A numerical method, suitable for the simulation of the time evolution of quantum spin models of arbitrary lattice dimension, is presented. The method combines sampling of the Wigner function with evolution equations obtained from the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Quantum correlations are treated through both the Wigner function sampling and the BBGKY evolution, bringing about highly accurate estimates of correlation functions. The method is particularly suitable for long-range interacting systems, and we demonstrate its power by comparing with exact results as well as other numerical methods. As an application we compute spin squeezing in a frustrated two-dimensional lattice with antiferromagnetic power-law interactions and a transverse field, making predictions pertinent to future ion trap experiments.
A numerical method, suitable for the simulation of the time evolution of quantum spin models of arbitrary lattice dimension, is presented. The method combines sampling of the Wigner function with evolution equations obtained from the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. Quantum correlations are treated through both the Wigner function sampling and the BBGKY evolution, bringing about highly accurate estimates of correlation functions. The method is particularly suitable for long-range interacting systems, and we demonstrate its power by comparing with exact results as well as other numerical methods. As an application we compute spin squeezing in a frustrated two-dimensional lattice with antiferromagnetic power-law interactions and a transverse field, making predictions pertinent to future ion trap experiments.
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.
The time evolution of -spin reduced density operators is studied for a class of Heisenberg-type quantum spin models with long-range interactions. In the framework of the quantum Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy, we introduce an unconventional representation, different from the usual cluster expansion, which casts the hierarchy into the form of a second-order recursion. This structure suggests a scaling of the expansion coefficients and the corresponding time scales in powers of N 1/2 with the system size N , implying a separation of time scales in the large-system limit. For special parameter values and initial conditions, we can show analytically that closing the BBGKY hierarchy by neglecting -spin correlations never leads to equilibration, but gives rise to quasi-periodic time evolution with at most independent frequencies. Moreover, for the same special parameter values and in the large- N limit, we solve the complete recursion relation (the full BBGKY hierarchy), observing a superexponential decay to equilibrium in rescaled time τ = tN − 1/2 .
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.
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.
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.
We introduce a variational method for simulating the dynamics of interacting open quantum spin systems. The method is based on the spin phase-space representation and variationally targets the Husimi-Q function with an ansatz based on a multi-dimensional mixture of spin-coherent states. Crucially, the mixture coefficients are allowed to take negative values, enabling the faithful capture of quantum correlations beyond semiclassical descriptions. The resulting equations of motion are derived from the Dirac-Frenkel variational principle and can be evaluated efficiently without resorting to Monte Carlo sampling by exploiting the analytical structure of the ansatz. As a first application, we demonstrate that this approach accurately captures both the full quantum dynamics and the non-equilibrium steady states of the transverse-field quantum Ising model, in excellent agreement with exact diagonalization. Furthermore, we show that the method scales efficiently to large two-dimensional lattices, a regime that remains challenging for other techniques.