Common Lindblad Operators

This page is a lookup table for standard Lindblad operators. It tells you what an operator usually means, what master-equation term it appears in, and what to check before assigning it a physical interpretation.

The convention used here is

$$\dot\rho = - \frac{i}{\hbar}[H,\rho] + \sum_k\gamma_k\mathcal D[L_k]\rho, \qquad \gamma_k\ge0,$$

with

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

Some authors absorb $\sqrt{\gamma_k}$ into $L_k$. Then $L_k$ has units of $\mathrm{time}^{-1/2}$ and the separate $\gamma_k$ is absent. Always check this convention before comparing rates.

For the canonical explanation of physical interpretation and nonuniqueness, see Lindblad Operators. For generator formulas, see Common Master Equations.

Operator Map

Operator	Usual meaning	Typical term	First caveat
$\sigma_-=\lvert g\rangle\langle e\rvert$	qubit relaxation, spontaneous emission	$\Gamma_\downarrow\mathcal D[\sigma_-]\rho$	assumes the lower state is retained
$\sigma_+=\lvert e\rangle\langle g\rvert$	excitation, incoherent pumping	$\Gamma_\uparrow\mathcal D[\sigma_+]\rho$	thermal pumping must satisfy detailed balance
$\sigma_z$	qubit pure dephasing in the $z$ basis	$(\Gamma_\phi/2)\mathcal D[\sigma_z]\rho$	factor-of-two conventions are common
$a$	oscillator loss, photon leakage, cavity damping	$\kappa(\bar n+1)\mathcal D[a]\rho$	infinite-dimensional domain or cutoff matters
$a^\dagger$	oscillator excitation, thermal pumping, idealized gain	$\kappa\bar n\,\mathcal D[a^\dagger]\rho$	gain without saturation is only an effective model
$n=a^\dagger a$	number dephasing, phase diffusion	$\gamma_n\mathcal D[n]\rho$	off-diagonal decay grows with number separation
$P=P^\dagger=P^2$	unread projective monitoring, measurement-induced dephasing	$\gamma_P\mathcal D[P]\rho$	a POVM alone does not fix this operator
$\sigma_x,\sigma_y,\sigma_z$ together	Pauli noise, depolarizing benchmark	$\sum_j\gamma_j\mathcal D[\sigma_j]\rho$	isotropic rates are rarely microscopic
$J_-=\sum_j\sigma_-^{(j)}$	collective emission	$\Gamma\mathcal D[J_-]\rho$	collective and independent decay are different models

Qubit Relaxation and Pumping

For a two-level system with $\lvert e\rangle$ above $\lvert g\rangle$,

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

The downward dissipator gives amplitude damping:

$$\Gamma_\downarrow\mathcal D[\sigma_-]\rho.$$

It transfers population from $\lvert e\rangle$ to $\lvert g\rangle$ and damps the coherence $\rho_{eg}$. Adding the upward term

$$\Gamma_\uparrow\mathcal D[\sigma_+]\rho$$

models incoherent excitation. For a single equilibrium bath,

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

in the usual weak-coupling convention. Without such a relation, the model may still be a pumped reservoir, but it should not be called an equilibrium thermal bath without qualification.

See Amplitude Damping Master Equation and Thermal Master Equations.

Qubit Dephasing

For pure dephasing in the $\sigma_z$ basis,

$$\frac{\Gamma_\phi}{2} \mathcal D[\sigma_z]\rho$$

gives

$$\dot\rho_{01} = -\Gamma_\phi\rho_{01}, \qquad \dot\rho_{00}=0, \qquad \dot\rho_{11}=0.$$

The same physics can be written with projectors. For $P_e=\lvert e\rangle\langle e\rvert$,

$$2\Gamma_\phi\mathcal D[P_e]\rho$$

also damps $\rho_{eg}$ at rate $\Gamma_\phi$ in this two-level convention. The operator is different, but the observable dephasing rate can be the same after convention matching.

See Pure Dephasing Master Equation.

Oscillator Loss, Pumping, and Number Dephasing

For a harmonic oscillator, $a$ lowers the excitation number:

$$a\lvert n\rangle = \sqrt n\,\lvert n-1\rangle.$$

Thus

$$\kappa(\bar n+1)\mathcal D[a]\rho$$

is oscillator loss into a bath, while

$$\kappa\bar n\,\mathcal D[a^\dagger]\rho$$

is thermal excitation from the bath. Together they relax the mean occupation toward $\bar n$.

The number operator

$$n=a^\dagger a$$

does not change populations in the number basis. Instead,

$$\left(\mathcal D[n]\rho\right)_{mn} = - \frac12(m-n)^2\rho_{mn}.$$

This is number-basis dephasing. It is useful for phase diffusion and quantum nondemolition monitoring of oscillator number, but it is not the same as energy loss.

For oscillator thermalization, see Thermal Master Equations. For simulation cautions with truncated oscillators, see Solving Lindblad Equations.

Projectors and Measurement-Induced Dephasing

A projector dissipator

$$\gamma_P\mathcal D[P]\rho, \qquad P^2=P=P^\dagger,$$

describes unread monitoring or dephasing between the $P$ and $I-P$ sectors. In the eigenbasis of $P$, coherences connecting the two sectors decay while populations remain fixed.

This is closely related to nonselective measurement: the apparatus has obtained which-sector information, but the observer has not conditioned on a particular outcome. The unconditional state loses coherence between distinguishable alternatives.

Main caution: a POVM effect $E_x$ does not by itself determine a Lindblad operator. The measurement instrument or continuous-monitoring model must be specified. See Quantum Instruments and Measurement Backaction.

Nonuniqueness Checks

The same master equation can be written with different operator lists.

If rates are absorbed into the operators and $U$ is unitary on the operator label space, then

$$M_\alpha = \sum_k U_{\alpha k}L_k$$

can give the same total dissipator after summing over $\alpha$. Operators can also sometimes be shifted by multiples of the identity if the Hamiltonian is adjusted.

Therefore:

• do not treat a Lindblad operator list as unique without extra physical information;
• do not identify a jump record unless a monitoring scheme is specified;
• do not infer microscopic bath operators from $L_k$ alone;
• do check whether rates are separate or absorbed into the operators.

Different unravelings of the same unconditional equation can correspond to different observed records. See Quantum-Jump Trajectories and Diffusive Trajectories.

Self-Checks

1. A term $\kappa\mathcal D[a]\rho$ appears in a cavity master equation. What physical process does it usually describe?

Solution

It describes oscillator or cavity loss: one quantum leaves the retained mode and enters unobserved environmental degrees of freedom.

2. Does $\gamma\mathcal D[\sigma_z]\rho$ change qubit populations in the $\sigma_z$ basis?

Solution

No. It damps coherences between the $\sigma_z$ eigenstates while leaving the corresponding populations fixed. The coherence decay rate depends on the factor convention.

3. If a paper writes only $\mathcal D[\sqrt{\Gamma}\sigma_-]\rho$, where is the rate?

Solution

The rate is absorbed into the operator. Since $\mathcal D[\sqrt{\Gamma}\sigma_-]\rho=\Gamma\mathcal D[\sigma_-]\rho$, this is equivalent to writing a separate rate $\Gamma$.

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. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.
• H. J. Carmichael, Statistical Methods in Quantum Optics 1, Springer, 1999.
• G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics 48, 119, 1976.