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arXiv:1706.03984 · SciPost Physics

Stochastic dissipative quantum spin chains (I) : Quantum fluctuating discrete hydrodynamics

Michel Bauer, Denis Bernard, Tony Jin

Published Nov 4, 2017• 71 citations
arXivHTMLPDF

Abstract

Motivated by the search for a quantum analogue of the macroscopic fluctuation theory, we study quantum spin chains dissipatively coupled to quantum noise. The dynamical processes are encoded in quantum stochastic differential equations. They induce dissipative friction on the spin chain currents. We show that, as the friction becomes stronger, the noise induced dissipative effects localize the spin chain states on a slow mode manifold, and we determine the effective stochastic quantum dynamics of these slow modes. We illustrate this approach by studying the quantum stochastic Heisenberg spin chain.

DOI
10.21468/scipostphys.3.5.033

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Motivated by the search for a quantum analogue of the macroscopic fluctuation theory, we study quantum spin chains dissipatively coupled to quantum noise. The dynamical processes are encoded in quantum stochastic differential equations. They induce dissipative friction on the spin chain currents. We show that, as the friction becomes stronger, the noise induced dissipative effects localize the spin chain states on a slow mode manifold, and we determine the effective stochastic quantum dynamics of these slow modes. We illustrate this approach by studying the quantum stochastic Heisenberg spin chain.
Three existing interpretations of quantum mechanics, given by Heisenberg, Bohm and Madelung, are examined to describe dissipative quantum systems as well. It is found that the Madelung quantum hydrodynamics is the only correct approach. A new stochastic reinterpretation of the quantum mechanics is proposed, which represents the microscopic face of the Madelung hydrodynamics. The main idea is that the vacuum fluctuates permanently, which explains the probabilistic character of the quantum mechanics. Thus, it is an objective theory independent of the human beings and their measurements. The effect of the thermal fluctuations in the surrounding is also accounted for via a heuristic Langevin equation with two random forces. Some statistical characteristics of these quantum and thermal noises are determined by reproducing known results for the system phase-space dynamics.
Three existing interpretations of quantum mechanics, given by Heisenberg, Bohm and Madelung, are examined to describe dissipative quantum systems as well. It is found that the Madelung quantum hydrodynamics is the only correct approach. A new stochastic reinterpretation of the quantum mechanics is proposed, which represents the microscopic face of the Madelung hydrodynamics. The main idea is that the vacuum fluctuates permanently, which explains the probabilistic character of the quantum mechanics. Thus, it is an objective theory independent of the human beings and their measurements. The effect of the thermal fluctuations in the surrounding is also accounted for via a heuristic Langevin equation with two random forces. Some statistical characteristics of these quantum and thermal noises are determined by reproducing known results for the system phase-space dynamics.
Abstract The theory of open quantum systems plays a fundamental role in several scientific and technological disciplines, from quantum computing and information science to molecular electronics and quantum thermodynamics. Despite its widespread relevance, a rigorous formulation of quantum dissipation in conjunction with thermal noise remains a topic of active research. In this work, we establish a formal correspondence between classical stochastic thermodynamics, in particular the Fokker–Planck and Klein–Kramers equations, and the quantum master equation. Building on prior studies of multiplicative noise in classical stochastic differential equations, we demonstrate that thermal noise at the quantum level manifests as a multidimensional geometric stochastic process. By applying canonical quantization, we introduce a novel Hermitian dissipation operator that serves as a quantum analogue of classical viscous friction. This operator allows for a well-defined expression of heat exchange between a system and its environment, enabling the formulation of an alternative quantum equipartition theorem. Our framework ensures a precise energy balance that aligns with the first law of thermodynamics and an entropy balance consistent with the second law. The theoretical formalism is applied to two prototypical quantum systems, the harmonic oscillator and a particle in an infinite potential well, for which it provides new insights into nonequilibrium thermodynamics at the quantum scale. Our results advance the understanding of dissipation in quantum systems and establish a foundation for future studies on stochastic thermodynamics in the quantum domain.
Stochastic quantum entanglements give a basic problem in the dynamics of the coherence effect and the randomness of the dynamics. The paper examines the entanglement evolution in stochastic partial differential equations (SPDEs) in terms of dynamical chaos theory. We solve a family of SPDEs modeling quantum systems under Gaussian and Lévy noise, both in the weak and strong coupling limits. Spectral spatial discretization and stochastic Runge-Kutta time integration allow the calculation of numerically accurate solutions and the reconstruction of density matrices and calculation of entanglement measures, such as von Neumann entropy and concurrence. Measurement of chaotic properties is done by maximal Lyapunov exponents, Poincaré-type diagnostics, and sensitivity to initial conditions measures of high-dimensional stochastic dynamics. Findings indicate that stochastic forcing is much more efficient to promote entanglement and weak noise spin-offs non-zero concurrence in more than 65 percent of realizations, and high-noise regimes increase von Neumann entropy means more than 200 percent compared to deterministic baselines. Entanglement amplification is strongly related to positive Lyapunov exponents (Γ≈0.58), and high-chaos flows have oscillatory entanglement with peak-to-peak variations of 45%. Such results define a quantitative correlation between stochastic chaos and quantum correlations, and it is possible to note that the noise-induced instabilities can serve as a constructive mechanism creation of entanglements. The work gives a solid framework for studying quantum systems with stochastic perturbations and has insight that is applicable in quantum information processing, complex network dynamics, and semiclassical quantum-chaotic systems.
We develop an interacting extension of the Double Covariance Model (DCM), a stochastic subquantum framework in which macroscopic quantum dynamics emerge through coarse-graining of correlated microscopic fluctuations. Starting from local stochastic differential equations on subsystem Hilbert spaces, we derive a closed evolution equation for a coarse-grained double covariance operator using multi-scale Itô calculus and sliding-window averaging. The construction explicitly incorporates two separated temporal scales: a fast microscopic fluctuation scale governing subquantum stochastic processes and a slower macroscopic observation scale associated with coarse-grained dynamics. Within the hydrodynamic limit, where the ratio between microscopic correlation time and averaging-window scale vanishes, rapidly fluctuating corrections disappear and the effective dynamics converges to a deterministic macroscopic transport equation. We show that the emergent macroscopic dynamics has the exact Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form: coherent Hamiltonian evolution arises from deterministic subquantum flow, while dissipative channels emerge from quadratic noise correlations. The framework further demonstrates how non-separable interaction Hamiltonians can arise from strictly local, state-dependent stochastic feedback fields. In the fluctuation-free limit, the model reduces naturally to the standard von Neumann equation, providing a unified stochastic foundation for both open and closed quantum dynamics.