Common Master Equations

This page is a compact guide to master equations that appear repeatedly in measurement theory and open quantum systems. It is not a derivation page. Use it to recognize a generator, identify the intended physical regime, and spot the first assumptions to check.

A master equation describes an infinitesimal time evolution,

$$\dot\rho(t) = \mathcal L_t[\rho(t)].$$

A finite-time channel describes the integrated map $\rho(0)\mapsto\rho(t)$. A generator can be useful and still fail to generate completely positive dynamics outside its approximation regime, so the equation’s assumptions matter as much as its algebraic form.

Throughout, the dissipator is

$$\mathcal D[L]\rho = L\rho L^\dagger - \frac12\{L^\dagger L,\rho\}.$$

For the canonical structure of completely positive Markovian semigroups, see Lindblad–GKSL Equation. For a practical derivation audit, see the Approximation Checklist.

Fast Map

Equation	Typical generator	Use	First caution
pure dephasing	$\Gamma_\phi\mathcal D[\text{diagonal operator}]$	phase coherence decay	basis and rate convention matter
amplitude damping	$\Gamma_1\mathcal D[\sigma_-]$	zero-temperature relaxation	not unital; coherences decay at half the population rate
thermal qubit damping	$\Gamma_\downarrow\mathcal D[\sigma_-]+\Gamma_\uparrow\mathcal D[\sigma_+]$	finite-temperature two-level relaxation	rates must satisfy thermal or reservoir constraints
damped harmonic oscillator	$\kappa(\bar n+1)\mathcal D[a]+\kappa\bar n\mathcal D[a^\dagger]$	oscillator thermalization	infinite-dimensional domain and cutoff care
quantum optical master equation	sum of optical loss, emission, pumping, and dephasing channels	atoms, cavities, light-matter systems	broadband Markov and secular assumptions
Caldeira–Leggett high-temperature equation	friction plus position diffusion	quantum Brownian motion	not automatically completely positive
Bloch–Redfield equation	weak-coupling nonsecular relaxation tensor	near-degenerate multilevel dynamics	may fail positivity
Pauli rate equation	classical transition-rate matrix for populations	incoherent populations	discards coherences

Lindblad-Form Workhorses

Pure Dephasing

For a qubit with Hamiltonian $H=(\hbar\omega_0/2)Z$, a common convention is

$$\dot\rho = - \frac{i}{\hbar}[H,\rho] + \frac{\Gamma_\phi}{2} \mathcal D[Z]\rho.$$

This convention gives

$$\rho_{01}(t) = e^{-i\omega_0 t} e^{-\Gamma_\phi t} \rho_{01}(0), \qquad \rho_{00}(t)=\rho_{00}(0).$$

The invariant physical statement is that populations in the dephasing basis are fixed while coherences decay. Equivalent equations may use projectors, such as $2\Gamma_\phi\mathcal D[\lvert e\rangle\langle e\rvert]\rho$, with the same coherence-decay rate after convention matching.

Use for slow frequency noise, unread which-path information, nonselective measurement in a basis, and elastic environment records. See Pure Dephasing Master Equation and Dephasing Channel.

Amplitude Damping

For a zero-temperature two-level system with excited state $\lvert e\rangle$ and ground state $\lvert g\rangle$,

$$\sigma_- = \lvert g\rangle\langle e\rvert, \qquad \dot\rho = - \frac{i}{\hbar}[H,\rho] + \Gamma_1\mathcal D[\sigma_-]\rho.$$

In the interaction picture,

$$\dot\rho_{ee} = -\Gamma_1\rho_{ee}, \qquad \dot\rho_{eg} = -\frac{\Gamma_1}{2}\rho_{eg}.$$

Use for spontaneous emission, qubit $T_1$ relaxation, and zero-temperature loss. Main caution: amplitude damping is not a Pauli channel and not unital. See Amplitude Damping Master Equation and Amplitude-Damping Channel.

Thermal Qubit Damping

At finite temperature or in a pumped reservoir, include upward and downward transitions:

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

For a true equilibrium bath at inverse temperature $\beta$ and transition frequency $\omega_0>0$,

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

in the usual weak-coupling convention. The stationary excited-state population is

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

Use for thermal relaxation of a two-level system. Main caution: positive rates are not enough to make an equation thermal; the rates must encode the reservoir state. See Thermal Master Equations and Detailed Balance.

Damped Harmonic Oscillator

For a harmonic oscillator coupled to a thermal bath with mean occupation $\bar n$,

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

The mean occupation relaxes as

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

At zero temperature, $\bar n=0$, only the loss dissipator remains. Use for damped cavities, vibrational modes, and truncated oscillator simulations. Main caution: numerical truncation can create artifacts; verify cutoff convergence. See Thermal Master Equations and Solving Lindblad Equations.

Quantum Optical Master Equation

The phrase “quantum optical master equation” usually refers to a Lindblad–GKSL equation derived for atoms, cavities, and light-matter systems coupled to broadband electromagnetic reservoirs:

$$\dot\rho = - \frac{i}{\hbar} [H_{\text{sys}}+H_{\mathrm{LS}},\rho] + \sum_j \Gamma_j\mathcal D[L_j]\rho.$$

Typical operators include $\sigma_-$ for spontaneous emission, $a$ for cavity loss, $a^\dagger$ for thermal excitation or gain, and projectors for dephasing. Use for resonance fluorescence, cavity damping, driven atoms, and photon-counting models. Main caution: the equation is unconditional; monitored photons require a trajectory or filtering description. See Quantum Optical Master Equation, Quantum-Jump Trajectories, and Diffusive Trajectories.

Handle-With-Care Equations

Caldeira–Leggett High-Temperature Equation

For a particle coordinate $q$ coupled to an Ohmic oscillator bath, the high-temperature Markovian Caldeira–Leggett equation is often written

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

The second term is friction. The third term is momentum diffusion and position-basis decoherence. In the $q$ representation it suppresses spatial coherences at a rate proportional to $(q-q')^2$.

Use for high-temperature quantum Brownian motion and coordinate decoherence estimates. Main caution: the displayed equation is not automatically in Lindblad form and can violate complete positivity outside its regime of validity. See Caldeira–Leggett Model and Fluctuation–Dissipation Relation.

Bloch–Redfield Equation

The Bloch–Redfield equation is a weak-coupling, usually Born–Markov, master equation before full secularization. In a system energy basis, it is often summarized as

$$\dot\rho_{ab} = -i\omega_{ab}\rho_{ab} + \sum_{cd} R_{ab,cd}\rho_{cd},$$

where the Redfield tensor $R_{ab,cd}$ is built from system coupling operators and bath correlation functions.

Use when coherence-population coupling or nearly degenerate Bohr frequencies matter and a full secular approximation would discard important physics. Main caution: nonsecular Redfield equations are not generally completely positive and may produce negative populations when pushed outside their controlled regime. See Redfield Equation.

Pauli Rate Equation

The Pauli rate equation keeps only populations in a preferred basis. See Pauli Rate Equations for the derivation, conventions, and limitations:

$$\dot p_n = \sum_m \left( W_{nm}p_m - W_{mn}p_n \right), \qquad W_{nm}\ge0.$$

It is the classical population limit of a quantum master equation after coherences have decayed or been neglected. It can be embedded in Lindblad form using jump operators

$$L_{nm} = \sqrt{W_{nm}}\, \lvert n\rangle\langle m\rvert \qquad (n\ne m),$$

which produce the same population equation and usually damp coherences as well.

Use for incoherent hopping, sequential tunneling, classical Markov models, and populations after secularization. Main caution: it is not a substitute for a density-matrix equation when coherences, interference, degeneracies, or measurement backaction are important.

Quick Consistency Checks

Before trusting a master equation, ask:

• Does it preserve trace and Hermiticity?
• If it is claimed to be Lindblad–GKSL, are all rates nonnegative?
• If it is claimed to be thermal, do the rates satisfy detailed balance or a stated nonequilibrium reservoir relation?
• If it is a Redfield or Caldeira–Leggett equation, what is the weak-coupling, high-temperature, or nonsecular domain where positivity problems are controlled?
• If it is solved numerically, are oscillator cutoffs, timestep convergence, and steady states checked?

For a fuller checklist, use the Approximation Checklist.

Self-Checks

1. A qubit’s excited-state population decays as $e^{-\Gamma_1 t}$ and its coherence decays as $e^{-\Gamma_1t/2}$, with no upward transitions. Which reference equation is this?

Solution

This is zero-temperature amplitude damping with generator $\Gamma_1\mathcal D[\sigma_-]\rho$, up to Hamiltonian phase rotation.

2. A two-level thermal equation has positive $\Gamma_\uparrow$ and $\Gamma_\downarrow$, but their ratio is arbitrary. Is it automatically an equilibrium thermal master equation?

Solution

No. It may be a valid pumped-reservoir model, but equilibrium thermal interpretation requires a detailed-balance relation such as $\Gamma_\uparrow/\Gamma_\downarrow=e^{-\beta\hbar\omega_0}$ in the standard convention.

3. Why is the high-temperature Caldeira–Leggett equation not listed as a generic Lindblad–GKSL workhorse?

Solution

The common high-temperature form is not automatically completely positive. It is an approximation useful in its Brownian high-temperature regime, but outside that regime one must check diffusion terms and positivity conditions.

References

• 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, 2004.
• H. J. Carmichael, Statistical Methods in Quantum Optics 1, Springer, 1999.
• U. Weiss, Quantum Dissipative Systems, World Scientific, 2012.
• A. O. Caldeira and A. J. Leggett, “Path integral approach to quantum Brownian motion,” Physica A 121, 587, 1983.
• A. G. Redfield, “On the theory of relaxation processes,” IBM Journal of Research and Development 1, 19, 1957.
• V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,” Journal of Mathematical Physics 17, 821, 1976.
• G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics 48, 119, 1976.