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.
© 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)·r is the dipole interaction energy, F(t)=F 0cos(ωt) 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 (φn), lower-energy resonance atomic states (φm) and continuum (φE):
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 (, pp. 366–367) the Schrödinger equation (1) can be reduced to a set of coupled equations for the functions an(t) and am(t) only:
Here Ωm,n=V m,n/2 is the tensor of Rabi frequencies and Γn,n′=(π/2)VnEVEn ′ is the tensor of ionization widths; V α,β=-F·r α,β and r α,β=〈α|r|β〉 are the dipole matrix elements. By definition, components of the tensor of Rabi frequencies Ωm,n are proportional to the square root of laser intensity √I, whereas components of the tensor of ionization widths Γn, n′ are proportional to I. For this reason, up to very high intensities |Ωm,n|≫Γn,n′ .
As it is well known, at high n (n≫1) Rydberg levels are almost equidistant, E n+1-En≡Δ≈n -3≈const., and matrix elements Ωn,m and Γn,n′ can be approximated [7–10, 12] by n-, n ′- and m-independent constants ΩR and Γ, correspondingly. Let us assume also that the light frequency ω is larger than binding energy |En| of all the initially populated Rydberg levels, where En=-1/2n 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 Em is relatively large, E m+1-Em≈m -3≫E n+1-En.≈n -3. For this reason, if the Rabi frequency ΩR obeys the conditions m -3≫ΩR≥n -3, and if one of the levels is close to a resonance with initially populated Rydberg levels , where , all the probability amplitudes am(t) except are small and can be dropped from Eqs. (4). This reduces Eqs. (4) to the form:
where , n 0 is the principal quantum number of the level closest to resonance with the level (in dependence on n, for n = n 0, is minimal), , and the initial conditions for Eqs. (5) are assumed to be given by
with arbitrary complex initial probability amplitudes an(0).
As the coefficients of Eqs. (5) do not depend on time, these equations have stationary solutions , where bn and b̃ are constants and γ is the quasienergy to be found from equations
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” corresponds to n=0.
where the expansion coefficients and C n,j are constant and obey the same algebraic equations as the constants b̃ and bn [Eqs. (7) with γ=γj] 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 γj
and the probability amplitudes and an(t) can be shown to have the form
Qusiclassical estimates of the constants ΩR and Γ [7, 12] show that, in a wide region of fields, resonance interaction of levels En and is much stronger than ionization broadening of levels En, ΩR≫Γ, and the strong-field criterion has the form ΩR≫Δ. In such a case, for strong fields, expansion in powers of Δ/ΩR and Γ/ΩR can be used to solve Eq.(9) approximately, and the solutions are given by
The second of two Eqs. (12) shows that at
the j 0-th quasienergy has a zero width, . 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, . 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 γj 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 . As it follows from Eqs. (12), the criterion of long pulses, |Im(γj)|τ≫1 for 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; TK=2π/Δ=2πn 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 (an(0)=1 for n=0 and 0 for n≠0) Eq. (15) coincides with one of the results of Ref. . 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 wion(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 an(0), population of the strong-field stable quasienergy state is different from zero and wion(τ→∞)≠1. In a general case, for a given realization of an(0), in dependence on a growing field strength amplitude F 0, the probability of ionization wion(τ→∞, F 0) falls and tends to its asymptotic value when F 0→∞,
Usually, 1>wion(τ→∞, F 0→∞)>0. However, in a special case when
wion(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 an(0) coinciding with amplitudes an of the strong-field stable quasienergy state (10). For δ=Δ/2 Eqs. (10) take the form
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 ΩR~Δ, wion(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/ω5/3 [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 and ½(En+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 wion vs. δ. The results of calculations (Fig. 3) show that in the limit of a strong field the dependence wion(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 τd. If an atom is excited initially to some Rydberg level , the first pulse provides efficient re-population of this and neighboring levels En. A mechanism of repopulation consists of Raman-type transitions via lower-energy resonance level . 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 an (18). During the time between the first and second pulses, phases of these Rydberg states evolve in accordance with the law exp(-iEnt), and at the initial probability amplitudes for the second pulse appear to be
If the delay τd is equal to s TK, where 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, τd=(s+½)TK, 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 τd 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. .
A purely periodical dependence wion(τd) 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 wion(τd) from a purely periodical one. However, it is possible to find such combinations of quantum numbers m 0 and n 0 and laser pulse parameters, for which the dependence wion(τd) is almost periodical (see Fig. 5). An example of parameters, corresponding to the picture of Fig. 5, is: n 0=25, ω=1,7×1014 sec-1, F 0=6×105 V/cm (I=1×109 W/cm2), τ=50·TK=120 ps.
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.
References and links
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