Adiabatic elimination and Wigner function approach in microscopic derivation of Open Quantum Brownian Motion

TL;DR

This work provides the first generic microscopic derivation of Open Quantum Brownian Motion (OQBM) for a Brownian particle with a single internal degree of freedom in a harmonic potential, interacting with a high-temperature Ohmic bath. Starting from a full system–bath Hamiltonian, the authors derive a Born-Markov master equation, recast it in the Wigner phase-space framework, and apply adiabatic elimination of the fast momentum variable in the large-damping limit to obtain the OQBM master equation with diffusion $\bar{\alpha}$ and drift $\bar{\beta}$, plus a quantum-coherent cross-coupling term. Numerical simulations show that, irrespective of Gaussian or non-Gaussian initial states, the position distribution evolves toward multiple Gaussians, with the asymmetry tied to the initial internal coherences and a ballistic-to-diffusive transition in the variance, while coherences decay over time. The study delivers a robust microscopic framework for OQBM, clarifies its connection to Caldeira–Leggett-type dynamics (non-CP in this setup), and sets the stage for completely positive generalizations in future work.

Abstract

Open Quantum Brownian Motion (OQBM) is a new class of quantum Brownian motion in which the dynamics of the Brownian particle depend not only on interactions with a thermal environment but also on the state of its internal degrees of freedom. For an Ohmic bath spectral density with a Lorentz-Drude cutoff frequency at a high-temperature limit, we derive the Born-Markov master equation for the reduced density matrix of an open Brownian particle in a harmonic potential. The resulting master equation is written in phase-space representation using the Wigner function, and due to the separation of associated timescales in the high-damping limit, we perform adiabatic elimination of the momentum variable to obtain OQBM. We numerically solve the derived master equation for the reduced density matrix of the OQBM for Gaussian and non-Gaussian initial distributions. In each case, the OQBM dynamics converge to several Gaussian distributions. To gain physical insight into the studied system, we also plotted the dynamics of the off-diagonal element of the open quantum Brownian particle and found damped coherent oscillations. Finally, we investigated the time-dependent variance in the position of the OQBM walker and observed a transition between ballistic and diffusive behavior.

Figures (5)

• Figure 1: The position probability distribution of the open quantum Brownian particle for different times. The initial distribution is given by Eqn. (\ref{['fn_1']}). Curves ($i$) through ($v$) correspond to times 0, 50, 100, 150, and 200, respectively. For subplot (a), the initial distribution is given by a Gaussian distribution with $k=2$, $\theta=\pi/6$, and $\phi=\pi$; the parameters are $\bar{\alpha}=\bar{\lambda}_2=8\times10^{-3}$, $\bar{\beta}=\bar{\lambda}_3=10^{-3}$, $\bar{\beta}_1=3\times10^{-3}$, $\bar{\beta}_2=5\times10^{-2}$, $\bar{\beta}_3=10^{-2}$, $\bar{\lambda}_1=5\times10^{-3}$, and $\Gamma(\Omega)=10^{-4}$. For subplot (b), the initial distribution is given by a non-Gaussian distribution with $k=10$, $\theta=\pi/6$, and $\phi=0$; the parameters are $\bar{\alpha}=\bar{\beta}_3=10^{-2}$, $\bar{\beta}=3\times10^{-3}$, $\bar{\beta}_1=\bar{\lambda}_1=5\times10^{-3}$, $\bar{\beta}_2=5\times10^{-2}$, $\bar{\lambda}_2=4\times10^{-2}$, $\bar{\lambda}_3=4\times10^{-3}$, and $\Gamma(\Omega)=8\times10^{-3}$.
• Figure 2: OQBM dynamics. In subplot (a), we plot 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 initial distribution is given by Eqn. (\ref{['fn_1']}) with $k=2$, $\theta=\pi/6$, and $\phi=\pi/4$. The parameters are set to $\bar{\alpha}=\bar{\beta}_1=\bar{\lambda}_2=5\times10^{-3}$, $\bar{\beta}=5\times10^{-4}$, $\bar{\beta}_2=4\times10^{-3}$, $\bar{\beta}_3=0.5$, $\bar{\lambda}_1=8\times10^{-4}$, and $\bar{\lambda}_3=\Gamma(\Omega)=10^{-3}$. Subplot (b) shows the variance $\sigma^2(t)$ as a function of time for different OQBM distributions. Curves ($i$)-($ii$) corresponds to Fig. \ref{['example1']}(a)-(b), respectively. Curve ($iii$) corresponds to the parameters, $k=10$, $\theta=\pi/4$, $\phi=0$, $\bar{\alpha}=\bar{\lambda}_3=10^{-2}$, $\bar{\beta}=2\times10^{-3}$, $\bar{\beta}_1=3.5\times10^{-2}$, $\bar{\beta}_2=\bar{\beta}_3=2\times10^{-4}$, $\bar{\lambda}_2=2.5\times10^{-2}$, and $\Gamma(\Omega)=\bar{\lambda}_1=10^{-3}$; and curve ($iv$) corresponds to $k=10$, $\theta=\pi$, $\phi=\pi/4$, $\bar{\alpha}=9\times10^{-3}$, $\bar{\beta}=\bar{\beta}_3=10^{-4}$, $\bar{\beta}_1=3.7\times10^{-2}$, $\bar{\beta}_2=3\times10^{-4}$, $\bar{\lambda}_2=10^{-2}$, $\bar{\lambda}_3=2\times10^{-2}$, and $\bar{\lambda}_1=\Gamma(\Omega)=10^{-3}$.
• Figure 3: The position probability distribution of the open quantum Brownian particle at different times. The initial distribution is given by Eqn. (\ref{['fn_2']}). Curves ($i$) through ($v$) correspond to times 0, 50, 100, 150, and 200, respectively. For subplot (a), the initial distribution is given by a Gaussian distribution with $k=2$, $\theta=\pi/4$, and $\phi=\pi/2$; the parameters are $\bar{\alpha}=\bar{\beta}_1=\bar{\lambda}_2=10^{-2}$, $\bar{\beta}=10^{-5}$, $\bar{\beta}_2=3\times10^{-2}$, $\bar{\beta}_3=5\times10^{-2}$, $\bar{\lambda}_1=\bar{\lambda}_3=10^{-3}$, and $\Gamma(\Omega)=10^{-4}$. For subplot (b), the initial distribution is given by a non-Gaussian distribution with $k=10$, $\theta=\pi/2$, and $\phi=\pi/6$; the parameters are $\bar{\alpha}=10^{-2}$, $\bar{\beta}=2\times10^{-4}$, $\bar{\beta}_1=10^{-4}$, $\bar{\beta}_2=5\times10^{-2}$, $\bar{\beta}_3=2\times10^{-2}$, $\bar{\lambda}_1=\bar{\lambda}_2=8\times10^{-3}$, $\bar{\lambda}_3=6\times10^{-3}$, and $\Gamma(\Omega)=10^{-4}$.
• Figure 4: OQBM dynamics. Subplot (a) shows the position probability distribution of the open quantum Brownian particle at different times. Equation (\ref{['fn_1']}) gives the initial position distribution with $\theta=\pi/2$ and $\phi=\pi$, which corresponds to an initial state in the excited state. Curves ($i$) through ($v$) in subplot (a) corresponds to times 0, 50, 100, 150, and 200, respectively; the parameters are $\bar{\alpha}=\bar{\lambda}_2=\bar{\lambda}_3=10^{-2}$, $\bar{\beta}=10^{-3}$, $\bar{\beta}_1=3\times10^{-3}$, $\bar{\beta}_2=\bar{\beta}_3=6\times10^{-2}$, and $\bar{\lambda}_1=\Gamma(\Omega)=10^{-4}$. Subplot (b) shows the variance $\sigma^2(t)$ as a function of time for different OQBM distributions; curves ($i$)-($ii$) in subplot (b) corresponds to Fig. \ref{['example3']}(a)-(b); the remaining curve ($iii$) correspond to Fig. \ref{['example4']}(a), respectively.
• Figure 5: The time evolution of the fourth-order moment with respect to the real part $\langle x^4 C_R (t) \rangle$ (dashed curve) and the imaginary part $\langle x^4 C_I (t) \rangle$ (solid curve) of the OQBM density matrix as a function of dimensionless time $\bar{\alpha}t$. The initial distribution is given by Eqn. (\ref{['iniit']}), where for subplot (a), we set $\theta=\pi/4$, and $\phi=\pi/4$; the parameters are $\bar{\alpha}=1$, $\bar{\beta}=5\times10^{-2}$, $\bar{\beta}_1=0.21$, $\bar{\beta}_2=3\times10^{-2}$, $\bar{\beta}_3=2\times10^{-2}$, $\bar{\lambda}_1=-2\times10^{-3}$, $\bar{\lambda}_2=4\times10^{-2}$, and $\bar{\lambda}_3=\Gamma(\Omega)=10^{-2}$. For subplot (b), the initial distribution of the internal degree of freedom is $\theta=\pi/6$, and $\phi=\pi$; the parameters are $\bar{\alpha}=1$, $\bar{\beta}=2.2\times10^{-2}$, $\bar{\beta}_1=0.26$, $\bar{\beta}_2=\bar{\beta}_3=10^{-2}$, $\bar{\lambda}_1=-10^{-2}$, $\bar{\lambda}_2=5.5\times10^{-2}$, $\bar{\lambda}_3=2.5\times10^{-3}$, and $\Gamma(\Omega)=10^{-3}$.