Quantum trajectories and reduced dynamics in time-correlated environments

TL;DR

This work develops a colored-noise stochastic Schrödinger equation approach to open quantum systems by using an Ornstein–Uhlenbeck drive to model time-correlated environments. Averaging pure-state trajectories yields master equations that are not generically Lindblad but remain positive, featuring a Lindblad term from white-noise components plus correlation terms from OU fluctuations; a Redfield-inspired closure provides an effective, physically transparent description of these memory effects. In a two-level system, the colored noise produces long-lived coherences and multi-timescale relaxation whose behavior depends on the noise correlation time and system symmetry, linking bath statistics to non-Markovian dissipation. The framework supports microscopically-inspired closures and offers a route to efficient quantum simulations of non-Markovian dynamics with potential applications in quantum technologies and molecular processes.

Abstract

The stochastic Schrödinger equation (SSE) provides a trajectory-level route to simulate the dynamics of open quantum systems with applications ranging from molecular processes to quantum technologies. We study a colored-noise extension of the SSE based on an Ornstein-Uhlenbeck (OU) noise drive, and benchmark its ensemble-averaged dynamics against the standard white-noise SSE and against a fluctuating OU random Hamiltonian. When the environment exhibits a finite correlation time, averaging over pure-state trajectories yields master equations that are generally open-form and not of Lindblad type, yet remain positive by construction. By considering the differential of the OU process, we define an effective correlated noise, whose properties we analyze and use to formulate an SSE unraveling of its associated open-form quantum master equation. We show that the averaged dissipator separates into a Lindblad contribution stemming from the white-noise component, and additional correlation terms arising from the fluctuations of the OU Hamiltonian. To obtain a practical closed description and physical intuition, we introduce a Redfield-inspired perturbative closure for these correlation terms, providing an effective master equation for the colored SSE. For a two-level system, the resulting dynamics exhibit long-lived coherences, nontrivial stationary (including oscillatory) states, and multi-timescale relaxation, rationalized through the components of a time-dependent Redfield tensor.

Figures (15)

• Figure 1: Spectral densities of white noise ($\xi$, dotted line), OU process ($X$, dashed line) and $\Upsilon_\mathrm{OU}$-noise ($\Upsilon$, solid line).
• Figure 2: Time-dependent coefficients of the Redfield equation computed for the different Hamiltonians, with $\gamma=0.45\sqrt{\hbar\theta}$ and $\theta=1$. The real part is on panel (a) and the imaginary part on panel (b). Negative frequency coefficients are not shown as they are the complex conjugate of the positive ones. The null frequency coefficient contribution is equal for all the Hamiltonians and labeled "pure decoherence". The characteristic frequencies of the different Hamiltonians are in different line styles as depicted in the legend.
• Figure 3: Comparison of the mean and single-trajectory evolution of the system under different driving stochastic processes. Swarms of 50 different stochastic trajectories (grey lines in transparency), the evolution of single trajectories (red lines), and the averaged evolution of the $\rho_{00}$ population---the observable under investigation (blue lines). The stochastic force for each unraveling: in the upper left panel white noise SSE, central OU process fluctuations and right $\Upsilon_\mathrm{OU}$-noise-driven SSE. Notice the difference in smoothness between a stochastic Hamiltonian propagation and the SSE propagations. Example for the the generic Hamiltonian $H_{\varepsilon,\Omega}$ system, with noise intensity $\gamma = 0.7 \sqrt{\hbar\theta}$ and with noise operator $R=\sigma_x$.
• Figure 4: Dynamics of the system described by the Hamiltonian $H_{\varepsilon,\Omega}$, with the stochastic forces applied through the Pauli $\sigma_z$ operator. In light grey, the closed-system Schrödinger equation evolution as the reference of the coherent beating frequency of the population.
• Figure 5: Dynamics of the system described by the different Hamiltonians, with the stochastic forces applied through the Pauli $\sigma_x$ operator.

Theorems & Definitions (2)

• Proposition 1
• proof