Exact Quantum Decay of an Interacting Many-Particle System: the Calogero-Sutherland model
EExact Quantum Decay of an Interacting Many-Particle System: theCalogero-Sutherland model
Adolfo del Campo Department of Physics, University of Massachusetts, Boston, MA 02125, USA
The exact quantum decay of a one-dimensional Bose gas with inverse-square interactions is pre-sented. The system is equivalent to a gas of particles obeying generalized exclusion statistics. Weconsider the expansion dynamics of a cloud initially confined in a harmonic trap that is suddenlyswitched off. The decay is characterized by analyzing the fidelity between the initial and the time-evolving states, also known as the survival probability. It exhibits early on a quadratic dependenceon time that turns into a power-law decay, during the course of the evolution. It is shown thatthe particle number and the strength of interactions determine the power-law exponent in the lat-ter regime, as recently conjectured. The nonexponential character of the decay is linked to themany-particle reconstruction of the initial state from the decaying products.
PACS numbers: 03.65.-w, 03.65.Ta, 67.85.-d
Understanding the decay dynamics of unstable isolatedsystems is of relevance to a wide variety of fields rangingfrom quantum science [1, 2] to statistical mechanics [3–5] and cosmology [6]. While the exponential decay lawis ubiquitous in Nature [7], quantum mechanics dictatesits breakdown when the time of evolution is large [8] orsmall [9]. The existence of these deviations follows fromthe linearity of unitary quantum dynamics. Consider thepreparation of a unstable quantum state. During its de-cay, the time-evolving state can be decomposed as a co-herent superposition of the initial state and a set of decayproducts (any state orthogonal to the initial state). Devi-ations from exponential decay result from the possibilityfor the decay products to reconstruct the initial state.To be precise, let | Ψ (cid:105) = | Ψ(0) (cid:105) be an unstable quan-tum state prepared at t = 0, that evolves into | Ψ( t ) (cid:105) whenthe dynamics is generated by a Hamiltonian H ( t ). Theprobability to find the time-evolving state | Ψ( t ) (cid:105) in itsinitial state | Ψ (cid:105) is referred to as the survival probability S ( t ), S ( t ) := |A ( t ) | = |(cid:104) Ψ | Ψ( t ) (cid:105)| , (1)where A ( t ) = (cid:104) Ψ | Ψ( t ) (cid:105) is the survival amplitude. Equiv-alently, the survival probability is given by the expec-tation value of the projector on the intial state P = | Ψ (cid:105)(cid:104) Ψ | , S ( t ) = tr[ | Ψ( t ) (cid:105)(cid:104) Ψ( t ) |P ] (2)and is identical to the fidelity between the ini-tial state and the time-evolving state, S ( t ) ≡ F [ | Ψ (cid:105)(cid:104) Ψ | , | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | ] [10]. The survival probability isas well closely related, but different, from the Loschmidtecho [3]. However the distinction is often omitted in re-cent literature.To appreciate that the decay dynamics of S ( t ) is gen-erally non exponential, it is convenient to use the Ersakequation, that relates the values of survival amplitude atdifferent times during the course of evolution [9], A ( t ) = A ( t − τ ) A ( τ ) + M ( t, τ ) . (3) Above, the memory term is given by M ( t, τ ) = (cid:104) Ψ | U ( t, τ ) Q U ( τ, | Ψ (cid:105) , (4)where U ( t, t (cid:48) ) = T exp( − i (cid:82) tt (cid:48) H ( s ) ds/ (cid:126) ) is the time evo-lution operator from time t (cid:48) to t , and Q = 1 − P isthe complement of the projector on the initial state P .Hence, M ( t, τ ) represents the probability amplitude forthe initial state to evolve into decay products at time τ ,and subsequently reconstruct the initial state at time t .For an exponential decay to hold, the memory term isto vanish. However, the memory term plays a dominantrole at short and long times of evolution. The short-timedecay is known to be governed by the energy fluctua-tions of the unstable state. Experimentally, it was firstdemonstrated in [11] and its existence sets the ground forthe quantum Zeno effect [12]. It is a consequence of uni-tary time-evolution, provided that the first and secondmoments of the Hamiltonian exist, see [13, 14] for excep-tional cases. That deviations from exponential decay areas well to be expected at long times was pointed out byKhalfin in 1957, for systems whose energy spectrum isbounded from below [8]. Measurements consistent withthese deviations were reported in [15].The decay dynamics of many-particle quantum sys-tems has recently received a great deal of attention [16–26]. These studies show the need to characterize quan-tum decay at a truly many-particle level, beyond its de-scription in terms of one-body observables such as, e.g.,the integrated density profile [27–30]. Achieving this is achallenging goal, due to the limitation of reliable numer-ical techniques. Even at the single-particle or mean-fieldlevel, propagation methods based on space discretizationin a finite spatial domain can introduce artifacts due tothe enhanced reflection from the boundaries of the nu-merical box, unavoidable for long-time expansions [31].An attempt to palliate this effect with complex absorbingpotentials [32] explicitly suppresses state reconstruction,and delays the onset of power-law behavior in an un-physical way [33]. By contrast, these issues are absent instudies of fidelity decay in spin systems [34]. As an out-come, analytical results in quantum decay, often based on a r X i v : . [ qu a n t - ph ] J a n time-dependent scattering theory, are highly desirable.Recent theoretical progress has been mainly restricted totwo particle systems [16–22] and quasi-free few-particlequantum fluids [23, 24, 26]. Experimentally, the roleof Pauli exclusion principle has been demonstrated us-ing analogue simulation in photonic lattices [35], whileinteraction-induced particle correlations have been mea-sured in optical lattices [36].In this article, we present the exact quantum decay dy-namics of an interacting many-body system that is equiv-alent to a gas of particles obeying generalized exclusionstatistics. We rigorously show that the survival probabil-ity decays as a power law at long times with an exponentthat depends on the strength of the interactions and theparticle number. In the non-interacting limit, this resultproves the scaling conjectured based on the study of fewparticles systems [16, 17, 22, 23, 25]. The non exponentialcharacter of the evolution is linked to the multi-particlereconstruction of the initial state. I. MODEL
The Hamiltonian model we consider is that of N bosonseffectively confined in one-dimension in a harmonic trapand interacting with each other through an inverse-square pairwise potential. This is the so-called Calogero-Sutherland (CS) model [37, 38] H ( t ) = N (cid:88) i =1 (cid:104) − (cid:126) m ∂ ∂q i + 12 mω ( t ) q i (cid:105) + (cid:88) i 0) = ω . From now on, we use dimension-less variables q → q/q , t → t/t with q = ( (cid:126) /mω ) / and t = ω − .To describe the decay of the survival probability, wefirst note that the CS model belongs to a broad classof systems for which the exact time-dependent coherentstates can be found [45, 46]. A stationary state Ψ of thesystem (5) at t = 0 with energy E , follows a self-similarevolution dictated by the SU (1 , 1) dynamical symmetrygroup,Ψ ( q , . . . , q N , t ) = 1 b N2 exp (cid:34) i ˙ b b N (cid:88) i =1 q i − i E (cid:126) τ ( t ) (cid:35) × Ψ (cid:16) q b , . . . , q b , (cid:17) , (5)where τ ( t ) = (cid:82) t dt (cid:48) /b ( t (cid:48) ). Here, the scaling factor b = b ( t ) > b + K ( t ) b = b − , (6)where K ( t ) = [ ω ( t ) /ω ] , and the boundary conditions b (0) = 1 and ˙ b (0) = 0 follow from the stationarity of theinitial state. Note that SU (1 , 1) and the scaling dynamicsare robust against the breakdown of integrability [46, 47].The ground-state of the CS model has a Bijl-Jastrowform [38]Ψ( q , . . . , q N , 0) = C − N ,λ N (cid:89) i =1 e − q i (cid:89) j>i | q i − q j | λ . (7)Here, the normalization constant C N ,λ reads C N ,λ = 2 − N2 [1+ λ (N − (2 π ) N2 N − (cid:89) j =0 Γ (1 + ( j + 1) λ )Γ (1 + λ ) ;it is related to the normalization constant of the prob-ability distribution function for the Gaussian ( β = 2 λ )-ensembles in random matrix theory and derived usingMehta’s integral [38, 48, 49]. Using the dynamics (5), itis found that S N ,λ ( t ) = C − ,λ b N[1+ λ (N − × (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R N N (cid:89) i =1 d q i e − q i (1+ b − i ˙ bb ) (cid:89) j>i | q i − q j | λ (cid:12)(cid:12)(cid:12)(cid:12) . Upon explicit computation, the following closed expres-sion is obtained S N ,λ ( t ) = [ α ( t ) b ( t )] − N[1+ λ (N − , (8)where the function α ( t ) is given by α ( t ) = 12 (cid:20) (cid:18) b (cid:19) + (cid:32) ˙ bb (cid:33) (cid:21) . (9)As a result of the boundary conditions, b (0) = 1 and˙ b (0) = 0, α ( t ) reduces to unity at t = 0. For N = 1,one recovers the survival probability of a single particlein a time-dependent harmonic trap S ,λ ( t ) = α ( t ) b ( t ).It follows that the survival probability of a N particleCS system is identical to that of N non-interacting parti-cles obeying Generalized Exclusion Statistics (GES) withexclusion parameter g = λ . GES was introduced by Hal-dane for systems with a finite Hilbert space [44], andextended by Wu to unbounded Hamiltonians [50]. It ac-counts for the number of available states excluded by aparticle in the presence of others, and it smoothly extrap-olates between bosonic and fermionic exclusion statistics,and beyond. Note that the many-body-wave functionis always symmetric under the permutation of particles,this is, the exchange statistics is always bosonic for arbi-trary λ . The exclusion parameter is defined as the ratio g = − ∆ d/ ∆N of the change in the available states ∆ d as the particle number is varied by ∆N. For the CSmodel the exclusion parameter is precisely given by λ [51]. Particles with fractional excitation λ (cid:54) = { , } areHaldane anyons. From (8), the survival probability for N non-interacting and hard-core bosons is recovered for λ = 0 , S N ,λ ( t ) = [ S N , ( t )] − λ [ S N , ( t )] λ , (10)that represents a signature on the quantum decay dy-namics imprinted by the transmutation of statistics ob-served in a CS system as a function of the GES param-eter λ . Indeed, Eq. (10) resembles the known relationfor the equilibrium partition functions obeyed by Haldaneanyons [51]. More generally, S N ,γ + λ ( t ) = [ S N ,γ ( t )] − λ [ S N ,γ +1 ( t )] λ . (11)Simly put, these mathematical identities emphasize thefact that the CS model smoothly extrapolates betweennon-interacting and hard-core bosons (or more generally,between different types of Haldane anyons). In addition,this duality is not only apparent in equilibrium proper-ties as discussed in [51], but can be clearly manifested innonequilibrium observables as well. II. SUDDEN EXPANSION Suddenly switching off the trap (e.g., K ( t ) = Θ( − t )),leads to an expansion dynamics with the scaling factorgiven by b ( t ) = √ t > 0. The survival probabil-ity decays monotonically as a function of time and thereis a smooth transition between the short and long timeasymptotics, as shown in Fig. 1 for different values of λ .At short-times, S N ,λ ( t ) is a quadratic function of time, S N ,λ ( t ) = 1 − 18 N[1 + λ (N − t + O ( t ) , (12)where all odd moments vanish identically. Given the gen-eral short-time asymptotics of the survival probability S N ,λ ( t ) = 1 − ∆ H t + O ( t ) , (13)where ∆ H = (cid:104) Ψ | ( H − E ) | Ψ (cid:105) with E = (cid:104) Ψ | H | Ψ (cid:105) ,the coefficient of t in (12) can be interpreted as thevariance of the energy in the initial state | Ψ (cid:105) , ∆ H = (cid:112) N[1 + λ (N − / (2 √ S N ,λ ( t ) ∼ (cid:18) t (cid:19) N[1+ λ (N − . (14)Hence, the survival probability decays as a power-law intime. The power law exponent depends linearly on theinteraction strength λ (the GES parameter) and exhibitsan at most quadratic dependence on the particle number - - ω t l n S N , λ FIG. 1. Exact decay dynamics of a Calogero-Sutherland gas. The early stage exhibits a quadratic depen-dence on time and is followed by a smooth transition to thelong-time power-law scaling. The dependence of the power-law exponent on the particle number (N = 2) changes fromlinear to quadratic as the interaction strength λ is tunedfrom the non-interacting case to the Tonks-Girardeau limit( λ = 1) and other values λ (cid:54) = { , } , describing Haldaneanyons with fractional exclusion statistics. From top to bot-tom, λ = 0 , / , , / , N. This is the main result of this manuscript. Its deriva-tion required (i) the scaling dynamics of the exact time-dependent coherent states, (ii) the use of the Bijl-Jastrowform of the initial state, and (iii) the identification of theleading term at long times. The scaling dynamics holdsexactly for the CS gas and simplifies the ensuing analy-sis by contrast to other many-body systems such as, e.g.,the 1D Bose gas, where only recently moderate progressaccounting for its dynamics has been reported [52–55].Comparing the first leading terms in a long-time asymp-totic expansion, it is found that the power-law (14) setsin when the time of evolution satisfies t (cid:29) (cid:112) λ (N − . (15)For free-bosons, the power-law exponent becomes lin-ear in the particle number S N , ( t ) ∼ (2 /t ) N . (16)The case of hard-core bosons correspond to λ = 1, andleads to a power-law exponent quadratic in the particlenumber S N , ( t ) ∼ (2 /t ) N . (17)The change in the scaling with N was conjectured an-alyzing quasi-free systems of N = 2 , III. ROBUSTNESS OF THE SCALING It is worth emphasizing that the power-law behavior(14) is observed in the long-time dynamics of other multi-particle observables such as the non-escape probabilityfrom a region of space, e.g., where the initial state isinitially localized. Explicitly, we define the N-particlenon-escape probability as P N ,λ ( t ) := (cid:90) ∆ N N (cid:89) n =1 d q n | Ψ( q , . . . , q N ; t ) | , (18)which is the probability for the N particles to be found si-multaneously in the ∆-region and can be extracted fromthe full-counting statistics [23]. Explicit computationshows that P N ,λ ( t ) = C − ,λ t N[1+ λ (N − I N ,λ ( t ) . (19)Taking ∆ = [ − a/ , a/ I N ,λ ( t ) becomestime-independent at long expansion times when b (cid:29)√ λa , i.e. t (cid:29) | λa − | / , I N ,λ ( t ) := (cid:90) ∆ N N (cid:89) i =1 d q i e − q ib (cid:89) j>i | q i − q j | λ , ∼ a N[1+ λ (N − S N (1 , , λ ) , (20)where S n ( α, β, γ ) is the Selberg integral [49]. As a result,the same power-law scaling sets in, i.e., P N ,λ ( t ) ∝ S N ,λ ( t ) ∝ t − N[1+ λ (N − . (21)However, the dependence of the power-law exponent onN and λ is lost when studying the decay in terms of one-body observables such as the one-particle density profile n ( x, t ) integrated over the region of interest ∆, p ( t ) := (cid:90) ∆ n ( q, t ) dq. (22)To illustrate this, let us consider the exact evolutionof the density profile that under scaling dynamics isgiven by n ( q, t ) = n ( q/b, /b . Although an explicitcomputation of the density profile n ( q, 0) is possible inthe CS model, it would suffice to consider the large Nlimit. Then, n ( q, 0) follows Wigner’s semicircular dis-tribution n ( q, 0) = (cid:112) − q /π which is already inde-pendent of λ . Under free expansion, it is found that p ( t ) ∼ a N / ( πt ), where the power-law exponent is inde-pendent of N. The same conclusion holds when using theexpressions for low N and large λ available in the litera-ture for n ( q, 0) [49, 56], but for the fact that the prefactoracquires a dependence on λ/ N. Generally, the 1 /t power-law decay of p ( t ) can be expected as the density profileflattens out at long expansion times, becoming approxi-mately constant over the region ∆, so that the integrateddensity profile p ( t ) is governed by the normalization fac-tor 1 /b ( t ). (a)
(b)
(c)
(d)
(b) ω τ - ω τ - ω τ - - ω τ FIG. 2. Multi-particle state reconstruction. The sur-vival probability at a given evolution time is the sum of theproduct of the survival probabilities S N ,λ ( t − τ ) S N ,λ ( τ ) (or-ange), the memory term M ( t, τ ) (green) and the interferenceterm I ( t, τ ) (red), as it follows from Eq. (24). All contribu-tions are normalized to normalized to the value at the finaltime S N ,λ ( t ) and the blue line is set at unity to identify thedominant term. (a) During the decay of a single particle allterms are significant. (b) As the particle number is increased,the memory term becomes dominant (N = 3, λ = 1). (c) Forlarger values of the interaction strength, state reconstructiondescribes accurately the decay dynamics except for values of τ which are small or comparable to the total evolution time t (N = 3, λ = 2). (d) Increasing the particle number (N = 6, λ = 2) further prolongs the interval governed by state recon-struction. The dynamics is induced by suddenly switching offa harmonic trap with initial frequency ω , t = 15 /ω . One might also wonder whether a non-sudden modu-lation of the trapping frequency will affect the long timepower-law behavior. We show next that as long as thefrequency of the trap is permanently switched off after agiven time t = t , the same power law scaling sets in. In-deed, assume that b ( t ) = b , ˙ b ( t ) = v , K ( t > t ) = 0.Then, b ( t ) = (cid:20) ( b + v ( t − t )) + ( t − t ) b (cid:21) , (23)which tends to b ( t ) ∼ t ( v + 1 /b ) / at large expansiontimes. As a result, only the prefactors of the survival andnonescape probability are affected, and the scaling is stilldictated by (21). IV. MANY-PARTICLE STATERECONSTRUCTION We next analyze the relevance of state reconstructionin the CS model as an example of a many-particle sys-tem. Using the Ersak equation (3) [9, 33], the followingdecomposition of survival probability is obtained S N ,λ ( t ) = S N ,λ ( t − τ ) S N ,λ ( τ ) + M ( t, τ ) + I ( t, τ ) , where the first two terms admit a classical interpreta-tion. In particular, S N ,λ ( t − τ ) S N ,λ ( τ ) is the proba-bility for the system to survive in the initial state attime t provided that it was in the initial state at time τ . Similarly, M ( t, τ ) = | M ( t, τ ) | accounts for statereconstruction in a classical sense, i.e., it is the prob-ability that the state has decayed at time τ and re-construct the initial state at time t . The last term in(24) represents the interference between the amplitudesfor the two histories just described, i.e., I N ,λ ( t, τ ) =2 (cid:60) [ M ( t, τ ) ∗ A N ,λ ( t − τ ) A N ,λ ( τ )]. Here, the survival am-plitude is A N ,λ ( t ) = (cid:20) ( b + 1 b − i ˙ b ) e − iτ ( t ) (cid:21) N2 [1+ λ (N − . (24)Figure 2 analyzes the relevance of each term in (24) nor-malized to S N ,λ ( t ) and after gauging away the dynami-cal phase E τ ( t ) / (cid:126) = N [1 + λ ( N − τ ( t ) / 2. Differentregimes can be distinguished as a function of the param-eter β = N[1 + λ (N − t andsmall values of β all terms in (24) play a role. For largervalues of β , achievable by increasing either the particlenumber or the interaction strength, M ( t, τ ) dominatesthe contribution to the survival probability. Thus, thestate reconstruction governs the long-time decay, exceptfor values of t/τ close to { , } , when all processes remainrelevant. In conclusion, we have characterized the the exact de-cay of an interacting many-body quantum fluid releasedfrom a harmonic trap. Exploiting the self-similarity ofthe ensuing dynamics, the long-time power-law behav-ior of the survival probability was shown to be dictatedby the strength of the interactions even at arbitrarilylarge expansion times. The scaling of the power-law ex-ponent is at most quadratic in the particle number. Thenon-exponential character of the decay can be attributedto the many-particle state reconstruction of the initialstate. Our results can be extended to other systemsincluding SU ( ν ) spin degrees of freedom and fermionicexchange statistics [39]. As an outlook, it is worth ex-ploring higher dimensional systems like the 2D Bose gas,for which self-similar dynamics holds [57] up to quantumanomalies [58] and the generalized exclusion parameter isknown [59]. In systems lacking self-similar dynamics, therole of interactions can be disentangled from that of theexclusion statistics and further studies will be illuminat-ing. A prominent example is the one-dimensional Bosegas with contact interactions, where recent advances indescribing its dynamics have been reported [52–55]. Acknowlegments.— It is a pleasure to dedicate this ar-ticle to Marvin D. Girardeau (1930-1915) and to thank S.B. Arnason, M. Beau, Y. 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