Diffusive Limit for a Quantum Linear Boltzmann Dynamics

We study a general class of translation invariant quantum Markov evolutions for a particle on $\bbZ^d$. The evolution consists of free flow, interrupted by scattering events. We assume spatial locality of the scattering events and exponentially fast relaxation of the momentum distribution. It is shown that the particle position diffuses in the long time limit. This generalizes standard results about central limit theorems for classical (non-quantum) Markov processes.Comment: 21 pages. Minor corrections of previous versio

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“…A quantum example that is very related to ours is in [15]. Also, the diffusive limit for another quantum master equation is studied in [4].…”

Section: Introductionmentioning

confidence: 68%

Diffusive behavior from a quantum master equation

Clark

Roeck

Maes

2011

Journal of Mathematical Physics

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“…This is simple to prove, since it can be reduced to showing exponential ergodicity for a classical Kolmogorov equation. I have discussed exponential dissipation in the article [11] for a three-dimensional case (hard-sphere interaction) for the purpose of studying diffusion. The discussion of the HamiltonianĤ is the same as before except for the inclusion of physical constants.…”

Section: Analogous Conjectures For a Dissipative Modelmentioning

confidence: 99%

Suppressed Dispersion for a Randomly Kicked Quantum Particle in a Dirac Comb

Clark

2013

J Stat Phys

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I study a model for a massive one-dimensional particle in a singular periodic potential that is receiving kicks from a gas. The model is described by a Lindblad equation in which the Hamiltonian is a Schrödinger operator with a periodic δ-potential and the noise has a frictionless form arising in a Brownian limit. I prove that an emergent Markov process in a semi-classical limit governs the momentum distribution in the extended-zone scheme. The main result is a central limit theorem for a time integral of the momentum process, which is closely related to the particle's position. When normalized by t 5 4 , the integral process converges to a time-changed Brownian motion whose rate depends on the momentum process. The scaling t 5 4 contrasts with t 3 2 , which would be expected for the case of a smooth periodic potential or for a comparable classical process. The difference is a wave effect driven by Bragg reflections that occur when the particle's momentum is kicked near the half-spaced reciprocal lattice.where the Hilbert spaces H φ are canonically identified with L 2 [−π, π) , and the restriction of H to the φ-fiber is a self-adjoint operator H φ . The operators H φ have a complete set of eigenvectors ψ n,φ , n ∈ N with eigenvalues E n,φ satisfying 0 ≤ E n,φ ≤ E n+1,φ , E n,φ −→ ∞ as n −→ ∞.

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“…In order to arrive at the quantum angular momentum diffusion equation we proceed in close analogy to the derivation of momentum diffusion of point particles from the master equation of collisional decoherence [41][42][43][44]. There one considers the limit of small momentum kicks and expands the Lindblad operators to leading order in the position operator.…”

Section: Master Equation Of Orientational Decoherencementioning

confidence: 99%

Quantum angular momentum diffusion of rigid bodies

2017

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We show how to describe the diffusion of the quantized angular momentum vector of an arbitrarily shaped rigid rotor as induced by its collisional interaction with an environment. We present the general form of the Lindblad-type master equation and relate it to the orientational decoherence of an asymmetric nanoparticle in the limit of small anisotropies. The corresponding diffusion coefficients are derived for gas particles scattering off large molecules and for ambient photons scattering off dielectric particles, using the elastic scattering amplitudes.The first term accounts for the free dynamics, the second part describes the effect of the environment. It follows from equation (2), similar to the derivation of the Fokker-Planck equation [27, 29], thatNote that the use of canonical momenta instead of J would render the free evolution in (4) phase space volume preserving, but it would complicate considerably the diffusive part (5).In order to observe that equation (5) indeed describes angular momentum diffusion, we note that the expectation value of J remains constant while its second moment increases linearly with time, J J J 0 and 2 D . 6 t t

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Diffusive Limit for a Quantum Linear Boltzmann Dynamics

Cited by 4 publications

References 43 publications

Diffusive behavior from a quantum master equation

Diffusive behavior from a quantum master equation

Suppressed Dispersion for a Randomly Kicked Quantum Particle in a Dirac Comb

Quantum angular momentum diffusion of rigid bodies

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