Non-Gaussian stochastic dynamics of spins and oscillators: A continuous-time random walk approach

Abstract

We consider separately a spin and an oscillator that are coupled to their environment. After a finite interval of random length, the state of the environment changes, and each change causes a random change in the resonance frequency of the spin or vibrational frequency of the oscillator. Mathematically, the evolution of these frequencies is described by a continuous-time random walk. Physically, the stochastic dynamics can be understood as non-Gaussian because the frequency of the system and state of the environment change on comparable time scales. These dynamics are also nonstationary, and so might apply to a nonequilibrium environment. The resonance and vibrational spectra of the spin and oscillator, as well as the ensemble-averaged displacement of the oscillator, are investigated in detail. We observe some distinct non-Gaussian features of the dynamics, such as the narrow, leptokurtic shape of the resonance spectrum of the spin and beating of the average oscillator displacement. The convergence to Gaussian dynamics as changes in the environment occur with increasing frequency is also considered. Among other results, we observe narrowing of the resonance and vibrational lines in the Gaussian limit due to a weakening of the system-environment interaction.

Dephasing by a continuous-time random walk process

“…The trends are the same as those predicted by approximate formulae in the previous paper [8]. Namely, that relaxation becomes faster as  and M increase.…”

“…We will then apply the model to study qubit dephasing in solid-state environments and to calculate two-dimensional spectra of a harmonic oscillator. This paper follows a previous paper, in which the properties of the continuous-time random walk were studied in detail and the relaxation equations evaluated approximately in certain limits [8].…”

Stochastic Boltzmann equation for magnetic relaxation in high-spin molecules

“…We suppose that the molecule-environment coupling causes a stochastic modulation to the vibrational mode frequencies. This 'stochastic frequency modulation' approach is used extensively in the field of spectroscopy to model optical relaxation processes [31,32], and can be described by the Kubo oscillator model [33][34][35][36][37][38],…”

Quantum dissipative systems beyond the standard harmonic model: Features of linear absorption and dynamics

“…Treatment of anharmonic behaviour has been tackled by stochastic environments 29,30 , molecular dynamics simulations [31][32][33] , and by including anharmonicity in the system potential [34][35][36] . In this study we use anharmonic system potentials whilst also including an interacting environment via a) cmlds@leeds.ac.uk b) A.G.Dijkstra@leeds.ac.uk the stochastic Schrödinger equation 22,37,38 .…”