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arXiv:2509.22604 · Journal of Physics A Mathematical and Theoretical

Exact solutions of open quantum Brownian motions on the real line for two-level systems

Manuel D. de la Iglesia, Carlos F. Lardizabal

Published Nov 19, 2025• 0 citations
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Abstract

Abstract We investigate open quantum Brownian motions as quantum analogues of classical diffusion processes under interaction with an external environment. Building upon the microscopic derivation by Sinayskiy and Petruccione (2015 Phys. Scr. T165 014017), we revisit the associated master equation and study its formulation as a generalized parabolic system. Employing Fourier transform methods, we derive exact analytical solutions for one-dimensional evolutions of particles with two-level internal degree of freedom.

DOI
10.1088/1751-8121/ae21a7

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Abstract We investigate open quantum Brownian motions as quantum analogues of classical diffusion processes under interaction with an external environment. Building upon the microscopic derivation by Sinayskiy and Petruccione (2015 Phys. Scr. T165 014017), we revisit the associated master equation and study its formulation as a generalized parabolic system. Employing Fourier transform methods, we derive exact analytical solutions for one-dimensional evolutions of particles with two-level internal degree of freedom.
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Open Quantum Brownian motion describes a Quantum Brownian particle with an additional internal degree of freedom.The quantum master equation for a free Open Quantum Brownian walker with two‐dimensional internal degree of freedom interacting decoherently with the dissipative environment is derived. Using the analytical solution for the moments of the position distribution of the Brownian walker, one finds that the steady state average position of the walker can be controlled by the strength of an external classical driving field.
We revisit the model of a system made up of a Brownian quantum oscillator under the influence of an external classical force and linearly coupled to an environment made up of many quantum oscillators at zero or finite temperature. We show that the HPZ master equation for the reduced density matrix derived earlier [B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)] with coefficients obtained from solutions of integro-differential equations can assume closed functional forms for a fairly general class of spectral densities of the environment at arbitrary temperature and coupling strength. As an illustration of these new results we solve the corresponding master equation and calculate, among other physical quantities, the uncertainty function whose late time behavior can be obtained fully. This produces a formula for investigating the standard quantum limit which is central to addressing many theoretical issues in macroscopic quantum phenomena and experimental concerns related to low temperature precision measurements. We find that any initial state always settles down to a Gaussian density matrix whose covariance is determined by the thermal reservoir and whose mean is determined by the external force. For more general spectra we show that the solution of the master equation can be reduced to solving for the motion of a classical parametric oscillator with parametric frequency determined by the unsolved for master equation coefficients. States in these systems experience evolution that is parametrically similar to the simpler evolution explicitly determined for in the case of Laurent-series spectra.
Open Quantum Brownian Motion (OQBM) is a new class of quantum Brownian motion in which the dynamics of the Brownian particle depend not only on interactions with a thermal environment but also on the state of its internal degrees of freedom. For an Ohmic bath spectral density with a Lorentz--Drude cutoff frequency at a high-temperature limit, we derive the Born--Markov master equation for the reduced density matrix of an open Brownian particle in a harmonic potential. The resulting master equation is written in phase-space representation using the Wigner function, and due to the separation of associated timescales in the high-damping limit, we perform adiabatic elimination of the momentum variable to obtain OQBM. We numerically solve the derived master equation for the reduced density matrix of the OQBM, using both Gaussian and non-Gaussian initial distributions. In each case, the OQBM dynamics converge to several Gaussian distributions. To gain physical insight into the studied system, we also plotted the dynamics of the off-diagonal element of the open quantum Brownian particle and found damped coherent oscillations. Finally, we investigated the time-dependent variance in the position of the OQBM walker and observed initial diffusive spreading, followed by a crossover to a finite steady-state value.