Light scattering in an open quantum system

Chaos, Solitons and Fractals 12 (2001) 2613±2618

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Light scattering in an open quantum system q Agapi Emmanouilidou *, L.E. Reichl Center for Studies in Statistical Mechanics and Complex Systems, The University of Texas at Austin, Austin, TX, 78212 USA

Abstract Using a complex spectral decomposition we study light scattering in a simple open quantum system. In particular, using Fermi's golden rule we express the photodetachment rate in terms of the quasibound states of the open system and we discuss some of its main features. Ó 2001 Published by Elsevier Science Ltd.

1. Introduction In the present paper, we explore light scattering in an open quantum system. In particular, we apply a weak external time-periodic ®eld in an open quantum system and using perturbation theory we ®nd the photodetachment rate. To describe the open quantum system, we consider a speci®c model that consists of an initially localized particle that tunnels out of a potential well into the continuum. The above model was ®rst introduced by Ludviksson [1] and describes single-particle states in the presence of a delta-potential well and a constant electric ®eld. The Ludviksson model is of particular interest because most of its dynamical properties can be described analytically and in addition it can o€er insight into physical processes like the photodetachment of H . More speci®cally, using Ludviksson's model, we ®nd that the photodetachment rate is strongly enhanced for certain frequencies of the external time-periodic ®eld while it is strongly suppressed for others. The above e€ect is observed experimentally [2,3] in the scattering cross-section when studying the e€ect of photodetaching the loosely bound second electron of the H ion in the presence of moderate static electric ®elds using light that is polarized parallel to the applied static electric ®eld. 2. Ludviksson's model In what follows we compute the photodetachment rate using perturbation theory. The unperturbed one-dimensional Hamiltonian is described by Ludviksson's model, H0 …x† ˆ

h2 o2 2m ox2

Fx

Xd…x†;

…1†

where F is the strength of the constant electric ®eld and X is the strength of the delta-potential well. In the rest of this paper we use the dimensionless variables introduced by Ludviksson [1], n ˆ x=x0 ;

q *

E0 ˆ E=0 ;

V ˆ X=x0 0 ;

x0 ˆ x

h  ; 0

…2†

Presented at the conference on ``Probability and Irreversibility in Quantum Mechanics'' at Les Treilles, France, 4±8 July 1999. Corresponding author.

0960-0779/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII: S 0 9 6 0 - 0 7 7 9 ( 0 1 ) 0 0 0 7 6 - 5

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where  x0 ˆ

h2 2mF

1=3 ;

0 ˆ Fx0

…3†

and the Hamiltonian is given by o2 on2

H0 …n† ˆ

n

V d…n†:

…4†

The above Hamiltonian has a continuum of energy eigenvalues over the interval 1 < E0 < 1. Ludviksson [1] found the exact form of the retarded and advanced energy Green functions of the system, GR=A …n; n0 ; z0 †, where z0 ˆ E0 ‡ id for the retarded energy Green function and z0 ˆ E0 id for the advanced energy Green function, respectively. In addition, he found that the retarded energy Green function has simple poles z0n on the lower energy plane that are given by 1 0 ‡ GR 1 …0; 0; zn † ˆ 0; V

…5†

0 0 with GR 1 …n; n ; z † the retarded energy Green function when V ˆ 0. The advanced energy Green function has simple poles on the upper half energy plane that are given by z0 n.

3. Photodetachment rate The Hamiltonian that describes the interaction of the particle and the weak perturbing ®eld, assuming dipole coupling, is of the following form [4]: H …n† ˆ

o2 on2

n

V d…n† ‡ E0 n cos…x0 t0 †;

…6†

where E0 , x0 are the strength and the frequency of the external time-periodic ®eld using dimensionless variables. q V In what follows, we compute the photodetachment rate considering as initial state of the particle, i.e., Wo ˆ V2 e 2 jnj , the eigenstate of the attractive delta-potential well. Actually, in the presence of the static electric ®eld and the attractive delta-potential well the particle has a continuum spectrum and eventually decays. However, Ludviksson [1] found that 3 the lifetime of the semibound state is given by V 2 eV =6 . Thus, if we consider that the attractive delta-potential well is very deep in comparison with the static electric ®eld, i.e., V is large, the lifetime of the semibound state is very large. So, the particle is essentially bound, i.e., the static electric ®eld does not ionize the particle for times longer than those of interest here. We therefore can consider as the particle's initial state the eigenstate of the attractive delta-potential well. Since the external time-periodic ®eld is a weak perturbation, we use Fermi's golden rule to ®nd that the photodetachment rate is given by !2 Z ‡1 o E0 W ˆ 2p jhWf jnjWo ij2 d…Ef0 Eo0 x0 †dEf0 h 2 1 !2 Z Z ‡1 ‡1 o E0 ˆ 2p dn hEf0 jninhnjWo i dn0 hWo jn0 in0 hn0 jEf0 id…Ef0 Eo0 x0 †; …7† h 2 1 1
2 where E00 ˆ V2 is the bound state energy. In Eq. (7), we identify hn0 jEf0 ihEf0 jni as the transition spectral density, qE …n; n0 †, of the unperturbed system Ho . Nickel and Reichl [5] found that the transition spectral density of Ludviksson's model is given by the following expression: qE …n; n0 † ˆ hnjEihEjn0 i i lim‰GR …n; n0 ; E0 ‡ id† GA …n; n0 ; E0 2p d!0 " # i X RR …n; n0 ; z0n † RA …n; n0 ; z0 n† ˆ ; 2p n E0 z0n E0 z0 n ˆ

id†Š …8†

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0

where RR …n; n ; z0n † is the residue of the retarded energy Green function GR …n; n ; z0 † evaluated at the poles z0n , i.e., RR …n; n0 ; z0n † ˆ Resz0 ˆz0n GR …n; n0 ; z0 †; 0 0 A 0 and, similarly, RA …n; n0 ; z0 n † is the residue of the advanced energy Green function G …n; n ; z † evaluated at the poles zn , i.e.,

RA …n; n0 ; z0n † ˆ Resz0 ˆz0n GA …n; n0 ; z0 †: Ludviksson [1] found that the residue of the energy Green function, at the complex poles, takes the form Resz0 ˆz0n GR …n; n0 ; z0 † ˆ /n …n†/n …n0 †; where

! o R 0 G …0; 0; z † 0 0 oz0 1 z ˆzn

/n …n† ˆ p

…9† 

1=2



C i‡ … z0n †A i… n A i… z0n †C i‡ … n

z0n †; z0n †;

n 6 0; n P 0:

…10†

The functions /n …n† are generalized eigenstates of the Hamiltonian with complex eigenvalues H0 …n†wn …n† ˆ z0n wn …n†;

…11†

and they are not square integrable. As Nickel and Reichl have shown, /n …n† belong to a larger space U0a , where   Z ‡1 a U0a :ˆ /; j/…n†j2 e jnj dn < 1 …12† 1

for any a > 1=2. If we use the expression for the transition spectral density given in Eq. (8) then Eq. (7) acquires the form !2 Z Z ‡1 ‡1 o E0 W ˆ 2p dn dn0 hnjWo inhWo jn0 in0 qEo0 ‡x0 …n0 ; n† h 2 1 1 !2 Z " # Z ‡1 ‡1 X RR …n; n0 ; z0 † o E0 RA …n; n0 ; z0 0 0 0 i n n† dn dn hnjWo inhWo jn in ˆ 2p 2p n Eo0 ‡ x0 z0n Eo0 ‡ x0 z0 h 2 1 1 n "Z # Z ‡1 0 ‡1 02 X /n …n†/n …n † o E Im dn dn0 hnjWo inhWo jn0 in0 : ˆ Eo0 ‡ x0 z0n h 2 1 1 n

…13†

In Eq. (13), we can interchange the in®nite sum and the integral only for a certain class of initial states Wo for which the integrals, Z ‡1 dnhnjWo in/n …n†; 1

are well de®ned. As Nickel and Reichl [5] have shown, these initial states must be elements of a space Ua where   Z ‡1 a Ua :ˆ W; jwj2 ejnj dn < ‡ 1 …14† 1

for any a > 1=2. The eigenstate of the attractive delta-potential well is an element of the space Ua , and thus we can safely interchange the in®nite sum and the integral in Eq. (13). The photodetachment rate is then given by " # X o ~ E02 Cn ~ W ˆ Im ; …15† W ˆ W; h 2 Eo0 ‡ x0 z0n n where Z Cn ˆ

‡1 1

2 dn nhWo jni/n …n†

:

The integral, Cn , is well de®ned only for states Wo …n† which belong to the space Ua .

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4. Numerical results We have computed the photodetachment rate for the case V ˆ 5. In Table 1, we present the numerical values for the positions of the poles z0n in the energy range 10 < E0 < 7 and the corresponding integrals Cn . In Fig. 1 we show schematically the location of the poles on the complex plane. In Fig. 2, we plot the photodetachment rate W~ as a function of x0 , using the complex poles in the energy range 10:5 < E < 32:5: As we can see from Fig. 2, the photodetachment rate is strongly enhanced and suppressed for certain frequencies of the weak time-periodic ®eld. We can actually identify the frequencies where the peaks of the photodetachment rate

Table 1 Numerical values for the poles z0n in the energy range

a

n

z0n

)8 )7 )6 )5 )4 )3 )2 )1 0 1 2 3 4 5 6 7 8 9 10 11 12 13

)9.32251 )8.95714 )8.58426 )8.20319 )7.81315 )7.41321 )7.00227 )6.57898 )6.25809 )6.14163 )5.68807 )5.21543 )4.71977 )4.19534 )3.63303 )3.01695 )2.31467 )1.44092 2.51848 4.25624 5.68101 6.94124

10 < E0 < 7 and the corresponding integrals Cn a Cn

16.14608I 15.50607I 14.85224I 14.18332I 13.49778I 12.79380I 12.06921I 11.32130I 10.54672I 9.74113I 8.89880I 8.01188I 7.06907I 6.05313I 4.93539I 3.66150I 2.09520I 0.05513I 0.06389I 0.06847I 0.07138I

)0.000311335 ‡ 0.000899006I )0.000323362 ‡ 0.00106289I )0.000327745 ‡ 0.00127049I )0.000317271 ‡ 0.00153771I )0.000278528 ‡ 0.00188673I )0.0000189445 ‡ 0.00234864I )0.0000139799 ‡ 0.00297552I )0.000331096 ‡ 0.00386946I 0.0000654702 0.00109418 ‡ 0.00519376I 0.00292376 ‡ 0.00700762I 0.00697213 ‡ 0.00861657I 0.0139294 ‡ 0.00753875I 0.0213013 ‡ 0.0000518591I 0.0221164 0.0141023I 0.0108941 0.0270179I )0.00755734 0.0280999I )0.0211296 0.0157336I 0.0143731 ‡ 0.00761277I 0.00634840 ‡ 0.00449177I 0.00367244 ‡ 0.00305638I 0.00240142 ‡ 0.00223819I

The pole labeled z00 has a very small imaginary part which we consider to be zero in the limits of our numerical accuracy.

Fig. 1. Complex poles in the energy range

7:5 < E0 < 10:5.

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Fig. 2. Photodetachment rate W~ …x0 † as a function of x0 , for 0 ˆ 0:01.

Fig. 3. Photodetachment rate W~ …x0 † as a function of x0 excluding poles z01 ; . . . ; z09 .

occur. If E00 is the energy of the initial state and En0 is the real part of the long-lived quasibound state z0n , i.e., a state with positive real part, then the peaks occur at x0 ˆ En0 E00 . Thus, for frequencies of the weak ®eld equal to x0 ˆ En0 E00 the system absorbs a photon and makes a transition from the initial state to the long-lived quasibound states. It is important to emphasize that the location of the maxima in the transition rate is the result of the presence of the static ®eld plus the delta-potential well. These results are in qualitative agreement with the results obtained by Cocke and Reichl [6], who computed the transition rate from an initial state to the continuum using a discrete set of energy eigenstates of a ®nite model (a wall was placed far down the hill from the delta potential). Another interesting feature of the photodetachment rate is that the system makes a transition even for frequencies that are below the ionization threshold of the bound state, which in our case is given by Eo0 ˆ …V =2†2 . This is not surprising. The lowering of the ionization threshold is due to the presence of the static electric ®eld. What is very interesting, though, is that we can show that the considerable transition rate below the threshold is due to the short-lived quasibound states. The largest contribution is due to the short-lived quasibound states that have real energy above the energy Eo0 , i.e., the poles labeled as z01 ; . . . ; z09 in Table 1. In Fig. 3 we plot the photodetachment rate excluding the poles labeled z01 ; . . . ; z09 in Table 1. If we compare Figs. 2 and 3 we ®nd that there is a considerable transition rate below the ionization threshold due to the short-lived quasibound states z01 ; . . . ; z09 . In addition, these short-lived quasibound states are responsible for a large percentage of the asymmetry that the peaks exhibit. 5. Conclusions We have computed the photodetachment rate for an open quantum system, using Ludviksson's model, in terms of a spectral complex decomposition. The advantage of using generalized eigenstates to describe decay phenomena is that it allows for a direct interpretation of the physical processes in terms of the quasibound states of the system. Speci®cally, in the case of the photodetachment rate, we directly associated the peaks observed with the long-lived quasibound states, while the strong enhancement below threshold as well as the asymmetric shape of the peaks are attributed to the short-lived quasibound states.

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References [1] [2] [3] [4] [5] [6]

Ludviksson A. J Phys A 1987;20:4733. Stewart JE, et al. Phys Rev A 1988;38:5628. Harris PG, et al. Phys Rev A 1990;41:5968. Emmanouilidou A, Reichl LE. Phys Rev A 2001;62:22709. Nickel JC, Reichl LE. Phys Rev A 1998;58:4210. Cocke S, Reichl LE. Phys Rev A 1995;52:4515.