Exact Floquet dynamics of strongly damped driven quantum systems

TL;DR

Problem: strongly damped, periodically driven open quantum systems challenge standard master equations due to non-Markovian effects and Floquet micromotion. Approach: introduce periodic Floquet influence functionals by representing enviroment-traced multi-time correlations as a periodic MPO with $H(t)=H(t+T)$ and construct a Floquet propagator $\mathcal{U}_{\mathrm{F}}=\mathcal{U}_M\cdots \mathcal{U}_1$, with uniTEMPO compressing the representation to a tractable bond dimension. Contributions: numerically exact treatment of stationary and transient dynamics in driven spin-boson models, characterization of Floquet heating and delta-peak heat currents, and demonstration that local driving stabilizes reservoir-mediated entanglement between two qubits; plus discussion of extensions to nonlocal driving and DMFT relevance. Significance: provides a transparent, scalable framework for dissipative Floquet engineering and the study of heat transport and entanglement in strongly damped, non-Markovian quantum systems.

Abstract

We present an approach for efficiently simulating strongly damped quantum systems subjected to periodic driving, employing a periodic matrix product operator representation of the influence functional. This representation enables the construction of a numerically exact Floquet propagator that captures the non-Markovian open system dynamics, thus providing a dissipative analogue to the Floquet Hamiltonian of driven isolated quantum systems. We apply this method to study the asymptotic heating of a reservoir in spin-boson models, characterizing the deviation from equilibrium conditions. Moreover, we show how a local driving of two qubits can be utilized to stabilize a transient entanglement buildup of the qubits originating from the interaction with a common environment. Our results make it possible to directly study both stationary and transient dynamics of strongly damped and driven quantum systems within a transparent theoretical and numerical framework.

Figures (6)

• Figure 1: Quantum circuit representation of a periodic Floquet influence functional. (a) Local multi-time correlation functions for open system dynamics with factorized initial conditions. Due to the periodic driving, the global unitaries $U_n$ are periodically repeating. (b) Periodic compressed MPS representation of the influence functional. The global unitaries are replaced by compressed tensors $Q_n$, but the periodic structure is preserved. (c) Coarse-graining the influence functional vilkoviskiyTemporalEntanglementTransition2025 by contracting all $Q_n$ tensors in one period, yields an effective Floquet propagator describing the stroboscopic dynamics.
• Figure 2: Spin-boson dynamics with transversal driving $H_{\mathrm{drive}}(t) = \epsilon_d\cos(\omega_d t)\sigma_z$ for $\alpha = 0.1$, $\omega_c = 2.5\Omega$, $\epsilon_d = 1\Omega$. We compare the exact uniTEMPO calculation with a Redfield master equation based on the Magnus expansion in the case of slow (left panel) and fast (right panel) driving. As expected, the Magnus expansion provides more accurate result when the driving frequency is high. IF simulations were performed with time-step $\delta t = \pi/(60\Omega)$ and bond dimension $\chi = 235$.
• Figure 3: Time-averaged asymptotic heat current density in a strongly driven spin-boson model ($\alpha = 0.1$, $\omega_c = 2.5\Omega$, $\epsilon_d = 1\Omega$) for longitudinal $H_{\mathrm{drive}}(t) = \epsilon_d\cos(\omega_d t)\sigma_x$ (top panel) and transversal $H_{\mathrm{drive}}(t) = \epsilon_d\cos(\omega_d t)\sigma_z$ (bottom panel) driving fields with different driving frequencies. In the transversal driving case, resonant excitations at odd multiples of the driving frequency lead to delta peaks, indicated by straight lines. Simulations were performed with time-step $\delta t \approx\pi/(60\Omega)$ and bond dimension $\chi = 235$.
• Figure 4: Entanglement dynamics in a driven two-spin-boson model Eq. \ref{['eq:two_spin_H']} with $\alpha = 0.2$, $\omega_c = 5\Omega$. Top panel: Entanglement dynamics after a quench for the initial state $\rho_0 = |00\rangle\!\langle00|$ in the undriven ($\epsilon_d = 0$) and driven ($\omega_d = 2.15\Omega$, $\epsilon_d = 1.15\Omega$, star in lower panel) cases. Bottom panel: period-averaged concurrence of the Floquet steady state as a function of the driving frequency and the driving amplitude. Simulations were performed with time-step $\delta t \approx \pi/(48\Omega)$ and bond dimension $\chi = 342$.
• Figure 5: Total time-averaged heat current \ref{['eq:curr_tot']} as a function of the driving frequency with the same parameters as in Fig. \ref{['fig:heat_current']}.