Scattering induced current in a tight-binding band

Bruneau, Laurent; Bièvre, Stephan De; Pillet, Claude-Alain

doi:10.1063/1.3555432

Cited by 12 publications

(16 citation statements)

References 14 publications

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“…These models were developed for they furnish toy models for quantum dissipative systems, they are at the same time Hamiltonian and Markovian, they spontaneously give rise to quantum stochastic differential equations in the continuous time limit. It has been proved in [7] and [8] that they constitute a good toy model for a quantum heat bath in some situations and that they can also give an account of the diffusive behavior of an electron in an electric field, when coupled to a heat bath. When adding to each step of the dynamics a measurement of the piece of the environment which has just interacted, we recover all the discrete-time quantum trajectories for quantum systems ( [15], [16], [17]).…”

Section: Introduction and Motivations 1generalitiesmentioning

confidence: 99%

Complex obtuse random walks and their continuous-time limits

Attal

Deschamps

Pellegrini

2015

Probab. Theory Relat. Fields

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We study a particular class of complex-valued random variables and their associated random walks: the complex obtuse random variables. They are the generalization to the complex case of the real-valued obtuse random variables which were introduced in [4] in order to understand the structure of normal martingales in R n .The extension to the complex case is mainly motivated by considerations from Quantum Statistical Mechanics, in particular for the seek of a characterization of those quantum baths acting as classical noises. The extension of obtuse random variables to the complex case is far from obvious and hides very interesting algebraical structures. We show that complex obtuse random variables are characterized by a 3-tensor which admits certain symmetries which we show to be the exact 3-tensor analogue of the normal character for 2-tensors (i.e. matrices), that is, a necessary and sufficient condition for being diagonalizable in some orthonormal basis. We discuss the passage to the continuous-time limit for these random walks and show that they converge in distribution to normal martingales in C N . We show that the 3-tensor associated to these normal martingales encodes their behavior, in particular the diagonalization directions of the 3-tensor indicate the directions of the space where the martingale behaves like a diffusion and those where it behaves like a Poisson process. We finally prove the convergence, in the continuous-time limit, of the corresponding multiplication operators on the canonical Fock space, with an explicit expression in terms of the associated 3-tensor again.

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Section: Introduction and Motivations 1generalitiesmentioning

confidence: 99%

Complex obtuse random walks and their continuous-time limits

Attal

Deschamps

Pellegrini

2015

Probab. Theory Relat. Fields

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“…Recalling the definition of ζ in (6.4) and of V and V (2) , we observe that, roughly speaking, M(z) contains all contributions to second order in λ from the correlation functions ζ, but higher orders of λ enter M(z) trough the field term λ 2 χ · X.…”

Section: Survey Of Expansionsmentioning

confidence: 81%

“…Thus, when taking the Laplace transform, it suffices to consider the Laplace transforms of U [0,t] and V [0,t] . The former being given by the resolvent of L S = ad(H S ), it suffices to consider the operators V [0,t] − V (2) [0,t] and V (2) [0,t] , respectively. The Laplace transform of V [0,t] − V (2) [0,t] can be computed explicitly (see below), and the claims concerning M in Lemma 6.1 can be checked easily.…”

Section: )mentioning

confidence: 99%

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Quantum diffusion with drift and the Einstein relation. I

Roeck

Fröhlich

Schnelli

2014

Journal of Mathematical Physics

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Abstract:We study the dynamics of a quantum particle hopping on a simple cubic lattice and driven by a constant external force. It is coupled to an array of identical, independent thermal reservoirs consisting of free, massless Bose fields, one at each site of the lattice. When the particle visits a site x of the lattice it can emit or absorb field quanta of the reservoir at x. Under the assumption that the coupling between the particle and the reservoirs and the driving force are sufficiently small, we establish the following results: The ergodic average over time of the state of the particle approaches a non-equilibrium steady state (NESS) describing a non-zero mean drift of the particle. Its motion around the mean drift is diffusive, and the diffusion constant and the drift velocity are related to one another by the Einstein relation.

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“…Since O Λ ≤ O , we conclude that the correlation functions in infinite volume satisfy (8.5), too. We set t 2 = 0 and define f a (t) := e λ 2 gt−at 2…”

Section: )mentioning

confidence: 99%

Quantum diffusion with drift and the Einstein relation. II

Roeck

Froehlich

Schnelli

2014

Journal of Mathematical Physics

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Diffusive behavior from a quantum master equationThis paper is a companion to Paper I [W. De Roeck, J. Fröhlich, and K. Schnelli, "Quantum diffusion with drift and the Einstein relation. I," J. Math. Phys. 55, 075206 (2014)]. The purpose of this paper is to describe and prove a certain number of technical results used in Paper I, but not proven there. Both papers concern long-time properties (diffusion, drift) of the motion of a driven quantum particle coupled to an array of thermal reservoirs. The main technical results derived in the present paper are: (1) an asymptotic perturbation theory applicable for small driving force, and (2) the construction of time-dependent correlation functions of particle observables. C 2014 AIP Publishing LLC. [http://dx.

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Scattering induced current in a tight-binding band

Cited by 12 publications

References 14 publications

Complex obtuse random walks and their continuous-time limits

Complex obtuse random walks and their continuous-time limits

Quantum diffusion with drift and the Einstein relation. I

Quantum diffusion with drift and the Einstein relation. II

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