(In words explained below:) **Quantum Brownian motion (QBM) is a class of possible dynamics for an open, quantum, continuous degree of freedom in which the reduced dynamics are specified by a quadratic Hamiltonian and linear Lindblad operators in the phase-space variables $x$ and $p$.**

Consider the arbitrary time-evolution of a system's density matrix when it is in contact with an environment:
\begin{align}
\rho = \rho_{\mathcal{S}}(t) = \mathrm{Tr}_{\mathcal{E}} [U_t \sigma^0_{\mathcal{SE}} U_t^\dagger],
\end{align}
where $\sigma^0_{\mathcal{SE}}$ is the initial global state (both $\mathcal{S}$ and $\mathcal{E}$) and $U_t$ is the unitary governing the global evolution. Then the system is said to evolve according to a special case of quantum Brownian motion — a QBM quantum dynamical semigroup — when the evolution of its density matrix obeys a Lindblad master equation
\begin{align}
\partial_t \rho = -i [\hat{H},\rho] + \sum_i \left(V_i \rho V_i^\dagger - \frac{1}{2} \{V_i^\dagger V_i, \rho\} \right),
\end{align}
generated by a time-independent Hamiltonian that is a quadratic polynomial in $x$ and $p$
\begin{align}
\hat{H} = \frac{1}{2m}\hat{p}^2 + \frac{\mu}{2} \{\hat{x},\hat{p}\} + \frac{m\omega^2}{2} \hat{x}^2,
\end{align}
with $\mu$, $m$, and $\omega^2$ real, and by time-independent Lindblad operators that are linear polynomials in the same
\begin{align}
V_i = a_i \hat{p} + b_i \hat{x}, \qquad (i=1,2)
\end{align}
with $a_i$ and $b_i$ complex. The master equation can be re-written as
\begin{align}
\partial_t \rho = -i &[\hat{H},\rho] + i (\lambda/2) [\hat{p},\{\hat{x},\rho\}] - i (\lambda/2) [\hat{x},\{\hat{p},\rho\}] \\
&- D_{pp}[\hat{x},[\hat{x},\rho]] - D_{xx}[\hat{p},[\hat{p},\rho]] + D_{xp}[\hat{p},[\hat{x},\rho]] + D_{px}[\hat{x},[\hat{p},\rho]]
\end{align}
with coefficients
\begin{align}
D_{xx} &= \frac{\vert a_1 \vert^2 + \vert a_2 \vert^2}{2} \quad , \quad & D_{pp} &= \frac{\vert b_1 \vert^2 + \vert b_2 \vert^2}{2},\\
D_{xp} &= D_{px} = -\mathrm{Re} \frac{a_1^* b_1 + a_2^* b_2}{2} \quad , \quad & \lambda &= \mathrm{Im} (a_1^* b_1 + a_2^* b_2),
\end{align}

**More generally, we say a system undergoes ***quantum Brownian motion* when it evolves according to the above master equation, regardless of whether it forms a quantum dynamical semigroup. If it obeys the master equation with time-independent coefficients then the QBM is *time-homogeneous* (in the sense of a Markov process); otherwise it is *time-inhomogeneous*. The class of all possible instantaneous QBM dynamics is parameterized by $\mu$, $m$, $\omega^2$, $a_i$, and $b_i$.

The resulting dynamics take a particularly beautiful form in the Wigner representation. The above master equation for $\rho$ is equivalent to the following dynamical equation for the Wigner function $W(x,p)$:
\begin{align}
\partial_t W = -\frac{p}{m}\partial_x W + m\omega^2 & x \partial_p W + (\lambda - \mu)\partial_x (x W) + (\lambda + \mu)\partial_p (p W)\\
&+D_{pp} \partial^2_x W + D_{xx} \partial^2_p + (D_{xp}+D_{px}) \partial_x \partial_p W.
\end{align}
More compactly:
\begin{align}
\partial_t W (\alpha) &= \left[ F_{ab} \partial_a \alpha_b + D_{ab} \partial_a \partial_b \right] W(\alpha)
\end{align}
where
\begin{align}
F_{ab} = \left( \begin{array}{cc} \lambda - \mu & -1/m \\ m \omega^2 & \lambda+\mu \end{array} \right) \quad, \quad D_{ab} = \left( \begin{array}{cc} D_{xx} & D_{xp} \\ D_{px} & D_{pp} \end{array} \right)
\end{align}
are matrices with real elements. Above, the phase-space indices $a,b$ take the values $x,p$, with Einstein summation assumed, so that $\alpha_a$ is a vector in phase space. (The directional derivative $\partial_a$ is just shorthand for $\partial_{\alpha_a}$.)

This is *identical* in form to a Klein-Kramers equation (more generally a Fokker-Planck-type equation) for the phase-space probability distribution of a *classical* point particle undergoing Brownian motion.

This is remarkable because such equations were originally derived for true probability distribution, but they also apply to the Wigner function. As a bonus, this gives us an immediate and simple physical interpretation for each of the terms in the QBM master equation

The best comprehensive modern statement of the above definition is probably

which includes comparisions to important special cases discussed by other authors. Here are some more references I found useful in compiling the above:

- Alicki and Lendi, arXiv:quant-ph/0205188.
- Alicki, Quantum Dynamical Semigroups and Applications.
- Lindblad, G. (1976). "On the generators of quantum dynamical semigroups". Commun. Math. Phys. 48 (2) 119.
- Breuer and Petruccione, The Theory Open Quantum Systems.
- C. Caves, Completely positive maps, positive maps, and the Lindblad form.
- G. Lindblad, "Brownian motion of a quantum harmonic oscillator".
- H. Dekker, "Quantization of the linearly damped harmonic oscillator"
- A. Sandulescu and H. Scutaru, "Open quantum systems and the damping of collective modes in deep inelastic collisions".
- A.O. Caldeira and A.J. Leggett, "Path integral approach to quantum Brownian motion".
- B. Hu, J. Paz, and Y. Zhang, "Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise".
- J. Halliwell and T. Yu, "Alternative derivation of the Hu-Paz-Zhang master equation of quantum Brownian motion".
- W Zurek, "Decoherence, einselection, and the quantum origins of the classical".

reviewhas nothing do with a non-commutative probabilistic description of quantum Brownian motion, I think that non-comm. prob. theory might be one framework in which to precisely describe QBM. I make no claims that this approach is necessary but it may be useful. $\endgroup$ – Tom LaGatta Feb 21 '13 at 16:18