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arXiv:quant-ph/0510171

Numerical Implementation of Non-Markovian Quantum State Diffusion

Joshua Wilkie, Ray Jia Hong Ng

Published Oct 21, 2005• 0 citations

Abstract

Non-Markovian quantum state diffusion (NMQSD) is a non-relativistic but otherwise exact theory which expresses the reduced density matrix of an arbitrary subsystem, interacting linearly with an uncoupled harmonic oscillator bath, as an average of diadics formed from state vectors which obey stochastic variational-differential equations. The vacuum radiation field can be represented as such an oscillator bath, and so this model is in widespread use in quantum optics. Prior to the development of NMQSD, exact subsystem solutions could only be obtained in a few special cases (e.g. spin-1/2, harmonic oscillator). Unfortunately, it has not yet been possible to obtain exact solutions to new problems using NMQSD due to the difficulty of solving the variational-differential equations. Here we show that these equations can be transformed into a pair of coupled nonlinear integrodifferential equations. We develop exact numerical methods for the integrodifferential equations and show that solutions can be readily obtained to good accuracy for quite general subsystems. We exactly solve various examples including tunneling in a double well representing molecular isomerization or racemization, suppression of fluorescence from a two-level atom in a band gap, and intermittent fluorescence from a driven three level system representing electronic states of singly ionized magnesium.

DOI: 10.48550/arxiv.quant-ph/0510171

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Non-Markovian quantum state diffusion (NMQSD) is a non-relativistic but otherwise exact theory which expresses the reduced density matrix of an arbitrary subsystem, interacting linearly with an uncoupled harmonic oscillator bath, as an average of diadics formed from state vectors which obey stochastic variational-differential equations. The vacuum radiation field can be represented as such an oscillator bath, and so this model is in widespread use in quantum optics. Prior to the development of NMQSD, exact subsystem solutions could only be obtained in a few special cases (e.g. spin-1/2, harmonic oscillator). Unfortunately, it has not yet been possible to obtain exact solutions to new problems using NMQSD due to the difficulty of solving the variational-differential equations. Here we show that these equations can be transformed into a pair of coupled nonlinear integrodifferential equations. We develop exact numerical methods for the integrodifferential equations and show that solutions can be readily obtained to good accuracy for quite general subsystems. We exactly solve various examples including tunneling in a double well representing molecular isomerization or racemization, suppression of fluorescence from a two-level atom in a band gap, and intermittent fluorescence from a driven three level system representing electronic states of singly ionized magnesium.

Non-Markovian Quantum State Diffusion: Application to Quantum Jumps in

^{24}

Mg

^+

Non-Markovian quantum state diffusion (NMQSD) is an exact method for calculating the reduced density matrix of an arbitrary subsystem interacting linearly with the radiation field. Applications of the theory have however been few due to the intractable nature of the variational-differential NMQSD evolution equation. Recently, we argued that the variational-differential equation can be rewritten as an integrodifferential equation which can be readily solved numerically. This manuscript provides an explicit derivation of the modified equations. Applications to intermittent fluorescence in

^{24}

Mg

^+

are discussed in detail. Earlier speculations that quantum jumps occur on all time scales are verified on a picosecond timescale. We show that a plot of the probability density of the signal vs signal strength shows the two characteristic peaks associated with the bright and dark manifolds, and that the ratio of the areas under the peaks is 16 as observed experimentally. We also show that the shape of this distribution is sensitive to bath memory, but has a mathematical form common to both the Markovian and non-Markovian cases.

The numerical simulation of non-Markovian open quantum systems based on the non-Markovian quantum state diffusion (NMQSD) equation is complicated by functional derivatives with respect to the stochastic process. A general numerical framework that directly treats these functional derivatives without relying on prescribed decompositions of the bath correlation function is still lacking. In this work, we derive an analytical solution of the linear NMQSD equation that reveals three elementary structures of the non-Markovian stochastic dynamics: stochastic propagation, functional-derivative insertion, and memory pairing. Based on this structure, we construct a general auxiliary-state framework for arbitrary bath correlation functions. The framework separates the numerical construction into time discretization, memory quadrature, and hierarchy truncation. We then construct first- and second-order schemes and provide diagrammatic transition rules for their explicit implementation. Numerical results verify the expected temporal accuracy and demonstrate the applicability of the proposed methods to different bath correlation functions and multi-level quantum systems.

Two perturbation methods for the non-Markovian quantum state diffusion (NMQSD) equation are investigated in this paper. The first perturbation method under investigation is based on a functional expansion of the NMQSD equation, while the second expands the NMQSD equation in terms of coupling strength between the system and its environment. We compare the two perturbation methods by solving the dynamics of a bipartite system in which the two perturbation methods can be compared with the exact NMQSD equation. In addition, as an application, we provide an analytical solution for a special family of the systemʼs initial states, and discuss the entanglement dynamics based on this solution.

Non-Markovian quantum state diffusion (NMQSD) provides a powerful approach to the dynamics of an open quantum system in bosonic environments. Here we develop a NMQSD method to study the open quantum system in fermionic environments. This problem involves anticommutative noise functions (i.e., Grassmann variables) that are intrinsically different from the noise functions of bosonic baths. We obtain the NMQSD equation for quantum states of the system and the non-Markovian master equation. Moreover, we apply this NMQSD method to single- and double-quantum-dot systems.

Based on the path integral approach, we derive a new realization of the exact non-Markovian stochastic Schrödinger equation (SSE). The main difference from the previous non-Markovian quantum state diffusion (NMQSD) method is that the complex Gaussian stochastic process used for the forward propagation of the wave function is correlated, which may be used to reduce the amplitude of the non-Markovian memory term at high temperatures. The new SSE is then written into the recently developed hierarchy of pure states scheme, in a form that is more closely related to the hierarchical equation of motion approach. Numerical simulations are then performed to demonstrate the efficiency of the new method.

Developing general-purpose numerical methods for non-Markovian quantum state diffusion has long been a challenge in the simulation of open quantum systems. This paper proposes a deterministic polynomial chaos Galerkin method, which is easy to implement, accurate, and efficient. By combining a low-rank decomposition of bath correlations with Galerkin projection onto a polynomial chaos basis, the method yields a deterministic formulation compatible with standard numerical solvers for partial differential equations. The reduced density matrix is obtained from a single deterministic solve at a cost comparable to that of computing one stochastic hierarchy trajectory. Replacing an ensemble of

N_{\mathrm{traj}}

trajectories therefore reduces the computational cost by approximately a factor of

N_{\mathrm{traj}}

and eliminates statistical sampling error. Benchmark comparisons show that the proposed method achieves smaller errors at lower computational cost than the tested stochastic hierarchy methods. Applications involving nonquadratic potentials, diffraction and interference, and coupled three-dimensional dynamics demonstrate its ability to handle challenging problems.