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Register a new userNon-Markovian Stochastic Schrodinger Equation: Matrix Product State Approach
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We derive a hierarchy of matrix product states (HOMPS) method which is numerically exact and efficient for general non-Markovian dynamics in open quantum system. This HOMPS is trying to attack the exponential wall issue in the recently developed hierarchy of pure states (HOPS) scheme with two steps: a. finding an effective time-dependent Schrodinger equation which is equivalent to HOPS, b. propagating this equation within matrix product states/operators (MPS/MPO) representation. HOMPS works in linear form and takes into account finite temperature effect straightforwardly from the initial pure state. Applications of HOMPS to spin-boson model covering both high and low temperatures are provided, demonstrating the validity and efficiency of the new approach.
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We show that the stochastic Schrodinger equation (SSE) provides an ideal way to simulate the quantum mechanical spin dynamics of radical pairs. Electron spin relaxation effects arising from fluctuations in the spin Hamiltonian are straightforward to include in this approach, and their treatment can be combined with a highly efficient stochastic evaluation of the trace over nuclear spin states that is required to compute experimental observables. These features are illustrated in example applications to a flavin-tryptophan radical pair of interest in avian magnetoreception, and to a problem involving spin-selective radical pair recombination along a molecular wire. In the first of these examples, the SSE is shown to be both more efficient and more widely applicable than a recent stochastic implementation of the Lindblad equation, which only provides a valid treatment of relaxation in the extreme-narrowing limit. In the second, the exact SSE results are used to assess the accuracy of a recently-proposed combination of Nakajima-Zwanzig theory for the spin relaxation and Schulten-Wolynes theory for the spin dynamics, which is applicable to radical pairs with many more nuclear spins. An appendix analyses the efficiency of trace sampling in some detail, highlighting the particular advantages of sampling with SU(N) coherent states.
We present a detailed microscopic derivation for a non-Markovian master equation for a driven two-state system interacting with a general structured reservoir. The master equation is derived using the time-convolutionless projection operator technique in the limit of weak coupling between the two-state quantum system and its environment. We briefly discuss the Markov approximation, the secular approximation and their validity.
Accurate and efficient simulation on quantum dissipation with nonlinear environment couplings remains nowadays a challenging task. In this work, we propose to incorporate the stochastic fields, which resolve just the nonlinear environment coupling terms, into the dissipaton-equation-of-motion (DEOM) construction. The stochastic fields are introduced via the Hubbard-Stratonovich transformation. After the transformation, the resulted stochastic-fields-dressed total Hamiltonian contains only linear environment coupling terms. On basis of that, a stochastic-fields-dressed DEOM (SFD-DEOM) can then be constructed. The resultant SFD-DEOM, together with the ensemble average over the stochastic fields, constitutes an exact and nonperturbative approach to quantum dissipation under nonlinear environment couplings. It is also of relatively high efficiency and stability due to the fact that only nonlinear environment coupling terms are dealt with stochastic fields while linear couplings are still treated as the usual DEOM. Numerical demonstrations are carried out on a two-state model system.
In stochastic modeling, there has been a significant effort towards finding predictive models that predict a stochastic process future using minimal information from its past. Meanwhile, in condensed matter physics, matrix product states (MPS) are known as a particularly efficient representation of 1D spin chains. In this Letter, we associate each stochastic process with a suitable quantum state of a spin chain. We then show that the optimal predictive model for the process leads directly to an MPS representation of the associated quantum state. Conversely, MPS methods offer a systematic construction of the best known quantum predictive models. This connection allows an improved method for computing the quantum memory needed for generating optimal predictions. We prove that this memory coincides with the entanglement of the associated spin chain across the past-future bipartition.
For a bosonic (fermionic) open system in a bath with many bosons (fermions) modes, we derive the exact non-Markovian master equation in which the memory effect of the bath is reflected in the time dependent decay rates. In this approach, the reduced density operator is constructed from the formal solution of the corresponding Heisenberg equations. As an application of the exact master equation, we study the active probing of non-Markovianity of the quantum dissipation of a single boson mode of electromagnetic (EM) field in a cavity QED system. The non-Markovianity of the bath of the cavity is explicitly reflected by the atomic decoherence factor.
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