Thermal Master Equations

A thermal master equation is a Markovian open-system equation for a system weakly coupled to an equilibrium heat bath. Its rates are constrained by bath temperature, and for a single undriven bath it should relax toward a Gibbs state, up to the approximations and Hamiltonian renormalizations used.

The minimal thermal consistency test is:

$$\mathcal L(\rho_\beta)=0, \qquad \rho_\beta = \frac{e^{-\beta H_S}} {\operatorname{Tr}(e^{-\beta H_S})}.$$

For a standard weak-coupling secular derivation, this stationarity is enforced by Detailed Balance. A master equation can be in Lindblad form and still fail to be the correct thermal equation if its rates do not satisfy the thermal relations.

For the general fixed-point and relaxation-mode language, see Steady States and Relaxation. For a small numerical contract that computes steady states and checks Liouvillian spectra, see Solving Lindblad Equations.

What Makes It Thermal

A Markovian master equation is thermal when the environment is modeled as a stationary equilibrium reservoir and the reduced generator reflects that equilibrium.

The usual ingredients are:

• a system Hamiltonian $H_S$ with resolved energy gaps;
• a bath Hamiltonian $H_B$ and thermal state $\rho_B=e^{-\beta H_B}/Z_B$;
• weak system–bath coupling;
• bath correlations satisfying the KMS condition;
• Markov and secular approximations;
• rates satisfying detailed balance;
• a Gibbs or appropriately renormalized thermal steady state.

The word “thermal” should not be attached merely because a dissipator has positive rates. The rates must know the bath temperature.

Standard Weak-Coupling Form

Start from an interaction

$$H_I = \sum_\alpha A_\alpha\otimes B_\alpha.$$

Decompose the system operators into Bohr-frequency components:

$$A_\alpha(\omega) = \sum_{\epsilon'-\epsilon=\hbar\omega} \Pi(\epsilon)A_\alpha\Pi(\epsilon').$$

With this convention, positive $\omega$ labels a system operator that lowers the system energy by $\hbar\omega$.

After Born, Markov, and secular approximations, a common thermal generator is

$$\begin{aligned} \frac{d\rho}{dt} =& - \frac{i}{\hbar} [H_S+H_{\mathrm{LS}},\rho] \\ &+ \sum_{\omega,\alpha,\beta} \gamma_{\alpha\beta}(\omega) \left( A_\beta(\omega)\rho A_\alpha^\dagger(\omega) - \frac12 \{A_\alpha^\dagger(\omega)A_\beta(\omega),\rho\} \right). \end{aligned}$$

The rate matrix $\gamma_{\alpha\beta}(\omega)$ is built from bath spectra. For an equilibrium bath, the KMS relation implies detailed-balance relations between the $\omega$ and $-\omega$ blocks. For one bath operator, the schematic relation is

$$\gamma(-\omega) = e^{-\beta\hbar\omega}\gamma(\omega), \qquad \omega>0.$$

This is what makes upward transitions thermally suppressed relative to downward transitions.

Two-Level System

Let $E_e-E_g=\hbar\omega_0$ with $\omega_0>0$. A finite-temperature two-level thermal master equation is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar}[H,\rho] + \Gamma_\downarrow\mathcal D[\sigma_-]\rho + \Gamma_\uparrow\mathcal D[\sigma_+]\rho,$$

where

$$\sigma_-=\lvert g\rangle\langle e\rvert, \qquad \sigma_+=\lvert e\rangle\langle g\rvert.$$

Thermal detailed balance requires

$$\frac{\Gamma_\uparrow}{\Gamma_\downarrow} = e^{-\beta\hbar\omega_0}.$$

The excited-state population obeys

$$\dot p_e = -\Gamma_\downarrow p_e + \Gamma_\uparrow(1-p_e),$$

This is the two-state instance of a Pauli Rate Equation.

The steady state is therefore

$$p_e^{\mathrm{ss}} = \frac{\Gamma_\uparrow} {\Gamma_\uparrow+\Gamma_\downarrow}.$$

Using detailed balance,

$$p_e^{\mathrm{ss}} = \frac{1} {1+e^{\beta\hbar\omega_0}},$$

which is the Gibbs excited-state population for a two-level system.

Zero-temperature amplitude damping is the special case $\Gamma_\uparrow=0$. It is not a finite-temperature thermal model unless upward excitation is negligible. See Amplitude Damping Master Equation for the two-level relaxation generator and its zero-temperature limit.

Harmonic Oscillator

For the occupation and vacuum-noise bookkeeping behind these terms, see Thermal and Vacuum Noise. For a harmonic oscillator coupled to a thermal bath with mean occupation

$$\bar n = \frac{1} {e^{\beta\hbar\omega}-1},$$

a standard thermal damping equation is

$$\frac{d\rho}{dt} = - \frac{i}{\hbar} [\hbar\omega a^\dagger a,\rho] + \kappa(\bar n+1)\mathcal D[a]\rho + \kappa\bar n\,\mathcal D[a^\dagger]\rho.$$

The $a$ term describes loss of one quantum. The $a^\dagger$ term describes absorption from the bath.

The mean occupation obeys

$$\frac{d}{dt}\langle n\rangle = -\kappa(\langle n\rangle-\bar n), \qquad n=a^\dagger a.$$

Thus the oscillator relaxes toward the thermal occupation:

$$\langle n(t)\rangle = \bar n + e^{-\kappa t} \left( \langle n(0)\rangle-\bar n \right).$$

The upward and downward adjacent-level rates satisfy

$$\frac{\kappa\bar n} {\kappa(\bar n+1)} = e^{-\beta\hbar\omega}.$$

Pure Dephasing Is Not Thermalization

A pure-dephasing term such as

$$\gamma_\phi\mathcal D[\sigma_z]\rho$$

may arise from thermal noise at zero frequency, but it does not by itself drive the system toward a Gibbs distribution. It damps coherences in the $\sigma_z$ basis while leaving populations unchanged.

Thermalization requires energy-exchange channels with rates that satisfy detailed balance. Dephasing can accompany thermal relaxation, but it is not a substitute for upward and downward transition terms.

Global Versus Local Thermal Equations

For a composite system with interacting parts, the safest weak-coupling thermal derivation usually uses the eigenbasis of the full system Hamiltonian $H_S$, not the bare Hamiltonians of the subsystems separately.

This is called a global master equation. It constructs jump operators from the Bohr frequencies of the complete $H_S$ and is the natural setting for detailed balance.

A local master equation uses dissipators such as

$$\mathcal D[a_1], \qquad \mathcal D[a_2],$$

on subsystems as if their couplings to the bath were independent of the interactions inside $S$. Local equations can be useful approximations, especially when internal couplings are weak or the modeling target is phenomenological, but they may violate detailed balance or predict heat currents in situations that should be equilibrium.

The practical question is:

Which Hamiltonian defines the transition frequencies seen by the bath?

If the answer is the interacting Hamiltonian, use the global energy basis.

Strong-Coupling Caveat

At stronger system–bath coupling, the bare Gibbs state

$$\frac{e^{-\beta H_S}} {\operatorname{Tr}(e^{-\beta H_S})}$$

need not be the correct reduced equilibrium state. The equilibrium reduced state of the coupled total system is

$$\rho_S^{\mathrm{eq}} = \operatorname{Tr}_B \frac{ e^{-\beta(H_S+H_B+H_I)} }{ \operatorname{Tr} e^{-\beta(H_S+H_B+H_I)} }.$$

This can be represented using a Hamiltonian of mean force, not simply $H_S$. A weak-coupling thermal master equation may still be useful, but one should not demand exact relaxation to the bare Gibbs state outside its regime. See Strong Coupling for the broader warning.

Multiple Baths and Nonequilibrium Steady States

If several baths are present, the generator may be a sum

$$\mathcal L = \sum_r\mathcal L_r,$$

where each $\mathcal L_r$ satisfies detailed balance at its own inverse temperature $\beta_r$.

If the temperatures or chemical potentials differ, there is generally no single Gibbs steady state. The system may reach a nonequilibrium steady state with heat or particle currents.

This is not a failure of Lindblad form. It is a different physical situation from a single equilibrium thermal bath.

Practical Derivation Checklist

For a proposed thermal master equation, identify:

• the Hamiltonian whose Gibbs state is expected;
• the bath temperature and spectral density;
• the Bohr-frequency convention;
• the downward and upward jump operators;
• the detailed-balance relation between rates;
• any Lamb-shift or renormalized Hamiltonian;
• whether secularization is valid near degeneracies;
• whether the equation is global or local;
• whether more than one bath or a drive is present.

The broader consistency checks are collected in the Approximation Checklist.

Quantum Annealing uses only a declared time-dependent generator or phenomenological rate reduction and records its regime, rates, schedule, endpoint distribution, and freeze-out hypothesis. This page retains the derivation and validity conditions for thermal generators: KMS or detailed balance, Gibbs stationarity, global-versus-local construction, secularization, and strong-coupling limits.

Common Mistakes

Calling arbitrary positive rates thermal

Positive Lindblad rates make a valid Markovian semigroup, but thermal rates must satisfy detailed balance with a specified temperature.

Forgetting upward transitions

Finite temperature generally requires both downward and upward jumps. Zero-temperature amplitude damping is only the low-temperature limit.

Expecting dephasing to thermalize populations

Pure dephasing can come from thermal fluctuations, but it does not change energy populations by itself.

Using a local equation outside its regime

For interacting systems, local dissipators can violate global thermal detailed balance. Check whether the bath resolves the energy gaps of the interacting Hamiltonian.

Ignoring Hamiltonian renormalization

The thermal steady state may involve a Lamb-shift-renormalized Hamiltonian in weak coupling or a Hamiltonian of mean force at stronger coupling. Strong Coupling explains the thermodynamic warning, and Reaction-Coordinate Mapping gives one constructive way to expose those corrections by enlarging the system.

Exercises

Two-level Gibbs population

Use detailed balance to show that the steady state of

$$\dot p_e = -\Gamma_\downarrow p_e + \Gamma_\uparrow(1-p_e)$$

has the Gibbs excited-state population.

Solution

The steady state is

$$p_e^{\mathrm{ss}} = \frac{\Gamma_\uparrow} {\Gamma_\uparrow+\Gamma_\downarrow}.$$

Detailed balance gives

$$\Gamma_\uparrow = e^{-\beta\hbar\omega_0}\Gamma_\downarrow.$$

Substitute:

$$p_e^{\mathrm{ss}} = \frac{e^{-\beta\hbar\omega_0}} {1+e^{-\beta\hbar\omega_0}} = \frac{1} {1+e^{\beta\hbar\omega_0}}.$$

Oscillator occupation relaxation

Given

$$\frac{d}{dt}\langle n\rangle = -\kappa(\langle n\rangle-\bar n),$$

solve for $\langle n(t)\rangle$.

Solution

This first-order equation has solution

$$\langle n(t)\rangle-\bar n = e^{-\kappa t} \left( \langle n(0)\rangle-\bar n \right).$$

Therefore

$$\langle n(t)\rangle = \bar n + e^{-\kappa t} \left( \langle n(0)\rangle-\bar n \right).$$

Zero-temperature limit

Take $\bar n\to0$ in the oscillator thermal damping equation. What remains?

Solution

When $\bar n=0$,

$$\kappa(\bar n+1)\mathcal D[a]\rho + \kappa\bar n\,\mathcal D[a^\dagger]\rho \longrightarrow \kappa\mathcal D[a]\rho.$$

Only loss remains. The oscillator relaxes toward the vacuum.

Why local can fail

Why can a local dissipator built from bare subsystem lowering operators fail detailed balance for an interacting composite system?

Solution

Detailed balance relates rates at the actual transition frequencies of the Hamiltonian whose Gibbs state is expected. If the subsystems interact, the eigenstates and energy gaps of the full $H_S$ need not match the bare subsystem transitions. A dissipator built from bare lowering operators can therefore assign rates to the wrong frequencies and fail to make the global Gibbs state stationary.

References

• E. B. Davies, Quantum Theory of Open Systems, Academic Press (1976).
• R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, Springer (1987).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• C. W. Gardiner and P. Zoller, Quantum Noise, Springer, 3rd ed. (2004).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press (2010).
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012).