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arXiv:1404.2286 · Physical Review A

Dynamics of noisy quantum systems in the Heisenberg picture: Application to the stability of fractional charge

Armin Rahmani

Published Oct 19, 2015• 10 citations
arXivHTMLPDF

Abstract

Based on the Heisenberg-picture analog of the master equation, we develop a method for computing the exact time dependence of noise-averaged observables for general noninteracting fermionic systems with noisy fluctuations. Upon noise averaging, these fluctuations generate effective interactions, limiting analytical approaches. While the short-time dynamics can be studied with Langevin-type numerical simulations, the long-time limit is not amenable to such simulations. Our results provide access to this long-time limit. As a simple example, we examine the fate of the fractional charge in cold-atom emulations of polyacetylene after stochastic driving. We find that in a quantum quench to a fluctuating hopping Hamiltonian, the fractional charge remains robust for hopping between different sublattices, while it becomes unstable in the presence of noisy hopping on the same sublattice.

DOI
10.1103/physreva.92.042110

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Europhysics Letters (EPL)•Apr 1, 2021•36 citations
Based on the Heisenberg-picture analog of the master equation, we develop a method for computing the exact time dependence of noise-averaged observables for general noninteracting fermionic systems with noisy fluctuations. Upon noise averaging, these fluctuations generate effective interactions, limiting analytical approaches. While the short-time dynamics can be studied with Langevin-type numerical simulations, the long-time limit is not amenable to such simulations. Our results provide access to this long-time limit. As a simple example, we examine the fate of the fractional charge in cold-atom emulations of polyacetylene after stochastic driving. We find that in a quantum quench to a fluctuating hopping Hamiltonian, the fractional charge remains robust for hopping between different sublattices, while it becomes unstable in the presence of noisy hopping on the same sublattice.
Noise-driven noninteracting fermions have an interacting effective action. This makes it exceedingly difficult to compute their noise-averaged density matrix. Nevertheless, here we show that the exact time dependence of noise-averaged quadratic observables (such as energy and the fermionic Green's function) can be obtained from a Heisenberg-picture approach. Our results provide access to a long-time limit, which is not amenable to numerical simulations. As a simple example, we examine the fate of fractional charge in a noisy dimerized lattice (relevant to cold-atom emulations of polyacetylene). We find that the fractional charge remains robust against noisy nearest-neighbor hopping processes, while it becomes unstable to fluctuations in the second-neighbor hopping.
We study how time-dependent energy fluctuations impact the dynamical quantum phase transitions (DQPTs) following a noisy ramped quench of the transverse magnetic field in a quantum Ising chain. By numerically solving the stochastic Schrödinger equation of the mode-decoupled fermionic Hamiltonian of the problem, we identify two generic scenarios: Depending on the amplitude of the noise and the rate of the ramp, the expected periodic sequence of noiseless DQPTs may either be uniformly shifted in time or else replaced by a disarray of closely spaced DQPTs. Guided by an exact noise master equation, we trace the phenomenon to the interplay between noise-induced excitations which accumulate during the quench and the near-adiabatic dynamics of the massive modes of the system. Our analysis generalizes to any one-dimensional fermionic two-band model subject to a noisy quench. Published by the American Physical Society 2024
A system of spins coupled to a bath is a traditional setup in open quantum systems. Through Heisenberg's equation, the spin dynamics can be modeled by a set of first-order differential equations. Interpreting the terms as colored noise and non-Markovian damping, one can write them as quantummechanical Heisenberg-Langevin (HL) equations. These are notoriously difficult to solve because of the high dimensionality of the Hilbert space. Classical generalized Langevin equations, involving non-Markovian damping and colored noise, are well understood and can be treated numerically with relative ease. Thus, a classical ansatz can be made by substituting quantum expectation values with classical functions. This allows the application of standard methods developed for classical stochastic dynamical systems to tackle spin dynamics. However, this approach is uncontrolled and should be benchmarked against known quantum dynamics. In this investigation, a Hamiltonian for spin dynamics is modified to obtain a setup analogous to the Weisskopf-Wigner (WW) theory of spontaneous emission, enabling a comparison of the results. This will be compared for T = 0 and with a slight adaptation in the high-temperature limit.
Following on from our recent work, we investigate a stochastic approach to non-equilibrium quantum spin systems. We show how the method can be applied to a variety of physical observables and for different initial conditions. We provide exact formulae of broad applicability for the time-dependence of expectation values and correlation functions following a quantum quench in terms of averages over classical stochastic processes. We further explore the behavior of the classical stochastic variables in the presence of dynamical quantum phase transitions, including results for their distributions and correlation functions. We provide details on the numerical solution of the associated stochastic differential equations, and examine the growth of fluctuations in the classical description. We discuss the strengths and limitations of the current implementation of the stochastic approach and the potential for further development.
We study the fluctuation statistics of integrated currents in noisy quantum diffusive systems, focusing on the Quantum Symmetric Simple Exclusion and Inclusion Processes (QSSEP/QSSIP). These one-dimensional fermionic (QSSEP) and bosonic (QSSIP) models feature stochastic nearest-neighbor hopping driven by Brownian noise, together with boundary injection and removal processes. They provide solvable microscopic settings in which quantum coherence coexists with diffusion. Upon noise averaging, their dynamics reduce to those of the classical SSEP/SSIP. We show that the cumulant-generating function of the integrated current, at large scales, obeys a large deviation principle. To leading order in system size and for each noise realization, it converges to that of the corresponding classical process, establishing a classical typicality of current fluctuations in these noisy quantum systems. We further demonstrate a direct connection with Macroscopic Fluctuation Theory (MFT), showing that the large-scale equations satisfied by biased quantum densities coincide with the steady-state Hamilton equations of MFT, thereby providing a microscopic quantum justification of the MFT framework in these models. Finally, we identify the leading finite-size corrections to the current statistics. We show the existence of subleading contributions of purely quantum origin, which are absent in the corresponding classical setting, and provide their explicit expressions for the second and third current cumulants. These quantum corrections are amenable to direct experimental or numerical verification, provided sufficient control over the noise realizations can be achieved. Their presence points toward the necessity of a quantum extension of MFT.
Abstract All quantum systems are subject to noise and imperfections due to stray fields, inhomogeneities or drifting experimental controls. An understanding of the effects of noise and decoherence is critical to the progress towards fully functional quantum devices. In this perspective, we focus on noise in quantum systems which are modelled by a dynamic stochastic parameter in the Hamiltonian. We will outline exact evolution equations describing the ensemble average dynamics for a variety of common noise types and their connections. We will also highlight an approximate evolution equation valid in the weak noise limit for an arbitrary classical stochastic field. This framework should serve as a starting point for identifying signatures of differing noise types and optimisation of robust control protocols.