Nonequilibrium dynamics of spin-boson models from phase-space methods

Abstract

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 [1]. Furthermore we go beyond the standard TWA approximation by applying a scheme based on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations [2] to our coupled spin-boson model. This allows 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 arbitrary number of bosonic modes.

Modeling near ground-state cooling of two-dimensional ion crystals in a Penning trap using electromagnetically induced transparency

“…Such a sampling and averaging procedure captures the build-up of non-trivial third and higher order cumulants that are neglected in the GM. We note that this approach is in the same spirit as the Truncated Wigner Approximation (TWA) used in calculating the dynamics of spin-spin and spin-boson models [21][22][23][24]. Moreover, we track separate phase space variables corresponding to system operators as well as operator pairs, and evolve these variables using the same equations of motion as in the GM (Appendix A), but for many trajectories.…”

“…Moreover, we track separate phase space variables corresponding to system operators as well as operator pairs, and evolve these variables using the same equations of motion as in the GM (Appendix A), but for many trajectories. We thereby perform beyond mean-field calculations [24] that capture those contributions to the covariances between system operators which develop as a result of the subsequent diffusive-dissipative dynamics. We refer to this method as the Sampling model (SM).…”

A generalized phase space approach for solving quantum spin dynamics

“…Our method thus allows us to study the TWA time-evolution not only of spin operators, but the full spin-density matrix and thus allows us to extract experimentally relevant time-dependent observables such as spin-state populations, and fundamentally relevant quantities such as entanglement. In the limit of S=1/2 our generalized discrete TWA approach (GDTWA), reduces to the previously proposed discrete TWA method (DTWA) [44], which has been remarkably successful in predicting S=1/2 model dynamics [16,[45][46][47][48][49].…”

Cooperative spontaneous emission via a renormalization approach: Classical versus semiclassical effects

“…Hence, exact numerical methods, such as diagonalization, are limited to a few tens of particles, even for supercomputers. This is the reason why truncation methods were developed, based on cumulant expansions [1][2][3][4][5][6], in which higher-order connected correlations are neglected to obtain a more favorable scaling of the complexity with the number of particles. As a consequence, nonlinearities emerge from the decomposition of higher-order correlations into products of lower-order ones, which turns even trickier predictions on collective quantum phenomena.…”

“…(4) for cooperative spontaneous emission as a particular case of Eq. (6). Replacing the general components x l by the expectation values σ α l , for α = ±, z, M becomes a 3N × 3N diagonal matrix, where the diagonal block of three single-atom eigenvalues, (Γ/2 − ∆i, Γ/2 + ∆i, Γ), is the same for all atoms.…”