Adjoint master equation for quantum Brownian motion

Quantum Brownian motion is a fundamental model for a proper understanding of open quantum systems in different contexts such as chemistry, condensed-matter physics, biophysics, and optomechanics. In this paper we propose a different approach to describe this model. We provide an exact and analytic equation for the time evolution of the operators and we show that the corresponding equation for the states is equivalent to well-known results in the literature. The dynamics is expressed in terms of the spectral density, regardless of the strength of the coupling between the system and the bath. Our derivation allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations. An example is explicitly studied.

Figure 1

Time evolution of the energy $E\left(t\right)$ (top panel) and diffusion in space ${\mathrm{\Lambda }}^{\text{dif}}\left(t\right)$ (bottom panel) for the first excited state of the harmonic oscillator with frequency ${\omega }_{\text{S}}=100$ centered in the origin $\left(\langle \widehat{x}\rangle =0=\langle \widehat{p}\rangle \right)$ with the parameters $M=1,\gamma =0.3,\mathrm{\Omega }=2000,\beta ={10}^{-1}$, and $\hslash =1$. The plot shows the behavior of $E\left(t\right)$ and ${\mathrm{\Lambda }}^{\text{dif}}\left(t\right)$ for the QBM model with the Drude-Lorentz spectral density, for the Caldeira-Leggett model (CL), and for its modification (MCL). In addition, $E_{\text{th,Q}},E_{\text{th,C}},{\mathrm{\Lambda }}_{\text{th,Q}}^{\text{dif}}$, and ${\mathrm{\Lambda }}_{\text{th,C}}^{\text{dif}}$ (see the main text) are also plotted.

Figure 2

Decoherence function $exp\left[{\mathrm{\Gamma }}^{\text{dec}}\left(t\right)\right]$ with the parameters $M=1,\gamma =0.3,\mathrm{\Omega }=200,{\omega }_{\text{S}}=100,\hslash =1,{\sigma }_0=\sqrt{\hslash /2M{\omega }_{\text{S}}},{\mathrm{\Delta }}_x=2{\sigma }_0$, and ${\mathrm{\Delta }}_p=0$. The plot shows the behavior of the decoherence function $exp\left[\mathrm{\Gamma }\right(t\left)\right]$ for the Drude-Lorentz spectral density (QBM), for the Caldeira-Leggett model (CL), and for its modification (MCL) at two different temperatures: $\beta ={10}^{-1}$ and $\beta ={10}^{-4}$. For $\beta ={10}^{-4}$ the differences between the three models are minimal and the three curves coincide with the dotted line.

Figure 3

Comparison of the solutions of the QBM model for two different initial states: the first excited state of the harmonic oscillator (HO) at frequency ${\omega }_{\text{S}}$ (dashed line) and the ground state of the square potential displayed in Eq. (51) (solid line). The chosen parameters are $M=1,\gamma =0.3,\mathrm{\Omega }=2000,{\omega }_{\text{S}}=100,\beta ={10}^{-1},a=0.23$, and $\hslash =1$. The top panel shows the evolution of the energy $E\left(t\right)$ for the two systems, compared with the equilibrium energy $E_{\text{th,Q}}$ for the quantum thermal state. The bottom panel shows the evolution of the diffusion function ${\mathrm{\Lambda }}^{\text{diff}}\left(t\right)$ for the two systems, compared with the equilibrium diffusion ${\mathrm{\Lambda }}_{\text{th,Q}}^{\text{diff}}$ for the quantum thermal state.