Quantum Brownian Motion

Quantum Brownian motion is the open quantum dynamics of a particle, oscillator, or collective coordinate coupled to many environmental degrees of freedom. It is the quantum version of Brownian motion, but with operator noise, commutation relations, decoherence, and complete-positivity constraints added to the classical ideas of friction and diffusion.

The canonical microscopic Hamiltonian is the Caldeira–Leggett Model. This page focuses on the resulting dynamics:

• generalized Langevin equations with memory;
• Markovian damping and diffusion limits;
• master equations and Wigner-space Fokker–Planck forms;
• position-basis decoherence;
• the relation to classical Brownian motion;
• low-temperature and positivity warnings.

The topic is a useful testing ground because it looks classically familiar while exposing nearly every subtlety of open quantum systems.

Friction and Force Fluctuations

The system is a coordinate $q$ with conjugate momentum $p$ and Hamiltonian

$$H_S = \frac{p^2}{2M} + V(q).$$

The environment is usually modeled as many oscillator modes coupled to $q$. The bath does two things:

• it damps the motion by absorbing energy and momentum;
• it fluctuates, producing random force, diffusion, and decoherence.

At thermal equilibrium these are not independent. The same bath spectral density controls both damping and force noise, with the relation fixed by the Fluctuation–Dissipation Relation.

Generalized Langevin Equation

Eliminating a linear oscillator bath gives a Heisenberg-picture equation with memory:

$$M\ddot q(t) + V'(q(t)) + \int_{t_0}^{t} ds\, \eta(t-s)\dot q(s) = \xi(t) + F_{\mathrm{slip}}(t).$$

Here:

• $\eta(t-s)$ is a damping kernel;
• $\xi(t)$ is a bath force operator;
• $F_{\mathrm{slip}}(t)=-\eta(t-t_0)q(t_0)$ for the uncoupled-bath force and factorized preparation defined in the Caldeira–Leggett model.

In many long-time discussions the slip term is absorbed into initial transients or avoided by choosing correlated thermal initial states. It should not be forgotten when short-time behavior matters.

The noise kernel is often written as the symmetrized correlation

$$\nu(t-s) = \frac12 \langle \{\xi(t),\xi(s)\} \rangle.$$

The commutator of $\xi(t)$ is also important. It is tied to dissipation and preserves the canonical commutator of $q$ and $p$. A purely classical random force may reproduce some high-temperature observables, but it does not carry the full operator structure.

Markov and High-Temperature Limit

If the bath memory time is short compared with the system dynamics, the damping kernel can become approximately local:

$$\int_{t_0}^{t} ds\, \eta(t-s)\dot q(s) \approx M\gamma\dot q(t).$$

At high temperature, the force noise can become approximately white:

$$\frac12 \langle \{\xi(t),\xi(s)\} \rangle \approx 2M\gamma k_BT\,\delta(t-s).$$

Then the familiar classical-looking Langevin equation appears:

$$M\ddot q + M\gamma\dot q + V'(q) = \xi(t).$$

This limit is useful, but it is a limit. It assumes high temperature, broad bath bandwidth, weak memory, and a frequency window where the bath looks Ohmic. Low-temperature noise, structured spectra, and short-time dynamics require the nonlocal kernels.

Brownian Master Equation

The reduced density operator can also be described by a master equation. A common high-temperature Caldeira–Leggett form is

$$\dot\rho = - \frac{i}{\hbar}[H_S,\rho] - \frac{i\gamma}{2\hbar} [q,\{p,\rho\}] - \frac{M\gamma k_BT}{\hbar^2} [q,[q,\rho]].$$

The terms have clear roles:

• $-[i/\hbar][H_S,\rho]$ gives unitary motion;
• $-[i\gamma/(2\hbar)][q,\{p,\rho\}]$ gives friction;
• the double commutator gives momentum diffusion and position-basis decoherence.

More general Brownian master equations include diffusion coefficients

$$D_{pp}, \qquad D_{qq}, \qquad D_{pq},$$

for momentum diffusion, position diffusion, and cross diffusion. We define their contributions by

$$- \frac{D_{pp}}{\hbar^2}[q,[q,\rho]] - \frac{D_{qq}}{\hbar^2}[p,[p,\rho]] + \frac{2D_{pq}}{\hbar^2}[q,[p,\rho]].$$

The compact high-temperature equation is historically important and often physically accurate for hot, weakly damped motion, but it is not a universal Lindblad equation.

Positivity Constraint

For a quadratic Hamiltonian and the constant coefficients defined above, complete positivity requires

$$D_{pp}\ge0,\qquad D_{qq}\ge0,\qquad D_{pp}D_{qq}-D_{pq}^2\ge\frac{\hbar^2\gamma^2}{16}.$$

Here $\gamma$ damps momentum, not half the momentum damping rate. A factor-of-two change in the definition of friction changes the right-hand side by a factor of four.

One way to understand this condition is to write the symmetrized covariance matrix of $(q,p)$ as $V$. Its evolution has the form

$$\dot V=AV+VA^{\mathsf T}+2D,\qquad D=\begin{pmatrix}D_{qq}&D_{pq}\\D_{pq}&D_{pp}\end{pmatrix}, \qquad \operatorname{tr}A=-\gamma.$$

An infinitesimal Gaussian channel must preserve the uncertainty principle even when the oscillator is entangled with another system. This is equivalent to positivity of the Hermitian matrix

$$2D+\frac{i\hbar\gamma}{2} \begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$

Its determinant gives the inequality above. The compact high-temperature equation sets $D_{qq}=D_{pq}=0$ and therefore fails this exact generator condition whenever $\gamma\ne0$. The omitted diffusion can be small in its intended hot-bath regime, but the equation must not be applied to arbitrary states and times as an exact quantum channel. See Carlesso and Bassi (2017) for comparison with the exact oscillator model.

Wigner-Space Form

The Wigner function gives a phase-space view of Brownian dynamics. For a harmonic potential, or for the leading classical part of a smooth potential, the high-temperature Brownian equation becomes a Kramers-type equation:

$$\partial_t W = - \frac{p}{M}\partial_q W + V'(q)\partial_p W + \gamma\partial_p(pW) + D_{pp}\partial_p^2 W + \cdots .$$

In the high-temperature Markov limit,

$$D_{pp} = M\gamma k_BT.$$

The ellipsis includes $D_{qq}\partial_q^2W+2D_{pq}\partial_q\partial_pW$ when these coefficients are present, and quantum Moyal corrections for nonlinear potentials. For harmonic systems, the Wigner equation remains Gaussian-preserving and closely resembles a classical Ornstein–Uhlenbeck process, with quantum constraints on covariance matrices.

For the phase-space formalism itself, see Wigner Function and Phase-Space Dynamics.

Spatial Decoherence

The same diffusion term that heats momentum also suppresses spatial coherence. In the position representation,

$$[q,[q,\rho]](q,q') = (q-q')^2\rho(q,q').$$

Thus the high-temperature diffusion term contributes

$$\partial_t\rho(q,q') \supset - \Gamma_{\mathrm{dec}}(q,q')\rho(q,q'),$$

with

$$\Gamma_{\mathrm{dec}}(q,q') = \frac{M\gamma k_BT}{\hbar^2} (q-q')^2.$$

Large spatial superpositions decohere much faster than nearby coherences. This is why Brownian environments are central examples of environment-induced decoherence and the emergence of approximately classical position records.

Decoherence is not the same as friction. The friction rate may be small while the spatial decoherence rate for macroscopic separations is enormous.

Classical Brownian Limit

In the classical high-temperature regime, the phase-space distribution obeys the Kramers equation. For a free particle, momentum relaxes on the timescale $1/\gamma$. At longer times, position diffuses with Einstein diffusion constant

$$D_x = \frac{k_BT}{M\gamma}.$$

The mean squared displacement grows as

$$\langle [q(t)-q(0)]^2 \rangle \approx 2D_x t,$$

after momentum has equilibrated.

The quantum model should reduce to this behavior when thermal occupation is large, action scales are large compared with $\hbar$, and the measured observables are insensitive to operator ordering. At low temperature or for coherent superpositions, the same model retains quantum noise, zero-point fluctuations, and decoherence.

Harmonic Oscillator Brownian Motion

For

$$V(q) = \frac12M\Omega^2q^2,$$

the dynamics is linear. Gaussian states remain Gaussian under linear Brownian evolution, so the problem can be tracked by first and second moments:

$$\langle q\rangle, \quad \langle p\rangle, \quad \langle q^2\rangle, \quad \frac12\langle qp+pq\rangle, \quad \langle p^2\rangle.$$

In a weak-coupling secular thermal master equation, the oscillator relaxes toward a Gibbs state with mean occupation

$$\bar n = \frac{1} {e^{\beta\hbar\Omega}-1}.$$

In a position-coupled Brownian equation, especially outside weak coupling or secular limits, the steady state may instead include cutoff dependence, squeezing-like covariance corrections, or Hamiltonian-of-mean-force effects. This is one reason Brownian motion is more subtle than simply adding the thermal damping master equation for $a$ and $a^\dagger$.

Regimes and Approximations

Useful regimes include:

Regime	Typical model	Main caution
high temperature, Ohmic, weak damping	local friction plus white force noise	positivity and cutoff assumptions
low temperature	colored quantum noise	classical noise intuition fails
structured bath	memory kernels or auxiliary modes	Markov damping misses recurrences
strong coupling	Hamiltonian of mean force or enlarged system	bare Gibbs steady state can fail
harmonic linear dynamics	covariance-matrix evolution	convention and diffusion coefficients matter
nonlinear potential	quantum Fokker–Planck with Moyal terms	classical phase-space intuition is incomplete

Choose an approximation using the bath correlation time, thermal scale, coupling, cutoff, and preparation. A short correlation time alone does not justify replacing low-temperature quantum noise by classical white noise.

Exercises

Position decoherence rate

Starting from

$$\dot\rho \supset - \frac{D_{pp}}{\hbar^2} [q,[q,\rho]],$$

derive the decay rate of $\rho(q,q')$.

Solution

In the position representation,

$$[q,\rho](q,q') = (q-q')\rho(q,q').$$

Applying the second commutator gives

$$[q,[q,\rho]](q,q') = (q-q')^2\rho(q,q').$$

Therefore

$$\partial_t\rho(q,q') = - \frac{D_{pp}}{\hbar^2} (q-q')^2 \rho(q,q').$$

The decoherence rate is

$$\Gamma_{\mathrm{dec}}(q,q') = \frac{D_{pp}}{\hbar^2} (q-q')^2.$$

For the high-temperature value $D_{pp}=M\gamma k_BT$, this reproduces the rate stated in the text.

Einstein diffusion constant

Use the classical Langevin equation for a free particle,

$$M\dot v = -M\gamma v+\xi(t),$$

with equilibrium velocity variance $\langle v^2\rangle=k_BT/M$, to motivate $D_x=k_BT/(M\gamma)$.

Solution

For a stationary Ornstein–Uhlenbeck velocity process,

$$\langle v(t)v(0)\rangle = \frac{k_BT}{M} e^{-\gamma |t|}.$$

The long-time position diffusion constant is

$$D_x = \int_0^\infty dt\, \langle v(t)v(0)\rangle.$$

Substituting the correlation function gives

$$D_x = \frac{k_BT}{M} \int_0^\infty dt\,e^{-\gamma t} = \frac{k_BT}{M\gamma}.$$

Diffusion and thermal momentum variance

For a free particle obeying $\partial_tW=\gamma\partial_p(pW)+D_{pp}\partial_p^2W$ after integrating over position, derive the stationary momentum variance and find the diffusion coefficient required by equipartition.

Solution

Multiply by $p^2$ and integrate twice by parts, assuming the distribution and its first derivatives vanish at large momentum:

$$\frac{d}{dt}\langle p^2\rangle=-2\gamma\langle p^2\rangle+2D_{pp}.$$

Thus $\langle p^2\rangle_{\mathrm{st}}=D_{pp}/\gamma$. Requiring $\langle p^2\rangle_{\mathrm{st}}=Mk_BT$ gives $D_{pp}=M\gamma k_BT$. Equivalently, the Langevin equation has noise amplitude $\sqrt{2D_{pp}}$, which reproduces $\langle\xi(t)\xi(s)\rangle=2M\gamma k_BT\delta(t-s)$.

Positivity warning

Consider a free particle under the constant-coefficient high-temperature equation, with $D_{pp}=M\gamma k_BT$, $D_{qq}=D_{pq}=0$, and $\gamma,T>0$. Prepare a centered minimum-uncertainty Gaussian with covariance

$$V(0)=\begin{pmatrix}s&0\\0&\hbar^2/(4s)\end{pmatrix},\qquad s>0.$$

Find the initial derivative of $\det V$. Which preparations immediately violate the uncertainty principle? Check what changes if the minimal completely positive correction $D_{qq}=\hbar^2\gamma/(16Mk_BT)$ is included.

Solution

The covariance equation uses

$$A=\begin{pmatrix}0&1/M\\0&-\gamma\end{pmatrix},\qquad \dot V=AV+VA^{\mathsf T}+2D.$$

For $D_{qq}=D_{pq}=0$, differentiation of the determinant gives

$$\frac{d}{dt}\det V=-2\gamma\det V+2D_{pp}V_{qq}.$$

Initially $\det V=\hbar^2/4$, so

$$\left.\frac{d}{dt}\det V\right|_0 =-\frac{\gamma\hbar^2}{2}+2M\gamma k_BT\,s.$$

Every $s<\hbar^2/(4Mk_BT)$ therefore gives $\det V(t)<\hbar^2/4$ for sufficiently small positive $t$. For example, $s=\hbar^2/(8Mk_BT)$ gives initial slope $-\gamma\hbar^2/4$. The formal Gaussian Wigner function remains nonnegative and normalized, but its implied purity $\hbar/(2\sqrt{\det V})$ exceeds one. Classical probability positivity does not ensure a physical quantum state.

With the stated $D_{qq}$ correction, the extra contribution to the initial determinant derivative is $2D_{qq}V_{pp}(0)$, giving

$$\left.\frac{d}{dt}\det V\right|_0 =\frac{2M\gamma k_BT}{s} \left(s-\frac{\hbar^2}{8Mk_BT}\right)^2\ge0.$$

This checks the squeezed family directly; complete positivity for all inputs follows from the generator matrix condition above. The correction repairs that condition but does not by itself reproduce the exact microscopic bath dynamics.

The counterexample extrapolates the approximate equation to arbitrarily short times and sharply localized pure states. Even at high temperature that extrapolation needs justification. A coarse-grained Brownian description can still be useful within its preparation and time-resolution limits; the exact factorized system–bath evolution is completely positive (Carlesso and Bassi, 2017, Sec. IV).

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
• M. Carlesso and A. Bassi, “Adjoint master equation for quantum Brownian motion,” Physical Review A 95, 052119 (2017). doi:10.1103/PhysRevA.95.052119; arXiv:1602.05116.
• A. Einstein, “On the movement of small particles suspended in stationary liquids required by the molecular-kinetic theory of heat,” Annalen der Physik 17, 549–560 (1905).
• H. Grabert, P. Schramm, and G.-L. Ingold, “Quantum Brownian motion: The functional integral approach,” Physics Reports 168, 115–207 (1988).
• B. L. Hu, J. P. Paz, and Y. Zhang, “Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise,” Physical Review D 45, 2843–2861 (1992).
• P. Langevin, “Sur la théorie du mouvement brownien,” Comptes Rendus 146, 530–533 (1908).
• U. Weiss, Quantum Dissipative Systems, 4th ed., World Scientific (2012).