P HYSICAL R EVIEW LETTERS VOLUME 81

6 JULY 1998

NUMBER 1

Noncyclic Geometric Phase Shift for Quantal Revivals Gonzalo Garcia de Polavieja* Physical and Theoretical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, United Kingdom (Received 7 January 1998) The theory of quantal revivals is extended to Hamiltonians that are time dependent through the slow noncyclic change of some parameters. It is shown by generalizing the cyclic geometric phase (Berry’s phase) to noncyclic Hamiltonians that there is a noncyclic geometric displacement for all quantum revivals. The treatment of the classical mechanical geometric angle is also extended to open paths in parameter space and its semiclassical relation with the noncyclic geometric phase derived. The displacement of the revivals is then shown to be given by the total angle shift on the final torus sum of the dynamical angle and the noncyclic geometric angle. [S0031-9007(98)06483-7] PACS numbers: 03.65.Bz, 31.50. + w

A quantal wave-packet revival takes place when a packet reassembles itself after having lost its localized structure, continues localized traveling along a classically stable orbit, and eventually delocalizes again until its next revival. Revivals have been observed in a multitude of experimental setups such as Rydberg systems even in the presence of external fields, femtosecond chemistry, semiconductor quantum-well systems, and micromasers [1]. The study of revivals also provides an ideal scenario for the study of the classical limit of quantum theory [2]. The theory of quantal revivals for time-independent Hamiltonians is based on ¯ of the an analysis of the dynamical phases exps2iEn ty hd eigenstates whose superposition makes the wave packet [3]. This theory has been extended to adiabatic cyclic Hamiltonians by Jarzynski [4]. It was shown that for those revivals that coincide with the cyclic time of the Hamiltonian the Berry [5] phase causes a displacement of the packet along the classical trajectory by an amount given by the Hannay [6] angle. Although revivals are generic for closed and open paths in parameter space, their study for open paths could not be performed by the apparent impossibility of extending the concept of the geometric phase to noncyclic Hamiltonians [4]. Recently, Garcia de Polavieja and Sjöqvist [7] have shown that all adiabatic wave functions must include a noncyclic geometric phase. This noncyclic geometric phase reduces to Berry’s phase for the case in which the path in parameter space is closed. In this paper we 0031-9007y 98y81(1)y1(5)$15.00

show that the effect of the noncyclic geometric phase in the theory of wave-packet revivals is to shift all revivals. The classical mechanical concept of geometric angle is also extended to the general case of open paths. The semiclassical relation of the noncyclic geometric angle and the noncyclic geometric phase is derived. The displacement of the quantum revival is then shown to be given by the total angle shift for open paths on the final torus as a sum of the dynamical angle and the noncyclic geometric angle. In order to study which are the new effects involved when the Hamiltonian changes noncyclically and adiabatically with time, we first briefly review the timeindependent case. Consider a one-dimensional system that is classically bounded and stable. We can express q l ­ q0 a quantal wave packet initially localized around kb b l ­ p0 [or equivalently its corresponding Wigner and k p function peaked at sq, pd ­ sq0 , pP 0 d] in terms of the energy eigenstates hjnlj as jCl ­ n exps2iEn tyhdc ¯ n jnl with cn peaked around n. In the semiclassical limit n ! ` we expect classical behavior to emerge for some time intervals. In order to separate the different time scales of the behavior of the wave packet, we Taylor expand the phase around n assuming a semiclassically continuous P j j n using En ­ j­0 En Dnj yj! with En ; d j En ydnj jn­n and Dn ; n 2 n. The term 2En tyh¯ is a global phase, 2En1 ty h¯ is of the order Osn0 d, and 2En2 tyh¯ is of the order Osn21 d. Therefore for times t , n 0 and for the © 1998 The American Physical Society

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semiclassical limit n ! ` the linear term dominates and we can write the wave packet as X ¯ exps2iDnEn1 ty hdc ¯ n jnl , jCstdl ­ exps2iEn tyhd n

(1)

and using the semiclassical WKB theory we can write 2DnEn1 tyh¯ ø 2Dns≠Hy≠Idt ­ 2DnDQ ,

(2)

where DQ is the variation of the angle variable along the classical trajectory. Thus the effect of the linear exponents 2DnDQ is to shift the location of the packet along the classical trajectory by an amount DQ ­ s≠Hy≠Idt. At times appreciable compared to the revival time TR ­ 4p hjE ¯ n2 j21 the quadratic term in the phase makes the wave packet to spread and at the revival time its value is 2psDnd2 , the linear term dominates again and a revival takes place with the wave packet shifted by DQ from the initial location. We now proceed to study general (cyclic or noncyclic) adiabatic Hamiltonians parametrized by the vector Rstd, HssRstddd. We might expect the instantaneous adiabatic state to have only a dynamical phase contribution and therefore the wave-packet solution to be given by √ ! X i Z t 0 0 exp 2 dt En s Rst ddd cn jn; Rstdl . jCstdl ­ h¯ 0 n (3) However, this form for the wave packet is shown in the following to be wrong in general as there is a geometric phase correction that must be taken into account. This geometric phase reduces to Berry’s phase for cyclic evolution. The form of the noncyclic geometric phase correction must be dictated by the time-dependent Schrödinger equation. Under adiabatic evolution the solution of the Schrödinger equation for a state initially in jCs0dl ­ jn; Rs0dl is given by ! √ i Z t 0 0 dt En s Rst ddd jn; Rstdl , jCstdl ­ exp ian std 2 h¯ 0 (4) where jn; Rstdl is the nth instantaneous energy eigenstate. Substitution of (4) in the time-dependent Schrödinger equation gives the phase an std as Z t Ù 0d an std 2 an s0d ­ i dt 0 Rst 0

3 kn; Rst 0 d j≠Rjn; Rst 0 dl ,

(5)

which is real as the basis jn; Rstdl is normalized. From (4) one finds that the total phase change is given by 1 Z t 0 argkCs0d j Cstdl ­ gn fCg 2 dt En s Rst 0 ddd , (6) h¯ 0 with gn fCg the adiabatic noncyclic geometric phase of the form 2

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gn fCg ­ argkn; Rs0d j n; Rstdl 1i

Z

t

0

Ù 0 d ? kn; Rst 0 d j≠Rjn; Rst 0 dl . (7) dt 0 Rst

Expression (7) is the general expression for the adiabatic noncyclic geometric phase that depends only on the open path C in parameter space. Its geometric nature is explained in the following. The adiabatic noncyclic geometric phase in (7) is independent of the phase choice of jn; Rstdl as it is invariant under the global phase transformation jn; Rstdl ! expfilssRstdddg jn; Rstdl. Therefore gn fCg cannot be eliminated by a global phase transformation and has to be included in any calculation for adiabatic evolution. The adiabatic noncyclic geometric phase is also independent of how the path is traversed as any differentiable monotonic transformation t ! nstd leaves (7) invariant. For the cyclic case, the geometric phase (7) reduces to Berry’s phase. To see this choose a singlevalued basis at the cyclic time TH of the Hamiltonian, argkn; Rs0djn; RsTH dl ­ 0, for which expression (7) reduces to Berry’s phase. Note that the derivation in (7) means that not only one can define a noncyclic geometric phase but also that one must include it in the adiabatic wave function to be a solution of the time-dependent Schrödinger equation for adiabatic motion. In order to study the revivals in the case of general (noncyclic) adiabatic Hamiltonians we then need to consider not only the dynamical phases of the eigenstates but also the noncyclic geometric phase. The wave packet can be written for time t in terms of a basis for which argkn; Rs0djn; Rstdl ­ 0 as X expfifn sRstdddgcn jn; Rstdl (8) jCstdl ­ n

with the phases fn sRstddd ­ gn fCg 2

Z

t 0

dt 0 En s Rst 0 dddyh¯

(9)

and gn fCg the noncyclic geometric phase in (7). Taylor expanding the phase around n we find fn sRstddd ­ P j j­0 sDnd fj std with √ fj std ;

j gn fCg

2

Z

t

dt 0

0

j En s Rst 0 dddyh¯

!¡ j! .

(10)

We now proceed to analyze which are the relevant terms for a wave-packet revival. The f0 term in (10) is a global phase, and for times t , n, the linear term f1 is of the order Osnd, f2 , Osn0 d, f3 , Osn 21 d, and so on. Therefore in the semiclassical limit n ! ` the terms of third order and higher are negligible. Let us denote the time at which the Hamiltonian is cyclic as TH . The conditions for a wave-packet revival in noncyclic Hamiltonians are the following. The cyclic time for

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the Hamiltonian, TH , is within the revival regime, that is, f2 sTH d of the order of unity but different from 2p, and the revival takes place at t ­ TR fi TH for which f2 sTR d ­ 2p. The relevant terms in the phase for the revival are then the linear ones. Using WKB theory for the dynamical phase, En1 yh¯ ø ≠Hy≠I, we can write the total phase fn sRstddd for the revival time TR as √ fn sRsTR ddd ­ Dn ≠gn fCgy≠n 2

Z 0

TR

! dt ≠HssRstdddy≠I

(11)

with C an open path in parameter space starting at Rs0d and ending at RsTR d that is in general different from Rs0d. The result obtained means that the noncyclic geometric phase correction to adiabatic wave functions has the effect of shifting all quantal revivals. The nature of this displacement and its connection to the classical mechanical solution is given in the following. First the noncyclic geometric phase is expressed using a closure of the quantum path. Second, we extend the classical mechanical treatment of the geometric angle to the case of noncyclic change of the Hamiltonian. Third, the semiclassical relation between the noncyclic geometric phase and the noncyclic geometric angle is derived. With this relation the displacement of the quantum revival is shown to be given by the total angle shift on the final torus with the noncyclic geometric angle shift coming from the noncyclic geometric phase contribution. The noncyclic geometric phase (7) is a functional of the open parameter path C. For interpretational purposes we want to express it in terms of the closure with the parameter path g with the prescription that the closure path leaves its value unmodified, that is, argkn; Rs0d jn; Rstdl 1 i

Z

Ù ds Rssd 3

g

kn; Rssd j≠Rjn; Rssdl ­ 0 .

(12)

A more general closure rule has the integral term R Ù ? kwssRssddd j≠R jwssRssdddl with wssRssddd as i g ds Rssd any parameter-dependent function, that is, wssRssddd ­ P a s Rssdd d jn; Rssdl with any an [8]. For our interpren n tational purposes, the closure rule (12) suffices. We can then use Stokes’ theorem to write gn fCg ­ i ­i

I Cg

Z Sg

We now turn to the classical treatment. For an adiabatic change through the open path C in parameter space, the variation of the action is DI ­ 0 and the variation of the angle variable is given by [6] Du ­

Z

t

dt ≠Hy≠I 1

Z C

0

dR ? k≠R ul ,

(14)

where the bracket stands for average around the HamiltonH ian contour on which the point lies, k fl ­ s2pd21 du f. The lines of constant angle are only specified once one is chosen as the origin of angles. Changing the parameter R arbitrarily changes the angle coordinates and the angle variable change Du then depends on the coordinates chosen for Rs0d and Rstd. R-independent statements about the angle variation can be achieved by a closure of the open path in the following manner that closely parallels the procedure used in the quantum case. First, use the R-dependent generating function S su, I; Rd ; S sbd s qsu, I, Rd, I; Rdd with S single valued and b labeling the branches of S and that u ­ s≠S sbd y≠Id to write for the second term in (14) Dk≠S y≠Il 2

Z C

dR ? k≠R ul ­

≠ Z dR ? kp≠R ql , ≠I C (15)

R with Dk≠S y≠Il ­ C dR ? ≠R k≠S y≠Il. To obtain a geometric angle on the final torus we send the initial point again to the final torus with a given rule through path c. The difference of the accumulated local change of the angles for C and c is geometric. We can adopt, for example, as closure rule the parameter path g obtained in the quantal case. A classical rule R can be obtained noting R that C dR ? k≠Rul ­ 2 ik 21 C dR ? fr p , ≠Rrgk 21 AC with [ , ] indicating integration over phase space and r the instantaneous classical Liouville eigenfunction, rI 0 ,k sq, p; Rd ­ de1y2 s Isq, p; Rd 2 I 0 d exp s ikusq, p; Rddd with eigenvalue kvsI 0 d. The closure path c that adds no geometric contribution is obtained taking into account the end-point contribution O ; argfr p s Rs0ddd, rssRstdddg as O 2 Ac ­ 0. In this way a meaningful comparison between the initial and final angle can then be given on the initial or final torus as Du ­

Z

t

dt ≠Hy≠I 1

I Cc

0

dR ? k≠Rul .

(16)

The first term is the dynamical angle Dud and the second one is the noncyclic geometric angle Dug . Using Stokes’ theorem we can rewrite the noncyclic geometric angle as

dR ? kn; Rj≠R jn; Rl dS ? ≠R ^ kn; Rj≠Rjn; Rl ,

6 JULY 1998

(13)

with Cg the closed parameter path formed by the open path C and its closure path g defined in (12).

Dug ­

Z Sc

­2

dS ? ≠R ^ k≠R ul

≠ Z dS ? k≠R p ^ ≠Rql , ≠I Sc

(17)

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which is clearly geometric, that is, invariant under any Rdependent shift of the origin of angles, u ! u 1 lsI; Rd and it is independent of how the parameter path is traversed as it is independent of time. It is also clear that for the cyclic case it reduces to the Hannay angle [6]. Figure 1(a) shows the total angle shift Du as measured in the final torus s0d where xc can be thought as the phase space point obtained from the initial point xssRs0ddd by the closure of the path discussed above and then subtracting the dynamical angle of the closure path. We now proceed to derive the semiclassical relation between the noncyclic geometric angle (17) and the noncyclic geometric phase (7). For the semiclassical wave function [9] X ab expfiS sbd sq, I; Rdyhg ¯ , (18) Csq; Rd ­ b

the quantum rule for closure in (12) reduces to the classical closure rule. Therefore in the semiclassical limit the path Cg reduces to Cc and we find gn fCg ­ h¯ 21

Z Sc

dS ? k≠R p ^ ≠Rql .

(19)

q (0)

xc

p

∆θ Final torus

x(R(t))

(a) x(R(0)) Initial torus

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The semiclassical connection between the noncyclic geometric phase and the noncyclic geometric angle is, comparing (17) and (19), ¯ n fCgy≠I ­ 2≠gn fCgy≠n . Dug fCg ­ 2h≠g

(20)

The nature of the displacement of the quantum revival can now be understood using the connection between the noncyclic geometric phase and the noncyclic geometric angle. The quantum wave function for the revival time TR with phases given by (11) is of the form " √ X jCsTR dl ­ exp iDn ≠gn fCgy≠n n !# Z TR 2 dt ≠HssRstdddy≠I 0

3 cn jn;RsTR dl , that can be written using (16) and (20) as X jCsTR dl ­ exps2iDnDudcn jn; RsTR dl

(21)

(22)

n

with Du the classical total angle shift for open paths. The revival wave packet is then shifted on the final torus by this classical total angle shift respect to the reference wave packet with no dynamical or geometrical phase contributions, that is, to the state obtained from the inital state by the closure of the quantum path in (12) to the final parameter value and then subtracting the dynamical phase of the closure, X cn jn; RsTR dl , (23) jCgs0d l ­ n

Ψg

(0)

∆θ

(b) Ψ(R(0))

Ψ(R(TR ))

FIG. 1. Total angle shift Du on the final torus for the adiabatic noncyclic change of a Hamiltonian system. The total angle shift is the sum of a dynamical and a geometrical angle shift, where the geometrical one has its origin in the quantum case in the noncyclic geometric phase. In (a) the total classical angle shift is measured on the final torus and it is defined between the final point xssRstddd and xs0d g , the point obtained from the initial classical point xssRs0ddd by the closure rule discussed in the text. In (b) the Wigner function of the wave packet at the revival time is displaced from the Wigner function of the reference state with no geometric or dynamical phase contributions. Note that the reference state is peaked on xgs0d at t ­ TR in the semiclassical limit.

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that corresponds in the semiclassical limit and for the revival time t ­ TR , as illustrated in Fig. 1(b), to the s0d classical point xg ; that is, the Wigner function of the refs0d erence state is peaked on xg . The noncyclic geometric angle contribution to this total shift has its origin in the quantal noncyclic geometric phase. Note that the effect of the geometric term in (11) is of the order Osn0 d whereas the dynamical term is of the order Osnd. However, their physical effects are comparable as they are, modulo 2p, of the same magnitude. A physical example is provided by a spin in a magnetic field of constant strength and slowly b for b ­ Bestd ? J, changing direction with Hamiltonian H which the geometric phase shift is obtained to be gn fCg ­ 2nVgc fCg, the solid angle defined by the open path of the field, and the geodesic closure in parameter space. The noncyclic geometric angle shift on the final torus is then given in this case by Dug fCg ­ Vgc fCg. C. Jarzynski is acknowledged for incisive questions and comments on the semiclassical result. M. S. Child and E. Sjöqvist are acknowledged for discussions. Financial support from a “Marie Curie” grant is also acknowledged.

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*Electronic address: [email protected] [1] A. ten Wolde et al., Phys. Rev. Lett. 61, 2099 (1988); J. A. Yeazell, M. Mallalieu, and C. R. Stroud, Jr., Phys. Rev. Lett. 64, 2007 (1990); J. C. Day et al., Phys. Rev. Lett. 72, 1612 (1994); L. Marmet et al., Phys. Rev. Lett. 72, 3779 (1994); J. Wals et al., Phys. Rev. Lett. 72, 3783 (1994); B. M. Garraway and K. A. Suominen, Rep. Prog. Phys 58, 365 (1990); E. O. Gobel et al., Phys. Rev. Lett. 64, 1801 (1990); G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. 58, 353 (1987). [2] Z. D. Gaeta, M. W. Noel, and C. R. Stroud, Jr., Phys. Rev. Lett. 73, 636 (1994). [3] J. Parker and C. R. Stroud, Phys. Rev. Lett. 56, 716 (1986); G. Alber, H. Ritsch, and P. Zoller, Phys. Rev.

[4] [5] [6] [7] [8]

[9]

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A 34, 1058 (1986); I. Sh. Averbukh and N. F. Perelman, Phys. Lett. A 139, 449 (1989). C. Jarzynski, Phys. Rev. Lett. 74, 1264 (1995). M. V. Berry, Proc. R. Soc. London A 392, 45 (1984). J. H. Hannay, J. Phys. A 18, 221 (1985). G. Garcia de Polavieja and E. Sjöqvist, Am. J. Phys. 66, 431 (1998). A particular case of this general closure rule is given by the shortest geodesic in the projective Hilbert space P , the space obtained by regarding all points in the Hilbert space H (formed by the vectors w) differing by a phase as a single point. M. V. Berry, J. Phys. A 18, 15 (1985).

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