Quantum Brownian motion: Definition

In this post I’m going to give a clean definition of idealized quantum Brownian motion and give a few entry points into the literature surrounding its abstract formulation. A follow-up post will give an interpretation to the components in the corresponding dynamical equation, and some discussion of how the model can be generalized to take into account the ways the idealization may break down in the real world.

I needed to learn this background for a paper I am working on, and I was motivated to compile it here because the idiosyncratic results returned by Google searches, and especially this MathOverflow question (which I’ve answered), made it clear that a bird’s eye view is not easy to find. All of the material below is available in the work of other authors, but not logically developed in the way I would prefer.

Preliminaries

Quantum Brownian motion (QBM) is a prototypical and idealized case of a quantum system $\mathcal{S}$ , consisting of a continuous degree of freedom, that is interacting with a large multi-partite environment $\mathcal{E}$ , in general leading to varying degrees of dissipation, dispersion, and decoherence of the system. Intuitively, the distinguishing characteristics of QBM is Markovian dynamics induced by the cumulative effect of an environment with many independent, individually weak, and (crucially) “phase-space local” components. We will defined QBM as a particular class of ways that a density matrix may evolve, which may be realized (or approximately realized) by many possible system-environment models. There is a more-or-less precise sense in which QBM is the simplest quantum model capable of reproducing classical Brownian motion in a $\hbar \to 0$ limit.

In words to be explained: QBM is a class of possible dynamics for an open, quantum, continuous degree of freedom in which the evolution is specified by a quadratic Hamiltonian and linear Lindblad operators.^a

Definition

Consider the arbitrary time-evolution of a system’s density matrix when it is in contact with an environment:

(1) $\begin{align*} \rho = \rho_{\mathcal{S}}(t) = \mathrm{Tr}_{\mathcal{E}} [U_t \sigma^0_{\mathcal{SE}} U_t^\dagger], \end{align*}$

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^b a Lindblad master equation

(2) $\begin{align*} \partial_t \rho = -i [H,\rho] + \sum_i \left(V_i \rho V_i^\dagger - \frac{1}{2} \{V_i^\dagger V_i, \rho\} \right), \end{align*}$

generated by a time-independent Hamiltonian that is a quadratic polynomial in $x$ and $p$

(3) $\begin{align*} \hat{H} = \frac{1}{2m}\hat{p}^2 + \frac{\mu}{2} \{\hat{x},\hat{p}\} + \frac{m\omega^2}{2} \hat{x}^2, \end{align*}$

with $\mu$ , $m$ , and $\omega^2$ real, and by time-independent Lindblad operators that are linear polynomials in the same

(4) $\begin{align*} V_i = a_i \hat{p} + b_i \hat{x} \qquad (i=1,2) \end{align*}$

with $a_i$ and $b_i$ complex. The master equation can be re-written as

(5) $\begin{align*} \partial_t \rho = -i &[\hat{H},\rho] + i (\lambda/2) [\hat{p},\{\hat{x},\rho\}] - i (\lambda/2) [\hat{x},\{\hat{p},\rho\}] \\ &- D_{pp}[\hat{x},[\hat{x},\rho]] - D_{xx}[\hat{p},[\hat{p},\rho]] + D_{xp}[\hat{p},[\hat{x},\rho]] + D_{px}[\hat{x},[\hat{p},\rho]] \end{align*}$

with coefficients

(6) $\begin{align*} D_{xx} &= \frac{\vert a_1 \vert^2 + \vert a_2 \vert^2}{2} \quad , \quad & D_{pp} &= \frac{\vert b_1 \vert^2 + \vert b_2 \vert^2}{2},\\ D_{xp} &= D_{px} = -\mathrm{Re} \frac{a_1^* b_1 + a_2^* b_2}{2} \quad , \quad & \lambda &= \mathrm{Im} (a_1^* b_1 + a_2^* b_2), \end{align*}$

satisfying the complete-positivity (CP) conditions^c

(7) $\begin{align*} D_{xx} \ge 0 \quad , \quad D_{pp} \ge 0 \quad , \quad D_{xx} D_{pp} - D_{xp}D_{px} \ge \frac{\lambda^2}{4}, \end{align*}$

which follow immediately (using the Cauchy–Schwarz inequality) from the definition of the Lindblad operators.

More generally, we say a system undergoes quantum Brownian motion when it evolves according to the above master equation, Eq. (5), regardless of whether it satisfies the CP condition, Eq. (7). 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 $b_i$ .

[Show remarks on the definition]

Remarks on the definition

When the CP condition is violated, the dynamics must arise from existing entanglement between the system and environment and so the master equation cannot be used to evolve an arbitrary density matrix without risking violation of positivity^d. As stated earlier, time-homogeneous QBM dynamics satisfying the CP condition form a quantum dynamical semigroup. Such evolution can be generated by a Lindblad superoperator ^e $\mathcal{L}$ as $\rho(t) = \phi_t (\rho_0) = e^{\mathcal{L}t} \rho_0$ , where the superoperator is just defined to give $\mathcal{L}\rho = \partial_t \rho$ using the right-hand side of Eq. (5).

Note that QBM is a Markov process as promised because $\partial_t \rho$ depends only on $\rho$ .

Isar 1994 is probably the best comprehensive modern statement of the above definition^f, including comparisions to important special cases discussed by other authors. In many of those cases the master equation coefficients — sometimes time-dependent and sometimes not — were obtained through a combination of intuition and phenomenology and do not satisfy the CP conditions. However, many of the modern treatments of QBM are derived explicitly by tracing out a well-defined model environment, leading to time-dependent coefficients that violate the CP restriction for some times but still produce physically possible dynamics.^g

The basic idea of a quantum dynamical semigroup is necessary for fully appreciating the above definition of QBM, for which I highly recommend the monograph by Alicki and Lendi^h, or Alicki’s closely related notesarXiv:quant-ph/0205188.ⁱ. Also useful are the original paper by Lindblad^j, the highly accessible notes by Caves^k, and the exhaustive treatment in the textbook by Breuer and PetruccioneThe Theory Open Quantum Systems.^l.

[Show remarks on the parameterization]

Remarks on the parameterization

An arbitrary Hamiltonian quadratic polynomial could of course contain linear terms in $\hat{x}$ and $\hat{p}$ , but these can be eliminated by simply translating our coordinates $x \to x+x_0$ , $p \to p+p_0$ . The cross term $\{\hat{x},\hat{p}\}$ in the Hamiltonian $\hat{H}$ is symmetric; an antisymmetric cross term $[\hat{x},\hat{p}]$ would just reduce to $i \hbar$ . The parameters $\mu$ , $m$ , and $\omega^2$ must be real for $\hat{H}$ to be Hermitian.

There are no constant terms in the Lindblad operators (proportional to the identity) because their contribution vanishes for all $\rho$ . We never need more than two Lindblad operators because the subspace of operator spanned by linear combinations of $\hat{x}$ and $\hat{p}$ is two dimensional, and we can always choose operators that form a basis for this space by diagonalizing the Lindblad equation as described by Wikipedia.

Thus, the above master equations describes the most general possible dynamics for a quadratic Hamiltonian and linear Lindblad operators.

Motivation

There are several complimentary strategies to motivate the above definition and arrive at an equivalent dynamical equation. I will simply identify the two most important ones and give references^m:

Look at the classical equations describing Brownian motion (e.g. the original Langevin equation or, equivalently, the Fokker-Planck equation) and try to quantize the variables in them while maintaining important features like conservation of Gaussianity. See Lindbladⁿ, Dekker^o, Sandulescu and Scutaru^p, and Isar^q.
Consider a model environment that generates reduced dynamics that plausibly mimic real-world examples of Brownian motion. Ssee Caldeira and Leggett^r; Hu, Paz, and Zhang^s; Halliwell and Yu^t; and Zurek^u.

The second method is more laborious and generally is found in later papers, but it is especially important for understanding how good of an approximation the QBM idealization is for any given system.

This is a situation where many independent cases all lead to the same dynamical equation, which is a good reason to study the equation in the abstract and then use that idealization to understand other, more complicated situations.

Position-space and phase-space representations

Path-integral techniques have been very important for calculating the QMB parameters in the case of an oscillator bath environment, so many authors work in position space. By inserting several copies of $I = \int \! \mathrm{d}x \, \vert x \rangle \langle x \vert$ into Eq. (5) we can write the position-space master equation as^v

(8) $\begin{align*} \partial_t \rho_{xx'} = \Big[ &i \frac{1}{2m}\left( \partial^2_x - \partial^2_{x'} \right) -i \frac{m \omega^2}{2} \left(x^2 - {x'}^2 \right) \\ &+ (\lambda+\mu) (x-x')\left( \partial_x - \partial_{x'} \right) - (\lambda-\mu) (x+x')\left( \partial_x + \partial_{x'} \right) \\ &- D_{pp} \left(x - {x'} \right)^2 + D_{xx} \left( \partial_x + \partial_{x'} \right)^2 - i (D_{xp}+D_{px}) \left(x - {x'} \right) \left( \partial_x + \partial_{x'} \right) \Big] \end{align*}$

But the simplicity and beauty of QBM is largely due to the fact that $x$ and $p$ are treated on equal footing and appear only at low order. When working with the standard density matrix representation, QBM equations are most symmetric when expressed in terms of $x$ and $p$ operators, as Eq. (5), rather than in position space (where $p$ is a derivative operation) or momentum space (where $x$ is).

The most beautiful form is in the Wigner representation, a formulation of quantum mechanics where a quantum state is represented by a quasiprobability distribution over phase space. In a rough sense (which I make more precise here), the Wigner function of a state acts like a probability distribution over scales large than $\hbar$ (a unit of area in phase space), but with sub- $\hbar$ structure that prevents it from being treated like a probability distribution, in particular taking negative values.

We can transform the position-space master equation above into the Wigner representation by changing coordinates $(x,'x)\to(\bar{x} = (x+x')/2,\Delta x=x-x')$ and then taking the Fourier tranform with respect to $\Delta x$ . The resulting dynamical equation for the Wigner function $W(x,p)$ takes the form

(9) $\begin{align*} \partial_t W = -\frac{p}{m}\partial_x W + m\omega^2 & x \partial_p W + (\lambda - \mu)\partial_x (x W) + (\lambda + \mu)\partial_p (p W)\\ &+D_{pp} \partial^2_x W + D_{xx} \partial^2_p + (D_{xp}+D_{px}) \partial_x \partial_p W. \end{align*}$

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^w.

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 master equation as well as the CP condition, Eq. (7), to be described in a later post. For now, let us write down the Wigner dynamics, Eq. (11), in a more compact form that emphasizes the “phase-space covariance”^x:

(10) $\begin{align*} \partial_t W (\alpha) &= \left[ F_{ab} \partial_a \alpha_b + D_{ab} \partial_a \partial_b \right] W(\alpha)\\ &= \left[ 2\lambda + F_{ab} \alpha_b \partial_a + D_{ab} \partial_a \partial_b \right] W(\alpha) \end{align*}$

where

(11) $\begin{align*} F_{ab} = \left( \begin{array}{cc} \lambda - \mu & -1/m \\ m \omega^2 & \lambda+\mu \end{array} \right) \quad, \quad D_{ab} = \left( \begin{array}{cc} D_{xx} & D_{xp} \\ D_{px} & D_{pp} \end{array} \right) \end{align*}$

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 $\partial_{\alpha_a}$ .) The second line of (10) follows from $\partial_a (\alpha_b W) = I_{ab} W + \alpha_b (\partial_a W)$ and $F_{aa} = 2\lambda$ , where $I$ is the identity matrix.

[Edited 2014-10-11 to reflect Isar et al.’s matrix form.]

Footnotes

(↵ returns to text)

As usual, the literature isn’t perfectly consistent with terminology and sometimes the term “Quantum Brownian motion” is used more broadly (e.g., with non-Markovian corrections or with higher-order terms in the Lindblad operators). Such extensions may better model the real-world, but they should probably be considered “generalized QBM”.↵
I use $\hbar=1$ , and hats denote operators. The commutator is $[\hat{A}, \hat{B}] = \hat{A}\hat{B}-\hat{B}\hat{A}$ and the anti-commutator is $\{\hat{A}, \hat{B}\} = \hat{A}\hat{B}+\hat{B}\hat{A}$ .↵
Some authors such as Dekker, and Sandulescu and Scutaru, argue for the requirement that both $D_{xx}$ and $D_{pp}$ be strictly greater than zero, but I’ve never been able to understand them unless they are implicitly assuming that $\lambda$ is strictly positive (which immediately implies that $D_{xx}$ and $D_{pp}$ are) in order for something to be considered true QBM. As a counter example, $D_{xx}$ and $D_{pp}$ will obviously be exactly zero for unitary evolution, and in the physical case of collisional decoherence of a high-mass particle by an environment of low-mass particles, we get that $D_{xx}=0$ (and hence $\lambda=0$ , i.e. exactly zero friction) as the mass ratio goes to zero.↵
Violation of the CP condition means the dynamics are not described by a completely positive map, i.e. there exists density matrices for a system entangled with some dummy ancilla such that the joints state of the system and ancilla fails to remain positive when the system evolves according to the master equation and the ancilla has trivial dynamics.↵
$\mathcal{L}$ is a superoperator because it linearly maps density matrices, which are themselves operators on the underlying Hilbert space, to other operators.↵
A definition nearly as general can be traced back to Lindblad 1976. This was one of the immediate results of Lindblad’s derivation of his epynynomous equation of the same year.↵
Several authors have considered version of the QBM Master equation that do not define CP maps. (See Isar et al. for some reference.) However, it’s well known that some linear maps aren’t CP but still accurately describe reduced dynamics; they arise when the assumption of zero initial correlations is violated, and hence these maps can only be applied to a certain subset of the density matrices that are sufficiently mixed.↵
Quantum Dynamical Semigroups and Applications.↵
arXiv:quant-ph/0205188.↵
Lindblad, G. (1976). “On the generators of quantum dynamical semigroups“. Commun. Math. Phys. 48 (2) 119.↵
Completely positive maps, positive maps, and the Lindblad form.↵
The Theory Open Quantum Systems.↵
There are other routes that arrive at the same equation, like Hasse’s “pure state theory”, connected to a (stochastic) nonlinear Schrodinger equation.↵
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. Isar et al, “Open Quantum Systems“.↵
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“.↵
As usual, this can be obtained through the substitution rules $\hat{x} \to x$ and $\hat{p} \to -i \partial_x$ when acting on $\rho$ from the right, $\hat{x} \to x'$ and $\hat{p} \to i \partial_{x'}$ when acting from the left, and finally sending $\rho \to \rho_{x x'}$ .↵
See, for instance, “The Fokker-Planck Equation: Methods of Solution and Applications by H. Risken.↵
Is this choice of notation a Freudian slip revealing that I secretly wish I was a field theorist or relativist? You be the judge.↵

Quantum Brownian motion: Definition

Preliminaries

Definition

Remarks on the definition

Remarks on the parameterization

Motivation

Position-space and phase-space representations

Footnotes

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