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arXiv:2206.13985

Semi-classical theory of quantum stochastic resistors

Tony Jin, João S. Ferreira, Michel Bauer, Michele Filippone, Thierry Giamarchi

Published Jun 28, 2022• 0 citations
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Abstract

We devise a semi-classical model to describe the transport properties of low-dimensional fermionic lattices under the influence of external quantum stochastic noise. These systems behave as quantum stochastic resistors, where the bulk particle transport is diffusive and obeys the Ohm/Fick's law. Here, we extend previous exact studies beyond the one-dimensional limit to ladder geometries and explore different dephasing mechanisms that are relevant to different physical systems, from solid-state to cold atoms. We find a non-trivial dependence of the conductance of these systems on the chemical potential of the reservoirs. We then introduce a semi-classical approach that is in good agreement with the exact numerical solution and provides an intuitive and simpler interpretation of transport in quantum stochastic resistors. Moreover, we find that the conductance of quantum ladders is insensitive to the coherence of the dephasing process along the direction transverse to transport, despite the fact that the system reaches different stationary states. We conclude by discussing the case of dissipative leads affected by dephasing, deriving the conditions for which they effectively behave as Markovian injectors of particles in the system.

DOI
10.48550/arxiv.2206.13985

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We devise a semi-classical model to describe the transport properties of low-dimensional fermionic lattices under the influence of external quantum stochastic noise. These systems behave as quantum stochastic resistors, where the bulk particle transport is diffusive and obeys the Ohm/Fick's law. Here, we extend previous exact studies beyond the one-dimensional limit to ladder geometries and explore different dephasing mechanisms that are relevant to different physical systems, from solid-state to cold atoms. We find a non-trivial dependence of the conductance of these systems on the chemical potential of the reservoirs. We then introduce a semi-classical approach that is in good agreement with the exact numerical solution and provides an intuitive and simpler interpretation of transport in quantum stochastic resistors. Moreover, we find that the conductance of quantum ladders is insensitive to the coherence of the dephasing process along the direction transverse to transport, despite the fact that the system reaches different stationary states. We conclude by discussing the case of dissipative leads affected by dephasing, deriving the conditions for which they effectively behave as Markovian injectors of particles in the system.
We devise a semiclassical model to describe the transport properties of low-dimensional fermionic lattices under the influence of external quantum stochastic noise. These systems behave as quantum stochastic resistors, where the bulk particle transport is diffusive and obeys the Ohm/Fick's law. Here, we extend previous exact studies beyond the one-dimensional limit to ladder geometries and explore different dephasing mechanisms that are relevant to different physical systems, from solid-state to cold atoms. We show how the semiclassical description is useful to explain the nontrivial dependence of the conductance of these systems on the chemical potential of the reservoirs. This description provides an intuitive and simpler interpretation of transport in quantum stochastic resistors in good quantitative agreement with the exact numerical solution. Moreover, we find that the conductance of quantum ladders is insensitive to the coherence of the dephasing process along the direction transverse to transport, despite the fact that the system reaches different stationary states.
We study the transport properties of generic out-of-equilibrium quantum systems connected to fermionic reservoirs. We develop a perturbation scheme in the inverse system size, named 1/N1/N1/N expansion, to study a large class of out of equilibrium diffusive/ohmic systems. The bare theory is described by a Gaussian action corresponding to a set of independent two level systems at equilibrium. This allows a simple and compact derivation of the diffusive current as a first-order pertubative term. In addition, we obtain exact solutions for a large class of quantum stochastic Hamiltonians (QSHs) with time and space dependent noise, using a self-consistent Born diagrammatic method in the Keldysh representation. We show that these QSHs exhibit diffusive regimes, which are encoded in the Keldysh component of the single particle Green's function. The exact solution for these QSHs models confirms the validity of our system size expansion ansatz, and its efficiency in capturing the transport properties. We consider in particular three fermionic models: (i) a model with local dephasing, (ii) the quantum simple symmetric exclusion process model, and (iii) a model with long-range stochastic hopping. For (i) and (ii) we compute the full temperature and dephasing dependence of the conductance of the system, both for two- and four-points measurements. Our solution gives access to the regime of finite temperature of the reservoirs, which could not be obtained by previous approaches. For (iii), we unveil a ballistic-to-diffusive transition governed by the range and the nature (quantum or classical) of the hopping. As a byproduct, our approach equally describes the mean behavior of quantum systems under continuous measurement.
Abstract We study the transport properties of a one-dimensional quantum system with disorder. We numerically compute the frequency dependence of the conductivity of a fermionic chain with nearest-neighbor interaction and a random chemical potential by using the Chebyshev matrix product state (CheMPS) method. As a benchmark, we investigate the noninteracting case first. Comparison with exact diagonalization and analytical solutions demonstrates that the results of CheMPS are reliable over a wide range of frequencies. We then calculate the dynamical conductivity spectra of the interacting system for various values of the interaction and disorder strengths. In the high-frequency regime, the conductivity decays as a power law, with an interaction-dependent exponent. This behavior is qualitatively consistent with the bosonized field theory predictions, although the numerical evaluation of the exponent shows deviations from the analytically expected values. We also compute the characteristic pinning frequency at which a peak in the conductivity appears. We confirm that it is directly related to the inverse of the localization length, even in the interacting case. We demonstrate that the localization length follows a power law of the disorder strength with an exponent dependent on the interaction, and find good quantitative agreement with the field theory predictions. In the low-frequency regime, we find a behavior consistent with the one of the noninteracting system ω2(ln⁡ω)2\omega ^{2}(\ln \omega )^{2}ω2(lnω)2 ω 2 ( ln ω ) 2 independently of the interaction. We discuss the consequences of our finding for experiments in cold atomic gases. Graphical abstract
We analyze quantal Brownian motion in ddd dimensions using the unified model for diffusion localization and dissipation, and Feynman-Vernon formalism. At high temperatures the propagator possess a Markovian property and we can write down an equivalent Master equation. Unlike the case of the Zwanzig-Caldeira-Leggett model, genuine quantum mechanical effects manifest themselves due to the disordered nature of the environment. Using Wigner picture of the dynamics we distinguish between two different mechanisms for destruction of coherence. The analysis of dephasing is extended to the low temperature regime by using a semiclassical strategy. Various results are derived for ballistic, chaotic, diffusive, both ergodic and non-ergodic motion. We also analyze loss of coherence at the limit of zero temperature and clarify the limitations of the semiclassical approach. The condition for having coherent effect due to scattering by low-frequency fluctuations is also pointed out. It is interesting that the dephasing rate can be either larger or smaller than the dissipation rate, depending on the physical circumstances.
The transport properties of a bosonic chain have been calculated by placing the ends of the chain in contact with thermal and particle reservoirs at different temperatures and chemical potentials. The contact with the reservoirs is described by the use of a quantum Fokker-Planck-Kramers equation, which is a canonical quantization of the classical Fokker-Planck-Kramers equation. From the quantum equation we obtain equations for the covariances of the creation and annihilation boson operators and solve them in the stationary state for small interactions. From the covariances we determine the Onsager coefficients and in particular the conductance, which was found to be finite for any chain size leading to an infinite conductivity and the absence of Fourier's law.
We analyse quantal Brownian motion in d dimensions using the unified model for diffusion localization and dissipation, and Feynman-Vernon formalism. At high temperatures the propagator possesses a Markovian property and we can write down an equivalent master equation. Unlike the case of the Zwanzig-Caldeira-Leggett model, genuine quantum mechanical effects manifest themselves due to the disordered nature of the environment. Using Wigner's picture of the dynamics we distinguish between two different mechanisms for destruction of coherence: scattering perturbative mechanism and smearing non-perturbative mechanism. The analysis of dephasing is extended to the low-temperature regime by using a semiclassical strategy. Various results are derived for ballistic, chaotic, diffusive, both ergodic and non-ergodic motion. We also analyse loss of coherence at the limit of zero temperature and clarify the limitations of the semiclassical approach. The condition for having coherent effect due to scattering by low-frequency fluctuations is also pointed out. It is interesting that the dephasing rate can be either larger or smaller than the dissipation rate, depending on the physical circumstances.