Semiclassical theory for decay and fragmentation processes in chaotic quantum systems

Semiclassical theory for decay and fragmentation processes in chaotic quantum systems

Martha Gutiérrez, Daniel Waltner, Jack Kuipers, and Klaus Richter Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany

共Received 12 November 2008; published 9 April 2009兲

We consider quantum decay and photofragmentation processes in open chaotic systems in the semiclassical limit. We devise a semiclassical approach which allows us to consistently calculate quantum corrections to the classical decay to high order in an expansion in the inverse Heisenberg time. We present results for systems with and without time-reversal symmetry, as well as for the symplectic case, and extend recent results to nonlocalized initial states. We further analyze related photodissociation and photoionization phenomena and semiclassically compute cross-section correlations, including their Ehrenfest-time dependence.

DOI:10.1103/PhysRevE.79.046212 PACS number共s兲: 05.45.Mt, 05.45.Pq, 03.65.Sq

I. INTRODUCTION

Physical phenomena involving decay processes have been addressed in many physical contexts. They play a central role in the study of excitation relaxation in semiconductor quan- tum dots and wires关1,2兴, photoionization via highly excited atomic关3兴or molecular关4兴Rydberg states, photodissociation of molecules 关5兴, atoms in optically generated lattices and cavities 关6兴, and optical microcavities 关7兴, to name a few examples.

For an open chaotic system it is well known that the clas- sical probability of finding a particle inside the system at a certain time, the classical survival probability, decays expo- nentially in time,␳^cl共t兲=e^−t/␶d, where␶dis the classical life or dwell time. Numerical calculations 关8兴, however, revealed that the quantum survival probability deviates from the clas- sical one at times comparable to t^ⴱ⬇

冑

= 2␲ប/⌬ is the Heisenberg time, and ⌬ is the mean level spacing. Theoretical calculations invoking supersymmetry techniques关9,10兴confirmed these findings. There it could be shown that in the random matrix theory 共RMT兲 limit, the quantum decay ␳共t兲 takes the form of a universal function, which only depends on the general symmetries of the sys- tem, the classical lifetime, and the Heisenberg time. The first successful semiclassical approach to deriving the RMT pre- dictions for quantum graphs was performed in Ref.关11兴, re- producing the first-order RMT quantum corrections for net- works with and without time-reversal symmetry.

Recently, it has been developed a semiclassical approach for calculating the decay of an initially localized wave func- tion inside an arbitrary chaotic system关12兴. The semiclassi- cal framework used there involves correlated trajectories which have been shown to be a powerful tool and the key to linking classical hyperbolic dynamics with universal quan- tum properties 关13兴. These semiclassical techniques have been recently extended and widely applied in the context of level statistics 关14–16兴, where multiple sums over periodic orbits 共POs兲have to be evaluated, as well as in the field of ballistic quantum transport involving Landauer-Büttiker for- mulas关17–23兴, where trajectories start and end at the open- ings where the chaotic conductor is attached to leads. In Ref.

关12兴 a unitarity problem was encountered when using these semiclassical techniques to evaluate the contribution of pairs of interfering trajectories starting and ending inside the sys-

tem. Therefore a new kind of diagram was considered, which is crucial for ensuring unitarity in problems involving open trajectories共OTs兲connecting two arbitrary points in the bulk.

A similar type of trajectory appears in the semiclassical de- scription of transport if the coupling between the chaotic conductor and the leads is not perfect, as shown in Ref.关24兴.

In this paper we generalize the approach presented in Ref.

关12兴 for localized initial wave functions to nonlocalized wave functions. We outline how to systematically obtain higher-order 共in t/t^ⴱ兲 quantum corrections to the classical decay and present terms up to the seventh order and eighth order, for systems with and without time-reversal symmetry, respectively. We further calculate the survival probability for systems with spin-orbit interaction, corresponding to the symplectic RMT ensemble.

Closely related to quantum decay are problems of atomic photoionization or molecular photodissociation where the fragmentation mechanism involves photoexcitation to an in- termediate excited resonant state 共with corresponding com- plex classical dynamics兲which then subsequently decays by sending out a particle, i.e., an electron, an atom, or an ion. In the semiclassical limit, spectral correlation functions for the related photoionization and photodissociation cross sections can be expressed through the spectral form factor and the survival probability. Earlier semiclassical treatments 关25,26兴 of photo-cross-sections were limited through the diagonal approximation used which was relaxed in this context only very recently关12兴. Here we will present a detailed semiclas- sical treatment of the brief account on photofragmentation given in关12兴and extend the results by including Ehrenfest- time effects for cross-section correlations and by computing higher-order contributions.

This paper is organized as follows: in Secs.IIandIIIwe present the semiclassical approach to the quantum survival probability, generalized to nonlocalized wave functions, by including a time average. In Sec.IVthis approach is further extended to derive higher-order corrections for systems with and without time-reversal symmetry as well as for the case of spin-orbit interaction which follows the universal RMT pre- diction for the symplectic case. In Sec. Vwe analyze fluc- tuations of the survival probability through its variance. In Secs.VIandVIIwe give a detailed semiclassical analysis of the statistics of photofragmentation, including higher-order corrections and the Ehrenfest-time dependence of the leading quantum contributions. We conclude with an outlook in Sec.

VIII.

II. SEMICLASSICAL APPROACH TO THE SURVIVAL PROBABILITY

The quantum-mechanical survival probability as a mea- sure of the decay is defined as

␳共t兲=

冕

dr␺共r,t兲␺^ⴱ共r,t兲, 共1兲 where␺共r,t兲is a wave function andAis the volume of the system we are considering. For a closed system ␳共t兲= 1, while for an open system this no longer holds and ␳共t兲 de- cays in time. Expressing ␺共r,t兲 in terms of the propagator K共r,r⬘^,t兲,

␺共r,t兲=

冕

dr⬘K共r,r⬘,t兲␺0共r⬘兲, 共2兲 we have

␳共t兲=

冕

drdr⬘^dr⬙^K共r,r⬘^{,t兲Kⴱ共r,r}⬙^,t兲␺0共r⬘^兲␺0ⴱ共r⬙^{兲, 共3兲} where␺0共r兲is the initial wave function at t= 0.

In order to calculate the semiclassical expression for␳共t兲, we replace the exact quantum propagatorK共r,r⬘^{,t兲with the} semiclassical Van Vleck propagator关27兴,

K^sc共r,r⬘,t兲= 1

共2␲iប兲^f/2

兺

␥

˜共r⬘^→r,t兲

D_␥˜e^{共i/ប兲S␥˜共r,r⬘,t兲}. 共4兲 Here f is the dimension of the system 共in the following we will considerf= 2兲,S_␥˜共r,r⬘^,t兲=兰₀^tdt⬘^L˜␥关r˙_␥˜,r˜_␥,t⬘兴is the clas- sical Hamilton’s principal function共withL_␥˜ as the Lagrang- ian兲along the path ^˜␥ connecting r⬘ ^{and r in a time t, and} D_␥˜=兩det共−^⳵

2S_␥˜共r,r⬘^,t兲

⳵r⳵r⬘ 兲兩^1/2e^{−i共␲/2兲␮␥˜}is the Van Vleck determinant

including the Morse index␮˜_␥.

The semiclassical survival probability is then given by

␳^sc共t兲= 1 共2␲ប兲²

冕

drdr⬘^dr␺0共r⬘^兲␺0ⴱ共r⬙^兲

⫻

兺

␥

˜共r⬘^→r,t兲

␥

˜⬘^共r⬙^→r,t兲

D˜_␥D_␥˜^ⴱ_⬘e^{共i/ប兲共S␥˜−S␥˜⬘兲}. 共5兲

In the following, we introduce a local time average in the survival probability which enables us to neglect highly oscil- lating terms in the above double sum. We define

␳

¯共t兲 ⬅ 具␳^sc共t兲典_⌬t⬅ 1

⌬t

冕

␳^sc共t⬘^兲dt⬘^{, 共6兲} with⌬tⰆt. We will later see that for a localized initial wave packet^¯␳共t兲⬇␳共t兲in the semiclassical limit, recalling the re- sult of Ref. 关12兴.

The phase difference in the double sum in Eq.共5兲rapidly oscillates unless the two related trajectories are correlated.

Therefore most of the contributions will disappear due to the time average. The contributions that prevail over the average are from pairs of correlated trajectories with action differ- ences on the order ofប, which implies that the trajectories^˜␥

and␥^˜⬘should be “similar”. This also implies that the initial points of the two trajectories should be almost the same. We can then expand trajectories␥^˜ 共or␥^˜⬘^{兲going from r}⬘^{共or r}⬙^兲 tor in a timet around trajectories␥ 共or ␥⬘兲 with the same topology going from r₀=共r⬘^+r⬙^{兲/2 to r in a time t. This} expansion amounts to approximating the classical prefactors D_␥˜共r,r⬘^,t兲⬇D␥共r,r₀,t兲 and D_␥˜⬘共r,r⬙^,t兲⬇D␥⬘共r,r₀,t兲, while expanding the phases in the exponents up to the first order, because the latter are more sensitive to small changes in their argument. The expansion of the actions yields

S_␥˜共r,r⬘,t兲 ⬇S_␥共r,r0,t兲−¹₂q·p_␥,0, 共7兲 S_␥˜⬘共r,r⬙,t兲 ⬇S_␥⬘共r,r0,t兲+¹₂q·p_␥⬘,0, 共8兲 whereq=r⬘^−r⬙ ^andp␥,0 共or p_␥⬘,0兲 is the initial momentum of the trajectory ␥ 共or␥⬘兲. The semiclassical survival prob- ability 关Eq.共5兲兴then reads

␳

¯共t兲=

冓

^冕

^冉

^冊

^冉

^冊

⫻

兺

␥,␥⬘^共r0→r,t兲

D_␥D_␥^ⴱ_⬘e^{共i/ប兲共S␥−S␥}⬘^兲e^{−共i/ប兲p}␥␥⁰ ⬘^·q

冔

wherep_␥␥⁰ _⬘=共p_␥,0+p_␥⬘,0兲/2. This can be written as

␳

¯共t兲=

冓

^冕

⫻

兺

␥,␥⬘^共r0→r,t兲

D_␥D_␥^ⴱ_⬘e^{共i/ប兲共S␥−S␥⬘兲}␳W共r0,p_␥␥⁰ _⬘兲

冔

共10兲 where

␳W共r,p兲=

冕

冉

冊

冉

冊

is the Wigner transformation of␺0共r兲. For an initial coherent state, the integrals over r₀ and r⬘ can easily be performed, and the result is consistent with that of Ref.关28兴.

Equation共10兲still involves rapidly oscillating phases, and again most of the contributions will cancel out, unless the trajectories in a pair are systematically correlated. The main contribution corresponds to the diagonal approximation, i.e.,

␥=␥⬘, which gives the classical survival probability. To- gether with the sum rule 关29兴for open systems, this yields

␳

¯^diag共t兲=具e^−t/␶d典r,p, 共12兲 where具¯典r,pindicates a phase-space average,

具F典r,p= 1

共2␲ប兲²

冕

and 1/␶d is the classical escape rate at the energy E

=H共r,p兲, where H共r,p兲 is the Hamiltonian of the system.

For a two-dimensional system, ␶d=⍀共E兲/共2wp兲, with⍀共E兲

=兰dr⬘^dp⬘␦^(E−H共r⬘^,p⬘^兲);w is the size of the opening; and p=兩p兩. For a chaotic billiard this reduces to␶d=m␲^A/共wp兲.

For an initial state with a well-defined energy E₀, we can write^¯␳^diag共t兲=e^{−t/␶d共E0兲}. In the following we will assume this to be the case and drop the brackets of the phase-space av- erage.

Equation 共12兲 has two restrictions. First, we have sup- posed that at timetthe trajectories can already be considered ergodic 共they have homogeneously explored the phase space兲. This is a good assumption as long as t␭Ⰷ1, with ␭ being the Lyapunov exponent. Second, we have assumed that the ergodicity of the corresponding closed system is not af- fected by the opening, meaning, classically the opening should be small, ␶d␭Ⰷ1, while quantum mechanically it is very large,␶dⰆtH.

III. SURVIVAL PROBABILITY: LEADING-ORDER WEAK LOCALIZATION-TYPE CONTRIBUTIONS

It was shown in Ref. 关12兴that the leading quantum cor- rections to the semiclassical survival probability关Eq.共5兲兴for systems with time-reversal symmetry come from orbits with a self-encounter 关Fig. 1共a兲兴, “two-leg-loops” 共2ll兲 共or 2-encounters兲 introduced in Ref. 关13兴, together with “one- leg-loops” 共1ll兲 关sketched in Figs. 1共b兲and 1共c兲兴, which to- gether preserve unitarity.

A. Two-leg-loops

In this section we will give a detailed derivation of these contributions to the survival probability following the phase- space approach 关15,19兴. The double sum over trajectories is replaced by the sum rule together with integrals over the stable and unstable manifolds along reference trajectories ␥ weighted by the density of 2-encounters in an orbit of length t, w^2ll共u,s,t兲, giving rise to a difference in action ⌬S共u,s兲

=us, whose absolute value is smaller than a classical value c². This density is given by

w^2ll共u,s,t兲=共t− 2t_enc兲²

2⍀t_enc , 共14兲

where the encounter time is t_enc=␭⁻¹ln共c²/兩us兩兲.

The classical survival probability is modified by a factor e^tenc/␶d, since the fact that the first stretch remains inside the cavity implies that the second will also be inside. Thus

␳

¯^2ll共t兲=e^−t/␶d

冕

du

冕

dsw^2ll共u,s,t兲e^tenc/␶de^{共i/ប兲us}. 共15兲 The integration can be performed by making the changes of variablesx=us/c²and␴⁼c/uas in Ref. 关20兴. The result is

␳

¯^2ll共t兲=e^−t/␶d

冉

− 2t

tH

冊

The quadratic term corresponds to the first-order quantum correction according to Ref.关9兴, while the linear term breaks unitarity, since it does not vanish as␶d→⬁共when the system is closed兲. As shown in Ref.关12兴another type of diagram has to be considered in order to solve this problem.

B. One-leg-loops

The relevant diagrams correspond to trajectories with an encounter at the beginning or at the end of the trajectory, as shown in Figs. 1共b兲and 1共c兲. Clearly, the latter only exists for initial and final points inside the cavity, since at the open- ings the exit of one stretch of the encounter implies the exit of the other one共with perfect coupling兲.

To evaluate these two contributions we define a Poincaré surface of section 共PSS兲 at some time t⬘ from the end or beginning of the trajectory 关20兴. The encounter time will be given by

t_enc共t⬘^,u兲=t⬘⁺_␭^{1ln共c/兩u兩兲, 共17兲} with the restrictiont⬘^⬍1_␭^ln共c/兩s兩兲, while the density of such encounters is given by

w^1ll共u,s,t兲= 2

冕

共1/␭兲ln共c/兩s兩兲

dt⬘

冕

dt₂ 1

⍀tenc共t⬘^,u兲

= 2

冕

共1/␭兲ln共c/兩s兩兲

dt⬘^{t− 2tenc共t}⬘^,u兲

⍀tenc共t⬘,u兲 . 共18兲 The factor of 2 is due to the possibility of having the encoun- ter at the beginning of the trajectory or at the end. The dif- ference in action will be⌬S⬇usat any point of the Poincaré surface of section. It is important to mention that this weight function automatically includes the situation where both end points are very close, i.e., coherent backscattering. We can now proceed to calculate this contribution to the survival probability in the same way as before, replacing w^2ll共u,s,t兲 withw^1ll共u,s,t兲in Eq.共15兲. In order to evaluate the integrals, we make the changes of variables关20兴

t⬙^=t⬘⁺¹_␭^ln

冉

冊

⬍¹_␭ln共_兩¹x兩兲. Here it is important to notice that the limits of t⬙ also include the situation where the point at which the orbits w

FIG. 1. 共Color online兲Schemes of共a兲“two-leg-loops”共2ll兲and 关共b兲 and 共c兲兴 “one-leg-loops” 共1ll兲 orbit pairs. The trajectories ␥ 共full line兲and␥⬘^共dashed line兲connect the pointsr₀withrin a time t, and they differ by a 2-encounter in共a兲. When the beginning or the end of the trajectory is inside the encounter, we have the situation plotted in共b兲.共c兲is a variation of共b兲where there is no self-crossing of either of the two trajectories.

start is after a possible self-crossing. This means that it is not necessary to have a true self-crossing in configuration space in order to give a contribution of this kind.

We define^¯␳^1ll共t兲=Ie^−t/␶d, where I= 2

冕

c

du

冕

ds

冕

共1/␭兲ln共c/兩s兩兲

dt⬘^{t− 2t}_⍀ ^enc

t_enc e^{共i/ប兲us}e^tenc/␶d, 共20兲 the integral over ␴ can be easily done after the changes of variables mentioned above, andI can be written as

I=4r␭

␲^tH

冕

dxcos共rx兲

冕

dt⬙共t− 2t⬙兲e^t⬙/␶d

=

冉

冊

冕

dxcos共rx兲x^{−1/共␭␶d兲}, 共21兲 wherer=c²/ប.

The integration overxcan be performed by parts, neglect- ing highly oscillating terms that will disappear after averag- ing关20兴, yielding

I=

冉

−1

冊

冉

^冕

冊

= 4t

␲^tH

冕

dysin共y兲 y ⬇ 4t

␲^tH

冕

⬁

dysin共y兲 y =2t

tH

. 共22兲

Then the 1ll contribution to the decay reads

␳

¯^1ll共t兲= 2 t tH

e^−t/␶d. 共23兲

This term exactly cancels the linear term in Eq.共16兲coming from the 2ll contribution, recovering unitarity. The leading semiclassical correction 共quadratic in time兲 to the classical survival probability is therefore关12兴

␳

¯^2ll+1ll= t² 2␶dtH

e^−t/␶d, 共24兲 which is consistent with the RMT prediction 关9兴. It can be interpreted as an interference-based weak localization-type enhancement of the survival probability.

In Sec.IVwe will extend this approach to include higher- order corrections, coming from semiclassical diagrams with multiple encounters or with one encounter involving multiple stretches.

IV. SURVIVAL PROBABILITY: HIGHER-ORDER CONTRIBUTIONS FOR THE GAUSSIAN UNITARY ENSEMBLE, GAUSSIAN ORTHOGONAL ENSEMBLE,

AND GAUSSIAN SYMPLECTIC ENSEMBLE CASES For the unitary case, the next-order contributions to ␳共t兲 are given by the diagrams shown in Fig. 2, as indicated in Ref. 关15兴. In a similar way, we can compute the next-order corrections for systems with time-reversal symmetry. Time- reversal symmetry, however, allows more structures, the cor- responding diagrams include the ones sketched in Fig. 2

共multiplied by a factor of 4 for Fig.2共a兲and a factor of 3 for Fig.2共b兲关15兴兲together with a structure including two copies of the encounter in Fig.1共a兲.

In general, an encounter region contains an arbitrary num- ber of lⱖ2 stretches of the trajectory, which are mutually linearizable, and one speaks of an l-encounter. In order to calculate higher-order corrections, we consider trajectory pairs with encounters described by the vectorv, whose ele- mentsvllist the number ofl-encounters in the trajectory pair.

The total number of encounters is then V=兺v_l, while the number of links, i.e., of parts of the orbit connecting the encounter stretches, isL+ 1 with L=兺lvlas in Ref.关19兴.

In order to analyze the possible configurations of trajec- tories, we consider the periodic orbit formed by joining the ends of the open orbit. We can generate the open trajectories by cutting this closed orbit along each of its links and mov- ing the ends of the cut to the required positions. Note that for systems with time-reversal symmetry, we must choose either the partner orbit or its time reversal so that the link, which is cut, is traversed in the same direction by both orbits. The contribution can then be separated into three cases: A, where the start and end points are outside of the encounters共2ll兲; B, where either the start or end point is inside an encounter 共1ll兲; and C, where both the start and end points are inside encounters共0ll兲.

A. Case A This contribution can be written as

␳

¯_v,A共t兲=N共v兲

冕

⫻

冋

冉

^兺

冊 册

where N共v兲 is the number of trajectory structures, i.e., the number of trajectories of different topologies, corresponding to each vector v 关15兴,␮= 1/␶d, and␣ ^{labels theV encoun-} ters, each being an l_␣-encounter. We have included the cor- rection to the survival probability of the trajectories due to the proximity of encounter stretches during the encounters.

In terms of an integral the weight is given by

γ γ

r

r

^γ

r

γ

r

(b) (a)

FIG. 2. 共Color online兲Scheme of orbit pairs that do not require time-reversal symmetry that give higher-order corrections: 共a兲 a single 3-encounter;共b兲a double 2-encounter. The trajectories␥共full line兲and␥⬘^共dashed line兲connect the pointsr₀withrin a timet, and they differ by the way they are connected in the encounter regions.

w_v,A共u,s,t兲=

冕

dt_L¯

冕

t−t_enc−t_L¯−t2

dt₁

⍀^L−V

兿

␣

t_enc^␣ , 共26兲 wheret_encis the total time that the trajectory spends in the encounterst_enc=兺_␣=1^V l_␣t_enc^␣ . Each of the links must have posi- tive duration and this restriction is included in the limits of integration. The weight is simply anL-fold integral over dif- ferent link timesti, withi= 1 , . . . ,L, while the last link time is fixed by the total trajectory time

t=

兺

i=1 L+1

ti+_␣=1

兺

When we perform the integrals the weight function becomes

w_v,A共u,s,t兲=

冉

^兺

冊

L!⍀^L−V

兿

␣

t_enc^␣

. 共28兲

To calculate the semiclassical contribution we will rewrite Eq. 共25兲as

␳

¯_v,A共t兲=N共v兲

冕

wherez_v,A共u,s,t兲is an augmented weight including the term from the survival probability correction of the encounters

z_v,A共u,s,t兲=w_v,A共u,s,t兲exp

冋 ^兺

册

⬇

冉

^兺

冊

^兿

L!⍀^L−V

兿

共30兲 where we have expanded in the second line the exponent to first order in the encounter times. We can now use the fact that the semiclassical contribution comes from terms where the encounter times in the numerator cancel those in the de- nominator exactly 关15兴. Keeping only those terms, we then obtain a factor of 共2␲ប兲^L−V from the integrals overs andu and obtain the result for trajectories described by the vector vof interest. This implies that for obtaining contributions to the survival probability of the orderបⁿ, we have to consider diagrams with n=L−V.

Consider for example a trajectory with a 3-encounter with two long legs, sketched in Fig. 2共a兲. The encounter has a duration given by

t_enc⬇ 1

␭ln c²

maxj兩sj兩⫻maxj兩uj兩, 共31兲 where j= 1 , 2 and u_j ands_j are the differences between the unstable and stable coordinates of the trajectory on PSS placed in the encounter region, respectively.

The density of this type of encounter, with an action dif- ference⌬S=u·s, is

w_共3兲¹_,A共u,s,t兲=共t− 3t_enc兲³

6⍀²t_enc , 共32兲 where we use the notation共l兲^vl to indicate that the trajectory hasvll-encounters. We can calculate the contribution of such orbits by replacing the sum over the partner trajectory ␥⬘ with an integral over the stable and unstable coordinates 共u,s兲with the densityw_共3兲¹_,A共u,s,t兲, modifying the classical survival probability entering the sum rule by a factor of e^2␮tenc. In the case of time-reversal symmetry there are four possible structures in this case 关19兴, and the final result is

␳

¯_共3兲1,A共t兲= 4e^−t/␶d

冉

3␶dt_H²

冊

For a double 2-encounter shown in Fig. 2共b兲, we define two encounter times:t_enc¹ ⬇¹_␭ln_兩u^c2

1s₁兩 andt_enc² ⬇_␭¹ln_兩u^c2

2s₂兩. The density of such a double encounter is given by

w_共2兲²_,A共u,s,t兲= 共t− 2t_enc兲⁴

24⍀²t_enc¹ t_enc² , 共34兲 witht_enc=t_enc¹ +t_enc² . In this case the number of possible struc- tures for systems with time-reversal symmetry is 5. The con- tribution of such orbits to the survival probability is

␳

¯_共2兲2,A共t兲= 5e^−t/␶d

冉

3␶dt_H² + t⁴ 24␶d

2t_H²

冊

The total contribution of structures with L−V= 2 of 2ll’s is then

␳

¯_2,A共t兲=e^−t/␶d

冉

␶dt_H² + 5t⁴ 24␶d

2t_H²

冊

B. Case B

Now we have to consider the corresponding one-leg-loops for the previous diagrams. This contribution can be written as

␳

¯_v,B共t兲=N共v兲

冕

Here one encounter overlaps with the start or end of the trajectory. We have therefore one link fewer共L in total兲and an extra integral over the position of the encounter relative to the starting point. Starting with a closed periodic orbit 共and dividing by the overcounting factor ofL兲, we can cut each of the Llinks in turn and move the encounter on either side of the cut to either the start or the end. In total we obtain l_␣⬘ copies of the same 1ll involving the encounter ␣⬘^{, and an} additional factor of 2 appears due to the possibilities of hav- ing the encounter at the beginning or at the end of the tra- jectory. The augmented weight can then be expressed as a sum over the different possibilities, each of which involves an integral over the distance from the PSS to the initial or final point,t_␣⬘,

z_v,B共u,s,t兲= 2

兺

␣⬘⁼¹

V

l_␣⬘

冕

^冉

^兺

_兿

^冊

␣

t_enc^␣

⫻exp

冋

^兺

册

Because of the integrals over the position of the encounter at the start or end of the trajectory, the semiclassical contribu- tion is calculated differently, using integrals of the type we encountered in Eq. 共21兲. However, it is easy to see in Eq.

共21兲 that after a suitable change of variables, the integral overt⬙ can be effectively replaced by at_enc. This change of variables can be done for each 共u_␣⬘,s_␣⬘,t_␣⬘兲, giving again a factor oft_enc^␣⬘ for each integral overt_␣⬘, so that the augmented weight can be written as

z_v,B共u,s,t兲 ⬇

2

冉 ^兺

冊冉

^兺

冊

L!⍀^L−V

兿

⫻

兿

and treated as before.

For a single 3-encounter, Eq.共39兲corresponds to z_共3兲1,B共u,s,t兲=共t− 3t_enc兲²

⍀² e^2␮tenc. 共40兲 Multiplying by the number of possible structures, the result- ing contribution for systems with time-reversal symmetry 共37兲is

␳

¯_共3兲1,B共t兲= 4e^−t/␶d

冉

冊

For the double 2-encounter the corresponding augmented weight of such pairs for systems with time-reversal symme- try is given by

z_共2兲²_,B共u,s,t兲=1 3

共t− 2t_enc兲³

⍀²t_enc¹ e^␮tenc, 共42兲 yielding

␳

¯_共2兲2,B共t兲= 5e^−t/␶d

冉

− 2t²

t_H²

冊

The total contribution of 1ll’s for L−V= 2 for systems with time-reversal symmetry is given by

␳

¯_2,B共t兲=e^−t/␶d

冉

− 6t²

t_H²

冊

C. Case C This contribution can be written as

␳

¯_v,C共t兲=

冕

Now that we have one encounter overlapping with the start of the trajectory, and a second 共different兲 encounter at the end of the trajectory, we have several additional compli- cations. First, there is again one link fewer共L− 1 in total兲and now we have two extra integrals over the position of the start and end encounters relative to the start and end points. Also the number of such structures is different. Starting with a closed periodic orbit, we can cut each of the L links in turn and move the encounters on either side of the cut to both the start and the end, as long as the link joins two different encounters. We therefore need to count the number of ways that this is possible for the different sizes of encounters that are linked. We record these numbers in a matrixN共v兲, where the elementsN_␣,␤共v兲record the number of links共divided by L兲 linking encounter ␣ with encounter ␤. In this case it is convenient to includeN_␣,␤共v兲in the augmented weight func- tion. The augmented weight, including these possibilities, can then be expressed as the following sum over the 0ll encounters:

z_v,C共u,s,t兲=

兺

␣⬘^,␤⬘N_{␣⬘,␤⬘}共v兲

冕

⫻exp

冋

^兺

册

^冉

^兺

_兿 ^冊

␣

t_enc^␣ .

共46兲 Again we can expand the exponent to first order in the en- counter times and write the augmented weight function as

z_v,C共u,s,t兲 ⬇

冉 ^兺

冊冉

^兺

l_␣t_enc^␣

冊

⫻

兿

共L− 2兲!⍀^L−V

兿

and treat it as before.

For a single 3-encounter there cannot be such a contribu- tion. For a double 2-encounter the augmented weight func- tion is given by

z_共2兲²_,C共u,s,t兲= 2e^␮tenc共t− 2t_enc兲²

⍀² . 共48兲 This gives the following contribution to survival probability 共45兲:

␳

¯_共2兲²_,C共t兲=e^−t/␶d

冉

2

冊

D. Unitary case

We can easily calculate the contribution for each vectorv for each of the three cases, as long as we know the numbers of possible trajectory structures. For cases A and B, these numbers can be found in Ref. 关15兴 and are repeated in the

first four columns of TableI. For case C we will go up to the sixth-order correction,L−V= 6, and for this we have at most three different types of l-encounters. It is useful to rewrite the sum over␣ and␤ as a sum over the components of the vector v. N_␣,␤共v兲 records the number of ways of cutting links that connect encounter ␣ ^and␤, in the periodic orbit structures described byv. However we can see that the im- portant quantities are the sizes of the encounter ␣ ^and ␤^. Instead we record inNk,l共v兲the number of links that join an encounter of sizekto an encounter of size l. If we number the encounters from 1 to V in order of their size, then we only need to know the numbers Nl₁,l₂共v兲, Nl₁,l_V共v兲, and Nl_V−1,l_V共v兲, as the maximal number of different sized encoun- ters is 3. MoreoverNk,lis symmetric; therefore we include in TableIbothNk,landNl,ktogether. Using a program to count and classify the possible permutation matrices, we obtain the remaining columns in Table I for systems without time- reversal symmetry. Note that Nl₁,l₂共v兲, Nl₁,l_V共v兲, and Nl_V−1,l_V共v兲might describe the same encounter combinations.

In this case we record their numbers in the leftmost column.

Table I allows us to obtain the following results for the quantum corrections to the classical decay for the unitary case:

␳

¯₂共t兲=e^−t/␶d t_H²

冉

2

冊

␳

¯₄共t兲=e^−t/␶d t_H⁴

冉

2− t⁷ 180␶d

3+ t⁸ 1920␶d

4

冊

␳

¯₆共t兲=e^−t/␶d

t_H⁶

冉

22680␶d3+ 31t¹⁰

30240␶d4− t¹¹ 10080␶d5

+ t¹² 322 560␶d

6

冊

These results enable us to calculate the decay up to eighth order in t, giving as the final result

␳

¯^GUE共t兲=e^−t/␶d

冋

+

冉

224␶d2t_H⁶

冊

册

E. Orthogonal case

Similarly, we can find all possible permutation matrices and obtain TableII共see Appendix A兲for systems with time- reversal symmetry. This gives us the following result up to seventh order int:

␳

¯^GOE共t兲=e^−t/␶d

冋

冉

冊

−

冉

5␶dt_H⁴

冊

冉

3t_H³ + 7 12␶d

2t_H⁴

+ 8

15␶dt_H⁵

冊

冉

3t_H⁴ + 14 15␶d

2t_H⁵ + 16 21␶dt_H⁶

冊

+¯

册

The predictions for the decay using supersymmetry tech- niques can be found in Ref.关10兴, where the integrals appear- ing there can be expanded in powers of t/tH, following the steps indicated in Ref. 关30兴. The results of these expansions agree with Eqs. 共53兲and共54兲.

F. Spin-orbit interaction and the symplectic case Along with the cases with and without time-reversal sym- metry, there has recently been interest in a semiclassical treatment corresponding to the symplectic RMT ensemble in different contexts, such as in spectral statistics 关15兴 and in the quantum transmission through mesoscopic conductors in the Landauer-Büttiker approach 关31,32兴. There the symplec- tic case is obtained by including in the Hamiltonian a clas- sically weak spin-orbit interaction.

In the following we study the effect of spin-orbit interac- tion on the survival probability. The spin-orbit interaction is accounted for by replacing the Hamiltonian for the orbital dynamics,Hˆ

0considered up to now, with Hˆ =Hˆ

0+sˆ·C共xˆ,pˆ兲, 共55兲 withC共xˆ,pˆ兲characterizing the coupling of the translational degrees of freedom to the spin operatorsˆ.

For weak spin-orbit interaction, the semiclassical propa- gator is similar to Eq.共4兲, where the classical trajectories are TABLE I. The numbers of trajectory pairs and the numbers

linking certain encounters for systems without time-reversal symmetry.

v L V N共v兲 Nl₁,l₂共v兲 Nl₁,l_V共v兲 Nl_V−1,l_V共v兲

共2兲² 4 2 1 1

共3兲¹ 3 1 1

共2兲⁴ 8 4 21 21

共2兲²共3兲¹ 7 3 49 12 32

共2兲¹共4兲¹ 6 2 24 16

共3兲² 6 2 12 8

共5兲¹ 5 1 8

共2兲⁶ 12 6 1485 1485

共2兲⁴共3兲¹ 11 5 5445 2664 2592

共2兲³共4兲¹ 10 4 3240 984 1920

共2兲²共3兲² 10 4 4440 464 2624 960

共2兲²共5兲¹ 9 3 1728 228 1080

共2兲¹共3兲¹共4兲¹ 9 3 2952 552 760 1080

共3兲³ 9 3 464 380

共2兲¹共6兲¹ 8 2 720 360

共3兲¹共5兲¹ 8 2 608 360

共4兲² 8 2 276 180

共7兲¹ 7 1 180