Jeremy Clark

Diffusive behavior from a quantum master equation

We study a general class of translation invariant quantum Markov evolutions for a particle on Zd. 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.

A Brownian particle in a microscopic periodic potential

We study a model for a massive test particle in a microscopic periodic potential and interacting with a reservoir of light particles. In the regime considered, the fluctuations in the test particle’s momentum resulting from collisions typically outweigh the shifts in momentum generated by the periodic force, so the force is effectively a perturbative contribution. The mathematical starting point is an idealized reduced dynamics for the test particle given by a linear Boltzmann equation. In the limit that the mass ratio of a single reservoir particle to the test particle tends to zero, we show that there is convergence to the Ornstein–Uhlenbeck process under the standard normalizations for the test particle variables. Our analysis is primarily directed towards bounding the perturbative effect of the periodic potential on the particle’s momentum.

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

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 an adiabatic 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

Diffusive Limit for a Quantum Linear Boltzmann Dynamics

t^{\frac{5}{4}}

Diffusive Behavior for Randomly Kicked Newtonian Particles in a Spatially Periodic Medium

the integral process converges to a time-changed Brownian motion whose diffusion rate depends on the momentum process. The scaling

An infinite-temperature limit for a quantum scattering process

t^{\frac{5}{4}}

Decoherence rates for Galilean covariant dynamics

contrasts with

Bounds for the state-modulated resolvent of a linear Boltzmann generator

t^{\frac{3}{2}}

A Ballistic Motion Disrupted by Quantum Reflections

, 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 momentum reflections that occur when the particle’s momentum is kicked near the half-spaced reciprocal lattice of the potential.

The reduced effect of a single scattering with a low-mass particle via a point interaction

In this article, I study the diffusive behavior for a quantum test particle interacting with a dilute background gas. The model that I begin with is a reduced picture for the test particle dynamics given by a quantum linear Boltzmann equation in which the gas particle scattering is assumed to occur through a hard-sphere interaction. The state of the particle is represented by a density matrix that evolves according to a translation-covariant Lindblad equation. The main result is a proof that the particle’s position distribution converges to a Gaussian under diffusive rescaling.