Stochastic Schr\xf6dinger equation derivation of non-Markovian two-time correlation functions

Rafael Carballeira,Peng Zhao,David Dolgitzer,Debing Zeng,Yusui Chen

doi:10.1038/s41598-021-91216-0

Rafael Carballeira, Peng Zhao + Show 3 more

Open Access

https://doi.org/10.1038/s41598-021-91216-0

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Abstract

We derive the evolution equations for two-time correlation functions of a generalized non-Markovian open quantum system based on a modified stochastic Schrödinger equation approach. We find that the two-time reduced propagator, an object that used to be characterized by two independent stochastic processes in the Hilbert space of the system, can be simplified and obtained by taking ensemble average over one single noise. This discovery can save the cost of computation, and accelerate the converging process when taking the average over noisy trajectories. As a result, our method can be widely applied to many open quantum models, especially large-scale systems and extend the quantum regression theory to the non-Markovian case. In the short-time simulations, it is observed a significant difference between Markovian and non-Markovian cases, which can be applied to realize the environmental spectrum detection and enhance the measurement sensitivity in varying open quantum systems.

Highlights

IntroductionWe briefly review the stochastic Schrödinger equation (SSE) approach and compare it with the quantum regression theorem (QRT)
We briefly review the stochastic Schrödinger equation (SSE) approach and compare it with the quantum regression theorem (QRT).Stochastic Schrödinger equation approach.The evolution of a general quantum open system (QOS), characterized by the Hamiltonian (1), is governed by the Schrödinger equation i∂t | (t) = − Htot | (t) . (3)Since the reservoir consists of a large number of bosonic oscillators, HR = k ωkbk†bk, so it is natural to choose the Bargmann coherent state basis to represent the environment, |z = ⊗k|zk, a tensor product of all the environmental oscillators, where|zk is defined as bk|z = zk|zk for the kth mode in the environment
By expanding the noisy term|η into a polynomial consisting of ascending orders of noise zs∗k, as shown in Eq (45), our method can be applied to systematically solve many models. For this two-level dissipative system, we have proved that the order of noise in the trajectory will not exceed 1 in our previous study

Summary

Introduction

We briefly review the stochastic Schrödinger equation (SSE) approach and compare it with the quantum regression theorem (QRT). The evolution of a general quantum open system (QOS), characterized by the Hamiltonian (1), is governed by the Schrödinger equation i. Since the reservoir consists of a large number of bosonic oscillators, HR = k ωkbk†bk , so it is natural to choose the Bargmann coherent state basis to represent the environment, |z = ⊗k|zk , a tensor product of all the environmental oscillators, where|zk is defined as bk|z = zk|zk for the kth mode in the environment. The state vector of the system in the Bargmann basis, |ψt (z∗) = z| (t) , can be expressed as a stochastic trajectory. Its evolution, in the interaction picture, is governed by the following equation (setting = 1).

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