Long Time Tails in Quantum Brownian Motion of a charged particle in a magnetic field

Abstract

We analyse the long time tails of a charged quantum Brownian particle in a harmonic potential in the presence of a magnetic field using the Quantum Langevin Equation as a starting point. We analyse the long time tails in the position autocorrelation function, position-velocity correlation function and velocity autocorrelation function. We study these correlations for a Brownian particle coupled to the Ohmic and Drude baths, via position coordinate coupling. At finite temperatures we notice a crossover from a power-law to an exponentially decaying behaviour around the thermal time scale k B T . We analyse how the appearance of the cyclotron frequency in our study of a charged quantum Brownian particle affects the behaviour of the long time tails and contrast it with the case of a neutral quantum Brownian particle.

Quantum Brownian Motion of a Charged Oscillator in a Magnetic Field Coupled to Aheat Bath Through Momentum Variables

“…We have also plotted the correlation functions at finite and T→ 0 temperatures and analyzed the long time behaviours of the correlation functions, similar to our previously reported work on the quantum Brownian motion of a particle coupled to a bath via a position coordinate coupling [20]. At finite temperatures, the correlation functions display an exponential decay at long times as discussed in the previous section.…”

“…At finite temperatures, the correlation functions display an exponential decay at long times as discussed in the previous section. However, at T → 0, we notice a transition from a power law to an exponential behaviour around a time scale set by 2πk B T [2,20]. This can be clearly seen from the above plots, where, the orange dashed lines in (Figs.…”

“…In contrast, at T−→ 0, we expand e πΩ th t as a Taylor series and we get a power law behaviour, similar to the case of a particle coupled to a bath via a position coordinate coupling [2,20], albeit, with higher power exponents.…”

“…and coth ω Ω th has poles at (-inπΩ th ), n being any positive integer. So, using the Cauchy's Residue Theorem, the position correlation function C(t) can be calculated by summing over the residues at the poles [2,20,23]:…”

“…It is evident from Eq. ( 43) that at long times the position autocorreation function falls off as t −4 , which is faster than the t −2 fall off of the position autocorrelation function of a particle coupled to the heat bath via a position coordinate coupling [20].…”