Common Master Equations

Master equations describe how a reduced state changes in time. The equations below are common starting points for qubits, oscillators, and Brownian particles. Their useful range depends on the bath, preparation, and timescales; a trace-preserving formula is not by itself a physically accurate model.

Required background. The Lindblad–GKSL Equation explains completely positive semigroups; The Redfield Equation explains the weak-coupling equation before full secular averaging.

Qubit and oscillator generators

Write $\mathcal H\rho=-i[H,\rho]/\hbar$ and

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

Unless stated otherwise, coefficients are constant and nonnegative. For qubits use $\sigma_-=|g\rangle\langle e|$, $Z=|g\rangle\langle g|-|e\rangle\langle e|$, and $H=-(\hbar\omega_0/2)Z$, with $\omega_0>0$. Thus $|e\rangle$ has the higher energy.

Pure dephasing:

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

Energy populations remain fixed and $\rho_{eg}(t)=e^{-(i\omega_0+\Gamma_\phi)t}\rho_{eg}(0)$. The constant-rate form describes exponential decay; slowly varying frequency noise often gives a nonexponential envelope and requires a different model. See Pure Dephasing Master Equation.

Relaxation, excitation, and additional dephasing:

$$\dot\rho={}\mathcal H\rho +\Gamma_\downarrow\mathcal D[\sigma_-]\rho +\Gamma_\uparrow\mathcal D[\sigma_+]\rho +\frac{\Gamma_\phi}{2}\mathcal D[Z]\rho.$$

Population deviations decay at $\Gamma_1=\Gamma_\downarrow+\Gamma_\uparrow$ and coherence at $\Gamma_2=\Gamma_1/2+\Gamma_\phi$. When $\Gamma_1>0$, the stationary excited population is $\Gamma_\uparrow/\Gamma_1$. Zero-temperature amplitude damping sets $\Gamma_\uparrow=0$. A single equilibrium reservoir requires $\Gamma_\uparrow/\Gamma_\downarrow=e^{-\beta\hbar\omega_0}$; positive rates alone do not specify a temperature. See Thermal Master Equations and Detailed Balance.

Thermal oscillator: with $H=\hbar\omega a^\dagger a$,

$$\dot\rho={}\mathcal H\rho +\kappa(\bar n+1)\mathcal D[a]\rho +\kappa\bar n\mathcal D[a^\dagger]\rho.$$

Here $\bar n=(e^{\beta\hbar\omega}-1)^{-1}$. The mean number obeys $d\langle N\rangle/dt=-\kappa(\langle N\rangle-\bar n)$, while the field amplitude decays at $\kappa/2$ and rotates at $\omega$. The mode has an infinite Hilbert space; numerical cutoffs require convergence checks.

These are common quantum optical master equations. Their microscopic derivation usually uses weak coupling, short bath memory, and suitable frequency averaging. Strong intermode coupling or strong driving can change the relevant transition operators; attaching the same local dissipators is an additional approximation. A photon record requires trajectories as well as the unconditional equation.

Brownian friction and diffusion

For a particle of mass $M$ with $[q,p]=i\hbar$, choose the friction convention $d\langle p\rangle/dt=-\gamma\langle p\rangle$ in the free case. The high-temperature, Ohmic Caldeira–Leggett equation is then

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

This convention gives momentum diffusion $D_{pp}=M\gamma k_BT$. Authors who put $-i\gamma[q,\{p,\rho\}]/\hbar$ in the friction term instead obtain $D_{pp}=2M\gamma k_BT$. Mixing these two conventions changes the equilibrium kinetic energy by a factor of two.

The displayed equation is not a completely positive semigroup at arbitrary short times. With additional terms $-D_{qq}[p,[p,\rho]]/\hbar^2$ and $2D_{pq}[q,[p,\rho]]/\hbar^2$, complete positivity requires

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

The uncorrected high-temperature equation has $D_{qq}=D_{pq}=0$ and fails this inequality when $\gamma\ne0$. Its usefulness is an approximation statement, not permission to treat it as a universal quantum channel. See Caldeira–Leggett Model and Quantum Brownian Motion.

Redfield dynamics and population equations

The Redfield equation retains weak-coupling terms connecting different system transition frequencies. In an energy basis it has the schematic form

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

It can retain near-degenerate coherence–population coupling lost by full secularization. It does not generally preserve positivity or complete positivity. Small negative eigenvalues need comparison with the approximation error and the intended time range; simply clipping them changes the dynamics. See Redfield Equation and Secular Approximation.

The Pauli rate equation retains populations:

$$\begin{gathered} \dot p_n=\sum_{m\ne n}(W_{nm}p_m-W_{mn}p_n),\\ W_{nm}\ge0. \end{gathered}$$

$W_{nm}$ is the rate from $m$ to $n$. Rate-absorbed jumps $L_{nm}=\sqrt{W_{nm}}|n\rangle\langle m|$ reproduce these populations when the Hamiltonian is diagonal in this basis. For this particular embedding, each coherence $\rho_{nm}$ decays at half the sum of the total outgoing rates from $n$ and $m$. Extra dephasing can leave the population equation unchanged, so population rates do not uniquely determine quantum dynamics. See Pauli Rate Equations.

Matching an equation to a calculation

Question	Useful check
Is the model structurally physical?	Trace, Hermiticity, and the relevant positivity or complete-positivity condition
Does it describe this reservoir?	Couplings, temperature, spectral shape, and detailed balance where applicable
Are approximations separated in scale?	Bath memory, relaxation, frequency splittings, and drive times
Is the numerical solution accurate?	Timestep, Hilbert-space cutoff, conservation laws, and a solvable limit

The Approximation Checklist explains these distinctions; Solving Lindblad Equations gives computational examples.

Exercises

Brownian equilibrium. The free-particle equation above implies $d\langle p^2\rangle/dt=-2\gamma\langle p^2\rangle+2D_{pp}$. Find its stationary kinetic energy.

Solution

$\langle p^2\rangle=D_{pp}/\gamma=Mk_BT$, so $\langle p^2\rangle/(2M)=k_BT/2$. Using $2M\gamma k_BT$ with the same friction coefficient would incorrectly double it.

Identical populations, different coherences. Add $\lambda\mathcal D[Z]$ to a thermal qubit equation. What changes?

Solution

The population equation and stationary excited population are unchanged. The coherence-decay rate increases by $2\lambda$. Population data alone cannot determine this additional dephasing.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Weak-coupling generators and Brownian motion.
• Caldeira, A. O., and A. J. Leggett. “Path integral approach to quantum Brownian motion.” Physica A 121, 587–616 (1983). doi:10.1016/0378-4371(83)90013-4.
• Gardiner, C. W., and P. Zoller. Quantum Noise. 3rd edition, Springer, 2004. Optical and oscillator equations.
• Redfield, A. G. “On the theory of relaxation processes.” IBM Journal of Research and Development 1, 19–31 (1957). doi:10.1147/rd.11.0019.