Microscopic derivation of a completely positive master equation for the description of Open Quantum Brownian Motion of a particle in a potential

TL;DR

This work addresses the need for physically valid open-system dynamics in Open Quantum Brownian Motion (OQBM) by deriving a completely positive Born–Markov master equation from a microscopic Hamiltonian for a weakly driven Brownian particle in a harmonic trap. The authors apply the Born–Markov and rotating wave approximations and then perform adiabatic elimination of fast variables in a coordinate representation, obtaining a diagonal, hybrid quantum–classical master equation that defines OQBM and preserves positivity (GKSL structure). They illustrate the dynamics with Gaussian and non-Gaussian initial spatial distributions, show a Gaussian-like long-time behavior arising from internal-state transfer to the external degree of freedom, and derive evolution equations for moments and cumulants, revealing a nonzero third cumulant and thus non-Gaussian features. The results provide a principled, first-principles framework for CP OQBM and set the stage for extensions to nonlinear couplings, anharmonic potentials, and higher-dimensional internal/position spaces, with potential relevance to broader hybrid quantum–classical dynamics and gravity-inspired models.

Abstract

Open Quantum Brownian Motion (OQBM) was introduced as a scaling limit of discrete-time open quantum walks. This limit defines a new class of quantum Brownian motion, which incorporates both the external and internal degrees of freedom of the Brownian particle. We consider a weakly driven Brownian particle confined in a harmonic potential and dissipatively coupled to a thermal bath. Applying the rotating wave approximation (RWA) to the system-bath interaction Hamiltonian, we derive a completely positive Born-Markov master equation for the reduced dynamics. We express the resulting master equation in the coordinate representation and, utilizing the adiabatic elimination of fast variables, derive a completely positive hybrid quantum-classical master equation that defines OQBM. We illustrate the resulting dynamics using examples of initial Gaussian and non-Gaussian distributions of the OQBM walker. Both examples reveal the emergence of Gaussian distributions in the limiting behavior of the OQBM dynamics, which closely matches that of the standard OQBM. With the help of the obtained OQBM master equation, we derive the equations for the $n$-th moments and the cumulants of the position distribution of the open Brownian walker. We subsequently solve these equations numerically for Gaussian initial distributions across various parameter regimes. Notably, we find that the third-order cumulant is nonzero, indicating that the dynamics' intrinsic generator is non-Gaussian.

Figures (6)

• Figure 1: The probability distribution $P(x,t)$ of the open quantum Brownian particle at position $x$ at various times. Curves ($a$) to ($e$) correspond to times 0, 50, 100, 150, and 200, respectively. The initial distribution is given by Eq. (\ref{['fn_1']}) with $j=2$, $\theta=\pi/5$, $\phi=\pi/4$ corresponding to subplot (a); and $j=10$, $\theta=\pi/4$, $\phi=\pi/4$ corresponding to subplot (b). Other parameters are $\Omega=0.5$, $\beta=10^{-4}$, $\Delta_1=10^{-5}$, $\Delta_2=0.01$, $\Delta_3=10^{-4}$, $\Delta_4=8\times10^{-3}$, $\lambda_2=10^{-4}$, $\lambda_3=5\times10^{-3}$, $\lambda_4=10^{-4}$, $\delta_1=10^{-4}$, $\delta_2=0.04$, $\delta_3=0.01$, $\tilde{a}_2=0.04$, $\tilde{a}_7=10^{-3}$, and $\tilde{a}_8=10^{-4}$.
• Figure 2: Illustration of OQBM dynamics. Subplot (b) shows the variance $\sigma^2(t)$ as a function of time $t$ for different OQBM probability distributions. The initial distribution is given by Eq. (\ref{['fn_1']}) with $j=2$. Curve ($i$) corresponds to the parameters, $\theta=\pi/2$, $\phi=\pi/4$, $\Omega=0.5$, $\beta=10^{-3}$, $\Delta_1=10^{-5}$, $\Delta_2=10^{-3}$, $\Delta_3=10^{-4}$, $\Delta_4=8\times10^{-3}$, $\lambda_2=10^{-4}$, $\lambda_3=5\times10^{-3}$, $\lambda_4=10^{-4}$, $\delta_1=10^{-4}$, $\delta_2=0.04$, $\delta_3=0.01$, $\tilde{a}_2=0.04$, $\tilde{a}_7=10^{-3}$, and $\tilde{a}_8=10^{-4}$; Curve ($ii$) corresponds to Fig. \ref{['example1']}(b); Curve ($iii$) correspond to $\theta=\pi/6$, $\phi=\pi/4$, $\Omega=0.5$, $\beta=10^{-4}$, $\Delta_1=10^{-5}$, $\Delta_2=10^{-3}$, $\Delta_3=10^{-4}$, $\Delta_4=8\times10^{-3}$, $\lambda_2=10^{-4}$, $\lambda_3=5\times10^{-3}$, $\lambda_4=10^{-4}$, $\delta_1=10^{-4}$, $\delta_2=0.04$, $\delta_3=0.01$, $\tilde{a}_2=-0.04$, $\tilde{a}_7=10^{-3}$, and $\tilde{a}_8=10^{-4}$; Curve ($iv$) correspond to $\theta=\pi$, $\phi=0$, $\Omega=0.1$, $\beta=0.01$, $\Delta_1=10^{-5}$, $\Delta_2=10^{-3}$, $\Delta_3=2\times10^{-4}$, $\Delta_4=0.01$, $\lambda_2=10^{-4}$, $\lambda_3=0.01$, $\lambda_4=10^{-3}$, $\delta_1=0.01$, $\delta_2=0.02$, $\delta_3=0.03$, $\tilde{a}_2=0.04$, $\tilde{a}_7=10^{-3}$, and $\tilde{a}_8=0.02$. Subplot (b) shows the time evolution of the imaginary part of the off-diagonal element $\bigl(C_I (t) = {\mathop{\mathrm{tr}}\nolimits}_x\bigl[C_I(x,t)\bigl]\bigl)$ (solid curve) and the expectation value of $\langle \hat{\sigma}_z (t) \rangle$ (dashed curve) of the open quantum Brownian particle. The parameters are $\theta=\pi/6$, $\phi=0$, $\Omega=0.5$, $\beta=0.01$, $\Delta_1=10^{-4}$, $\Delta_2=10^{-3}$, $\Delta_3=10^{-4}$, $\Delta_4=8\times10^{-3}$, $\lambda_2=10^{-4}$, $\lambda_3=4\times10^{-3}$, $\lambda_4=10^{-4}$, $\delta_1=0.01$, $\delta_2=0.06$, $\delta_3=0.01$, $\tilde{a}_2=10^{-3}$, $\tilde{a}_7=0.04$, and $\tilde{a}_8=10^{-4}$.
• Figure 3: The time evolution of the $n$-th moments as a function of dimensionless time: the $n$-th moment with respect to the imaginary part of the off-diagonal element of the OQBM density matrix, $\langle \xi^n \tilde{C}_I(t) \rangle$ (solid curve), and the $n$-th moment of the internal degree of freedom of the open quantum Brownian particle, $\langle \xi^n \hat{\sigma}_z(t) \rangle$ (dashed curve). The initial distribution is given by Eq. (\ref{['iniit']}). Subplot (a) corresponds to the second-order moment $n=2$, and $\theta=\phi=\pi/2$, while subplot (b) corresponds to the fifteenth-order moment $n=15$, and $\theta=\phi=\pi/6$. Other parameters are $\Omega=0.5$, $\bar{\beta}=\Delta_3=\Delta_4=\bar{\lambda}_2=\bar{\lambda}_3=\bar{\lambda}_4=\bar{\delta}_1=\bar{\delta}_2=\bar{a}_7=\bar{a}_8=0.01$, $\Delta_1=0.04$, $\Delta_2=\bar{\delta}_3=\bar{a}_2=0.02$, and $\tilde{x}_0=1$. These plots illustrate the differences in the behavior of lower and higher-order moments for different initial states of the internal degree of freedom of the OQBM walker.
• Figure 4: The time evolution of the $n$-th moments as a function of dimensionless time: the $n$-th moments with respect to the real part, $\langle \xi^n \tilde{C}_R(t) \rangle$, the imaginary part of the off-diagonal element of the OQBM density matrix, $\langle \xi^n \tilde{C}_I(t) \rangle$, and the $n$-th moment with respect to the inverse population in the internal degree of freedom of the open quantum Brownian particle, $\langle \xi^n \hat{\sigma}_z(t) \rangle$. The initial distribution is given by Eq. (\ref{['iniit']}). Subplot (a) corresponds to the tenth-order moment $n=10$, and the parameters are $\theta=\pi/6$, $\phi=\pi/4$, $\Omega=0.17$, $\bar{\beta}=\Delta_1=\Delta_2=\Delta_3=\Delta_4=\bar{\lambda}_2=\bar{\lambda}_4=\bar{\delta}_1=\bar{\delta}_2=\bar{a}_7=\bar{a}_8=0.01$, $\bar{\lambda}_3=0.05$, $\bar{\delta}_2=\bar{a}_2=0.02$, and $\tilde{x}_0=1$. Subplot (b) corresponds to $n=2, 8, 15$, respectively, as shown in the legend, and the parameters are $\theta=\phi=\pi/2$, $\Omega=0.1$, $\bar{\beta}=0.001$, $\Delta_1=\bar{\delta}_3=0.02$, $\Delta_2=\Delta_3=\Delta_4=\bar{\lambda}_2=\bar{\lambda}_3=\bar{a}_7=0.01$, $\bar{\lambda}_4=0.04$, $\bar{\delta}_1=\bar{a}_8=0.05$, $\bar{\delta}_2=0.03$, $\bar{a}_2=0.008$, and $\tilde{x}_0=1$.
• Figure 5: The time evolution of the first-order cumulant $\langle x \rangle_c$ and the second-order cumulant $\langle x^2 \rangle_c$ as functions of time. Curves ($i$) correspond to $\langle x^n \rangle_c=0$ with $n\geq 3$, while curves ($ii$) correspond to $\langle x^n \rangle_c=0$ with $n\geq 4$. The initial distribution is given by Eq. (\ref{['cum1']}) with $\theta = \pi/8$ and $\phi = \pi/4$. Other parameters are $\Omega=0.01, \beta=\bar{\lambda}_2=\Delta_3=0.001, \chi=0.25, \Delta_1=0.05, \bar{\lambda}_3=0.02, \Delta_4=0.1, \bar{a}_2=0.004, \bar{a}_7=0.02, \bar{a}_8=\bar{\delta}_1=0.01,$ and $\bar{\delta}_3=0.002$.