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Optics Express

  • Editor: J. H. Eberly
  • Vol. 6, Iss. 5 — Feb. 28, 2000
  • pp: 117–123
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Coherent control of strong-field two-pulse ionization of Rydberg atoms

M.V. Fedorov and N.P. Poluektov  »View Author Affiliations


Optics Express, Vol. 6, Issue 5, pp. 117-123 (2000)
http://dx.doi.org/10.1364/OE.6.000117


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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.

© Optical Society of America

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

iΨt=[H0+V(t)]Ψ,
(1)

where H 0 is the field-free atomic Hamiltonian, V(t)=-F(tr 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):

Ψ(t)=mam(t)φm+nan(t)φn+dEaE(t)φE.
(2)
Fig.1 . scheme of V-type Raman transitions in a Rydberg atom.

The probability of photoionization is determined as

wion=1nan(τ)2mam(τ)2,
(3)

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

10. M.V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific: Singapore, 1997.

], 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:

ia˙m=Emam+nΩm,nexp(iωt)an,
ia˙n=mΩn,mexp(iωt)am+Enanin'Γn,n'2an'.
(4)

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 [9

9. N.P. Poluektov and M.V. Fedorov, “Stabilization of a Rydberg atom and competition between the Λ and V transition channels”, JETP , 87, 445 (1998). [CrossRef]

].

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

7. M.V. Fedorov and N.P. Poluektov, “Λ- and V-Type Transitions and Their Role in the Interference Stabilization of Rydberg Atoms”, Laser Physics , 7, 299 (1997).

10

10. M.V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific: Singapore, 1997.

, 12

12. N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, “Quasiclassical dipole matrix elements for atomic continuum states”, J. Phys. B 22, 2941 (1989). [CrossRef]

] 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, mn (though we assume that m≫1), and spacing between neighboring levels Em is relatively large, E m+1-Emm -3E n+1-En.≈n -3. For this reason, if the Rabi frequency ΩR obeys the conditions m -3≫ΩRn -3, and if one of the levels Em(Em0) is close to a resonance with initially populated Rydberg levels En(δn,m0Δ=n3, where δn,m0=EnEm0ω), all the probability amplitudes am(t) except am0(t) are small and can be dropped from Eqs. (4). This reduces Eqs. (4) to the form:

ia˜˙m0=δa˜m0+ΩRnan,
ia˙n=ΩRa˜m0+(nn0)ΔaniΓ2nan,
(5)

where a˜m0(t)=am0(t)exp(iωt), n 0 is the principal quantum number of the level En0 closest to resonance with the level Em0 (in dependence on n, for n = n 0, δn,m0 is minimal), δδn0,m0, and the initial conditions for Eqs. (5) are assumed to be given by

a˜m0(t=0)=0,an(t=0)=an(0)
(6)

with arbitrary complex initial probability amplitudes an(0).

As the coefficients of Eqs. (5) do not depend on time, these equations have stationary solutions {an(t),a˜m0(t)}={bn,b˜}exp(-iγt), where bn and b̃ are constants and γ is the quasienergy to be found from equations

δb˜ΩRnbn=γb˜,
ΩRb˜nΔbn+iΓ2n'bn'=γbn.
(7)

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” En0 corresponds to n=0.

If γj 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

a˜m0=jC˜m0,jexp(iγjt),an=jCn,jexp(iγjt),
(8)

where the expansion coefficients C˜m0,j 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

(γδ){1+iΓ2n1γnΔ}ΩR2n1γnΔ=0,
(9)

and the probability amplitudes a˜m0(t) and an(t) can be shown to have the form

a˜m0(t)=ΩRjB(γj)A(γj)exp(iγjt),
an(t)=jΩR2+iΓ2(γjδ)(γjnΔ)A(γj)B(γ)exp(iγjt),
(10)

where

A(γj)=1+iΓ2n1γjnΔ[(γjδ)iΓ2ΩR2]n1(γjnΔ)2,
B(γj)=nan(0)γjnΔ).
(11)

Qusiclassical estimates of the constants ΩR and Γ [7

7. M.V. Fedorov and N.P. Poluektov, “Λ- and V-Type Transitions and Their Role in the Interference Stabilization of Rydberg Atoms”, Laser Physics , 7, 299 (1997).

, 12

12. N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, “Quasiclassical dipole matrix elements for atomic continuum states”, J. Phys. B 22, 2941 (1989). [CrossRef]

] show that, in a wide region of fields, resonance interaction of levels En and Em0 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

Re[γj]=(j+12)Δ(ΔπΩR)2[(j+12)Δδ],
Im[γj]=Γ2(Δ2πΩR2)2[j+12δΔ]2.
(12)

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

δ=δj0=(j0+12)Δ
(13)

the j 0-th quasienergy has a zero width, Im[γj0]=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, Re[γj0]=δj0. 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 jj 0 are assumed to decay (to be ionized) completely and all the remaining bound-state population is concentrated at the stable quasienergy level γj0. As it follows from Eqs. (12), the criterion of long pulses, |Im(γj)|τ≫1 for jj 0, has the form

τ>1Γ(ΩRΔ)4TK(ΩRΔ)4,
(14)

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

wion(τ)=1(ΩRΔ)21+π2(ΩRΔ)2nan(0)n122.
(15)

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. [6

6. A. Wojcik and R. Parzinski, “Dark-state effect in Rydberg-atom stabilization” J.Opt.Soc.Am.B , 12, 369 (1995). [CrossRef]

]. 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→∞,

wion(τ,F0)=11π2nan(0)n122.
(16)

Usually, 1>wion(τ→∞, F 0→∞)>0. However, in a special case when

nan(0)n122=π2,
(17)

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

a˜m0=1[1+π2(ΩRΔ)2]12,an=ΩRΔ[1+π2(ΩRΔ)2]121n12.
(18)

Asymptotically, in the limit F 0→∞, this yields

a˜m0=0,an=1π1n12,
(19)

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

wion(τ)=11+π2(ΩRΔ)2.
(20)

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 05/3 [10

10. M.V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific: Singapore, 1997.

, 12

12. N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, “Quasiclassical dipole matrix elements for atomic continuum states”, J. Phys. B 22, 2941 (1989). [CrossRef]

]. 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 Em0 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.

Fig.2 . robability of ionization of “the most stable” state (19) vs. the field-strength parameter V calculated in a model of 17 equidistant Rydberg levels for τ=TK (blue); 5 TK (green); 15 TK (red); and ∞ (black), δ=Δ/2, ΩR=3Δ·V.
Fig.3 . robability of ionization of “the most stable” state (16) vs. detuning δ for τ=TK and V=0,3 (green), 1,0 (red), and 3,0 (black); ΩR=3Δ·V and 17 equidistant Rydberg levels are taken into account.

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 En0, 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 Em0. 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

an(0)=anexp(iEnτd)=anexp[i(EnEn0)τd]exp(iEn0τd)
anexp[i(nn0)Δτd]exp(iEn0τd).
(21)

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. [13

13. M.B. Campbell, T.J. Bensky, and R.R. Jones, “Single-shot detection of wavepacket evolution”, Optics Express , 1, 197 (1997). http://www.opticsexpress.org/oearchive/source/2217.htm [CrossRef] [PubMed]

].

Fig. 4. Probability of ionization by two subsequent laser pulses vs. the delay time between them τd (in units of 1/Δ) for n 0=25, m 0=10, V=0,5, τ=50 TK, and δ=Δ/2 in a model of 17 equidistant Rydberg levels and Ωn, m≡Ω and Γn, n≡Γ
Fig. 5. The same as in Fig. 4, but for the realistic non-equidistant atomic spectrum and realistic n-dependent quasiclassical [10, 12] matrix elements Ωn, m and Γn, n .

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

1.

M.V. Fedorov and A.M. Movsesian, “Field-Induced Effects of Narrowing of Photoelectron Spectra and Stabilization of Rydberg Atoms” J. Phys. B 21, L155 (1988). [CrossRef]

2.

L. Noordam, H. Stapelfeldt, D.I. Duncan, and T.F. Gallagher, “Redistribution of Rydberg States by Intense Picosecond Pulses”, Phys. Rev. Lett. 68, 1496 (1992). [CrossRef] [PubMed]

3.

J.H. Hoogenraad, R.B. Vrijen, and L.D. Noordam, “Ionization suppression of Rydberg atoms by short laser pulses” Phys. Rev. A 50, 4133 (1994). [CrossRef] [PubMed]

4.

M.Yu. Ivanov, “Suppression of resonant multiphoton ionization via Rydberg states” Phys.Rev.A , 49, 1165 (1994). [CrossRef] [PubMed]

5.

A. Wojcik and R. Parzinski, “Rydberg-atom stabilization against photoionization: an analitically solvable model with resonance” Phys. Rev. A , 50, 2475 (1994). [CrossRef] [PubMed]

6.

A. Wojcik and R. Parzinski, “Dark-state effect in Rydberg-atom stabilization” J.Opt.Soc.Am.B , 12, 369 (1995). [CrossRef]

7.

M.V. Fedorov and N.P. Poluektov, “Λ- and V-Type Transitions and Their Role in the Interference Stabilization of Rydberg Atoms”, Laser Physics , 7, 299 (1997).

8.

M.V. Fedorov and N.P. Poluektov, “Competition between Λ- and V-type transitions in interference stabilization of Rydberg atoms”, Optics Express , 2, 51 (1998). http://www.opticsexpress.org/oearchive/source/2982.htm [CrossRef] [PubMed]

9.

N.P. Poluektov and M.V. Fedorov, “Stabilization of a Rydberg atom and competition between the Λ and V transition channels”, JETP , 87, 445 (1998). [CrossRef]

10.

M.V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific: Singapore, 1997.

11.

D.I. Duncan and R.R. Jones, “Interferometric characterization of Raman redistribution among perturbed Rydberg states of barium”, Phys. Rev. A , 53, 4338 (1996). [CrossRef] [PubMed]

12.

N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, “Quasiclassical dipole matrix elements for atomic continuum states”, J. Phys. B 22, 2941 (1989). [CrossRef]

13.

M.B. Campbell, T.J. Bensky, and R.R. Jones, “Single-shot detection of wavepacket evolution”, Optics Express , 1, 197 (1997). http://www.opticsexpress.org/oearchive/source/2217.htm [CrossRef] [PubMed]

OCIS Codes
(000.6800) General : Theoretical physics
(020.5780) Atomic and molecular physics : Rydberg states
(140.3550) Lasers and laser optics : Lasers, Raman

ToC Category:
Research Papers

History
Original Manuscript: January 11, 2000
Published: February 28, 2000

Citation
Mikhail Fedorov and Nikolay Poluektov, "Coherent control of strong-field two-pulse ionization of Rydberg atoms," Opt. Express 6, 117-123 (2000)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-6-5-117


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References

  1. M.V. Fedorov and A.M. Movsesian, "Field-Induced Effects of Narrowing of Photoelectron Spectra and Stabilization of Rydberg Atoms," J. Phys. B 21, L155 (1988). [CrossRef]
  2. L. Noordam, H. Stapelfeldt, D.I. Duncan, and T.F. Gallagher, "Redistribution of Rydberg States by Intense Picosecond Pulses," Phys. Rev. Lett. 68, 1496 (1992). [CrossRef] [PubMed]
  3. J.H. Hoogenraad, R.B. Vrijen, and L.D. Noordam, "Ionization suppression of Rydberg atoms by short laser pulses," Phys. Rev. A 50, 4133 (1994). [CrossRef] [PubMed]
  4. M.Yu. Ivanov, "Suppression of resonant multiphoton ionization via Rydberg states," Phys. Rev. A, 49, 1165 (1994). [CrossRef] [PubMed]
  5. A. Wojcik and R. Parzinski, "Rydberg-atom stabilization against photoionization: an analitically solvable model with resonance," Phys. Rev. A, 50, 2475 (1994). [CrossRef] [PubMed]
  6. A. Wojcik and R. Parzinski, "Dark-state effect in Rydberg-atom stabilization," J. Opt. Soc. Am. B, 12, 369 (1995). [CrossRef]
  7. M.V. Fedorov and N.P. Poluektov, "_- and V-Type Transitions and Their Role in the Interference Stabilization of Rydberg Atoms," Laser Physics, 7, 299 (1997).
  8. M.V. Fedorov and N.P. Poluektov, "Competition between _- and V-type transitions in interference stabilization of Rydberg atoms," Opt. Express, 2, 51 (1998). http://www.opticsexpress.org/oearchive/source/2982.htm [CrossRef] [PubMed]
  9. N.P. Poluektov and M.V. Fedorov, "Stabilization of a Rydberg atom and competition between the _ and V transition channels," JETP, 87, 445 (1998). [CrossRef]
  10. M.V. Fedorov, Atomic and Free Electrons in a Strong Light Field, World Scientific: Singapore, 1997.
  11. D.I. Duncan and R.R. Jones, "Interferometric characterization of Raman redistribution among perturbed Rydberg states of barium," Phys. Rev. A, 53, 4338 (1996). [CrossRef] [PubMed]
  12. N.B. Delone, S.P. Goreslavsky, and V.P. Krainov, "Quasiclassical dipole matrix elements for atomic continuum states," J. Phys. B 22, 2941 (1989). [CrossRef]
  13. M.B. Campbell, T.J. Bensky, and R.R. Jones, "Single-shot detection of wavepacket evolution," Opt. Express, 1, 197 (1997). http://www.opticsexpress.org/oearchive/source/2217.htm [CrossRef] [PubMed]

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