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arXiv:1809.11016 · Physical Review Letters

Stochastic Representation of Non-Markovian Fermionic Quantum Dissipation

Lu Han, Vladimir Chernyak, Yun‐An Yan, Xiao Hong Zheng, YiJing Yan

Published Jul 31, 2019• 35 citations
arXivHTMLPDF

Abstract

Quantum Brownian motion plays a fundamental role in many areas of modern physics. In the path-integral formulation, environmental fluctuations can be characterized by auxiliary stochastic fields. Intriguingly, for fermionic environments the stochastic fields must be Grassmann valued so as to memorize the order of the random forces exerted on the system. Such nonclassical fields cannot be represented by conventional means. We propose a strategy to map the Grassmann-number fields to conventional c-number noises and a set of quantized pseudolevels. The resulting stochastic equation of motion (SEOM) enables direct stochastic simulation of the fermionic dissipative dynamics. The SEOM gives exact physical observables of noninteracting systems, and yields accurate approximate results for interacting systems. The practicality and accuracy of the proposed strategy and the SEOM are exemplified by numerical studies conducted on a single-impurity Anderson model.

DOI
10.1103/physrevlett.123.050601

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Quantum Brownian motion plays a fundamental role in many areas of modern physics. In the path-integral formulation, environmental fluctuations can be characterized by auxiliary stochastic fields. Intriguingly, for fermionic environments the stochastic fields must be Grassmann valued so as to memorize the order of the random forces exerted on the system. Such nonclassical fields cannot be represented by conventional means. We propose a strategy to map the Grassmann-number fields to conventional c-number noises and a set of quantized pseudolevels. The resulting stochastic equation of motion (SEOM) enables direct stochastic simulation of the fermionic dissipative dynamics. The SEOM gives exact physical observables of noninteracting systems, and yields accurate approximate results for interacting systems. The practicality and accuracy of the proposed strategy and the SEOM are exemplified by numerical studies conducted on a single-impurity Anderson model.
This paper provides a detailed account of the numerical implementation of the stochastic equation of motion (SEOM) method for the dissipative dynamics of fermionic open quantum systems. To enable direct stochastic calculations, a minimal auxiliary space (MAS) mapping scheme is adopted, with which the time-dependent Grassmann fields are represented by c-number noises and a set of pseudo-operators. We elaborate on the construction of the system operators and pseudo-operators involved in the MAS-SEOM, along with the analytic expression for the particle current. The MAS-SEOM is applied to study the relaxation and voltage-driven dynamics of quantum impurity systems described by the single-level Anderson impurity model, and the numerical results are benchmarked against those of the highly accurate hierarchical equations of motion method. The advantages and limitations of the present MAS-SEOM approach are discussed extensively.
In this work, we establish formally exact stochastic equation of motion (SEOM) theory to describe the dissipative dynamics of fermionic open systems. The construction of the SEOM is based on a stochastic decoupling of the dissipative interaction between the system and fermionic environment, and the influence of environmental fluctuations on the reduced system dynamics is characterized by stochastic Grassmann fields. Meanwhile, numerical realization of the time-dependent Grassmann fields has remained a long-standing challenge. To solve this problem, we propose a minimal auxiliary space (MAS) mapping scheme with which the stochastic Grassmann fields are represented by conventional c-number fields along with a set of pseudo-levels. This eventually leads to a numerically feasible MAS-SEOM method. The important properties of the MAS-SEOM are analyzed by making connection to the well-established time-dependent perturbation theory and the hierarchical equations of motion theory. The MAS-SEOM method provides a potentially promising approach for the accurate and efficient simulation of fermionic open systems at ultra-low temperatures.
This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath to the models in which the system operators commute with the fermionic bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schr\"odinger equation (SSE), called the fermionic quantum state diffusion (QSD) equation, from the first principle by using the fermionic coherent state representation. The reduced density operator for the open system can be recovered from the stochastic average of the solutions to the QSD equation over the Grassmann-type noise. By employing the exact fermionic QSD equation, we can derive the corresponding exact master equation. The power of our approach is illustrated by the applications of our stochastic approach to several models of interest including the one-qubit dissipative model, the coupled two-qubit dissipative model, the quantum Brownian motion model, and the NNN-fermion model coupled to a fermionic bath. Different effects caused by the fermionic and bosonic baths on the dynamics of open systems are also discussed.
A quantum dissipation theory is formulated in terms of hierarchically coupled equations of motion for an arbitrary fermionic system coupled with grand canonical fermionic bath ensembles.1 The theoretical construction starts with the second-quantization influence functional in path integral formalism, in which the fermionic creation and annihilation operators are represented by Grassmann variables. Temporal derivatives on influence functionals are then performed in a hierarchical manner, on the basis of calculus-on-path-integral algorithm. Both the multiple-frequency-dispersion and the non-Markovian reservoir parametrization schemes are considered for the desired hierarchy construction. The resulting formalism is in principle exact, applicable to interacting systems, with arbitrary time-dependent external fields. It renders an exact tool to evaluate various transient and stationary quantum transport properties of many-electron systems. At the second-tier truncation level the present theory recovers the real-time diagrammatic formalism developed by Schön and coworkers.2 For a single-particle system, the hierarchical formalism terminates at the second tier exactly, and the Landuer-Büttiker's transport current expression is readily recovered. Numerical studies will be presented to highlight the richness of transient current through both interacting and noninteracting model systems.
We extend a standard stochastic theory to study open quantum systems coupled to a generic quantum environment. We exemplify the general framework by studying a two-level quantum system coupled bilinearly to the three fundamental classes of non-interacting particles: bosons, fermions, and spins. In this unified stochastic approach, the generalized stochastic Liouville equation (SLE) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied. Anharmonic effects of a non-Gaussian bath are precisely encoded in the bath multi-time correlation functions that noise variables have to satisfy. Starting from the SLE, we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi-time correlation functions in a complete basis of orthonormal functions. The general hierarchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics. For bosonic bath models, our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes. Similar efficiency and flexibility can be achieved for the fermionic bath models within our formalism. The spin bath models can be simulated with two complementary approaches in the present formalism. (I) They can be viewed as an example of non-Gaussian bath models and be directly handled with the general hierarchical equation approach given their multi-time correlation functions. (II) Alternatively, each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism.
The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schrödinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born-Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system.