Dynamics of Open Quantum Systems with Initial System-Environment Correlations via Stochastic Unravelings

TL;DR

This work addresses open quantum systems with initial system–environment correlations by extending stochastic unravelings to the OPD/B+ framework. It develops a practical protocol to handle non-positive OPD components by decomposing them into positive parts and evolving each part with CPTP maps, then recombining to obtain the reduced state dynamics. The Adapted Projection Operator (APO) technique is introduced to systematically derive second-order generators for each OPD component, enabling tractable unravelings in Jaynes–Cummings, dephasing, and damped two-qubit models, including non-Markovian regimes. The results show that OPD unravelings can describe a large subspace of states obtainable via system-only repreparations, outperforming fixed-correlation approaches in scope and robustness, and providing powerful tools for accurate simulations and deeper understanding of correlated open-system dynamics.

Abstract

In standard treatments of open quantum systems, the reduced dynamics is described starting from the assumption that the system and the environment are initially uncorrelated. This assumption, however, is not always guaranteed in realistic scenarios and several theoretical approaches to characterize initially correlated dynamics have been introduced. For the uncorrelated scenario, stochastic unravelings are a powerful tool to simulate the dynamics, but so far they have not been used in the most general case in which correlations are initially present. In our work, we employ the bath positive (B+) or one-sided positive decomposition (OPD) formalism as a starting point to generalize stochastic unraveling in the presence of initial correlations. Noticeably, our approach doesn't depend on the particular unraveling technique, but holds for both piecewise deterministic and diffusive unravelings. This generalization allows not only for more powerful simulations for the reduced dynamics, but also for a deeper theoretical understanding of open system dynamics.

Figures (5)

• Figure 1: For the uncorrelated case, a single Lindblad master equation, depending on $\rho_E$, is sufficient to describe the reduced evolution. If initial correlations are present, then one needs a set of up to $d^2-1$ master equations, one for each term of Eq. \ref{['eq:OPD_state']}, to describe the reduced evolution. Such master equations act on non-positive operators $Q_\alpha$, and therefore cannot be directly unraveled. However, as discussed in Sec. \ref{['sec:unr_init_corr']}, they can be reconstructed from positive operators $\Sigma_\alpha^\pm$, which can be unraveled.
• Figure 2: Unravelings of $\Phi^x_t(Q_x)$ for the dephasing dynamics of Eqs. \ref{['eq:dephasing_Hamiltonian_d=4_free']}-\ref{['eq:dephasing_Hamiltonian_d=4_coupling']}; showing the real (solid) and imaginary (dashed) components of $\braket{0\vert \Phi^x_t(Q_x)\vert1}$. Left: unraveling of the positive part $\Sigma_x^+$. Middle: unraveling of the negative part $\Sigma_x^-$. Right: unraveling of $Q_x$, obtained as the difference between the two, and dynamics of the coherence of $\rho_S(t)$ (green). Inset: dephasing rates, i.e. eigenvalues of $K_{k\ell}^x(t)$. The unravelings are done using MCWF (red lines and circles) and QSD (blue lines and crosses) and for both 6 stochastic trajectories (piecewise deterministic using MCWF and diffusive using QSD) are shown in lighter shades. Parameters: $g=0.5$, $10^3$ stochastic realizations have been used.
• Figure 3: Unraveling of $\Phi^x_t[Q_x]$ for the Jaynes-Cummings dynamics in the continuum limit, for the maximally initial state of Eq. \ref{['eq:JC_cont_init_state']}. Left: unravelings for the positive part of $Q_x$; middle: for the negative part; right: $Q_x$ is obtained as the difference between the two. The unravelings are done both with the MCWF (red circles) and QSD (blue crosses), and 10 trajectories are shown. Inset: rates $\gamma^x_\pm$ for $\mathcal{L}^x_t$ of Eq. \ref{['eq:ME_JC_APO']}. Parameters: $g=0.05$, $N=10$. $10^4$ stochastic realizations have been used.
• Figure 4: Unravelings of the single-mode Jaynes-Cummings dynamics for the maximally entangled initial state of Eq. \ref{['eq:JC_single_mode_init_state-ent']}, obtained using NMQJ. Left panel: $z$ component of the Bloch vector for $\Phi^\alpha_t[Q_\alpha]$. Right panel: reduced dynamics for the maximally entangled state (blue) and of states obtained via the system-side repreparations: zero discord states $\rho_{SE}^{\mathcal{R}_0^p}$ of Eq. \ref{['eq:JC_single_mode_rep_0_disc']} with $p=0.5$ (red solid), $p=0.9$ (red dashed), factorized initial state $\rho_{SE}^{\mathcal{R}_{\text{f}}}$ of Eq. \ref{['eq:JC_single_mode_rep_fact']} (green). Parameters: $n_0=1$, $n_1=0$, $\omega_0=1$, $\omega=0.1$, $g=0.5$. $10^4$ stochastic realizations have been used.
• Figure 5: Unravelings of the system qubit dynamics obtained from Eq. \ref{['eq:H_sigma_xz']}. Left: dynamics of $\Phi^\alpha_t[Q_\alpha^+]$, showing (in lighter shade) 5 trajectories; for $t\le1$, no reverse jumps are needed even if $\gamma_-<0$. Right: dynamics of $\Phi^\alpha_t[Q_\alpha]$ and of $\rho_S(t)$. Inset: rates $\gamma_\pm$ of Eq. \ref{['eq:sigma_xz_rates']}. Parameters: $g=\omega_1=\omega_2 = \omega=\mu=1$.

Theorems & Definitions (2)

• Proposition 1
• proof