Using a recently developed extension of the time-dependent variational principle for matrix product states, we evaluate the dynamics of 2D power-law interacting XXZ models, implementable in a variety of state-of-the-art experimental platforms. We compute the spin squeezing as a measure of correlations in the system, and compare to semiclassical phase-space calculations utilizing the discrete truncated Wigner approximation (DTWA). We find the latter efficiently and accurately captures the scaling of entanglement with system size in these systems, despite the comparatively resource-intensive tensor network representation of the dynamics. We also compare the steady-state behavior of DTWA to thermal ensemble calculations with tensor networks. Our results open a way to benchmark dynamical calculations for two-dimensional quantum systems, and allow us to rigorously validate recent predictions for the generation of scalable entangled resources for metrology in these systems.
Using a recently developed extension of the time-dependent variational principle for matrix product states, we evaluate the dynamics of 2D power-law interacting XXZ models, implementable in a variety of state-of-the-art experimental platforms. We compute the spin squeezing as a measure of correlations in the system, and compare to semiclassical phase-space calculations utilizing the discrete truncated Wigner approximation (DTWA). We find the latter efficiently and accurately captures the scaling of entanglement with system size in these systems, despite the comparatively resource-intensive tensor network representation of the dynamics. We also compare the steady-state behavior of DTWA to thermal ensemble calculations with tensor networks. Our results open a way to benchmark dynamical calculations for two-dimensional quantum systems, and allow us to rigorously validate recent predictions for the generation of scalable entangled resources for metrology in these systems.
Interacting quantum spin models are remarkably useful for describing different types of physical, chemical, and biological systems.Significant understanding of their equilibrium properties has been achieved to date, especially for the case of spin models with short-range couplings.However, progress toward the development of a comparable understanding in long-range interacting models, in particular out-of-equilibrium, remains limited.In a recent work, we proposed a semiclassical numerical method to study spin models, the discrete truncated Wigner approximation (DTWA), and demonstrated its capability to correctly capture the dynamics of one-and two-point correlations in one-dimensional (1D) systems.Here we go one step forward and use the DTWA method to study the dynamics of correlations in two-dimensional (2D) systems with many spins and different types of long-range couplings, in regimes where other numerical methods are generally unreliable.We compute spatial and time-dependent correlations for spin-couplings that decay with distance as a power-law and determine the velocity at which correlations propagate through the system.Sharp changes in the behavior of those velocities are found as a function of the power-law decay exponent.Our predictions are relevant for a broad range of systems including solid state materials, atom-photon systems and ultracold gases of polar molecules, trapped ions, Rydberg, and magnetic atoms.We validate the DTWA predictions for small 2D systems and 1D systems, but ultimately, in the spirt of quantum simulation, experiments will be needed to confirm our predictions for large 2D systems.
A phase-space method, suitable to study the many-body spin dynamics in generic spin lattice models is introduced, discussed, and tested. This technique, which we refer to as the dTWA, is based on the discrete sampling of the Wigner function instead of the continuous sampling in traditional Truncated Wigner Approximation (TWA) methods. The power of the method is demonstrated by comparisons with analytical and numerically exact calculations, which show a significant improvement compared to the TWA, e.g. by capturing revivals of spin-observables. The dTWA is demonstrated to not only correctly reproduce dynamics of one-particle observables but also two-point correlations and spin squeezing, a property relying on entanglement. This method thus opens the possibility to study the quantum dynamics in long-range spin models, accessible to recent state-of-the-art experiments, in regimes where other numerical techniques are inapplicable.
Nonequilibrium dynamics of highly controlled quantum systems is a challenging issue in statistical physics and quantum many-body physics, relevant to recent experimental developments of analog and digital quantum simulations. In this work, we develop a discrete phase-space approach for general SU(N) spin systems that utilizes cluster mean-field equations, which capture nontrivial quantum correlations inside each cluster, beyond the capability of the standard discrete truncated Wigner approximation for individual classical spins. Our formalism, based on a cluster phase-point operator, enables efficient numerical samplings of cluster phase-space variables, where the total number of noise variables for a direct product state is independent of the specific way in which the entire system is divided into multiple equally sized finite clusters. We numerically demonstrate that the cluster discrete truncated Wigner approximation (C-dTWA) method can reproduce key results in a recent experiment on correlation propagation dynamics in a two-dimensional Bose-Hubbard system. We further compare the results of C-dTWA for clusters of $2\ifmmode\times\else\texttimes\fi{}2$ sites with those from a two-dimensional tensor network method and discuss that both approaches agree very well in the short-time region, where the energy conservation is well maintained in the tensor network simulations. Since we formulate the C-dTWA method in a general form, it has the potential for application to various dynamical problems in isolated and open quantum systems, even in higher dimensions.
We review the truncated Wigner approximation (TWA) for spins as a computationally inexpensive numerical approximation method to describe interacting and / or dissipative many-body spin systems. Using the Wigner-Moyal mapping from Hilbert space to a suitable phase space, the many-body density matrix is represented by a c-number distribution, the Wigner function. The gauge freedom in continuous phase space can be exploited to find positive Wigner functions for a large class of spin states, including entangled ones. Employing different sets of correspondence rules, we derive equations of motion for the Wigner function, which, applying controlled approximations, can be mapped to stochastic differential equations. This allows a computationally inexpensive simulation of expectation values. Using a phase-space analog of the quantum regression theorem also multi-time correlations and spectra can be obtained. To illustrate the potential of the method, we benchmark the TWA for spins with some exactly solvable problems of interacting, dissipative spin systems, and then discuss its application to collective processes, such as the superradiant emission of light. Extending the TWA to imaginary time furthermore provides a tool to approximately calculate thermal and ground states of spin Hamiltonians. Finally, we show that the TWA stochastic equations can equivalently be derived within a path-integral approach, provided that the operator products in the dissipator are rigorously mapped onto the curved phase space.
Abstract Numerical techniques to efficiently model out-of-equilibrium dynamics in interacting quantum many-body systems are key for advancing our capability to harness and understand complex quantum matter. Here we propose a new numerical approach which we refer to as generalized discrete truncated Wigner approximation (GDTWA). It is based on a discrete semi-classical phase space sampling and allows to investigate quantum dynamics in lattice spin systems with arbitrary S ≥ 1/2. We show that the GDTWA can accurately simulate dynamics of large ensembles in arbitrary dimensions. We apply it for S > 1/2 spin-models with dipolar long-range interactions, a scenario arising in recent experiments with magnetic atoms. We show that the method can capture beyond mean-field effects, not only at short times, but it also can correctly reproduce long time quantum-thermalization dynamics. We benchmark the method with exact diagonalization in small systems, with perturbation theory for short times, and with analytical predictions made for models which feature quantum-thermalization at long times. We apply our method to study dynamics in large S > 1/2 spin-models and compute experimentally accessible observables such as Zeeman level populations, contrast of spin coherence, spin squeezing, and entanglement quantified by single-spin Renyi entropies. We reveal that large S systems can feature larger entanglement than corresponding S = 1/2 systems. Our analyses demonstrate that the GDTWA can be a powerful tool for modeling complex spin dynamics in regimes where other state-of-the art numerical methods fail.
The discrete truncated Wigner approximation (DTWA) is a semiclassical phase-space method useful for the exploration of many-body quantum dynamics. In this work we investigate many-body localization (MBL) and thermalization using DTWA and compare its performance to exact numerical solutions. By taking as a benchmark case a one-dimensional random field Heisenberg spin chain with short-range interactions, and by comparing to numerically exact techniques, we show that DTWA is able to reproduce dynamical signatures that characterize both the thermal and the MBL phases. It exhibits the best quantitative agreement at short times deep in each of the phases and larger mismatches close to the phase transition. The DTWA captures the logarithmic growth of entanglement in the MBL phase, even though a pure classical mean-field analysis would lead to no dynamics at all. Our results suggest the DTWA can become a useful method to investigate MBL and thermalization in experimentally relevant settings intractable with exact numerical techniques, such as systems with long-range interactions and/or systems in higher dimensions.