Nonequilibrium quantum dynamics of partial symmetry breaking for ultracold bosons in an optical lattice ring trap

A vortex in a Bose-Einstein condensate on a ring undergoes quantum dynamics in response to a quantum quench in terms of partial symmetry breaking from a uniform lattice to a biperiodic one. Neither the current, a macroscopic measure, nor fidelity, a microscopic measure, exhibit critical behavior. Instead, the symmetry memory succeeds in identifying the point at which the system begins to forget its initial symmetry state. We further identify a symmetry energy difference in the low lying excited states which trends with the symmetry memory.

Nonequilibrium quantum dynamics is a rapidly growing field of study in part due to the emergence of hundreds of quantum simulator platforms build on multiple architectures, presenting enormous flexibility to explore new problems with detailed control of lattice structure, interaction strength, and bosonic or fermionic statistics [1][2][3][4]. For example, global quantum quench dynamics have led to a deep understanding of the Kibble-Zurek mechanism relating non-equilibrium dynamics to critical exponents in quantum phase transitions [5]. Likewise, the study of local quenches or perturbations have taught us the role of short and long-range interactions in establishing a quantum speed limit on the propagation of correlations [6,7]. Use of a biperiodic optical lattice quenched to a uniform lattice has resulted in the first experimental demonstration of many-body localization [8,9]. Quantum simulators offer unusually isolated systems and long quantum coherence times, allowing careful exploration of the memory of initial conditions, and have resulted in e.g. approach to a new kind of "thermal" equilibrium under the eigenstate thermalization hypothesis, called the generalized Gibbs ensemble [10]. By taking the opposite route from the many-body localization experiment, i.e., quenching from a uniform lattice to a bi-periodic one and thereby partially breaking the discrete rotational symmetry of a ring lattice, we find a completely different kind of longlived robust dynamics in which newly identified quantum measures, the symmetry gap and the symmetry memory, reveal that the system only "remembers" its initial symmetry state below a critical partial symmetry breaking strength.
Entanglement growth under a quantum quench is too rapid to capture long-time dynamics with tensor network methods [11,12], as is necessary for large systems; therefore we employ exact diagonalization in small ring systems of N = 2 to 10 bosons on L = 6 to 10 sites [13], as well as perturbation theory to corroborate results and extrapolate trends in large interaction. Partial symmetry breaking in the 6-site case in particular corresponds to breaking the A-B sublattice symmetry in graphene, creating a gap at the Dirac point [14][15][16]. We focus on the bosonic cold-atom-based quantum simulator architectures where much of the groundbreaking work on quantum dynamics has been performed and is frequently modeled with the Bose-Hubbard Hamiltonian (BHH) [17]. In laser trapping of Bose-Einstein condensates, the discrete rotational symmetry ring trap [18] is an ideal potential to investigate dynamical symmetry breaking produced by a fast potential quench. Such a trap, shown in Fig. 1, can be achieved by the interference of XX and Y Y Laguerre-Gaussian beams with the introduction of a quench to change the trap depth of even or odd sites thereafter. In addition "painted" potentials with ultrafast lasers can achieve the same end [19][20][21]. We find that in order to correctly characterize the quantum dynamics of rotational states, or vortices in such potentials, it is necessary to go beyond fidelity, current, etc. and introduce the symmetry gap, a measure drawn from a cluster of low-lying excited states, and the symmetry memory, based on a time average over projections into rotational quantum numbers. Such projections correspond to measurement of the winding number in the discretized ring system, exactly as occurs in BEC experiments. Although rotational measurements have typically been performed in the past on large continuous systems [22], with the advent of ultracold microscopy [23] and other precision techniques together with precise control over small systems [10], rotational projections present an accessible avenue of exploration for upcoming quantum dynamics experiments. For example, our 6 site system may be taken as a study of the subsystem in a honeycomb lattice [24], with an experiment performing an average over many such subsystems to do "one-shot" emulation of quantum averages. The partial symmetry breaking Hamiltonian (PSBH), a rescaling of the usual BHH incorporating a two-period potential, takes the form where U determines the on-site two-particle interaction; i, j denotes summation over the nearest neighbors; where the plus (minus) sign is taken for site i even (odd). The hopping energy J e (J o ) encapsulates the biperiodic lattice through the usual overlap integral [25]. We scale our study to the average hopping energyJ ≡ (J e + J o )/2, so that energies are in units ofJ and times in units of /J. Finally, we further define symmetry breaking strength ε = |ε ij |. As we will show, there exists a critical ε c determining the vortex dynamics on the ring. The case of ε = 0 restores the Lfold discrete rotationally symmetric lattice and the usual BHH whereas the introduction of J e , J o enforces L/2-fold discrete rotational symmetry (we consider only even L for simplicity).
Considering rotational eigenstates on the ring, the unitary n-fold discrete rotational symmetry operator satisfieŝ C n |m n = e i2πmn/n |m n , where m n is the corresponding rotational quantum number, or winding number. Consider the honeycomb case L = 6. Then the quench procedure begins with n = 6, ε = 0 for t < 0, and we take n = 3, ε = 0, for t ≥ 0. The PSBH has time reversal invariance symmetry. Thus the eigenstates characterized by ±m n are degenerate, as shown in Fig. 2, and the energy of the system depends only on |m n |. For time t > 0, i.e., after the quench, the 6-fold symmetry is partially reduced to 3-fold, and the energy eigenstates have a well-defined m 3 discrete rotational number. Such partial symmetry reduction via quench has been shown in group theory [26] to satisfy m − m n = p, where p is an integer; here m 6 − m 3 = 3p. Each m 3 corresponds to a pair of m 6 with distinct 2pπ phase differences under the action of the operatorĈ 6 .
In the limit in which the PSBH reduces to the usual BHH, there are two well known distinct quantum phases, a Mott insulator and a superfluid; mesoscopic analogs of these phases exist in both canonical and grand canonical ensembles [27], where "quantum phase" is determined by a sharp change in a quantum observable, rather than a singularity, as commonly observed in quantum simulator experiments [2,[6][7][8]. A Berzinskii-Kosterlitz-Thouless (BKT) transition occurs for integer filling and a mean field U (1) transition otherwise. At unit filling the BKT transitions occurs for (U/J) crit 1/0.305 3.28 [28,29]; for our ring system of 6 to 10 sites, the mesoscopic analog of the critical point is between (U/J) crit 5 to 10, depending on the choice of quantum measure used to determine the extremal behavior signifying the quantum phase transition [27]. We refer to regimes below (above) the effective critical point as weakly (strongly) interacting. The eigen-energy spectra of the BHH determined by exact diagonalization are shown in Fig. 2, for unit filling and the strongly-interacting case, with N = 6 particles on 6 sites. The eigenstates occur in clusters, in which several states are nearly degenerate; we refer to this as energy clustering. The energy difference between the ground state and the first cluster corresponds to the Mott gap, separating the ground state from particle-hole excitations [30]. We can think of the partial symmetry breaking from 6-fold to 3-fold discrete rotational symmetry as mixing a definedĈ 6 eigenstate |m 6 in the L = 6 BHH with all states having the same value of m 3 underĈ 3 acting on the 6 site ring, including compatible pairs of m 6 . Thus, for example, under time evolution, for ε = 0 any m 3 = +1 eigenstate ofĈ 3 will evolve as a linear combination of all the m 6 = +1, −2 states. To capture this kind of evolution in partial symmetry breaking, it is expedient to trace its origin to the energy eigenspectrum in terms of the symmetry gap, which is the energy difference between the nearest compatible pair of m 6 states. A similar expression is obtained for larger system size. Especially as system size grows, there are more such symmetry gaps in the spectrum: we find the lowest energy symmetry gap suffices to characterize the discrete rotational dynamics, similar in spirit to the use of Yrast states for the continuous rotational symmetry case [31]. Note that the definition holds independent of filling factor and interaction regime. We take our initial state to be a vortex of winding number +1: for L = 6 this is m 6 = +1, also the lower energy state in the symmetry gap. The time-dependent quantum average of the rotation operator is If ψ(t) is an eigenstate ofĈ 6 , then η(t) = 1 and m 6 (t) is a time-independent constant. Once we quench to ε = 0, ψ(t) retains its initial m 3 quantum number in terms of a superposition of m 6 = +1, −2 states. At each time step t, when the projection of one of these two portions exceeds 50%, then the total six-fold rotation number m 6 (t) of ψ(t) takes its m 6 value; m 6 (t) is consequently time-dependent. Critical symmetry breaking strength εc (right axis, red curves) determined from the symmetry memory (see Fig. 3(f)) trends with the symmetry gap ∆s/J (left axis, black curves) determined from the spectrum (see Fig. 2) for strong interactions U/J. Shown are N = 2, 3, 4, 5 particles on 6 sites. Perturbation theory (blue dashed curves) helps explain the two classes of asymptotic behavior in large U/J for different N , see text. The green regions indicate convergence error based on quadrupling total simulation time τ and higher resolution of εc.
It is natural then to define the symmetry memory As shown in Fig. 3(f), critical behavior appears in this symmetry-based quantum measure, in strong contrast to the more typical microscopic measure, the time-averaged fidelityf /f (ε = 0) ≡ τ −1 τ 0 dt| ψ(t)| |ψ(0) | in Fig. 3(b), or the macroscopic measure, the time-averaged current Fig. 3(d) (where we've taken the lattice constant and equal to unity.) We take the total simulation time τ as hundreds of the typical oscillation periods shown in Fig. 3(a),(c),(e): this corresponds physically to hundreds of circuits of atoms around the ring. Typical hopping frequencies in BECs are kHz; thus τ is on the order of tens of milliseconds. The time evolution of m 6 (t) in Fig. 3(e) is distinct in the weakly and strongly-interacting cases at ε = 0.5. In the strongly-interacting case, m 6 (t) keeps its initial value of m 6 (t = 0) = +1 for all times and thus M s = 1; however, in the weakly-interacting case, it occasionally loses the memory of its inital state and jumps to m 6 = −2, generating M s < 1. This difference exhibited in Fig. 3(f) suggests that for each value of U/J a critical symmetry breaking strength ε c presents a cusp beyond which the symmetry memory M s dips below unity, that is, beyond ε c the system begins to lose the ability to retain its initial symmetry features. Figure 4 shows that indeed critical behavior is exhibited for all interaction strengths. What is the origin of this effect? As shown in Fig. 4, the symmetry gap from Fig. 2 trends overall with ε c . Thus the low lying excited states capture much of the dynamics of vortices under a potential quench.
In these calculations there are two main practical issues affecting the numerics. First, the symmetry gap can move within the lowest energy cluster in Fig. 2 for smaller interaction strengths, and this effect must be carefully accounted for, as we have done to obtain the curves in Fig. 4. Second, we are restricted to exact diagonalization. We clarify that not only is there a failure of approximation methods based on tensor networks at long times under quantum quenches [6,12], a well known problem making a strong case for nonequilibrium dynamics quantum simulator experiments, but also our potential quench in particular causes large fluctuation in on-site particle number. Thus the usual truncation of the local Hilbert space to an occupation number of n max i used in tensor network simulations [32,33] is not effective, and we find Fock states with up to n max i = N have significant weight in the dynamics. However, by calculating the complete eigenspectrum in exact diagonalization we do find the lowest energy cluster dominates the dynamics, a significant advantage for perturbation theory, as we now describe.
We observe for both L = 6 and for larger systems in Fig. 5 two asymptotic trends for large interaction strength U/J: ε c either (i) ascends to a non-zero constant or (ii) decreases toward zero, depending on N . A brief study of second order degenerate perturbation theory on 6 sites reveals these two cases, taking the hopping term as a perturbation of the PSBH before the quench, and focusing on the symmetry gap ∆ s in Eq. (2). Under perturbation inJ, ∆ s a(N )U + b(N )J + c(N )J 2 /U to orderJ 2 . Rescaling toJ to match the units used throughout this Letter, ∆ s /J a(N )U/J + b(N ) + c(N )J/U . Because the upper and lower states in ∆ s are degenerate in the same energy cluster (see Fig. 2), the zeroth order term a(N ) = 0 for all N . We find b(N ) = 2, 0, 1, 0, 2, 1.58 for N = 1 to 6. Nonzero b(N ) matches case (i), as seen in Fig. 4(b) and (d). In contrast, for case (ii) in Fig. 4(a) and (c) it is necessary to calculate to second order. We find the dominant contributions are obtained from the lowest energy clusters seen in Fig. 2. Thus energy clustering overall determines the trend in the dynamics. We find c(N ) = 0, 8/3, −2, 8/3, 0, 3.15 for N = 1 to 6, correctly reproducing and verifying the numerically determined symmetry gap from exact diagonalization in strongly-interacting regions as seen in Fig. 4.
A key question remains: how can our study of such small systems be extrapolated to the much larger system sizes present in quantum simulator experiments? First, Fig. 5 indicates that larger system sizes also display symmetry breaking with the same two asymptotic trends in interaction strength as the 6-site case; thus we expect that large ring optical lattices will lose memory of an initial vortex state for a critical symmetry breaking strength. Second, taking our 6-site case as a plaquette in a honeycomb lattice (see Fig. 1) we can study the effect of vorticity distribu- Although exact diagonalization provides only limited access to larger systems, these results indicate critical behavior is pervasive and will be present even in the thermodynamic limit. The same two classes of asymptotic trends for strong interactions are seen as in Fig. 4. The green error is determined in the same way as in Fig. 4.
tion in a lattice system and how it depends on A-B sublattice symmetry breaking; such biperiodic lattices were used previously for C-NOT gates in a square lattice [34]. This kind of larger scale experiment offers the advantage of performing the quantum average in a single shot since many plaquettes can display oscillation (or lack thereof) to different local winding number. Using the standard tool of interference with a non-rotating condensate, such a distribution should appear as an array of bifurcations in the interference pattern [22]. Quantum microscopy may also prove useful for close observation [23] in order to determine winding number on hexagonal plaquettes in the honeycomb lattice [24]. This kind of study presents a totally new kind of quantum dynamics experiment requiring quantum supremacy [35], that is, inaccessible to quantum simulations on classical computers but perfectly accessible to quantum simulator experiments.
In conclusion, we studied the nonequilibrium dynamics of bosons in a discrete optical ring trap or honeycomb lattice plaquette with an initial vortex state. After quenching to a partial-symmetry broken lattice, we found critical behavior in the ensuing dynamics as determined by projection onto different rotational quantum numbers in the discretized system. Up to a critical value of symmetry breaking, an initial winding number persists to long times; beyond this point, memory of the initial state is periodically lost and overall gradually decreases. The symmetry memory, or time average over such projections, was found to trend with the symmetry gap, identified in the lowest lying cluster of excited states in the energy eigenspectrum. Our exact diagonalization studies lay the groundwork for larger scale exploration of novel symmetry-based quantum dynamics in quantum simulator experiments.