Impurity induced quantum chaos for an ultracold bosonic ensemble in a double-well
IImpurity induced quantum chaos for an ultracold bosonic ensemble in a double-well
Jie Chen, ∗ Kevin Keiler, Gao Xianlong, and Peter Schmelcher
1, 3 Zentrum f¨ur Optische Quantentechnologien, Fachbereich Physik,Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Department of Physics, Zhejiang Normal University, Jinhua 321004, China The Hamburg Centre for Ultrafast Imaging, Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany (Dated: January 28, 2021)We demonstrate that an ultracold many-body bosonic ensemble confined in a one-dimensional (1D) double-well (DW) potential can exhibit chaotic dynamics due to the presence of a single impurity. The non-equilibriumdynamics is triggered by a quench of the impurity-Bose interaction and is illustrated via the evolution of thepopulation imbalance for the bosons between the two wells. While the increase of the post-quench interac-tion strength always facilitates the irregular motion for the bosonic population imbalance, it becomes regularagain when the impurity is initially populated in the highly excited states. Such an integrability to chaos (ITC)transition is fully captured by the transient dynamics of the corresponding linear entanglement entropy, whoseinfinite-time averaged value additionally characterizes the edge of the chaos and implies the existence of ane ff ective Bose-Bose attraction induced by the impurity. In order to elucidate the physical origin for the observedITC transition, we perform a detailed spectral analysis for the mixture with respect to both the energy spectrumas well as the eigenstates. Specifically, two distinguished spectral behaviors upon a variation of the interspeciesinteraction strength are observed. While the avoided level-crossings take place in the low-energy spectrum, theenergy levels in the high-energy spectrum possess a band-like structure and are equidistant within each band.This leads to a significant delocalization of the low-lying eigenvectors which, in turn, accounts for the chaoticnature of the bosonic dynamics. By contrast, those highly excited states bear a high resemblance to the non-interacting integrable basis, which explains for the recovery of the integrability for the bosonic species. Finally,we discuss the induced Bose-Bose attraction as well as its impact on the bosonic dynamics. I. INTRODUCTION
Trapping of an ultracold many-body bosonic ensemble ina one-dimensional (1D) double-well (DW) potential consti-tutes a prototype system for the investigations of the corre-lated quantum dynamics [1–3]. Such a system represents abosonic Josephson junction (BJJ), an atomic analogy of theJosephson e ff ect initially predicted for Cooper pair tunnelingthrough two weakly linked superconductors [4, 5]. Owing tothe unprecedented controllability of the trapping geometriesas well as the atomic interaction strengths [6], studies of theBJJ unveil various intriguing phenomena which are not acces-sible for conventional superconducting systems [7–13]. Ex-amples are the Josephson oscillations [7–9], fragmentations[10, 11], macroscopic quantum self trapping [3, 7, 8], collapseand revival sequences [9] as well as the atomic squeezing state[12, 13].Under the explicit time-dependent driving forces, the BJJcan alternatively turn into the quantum kicked top (QKT), a fa-mous platform for the investigations of quantum chaos as wellas the classical-quantum correspondence [14–26]. To date, re-lated studies include the spectral statistics [15], the entangle-ment entropy production [16–23], the quantum decoherenceand quantum correlations [24, 25] as well as the border be-tween regular and chaotic dynamics [26]. Moreover, by view-ing the QKT as a collective N -qubit system, the e ff ects of thequantum chaos on the digital quantum simulations have alsobeen detailed discussed recently [27, 28]. ∗ [email protected] On the other hand, stimulated by the experimental pro-gresses on few-body ensembles [29–34], significant theoret-ical e ff ort also focuses on the 1D few-body atomic systems[35–46], revealing for example the ground state [35–43] aswell as the dynamical properties [44–46], which pave the wayfor the studies of the binary mixtures with large particle num-ber imbalance. Such hybridized systems are deeply relatedto the polaron physics [47–49] as well as the open quantumsystems [50] and are particularly interesting owing to the factthat one subsystem is in the deep quantum regime while theother one can more or less be described by the semi-classicalphysics. Note, however, that while most of the discussionsfocus on impacts on the minority species from the majoritybath, studies which alternatively explore the feedback to themajority species due to the presence of the minority one arestill rare.In the present paper, we investigate a binary ultracoldatomic mixture made of a single impurity and a non-interacting many-body bosonic ensemble that are confinedwithin a 1D DW potential. Unlike most of the previous stud-ies where the focuses are put on the weak-interacting regime,rendering the impurity being restricted into the lowest twomodes of the DW potential [51–56], our discussions are notrestricted to such a scenario. Specifically, we study the on-set of the chaos for the majority bosonic species due to thepresence of the impurity and put particular emphasis on theits dynamical response upon a sudden quench of the impurity-Bose interaction strength. As an exemplary observable, wemonitor the quantum evolution of the population imbalancefor the bosons between the two wells starting from a balancedparticle population. While the increase of the post-quench in-teraction strength always facilitates a chaotic motion for the a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n bosonic population imbalance, it becomes regular again whenthe impurity initially is prepared in the highly excited states.In order to characterize such an integrability to chaos (ITC)transition, we employ the linear entanglement entropy as asignature of quantum chaos, which alternatively measures thedecoherence for the bosonic species. Depending on the de-gree of chaos, the transient dynamics of the corresponding lin-ear entanglement entropy can behave as either a rapid growthor a slow variation with increasing time, whereas, its infinite-time averaged value, in addition, captures the edge of quantumchaos, i.e., the border between the integrable and the chaoticregions in the corresponding classical phase space. Further-more, by computing the infinite-time averaged values of thelinear entanglement entropy for various initial conditions, wefind a striking resemblance between its profile and a classi-cal phase space with attractive Bose-Bose interaction, whichimplies the existence of an attractive interaction among thebosons induced by the impurity.In order to elucidate the physical origin for the above ob-served ITC transition, we perform a detailed spectral analy-sis with respect to both the energy spectrum as well as theeigenstates of the mixture. Two distinguished spectral behav-iors upon a variation of the interspecies interaction strengthare observed. While the avoided level-crossings take place inthe low-energy spectrum, the energy levels in the high-energyspectrum possess a band-like structure and are equidistantwithin each band. Consequently, this results in a significantdelocalization for those low-lying eigenstates which, in turn,accounts for the chaotic nature of the bosonic non-equilibriumdynamics. Remarkably, those highly excited states bear astriking resemblance to the non-interacting integrable basis,which explains the recovery of the integrability for the bosonicspecies. Finally, we also discuss the induced Bose-Bose at-traction and its impact on the bosonic dynamics.This paper is organized as follows. In Sec. II, we introduceour setup including the Hamiltonian, the initial conditions aswell as the quantities of interests. In Sec. III, we present ourmain observation: the ITC transition for the bosonic species.In Sec. IV, we perform a detailed spectral analysis for themixture with respect to both the energy spectrum as well asthe eigenstates, so as to elucidate the physical origin for theabove observed ITC transition. Finally, our conclusions andoutlook are provided in Sec. V. II. SETUPA. Hamiltonian and angular-momentum representation
The Hamiltonian of our 1D ultracold impurity-Bose mix-ture is given by ˆ H = ˆ H I + ˆ H B + ˆ H IB , whereˆ H σ = (cid:90) dx σ ˆ ψ † σ ( x σ ) h σ ( x σ ) ˆ ψ σ ( x σ ) , ˆ H IB = g IB (cid:90) dx ˆ ψ † I ( x ) ˆ ψ † B ( x ) ˆ ψ B ( x ) ˆ ψ I ( x ) , (1) and h σ ( x σ ) = − (cid:126) m σ ∂ ∂ x σ + V DW ( x σ ) is the single-particle Hamil-tonian for the σ = I ( B ) species being confined within a 1Dsymmetric DW potential V DW ( x σ ) = a σ ( x σ − b σ ) . For sim-plicity, we consider the atoms for both species are of the samemass ( m I = m B = m ) and are trapped by the same potentialgeometry, i.e., a I = a B = a DW and b I = b B = b DW . ˆ ψ † σ ( x σ )[ ˆ ψ σ ( x σ )] is the field operator that creates (annihilates) a σ -species particle at position x σ . Moreover, we neglect the in-teractions among the bosons and assume the impurity-Boseinteraction is of zero range and can be modeled by a contactpotential of strength [46, 57–59] g IB = (cid:126) a D µ a ⊥ [1 − C a D a ⊥ ] − . (2)Here a D is the 3D impurity-Bose s -wave scattering lengthand C ≈ . a ⊥ = (cid:112) (cid:126) /µω ⊥ describes the transverse confinement with µ = m / ω ⊥ to be equal for both species. In the following dis-cussions, we rescale the Hamiltonian of the mixture ˆ H for theunits of the energy, length and time as η = (cid:126) ω ⊥ , ξ = √ (cid:126) / m ω ⊥ and τ = /ω ⊥ , respectively. We focus on the repulsive inter-action regime, i.e., g IB (cid:62) a DW = . b DW = . V DW ( x ) (black dashed line) as well as the lowest sixsingle-particle energy levels (grey solid lines)]. Throughoutthis work, we explore a binary mixture made of a single im-purity and 100 bosons ( N I = N B = ff erent kinds of atoms [60, 61] or a Bose-Bose mixturemade of the same atoms with two di ff erent hyperfine states[62, 63]. The DW potential can also be readily constructed byimposing a 1D optical lattice on top of a harmonic trap [3, 5].Moreover, the contact interaction strength g IB can be con-trolled experimentally by tuning the s -wave scattering lengthsvia Feshbach or confinement-induced resonances [57–59].Noticing further that the bosonic species is confined withina tight DW potential with δ (cid:29) δ [c.f. Fig. 1 (a)], here δ i de-notes the energy di ff erence between the i -th and the ( i + ψ B ( x ) = u L ( x )ˆ b L + u R ( x )ˆ b R , (3)with u L , R ( x ) being the Wannier-like states localized in the leftand right well, respectively. This leads to the low-energy ef-fective Hamiltonian for the bosonic speciesˆ H B = − J (ˆ b † L ˆ b R + ˆ b † R ˆ b L ) , (4)corresponding to the two-site Bose-Hubbard (BH) model with J = .
071 being the hopping amplitude.Before proceeding, it is instructive to express the above BHHamiltonian in the angular-momentum representation. To seethis, we introduce three angular-momentum operators [9, 64]ˆ J x =
12 (ˆ b † L ˆ b R + ˆ b † R ˆ b L ) , ˆ J y = − i b † L ˆ b R − ˆ b † R ˆ b L ) , ˆ J z =
12 (ˆ b † L ˆ b L − ˆ b † R ˆ b R ) , (5)obeying the SU(2) commutation relation [ ˆ J α , ˆ J β ] = i (cid:15) αβγ ˆ J γ .The BH Hamiltonian in Eq. (4) thus can be rewritten asˆ H B = − J ˆ J x , (6)which describes the angular momentum precession of a singleparticle whose spatial degrees of freedom (DOFs) are frozen.According to definitions for ˆ J x and ˆ J z in Eq. (5), we note thatthe kinetic energy in the BH model as well as the populationimbalance for the bosons between the two wells are in anal-ogy to the magnetizations of this single particle along the x and the z axes. Moreover, the particle number conservationin the Hamiltonian (4) corresponds to the angular momentumconservation ˆ J = ˆ J x + ˆ J y + ˆ J z = N B N B +
1) (7)for the Hamiltonian (6).For the case g IB =
0, the angular momentum dynamics canbe simply integrated out from the corresponding Heisenbergequations of motion, in whichˆ J y ( t ) = ˆ J z (0)cos(2 J t ) − ˆ J y (0)sin(2 J t ) , ˆ J z ( t ) = ˆ J y (0)cos(2 J t ) + ˆ J z (0)sin(2 J t ) , (8)being the harmonic oscillations with the frequency ω = J and ˆ J x ( t ) = ˆ J x (0) is time-independent since [ ˆ J x , ˆ H B ] =
0. Fur-ther introducing the normalized vector ˆ (cid:126) S ( t ) = ˆ S x ( t ) (cid:126) i + ˆ S y ( t ) (cid:126) j + ˆ S z ( t ) (cid:126) k with ˆ S γ ( t ) = ˆ J γ ( t ) / J for γ = x , y , z and J = N B / (cid:88) γ = x , y , z (cid:104) ˆ S γ (cid:105) ( t ) = (cid:88) γ = x , y , z (cid:104) ˆ S γ (cid:105) (0) , (9)one can readily show that the motion of the vector ˆ (cid:126) S ( t ) alwayslies on the Bloch sphere with unit radius if, in addition, wechoose the initial state as the atomic coherent state (ACS) (seebelow). B. Classical dynamics
The above angular momentum dynamics can alternativelybe understood in a classical manner. As we will show below,the periodic motions for ˆ J y ( t ) and ˆ J z ( t ) [equivalently ˆ S y ( t ) andˆ S z ( t )] correspond to the periodic oscillation of a classical non-rigid pendulum around its equilibrium position, while the con-servation of ˆ J x ( t ) [ ˆ S x ( t )] relates to the energy conservation ofthis pendulum. To this end, we first adopt the mean-field ap-proximation as ˆ b β = b β ( β = L , R ) with b β being a c -number [65]. The quantum operators ˆ S x , ˆ S y and ˆ S z then should berewritten as S x = J ( b ∗ L b R + b ∗ R b L ) , S y = − i J ( b ∗ L b R − b ∗ R b L ) , S z = J ( b ∗ L b L − b ∗ R b R ) . (10)Employing the phase-density representation for b β as b β = (cid:113) N B β e i θ β and further introducing the two conjugate variables Z = ( N BL − N BR ) / N B , ϕ = θ R − θ L , (11)representing the relative population imbalance between thetwo wells and the relative phase di ff erence, respectively, wearrive at S x = √ − Z cos ϕ, S y = √ − Z sin ϕ, S z = Z , (12)whose dynamics are governed by the Hamiltonian H cl = − J √ − Z cos ϕ, (13)which, as aforementioned, describes a non-rigid pendulumwith angular momentum Z whose length is proportional to √ − Z [7–9, 66]. Comparing the Eq. (12) to the Eq. (13),we note that S y and S z , being the classical counterpart of thequantum operators ˆ S y and ˆ S z , now represent the horizontaldisplacement and the angular momentum of this classical pen-dulum, while the S x ( ˆ S x ) proportions to its total energy whichis conserved during the dynamics.In this way, an one-to-one correspondence between thequantum and classical dynamics is established in which theperiodic motions for ˆ S y ( t ) and ˆ S z ( t ) are mapped to the peri-odic oscillations for this classical pendulum around its equi-librium position. Since our focus is put on the dynamics of thepopulation imbalance of the bosons, we compare the quantumevolution ˆ S z ( t ) for the case g IB = Z ( t ) in Fig. 1 (b) and no discrepancies are observed amongthem. Hence, for the case g IB =
0, we will always refer theclassical Z ( t ) dynamics as the quantum S z ( t ) evolutions. How-ever, it should also be emphasized that the agreement betweenˆ S z ( t ) and Z ( t ) takes place only for this non-interacting case.For g IB >
0, on one side, the mixture has no classical map-ping, on the other side, the quantum correlations among thebosons come into play, and, as a result, one can witness evena completely di ff erent quantum dynamics as compared to theclassical one, albeit the fact that the bare Bose-Bose interac-tion always vanishes (see below).The above classical interpretation provides us not only witha vivid picture for visualizing the quantum dynamics in a clas-sical manner, but also with the profound physical insights withrespect to its overall dynamical properties. In particular, theperiodic motions for ˆ S y ( t ) and ˆ S z ( t ) obtained from the quan-tum simulations are a direct consequence of the integrabil-ity of the classical Hamiltonian H cl . Owing to the energyconservation for the case g IB = H cl is completely inte-grable with all the corresponding classical trajectories, char-acterized by [ Z ( t ) , ϕ ( t )], being periodic in time [67]. Such an FIG. 1. (Color online) (a) Single-particle spectrum for the double-well potential, in which the gray horizontal lines denote the lowestsix energy levels and the blue (red) arrows represent possible tran-sitions that reverse (preserve) the spatial parity of the impurity. (b)Real-time dynamics for the bosonic population imbalance S z ( t ) forthe initial state | Ψ (0) (cid:105) = | φ (cid:105)⊗| π/ , π/ (cid:105) and for the case g IB = Z ( t ) dynamics starting from thephase point ( Z = , ϕ = π/
4) (blue dashed line). (c) Classical phasespace for J = . Z = , ϕ = π/
4) corresponding to the ACS | θ, ϕ (cid:105) = | π/ , π/ (cid:105) . integrability is also transparently shown in the classical phasespace [see Fig. 1 (c)]. Depending on the initial condition, twodistinguished types of motions are clearly observed: a peri-odic trajectory orbiting around the fix point either located at( Z = , ϕ =
0) or ( Z = , ϕ = π ), referred as the zero- and the π -phase mode for a 1D BJJ [5]. C. Breaking of the integrability
In contrast to the above integrable limit, the presence ofthe impurity-Bose interaction leads to the energy transport be-tween the two species and, hence, breaks the integrability forthe bosonic species. In order to elaborate on this process inmore detail, we decompose the interspecies interaction intovarious impurity-boson pair excitationsˆ H IB = ∞ (cid:88) i , j = (cid:88) α,β = L , R U i j αβ ˆ a † i ˆ a j ˆ b † α ˆ b β , (14)with U i j αβ = g IB (cid:82) dx φ i ( x ) φ j ( x ) u α ( x ) u β ( x ) and { φ i ( x ) } be-ing the single-particle basis for the DW potential. Moreover, u L / R ( x ), being the above mentioned localized Wannier-likestates, are constructed via a linear superposition of the lowesttwo eigenstates φ ( x ) and φ ( x ). Note that Eq. (14) is obtainedby means of an expansion of the field operator for the impurityˆ ψ I ( x ) = (cid:80) ∞ i = φ i ( x )ˆ a i , meanwhile, by employing the two-modeapproximation in Eq. (3) for the bosonic species. Besides, allthe eigenstate wavefunctions { φ i ( x ) } are chosen to be real due to the preserved time-reversal symmetry in the single-particleHamiltonian h σ .Next, we group di ff erent pair excitations with respect totheir bosonic indices asˆ H IB = ∞ (cid:88) i , j = U i jLR ˆ a † i ˆ a j ˆ b † L ˆ b R + ∞ (cid:88) i , j = U i jRL ˆ a † i ˆ a j ˆ b † R ˆ b L + ∞ (cid:88) i , j = U i jLL ˆ a † i ˆ a j ˆ b † L ˆ b L + ∞ (cid:88) i , j = U i jRR ˆ a † i ˆ a j ˆ b † R ˆ b R . (15)By noticing the fact that U i jLR = U i jRL , U i jLL = η U i jRR (16)with η = η = −
1) for n e , o = | i − j | being an even (odd) num-ber, together with the definitions given in Eq. (5), we finallyarrive atˆ H IB = ∞ (cid:88) i , j = U (1) i j ˆ a † i ˆ a j J x + ∞ (cid:88) | i − j | = n e U (2) i j ˆ a † i ˆ a j ˆ N B + ∞ (cid:88) | i − j | = n o U (3) i j ˆ a † i ˆ a j J z = ˆ H (1) IB + ˆ H (2) IB + ˆ H (3) IB . (17)Here U (1) i j = U i jLR = U i jRL , U (2) i j = U i jLL = U i jRR and U (3) i j = U i jLL = − U i jRR . Let us emphasize that the Eq. (16) relieson the fact that the DW potential is spatially symmetric, as aresult, all its single-particle eigenstates { φ i } respect the spatialparity symmetry.Equation (17) transparently elaborates how the interspeciesinteraction ˆ H IB breaks the integrability for the Hamiltonianˆ H B . Since both ˆ H (1) IB and ˆ H (2) IB commute with ˆ H B [c.f. Eq. (6)],it is the non-commutativity between ˆ H (3) IB and ˆ H B that resultsin the energy non-conservation for the bosonic species, andbreaks its integrability for g IB =
0. Further inspecting the ˆ H (3) IB term in more detail, we notice that it corresponds to all thedi ff erent single-particle excitations that reverse the impurity’sspatial parity [see Fig. 1 (a) for a schematic illustration]. Withthis, we conclude that those parity non-conservation transi-tions of the impurity leads to the integrability breaking for themajority bosonic species. D. Initial condition
We prepare our impurity-Bose mixture initially as | Ψ (0) (cid:105) = | φ n (cid:105) ⊗ | θ, ϕ (cid:105) , being a product state between the two species.Here | φ n (cid:105) is the n -th single-particle eigenstate for the impurityand | θ, ϕ (cid:105) denotes the ACS given by [68, 69] | θ, ϕ (cid:105) = √ N B ! (cid:20) cos( θ b † L + sin( θ e i ϕ ˆ b † R (cid:21) N B | vac (cid:105) = N B (cid:88) N BL = (cid:32) N B N BL (cid:33) / cos N BL ( θ/
2) sin N BR ( θ/ e iN BR ϕ | N BL , N BR (cid:105) , (18)which is the linear superposition of all the number states {| N BL , N BR (cid:105)} and fulfills the completeness relation( N B + (cid:90) d Ω π | θ, ϕ (cid:105)(cid:104) θ, ϕ | = d Ω = sin θ d θ d ϕ being the volume element. Physically,the ACS | θ, ϕ (cid:105) corresponds to the classical state ( Z , ϕ ) in sucha way that cos θ = ( N BL − N BR ) / N B = Z controls the initial pop-ulation di ff erence for the bosons and ϕ , possessing the samemeaning with its classical counterpart, determines the phasedi ff erence between the two wells [64]. For a given ACS | θ, ϕ (cid:105) ,the mean values for the angular-momentum operators intro-duced in Eq. (5) are [9] (cid:104) ˆ S x (cid:105) = sin θ cos ϕ, (cid:104) ˆ S y (cid:105) = sin θ sin ϕ, (cid:104) ˆ S z (cid:105) = cos θ, (20)which satisfies the normalization condition (cid:104) ˆ S x (cid:105) + (cid:104) ˆ S y (cid:105) + (cid:104) ˆ S z (cid:105) =
1. Together with the Eqs. (8) and (9), we concludethat, for the case g IB =
0, the motion of the ˆ (cid:126) S ( t ) vector start-ing from an arbitrary ACS always lies on a Bloch sphere withunit radius. Even for the case g IB >
0, where the vector ˆ (cid:126) S ( t )can jump out of the Bloch sphere significantly, the use of theACS still allows us to visualize the quantum trajectory in aclassical manner (see below), which simplifies the analysis ofthe complex quantum dynamics to a large extent, meanwhile,provides insights for the classical-quantum correspondence.Finally, let us note that the ACS has been implemented inrecent ultracold experiments in a controllable manner. Tun-ing a two-photon transition between two hyperfine states of Rb atoms, allows for preparing an ACS with arbitrary | θ, ϕ (cid:105) [70, 71].In this paper, we aim at exploring the dynamical responseof the majority bosonic species to the presence of the impurity.To this end, we quench at t = g IB = g IB > | Ψ (0) (cid:105) = | φ n (cid:105) ⊗ | θ, ϕ (cid:105) , with-out other specifications, we always choose the bosonic partbeing | θ, ϕ (cid:105) = | π/ , π/ (cid:105) . The corresponding S z ( t ) dynamicsfor this initial ACS and for the case g IB = | φ n (cid:105) , so as to explore its impact on the bosonicdynamics. III. BOSONIC ITC TRANSITIONA. Onset of quantum chaos
Let us first focus on the case where the impurity is initiallyprepared in its ground state. The many-body initial state forthe mixture is then given by | Ψ (0) (cid:105) = | φ (cid:105) ⊗ | π/ , π/ (cid:105) . Fig. 2depicts the real-time population imbalance for the bosonicspecies S z ( t ) for various fixed postquench impurity-Bose in-teraction strengths g IB = .
01 [Fig. 2 (a)], g IB = . g IB = . Z ( t )dynamics (all blue dashed lines) which, as aforementioned,equivalents to the S z ( t ) for g IB =
0. For a weak impurity-Bose interaction ( g IB = . S z ( t ) dynamics is onlyslightly perturbed by the presence of the impurity, as a result,it leads to the small deviations of the population imbalancebetween the quantum and the classical simulations [c.f. Fig. 2(a), red solid line and blue dashed line]. For a larger timescale ( t > S z ( t )is observed (result is not shown here), manifesting its nearintegrability in this weak interacting regime [9, 66]. Furtherincreasing the interaction strength, the quantum S z ( t ) evolu-tion becomes much more complicated and large discrepan-cies between S z ( t ) and Z ( t ) are observed with respect to boththe oscillation amplitude and the frequencies. For the case g IB = .
0, the quantum S z ( t ) dynamics finally becomes com-pletely irregular [c.f. Fig. 2 (c), red solid line], signifying theonset of quantum chaos for the bosonic species.In order to diagnose such an ITC transition, meanwhile, toquantify the degree of the above observed quantum chaos, weemploy the linear entanglement entropy (EE) S L = − tr ˆ ρ B (21)for the bosonic species, which represents the bipartite entropybetween the single boson and the N B − ρ B stands for the reducedone-body density matrix for the bosonic species [64, 72, 73].Before proceeding, let us point out the reason for not usingthe spectral statistics as an indicator for the quantum chaos.Similar to the situation for a single particle in a 1D harmonictrap, the single DOF of the Hamiltonian ˆ H B for a fixed parti-cle number violates the Berry-Tabor conjecture, which statesthat the energy level spacing distribution follows the univer-sal Poisson form for an integrable system [74–76]. As a re-sult, the variation of the level distribution for our mixtureupon the increase of g IB can behave largely di ff erent as com-pared to other systems [76], and hence, it is insu ffi cient tocapture the quantum chaos. Upon a spectral decompositionof the reduced density matrix ˆ ρ B , S L in Eq. (21) can be ex-pressed, with respect to the natural populations { n , n } , as S L = − (cid:80) i = n i . In this way, the linear EE alternatively mea-sures the degree of the decoherence for the bosonic species.Note that the two-mode expansion employed in the Eq. (3)renders the single-particle Hamiltonian h ( x ) being restrictedto a two-dimensional Hilbert space and thus gives rise to onlytwo natural populations obtained from the spectral decompo-sition [64]. For the case where all the bosons reside in thesame single-particle state, the bosonic species is of completecoherence, as a result, we have S L =
0. By contrast, for thecase of maximal decoherence we have n = n = /
2, whichgives rise to the upper bound for the linear EE as S L = / t < S L ( t ) evolution for astronger interaction exhibits a more rapid growth as comparedto the cases with a smaller g IB . This is particularly obviousfor the case g IB = .
0, where we observe the linear EE surgesto the value S L = .
38 at t =
10, while it only reaches to S L = .
02 ( S L = . g IB = . g IB = . ff erent transientdynamical behaviors of the linear EE fully capture the ITCtransition that is observed in the dynamics of the bosonic pop-ulation imbalance. Besides, we shall also note that the linearEE for t = ff erent ACSs, which, as aforementioned,characterize the edge of the quantum chaos. To this end,we compute the infinite-time averaged value of the linear EE(ITEE) for the initial state | Ψ (0) (cid:105) = | φ (cid:105) ⊗ | θ, ϕ (cid:105) , S L ( θ, ϕ ) = lim T →∞ T (cid:90) T dt S L ( t ) . (22)Note that, the impurity initially always occupies the groundstate | φ (cid:105) and in our practical numerical simulations the timeaverage is performed up to t = , being much larger thanany other time scales involved in the dynamics. Before pro-ceeding, let us point out the geometrical interpretation of theITEE value. To show it, we first of all rewrite the linear EE inEq. (21) for time t as [20, 21] S L ( t ) = − (cid:88) γ = x , y , z (cid:104) ˆ S γ (cid:105) ( t ) , (23)where we have used the relationˆ ρ B = + (cid:88) γ = x , y , z (cid:104) ˆ S γ (cid:105) ˆ σ γ , (24)with { ˆ σ γ } being the Pauli matrices. Since S L ( t ) is propor-tional to the instant distance of the vector ˆ (cid:126) S ( t ) to the Blochsphere, S L thus measures its averaged distance for the en-tire dynamics. From the results in the QKT systems [20, 21],we note that there exists a clear correspondence between theITEE values and the classical phase space structure. Regions FIG. 2. (Color online) Time evolution of the bosonic population im-balance S z ( t ) (red solid lines) for the initial state | Ψ (0) (cid:105) = | φ (cid:105) ⊗| π/ , π/ (cid:105) and for various fixed impurity-Bose interaction strengths,in which (a) g IB = .
01, (b) g IB = . g IB = .
0. For compar-isons, the classical Z ( t ) dynamcis is depicted as well (all blue dashedlines). of low ITEE correspond to regular trajectories, while regionsof high EE correspond to the chaotic trajectories. More-over, a sudden change of the ITEE value takes place as onecrosses the border between the integrable and the chaotic re-gion, which, as aforementioned, characterizes the edge of thequantum chaos. Fig. 3 (b) depicts the computed ITEE valuesfor various ACSs for the case g IB = .
0. Note that, we haverescaled the θ axis to cos θ since cos θ = Z [see discussions inSec. II D]. Varying the initial ACS, the ITEE value varies ac-cordingly. In particular, regions close to (cos θ = , ϕ = π ) and(cos θ = ± . , ϕ = , π ) possess significant low ITEE val-ues as compared to the other places. Such a S L ( θ, ϕ ) profilesignificantly deviates from the structure of the non-interactingclassical phase space. Instead, it bears a striking resemblanceto the phase space with an attractive Bose-Bose interactionwith the positions for those fixed points precisely match withthose low ITEE regions [c.f. Fig. 3 (c), red stars]. Hence, wenote that it indicates an e ff ective Bose-Bose attraction existingamong the bosons. In Sec. IV C, we will discuss this inducedinteraction in detail as well as its impact on the bosonic dy-namics. FIG. 3. (Color online) (a) The linear EE evolutions for the initialstate | Ψ (0) (cid:105) = | φ (cid:105) ⊗ | π/ , π/ (cid:105) and for the post-quench interactionstrengths g IB = .
01 (red solid line), g IB = . g IB = . g IB = . B. Recovery of the integrability
In this section, we investigate the scenario where the impu-rity is initially pumped into a highly excited state. The out-of-equilibrium dynamics again is triggered by a sudden quenchof the impurity-Bose interaction strength. Here, our main aimis to show that the integrability of the bosonic species is recov-ered by means of preparing the impurity in a highly excitedstate. The initial condition of the impurity, therefore, providesan additional DOF for controlling the ITC transition of themajority bosonic species. Here, we note that the employednotion of “integrability” specifically refers to how close thebosonic dynamics in the interacting cases ( g IB >
0) is to theone in the non-interacting integrable case ( g IB = ff erent from the commonly used context in which it isuniquely associated to the system’s Hamiltonian.For an illustrative purpose, we consider the impurity is ini-tially at | φ (cid:105) , being the 150-th excited state, and focus onthe case for the post-quench interaction strength g IB = . t = | Ψ (0) (cid:105) = | φ (cid:105) ⊗ | π/ , π/ (cid:105) . The corresponding quantum evolution ofthe bosonic population imbalance S z ( t ) is depicted in Fig. 4 (a)(red solid line). As compared to the classical Z ( t ) dynamics[Fig. 4 (a), blue dashed line], we find a good agreement be-tween them with negligible discrepancies. Interestingly, thesediscrepancies are even much smaller than the ones between S z ( t ) and Z ( t ) for the case g IB = .
01 [c.f. Fig. 2 (a)]. Besides,we also note that the negligible increment of the correspond- -3 FIG. 4. (Color online) Time evolution of the bosonic populationimbalance S z ( t ) for g IB = . | Ψ (0) (cid:105) = | φ (cid:105) ⊗ | π/ , π/ (cid:105) (red solid line), together with the classical Z ( t ) dy-namics (blue dashed line) which corresponds to the S z ( t ) dynamicsfor g IB =
0. (b) The evolution of the linear EE for the correspondingcase. ing linear EE in the course of the dynamics alternatively sig-nifies the recovery of the integrability for the bosonic species[c.f. Fig. 4 (b)].
IV. SPECTRAL ANALYSIS AND INDUCEDINTERACTION
In order to shed light on the physics for the above-analyzedbosonic dynamics, hereafter, we perform a detailed spectralanalysis for the mixture with respect to both the energy spec-trum and the eigenstates via a numerically exact diagonaliza-tion (ED). In particular, we would like to unveil the physicalorigin for the observed ITC transition for the bosonic speciesmanifested by the corresponding dynamics of the populationimbalance. Moreover, we will discuss the presented Bose-Bose attraction induced by the impurity as well as its impacton the bosonic dynamics.
A. Spectral structure
Let us begin with the case for g IB =
0. In the absenceof the interspecies interaction, the two species are completelydecoupled. As a result, the eigenenergy of the mixture is triv-ially given by E = (cid:15) k + (cid:15) Bl with (cid:15) k and (cid:15) Bl being the k -th and l -th eigenvalue for the subsystem Hamiltonians ˆ H I and ˆ H B , re-spectively. Owing to the neglected Bose-Bose interaction, themany-body spectrum for ˆ H B is always equidistant with the en-ergy di ff erence 2 J between the two successive levels, whichaccounts for the harmonic oscillation of the S z ( t ) dynamicsfor the case g IB = ff erence between two suc-cessive eigenstates, the single-particle spectrum is inhomoge-neous in which the high-energy part is much more sparse ascompared to the low-energy one [c.f. Fig. 1 (a)]. An im-portant consequence for such a spectral structure on the mix-ture’s many-body spectrum is the following. For δ i > ∆ B ,with δ i = (cid:15) i + − (cid:15) i being the energy di ff erence between the i -th and the ( i + ∆ B repre-senting the width of the spectrum for the Hamiltonian ˆ H B , aband-like structure is naturally formed in the high-energy partof the many-body spectrum with the band gap being δ i − ∆ B ,meanwhile, the energy levels within each band are equidistant.This simple picture, however, ceases to be valid upon thevariation of the impurity-Bose interaction. Indeed, the inclu-sion of the interspecies interaction introduces additional cou-pling between the two subsystems and, as a result, our spectralanalysis needs to be performed with respect to the completemixture. Figure 5 showcases the many-body spectrum as afunction of the interspecies interaction strength g IB . Owing tothe preserved spatial parity symmetry in the Hamiltonian ˆ H ,we present here only half of the spectrum which correspondsto the even parity eigenstates. With the increase of g IB , thelow-energy spectrum shows many avoided-crossings amongthe energy levels, which is in sharp contrast to the high-energyspectrum where only a linear growth of their values is ob-served [c.f. Figs. 5 (a) and (c)]. Moreover, for the high-energyspectrum, features like the band-like structure as well as theequidistant energy levels within each band that are present inthe non-interacting limit are retained in the interacting casesas well.The above two distinguished spectral behaviors can roughlybe understood via the structure of the impurity’s single-particle spectrum [c.f. Fig. 1 (a)]. Owing to the large en-ergy separations among those highly excited states, the tran-sitions for the impurity among those states are significantlyprohibited. From a many-body perspective, the resultinghigh-energy e ff ective Hamiltonian of the mixture reads ˆ H (cid:48) = ˆ H I + ˆ H B + ˆ H (cid:48) IB , withˆ H I = (cid:88) i (cid:29) (cid:15) i ˆ a † i ˆ a i , ˆ H B = − J ˆ J x , ˆ H (cid:48) IB ≈ (cid:88) i (cid:29) U (1) i ˆ J x + U (2) i ˆ N B ≈ (cid:88) i (cid:29) U (2) i ˆ N B . (25)Here U (1) i = U iiLR = U iiRL , U (2) i = U iiLL = U iiRR and we noticethat U (1) i = g IB (cid:82) dx φ i ( x ) φ i ( x ) u L ( x ) u R ( x ) ≈
0, due to the neg-ligible spatial overlap between the two localized states u L ( x )and u R ( x ). Before proceeding, we note the validity conditionfor the above high-energy e ff ective Hamiltonian as: δ i (cid:29) (cid:15) IB and δ i (cid:29) ∆ B with (cid:15) IB being the interspecies interaction en-ergy per particle. Eq. (25) explains the observed high-energyspectral behaviors as follows: since the interspecies interac-tion ˆ H (cid:48) IB now becomes the “zero-point” energy of the mixture,the increment of the g IB thus only raises the energy level forthose highly excited states. As a result, the band-like struc-ture as well as equidistant nature that are formed in the non-interacting case are naturally preserved. In contrast, the densely distributed low-energy (single-particle) spectrum of the impurity facilitates the transitionsamong di ff erent (low-lying) many-body eigenstates causedby the interspecies interaction ˆ H IB [c.f. Eq. (17)]. With in-creasing g IB , this results in the above observed avoided level-crossings among the low-energy many-body spectrum [14]. B. Eigenstate delocalization
The avoided level-crossings in the low-energy spectrum im-pact the characteristics of the corresponding eigenstates aswell. Specifically, it results in a significant delocalization forthose low-lying eigenvectors with respect to an integrable ba-sis (see below), which, in turn, accounts for the chaotic natureof the bosonic non-equilibrium dynamics. To demonstratethis, we introduce the Shannon entropy S Sj = − (cid:88) k c kj ln c kj (26)for a many-body eigenstate | Φ j (cid:105) of the mixture as a measureof the delocalization [77, 78]. Here c kj = |(cid:104) ψ k | Φ j (cid:105)| with {| ψ k (cid:105)} being the eigenbasis for the Hamiltonian ˆ H B that are used asthe “integrable basis”. The Shanon entropy thereby measuresthe number of this integrable basis vectors that contribute toeach eigenstate. As a result, the lower the Shanon entropyvalue is the closer this eigenstate | Φ j (cid:105) is to a non-interactingeigenvector. From the random matrix theory (RMT), for achaotic system described by the gaussian orthogonal ensemble(GOE), the amplitudes c kj are independent random variablesand all eigenstates are completely delocalized [14]. How-ever, due to the spectral fluctuations the weights { c kj } fluctuatearound 1 / D , yielding the averaged value S GOE = ln (0 . D )[77, 78]. Here, we refer to D = N B + ff erent from thesingle-species cases [77, 78].Figure 6 (a) presents the Shannon entropy of the many-bodyeigenstates as a function of their quantum numbers j (sortedin the ascending order with respect to the energy) for the case g IB = .
0. The distinguished localization nature between thelow-lying and the highly excited eigenvectors are clearly ex-hibited. While those low-energy eigenvectors are delocalizedwith the corresponding Shannon entropy values close to theresult from the GOE S GOE = . j , a de-crease of the S Sj value is clearly observed, indicating thosehigh-energy eigenvectors are significantly localized. Thus,we may further conjecture that S Sj → j → ∞ . Physi-cally, the avoided level-crossings in the low-energy spectrumresults in a strong mixing of di ff erent eigenstates with respectto their physical properties. In this way, an eigenstate fromthe non-interacting basis can be largely delocalized after ex-periencing a serious of avoided level-crossings [14]. On theother hand, the localized nature for those high-lying excitedstates can also be readily seen from the e ff ective Hamiltonianin Eq. (25). Since here ˆ H (cid:48) IB corresponds to the “zero-point ”energy of the mixture, it is not surprising that the interactingbasis (eigenstates of the mixture for g IB >
0) is similar to thenon-interacting integrable basis.Before proceeding, let us highlight that the degree of the lo-calization for an eigenstate | Φ j (cid:105) also reflects the degree of theencoded entanglement between the impurity and the majoritybosons. To see this, we employ the von Neumann entropy foran eigenstate | Φ j (cid:105) [41, 79], S Vj = − tr( ˆ ρ j ln ˆ ρ j ) (27)with ˆ ρ j = | Φ j (cid:105)(cid:104) Φ j | being the corresponding density matrix.For the case where the two species are non-entangled, theeigenstate | Φ j (cid:105) is simply of a product form with respect to thewavefunctions of the two species. Correspondingly, it givesrise to the von Neumann entropy S Vj =
0. By contrast, anyexisting entanglement between the two species will lead toan increase of the von Neumann entropy, therefore, one cananticipate large S Vj values for those highly entangled eigen-states. The corresponding von Neumann entropies for variouseigenstates for the case g IB = . S Vj and S Sj distributions are transparently observed, manifest-ing the existence of the correspondence between a delocalized(localized) eigenstate to a large (small) von Neumann entropyvalue. Based on this knowledge, we refer to the above eigen-state delocalization as the entanglement induced delocaliza-tion.Finally, let us discuss the impact of the eigenstate delocal-ization to the bosonic non-equilibrium dynamics. For the case | Ψ (0) (cid:105) = | φ (cid:105) ⊗ | π/ , π/ (cid:105) , the initial state is mainly a linearsuperposition of those low-lying eigenvectors for both g IB = g IB = . ffi cients { A j } for g IB = . { A j } ) for g IB = .
0, reflecting the fact that much more eigen-states are involved in the bosonic dynamics. Since the energylevels for the interacting case are no longer equidistant, it thusgives rise to the completely irregular behaviors for the above S z ( t ) dynamics [c.f. Fig. 2 (c)]. In contrast, those highly ex-cited states in the interacting basis preserve the main featuresof the non-interacting basis, leaving a similar distribution ofthe corresponding expansion coe ffi cients [c.f. Fig. 6 (c), theright part]. Together with the equidistant nature for thosehigh-lying energy levels, it thereby accounts for the integrable S z ( t ) motion for the initial state | Ψ (0) (cid:105) = | φ (cid:105)⊗| π/ , π/ (cid:105) andfor the case g IB = . C. Induced Bose-Bose attraction
The presence of the impurity not only brings the bosonicspecies into the chaotic regime, yielding an irregular behaviorfor the corresponding S z ( t ) motion, but also fundamentallychanges its dynamical properties. As we will show below,the impurity e ff ectively induces an attractive Bose-Bose inter-action, which, in turn, leads to a completely di ff erent quan-tum trajectory as compared to the integrable case. To showit, we employ the time-averaged Husimi distribution (TAHD) FIG. 5. (Color online) Energy spectrum of the mixture as a functionof impurity-Bose interaction strength g IB . (a) High-energy part of thespectrum, (b) A zoom-in view of the high-energy spectrum, (c) Low-energy part of the spectrum, (d) A zoom-in view of the low-energyspectrum. [20, 64, 80] Q H ( θ, ϕ ) = lim T →∞ T (cid:90) T Q H ( θ, ϕ, t ) dt , (28)with Q H ( θ, ϕ, t ) = N B + π (cid:104) θ, ϕ | ˆ ρ B ( t ) | θ, ϕ (cid:105) , (29)and ˆ ρ B ( t ) being the reduced density matrix for the bosonicspecies after tracing out the impurity. According to theEq. (19), Q H ( θ, ϕ, t ) satisfies the normalization condition (cid:82) Q H ( θ, ϕ, t ) d Ω =
1. Physically, the TAHD represents theprobability for the bosons to locate at a specific ACS | θ, ϕ (cid:105) averaged over the entire dynamics, which, with respect to itsphysical meaning, resembles to the probability density func-tion (PDF) for a classical trajectory. In this sense, we notethat the TAHD represents a quantum trajectory in an averagedmanner.The computed TAHD for the initial state | Ψ (0) (cid:105) = | φ (cid:105) ⊗| π/ , π/ (cid:105) and for the case g IB = H cl and starting from the phase point ( Z = , ϕ = π/ Q H ( θ, ϕ ) regions precisely matching thepositions for this classical trajectory, which additionally man-ifests the agreement between the quantum S z ( t ) and classical0 FIG. 6. (Color online) (a) Shannon entropy S Sj for the many-bodyeigenstates as a function of quantum number j for the case g IB = .
0. The red dashed line denotes the Shannon entropy from the GOE S GOE = . S Vj for the eigenstates forthe case g IB = .
0. (c) Expansion coe ffi cients A j = |(cid:104) Ψ (0) | Φ j (cid:105)| withrespect to eigenstates for initial states | Ψ (0) (cid:105) = | φ (cid:105) ⊗ | π/ , π/ (cid:105) (leftpart) and | Ψ (0) (cid:105) = | φ (cid:105) ⊗ | π/ , π/ (cid:105) (right part) and for the cases g IB = . A j ) and g IB = . A j ). Z ( t ) dynamics for the case g IB = g IB = .
0, however, deviates from the non-interacting casesignificantly and bears a striking resemblance to the classi-cal trajectory corresponding to the BH Hamiltonian in Eq. (4)with an on-site attraction [c.f. Figs. 7 (b) and 3 (c)]. In thissense, we conjecture an e ff ective Bose-Bose attraction is in-duced by the impurity in the dynamics which, in turn, altersthe corresponding quantum trajectory.This expectation is indeed confirmed by analyzing the pair-correlation function [40, 41, 65] g ( α, β ) = ρ B ( α, β ) ρ B ( α ) ρ B ( β ) , (30)for the bosons, with ρ B ( α, β ) and ρ B ( α ) being the reduced two-and one-body density for the bosonic species and α, β = L , R .Physically, ρ B ( L , R ) denotes a measure for the joint proba-bility of finding one boson at the left well while the secondis at the right well. Through the division by the one-bodydensities, the g function excludes the impact of the inho-mogeneous density distribution and thereby directly revealsthe spatial two-particle correlations induced by the interaction[40, 41]. Based on this knowledge, let us first elaborate the g function for the non-interacting case, which corresponds tothe TAHD depicted in Fig. 7 (a). Since there is no interactionamong the particles, all the bosons thus can independently hopbetween the two wells, hence, it always results in g o = g d = g o = g ( α, α ) [ g d = g ( α, β (cid:44) α )] being the two-particlecorrelations within the same well (between the two wells). Bycontrast, the presence of the impurity-Bose interaction largelychanges the above g profile. As shown in Fig.7 (c), the g function quickly deviates from the initial values g o = g d = g o > g d < t < g o = .
28 and g d = .
72, respectively.Physically, such an evolution of the g function indicates thatthe bosons are in favor of bunching together with a collectivetunneling between the wells in the dynamics, which evidentlymanifests the existence of the Bose-Bose attraction inducedby the impurity-Bose repulsion. V. CONCLUSIONS AND OUTLOOK
We have demonstrated that a non-interacting ultracoldmany-body bosonic ensemble confined in a 1D DW poten-tial can exhibit a chaotic nature due to the presence of a singleimpurity. We trigger the non-equilibrium dynamics by meansof a quench of the impurity-Bose interaction and monitor theevolution of the population imbalance for the bosons betweenthe two wells. While the increase of the post-quench inter-action strength always facilitates the chaotic motion for thebosonic population imbalance, it becomes regular again forthe cases where the impurity is initially prepared in a highlyexcited state. Employing the linear entanglement entropy, itnot only enables us to characterize such an ITC transition butalso implies the existence of an e ff ective Bose-Bose attrac-tion in the dynamics induced by the impurity. In order toelucidate the physical origin for the above observed ITC tran-sition, we perform a detailed spectral analysis for the mix-ture with respect to both the energy spectrum as well as theeigenstates. In particular, two distinguished spectral behav-iors upon a variation of the interspecies interaction strengthare observed: while the avoided level-crossings take place inthe low-energy spectrum, the energy levels in the high-energyspectrum possess the main features of the integrable limit.Consequently, it results in a significant delocalization for thelow-lying eigenvectors which, in turn, accounts for the chaoticnature of the bosonic dynamics. In contrast, those highly ex-cited states bear a high resemblance to the non-interacting in-tegrable basis, rendering the recovery of the integrability forthe bosonic species. Finally, we discuss the induced Bose-Bose attraction as well as its impact on the bosonic dynamics.Possible future investigations include the impact on thebosonic dynamics with the inclusion of several additional im-purities and / or the bare Bose-Bose repulsion. Since for the lat-ter there exists a competition between the bare Bose-Bose re-pulsion and the induced attractive interaction, this may signif-icantly a ff ect the bosonic ITC transition. Another interestingperspective is the study of the chaotic dynamics for an atomicmixture consisting of atomic species with di ff erent masses.The impact of the higher bands of the DW potential, beyond1 FIG. 7. (Color online) Time-averaged Husimi distribution for the initial state | Ψ (0) (cid:105) = | φ (cid:105) ⊗ | π/ , π/ (cid:105) and for (a) g IB = . g IB = . Z = , ϕ = π/ g o ( t ) (blue solid line) and g d ( t ) (red solid line) for the case examined in (b). the two-site BH description for the bosonic species, is also aninteresting perspective. ACKNOWLEDGMENTS
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