coherent control of strong-field two-pulse ionization of rydberg atoms

Coherent control of strong-field two-pulseionization of Rydberg atoms

M.V. Fedorov and N.P. Poluektov

General Physics Institute, Russian Academy of Sciences38 Vavilov St., 117942 Moscow, Russia

[email protected], [email protected]

Abstract: Strong-field ionization of Rydberg atoms is investigated in itsdependence on phase features of the initial coherent population ofRydberg levels. In the case of a resonance between Rydberg levels andsome lower-energy atomic level (V-type transitions), this dependence isshown to be very strong: by a proper choice of the initial population anatom can be made either completely or very little ionized by a stronglaser pulse. It is shown that phase features of the initial coherentpopulation of Rydberg levels and the ionization yield can be efficientlycontrolled in a scheme of ionization by two strong laser pulses with avarying delay time between them. 2000 Optical Society of AmericaOCIS codes: (000.6800) Theoretical physics; (020.5780) Rydberg states; (140.3550)Lasers, Raman

References and links1. 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).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).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).4. M.Yu. Ivanov, "Suppression of resonant multiphoton ionization via Rydberg states" Phys.Rev.A, 49, 1165

(1994).5. A.Wojcik and R.Parzinski, "Rydberg-atom stabilization against photoionization: an analitically solvable

model with resonance" Phys. Rev. A, 50, 2475 (1994).6. A.Wojcik and R.Parzinski, "Dark-state effect in Rydberg-atom stabilization" J.Opt.Soc.Am.B, 12, 369

(1995).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

9. N.P. Poluektov and M.V. Fedorov, "Stabilization of a Rydberg atom and competition between the Λ and Vtransition channels", JETP, 87, 445 (1998).

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).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).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

Properties of Rydberg atoms in a strong light field have been studied for decades. Inparticular, such an interesting phenomenon as interference stabilization of Rydberg atomsin a strong laser field has been thoroughly studied since 1988 [1–11]. A concept ofstabilization is used to characterize a regime in which, counter-intuitively, the probabilityof 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 Rydberglevels via Raman-type transitions between them in the process of strong-fieldphotoionization. Two main channels of Raman-type transitions between close Rydberg

#19077 - $15.00 US Received January 11, 2000; Revised February 25, 2000

(C) 2000 OSA 28 February 2000 / Vol. 6, No. 5 / OPTICS EXPRESS 117

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levels are Λ- and V-type transitions via the continuum [1, 3, 10, 11] and via lower-energyresonance atomic levels [4-9], correspondingly. The effect of strong-field Λ-typeinterference stabilization is known to exist independently of the initial population ofRydberg levels, i.e., independently of whether initially a single or many Rydberg levelsare populated [1, 10, 11]. In the case of coherent initial population of several Rydberglevels, the dependence of photoionization on phase distribution of initial Rydbergprobability amplitudes was not investigated. As it is shown below, such a dependence isstrong 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 theSchrödinger equation

( )[ ]Ψ+=∂

Ψ∂tVH

ti 0 , (1)

where 0H is the field-free atomic Hamiltonian, ( ) rF ⋅−= ttV )( is the dipole interaction

energy, ( ) ( )tt ω= cos0FF is the electric field strength of a linearly polarized light wave

( 0F and ω are its filed-strength amplitude and frequency, respectively); here and

everywhere below atomic units 1=== em! are used.

The field-dependent atomic wave function Ψ can be expanded in a series of thefield-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 Rydberglevels (3n), lower-energy resonance atomic states (3m) and continuum (3E):

∫ ϕ+ϕ+ϕ=Ψ ∑∑ EEn

nnm

mm tadEtatat )()()()( . (2)

Fig.1 A scheme of V-type Raman transitions in a Rydberg atom.The probability of photoionization is determined as

∑∑ τ−τ−=m

mn

nion aaw 22 |)(||)(|1 , (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], pp. 366-367) the Schrödinger equation (1) can be reduced to a set of

coupled equations for the functions an(t) and )(tam only:

Continuum

ω

ω

En

Em



∑ ωΩ+=n

nnmmmm atiaEai )exp(, ,

∑Γ

−+ω−Ω= ∑'

'

',

,2

)exp(n

n

nn

nnm

mmnn aiaEatiai . (4)

Here Ωm, n = 2,nmV is the tensor of Rabi frequencies and ( ) ',,', 2 nEEnnn VVπ=Γ is the

tensor of ionization widths; βαβα ⋅−= ,, rFV and βα=βα rr , 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 tensorof ionization widths Γn, n' are proportional to I. For this reason, up to very high intensities

nm ,Ω >> Γn, n' [9].

As it is well known, at high n (n >> 1) Rydberg levels are almost equidistant,En+1 – En ≡ ∆ ≈ 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 assumealso that the light frequency ω is larger than binding energy |En| of all the initiallypopulated Rydberg levels, where En = − 1/2n2. Under this condition, inevitably, principalquantum 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 relativelylarge, Em+1 − Em ≈ m− 3 >> En+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 Em (0mE ) is close to a

resonance with initially populated Rydberg levels En (0

3

, ~n m nδ −∆ = , where

ω−−=δ00, mnmn EE ) , 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:

∑Ω+δ=n

nRmm aaai00

~~ ,

∑Γ−∆−+Ω=

nnnmRn aiannaai

2)(~

00 , (5)

where )exp()()(~00

titata mm ω−= , n0 is the principal quantum number of the level 0nE

closest to resonance with the level 0mE (in dependence on n, for n = n0,

0,mnδ is

minimal), 00 ,mnδ≡δ , and the initial conditions for Eqs. (5) are assumed to be given by

)0()0(,0)0(~0 nnm atata ==== (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), )(~0

tam = bn, b~

exp(−i γ t ), where bn and b~

are constants

and γ is the quasienergy to be found from equations

bbbn

nR

~~ γ=Ω−δ ∑ ,

nn

nnR bbibnb γ=Γ

+∆Ω− ∑−'

'2

~. (7)

where, to shorten notations, we drop the term n0 in the difference n − n0. In thesenotations, n can take both positive and negative integer values and the "closest to

resonance level" 0nE corresponds to n = 0.

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



∑ γ−=j

jjmm tiCa )exp(~~

,00, ∑ γ−=

jjjnn tiCa )exp(, , (8)

where the expansion coefficients jmC ,0

~ and jnC , are constant and obey the same

algebraic equations as the constants b~

and bn [Eqs. (7) with γ = γj] plus equationsidentical to (6) (initial conditions). Solutions of these equations can be found in a generalform. In particular, from Eqs. (7) one can find the equation for quasienergies γj

011

21)( 2 =

∆−γΩ−

∆−γΓ

+δ−γ ∑

∑n

R

nnn

i , (9)

and the probability amplitudes )(~0

tam and an(t) can be shown to have the form

∑ γ−γ

γΩ−=

jj

j

j

Rm tiA

Bta )exp(

)(

)()(~

0,

∑ γ−γγ∆−γ

δ−γΓ

+Ω−=

jj

jj

jR

n tiBAn

ita )exp()(

)()(

)(2)(

2

, (10)

where

∑∑∆−γ

Ω−Γ

δ−γ−∆−γ

Γ+=γ

n j

Rjn j

jn

in

iA2

2

)(

1

2)(

1

21)( ,

∑∆−γ

γ =n j

nj

n

aB

)0()( ). (11)

Qusiclassical estimates of the constants ΩR and Γ [7, 12] show that, in a wide

region of fields, resonance interaction of levels En and 0mE is much stronger than

ionization broadening of levels En, ΩR >> Γ, and the strong-field criterion has the formΩR >> ∆ . In such a case, for strong fields, expansion in powers of ∆/ΩR and Γ/ΩR can beused to solve Eq.(9) approximately, and the solutions are given by

δ−∆+

Ωπ

∆−∆+=γ

2

1

2

1]Re[

2

jjR

j ,

22

2

2

2

1

2]Im[

∆

δ−+

Ωπ

∆Γ−=γ j

R

j . (12)

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

∆+=δ=δ )2/1( 00jj (13)

the j0-th quasienergy has a zero width, 0]Im[0

=γ j . This is an absolutely stable

quasienergy level, the population of which is completely “trapped” at any values of thepulse duration τ and field-strength amplitude F0 (limited only by the applicabilityconditions of the used equations (4), (5)). Position of this stable quasienergy level

coincides with the resonance detuning, 00

]Re[ jj δ=γ . Let the condition (13) be satisfied

at j0 = 0 (δ = ∆/2). From Eqs. (3) and (10)-(12) we can find in this case a rather simpleformula for the long-time limit of the ionization probability. The "long-time limit" meansthat all the quasienergy levels γj with j ≠ j0 are assumed to decay (to be ionized)completely and all the remaining bound-state population is concentrated at the stable

quasienergy level 0j

γ . As it follows from Eqs. (12), the criterion of long pulses,

1)Im( >>τγj

for j ≠ j0, has the form



44

~1

∆

Ω

∆

Ω

Γ>τ R

KR T , (14)

where in the last estimate the ionization width Γ is assumed to be on the order of spacingbetween Rydberg levels ∆; TK = 2π/∆=2π n3 is the classical Kepler period. Under theseassumptions the long-time limit of the ionization probability is given by

( )( )

2

22

2

2/1

)0(

11)( ∑

−∆Ωπ+

∆Ω−=∞→τ

n

n

R

Rion

n

aw . (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]. In the case of initial coherentpopulation symmetric with respect to the point n = ½ (i.e., if a–n = an+1 for n = 0, 1, 2, …)

the sum on the right-hand side of Eq. (15) turns zero and 1)( =∞→τionw , 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 ofthe initial probability amplitudes an(0), population of the strong-field stable quasienergy

state is different from zero and 1)( ≠∞→τionw . In a general case, for a given realization

of an(0), in dependence on a growing field strength amplitude F0, the probability of

ionization ),( 0Fwion ∞→τ falls and tends to its asymptotic value when 0F → ∞ ,

2

202/1

)0(11),( ∑

−π−=∞→∞→τ

n

nion

n

aFw . (16)

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

2

2

2/1

)0(π=

−∑

n

n

n

a, (17)

0),( 0 =∞→∞→τ Fwion , 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). Forδ = ∆/2 Eqs. (10) take the form

( )[ ] 2/1221

1~0

∆Ωπ+−=

R

ma ,( )[ ] 2

1

1

12/122 −∆Ωπ+

∆Ω=

na

R

Rn . (18)

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

0~0

=ma ,

21

11

−π=

nan , (19)

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

( )221

1)(

∆Ωπ+=∞→τ

R

ionw . (20)

It is interesting to notice that even in a weaker field, at ΩR ∼ ∆, ( )ionw τ → ∞ (20) does

not exceed 10%. This indicates a rather high degree of stabilization that occurs atmoderate 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 verylong pulse duration (14). For shorter pulses, the probability of ionization (3) can becalculated numerically with the help of Eqs. (9)-(11), and the results of calculations areshown in Fig. 2 for several different values of the pulse duration τ. It is assumed thatinitially an atom is excited to the state determined by Eqs. (19). The field strength F0 is



characterized by the well known quasiclassical parameter V = F0/ω5/3 [10, 12]. As it isseen 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 theresonance detuning δ, which is assumed to obey the condition (13) of an exact resonance

between the level 0mE and ½(En + En+1) for some n. The question is how strictly this

assumption should be fulfilled? To answer this question, we have calculated numericallythe probability of ionization wion vs. δ. The results of calculations (Fig. 3) show that in thelimit 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 canoccur in a wide range of values of the detuning δ.

Fig.2 Probability of ionization of "the most stable"state (19) vs. the field-strength parameter V calculatedin a model of 17 equidistant Rydberg levels for τ=TK

(blue); 5 TK (green); 15 TK (red); and ∞ (black),δ=∆/2, ΩR=3∆⋅V.

Fig.3 Probability 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=3∆⋅V and 17 equidistantRydberg levels are taken into account.

As a resume, we conclude that in the V-scheme of interference stabilization, theprobability of ionization depends strongly on the phases of the initial coherent populationof Rydberg levels. By a proper choice of these phases one can provide either complete oralmost zero ionization of an atom by a sufficiently long and strong laser pulse. The moststable initial state corresponds to that given by Eqs. (18). In practice, such a state arises ina 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 0nE , the first pulse

provides efficient re-population of this and neighboring levels En. A mechanism of re-

population consists of Raman-type transitions via lower-energy resonance level 0mE . If

the pulse duration is long enough, by the end of the first pulse all the quasienergy states ofan atom in the field decay. Under these conditions, the remaining population of a neutralatom is concentrated in the absolutely stable quasienergy state of a system, whichcorresponds to the probability amplitudes an (18). During the time between the first andsecond pulses, phases of these Rydberg states evolve in accordance with the lawexp(−iEnt), and at the initial probability amplitudes for the second pulse appear to be

.)exp(])(exp[

)exp(])(exp[)exp()0(

0

00

0 dndn

dndnnndnnn

Einnia

EiEEiaEiaa

τ−τ∆−−≈

τ−τ−−=τ−=. (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

wionwion

V δ/∆



pulse is expected to be close to zero. If, however, τd = (s+½)TK, the probability ofionization by the second pulse is expected to be close to one. Such a periodicaldependence of the probability of ionization on the delay time τd is confirmed by theresults of numerical calculations shown in Fig. 4. A specific experimental scheme forobservation of such an effect can be similar to that of Ref. [13].

Fig. 4 Probability of ionization by two subsequentlaser pulses vs. the delay time between them τd (inunits of 1/∆) for n0=25, m0=10, V=0,5, τ=50 TK , andδ=∆/2 in a model of 17 equidistant Rydberg levelsand 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-dependentquasiclassical [10, 12] matrix elements Ωn, m and Γn, n'.

A purely periodical dependence wion(τd) is a specific feature of a system withequidistant spectrum of levels. Rydberg atoms only approximately satisfy this condition,and the energy spectrum is not purely equidistant. This difference gives rise tocomplications and deviations of the dependence wion(τd) from a purely periodical one.However, it is possible to find such combinations of quantum numbers m0 and n0 andlaser 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: n0 = 25,ω = 1,7×1014 sec–1, F0 = 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 coherentfeatures of photoionization process in the V-type scheme of interference stabilization. Inaccordance with the results of our analysis, such a scheme provides possibilities of thequantum control of the photoionization yield: almost total or almost zero probability ofionization in dependence on the delay time between two pulses. Physical reasons of suchdrastic variations consist in coherent re-population of Rydberg levels owing to V-typetransitions via a lower-energy resonance atomic level. We think that this effect is ratherinteresting 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 Basic5HVHDUFK JUDQWV DQG DQG DOVR E\ WKH &5')

wion

∆⋅τd

wion

∆⋅τd



coherent control of strong-field two-pulse ionization of rydberg atoms

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