Caldeira–Leggett Model

The Caldeira–Leggett model is the canonical oscillator-bath model for a quantum coordinate coupled to a dissipative environment. It is the standard starting point for quantum Brownian motion, dissipative tunneling, coordinate decoherence, damping kernels, and the relation between a bath spectral density and friction.

The Hamiltonian and its high-temperature master equation are different levels of description. The Hamiltonian can describe memory and strong coupling; the compact master equation requires additional assumptions about bandwidth, temperature, and coupling. Keeping this distinction in view helps explain when ordinary friction emerges.

A Coordinate Coupled to Oscillators

Let $q$ and $p$ be the system coordinate and momentum, with

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

The bath is a collection of harmonic oscillators with coordinates $x_j$, momenta $p_j$, masses $m_j$, and frequencies $\omega_j$. A compact form of the Caldeira–Leggett Hamiltonian is

$$H = \frac{p^2}{2M} + V(q) + \sum_j \left[ \frac{p_j^2}{2m_j} + \frac12 m_j\omega_j^2 \left( x_j - \frac{c_j q}{m_j\omega_j^2} \right)^2 \right].$$

Expanding the square gives

$$H = H_S + \sum_j \left( \frac{p_j^2}{2m_j} + \frac12 m_j\omega_j^2 x_j^2 \right) - q\sum_j c_j x_j + q^2 \sum_j \frac{c_j^2}{2m_j\omega_j^2}.$$

The interaction is linear in the system coordinate and bath coordinates:

$$H_I = - q\sum_j c_j x_j.$$

The final $q^2$ term is the counterterm. It cancels the static potential shift obtained by minimizing the bath coordinates at fixed $q$. This fixes the meaning of $V(q)$; it does not remove every dynamical frequency shift or strong-coupling correction.

Spectral Density

The oscillator bath is summarized by a spectral density. A common convention is

$$J(\omega) = \frac{\pi}{2} \sum_j \frac{c_j^2}{m_j\omega_j} \delta(\omega-\omega_j), \qquad \omega>0.$$

This is not merely a density of frequencies. It includes the coupling strengths $c_j$ and controls both noise and damping. Different communities place different factors of $\pi$ in $J(\omega)$, so formulas must be read together with their convention. The convention issue is summarized in Spectral Densities.

The bath force operator is

$$B = \sum_j c_j x_j.$$

Its equilibrium correlations are determined by $J(\omega)$ and the bath temperature. This is why the model is a clean bridge between Noise Spectra and the Fluctuation–Dissipation Relation.

Damping and Noise Kernels

Solving each driven bath oscillator and substituting its solution into the equation for $q$ gives the exact operator equation

$$M\ddot q(t)+V'(q(t))+\int_0^t\eta(t-s)\dot q(s)\,ds =\xi(t)-\eta(t)q(0),$$

where, with the spectral density defined above,

$$\begin{aligned} \eta(t)&=\frac{2}{\pi}\int_0^\infty\frac{J(\omega)}{\omega}\cos(\omega t)\,d\omega,\\ \xi(t)&=\sum_j c_j\left[x_j(0)\cos\omega_jt +\frac{p_j(0)}{m_j\omega_j}\sin\omega_jt\right]. \end{aligned}$$

The term $-\eta(t)q(0)$ is the initial slip in this choice of free bath force. For a factorized initial state with the bath thermal about its uncoupled origin, it must be retained. Preparing a displaced or correlated bath changes the initial force statistics and can absorb this term into a different definition of noise. It is not an extra phenomenological force.

For the uncoupled thermal bath, the symmetrized force correlation is

$$\nu(t-s)\equiv\frac12\langle\{\xi(t),\xi(s)\}\rangle =\frac{\hbar}{\pi}\int_0^\infty J(\omega) \coth\!\left(\frac{\beta\hbar\omega}{2}\right) \cos[\omega(t-s)]\,d\omega.$$

Thus the same spectral density determines friction and noise. The factor of $\hbar$ makes $\nu$ a force-squared correlation and gives the classical limit $\nu(t)\simeq k_BT\eta(t)$. The force operators also have a nonzero commutator; replacing them by classical noise loses that part of the dynamics.

Ohmic Bath

An Ohmic bath has low-frequency spectral density proportional to frequency:

$$J_{\mathrm{Ohmic}}(\omega) \propto M\gamma\omega.$$

A cutoff is required at high frequency. Two common examples are

$$J(\omega) = M\gamma\omega e^{-\omega/\Omega_c}$$

and the Drude form

$$J(\omega) = M\gamma\omega \frac{\Omega_c^2}{\omega^2+\Omega_c^2}.$$

Here $\gamma$ is the momentum damping rate in the local limit and $\Omega_c$ is a cutoff frequency. The Drude spectrum gives

$$\eta(t)=M\gamma\Omega_c e^{-\Omega_c t},\qquad t\ge0, \qquad \int_0^\infty\eta(t)\,dt=M\gamma.$$

For times well beyond $1/\Omega_c$, with $\dot q$ varying slowly over that interval, the convolution therefore approaches

$$\int_0^t ds\, \eta(t-s)\dot q(s) \approx M\gamma\dot q(t),$$

In an even, two-sided distribution notation the same limit is $\eta(t)\to2M\gamma\delta(t)$; the integral ending at the delta peak supplies half its weight.

The cutoff is not a cosmetic detail. It affects short-time behavior, renormalization, initial slips, and whether a Markov approximation is meaningful.

High-Temperature Master Equation

The famous Caldeira–Leggett master equation is a high-temperature, weak-coupling, Markovian approximation. In one common convention it reads

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

The second term gives $d\langle p\rangle/dt=-\gamma\langle p\rangle$ for a free particle. The third has momentum diffusion coefficient $D_{pp}=M\gamma k_BT$: it gives $d\langle p^2\rangle/dt=-2\gamma\langle p^2\rangle+2D_{pp}$ and hence the thermal value $\langle p^2\rangle=Mk_BT$. This equilibrium check fixes the factor of two. The same diffusion term produces position-basis decoherence. In the $q$ representation,

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

so the diffusion term suppresses spatial coherences:

$$\partial_t\rho(q,q') \supset - \frac{M\gamma k_B T}{\hbar^2} (q-q')^2\rho(q,q').$$

This formula explains why macroscopic position superpositions decohere rapidly in a hot dissipative environment.

When the Approximate Generator Is Not Completely Positive

The high-temperature equation omits small quantum diffusion terms and is not an exact Lindblad generator when $\gamma\ne0$. It can approximate coarse-grained hot-bath dynamics while giving an unphysical state for sufficiently narrow initial wave packets or sufficiently short times.

With the friction convention above, adding

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

gives a constant-coefficient Gaussian generator that is completely positive precisely when

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

For $D_{pq}=0$ and $D_{pp}=M\gamma k_BT$, the smallest permitted position diffusion is $D_{qq}=\hbar^2\gamma/(16Mk_BT)$. This is one possible positive completion, not a derivation of the exact low-temperature dynamics or a guarantee of an exact Gibbs stationary state. Quantum Brownian Motion explains the covariance form of this condition. Carlesso and Bassi (2017) compare exact oscillator dynamics with the high-temperature equation and a positive completion.

Physical Interpretations

The model has several equivalent readings.

As a Brownian model, $q$ is a particle coordinate and the bath oscillators represent many environmental modes. The bath both kicks and damps the particle.

As a condensed-phase model, $q$ may be a reaction coordinate, tunneling coordinate, or collective nuclear coordinate coupled to solvent or phonon modes.

As a circuit model, $q$ may be a flux or charge coordinate coupled to electromagnetic environmental modes, transmission lines, or impedances.

As a decoherence model, the environment monitors position-like alternatives because the coupling operator is $q$.

These readings share the same mathematical structure but differ in spectral density, cutoff, initial state, and interpretation of $q$. At strong coupling the reduced thermal state is generally not the bare Gibbs state; see strong-coupling equilibrium.

Relation to Other Models

The Caldeira–Leggett model is the coordinate-coupled oscillator-bath model. It is distinct from the Spin-Boson Model, where a two-level system couples to a bosonic bath. The spin-boson model is better for tunneling between two localized states, qubit dephasing and relaxation, and dissipative two-state dynamics.

For structured oscillator baths, Reaction-Coordinate Mapping can move a collective bath coordinate into the system. For Gaussian baths with correlation functions expanded as exponentials, Hierarchical Equations of Motion provide a systematic numerical route.

Exercises

Expand the Counterterm Form

Expand

$$\frac12m_j\omega_j^2 \left( x_j-\frac{c_jq}{m_j\omega_j^2} \right)^2$$

and identify the interaction and counterterm.

Solution

The square gives

$$\frac12m_j\omega_j^2x_j^2 - c_jq x_j + \frac{c_j^2q^2}{2m_j\omega_j^2}.$$

Thus the interaction contribution is

$$-c_j qx_j,$$

and the counterterm contribution is

$$\frac{c_j^2q^2}{2m_j\omega_j^2}.$$

Summing over $j$ gives the full interaction and counterterm.

Position Decoherence Rate

Using the high-temperature Caldeira–Leggett master equation, find the decay rate of the off-diagonal element $\rho(q,q')$ due to the double-commutator term.

Solution

In the coordinate 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 the diffusion term contributes

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

with

$$\Gamma_{\mathrm{dec}}(q,q') = \frac{M\gamma k_B T}{\hbar^2} (q-q')^2.$$

High-Temperature Noise Factor

Show that when $\beta\hbar\omega\ll1$,

$$\coth \left( \frac{\beta\hbar\omega}{2} \right) \approx \frac{2}{\beta\hbar\omega}.$$

Solution

For small $x$,

$$\coth x = \frac{1}{x} + O(x).$$

Set

$$x=\frac{\beta\hbar\omega}{2}.$$

Then

$$\coth \left( \frac{\beta\hbar\omega}{2} \right) \approx \frac{2}{\beta\hbar\omega}.$$

This is the classical high-temperature limit of the quantum thermal factor.

Why Positivity Is Not Automatic

Why does the high-temperature Caldeira–Leggett equation require caution as a quantum channel generator?

Solution

The compact equation contains friction and a dominant momentum-diffusion term, but it is not generally written in Lindblad form. Complete positivity imposes constraints on diffusion coefficients. In particular, position diffusion and cross diffusion terms that are negligible in a high-temperature approximation may still be needed to make the generator completely positive. Thus the equation can be a good asymptotic approximation for some observables while failing as an exact quantum channel generator.

References

• A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587–616 (1983).
• A. O. Caldeira and A. J. Leggett, “Quantum tunnelling in a dissipative system,” Annals of Physics 149, 374–456 (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.
• 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).
• A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, “Dynamics of the dissipative two-state system,” Reviews of Modern Physics 59, 1–85 (1987).
• U. Weiss, Quantum Dissipative Systems, 4th ed., World Scientific (2012).