## Abstract

Strong-field ionization of Rydberg atoms is investigated in its dependence on
phase features of the initial coherent population of Rydberg levels. In the case
of a resonance between Rydberg levels and some lower-energy atomic level
(*V*-type transitions), this dependence is shown to be very
strong: by a proper choice of the initial population an atom can be made either
completely or very little ionized by a strong laser pulse. It is shown that
phase features of the initial coherent population of Rydberg levels and the
ionization yield can be efficiently controlled in a scheme of ionization by two
strong laser pulses with a varying delay time between them.

©2000 Optical Society of America

Properties of Rydberg atoms in a strong light field have been studied for decades. In particular, such an interesting phenomenon as interference stabilization of Rydberg atoms in a strong laser field has been thoroughly studied since 1988 [1–11]. A concept of stabilization is used to characterize a regime in which, counter-intuitively, the probability of photoionization of an atom stops growing or even falls with increasing light intensity. Interference stabilization is known to be related to coherent re-population of Rydberg levels via Raman-type transitions between them in the process of strong-field photoionization. Two main channels of Raman-type transitions between close Rydberg levels are Λ- and V-type transitions via the continuum [1, 3, 10, 11] and via lower-energy resonance atomic levels [4–9], correspondingly. The effect of strong-field Λ-type interference stabilization is known to exist independently of the initial population of Rydberg levels, i.e., independently of whether initially a single or many Rydberg levels are populated [1, 10, 11]. In the case of coherent initial population of several Rydberg levels, the dependence of photoionization on phase distribution of initial Rydberg probability amplitudes was not investigated. As it is shown below, such a dependence is strong enough and mostly pronounced in the scheme of V-type interference stabilizatio.

The wave function Ψ of an atom interacting with a light field obeys the Schrödinger equation

where *H*
_{0} is the field-free atomic Hamiltonian,
*V*(*t*)=-* F*(

*t*)·

*is the dipole interaction energy,*

**r***(*

**F***t*)=

**F**_{0}cos(ω

*t*) is the electric field strength of a linearly polarized light wave (

**F**_{0}and ω are its filed-strength amplitude and frequency, respectively); here and everywhere below atomic units

*ħ*=

*m*=|

*e*|=1 are used.

The field-dependent atomic wave function Ψ can be expanded in a series of
the field-free atomic eigenstates. By considering here only the scheme of V-type
transitions (Fig. 1), let us take into account only the most important
terms corresponding to Rydberg levels (φ* _{n}*), lower-energy resonance atomic states (φ

*) and continuum (φ*

_{m}*):*

_{E}The probability of photoionization is determined as

where t=τ corresponds to the time when he field is turned-off.

With the help of the well-known procedure of adiabatic elimination of the continuum ([10], pp. 366–367) the Schrödinger equation (1) can be reduced to a set of coupled equations for
the functions *a _{n}*(

*t*) and

*a*(

_{m}*t*) only:

$$i{\dot{a}}_{n}=\sum _{m}{\Omega}_{n,m}\mathrm{exp}\left(-i\omega t\right){a}_{m}+{E}_{n}{a}_{n}-i\sum _{n\text{'}}\frac{{\Gamma}_{n,n\text{'}}}{2}{a}_{n\text{'}}.$$

Here
Ω_{m,n}=*V*
_{m,n}/2
is the tensor of Rabi frequencies and
Γ_{n,n′}=(π/2)*V _{nE}V_{En}*

_{′}is the tensor of ionization widths;

*V*

_{α,β}=-

*·*

**F**

**r**_{α,β}and

*r*

_{α,β}=〈

*α*|

*|β〉 are the dipole matrix elements. By definition, components of the tensor of Rabi frequencies Ω*

**r**_{m,n}are proportional to the square root of laser intensity √

*I*, whereas components of the tensor of ionization widths Γ

_{n, n′}are proportional to

*I*. For this reason, up to very high intensities |Ω

_{m,n}|≫Γ

_{n,n′}[9].

As it is well known, at high *n* (*n*≫1)
Rydberg levels are almost equidistant,
*E*
_{n+1}-*E _{n}*≡Δ≈

*n*

^{-3}≈const., and matrix elements Ω

_{n,m}and Γ

_{n,n′}can be approximated [7–10, 12] by

*n*-,

*n*

^{′}- and

*m*-independent constants Ω

*and Γ, correspondingly. Let us assume also that the light frequency ω is larger than binding energy |*

_{R}*E*| of all the initially populated Rydberg levels, where

_{n}*E*=-1/2

_{n}*n*

^{2}. Under this condition, inevitably, principal quantum numbers

*m*of lower-energy resonance levels are relatively small,

*m*≪

*n*(though we assume that

*m*≫1), and spacing between neighboring levels

*E*is relatively large,

_{m}*E*

_{m+1}-

*E*≈

_{m}*m*

^{-3}≫

*E*

_{n+1}-

*E*.≈

_{n}*n*

^{-3}. For this reason, if the Rabi frequency Ω

*obeys the conditions*

_{R}*m*

^{-3}≫Ω

*≥*

_{R}*n*

^{-3}, and if one of the levels ${E}_{m}\left({E}_{{m}_{0}}\right)$ is close to a resonance with initially populated Rydberg levels ${E}_{n}(\mid {\delta}_{n,{m}_{0}}\mid \sim \Delta ={n}^{-3}$, where ${\delta}_{n,{m}_{0}}={E}_{n}-{E}_{{m}_{0}}-\omega )$, all the probability amplitudes

*a*(

_{m}*t*) except ${a}_{{m}_{0}}\left(t\right)$ are small and can be dropped from Eqs. (4). This reduces Eqs. (4) to the form:

$$i{\dot{a}}_{n}={\Omega}_{R}{\tilde{a}}_{{m}_{0}}+\left(n-{n}_{0}\right)\Delta {a}_{n}-i\frac{\Gamma}{2}\sum _{n}{a}_{n},$$

where ${\tilde{a}}_{{m}_{0}}\left(t\right)={a}_{{m}_{0}}\left(t\right)\mathrm{exp}\left(-i\omega t\right)$, *n*
_{0} is the principal quantum number of
the level ${E}_{{n}_{0}}$ closest to resonance with the level ${E}_{{m}_{0}}$ (in dependence on *n*, for *n* = *n*
_{0}, $\mid {\delta}_{n,{m}_{0}}\mid $ is minimal), $\delta \equiv {\delta}_{{n}_{0},{m}_{0}}$, and the initial conditions for Eqs. (5) are assumed to be given by

with arbitrary complex initial probability amplitudes
*a _{n}*(0).

As the coefficients of Eqs. (5) do not depend on time, these equations have stationary
solutions $\left\{{a}_{n}\left(t\right),{\tilde{a}}_{{m}_{0}}\left(t\right)\right\}=\{{b}_{n},\tilde{b}\}exp\left(\mathrm{-i}\gamma t\right)$, where *b _{n}* and

*b*̃ are constants and

*γ*is the quasienergy to be found from equations

$$-{\Omega}_{R}\tilde{b}-n\Delta {b}_{n}+i\frac{\Gamma}{2}\sum _{n\text{'}}{b}_{n\text{'}}=\gamma {b}_{n}.$$

where, to shorten notations, we drop the term *n*
_{0} in the
difference *n*-*n*
_{0}. In these notations,
*n* can take both positive and negative integer values and the
“closest to resonance level” ${E}_{{n}_{0}}$ corresponds to *n*=0.

If *γ _{j}* are eigenvalues of the set of equations (7) (j=0, ±1, ±2,
…), the solution of the initial-value problem for Eqs. (5) can be presented in the form of a superposition

where the expansion coefficients ${\tilde{C}}_{{m}_{0},j}$ and
*C*
_{n,j} are constant and
obey the same algebraic equations as the constants *b*̃
and *b _{n}* [Eqs. (7) with

*γ*=

*γ*] plus equations identical to (6) (initial conditions). Solutions of these equations can be found in a general form. In particular, from Eqs. (7) one can find the equation for quasienergies

_{j}*γ*

_{j}and the probability amplitudes ${\tilde{a}}_{{m}_{0}}\left(t\right)$ and *a _{n}*(

*t*) can be shown to have the form

$${a}_{n}\left(t\right)=\sum _{j}\frac{-{\Omega}_{R}^{2}+i\frac{\Gamma}{2}\left({\gamma}_{j}-\delta \right)}{\left({\gamma}_{j}-n\Delta \right)A\left({\gamma}_{j}\right)}B\left(\gamma \right)\mathrm{exp}\left(-i{\gamma}_{j}t\right),$$

where

$$B\left({\gamma}_{j}\right)=\sum _{n}\frac{{a}_{n}\left(0\right)}{{\gamma}_{j}-n\Delta}).$$

Qusiclassical estimates of the constants Ω* _{R}* and Γ [7, 12] show that, in a wide region of fields, resonance
interaction of levels

*E*and ${E}_{{m}_{0}}$ is much stronger than ionization broadening of levels

_{n}*E*, Ω

_{n}*≫Γ, and the strong-field criterion has the form Ω*

_{R}*≫Δ. In such a case, for strong fields, expansion in powers of Δ/Ω*

_{R}*and Γ/Ω*

_{R}*can be used to solve Eq.(9) approximately, and the solutions are given by*

_{R}$$\mathrm{Im}\left[{\gamma}_{j}\right]=-\frac{\Gamma}{2}{\left(\frac{{\Delta}^{2}}{\pi {\Omega}_{R}^{2}}\right)}^{2}{\left[j+\frac{1}{2}-\frac{\delta}{\Delta}\right]}^{2}.$$

The second of two Eqs. (12) shows that at

the *j*
_{0}-th quasienergy has a zero width, $\mathrm{Im}\left[{\gamma}_{{j}_{0}}\right]=0$. This is an absolutely stable quasienergy level, the population of
which is completely “trapped” at any values of the pulse
duration τ and field-strength amplitude *F*
_{0}
(limited only by the applicability conditions of the used equations (4), (5)). Position of this stable quasienergy level coincides with
the resonance detuning, $\mathrm{Re}\left[{\gamma}_{{j}_{0}}\right]={\delta}_{{j}_{0}}$. Let the condition (13) be satisfied at
*j*
_{0}=0(δ=Δ/2). From Eqs. (3) and (10)–(12) we can find in this case a rather simple formula for the
long-time limit of the ionization probability. The “long-time
limit” means that all the quasienergy levels
*γ _{j}* with

*j*≠

*j*

_{0}are assumed to decay (to be ionized) completely and all the remaining bound-state population is concentrated at the stable quasienergy level ${\gamma}_{{j}_{0}}$. As it follows from Eqs. (12), the criterion of long pulses, |Im(

*γ*)|τ≫1 for

_{j}*j*≠

*j*

_{0}, has the form

where in the last estimate the ionization width Γ is assumed to be on the
order of spacing between Rydberg levels Δ;
*T _{K}*=2π/Δ=2π

*n*

^{3}is the classical Kepler period. Under these assumptions the long-time limit of the ionization probability is given by

In a special case of a single initially populated level
(*a _{n}*(0)=1 for

*n*=0 and 0 for

*n*≠0) Eq. (15) coincides with one of the results of Ref. [6]. In the case of initial coherent population symmetric with respect to the point

*n*=½ (i.e., if

*a*

_{-n}=

*a*

_{n+1}for

*n*=0, 1, 2, …) the sum on the right-hand side of Eq. (15) turns zero and

*w*(

_{ion}*τ*→∞)=1, i.e., in the long-time limit, ionization of an atom is complete. This is the case when the above-mentioned stable quasienergy state is not populated at all. For any other distributions of the initial probability amplitudes

*a*(0), population of the strong-field stable quasienergy state is different from zero and

_{n}*w*(

_{ion}*τ*→∞)≠1. In a general case, for a given realization of

*a*(0), in dependence on a growing field strength amplitude

_{n}*F*

_{0}, the probability of ionization

*w*(τ→∞,

_{ion}*F*

_{0}) falls and tends to its asymptotic value when

*F*

_{0}→∞,

Usually,
1>*w _{ion}*(

*τ*→∞,

*F*

_{0}→∞)>0. However, in a special case when

*w _{ion}*(

*τ*→∞,

*F*

_{0}→∞)=0, i.e., asymptotically, an atom appears to be absolutely stable. This special case corresponds to a choice of the initial probability amplitudes

*a*(0) coinciding with amplitudes

_{n}*a*of the strong-field stable quasienergy state (10). For δ=Δ/2 Eqs. (10) take the form

_{n}Asymptotically, in the limit *F*
_{0}→∞,
this yields

and this is just that special form of the initial probability amplitudes for which the condition (17) is fulfilled and Eq. (15) takes the form

It is interesting to notice that even in a weaker field, at Ω* _{R}*~Δ,

*w*(

_{ion}*τ*→∞) (20) does not exceed 10%. This indicates a rather high degree of stabilization that occurs at moderate fields, if only the initial state of an atom is determined by Eqs. (19).

Analytical formulas (15), (16), (20) are derived under the assumption about very long
pulse duration (14). For shorter pulses, the probability of ionization (3) can be
calculated numerically with the help of Eqs. (9)–(11), and the results of calculations are shown in Fig. 2 for several different values of the pulse duration
τ. It is assumed that initially an atom is excited to the state
determined by Eqs. (19). The field strength *F*
_{0} is
characterized by the well known quasiclassical parameter
*V*=*F*
_{0}/ω^{5/3} [10, 12]. As it is seen from the picture of Fig. 2, finiteness of the pulse duration does not change the
above-described results qualitatively: still, an atom shows strong resistance to
photoionization, and this effect becomes more and more pronounced with increasing
field strength.

Another important assumption of the derivation given above concernes the resonance
detuning *δ*, which is assumed to obey the condition (13)
of an exact resonance between the level ${E}_{{m}_{0}}$ and
½(*E _{n}*+

*E*

_{n+1}) for some

*n*. The question is how strictly this assumption should be fulfilled? To answer this question, we have calculated numerically the probability of ionization

*w*vs.

_{ion}*δ*. The results of calculations (Fig. 3) show that in the limit of a strong field the dependence

*w*(

_{ion}*δ*) appears to be rather smooth and, hence, stabilization of an atom due to a proper choice of the initial coherent population can occur in a wide range of values of the detuning

*δ*.

As a resume, we conclude that in the V-scheme of interference stabilization, the
probability of ionization depends strongly on the phases of the initial coherent
population of Rydberg levels. By a proper choice of these phases one can provide
either complete or almost zero ionization of an atom by a sufficiently long and
strong laser pulse. The most stable initial state corresponds to that given by Eqs. (18). In practice, such a state arises in a rather natural
way in a scheme of two (pump-probe) identical laser pulses separated by a
time-interval *τ _{d}*. If an atom is excited
initially to some Rydberg level ${E}_{{n}_{0}}$, the first pulse provides efficient re-population of this and
neighboring levels

*E*. A mechanism of repopulation consists of Raman-type transitions via lower-energy resonance level ${E}_{{m}_{0}}$. If the pulse duration is long enough, by the end of the first pulse all the quasienergy states of an atom in the field decay. Under these conditions, the remaining population of a neutral atom is concentrated in the absolutely stable quasienergy state of a system, which corresponds to the probability amplitudes

_{n}*a*(18). During the time between the first and second pulses, phases of these Rydberg states evolve in accordance with the law exp(-

_{n}*iE*), and at the initial probability amplitudes for the second pulse appear to be

_{n}t$$\approx {a}_{n}\mathrm{exp}\left[-i\left(n-{n}_{0}\right)\Delta {\tau}_{d}\right]\mathrm{exp}\left(-i{E}_{{n}_{0}}{\tau}_{d}\right).$$

If the delay *τ _{d}* is equal to

*s*

*T*, where

_{K}*s*is an integer, the phase distribution of Eq. (21) repeats that of (18) and, hence, the probability of ionization of an atom by the second pulse is expected to be close to zero. If, however,

*τ*=(

_{d}*s*+½)

*T*, the probability of ionization by the second pulse is expected to be close to one. Such a periodical dependence of the probability of ionization on the delay time

_{K}*τ*is confirmed by the results of numerical calculations shown in Fig. 4. A specific experimental scheme for observation of such an effect can be similar to that of Ref. [13].

_{d}A purely periodical dependence
*w _{ion}*(

*τ*) is a specific feature of a system with equidistant spectrum of levels. Rydberg atoms only approximately satisfy this condition, and the energy spectrum is not purely equidistant. This difference gives rise to complications and deviations of the dependence

_{d}*w*(

_{ion}*τ*) from a purely periodical one. However, it is possible to find such combinations of quantum numbers

_{d}*m*

_{0}and

*n*

_{0}and laser pulse parameters, for which the dependence

*w*(

_{ion}*τ*) is almost periodical (see Fig. 5). An example of parameters, corresponding to the picture of Fig. 5, is:

_{d}*n*

_{0}=25, ω=1,7×10

^{14}sec

^{-1},

*F*

_{0}=6×10

^{5}V/cm (

*I*=1×10

^{9}W/cm

^{2}), τ=50·

*T*=120 ps.

_{K}So, the pump-probe scheme looks very promising for investigation of coherent features of photoionization process in the V-type scheme of interference stabilization. In accordance with the results of our analysis, such a scheme provides possibilities of the quantum control of the photoionization yield: almost total or almost zero probability of ionization in dependence on the delay time between two pulses. Physical reasons of such drastic variations consist in coherent re-population of Rydberg levels owing to V-type transitions via a lower-energy resonance atomic level. We think that this effect is rather interesting for physics of laser-atom interactions, physics of electron wave packets, and, maybe, for applications.

This work was supported partially by the Russian Foundation for Basic Research, grants №№ 99-02-18034 and 97-02-71024, and also by the CRDF.

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