Spectra of Hamiltonians with Generalized Single-Site Dynamical Disorder

TL;DR

This paper introduces a family of stochastic processes derived from deformed commutation relations with a covariance function, enabling interpolation between classical and quantum noise models, and analyzes the spectra of Hamiltonians with dynamical disorder.

Contribution

It constructs a continuous deformation of classical stochastic processes using deformed commutation relations and computes Hamiltonian spectra with dynamical disorder using a novel formalism.

Findings

Interpolates between classical and quantum stochastic processes.

Calculates spectra of disordered Hamiltonians with colored noise.

Provides a unified framework for various noise models.

Abstract

Starting from the deformed commutation relations \ba a_q(t) \,a_q^{\dag}(s) \ - \ q\,a_q^{\dag}(s)\,a_q(t) \ = \ \Gam(t-s) {\bf 1} , \quad -1\ \le \ q\ \le\ 1\nn \ea with a covariance $\Gam(t-s)$ and a parameter $q$ varying between $-1$ and $1$, a stochastic process is constructed which continuously deforms the classical Gaussian and classical compound Poisson process. The moments of these distinguished stochastic processes are identified with the Hilbert space vacuum expectation values of products of $\hat{\om}_q (t) = \gam\,\big(\, a_q(t) + a_q^{\dag} (t) \,\big)\;+ \; \xi\, a_q^{\dag} (t) a_q(t)$ with fixed parameters $q$, $\gam$ and $\xi$. Thereby we can interpolate between dichotomic, random matrix and classical Gaussian and compound Poisson processes. The spectra of Hamiltonians with single-site dynamical disorder are calculated for an exponential covariance (coloured noise) by means of the time convolution generalized master equation formalism (TC-GME) and the