A semiclassical theory of phase-space dynamics of interacting bosons
AA semiclassical theory of phase-space dynamics of interacting bosons
R. Mathew
Joint Quantum Institute, University of Maryland and National Instituteof Standards and Technology, College Park, Maryland 20742, USA
E. Tiesinga
Joint Quantum Institute and Joint Center for Quantum Information and Computer Science,National Institute of Standards and Technology and University of Maryland, Gaithersburg, Maryland 20899, USA
We study the phase-space representation of dynamics of bosons in the semiclassical regime wherethe occupation number of the modes is large. To this end, we employ the van Vleck-Gutzwillerpropagator to obtain an approximation for the Green’s function of the Wigner distribution. Thesemiclassical analysis incorporates interference of classical paths and reduces to the truncated Wignerapproximation (TWA) when the interference is ignored. Furthermore, we identify the Ehrenfest timeafter which the TWA fails. As a case study, we consider a single-mode quantum nonlinear oscillator,which displays collapse and revival of observables. We analytically show that the interference ofclassical paths leads to revivals, an effect that is not reproduced by the TWA or a perturbativeanalysis.
I. INTRODUCTION
The crucial difference between quantum mechanics anda statistical theory based on classical mechanics is themethod of computing the transition probability betweenan initial and a final state [1]. In the classical theory, thetransition probability is the sum over probabilities of thepaths connecting the two states. In contrast, in quan-tum mechanics, the transition probability is obtained byfirst summing the amplitudes of all the connecting pathsand then squaring the sum. This procedure leads to in-terference, a feature absent in the classical theory. Anarchetypal example of interference is a double-slit exper-iment in which a beam of particles after passing throughtwo slits forms an oscillating intensity pattern on a screen[1].The aforementioned difference between the theoriescan be systematically studied in the semiclassical regime(where the typical action (cid:29) (cid:126) , the reduced Planck’s con-stant). In this regime, a probability amplitude can beapproximated by the contributions from a subset of allconnecting paths: the classical paths [2, 3]. (This is thecase with the textbook treatment of the double-slit ex-periment.) Crucially, within this semiclassical approxi-mation, the transition probability retains interference ofpaths, albeit classical ones. The role of classical trajec-tories in quantum dynamics was first elucidated by vanVleck [4]. Later, Gutzwiller extended the van Vleck prop-agator by including Maslov indices and used it to derivehis trace formula [5]. The role of classical paths in quan-tum mechanics has been extensively studied; for example,in scattering [6], localization [7, 8], quantum kicked ro-tor [9], level statistics [10, 11], quantum work [12], theHelium atom [13] and quantum transport [14, 15].In this paper, we study a semiclassical approximationof the phase-space dynamics of interacting bosons in theWigner-Weyl representation. In this representation, uni-tary evolution of an initial quantum state in the Hilbert space is equivalent to evolution of an initial Wigner dis-tribution in phase space in accordance to the Moyal’sequation [16–18]. The reduction of the state space froma high-dimensional Hilbert space to a lower-dimensionalphase space makes the phase-space picture particularlyuseful for implementing approximations of quantum dy-namics. An approximation that is usually made is amean-field approach. In this case, the distribution is ap-proximated at all times by a delta function whose loca-tion is determined by the classical Hamilton’s equations.The Gross-Pitaevskii equation and its discrete versionsfall under this category.An improvement over the mean-field description is thetruncated Wigner approximation (TWA) [19–21], wherethe initial distribution is extended and is the Wignertransform of a quantum state. The subsequent dynamicsof the Wigner distribution is still classical. Equivalently,the Moyal’s equation is replaced by the classical Liou-ville’s equation. In the literature, the TWA is sometimescalled a semiclassical method eventhough it lacks inter-ference effects. Quantum corrections to the TWA for in-teracting bosons were studied by A. Polkovnikov [22, 23]using a perturbation theory with the TWA as its zeroth-order approximation. In particular, a nonlinear oscillatorwas studied whose quantum dynamics exhibits collapseand revival of coherences. The perturbative analysis de-scribes the initial collapse, with increasing accuracy withthe order of the perturbation parameter. It fails to de-scribe revivals in the system because the analysis stilllacks interference of classical paths.We study semiclassical dynamics of a general Bose sys-tem in phase space that incorporates interference of clas-sical paths and makes comparison with the TWA trans-parent. In particular, our analysis identifies the Ehren-fest time associated with the TWA as the time wheninterference of classical paths becomes important. As acase study, we investigate the nonlinear oscillator andshow that the semiclassical dynamics leads to revivals.Recently, others have also applied semiclassical methods a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a r to bosons. For example, these methods have been appliedto coherent backscattering [24] and autocorrelation func-tions [25] in the Bose-Hubbard model. In addition, thesemiclassical Herman-Kluck propagator has been used tostudy boson dynamics [26, 27].The remainder of the paper is organized as follows.First, we define the phase space of a bosonic system andthe Green’s function of a Wigner distribution in Sec. IIand Sec. III, respectively. A semiclassical approxima-tion of this Green’s function is obtained in Sec. IV. InSec. IV A, we find that our semiclassical formalism re-duces to the TWA when the interference terms are ig-nored. Next, we discuss Ehrenfest times associated withthe TWA and semiclassical approximation in Sec. IV B.Subsequently, we apply our formalism to analyticallystudy of a nonlinear oscillator in Sec. V and concludein Sec. VI. II. PHASE-SPACE FORMULATION OF ABOSONIC SYSTEM
A bosonic system with a finite number of modes canbe described in terms of annihilation and creation opera-tors ˆ a j and ˆ a † j , respectively, with j = 1 , . . . , d , where d isthe number of modes. For example, the modes could bethe sites of a Bose-Hubbard model or spin componentsof a single-mode Bose-Einstein condensate. The oper-ators satisfy the commutation relations [ a j , a † k ] = δ jk ,where δ jk is the Kronecker delta function. To constructthe phase space, we first define the quadrature operatorsˆ x j = (cid:112) (cid:126) / a j + ˆ a † j ) and ˆ p j = − i (cid:112) (cid:126) / a j − ˆ a † j ) satisfy-ing the canonical commutation relations [ˆ x j , ˆ p k ] = i (cid:126) δ jk .The eigenstates of ˆ x j satisfy ˆ x j | x (cid:105) = x j | x (cid:105) for all j ∈ { , · · · , d } , with “position” x = ( x , x , . . . , x d ).Similarly, the eigenstates of ˆ p j satisfy ˆ p j | p (cid:105) = p j | p (cid:105) ,with “momentum” p = ( p , p , . . . , p d ). The eigen-states form a complete basis with (cid:104) x (cid:48) | x (cid:105) = δ ( x − x (cid:48) ), (cid:104) p (cid:48) | p (cid:105) = δ ( p − p (cid:48) ) and (cid:82) d x | x (cid:105) (cid:104) x | = (cid:82) d p | p (cid:105) (cid:104) p | = 1,where δ ( z ) is a Dirac delta function and the integralsare over R d . We construct a phase space by imposing { x i , p j } = δ ij , where { ., . } is the Poisson bracket. Wewill refer to r = ( x , p ) as a phase-space point. Thus, byintroducing quadrature operators, we have mapped thekinematics of a many-body boson system with d modesto that of a single particle in d -dimensional position orconfiguration space.The Wigner transform [18, 28] maps an operator ˆ O , afunction of ˆ a j and ˆ a † j or ˆ x j and ˆ p j , to its Weyl symbol O in the phase space. In fact, O ( r ) = (cid:90) d q (cid:68) x + q | ˆ O| x − q (cid:69) e − i p · q / (cid:126) , (1)where r = ( x , p ), p · q is the dot product between p and q , and the integral is over the configuration space q ∈ R d .In particular, the Wigner distribution W ( r , t ) at time t is the Weyl symbol of the density operator ˆ ρ ( t ), up to a factor of 1 / (2 π (cid:126) ) d , i.e., W ( r , t ) = 1(2 π (cid:126) ) d (cid:90) d q (cid:10) x + q | ˆ ρ ( t ) | x − q (cid:11) e − i p · q / (cid:126) , (2)These definitions imply that (cid:82) d r W ( r , t ) = 1 and in theSchr¨odinger picture, the expectation value of an operatorˆ O at a time t is (cid:68) ˆ O ( t ) (cid:69) ≡ Tr[ˆ ρ ( t ) ˆ O ] = (cid:90) d r W ( r , t ) O ( r ) , (3)where the integrals are over the phase space R d . Equiv-alently, in the Heisenberg picture, (cid:68) ˆ O ( t ) (cid:69) ≡ Tr[ˆ ρ ˆ O ( t )] = (cid:90) d r W ( r ) O ( r , t ) , (4)where W ( r ) is the initial Wigner distribution. III. GREEN’S FUNCTION OF THE WIGNERDISTRIBUTION
The Green’s function G ( r f , r i , t ) of the Wigner distri-bution in the Schr¨odinger picture is defined by [17, 29–31] W ( r f , t ) = (cid:90) d r i G ( r f , r i , t ) W ( r i , , (5)for t ≥ r f = ( x f , p f ), r i = ( x i , p i ) and G ( r f , r i ,
0) = δ ( r f − r i ). In a seminal paper on quan-tum dynamics in phase space, Moyal called G ( r f , r i , t )the “temporal transformation function” [17]. He derivedan expression for G ( r f , r i , t ) in terms of Feynman prop-agators. We give a short and direct derivation.The time evolution of the density operator is ˆ ρ ( t ) =ˆ U ( t )ˆ ρ ˆ U † ( t ), where ˆ U ( t ) and ˆ ρ are the unitary time-evolution and the initial density operator, respectively.We insert (cid:82) d y | y (cid:105) (cid:104) y | = 1 and (cid:82) d y | y (cid:105) (cid:104) y | = 1,with y and y in configuration space, into Eq. 2 andfind W ( r f , t ) = 1(2 π (cid:126) ) d (cid:90) d q d y d y K (cid:0) x f + q , y , t (cid:1) (6) × (cid:104) y | ˆ ρ | y (cid:105) K ∗ (cid:0) x f − q , y , t (cid:1) e − i p f · q / (cid:126) , where K ( x f , x i , t ) = (cid:104) x f | ˆ U ( t ) | x i (cid:105) is the Feynman prop-agator in the configuration space. For notational sim-plicity, we have and will hereafter set (cid:126) = 1. Next, weexpress the initial condition (cid:104) y | ˆ ρ | y (cid:105) in terms of the ini-tial Wigner distribution. To this end, we multiply Eq. 2,evaluated at t = 0 and r = r i , by e i p i · q (cid:48) and integrateover p i to find (cid:90) d p i e i p i · q (cid:48) / (cid:126) W ( r i ,
0) = (cid:10) x i + q (cid:48) | ˆ ρ | x i − q (cid:48) (cid:11) (7)We substitute this expression in Eq. 6 and identify y = x i + q (cid:48) and y = x i − q (cid:48) . From the definition of Green’sfunction in Eq. 5 we find G ( r f , r i , t ) = 1(2 π (cid:126) ) d (cid:90) d q d q (cid:48) K (cid:0) x f + q , x i + q (cid:48) , t (cid:1) × K ∗ (cid:0) x f − q , x i − q (cid:48) , t (cid:1) e − i [ p f · q − p i · q (cid:48) ] / (cid:126) . (8)Thus, the exact Green’s function of the Wigner distri-bution involves the product of two Feynman propagatorsin configuration space. We expect that this product willhave interference terms. IV. SEMICLASSICAL APPROXIMATION OFTHE GREEN’S FUNCTION
A quantum system is said to be in the semiclassicalregime when the typical action (in units of (cid:126) ) that ap-pears in the path integral description of the Feynmanpropagator is much greater than one. For bosonic modes,this regime corresponds to large occupation numbers. Infact, the semiclassical approximation of the propagator,also known as the van Vleck-Gutzwiller propagator, is[2, 32] K SC ( x f , x i , t ) = (cid:88) b e − iµ b π/ (2 πi (cid:126) ) d/ (cid:113) D b ( x f , x i , t ) e iS b ( x f , x i ,t ) / (cid:126) , (9)where the sum is over all classical paths, indexed by b ,that start from position x i and reach x f in time t . The action S b ( x f , x i , t ) = (cid:82) t dτ L [ x b cl ( τ ) , d x b cl ( τ ) /dτ ], where L is the system Lagrangian and x b cl ( τ ) is the positionas a function of time τ of the b -th classical path with x b cl (0) = x i and x b cl ( t ) = x f . Finally, µ b is the Maslov in-dex and D b ( x f , x i , t ) = | det (cid:2) ∂ S b ( x f , x i , t ) / ( ∂ x f ∂ x i ) (cid:3) | is the absolute value of the determinant of a d × d matrix.The number of classical paths contributing to K SC ( x f , x i , t ) can be found by studying the dynamicsof the initial Lagrangian manifold M i , the hyperplane x = x i . Classical evolution of each point of M i yieldsa final manifold M f . Figure 1 shows M i and M f fora two-dimensional example. The final manifold folds atsingular positions x f on M f where the number of mo-menta p f on M f as a function of x f changes. Partsof the manifold M f between these singular regions are“branches”. The example in Fig. 1 has the three suchbranches. Crucially, the final momentum p f ( x f , x i , t ),which, in general, is a multivalued function of x f at fixed x i and t , is unique on each branch. Therefore, classi-cal paths connecting x i and x f can be indexed by thebranches that intersect the manifold x = x f . It is thesebranches which contribute to the van Vleck-Gutzwillerpropagator in Eq. 9. For example, for the position x f shown in Fig. 1 has three paths that contribute to thepropagator.Substitution of van Vleck-Gutzwiller propagator inEq. 8 yields the semiclassical approximation to theGreen’s function G SC ( r f , r i , t ) = 1(2 π (cid:126) ) d (cid:90) d q d q (cid:48) e − i [ p f · q − p i · q (cid:48) ] / (cid:126) (cid:88) b (cid:113) D b ( x f + q / , x i + q (cid:48) / , t ) e iS b ( x f + q / , x i + q (cid:48) / ,t ) / (cid:126) − iµ b π/ × (cid:88) b (cid:48) (cid:113) D b (cid:48) ( x f − q / , x i − q (cid:48) / , t ) e − iS b (cid:48) ( x f − q / , x i − q (cid:48) / ,t ) / (cid:126) + iµ b (cid:48) π/ . (10)The expression is cumbersome for our analytical study.To proceed, we assume that in Eq. 10 only the contri-butions from small regions Q and Q (cid:48) around q = 0 and q (cid:48) = 0, respectively, are important; and, secondly, theTaylor expansion of the action S b (cid:0) x f + q , x i + q (cid:48) , t (cid:1) ≈ S b ( x f , x i , t ) + p bf · q − p bi · q (cid:48) , (11)up to linear terms is sufficient in these regions. Here, p bi = − ∂S b ( x f , x i , t ) /∂ x i and p bf = ∂S b ( x f , x i , t ) /∂ x f , respectively, are the initial and final momenta of theclassical path along which the action is computed (seeAppendix A for a derivation). We further assume thatthe extent of the small regions Q and Q (cid:48) in each direc-tion in position space is much greater than √ (cid:126) . (Notethat from Sec. II, both position and momentum havethe same units as √ (cid:126) ). Furthermore, we approximate D b (cid:0) x f ± q , x i ± q (cid:48) , t (cid:1) by D b ( x f , x i , t ). Substitutingthese approximations for S b and D b in Eq. 10 and inter-changing the sum and the integral, we find FIG. 1: Classical dynamics of a manifold in a two-dimensional phase space ( x, p ). An initial manifold M i , theline x = x i shown in black, evolves into the manifold M f ,the curve shown in blue, at time t . For example, points A , B , C , and D on M i are mapped to A (cid:48) , B (cid:48) , C (cid:48) and D (cid:48) on M f ,respectively. The classical path connecting C and C (cid:48) is alsoshown. The manifold M f has three branches I, II and III sep-arated by caustics B (cid:48) and C (cid:48) . The line x = x f intersects M f thrice; therefore, three paths start on M i and reach position x f at time t . G ( r f , r i , t ) = 1(2 π (cid:126) ) d (cid:88) b,b (cid:48) (cid:113) D b ( x f , x i , t ) D b (cid:48) ( x f , x i , t ) e i [ S b ( x f , x i ,t ) − S b (cid:48) ( x f , x i ,t )] / (cid:126) − i ( µ b − µ b (cid:48) ) π/ × (cid:90) Q d q (cid:90) Q (cid:48) d q (cid:48) e − i (cid:16) p f − p bf / − p b (cid:48) f / (cid:17) · q / (cid:126) + i (cid:16) p i − p bi / − p b (cid:48) i / (cid:17) · q (cid:48) / (cid:126) . (12)Implicit in the existence of Q is the assumption that x f is away from the position of any caustics, where twobranches meet. The example in Fig. 1 has two caustics.The integration over q and q (cid:48) yields functions of p f and p i localized around p i = ( p bi + p b (cid:48) i ) and p f = ( p bf + p b (cid:48) f ), whose characteristic widths in momentumspace are much less than √ (cid:126) . (For “rectangular” re-gions Q and Q (cid:48) we obtain multidimensional sinc func- tions.) Typically, observables are smooth functions inphase space, i.e., they vary slowly on the scale of √ (cid:126) .Moreover, initial states of interest are classical states(coherent states) whose width is of the order of √ (cid:126) . (We do not consider initial Wigner distributions with finesub-Planck structures.) Then we can approximate thelocalized functions by δ -functions to find G SC ( r f , r i , t ) ≈ (cid:88) b,b (cid:48) √D b D b (cid:48) e i ( S b − S b (cid:48) ) / (cid:126) − i ( µ b − µ b (cid:48) ) π/ δ (cid:104) p f − ( p bf + p b (cid:48) f ) (cid:105) δ (cid:104) p i − ( p bi + p b (cid:48) i ) (cid:105) , (13)where, for clarity, we suppress the dependence of S b , D b ,etc., on x i , x f and t . This is the main result of this pa-per and relates the Green’s function of the Wigner dis-tribution to a double sum over classical paths connectingpositions x i and x f in time t . A. The truncated Wigner approximation
In the TWA, the Wigner distribution is propagatedclassically, i.e., it obeys the Liouville’s equation. TheGreen’s function according to the Liouville’s equation is G TWA ( r f , r i , t ) = δ [ x f − x cl ( t ; r i )] δ [ p f − p cl ( t ; r i )] , (14)where [ x cl ( t ; r i ) , p cl ( t ; r i )] is the classical path startingfrom r i = ( x i , p i ). We now show that the “diagonal”part of the double sum in Eq. 13, i.e., when b = b (cid:48) , isequal to G TWA . To this end, we change the independentvariables ( x i , p i , t ) of Eq. 14 to ( x f , x i , t ) and find G TWA ( r f , r i , t ) = (cid:88) b δ (cid:0) p i − p bi (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:34) ∂ x cl ( t ; r i ) ∂ p i (cid:12)(cid:12)(cid:12)(cid:12) p i = p bi (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ( p f − p bf ) , (15)where the sum is over all roots p bi (enumerated by b ) ofequation x cl ( t ; x i , p i ) = x f , and p bf = p cl ( t ; x i , p bi ) [52].We have suppressed the dependence of p bi and p bf on( x f , x i , t ). Next, we apply the inverse function theorem,which states that the matrix inverse of a Jacobian is theJacobian of the inverse mapping, to find1det (cid:34) ∂ x cl ( t ; r i ) ∂ p i (cid:12)(cid:12)(cid:12)(cid:12) p i = p bi (cid:35) = det (cid:34) − ∂ S b ( x f , x i , t ) ∂ x f ∂ x i (cid:12)(cid:12)(cid:12)(cid:12) x f = x bf (cid:35) , (16)where we used that p bi = − ∂S b ( x f , x i , t ) /∂ x i . Substitut-ing the expression in Eq. 15, we arrive at G TWA ( r f , r i , t ) = (cid:88) b D b ( x f , x i , t ) δ (cid:0) p f − p bf (cid:1) δ (cid:0) p i − p bi (cid:1) , (17)which is the diagonal part of Eq. 13. Thus, TWA ignoresinterference of classical paths. For the special cases of theharmonic oscillator and free particle, the TWA matcheswith the quantum motion because only a single path con-tributes to the sum in Eq. 13 and, hence, there are nointerference terms. B. Ehrenfest times
An Ehrenfest time is the time scale when an approxi-mation to the quantum motion deviates appreciably fromexact evolution [33, 34]. In fact, there is a hierarchyof Ehrenfest times based on the approximations to thequantum dynamics [35, 36]. For the mean-field approxi-mation, the Ehrenfest time τ MF is the time scale when aninitially localized Wigner distribution becomes distortedand stretched due to nonlinear (not necessarily chaotic)classical dynamics. From Sec. IV A, we find that theEhrenfest time τ TWA associated with the TWA occurswhen interference of classical paths becomes important.This time scale is greater than τ MF because interferenceof paths occurs when the Wigner distribution becomesso distorted that it fills up the accessible phase space. Finally, there is τ SC , the Ehrenfest time for the break-down of semiclassical approximation based on van Vleck-Gutzwiller propagator, which is greater than τ TWA . Nu-merical studies have shown that the breakdown occurswhen diffraction becomes important [37, 38].
V. CASE STUDY: A NONLINEAROSCILLATOR
We consider a single-mode nonlinear oscillator whosequantum Hamiltonian isˆ H NO = U a † ˆ a † ˆ a ˆ a, (18)where U is the interaction strength and ˆ a (ˆ a † ) is the an-nihilation (creation) operator of the associated bosonicmode. As the number operator ˆ a † ˆ a commutes with H NO , the energy eigenstates are | n (cid:105) with eigen-energies E n = U n ( n − /
2, where n is the occupation num-ber of the mode. Decomposing an arbitrary initial state | ψ (cid:105) = (cid:80) ∞ n =0 c n | n (cid:105) and noting that n ( n − / | ψ ( t ) (cid:105) periodically revives, i.e., | ψ ( t ) (cid:105) = | ψ (cid:105) when t is aninteger multiple of the period t rev = 2 π (cid:126) /U .The nonlinear oscillator has been studied in experi-ments with a BEC in an optical lattice [39] and withphotons using Kerr nonlinearity [40]. In these experi-ments, the initial state is well-described by a coherentstate, | ψ (cid:105) = e −| α | / (cid:80) ∞ n =0 α n / √ n ! | n (cid:105) , where α , in gen-eral, is a complex number and | α | = N is the averagenumber of atoms or photons. Using interference, the col-lapse and revival of the absolute value of the expectationvalue of ˆ a and a generalized Husimi function, respectively,were measured in these experiments. We find that theexpectation value of ˆ a evolves as (cid:104) ˆ a ( t ) (cid:105) = (cid:104) ψ ( t ) | ˆ a | ψ ( t ) (cid:105) = αe | α | ( e − iUt/ (cid:126) − . (19)Its absolute value is shown in Fig. 2. At short times U t/ (cid:126) (cid:28) (cid:104) ˆ a ( t ) (cid:105) ≈ αe −| α | U t / (2 (cid:126) ) − iU | α | t/ (cid:126) , (20)whose decay in time is Gaussian with time constant (cid:126) / ( U √ N ). The collapse time t col is a few times this timeconstant, as shown in Fig. 2, and is much smaller t rev forlarge N . In the experiments with a BEC in an opticallattice, three-body effects proportional to (ˆ a † ) ˆ a changethe nature of the collapse and revival in an interestingmanner [41–43]. A. Dynamics according to the TWA
Next, we study the time dynamics of (cid:104) ˆ a ( t ) (cid:105) within theTWA. First, we need to write down the classical Hamil-tonian corresponding to ˆ H NO [53]. It is H NO = U (cid:126) ( x + p ) (21) Q , TWA, SC Q , SCTWAMF Q SC MFSC,TWA (a)(b)
FIG. 2: Collapse and revival in a nonlinear oscillator. Pan-els (a) and (b) show | (cid:104) a ( t ) (cid:105) | and ln | (cid:104) a ( t ) (cid:105) | , respectively, asa function of time t for an initial coherent state whose meanatom number is four. Exact quantum dynamics (labeled Q)displays collapse and revival of | (cid:104) a ( t ) (cid:105) | with collapse and re-vival times t col and t rev , respectively. The mean-field solution,labeled MF, is time independent. The TWA result, Eq. 24,closely replicates the first collapse but shows no revival anddeviates appreciably from the quantum dynamics after a time τ TWA . On the other hand, the semiclassical approximation(labeled SC), Eq. 27, agrees well with the quantum evolu-tion for all times. In panel (a) the semiclassical and quantumcurves are indistinguishable. with classical equations of motion dxdt = ∂ H NO ∂p = U (cid:126) ρ p , dpdt = − ∂ H NO ∂x = − U (cid:126) ρ x, (22)where ρ = x + p . Hence, classical paths lie on circlesin phase space centered around the origin ( x, p ) = (0 , ω = U ρ / (2 (cid:126) ). The sys-tem is integrable as the phase space is two-dimensionaland energy is conserved. The angle of the action-anglecoordinates is the polar angle ϕ measured in clockwise di-rection of motion and evolves as ϕ ( t ) = ωt + ϕ i , where ϕ i is the initial angle. Using the definition ω = ∂ H NO /∂I ,we find that the action coordinate I = ρ /
2. Hence, ρ isa constant of motion.For concreteness, let the initial coherent state | ψ (cid:105) ,with occupation number N (cid:29)
1, be centered along the p -axis, i.e., α = i √ N . Its Wigner distribution W ( x, p ) = 1 π (cid:126) e − [ x +( p −√ (cid:126) N ) ] / (cid:126) , (23)is centered at ( x, p ) = (0 , √ (cid:126) N ) and width O ( √ (cid:126) ).Next, we calculate (cid:104) ˆ a ( t ) (cid:105) TWA , the expectation value ofˆ a within the TWA. Instead of using the Green’s func-tion G TWA of Eq. 14, it is more convenient to work inthe Heisenberg picture. In this picture, the Wigner-Weyl transform of the operator ˆ a ( t ) is a ( t ) = [ x ( t ) + ip ( t )] / √ (cid:126) = iρe − iϕ ( t ) / √ (cid:126) with ϕ ∈ ( − π, π ] and ϕ = 0along the p -axis. Then using Eq. 4 and writing W ( x, p )in polar coordinates, we find (cid:104) ˆ a ( t ) (cid:105) TWA = (cid:90) ∞ ρdρ (cid:90) π − π dϕ i iρ √ (cid:126) e − iϕ ( t ) W ( ρ, ϕ i ) . (24)For N (cid:29)
1, it is sufficient to expand the exponent of W ( x, p ) to second order in ρ and ϕ around the locationof the maximum of the Wigner distribution, i.e., W ( ρ, ϕ i ) ≈ π (cid:126) e − [ ( ρ −√ (cid:126) N ) − (cid:126) Nϕ i ] / (cid:126) . (25)Substituting this expression and ω = U ρ / (2 (cid:126) ) in Eq. 24,we derive (cid:104) ˆ a ( t ) (cid:105) TWA ≈ i √ N e − U Nt / (2 (cid:126) ) − iUNt/ (cid:126) , (26)which matches the initial collapse of the coherent state inEq. 20, but has no revival. A comparison of Eq. 26 withthe exact quantum result of Eq. 19 for the absolute valueof (cid:104) ˆ a ( t ) (cid:105) is shown in Fig. 2. The figure also shows themean-field value | (cid:104) ˆ a ( t ) (cid:105) MF | , which is | a ( t ) | along the sin-gle circular trajectory starting from ( x, p ) = (0 , √ (cid:126) N ).Thus, | (cid:104) ˆ a ( t ) (cid:105) MF | = √ N is a constant.The classical phenomenon of phase-space mixing ex-plains the collapse of (cid:104) ˆ a ( t ) (cid:105) [44]. For an integrablesystem, the coarse-grained long-time Wigner distribu-tion is uniformly distributed in the angle coordinates ofthe action-angle variables. For the nonlinear oscillator, a ( t ) ∝ e iϕ ( t ) and its expectation value goes to zero asthe Wigner distribution mixes in the angle ϕ . Further-more, within the TWA, the coarsened Wigner distribu-tion reaches a steady state; hence, there is no revival.This latter observation indicates that quantum interfer-ence reverses phase-space mixing and revives the quan-tum state. In the next section, we find that applying thesemiclassical formalism, indeed, leads to revival. B. Dynamics according to the semiclassicalapproximation
The calculation of (cid:104) ˆ a ( t ) (cid:105) SC according to the semiclas-sical approximation is lengthy and has been relegatedto Appendices B and C. We first calculate the action interms of the polar angles and winding number of classicalpaths around the origin in Appendix B. We carry out theremainder the calculation in Appendix C. Here, we listthe main steps:1. The time evolution of an observable with Weyl sym-bol O ( x, p ) in the Schr¨odinger picture is given byEq. 3. We replace G ( r f , r i , t ) by G SC ( r f , r i , t ) asgiven in Eq. 13 and carry out the integrals over p i and p f to arrive at Eq. C2.2. The classical equations of motion are simplest inthe action-angle coordinates. Therefore, we con-vert the integrals over x i , x f and the double sumover b , b (cid:48) in Eq. C2 into integrals over the initialand final angles ϕ i and ϕ f , respectively, and a dou-ble sum over winding numbers of classical pathsaround the origin. We also express the observable, G SC ( r f , r i , t ), and the initial Wigner distributionin terms of ϕ i , ϕ f and winding numbers.3. Next, we note that the classical motion in thephase space is restricted in an annulus of ra-dius √ N and width of O (1). Then, O ( x, p ) ≈O ( √ N sin ϕ, √ N cos ϕ ); in particular, a ( x, p ) ≈ i √ N e − iϕ . We make similar approximations for thedeterminants D b ( x f , x i , t ). The initial Wigner dis-tribution, however, varies sharply with ρ and re-quires a more careful approximation. We then solvethe remaining integrals.Finally, we find (cid:104) ˆ a ( t ) (cid:105) SC = ∞ (cid:88) v = −∞ i √ N e − iUNt/ (cid:126) e − (2 πv − Ut/ (cid:126) ) N/ , (27)where v is difference of the winding number of the inter-fering paths. This expression corresponds to a train of lo-calized Gaussians and is invariant under the transforma-tion t → t +2 π (cid:126) /U and v → v −
1; hence, is periodic withtime period t rev = 2 π (cid:126) /U . Figure 2 shows that (cid:104) ˆ a ( t ) (cid:105) SC agrees with the exact quantum average (cid:104) ψ ( t ) | ˆ a | ψ ( t ) (cid:105) forall times.Finally, we discuss the Ehrenfest times of the nonlin-ear oscillator for initial coherent states. From Fig. 2(a),we see that the mean-field prediction deviates from thequantum evolution well before the collapse time t col , i.e., τ MF < t col . In contrast, the deviation of the TWAfrom quantum evolution (ignoring exponentially smalldifferences) occurs abruptly after a finite time τ TWA = t rev − t col before the first revival of (cid:104) ˆ a ( t ) (cid:105) . The interfer-ence of classical paths starts at τ inter when the Wignerdistribution fills up the annular accessible phase space,i.e., when paths starting from the initial localized distri-bution with winding numbers zero and one terminate inthe same small region of phase space. We can estimate τ inter by noting that for a coherent state the distributionof classical frequencies ω has a mean U N/ (cid:126) and width∆ ω = U √ N / (cid:126) . Therefore, τ inter ∼ π/ ∆ ω is of the or-der of t col and, hence, is much smaller than τ TWA . In other words, it takes time for the interference of pathsto affect (cid:104) ˆ a ( t ) (cid:105) appreciably. In fact, at τ TWA the numberof interfering classical paths is of the order of √ N . Onthe other hand, the Ehrenfest time τ SC is infinite for thenonlinear oscillator.The Ehrenfest time τ TWA depends on the observableunder consideration. For example, for (cid:10) ˆ a ( t ) (cid:11) the col-lapse and revival times are t rev / t col /
2, respectively,and the TWA fails after ( t rev − t col ) /
2. Nevertheless, τ TWA is still greater than τ inter for all observables (thatare polynomials in a and a † with a degree smaller than N ). We also expect the delay in effects of interferenceand the dependence of τ TWA on the observable to holdtrue for generic integrable systems (where the dynamicsis away from singularities like a saddle point of the clas-sical Hamiltonian). In contrast, in a chaotic system andfor motion near a saddle point of an integrable system,the Ehrenfest time τ TWA ∼ τ inter [44–46]. VI. CONCLUSION AND OUTLOOK
In conclusion, we presented a semiclassical theory ofphase-space dynamics of bosons. We derived a semiclas-sical approximation, Eq. 10, to the exact Green’s functionof the Wigner distribution. Crucially, the approximationpreserves the quantum interference of classical trajecto-ries. In fact, we have shown that the formalism reducesto the TWA when the interference terms are ignored.Hence, the Ehrenfest time associated with the break-down of the TWA occurs when interference of classicalpaths becomes important. As a case study, we examineda single-mode nonlinear oscillator whose exact quantumdynamics exhibits collapse and revival. We investigatedthe dynamics of an observable of this oscillator using theTWA and our semiclassical formalism. Within TWA,the expectation value of an observable collapses due tophase mixing, and there is no revival. The semiclassicalapproximation, however, reproduces revivals and accu-rately matches the exact quantum dynamics for all times.Finally, we comment on the long-time validity of oursemiclassical approximation. For the nonlinear oscillator,the semiclassical approach is valid for all times [54]. Weexpect this to be true for generic integrable systems asthey can be quantized by the Einstein-Brillouin-Kellermethod [47]. The situation, however, is not straight-forward for chaotic systems. For example, the semi-classical evolution (based on the van Vleck-Gutzwillerpropagator) of an initial wavefunction defined on a La-grangian manifold, whose Wigner distribution is not lo-calized, breaks down after a time of the order of theEhrenfest time associated with interference of classicalpaths [29, 48]. For localized initial Wigner distributions,however, numerical studies and heuristic arguments haveshown that the van Vleck-Gutzwiller propagator worksfor rather longer times [37, 38, 49] and only breaks downdue to diffraction. The validity of our semiclassical ap-proach for chaotic systems will require further study.
Appendix A: Derivatives of action
We evaluate the partial derivatives of the action S b ( x f , x i , t ) with respect to the initial and final positions.The action satisfies the Hamilton-Jacobi equation and, inprinciple, its derivates are well known [50, 51]. Here, wegive a derivation for the sake of completeness. For nota-tional simplicity, we assume that the configuration spaceis one-dimensional; generalization to higher dimensions isstraightforward. Consider a classical path [ x b cl ( τ ) , p b cl ( τ )],which starts from the phase-space point ( x i , p bi ) and endsat ( x f , p bf ). Next, consider another classical path whoseposition in time, x b cl ( τ ) + δx b cl ( τ ), is infinitesimally closeto x b cl ( τ ) such that δx b cl (0) = ∆ x i and δx b cl ( t ) = 0. Thenthe change in the action is∆ S b = (cid:90) t dτ (cid:18) ∂L∂x cl δx b cl ( τ ) + ∂L∂ ˙ x cl δ ˙ x b cl ( τ ) (cid:19) = ∂L∂ ˙ x cl δx b cl ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) t + (cid:90) t dt (cid:18) ∂L∂x cl − ddt ∂L∂ ˙ x cl (cid:19) δx b cl ( τ ) , where ˙ x cl = dx cl /dτ and we have suppressed the ar-guments of L . Now, the second term vanishes because x b cl ( τ ) satisfies the Euler-Lagrange equations of motion.Using the fact that p = ∂L ( x, ˙ x ) /∂ ˙ x , we have ∆ S b = − p bi ∆ x i or ∂S b ( x f , x i , t ) ∂x i = − p bi . (A1)Similarly, we can prove that ∂S b ( x f , x i , t ) ∂x f = p bf . (A2) Appendix B: Action of the nonlinear oscillator
We compute the action S b ( x f , x i , t ) of the nonlinearoscillator described in Sec. V. The action depends on theindex b , which we have not yet quantified. A naturalguess is the winding number w of a circular path aroundthe phase-space origin. The winding number is a non-negative integer as the motion in phase space is unidirec-tional. For a given ( x f , x i , w, t ), however, more than oneclassical path can exist. For example, two such paths areshown in Fig. 3. In contrast, a given ( ϕ f , ϕ i , w, t ), where ϕ i and ϕ f are the initial and final angles, respectively,uniquely determines a classical path. The reason is thatthe oscillation frequency is specified by ω = ( ϕ f − ϕ i ) mod 2 π + 2 πwt (B1)and, hence, uniquely determines the radius ρ = (cid:126) (cid:112) ω/U (see Sec. V A) of the classical path.It is convenient to define the action S w ( ϕ f , ϕ i , t ) = S b [ x f ( ϕ f , ϕ i , w, t ) , x i ( ϕ f , ϕ i , w, t ) , t ], indexed by the FIG. 3: Classical paths of a nonlinear oscillator in phasespace ( x, p ) starting from a Wigner distribution initially lo-calized in region Ω shown by the solid blue circle. Polarcoordinates ( ρ, ϕ ), with angle ϕ measured from the p -axis ina clockwise direction, are also shown. The region Ω is cen-tered at ( ρ, ϕ ) = ( √ N,
0) and has a width of O (1). Trajecto-ries starting from Ω lie within the gray annulus. Two pathswith traversal time t and winding number zero that start from x = 0 and end at x = x f are shown. winding number w of path b and S w ( ϕ f , ϕ i , t ) = (cid:90) t dτ (cid:20) p cl dx cl dτ − H NO (cid:21) , (B2)where H NO is given by Eq. 21 and we have suppressed thearguments ( ϕ f , ϕ i , w, t ). Substituting x cl ( τ ) = ρ sin ϕ ( τ )and p cl ( τ ) = ρ cos ϕ ( τ ), we find S w ( ϕ f , ϕ i , t ) = (cid:90) t dτ (cid:18) ωρ cos ϕ ( τ ) − U ρ (cid:19) , where we used dρ/dτ = 0, dϕ/dτ = ω and have set (cid:126) = 1.The integration over τ yields S w ( ϕ f , ϕ i , t ) = [( ϕ f − ϕ i ) mod 2 π + 2 πw ]2 U t (B3) × [( ϕ f − ϕ i ) mod 2 π + 2 πw + sin(2 ϕ f ) − sin(2 ϕ i )] . Appendix C: Calculation of (cid:104) ˆ a ( t ) (cid:105) We calculate the expectation value of ˆ a ( t ) within thesemiclassical approximation and follow the outline pre-sented in Sec. V B.1. The semiclassical evolution of the expectation valueof an observable of the nonlinear oscillator ˆ O with Weylsymbol O ( x, p ) is (cid:68) ˆ O ( t ) (cid:69) SC = (cid:90) dr i dr f O ( r f ) G SC ( r f , r i , t ) W ( r i ) . (C1)Substituting G SC ( r f , r i , t ) from Eq. 10 and integratingover the momenta p i and p f , we find (cid:68) ˆ O ( t ) (cid:69) SC = (cid:90) dx i dx f (cid:88) b,b (cid:48) O (cid:32) x f , p bf + p b (cid:48) f (cid:33) (C2) × W (cid:32) x i , p bi + p b (cid:48) i (cid:33) √D b D b (cid:48) e iS b − iS b (cid:48) − i ( µ b − µ b (cid:48) ) π/ , where we suppress the dependence of p bi , D b , S b , etc.,on ( x f , x i , t ) and set (cid:126) = 1. The range of integration is( −∞ , ∞ ) for both x i and x f .2. The action has a simpler form in terms of the angles(see Eq. B3). Hence, we proceed to change the integra-tion variables in Eq. C2 to the angle coordinates. To thisend, we first introduce a set of initial and final positions x (cid:48) i and x (cid:48) f , respectively, and write a symmetric expression (cid:104)O ( t ) (cid:105) SC = (cid:90) dx i dx f dx (cid:48) i dx (cid:48) f δ ( x i − x (cid:48) i ) δ ( x f − x (cid:48) f ) (C3) × (cid:88) b,b (cid:48) O (cid:34) x f + x (cid:48) f , p bf ( x f , x i , t ) + p b (cid:48) f ( x (cid:48) f , x (cid:48) i , t )2 (cid:35) W (cid:34) x i + x (cid:48) i , p bi ( x f , x i , t ) + p b (cid:48) i ( x (cid:48) f , x (cid:48) i , t )2 (cid:35) × (cid:113) D b ( x f , x i , t ) D b (cid:48) ( x (cid:48) f , x (cid:48) i , t ) e iS b ( x f ,x i ,t ) − iS b (cid:48) ( x (cid:48) f ,x (cid:48) i ,t ) − i [ µ b ( x f ,x i ,t ) − µ b (cid:48) ( x (cid:48) f ,x (cid:48) i ,t )] π/ , where the explicit dependence of the quantities is shownto avoid any confusion. The two sets of paths indexed by b and b (cid:48) now have different boundary conditions ( x f , x i , t )and ( x (cid:48) f , x (cid:48) i , t ), respectively, enabling us to interchangethe sum over b and integrals over x (cid:48) i and x (cid:48) f . The nextstep is to change the integration measure in terms of onefor the angles. This step is carried out in Appendix C 1and we find (cid:90) dx i dx f (cid:88) b ( . . . ) = w max (cid:88) w = w min (cid:90) π − π dϕ f (cid:90) π − π dϕ i | det J | ( . . . ) , (C4) where ( . . . ) is a function of ( x f , x i , b, t ) and the Ja-cobian matrix J = ∂ ( x wi , x wf ) /∂ ( ϕ i , ϕ f ) with x wi = x i ( ϕ f , ϕ i , w, t ) and x wf = x f ( ϕ f , ϕ i , w, t ). The nonneg-ative integers w min and w max are minimum and maxi-mum winding numbers, respectively, of trajectories start-ing from region Ω , as shown in Fig. 3. An equationanalogous to Eq. C4 holds for measures of x (cid:48) i and x (cid:48) f .Substitution of these measure changes in Eq. C3 yields (cid:104)O ( t ) (cid:105) SC = w max (cid:88) w,w (cid:48) = w min (cid:90) dϕ i dϕ f dϕ (cid:48) i dϕ (cid:48) f (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20) ∂ ( x wi , x wf ) ∂ ( ϕ i , ϕ f ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det (cid:34) ∂ ( x w (cid:48) i , x w (cid:48) f ) ∂ ( ϕ (cid:48) i , ϕ (cid:48) f ) (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ( x wi − x w (cid:48) i ) δ ( x wf − x w (cid:48) f ) × O (cid:32) x wf + x w (cid:48) f , p wf + p w (cid:48) f (cid:33) W (cid:32) x wi + x w (cid:48) i , p wi + p w (cid:48) i (cid:33) √ D w D w (cid:48) e i S w − i S w (cid:48) − i ( µ w − µ w (cid:48) ) π/ , (C5)where the arguments of quantities with superscript w and w (cid:48) are ( ϕ i , ϕ f , t ) and ( ϕ (cid:48) i , ϕ (cid:48) f , t ), respectively. Moreover,we have introduced D w ( ϕ f , ϕ i , t ) = D b ( x wf , x wi , t ) and S w ( ϕ f , ϕ i , t ) is given by Eq. B3.3. We explicitly write all quantities appearing in Eq. C5in terms of ( ϕ f , ϕ i , w, t ). We do so by noting that the relevant classical motion is restricted in an annulus ofwidth O (1) around ρ = √ N (see Fig. 3). In the annulus,we approximate the radius by its mean √ N , i.e., x wi ≈√ N sin ϕ i , p wi ≈ √ N sin ϕ i , etc., which leads to (cid:12)(cid:12)(cid:12)(cid:12) det (cid:20) ∂ ( x wi , x wf ) ∂ ( ϕ i , ϕ f ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≈ N | cos ϕ i cos ϕ f | (C6)0and D w ( ϕ f , ϕ i , t ) ≈ U N t | cos ϕ i cos ϕ f | , (C7)etc. Moreover, as the initial Wigner distribution is lo-calized around angle ϕ = 0, δ ( x wi − x w (cid:48) i ) ≈ δ ( ϕ f − ϕ (cid:48) f ) / ( √ N cos ϕ f ) . The other delta function becomes δ ( x wf − x w (cid:48) f ) ≈ δ ( ϕ f − ϕ (cid:48) f ) + δ ( ϕ f + ϕ (cid:48) f − π ) √ N cos ϕ f . (C8)The two contributions reflect the fact that a line at fixedvalue of x f intersects the thin annulus in two regions,whose respective angles are approximated by the angleof the intersection with the circle of radius ρ = √ N .Substituting these approximations into Eq. C5 and in-tegrating over ϕ (cid:48) i and ϕ (cid:48) f , we find (cid:104)O ( t ) (cid:105) SC = 1 U t w max (cid:88) w,w (cid:48) = w min (cid:90) dϕ i dϕ f O W (C9) × e i S w − i S w (cid:48) − i ( µ w − µ w (cid:48) ) π/ , where we suppress the arguments of O and W , and ne-glect the contribution from the second term in Eq. C8.This term leads to a highly oscillating integrand whoseintegral is small. The arguments of quantities in the in-tegrand with either superscript w or w (cid:48) are now ϕ f , ϕ i and t .Next, we note that O ( x, p ) is a slowlyvarying function of x, p and within the an-nulus O (cid:104) (cid:16) x wf + x w (cid:48) f (cid:17) , (cid:16) p wf + p w (cid:48) f (cid:17)(cid:105) ≈O ( √ N sin ϕ f , √ N cos ϕ f ). In particular, a (cid:104) (cid:16) x wf + x w (cid:48) f (cid:17) , (cid:16) p wf + p w (cid:48) f (cid:17)(cid:105) ≈ i √ N e − iϕ f . (C10) We cannot make a similar approximation for the initialWigner distribution, i.e., replace ρ w and ρ w (cid:48) by √ N ,because the distribution varies sharply around ρ = √ N .Instead, we write ρ w ( ϕ f , ϕ i , t ) √ N = (cid:20) ( ϕ f − ϕ i ) mod 2 π + 2 πwU N t (cid:21) / ≈ (cid:20) ( ϕ f − ϕ i ) mod 2 π + 2 πwU N t − (cid:21) , (C11)where we used the relation ρ = (cid:112) ω/U (see Sec. V A),Eq. B1 and performed a Taylor expansion around ρ w / √ N = 1. We substitute ρ in the initial Wignerdistribution of Eq. 25 by the Taylor approximation for( ρ w + ρ w (cid:48) ) /
2, to find W (cid:32) x wi + x w (cid:48) i , p wi + p w (cid:48) i (cid:33) ≈ (C12)1 π e − [ ( ϕ f − ϕ i ) mod 2 π + π ( w + w (cid:48) ) − UNt ] / U Nt − Nϕ i . Also, from Eq. B3, we have S w − S w (cid:48) = 2 πU t ( w − w (cid:48) )[( ϕ f − ϕ i ) mod 2 π + π ( w + w (cid:48) )](C13)Finally, the Maslov index, which is the number of turn-ing points of a classical path, increases by two for everywinding. Therefore, µ w − µ w (cid:48) = 2( w − w (cid:48) ) . (C14)After substituting O ( x, p ) = a ( x, p ), Eqs. C10, C12,C13, and C14 in Eq. C9, we find (cid:104) a ( t ) (cid:105) SC = i √ NπU t w max (cid:88) w,w (cid:48) = w min (cid:90) π − π dϕ i (cid:90) π − π dϕ f e − iϕ f − i ( w − w (cid:48) ) π e − [ ( ϕ f − ϕ i ) mod 2 π + π ( w + w (cid:48) ) − UNt ] / (2 U Nt ) × e − Nϕ i e i π ( w − w (cid:48) )[( ϕ f − ϕ i ) mod 2 π + π ( w + w (cid:48) )] / ( Ut ) . (C15)Next, we extend the limits on w and w (cid:48) to [0 , ∞ ) andwrite the sums over w and w (cid:48) in terms of u = w + w (cid:48) and v = w − w (cid:48) . We combine the sum over u and theintegral over ϕ f by defining y = ( ϕ f − ϕ i ) mod 2 π + πu ,whose range is [0 , ∞ ). We realize that e − iϕ f − i ( w − w (cid:48) ) π = e − i ( y + ϕ i ) and the integrand is separable in ϕ i and y . Af- ter evaluating the integrals, we arrive at (cid:104) a ( t ) (cid:105) SC = ∞ (cid:88) v = −∞ i √ N e − iUNt e − (2 πv − Ut ) N/ e − / (8 N ) which becomes Eq. 27 of the main text for large N .1
1. Derivation of Eq. C4
Here, we derive Eq. C4. We restrict our attention topaths that start from the phase-space region Ω , in whichthe initial Wigner distribution is concentrated. Figure 3shows the region Ω for the nonlinear oscillator. Thepaths starting within Ω lie on the annulus shown in thefigure. Now, the winding number of a circular path at afixed traversal time is a stepwise increasing function ofthe radius. Let the (time-dependent) winding numbers ofpaths that lie on the inner and outer circles of the annulusbe w min and w max , respectively, with w min ≤ w max . Fora given winding number, there can be two paths thatstart from Ω with position x i and reach position x f intime t . Figure 3 shows a pair of such paths with windingnumber zero and x i = 0. Moreover, the paths end inthe upper ( p >
0) and lower ( p ≤
0) halves of the phasespace. Therefore, we can interchange the integrals overboundary conditions and sum over paths to find (cid:90) dx i dx f (cid:88) b ( . . . ) = w max (cid:88) w = w min , upper (cid:90) dx i dx f ( . . . )+ w max (cid:88) w = w min , lower (cid:90) dx i dx f ( . . . ) , (C16) where the labels “upper” and “lower” indicate paths thatend in the corresponding half of phase space.In each half of the phase space, the final angle isuniquely determined given ( x f , x i , w, t ). Therefore, wecan transform the integrals over x i and x f in Eq. C16to one over angles and combine the “upper” and “lower”contributions to arrive at Eq. C4. (cid:3) [1] R. Feynman, R. Leighton, and M. Sands, The FeynmanLectures on Physics , no. v. 3 in The Feynman Lectureson Physics (Pearson/Addison-Wesley, 1963).[2] L. S. Schulman,
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