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

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.

Figure 1

Comparison of different observables using TWA and the exact solution for the single-mode case in one dimension and for $N_s=10$ spins, $\delta =1\text{kHz}$ and $\mathrm{\Omega }=0.65\text{kHz}$. (a) and (c) show the total magnetization and a spin-spin correlator, whereas (b) and (d) show the real and imaginary parts of the two-point correlator between a single-spin and the bosonic mode, respectively. The deviation of the TWA solution from the exact behavior is due to the self-interaction terms.

Figure 2

Comparison of collective, single-spin, and mixed spin-boson observables using TWA, BBGKY and the exact solution for the many-mode case with $N_s=N_b=10, \delta =80\text{kHz}$ and $\mathrm{\Omega }=19.42\text{kHz}$. In the above panels we demonstrate how using the BBGKY extension allows one to greatly improve on the TWA in capturing the dynamics of (a) collective spin length and (d) variance, (b) and (e) spin-spin correlators, as well as (c) and (f) spin-phonon correlators.

Figure 3

Evolution of the collective spin $\langle S_x\rangle$ as obtained from BBGKY with and without sampling, TWA and the exact solution for the many-mode case with $N_s=N_b=10$. Initial conditions: the bosons start in the vacuum and the spins along ${[cos(\theta /2)|\uparrow \rangle +sin(\theta /2)|\downarrow \rangle ]}^{\otimes N_s}$ with (a) $\theta =\pi /2$ and (b) $\theta =\pi /4$.

Figure 4

(a)–(d) Comparison of the collective spin $\langle S_x\rangle$ using TWA and BBGKY for thermal initial conditions in two spatial dimensions. The different plots correspond to the following values $\{\delta ,\mathrm{\Omega }\}$ of detuning and force: (a) $\{1,1\}$ kHz, (b) $\{100,12\}$ kHz, (c) $\{-26,5\}$ kHz, and (d) $\{-250,2\}$ kHz. The insets show the corresponding values of $J_{ij}$ as a function of the interparticle distance $r_{ij}\equiv |r_i-r_j|/{\ell }_0$ in units of ${\ell }_0={(2e^2/\left(4\pi {\varepsilon }_{0}M\omega \right))}^{1/3}$. (e) Phonon frequencies for two dimensions. The dashed lines correspond to the position of the beat-note frequency ${\omega }_R$ for the different detunings used.

Figure 5

(a) Phonon frequencies for one dimension. The dashed lines correspond to the position of the beat-note frequency ${\omega }_R$ for the different detunings $\delta$ used in the main text. (b), (c) Effective spin-spin interaction $J_{ij}$ as a function of the interparticle distance $r_{ij}\equiv |i-j|$ (dimensionless) for the two different detunings. The gray dashed lines show the result of a fit to $1/r_{ij}^{\alpha }$.