Non-Markovian Quantum Jump Method for Driven-Dissipative Two-Level Systems

We propose a modified non-Markovian quantum jump method to overcome the obstacle of dramatically increased trajectory number in conventional quantum trajectory simulations. In our method the trajectories are classified into the trajectory classes characterized by the number of quantum jumps. We derive the expression of the existence probability of each trajectory (class), which is essential to construct the density matrix of the open quantum system. This modified method costs less computational resources and is more efficient than the conventional quantum trajectory approach. As applications we investigate the dynamics of spin-1/2 systems subject to Lorentzian reservoirs with considering only the no-jump and one-jump trajectories. The revival of coherence and entanglement induced by the memory effect is observed.

💡 Research Summary

This paper addresses a significant computational challenge in simulating the dynamics of non-Markovian open quantum systems: the explosive growth in the number of quantum trajectories required in conventional Non-Markovian Quantum Jump (NMQJ) simulations, especially for driven or many-body systems. To overcome this, the authors propose a modified NMQJ method centered on two key innovations: the classification of trajectories and the calculation of trajectory existence probabilities.

The core idea is to group individual quantum trajectories into “trajectory classes” based solely on the number of quantum jumps (n) they have undergone, rather than tracking the exact sequence of jump times. All trajectories that have experienced exactly n jumps belong to the class {Hα_n}. This classification is powerful because, under the assumption of infinite memory time (a common simplification), a reversed quantum jump acting on any trajectory within a class will map it back to the same “mother trajectory” in the class with n-1 jumps. This simplifies the stochastic algorithm considerably.

The second pillar of the method is the derivation and systematic calculation of the “existence probability” Kα_n(t) for each trajectory class. This probability represents the statistical weight of a given class in an infinite ensemble of realizations. The authors derive an expression for the existence probability of the no-jump trajectory class and establish a recursive, top-down relation to compute the probabilities for classes with higher jump counts. The system’s density matrix at any time is then reconstructed as a weighted sum over states from all trajectory classes, using these existence probabilities as weights. This framework allows for computational efficiency by enabling truncation at a desired jump order without significant loss of accuracy, as higher-order classes typically have diminishing probabilities.

The authors demonstrate the efficacy of their method by applying it to two toy models: a single driven spin-1/2 particle and a coupled two-spin system, both interacting with Lorentzian reservoir(s). The interaction generates a time-local master equation with a decay rate that can become negative, signaling non-Markovian memory effects. Using the modified NMQJ method and truncating after the one-jump trajectory class, they efficiently simulate the systems’ dynamics. The results clearly show phenomena hallmark to non-Markovianity: a temporary revival of quantum coherence in the single-spin system and non-monotonic dynamics of quantum entanglement (quantified by concurrence) in the two-spin system. These are direct manifestations of information backflow from the environment, facilitated by the reversed quantum jumps during negative decay rate periods.

In summary, this work presents a computationally efficient reformulation of the NMQJ technique. By shifting the focus from innumerable individual trajectories to a manageable set of trajectory classes and their statistical weights, it significantly reduces the resource overhead for simulating non-Markovian dynamics, paving the way for more practical investigations of memory effects in complex open quantum systems.