We compare the continuous and discrete truncated Wigner approximations of various spin models' dynamics to exact analytical and numerical solutions. We account for all components of spin-spin correlations on equal footing, facilitated by a recently introduced geometric correlation matrix visualization technique [Mukherjee et al., Phys. Rev. A 97, 043606 (2018)]. We find that at modestly short times, the dominant error in both approximations is to substantially suppress spin correlations along one direction.
We compare the continuous and discrete truncated Wigner approximations of various spin models' dynamics to exact analytical and numerical solutions. We account for all components of spin-spin correlations on equal footing, facilitated by a recently introduced geometric correlation matrix visualization technique [Mukherjee et al., Phys. Rev. A 97, 043606 (2018)]. We find that at modestly short times, the dominant error in both approximations is to substantially suppress spin correlations along one direction.
We compare the continuous and discrete truncated Wigner approximations - TWA and dTWA, respectively - of various spin models' dynamics to exact analytical and numerical solutions. We account for all components of spin-spin correlations on equal footing, facilitated by a recently introduced geometric correlation matrix visualization (CMV) technique. We find that the two approximations capture different aspects of the dynamics of correlations: dTWA captures periodic partial revivals of spin-spin correlations missed in the TWA; but at least for some conditions, TWA better captures the directions in which spins have correlations. We also find that at modestly short times, the dominant error in both approximations is to substantially suppress spin correlations along one direction. In addition to assessing the accuracy of approximations, our comparisons illustrate the utility of CMVs as an intuitive visual tool to understand the full complexity of spin correlations.
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.
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.
An accurate description of the nonequilibrium dynamics of systems with coupled spin and bosonic degrees of freedom remains theoretically challenging, especially for large system sizes and in higher than one dimension. Phase-space methods such as the truncated Wigner approximation (TWA) have the advantage of being easily scalable and applicable to arbitrary dimensions. In this work we adapt the TWA to generic spin-boson models by making use of recently developed algorithms for discrete phase spaces [J. Schachenmayer, A. Pikovski, and A. M. Rey, Phys. Rev. X 5, 011022 (2015)]. Furthermore we go beyond the standard TWA approximation by applying a scheme based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations to our coupled spin-boson model. This allows us, in principle, to study how systematically adding higher-order corrections improves the convergence of the method. To test various levels of approximation we study an exactly solvable spin-boson model, which is particularly relevant for trapped-ion arrays. Using TWA and its BBGKY extension we accurately reproduce the time evolution of a number of one- and two-point correlation functions in several dimensions and for an arbitrary number of bosonic modes.
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.
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.