Stochastic unravelings for trace-nonpreserving open quantum system dynamics

Abstract

Stochastic unravelings allow to efficiently simulate open system dynamics, yet their application has traditionally been restricted to master equations that preserve both Hermiticity and trace. In this work, we introduce a general framework that extends piecewise-deterministic unravelings to arbitrary trace-nonpreserving master equations, requiring only positivity and Hermiticity of the dynamics. Our approach includes, as special cases, unravelings of arbitrary dynamics in the Heisenberg picture, evolutions interpolating between fully Lindblad and non-Hermitian Hamiltonian generators, and equations employed in the derivation of full counting statistics, for which we show it can be used to obtain the moments of the associated probability distribution. The framework is suitable for both trace-decreasing and trace-increasing processes through stochastic disappearance and replication of the stochastic realizations, and it is compatible with different unraveling schemes and with reverse jumps in the non-Markovian regime. Thereby, our approach provides a powerful and versatile simulation method that significantly broadens the applicability of stochastic techniques for open system dynamics.

Figures (4)

• Figure 1: At any given moment of time, there are 4 possibilities for the evolution of the realization, as given in Eqs. \ref{['eq:prob_all_1']}-\ref{['eq:prob_all_2']}. It can either evolve deterministically, jump, create a copy of itself or vanish.
• Figure 2: Moments of $P(n,t)$. The exact solution is obtained by solving Eq. \ref{['eq:ME_photon_counting']} and taking the derivative of the trace; the unravelings are performed on Eq. \ref{['eq:ME_tau_k']}. Inset: ${\operatorname{tr}}\rho_\zeta - 1$, for $\zeta=-0.02$ (bottom) to $\zeta=0.02$ (top). Parameters: $\gamma = \Omega = 1$, $\bar{n} = 0.5$, $\phi = 0.2$, $dt = 10^{-2}$, initial number of trajectories: $N = 10^4$, initial state: $\ket{\psi_0} = (\ket0+\ket1)/\sqrt{2}$. Notice that the final time is chosen to be comparable to the typical timescale $1/\gamma$ of the dynamics.
• Figure 3: Dynamics of the Heisenberg picture ME \ref{['eq:Heis_ex_ME']}, $x$ and $z$ components of the Bloch vector. The unraveling match the exact solution (dark lines); in lighter shade $7$ stochastic trajectories are also shown. Lower left inset: rates $\gamma_\pm$ and $\epsilon$ (logarithmic scale). Upper right inset: dynamics of the trace ${\operatorname{tr}}[X(t)]$, obtained as the ratio $\sum_iN_i(t)/N$. The timestep used in the simulations is $dt = 10^{-3}$; $N=2\cdot10^4$ stochastic realizations are present at $t=0$, the number at time $t$ follows the same dynamics as ${\operatorname{tr}}[X(t)]$ (upper right inset).
• Figure S1: Left panel: the three smallest eigenvalues of the Choi state of Eq. \ref{['eq:app_Choi']}, the fourth eigenvalue is such that ${\operatorname{tr}} J_t=1$. Negativity of one eigenvalue implies violations of CP divisibility. Right panel: maximal norm of the Bloch vector under the action of $\Lambda_{t+dt,t}$. The fact that such norm is greater than one implies that the dynamics is not P divisible.