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arXiv:1711.08957 · AIP conference proceedings

Stochastic wave-function simulation of irreversible emission processes for open quantum systems in a non-Markovian environment

Evgeny A. Polyakov, Alexey N. Rubtsov

Published Jan 1, 2018• 2 citations
arXivHTMLPDF

Abstract

When conducting the numerical simulation of quantum transport, the main obstacle is a rapid growth of the dimension of entangled Hilbert subspace. The Quantum Monte Carlo simulation techniques, while being capable of treating the problems of high dimension, are hindered by the so-called “sign problem”. In the quantum transport, we have fundamental asymmetry between the processes of emission and absorption of environment excitations: the emitted excitations are rapidly and irreversibly scattered away. Whereas only a small part of these excitations is absorbed back by the open subsystem, thus exercising the non-Markovian self-action of the subsystem onto itself. We were able to devise a method for the exact simulation of the dominant quantum emission processes, while taking into account the small backaction effects in an approximate self-consistent way. Such an approach allows us to efficiently conduct simulations of real-time dynamics of small quantum subsystems immersed in non-Markovian bath for large times, reaching the quasistationary regime. As an example we calculate the spatial quench dynamics of Kondo cloud for a bozonized Kodno impurity model.

DOI
10.1063/1.5025466

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When conducting the numerical simulation of quantum transport, the main obstacle is a rapid growth of the dimension of entangled Hilbert subspace. The Quantum Monte Carlo simulation techniques, while being capable of treating the problems of high dimension, are hindered by the so-called “sign problem”. In the quantum transport, we have fundamental asymmetry between the processes of emission and absorption of environment excitations: the emitted excitations are rapidly and irreversibly scattered away. Whereas only a small part of these excitations is absorbed back by the open subsystem, thus exercising the non-Markovian self-action of the subsystem onto itself. We were able to devise a method for the exact simulation of the dominant quantum emission processes, while taking into account the small backaction effects in an approximate self-consistent way. Such an approach allows us to efficiently conduct simulations of real-time dynamics of small quantum subsystems immersed in non-Markovian bath for large times, reaching the quasistationary regime. As an example we calculate the spatial quench dynamics of Kondo cloud for a bozonized Kodno impurity model.
In the dynamics of driven impurity models, there is a fundamental asymmetry between the processes of emission and absorption of environment excitations: most of the emitted excitations are rapidly and irreversibly scattered away, and only a small amount of them is reabsorbed back. We propose to use a stochastic simulation of the irreversible quantum emission processes in real-time dynamics, while taking into account the reabsorbed virtual excitations by the bath discretization. The resulting method delivers a fast convergence with respect to the number of bath sites, on a wide time interval, without the sign problem.
We present a real-time quantum Monte Carlo algorithm that simulates the dynamics of open quantum systems by stochastically compressing and evolving the density matrix under both Markovian and non-Markovian master equations. Our algorithm uses population dynamics to continuously suppress the sign problem, preventing its accumulation throughout the evolution for a broad class of noisy-circuit Liouvillians of practical and experimental interest. We apply it to a variety of quantum circuits and demonstrate significant speedups and scaling improvements over state-of-the-art quantum trajectory methods and convergence to exact solutions even in non-Markovian regimes where trajectory methods fail. Our approach improves the efficiency of classical simulation of gate-based quantum computing, quantum annealing, and general open system dynamics.
This thesis focuses on the study of dissipative quantum impurity models. Quantum impurity models describe a localized quantum system, known as the impurity, that interacts with a large quantum bath, enabling the exploration of essential phenomena such as the Kondo effect in magnetic impurities and the transport properties in quantum dots. An intriguing extension of these models is the exploration of the interplay between strong correlations and dissipation. Dissipation, often described by the Lindblad master equation in its Markovian limit, introduces challenges and opportunities as it interacts with coherent many-body effects. In this thesis, we want to investigate the non-equilibrium dynamics of quantum impurity systems influenced by both strong correlations and dissipation across four main parts. We begin by developing numerical methods, specifically a diagrammatic Monte Carlo technique, to analyze the real-time dynamics of dissipative quantum impurity models. Our findings demonstrate that local Markovian dissipation enhances the convergence of Monte Carlo sampling by constraining the process to a reduced subspace of diagrams. This effectively mitigates the sign problem and facilitates the exploration of longer time scales compared to conventional unitary approaches. In the second part, we derive a self-consistent dynamical map using the Non-Crossing Approximation to study the long-term behavior of dissipative quantum impurity models. This method effectively resums diagrams into a self-consistent Dyson equation, capturing essential non-Markovian environmental effects, including the Kondo effect. In a third part, we examine the relationship between Kondo physics and Markovian dissipation by investigating an interacting quantum dot coupled to a metallic bath under continuous charge monitoring. We identify a crossover from Kondo screening to the Quantum Zeno effect, effectively described by a non-Hermitian Kondo model in the weak dissipation regime. Finally, the last part explores the effects of particle losses on the Kondo effect and the Kondo-Zeno crossover. Preliminary results indicate that while single-particle losses reduce many-body effects, two-body losses introduce intriguing phenomena, such as spin decay and hints of the Kondo-Zeno crossover, along with an unusual spectral function exhibiting a low-frequency lineshape.
Simulation of open quantum systems is undoubtedly critical to design of quantum devices and applications of quantum technologies since dynamics of open quantum systems cannot be solved analytically in general. However, the dimension of open quantum systems grows exponentially as the number of their subsystems increases or as involving complicated environments, which reduces the computational efficiency of the simulation. In this paper, we boost numerical simulation of master equations of open quantum systems from several aspects with which the total time and the occupied memory in the simulation are significantly reduced without losing precision. Our method can be applied to the simulation of non-Markovian quantum systems described by an augmented system model as well. We conduct numerical experiments on simulation of a non-Markovian spin chain. Compared with the algorithm in a well-known toolbox, QuTiP, our method can take 26.3% time and less memory under fair conditions.
Various Monte Carlo methods allow the simulation of the dynamics of open quantum systems. We highlight the central features a commonly used method for Markovian systems, i.e., Monte Carlo Wave Function method and another one used for non-Markovian systems, i.e., the doubled Hilbert space method. One of the open problems in the field has been the lack of the existence of Monte Carlo method with quantum jumps for non-Markovian systems. We characterize the key problems that should be solved in this context and give indications for the way towards the development of quantum jump method without auxiliary states for non-Markovian systems.
We develop a systematic and efficient approach for numerically solving the non-Markovian quantum state diffusion equation for an open quantum system that can be strongly coupled to an environment. As an important application, we consider a real-time simulation of a spin-boson model in a strong-coupling regime that is difficult to deal with using conventional methods. We show that the non-Markovian stochastic Schr\"odinger equation can be efficiently implemented as a real-time simulation for this model, so as to give an accurate description of spin-boson dynamics beyond the rotating-wave approximation.