February 1985
(7a)
Eq. (6) together with (7), after some algebra, gives our final, analytical result (too lengthy to be quoted here) for electron SP. We present the electron SP as a function of the de-
tuning A in figs. 2--4. For all three figures the typical value of Fano parameter q = 5 is chosen. We denote in the figures the relative strength o f the coupling to the B continuum by f, which is defined in eq. (7a). As the simplest choice we take PA = 1, PB = --1 (the curves remain the same for other values o f P i with appropriate adjustment of the vertical axis units). Before discussing strong field effects, we quote the known, first order perturbation theory result
p(A)
PA 1~2A(WL)12 + PB I~B(COL) 12 -
IFZA(COL)I2 + 1~2B(WL)I2
1
1
P
P
32. -.5
-1
-6
I
o
I
2
Fig. 2. Long time (T = 100 3,0] ) SP curves for f = 1.1. The 2 2 values of Do/3, o are indicated in the figure.
-1
I
I
I
I
I
I
-6
-4
-2
0
2
4 Alto
Fig. 4. Long-time SP for ~ / ' r ~ = 64 and different coupling parameters f. 101
Volume 53, number 2
OPTICS COMMUNICATIONS
As I~2A(¢OL)[2 = 0 at A - --q'/0 (Fano zero) we note that the weak field curve has a deep minimum at this point, ionisation goes via B channel only. In the strong field case we expect that the rapid increase of the B-electrons will take place near the CC point, when the A channel is closed due to interference effects. As the CC occurs when one of the Autler-Townes doublet coincides with the Fano zero we expect the appropriate, intensity-dependent shift of the polarisation curve minimum from its weak field position. The long-time SP curves for four values of Rabi frequency ~20 are plotted in fig. 2. Apart from the expected shift they show that with the increase of Y20 the minima become broadened and less pronounced. In fig. 3 the SP curves are drawn for different observation times T. For very short times the SP is almost independent of detuning. It is merely the manifestation of time-energy uncertainty relation. If the field is stronger, more electrons will be produced at that early stage, which explains the flattening of the SP minimum for larger ~20 (fig. 2). For longer times (fig. 3) the minimum appears and moves to its long time position as T goes to infinity. The influence of the relative strength f of the coupling to the flat B channel is presented in fig. 4. For larger f the minimum broadens rapidly and shifts towards its weak field position. This shift can be associated with weakening of the coupling on the resonant 10 ) ~ Iw; A) transition due to additional coupling to the flat 1¢o; B) continuum. The term "confluence of coherences" was originally attributed to the effect of "closing" the resonant channel by interference of two coherent processes [7]. The usually discussed model [7] involves one, resonant continuum only. Thus CC occurs simultaneously with trapping of the initial state population. In our, two continua model, the additional decay via flat channel makes the complete trapping or "closing" processes impossible. One can define the point of CC by a requirement that the number of electrons produced in the resonant channel (NA) is minimal. On the other hand, the detuning for which the sum N A + N B is minimal corresponds to the point of population trapping. This point will be later called the "maximal slowing down of ionisation process" (MSIP) point, as for f > 1 ( r / ~ 0) no population is trapped in T ~ oo limit. It comes as a surprise that 102
15 February 1985
Table 1 Separation of CC and MSIP effects. Detuning A in/~ units. S22 -3'0 -~
f= 1 AMSIP = ACC
f = 1.1 AMSIP
f = 1.1 ACC
4
-4.81
~ .80
-4.85
16 32 64
4.23 -3.46 -1.92
-4.20 -3.40 -1.82
-4.33 -3.60 -2.20
both points i.e. CC point and MSIP point do not coincide. There exists no simple analytical expression for either CC or MSIP point (contrary to one continuum, f--- 1 model). The numerical values are given in table 1. In the second column the critical detuning for CC (MSIP) is given for the f = 1 model. Separation of both effects can be understood on the basis of decrease of the effective coupling in the resonant channel, which gives the position of CC point. Thus this point is shifted towards weak-field limit from its position given by one-continuum model. The MSIP point is shifted a little in the opposite direction. Thus for MSIP rather the total coupling strength is relevant (as it should be) than the coupling in the resonant channel only. The separation of CC and MSIP points could not be established on the basis of total ionisation, timedependent rate investigation, which gives the total number of ejected electrons (N A + NB). The additional SP measurement gives also the difference N A - N B, which allows for obtaining N A and N B separately. It is worth stressing that the SP curve minimum coincides with the CC point (at least for weak couplings to the fiat channel). In this communication we restricted ourselves to discussion of electron SP. It is quite obvious, however, that similar results may be obtained for the electron angular distribution, provided, the resonant A and the flat B channel lead to different electron angular distributions. Finally it should be stressed that in order to observe the shift of the SP curve minimum (fig. 2), the Rabi frequency ~0 must be at least of the order of the AS width 70- It may be difficult to achieve in our, particular T1 example, however much narrower AS or autoionising-like structures in the c o n t i n u u m has been already observed [4].
Volume 53, number 2
OPTICS COMMUNICATIONS
To conclude: we have discussed in the simple, analytically soluble m o d e l the SP o f electrons produced in strong field auto-ionisation. We have shown that the confluence o f coherences m a y be possible to verify in a SP measurement. The separation o f CC and MSIP effects has b e e n established and explained. I thank Kazimierz Rzazewski and T h o m a s z Dohnalik for helpful discussions. The w o r k presented was partially supported by Polish Ministry o f Sciences under contract M R I / 5 .
References [ 1 ] J. Kessler, Polarized electrons, Chap. 5 (Springer-Verlag, Berlin, Heidelberg, New York 1976); U. Heinzmann, A. Wolke and J. Kessler, J. Phys. B13 (1980) 3149; U. Heinzemann, H. Hauer and J. Kessler, Phys. Rev. Lett. 34 (1975) 441. [2] e.g.G. Leuchs, in: Laser physics, Proc., New Zealand 1983, eds. J.D. Harvey and D.F. Walls (Springer-Verlag 1983).
15 February 1985
[ 3 ] S.N. Dixit, P. Lambropoulos and P. Zoner, Phys. Rev. A24 (1981) 318; S.N. Dixit and P. Lambropoulos, Phys. Rev. A27 (1983) 861. [4 ] S. Feneuille, S. Liberman, J. Pinard and A. Taleb, Phys. Rev. Lett. 42 (1979) 1404. [5] Yu.l. HeUer and A.K. Popov, Zh. Eks. Teor. Fiz. 78 (1980) 5O6. [6] P. Lambropoulos and P. Zoller, Phys. Rev. A24 (1981) 379. [7] K. Rzazewski and J.H. Eberly, Phys. Rev. Lett. 47 (1981) 4O8. [8] K. Rzazewski and J.H. Eberly, Phys. Rev. A27 (1983) 2026. [9] M. Lewenstein, J. Haus and K. Rzazewski, Phys. Rev. Lett. 50 (1983)417; J. Haus, M. Lewenstein and K. Rzazewski, Phys. Rev. A28 (1983) 2269; G.S. Agarwal, S.L. Haan, K. Burnett and J. Cooper, Phys. Rev. Lett. 48 (1982) 116; Phys. Rev. A26 (1982) 2277; see also G.S. Agarwal, in: Coherence and quantum optics V (Plenum, New York 1984). [10] J.W. Haus, K. Rzazewski and J.H. Eberly, Optics Comm. 46 (1983) 191. [11 ] J. Zakrzewski, J. Phys. B17 (1984) 719. [12] S.H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703. [13] U. Fano, Phys. Rev. 124 (1961) 1866.
103
OPTICS COMMUNICATIONS
15 February 1985
ELECTRON SPIN POLARISATION IN LASER INDUCED AUTOIONISATION Jakub ZAKRZEWSKI * Institute for Theoretical Physics, Polish Academy of Sciences, Warsaw,Poland
Received 18 July 1984 Revised manuscript received 5 December 1984
The electron spin polarisation in the strong laser light induced autoionisation is discussed in the simple model. It is shown that confluence of coherences manifests itself in the dependence of polarisation on the laser detuning as a field strength dependent resonance, which enables verification of the phenomenon by means of the spin polarisation measurement. The separation of the confluence of coherences point and the point at which the maximal slowing down of the ionisation process (population trapping) occurs is showed.
The investigations of the spin polarisation (SP) and angular distribution of electrons obtained in photoionisation are of considerable recent interest [ 1 - 3 ] . It stems from the fact, that measurements of these quantities provide a lot of information about the atomic structure as well as about the excitation process itself. Traditionally, the ionisation step of the excitation is included in the theoretical studies perturbatively i.e. continuum of electron states is treated as a sink for the loss of the bound states populations [3]. Recently attention has been paid to strong field effects arising due to saturation of bound-free transition, possible in the vicinity o f the narrow, autoionisation (AS) (or externally induced autoionising-like [4,5]) resonance [5,11]. Particularly interesting was the discovery of drastic narrowing of the photoelectron energy spectra for strong excitation of AS resonance [7]. It appeared when one of the AutlerTownes [12] doublet was tuned to the Fano zero [ 13 ] o f the dipole matrix element o f bound-free transition and therefore the phenomenon was termed [7] the confluence o f coherences (CC). Further research incorporated incoherence effects such as finite laser bandwidth [8], radiative damping [9], inhomogeneous broadening [10] or the existence of other atomic continua [ 11 ]. * Present address: Institute of Physics, Jagellonian University Reymonta 4, Krakow, Poland. 0 030-4018/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
However, the phenomenon has not been yet verified experimentally and its direct observation by electron spectra measurement will be impossible for some time due to insufficient electron energy resolution. Therefore the interest arose for finding other experimental arrangement which will make observation of CC possible. Up till now it was shown that CC manifests itself in the spectra of photons emitted during recombination accompanying autoionisation [9]. The aim of this communication is to propose another experiment capable of verifying the CC phenomenon, namely SP experiment. To this end we discuss the simple atomic model introduced before [11 ]. It consists (fig. 1) of the initial state 10 ) coupled by a laser light of frequency o3 L to two atomic continua I~o; A) and I~; B). In ref. [11] we have shown that the existence o f additional (as compared with the standard [7 ] model) ionisation channel may diminish the CC by irreversible decay. On the other hand the very existence of the additional channel forms the basis for SP experiment (provided the resonant, AS channel and the non-resonant one have different, appropriate quantum numbers). As a simple example o f the atomic structure we have in mind may serve excitation of 6 2D3/2 ( m / = 1/2) state of T1 autoionising to e 2D3/2 ( m / = 1/2) continuum [1 ]. Assuming n 2P1/2(m/ = -1/2)initial atomic state o + excitation couples it to 6 2S1/2(m/ = 1/2) continuum also. 99
Volume 53, number 2
OPTICS COMMUNICATIONS
)
I0)
Fig. 1. Scheme of the discussed system.
The hamiltonian of our system (in ~ = c = 1 units) reads H=co010X01 + f dcoco IcO;A)(cO;AI
i=A,B
of the AS resonance, COL its position, q is the Fano asymmetry parameter [ 13], f20 is a bound-free equivalence of the Rabi frequency and 77 denotes the relative strength of the 10) ~ IcO; B) coupling. We denote the SP of electrons that belong to rco; A)/Iw; B) continuum (A/B channel) by PA(co)/PB(co). Their w-dependence is determined in general by the detailed atomic structure. Here we assume the simplest possible choice making Pi(w), i = A, B independent of co (in the vicinity of 6 2D3/2 resonance this condition is satisfied in Tt example). In ref. [11] we discussed the long time (t -+ oo) photoelectron spectra resulting from the model defined by refs. [1] and [2]. Now we are interested in full time evolution of the system. We expand the Schrodinger wave vector in the basis of states { 10), Iw; A), Ico; B) } I~(t)) = ~(t) 10) + exp(--icoLt )
+ fdco co IcO;B)(w;BI f d~o[s-zAco) exp(icoLt) 10)(cO; il
+ h.c.].
(1)
Further the arbitrary zero on the energy scale is chosen to coincide with the initial state energy, thus cOo = 0. We assume the following form of the couplings ~2A(cO), ~2B(cO)
X f dco[t3A(co,t)lcO; A) +/~B(cO't)lco; B ) ] .
Because I/3i(co, T) I2 is the probability density for the electron with energy 6o to be ejected via i-channel between t = 0 and t = T, the total SP of electrons emitted during this time interval reads
P(T)=
fd6o [ [/3A(co, 7")12 + [/3B(cO,T)I 2]
~A(C°) - (47r70)1/2 70 co-Col-iT0 g2o S2B(~O ) = n -
1 i71 ). q +i cO-Wl-iT1
(2)
which as Pi(co) are assumed to be independent of co can be cast into the form
-i~' 1
-
(47r70) 1/2 co - co1 - i71"
Note that ~A(co) has the frequently used [7-11 ] two-lorentzian shape which takes into account the autoionising character of the 10) ~ Ico; A) transition and ~2B(cO) is essentially "flat in energy". (In our T1 example e 2S1/2(m / = 1/2) continuum serves as a flat channel. It couples to different AS state 6 2P1/2(m/ = 1/2) which however is well off resonance, thus "flat in energy" approximation is well justified.) In ref. [2] 71 is the width of the fiat background (taken to be infinite at the end of calculation), 70 is the width I00
(3)
fdco PA(co) I/~A(co, T) I 2 + fdcO eB(cO) lflB(W, T) 12
~20
X(
15 February 1985
P(T, A) =
PANA(T,A) +PBNB(T, A ) NA(T,A) +NB(T , A) '
(5)
where by Ni(T, &) --- fd~ol/3i(co, T) I 2 we denoted the number of electrons ejected via A/B channel and by A = cOL -- cO1 the detuning of the laser light from AS resonance. In the usual SP experiments [1 ] the quantity P(T, A) is measured. We shall discuss below how the confluence of coherences affects the shape of the polarisation P(T, A). From the set of coupled differential equations for the Schrodinger amplitudes a(t),/3i(co, t), i = A, B follows that due to the flat character of the B channel a
Volume 5 3, number 2
OPTICS COMMUNICATIONS
simple rate equation can be found for NB(t, A)
(d/dt~B(t, A) = P la(t, A) I2,
(5)
,I
T = .1
P
where F = ~22 [r/I 2/2. Eq. (5) combined with the requirement of the probability conservation Ic~(T, A) I2 + NA(T, A) + NB(T, A) = 1 allows expressing (4) in the form
~
FofTdt la(t, A)] 2 P(T,
= PA + (PB - PA)
A)
-
I
I
I
I
[
I
-6
-4
-2
0
2
4 a l ~o
(7)
where z 1 and z 2 are the roots of the complex quadratic equation
(z + Cf)(z - iA + 3'0) -- C(1 + iq) 2 = 0, and f = 1 + It/12(1 + q2)
-I
Fig. 3. SP for different observation times (denoted in the 2 figure in the 3,01 units);f= 1.1, S2o/3` 02 = 64.
z2 Zl - exp(z 1 t) + exp(z2t), z 2 -- z 1 z 1 -- z 2
C = ~22/(43'0(1 + q2)),
T:IO0.
(6)
1 - I o ~ ( T , A) 12
By means o f the Laplace transform of the set of equations for a(t),/3i(6o, t) we obtained in ref. [11 ] the resolvent of the model defined by (1) and (2) ([11] eq. 11). Making use of it we find that
a(t, A)
15 February 1985
(7a)
Eq. (6) together with (7), after some algebra, gives our final, analytical result (too lengthy to be quoted here) for electron SP. We present the electron SP as a function of the de-
tuning A in figs. 2--4. For all three figures the typical value of Fano parameter q = 5 is chosen. We denote in the figures the relative strength o f the coupling to the B continuum by f, which is defined in eq. (7a). As the simplest choice we take PA = 1, PB = --1 (the curves remain the same for other values o f P i with appropriate adjustment of the vertical axis units). Before discussing strong field effects, we quote the known, first order perturbation theory result
p(A)
PA 1~2A(WL)12 + PB I~B(COL) 12 -
IFZA(COL)I2 + 1~2B(WL)I2
1
1
P
P
32. -.5
-1
-6
I
o
I
2
Fig. 2. Long time (T = 100 3,0] ) SP curves for f = 1.1. The 2 2 values of Do/3, o are indicated in the figure.
-1
I
I
I
I
I
I
-6
-4
-2
0
2
4 Alto
Fig. 4. Long-time SP for ~ / ' r ~ = 64 and different coupling parameters f. 101
Volume 53, number 2
OPTICS COMMUNICATIONS
As I~2A(¢OL)[2 = 0 at A - --q'/0 (Fano zero) we note that the weak field curve has a deep minimum at this point, ionisation goes via B channel only. In the strong field case we expect that the rapid increase of the B-electrons will take place near the CC point, when the A channel is closed due to interference effects. As the CC occurs when one of the Autler-Townes doublet coincides with the Fano zero we expect the appropriate, intensity-dependent shift of the polarisation curve minimum from its weak field position. The long-time SP curves for four values of Rabi frequency ~20 are plotted in fig. 2. Apart from the expected shift they show that with the increase of Y20 the minima become broadened and less pronounced. In fig. 3 the SP curves are drawn for different observation times T. For very short times the SP is almost independent of detuning. It is merely the manifestation of time-energy uncertainty relation. If the field is stronger, more electrons will be produced at that early stage, which explains the flattening of the SP minimum for larger ~20 (fig. 2). For longer times (fig. 3) the minimum appears and moves to its long time position as T goes to infinity. The influence of the relative strength f of the coupling to the flat B channel is presented in fig. 4. For larger f the minimum broadens rapidly and shifts towards its weak field position. This shift can be associated with weakening of the coupling on the resonant 10 ) ~ Iw; A) transition due to additional coupling to the flat 1¢o; B) continuum. The term "confluence of coherences" was originally attributed to the effect of "closing" the resonant channel by interference of two coherent processes [7]. The usually discussed model [7] involves one, resonant continuum only. Thus CC occurs simultaneously with trapping of the initial state population. In our, two continua model, the additional decay via flat channel makes the complete trapping or "closing" processes impossible. One can define the point of CC by a requirement that the number of electrons produced in the resonant channel (NA) is minimal. On the other hand, the detuning for which the sum N A + N B is minimal corresponds to the point of population trapping. This point will be later called the "maximal slowing down of ionisation process" (MSIP) point, as for f > 1 ( r / ~ 0) no population is trapped in T ~ oo limit. It comes as a surprise that 102
15 February 1985
Table 1 Separation of CC and MSIP effects. Detuning A in/~ units. S22 -3'0 -~
f= 1 AMSIP = ACC
f = 1.1 AMSIP
f = 1.1 ACC
4
-4.81
~ .80
-4.85
16 32 64
4.23 -3.46 -1.92
-4.20 -3.40 -1.82
-4.33 -3.60 -2.20
both points i.e. CC point and MSIP point do not coincide. There exists no simple analytical expression for either CC or MSIP point (contrary to one continuum, f--- 1 model). The numerical values are given in table 1. In the second column the critical detuning for CC (MSIP) is given for the f = 1 model. Separation of both effects can be understood on the basis of decrease of the effective coupling in the resonant channel, which gives the position of CC point. Thus this point is shifted towards weak-field limit from its position given by one-continuum model. The MSIP point is shifted a little in the opposite direction. Thus for MSIP rather the total coupling strength is relevant (as it should be) than the coupling in the resonant channel only. The separation of CC and MSIP points could not be established on the basis of total ionisation, timedependent rate investigation, which gives the total number of ejected electrons (N A + NB). The additional SP measurement gives also the difference N A - N B, which allows for obtaining N A and N B separately. It is worth stressing that the SP curve minimum coincides with the CC point (at least for weak couplings to the fiat channel). In this communication we restricted ourselves to discussion of electron SP. It is quite obvious, however, that similar results may be obtained for the electron angular distribution, provided, the resonant A and the flat B channel lead to different electron angular distributions. Finally it should be stressed that in order to observe the shift of the SP curve minimum (fig. 2), the Rabi frequency ~0 must be at least of the order of the AS width 70- It may be difficult to achieve in our, particular T1 example, however much narrower AS or autoionising-like structures in the c o n t i n u u m has been already observed [4].
Volume 53, number 2
OPTICS COMMUNICATIONS
To conclude: we have discussed in the simple, analytically soluble m o d e l the SP o f electrons produced in strong field auto-ionisation. We have shown that the confluence o f coherences m a y be possible to verify in a SP measurement. The separation o f CC and MSIP effects has b e e n established and explained. I thank Kazimierz Rzazewski and T h o m a s z Dohnalik for helpful discussions. The w o r k presented was partially supported by Polish Ministry o f Sciences under contract M R I / 5 .
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15 February 1985
[ 3 ] S.N. Dixit, P. Lambropoulos and P. Zoner, Phys. Rev. A24 (1981) 318; S.N. Dixit and P. Lambropoulos, Phys. Rev. A27 (1983) 861. [4 ] S. Feneuille, S. Liberman, J. Pinard and A. Taleb, Phys. Rev. Lett. 42 (1979) 1404. [5] Yu.l. HeUer and A.K. Popov, Zh. Eks. Teor. Fiz. 78 (1980) 5O6. [6] P. Lambropoulos and P. Zoller, Phys. Rev. A24 (1981) 379. [7] K. Rzazewski and J.H. Eberly, Phys. Rev. Lett. 47 (1981) 4O8. [8] K. Rzazewski and J.H. Eberly, Phys. Rev. A27 (1983) 2026. [9] M. Lewenstein, J. Haus and K. Rzazewski, Phys. Rev. Lett. 50 (1983)417; J. Haus, M. Lewenstein and K. Rzazewski, Phys. Rev. A28 (1983) 2269; G.S. Agarwal, S.L. Haan, K. Burnett and J. Cooper, Phys. Rev. Lett. 48 (1982) 116; Phys. Rev. A26 (1982) 2277; see also G.S. Agarwal, in: Coherence and quantum optics V (Plenum, New York 1984). [10] J.W. Haus, K. Rzazewski and J.H. Eberly, Optics Comm. 46 (1983) 191. [11 ] J. Zakrzewski, J. Phys. B17 (1984) 719. [12] S.H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703. [13] U. Fano, Phys. Rev. 124 (1961) 1866.
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