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Non-equilibrium transition from dissipative quantum walk to classical random walk

Marco Nizama, Manuel O. Cáceres

Published Jul 1, 2012• 8 citations
OpenAlex

Abstract

We have investigated the time evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson’s boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann’s entropy and quantum purity. In addition, we have studied Wigner’s function. The timedependent behavior of the quantum entanglement between a free particle—in the lattice—and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.

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We have investigated the time evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson’s boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann’s entropy and quantum purity. In addition, we have studied Wigner’s function. The timedependent behavior of the quantum entanglement between a free particle—in the lattice—and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.
We have investigated the time evolution of a free particle in interaction with a phonon thermal bath, using the tight-binding approach. A dissipative quantum walk can be defined and many important non-equilibrium decoherence properties can be investigated analytically. The non-equilibrium statistics of a pure initial state have been studied. Our theoretical results indicate that the evolving wave-packet shows the suppression of Anderson's boundaries (ballistic peaks) by the presence of dissipation. Many important relaxation properties can be studied quantitatively, such as von Neumann's entropy and quantum purity. In addition, we have studied Wigner's function. The time-dependent behavior of the quantum entanglement between a free particle—in the lattice—and the phonon bath has been characterized analytically. This result strongly suggests the non-trivial time dependence of the off-diagonal elements of the reduced density matrix of the system. We have established a connection between the quantum decoherence and the dissipative parameter arising from interaction with the phonon bath. The time-dependent behavior of quantum correlations has also been pointed out, showing continuous transition from quantum random walk to classical random walk, when dissipation increases.
Bath-induced correlations in an infinite dimension Hilbert space has been shown. Quantum correlations between two free spinless dissipative distinguishable particles (initially uncorrelated) are studied analytically using the quantum master equation. We show that for temperature T>0T>0T>0 the time-evolution of the reduced density matrix cannot be written as the direct product of two independent particles. We found a time-scale that characterizes the time when the bath-induced correlation is maximum before being wipe out by dissipation (purity, spatial, and mirror correlations are studied). We introduce a Wigner function associated to the Wannier lattice where the dissipative quantum walks move. Thus we discuss the peculiarities of the scenario of enlarging the lattice to the semi-integers in order to fulfilll all the fundamental properties of the Wigner function. We have supported the quantum character of the correlations by analyzing the negativity of the Wigner function.
Quantum correlations and coherence generated between two free spinless particles in the lattice, interacting with a common quantum phonon bath, are studied. The reduced density matrix is solved using the Markov approach. We show that the bath induces correlations between the particles. The coherence induced by the bath is studied, calculating off-diagonal elements of the density matrix, spatiotemporal dispersion, purity and quantum mutual information. We find a characteristic time-scale pointing out when this coherence is maximum. In addition, a Wigner-like distribution in the phase-space (lattice) is introduced as an indirect indicator of the quantumness of total correlations and coherence induced by the thermal bath. The negative volume of the Wigner function also shows a behavior which is in agreement with the time-scale that we have found. A Gaussian distribution for the profile of particles is not obtained, and interference patterns are observed as the result of bath-induced coherence. As the temperature of the bath vanishes, the ballistic behavior of the tight-binding model is recovered. The geometric quantum discord is calculated, to characterize the nature of the correlations.
We introduce the quantum Lévy walk to study transport and decoherence in a quantum random model. We have derived from second-order perturbation theory the quantum master equation for a Lévy-like particle that moves along a lattice through scale-free hopping while interacting with a thermal bath of oscillators. The general evolution of the quantum Lévy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation and the probability to be in a lattice site, all of them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general, our approach allows us to study analytically quantum fluctuations and decoherence in a long-range hopping model.
Abstract Quantum correlations and coherence generated between two free spinless particles in the lattice, interacting with a common quantum phonon bath, are studied. The reduced density matrix is solved using the Markov approach. We show that the bath induces correlations between the particles. The coherence induced by the bath is studied, calculating off-diagonal elements of the density matrix, spatiotemporal dispersion, purity and quantum mutual information. We find a characteristic time-scale pointing out when this coherence is maximum. In addition, a Wigner-like distribution in the phase-space (lattice) is introduced as an indirect indicator of the quantumness of total correlations and coherence induced by the thermal bath. The negative volume of the Wigner function also shows a behavior which is in agreement with the time-scale that we have found. A Gaussian distribution for the profile of particles is not obtained, and interference patterns are observed as the result of bath-induced coherence. As the temperature of the bath vanishes, the ballistic behavior of the tight-binding model is recovered. The geometric quantum discord is calculated, to characterize the nature of the correlations.
We introduce the quantum Levy walk to study transport and decoherence in a quantum random model. We have derived from second order perturbation theory the quantum master equation for a extit{Levy-like particle }that moves along a lattice through hopping scale-free while interacting with a thermal bath of oscillators. The general evolution of the quantum Levy particle has been solved for different preparations of the system. We examine the evolution of the quantum purity, the localized correlation, and the probability to be in a lattice site, all them leading to important conclusions concerning quantum irreversibility and decoherence features. We prove that the quantum thermal mean-square displacement is finite under a constraint that is different when compared to the classical Weierstrass random walk. We prove that when the mean-square displacement is infinite the density of state has a complex null-set inside the Brillouin zone. We show the existence of a critical behavior in the continuous eigenenergy which is related to its non-differentiability and self-affine characteristics. In general our approach allows to study analytically quantum fluctuations and decoherence in a long-range hopping model.