Tensor network influence functionals for open quantum systems with general Gaussian bosonic baths

Abstract

Dynamics of open quantum systems with structured reservoirs can often be simulated efficiently with tensor network influence functionals. The standard variants of the time-evolving matrix product operator (TEMPO) method are applicable when the systems is coupled to Gaussian bosonic baths via hermitian coupling operators that mutually commute. In this work we introduce a generalization to cases where the system is coupled to a single reservoir through multiple non-commuting operators, representing the most general form of linear system-bath coupling. We construct a Gaussian influence functional that properly handles Trotter errors arising from a finite evolution time step, thus ensuring convergence for long evolution times. Based on this result, the uniform TEMPO scheme can be employed to obtain a matrix product operator form of the influence functional, enabling efficient simulations of the real-time dynamics of the open system. As a demonstration, we simulate the time evolution of driven two-level emitters coupled to a bosonic lattice at different lattice sites.

Figures (6)

• Figure 1: Simulation of the driven Jaynes-Cummings model with $H_\mathrm{sys}=\Omega\sigma_x/2$ and the bath \ref{['eq:bcff_lorenz']}$\omega=g=\Gamma=2\Omega,\, \bar{n}=0.25$. Left: The dynamics computed with uniTEMPO is in perfect agreement with the reference solution (black markers) computed via the full system \ref{['eq:HJC']}. Right: Convergence of the steady state with respect to the Trotter time step $\delta t$. Using the symmetric splitting \ref{['eq:Utrotter']} yields quadratic error scaling, whereas the error is linear when omitting the Trotter corrections in Eq. \ref{['eq:infl_gauss0']}. Simulations were performed at bond dimension $\chi\approx 28$ ($N_c=2048$). The error is computed via the matrix norm of the difference of the steady state computed with uniTEMPO and the numerically exact reference state $||\rho-\rho_\mathrm{ref}||_2$.
• Figure 2: Symmetrized power spectral density \ref{['eq:PSDzz']} for the spin boson model with Jaynes-Cummings-type coupling \ref{['eq:SB_JC']} with $H_\mathrm{sys}=\varepsilon\sigma_z/2+\Omega\sigma_x/2$ and an ohmic bath \ref{['eq:bcff_ohmic']}$\omega_c=5\varepsilon,\,\beta\varepsilon=1,\alpha=0.05$. The two plots show spectra for different values of $\Omega$ and their fitting to the fluctuation-dissipation relation \ref{['eq:KMS']}. Simulations were performed at $\delta t\,\varepsilon=0.05$ and bond dimension $\chi=170$ ($N_c=65536$).
• Figure 3: Left: Sketch of the model. Two emitters are coupled to a non-interacting bosonic lattice at different lattice sites. One of the emitters is driven. Right: Local spectral density \ref{['eq:sds']} for a cubic lattice in two and three dimensions.
• Figure 4: Dynamics of the emitter occupation ($\sigma_+\sigma_-$) for two emitters coupled to a three-dimensional cubic bosonic lattice at neighboring sites $g^2=0.1 J^2$, $\Delta=0$ (Eq. \ref{['eq:emitter_model1']},\ref{['eq:emitter_model2']},\ref{['eq:emitter_model3']}). The emitter $a$ is initially occupied while emitter $b$ is not. Simulation results for $\delta tJ=0.025$ and $\chi=117$ ($N_c=2048$).
• Figure 5: Dynamics of the emitter occupation ($\sigma_+\sigma_-$) for two emitters coupled to a two-dimensional cubic bosonic lattice at neighboring sites $g^2=0.1 J^2$, with nonzero detuning $\Delta=0.5J$ (Eq. \ref{['eq:emitter_model1']},\ref{['eq:emitter_model2']},\ref{['eq:emitter_model3']}). The emitter $a$ is initially occupied while emitter $b$ is not. Simulation results for $\delta tJ=0.05$ and $\chi=357$ ($N_c=131072$).