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 model is simple enough to write in one line and rich enough to be dangerous. Depending on the spectral density, cutoff, temperature, coupling strength, and initial state, it can describe Markovian damping, non-Markovian memory, strong-coupling equilibrium, or an invalid high-temperature approximation pushed outside its domain.

Hamiltonian

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 prevents the bath coupling from unintentionally changing the bare potential $V(q)$ under this convention.

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

In the corresponding Langevin description, the coordinate feels both a retarded friction force and a fluctuating force. The damping kernel is often written, up to convention-dependent factors, as

$$\eta(t) = \frac{2}{\pi} \int_0^\infty d\omega\, \frac{J(\omega)}{\omega} \cos(\omega t).$$

The symmetrized noise kernel has the form

$$\nu(t) = \frac{1}{\pi} \int_0^\infty d\omega\, J(\omega) \coth \left( \frac{\beta\hbar\omega}{2} \right) \cos(\omega t).$$

The two kernels are not independent at thermal equilibrium. The same $J(\omega)$ controls both dissipation and fluctuations, with the thermal factor $\coth(\beta\hbar\omega/2)$ supplying quantum and thermal noise.

For a classical-looking coordinate equation, the corresponding structure is

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

where $\xi(t)$ is the fluctuating force. This equation is an interpretation aid; the fully quantum problem requires operator ordering or an equivalent influence-functional, master-equation, or phase-space treatment.

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 a damping scale and $\Omega_c$ is a cutoff frequency. In a broad-band high-cutoff limit, the damping kernel becomes sharply peaked and one recovers approximately local friction,

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

up to the convention used for $\gamma$.

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

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

The second term is friction. The third term is momentum diffusion and 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:

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

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

Positivity Warning

The high-temperature Caldeira–Leggett equation above is not automatically in Lindblad form. As written, it can violate complete positivity outside its regime of validity, especially at short times or low temperature.

A more general Brownian master equation includes diffusion coefficients such as

$$D_{pp}, \qquad D_{xx}, \qquad D_{xp},$$

and complete positivity imposes inequalities among diffusion and friction coefficients. A commonly stated condition has the schematic form

$$D_{pp}D_{xx}-D_{xp}^2 \ge \frac{\hbar^2\gamma^2}{4},$$

with details depending on the friction convention.

The moral is simple: the compact high-temperature equation is historically and physically important, but it should not be used as a universal Lindblad generator. If positivity matters, derive or choose a completely positive form and state the approximation.

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 skeleton but differ in spectral density, cutoff, initial state, and interpretation of $q$.

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.

Common Mistakes

Dropping the counterterm without noticing

The counterterm fixes which potential is being called $V(q)$. Omitting it changes the model unless the potential has already been renormalized.

Treating Ohmic as cutoff-free

A strictly Ohmic spectrum to infinite frequency is an idealization with short-time pathologies. A cutoff or physical bandwidth is part of the model.

Using the high-temperature equation at low temperature

The compact Caldeira–Leggett master equation assumes high temperature compared with relevant quantum frequencies. Low-temperature quantum noise requires more careful treatment.

Confusing damping with decoherence

Friction and position decoherence are linked by fluctuation–dissipation in equilibrium, but they are not the same term in the master equation.

Assuming the steady state is bare Gibbs at strong coupling

At strong coupling, equilibrium may involve a Hamiltonian of mean force rather than $e^{-\beta H_S}$. Weak-coupling thermal intuition can fail.

Ignoring initial slips

Factorized initial states and sharp cutoffs can produce short-time transients. They should not be mistaken for universal long-time damping.

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

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

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.

Cross-Links

• System–Bath Hamiltonians
• Quantum Langevin Equations
• Quantum Brownian Motion
• Noise Spectra
• Fluctuation–Dissipation Relation
• Spin-Boson Model
• Memory Kernels
• Thermal Master Equations
• Reaction-Coordinate Mapping
• Hierarchical Equations of Motion
• Approximation Checklist

References

• A. O. Caldeira and A. J. Leggett, “Quantum tunnelling in a dissipative system,” Annals of Physics 149, 374-456 (1983).
• A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587-616 (1983).
• 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).
• 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).
• U. Weiss, Quantum Dissipative Systems, World Scientific (2012).