Quantum diffusion with drift and the Einstein relation. I

Communications in Mathematical Physics, 2009

We consider a quantum particle coupled (with strength λ) to a spatial array of independent non-interacting reservoirs in thermal states (heat baths). Under the assumption that the reservoir correlations decay exponentially in time, we prove that the motion of the particle is diffusive at large times for small, but finite λ. Our proof relies on an expansion around the kinetic scaling limit (λ 0, while time and space scale as λ −2) in which the particle satisfies a Boltzmann equation. We also show an equipartition theorem: the distribution of the kinetic energy of the particle tends to a Maxwell-Boltzmann distribution, up to a correction of O(λ 2).

The European Physical Journal B, 2007

A fresh and critical look has been given to the long-time behaviour of the quantum diffusion problem and a marginally more accurate solution has been obtained as compared to the one reported in the recent literature. Attempt has also been made to bring out a few interesting generic features of this problem which may have important bearing on real systems in the quantum domain.

Journal of Physics A: Mathematical and Theoretical, 2012

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.

Europhysics Letters (EPL), 1986

The mapping of classical diffusion onto a quantum system is used to explore the connections between diffusion phenomena in random media, llw noise and the motion of a quantum particle in a random potential.

Journal of Physics A: Mathematical and Theoretical, 2007

We apply Stochastic Quantization to a system of N interacting identical Bosons in an external potential Φ, by means of a general stationary-action principle. The collective motion is described in terms of a Markovian diffusion on R 3N , with joint densityρ and entangled current velocity fieldV , in principle of non-gradient form, related one to the other by the continuity equation. Dynamical equations relax to those of canonical quantization, in some analogy with Parisi-Wu stochastic quantization. Thanks to the identity of particles, the one-particle marginal densities ρ, in the physical space R 3 , are all the same and it is possible to give, under mild conditions, a natural definition of the single-particle current velocity, which is related to ρ by the continuity equation in R 3. The motion of single particles in the physical space comes to be described in terms of a non-Markovian three-dimensional diffusion with common density ρ and, at least at dynamical equilibrium, common current velocity v. The three-dimensional drift is perturbed by zero-mean terms depending on the whole configuration of the N-bosons interacting system. Finally we discuss in detail under which conditions the one-particle dynamical equations, which in their general form allow rotational perturbations, can be particularized, up to a change of variables, to Gross-Pitaevskii equations.

Physical Review E, 2006

Diffusive transport properties of a quantum Brownian particle moving in a tilted spatially periodic potential and strongly interacting with a thermostat are explored. Apart from the average stationary velocity, we foremost investigate the diffusive behavior by evaluating the effective diffusion coefficient together with the corresponding Peclet number. Corrections due to quantum effects, such as quantum tunneling and quantum fluctuations, are shown to substantially enhance the effectiveness of diffusive transport if only the thermostat temperature resides within an appropriate interval of intermediate values.

AIP Conference Proceedings, 2014

We use ultracold atoms in a quasiperiodic lattice to study two outstanding problems in the physics of disordered systems: a) the anomalous diffusion of a wavepacket in the presence of disorder, interactions and noise; b) the transport of a disordered superfluid. a) Our results show that the subdiffusion, observed when interaction alone is present, can be modelled with a nonlinear diffusion equation and the peculiar shape of the expanding density profiles can be connected to the microscopic nonlinear diffusion coefficients. Also when noise alone is present we can describe the observed normal diffusion dynamics by existing microscopic models. In the unexplored regime in which noise and interaction are combined, instead, we observe an anomalous diffusion, that we model with a generalized diffusion equation, where noise-and interaction-induced contributions add each other. b) We find that an instability appearing at relatively large momenta can be employed to locate the fluid-insulator crossover driven by disorder. By investigating the momentum-dependent transport, we observe a sharp crossover from a weakly dissipative regime to a strongly unstable one at a disorder-dependent critical momentum. The set of critical disorder and interaction strengths for which such critical momentum vanishes, can be identified with the separation between a fluid regime and an insulating one and can be related to the predicted zero-temperature superfluid-Bose glass transition.

Physica Scripta, 1988

Phys. Rev. E, 2019

This article discusses the numerical result predicted by the quantum Langevin equation of the generalized diffusion function of a Brownian particle immersed in an Ohmic quantum bath of harmonic oscillators. The time dependence of the standard deviation of the reduced Wigner function of the system, obtained by integrating the whole function in the momentum space, is determined as well. The complexity of the equations leads to resort to a much simpler calculation based in the position correlation function. They are done for the three possible regimes of the system, namely, periodic, aperiodic, and overdamped. It is found in the periodic case that the generalized diffusion is a discontinuous function exhibiting negative values during short time periods of time. This counterintuitive result, found theoretically in other systems and waiting for its experimental confirmation, can be perfectly explained in the framework of the quantum Langevin equation. Its oscillatory behavior is primordially due to the response to the external field while its quantum origin contributes only in its magnitude. The results are compared to those of the continuum limit which exhibits similar behavior.

2011

We prove diffusion for a quantum particle coupled to a field of bosons (phonons or photons). The importance of this result lies in the fact that our model is fully Hamiltonian and randomness enters only via the initial (thermal) state of the bosons. This model is closely related to the one considered in [9] but various restrictive assumptions of the latter have been eliminated. In particular, depending on the dispersion relation of the bosons, the present result holds in dimension d ≥ 3 and no severe infrared conditions on the coupling are necessary.