Lindblad Operators

A Lindblad operator is an operator that labels one dissipative, noisy, or monitored channel in a Markovian master equation. In the convention

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

the dissipator is

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

The canonical page for the full generator is Lindblad–GKSL Equation, and the structural statement is the Lindblad Theorem. This page explains how to read the operators $L_k$: what they say physically, what they do not say, and why the same master equation can have more than one operator representation.

First Interpretation

A Lindblad operator answers the question:

Which system operator is being acted on by the ignored degrees of freedom?

Different choices of $L$ describe different environmental actions.

Operator	Common meaning
$\sigma_-=\lvert g\rangle\langle e\rvert$	decay from excited to ground state
$\sigma_+=\lvert e\rangle\langle g\rvert$	incoherent excitation or pumping
$\sigma_z$	qubit pure dephasing in the $z$ basis
$a$	oscillator photon loss or damping
$a^\dagger$	oscillator thermal excitation or gain
$n=a^\dagger a$	number dephasing
$\sigma_x,\sigma_y,\sigma_z$	isotropic Pauli noise or depolarization
$J_-=\sum_j\sigma_-^{(j)}$	collective decay of several emitters

The rate may be written separately as $\gamma_k$ or absorbed into the operator. Thus

$$\gamma\,\mathcal D[L]\rho = \mathcal D[\sqrt{\gamma}\,L]\rho.$$

When the rate is separated, $L$ is often dimensionless. When the rate is absorbed, $L$ has units of $\mathrm{time}^{-1/2}$.

What the Operator Does

The term $L\rho L^\dagger$ transfers weight through the action of $L$. The anticommutator term removes the corresponding weight needed to preserve trace. Both pieces are required.

For a pure input $\rho=\lvert\psi\rangle\langle\psi\rvert$, the positive part gives

$$L\rho L^\dagger = L\lvert\psi\rangle\langle\psi\rvert L^\dagger.$$

If $L\lvert\psi\rangle$ points toward a different state, the dissipator can move population. If $L$ is diagonal in a basis, the dissipator tends to suppress coherences between different eigenvalues without moving populations.

The operator $L$ therefore identifies a preferred structure: a decay direction, a measured observable, a noise axis, a loss mode, or a collective channel.

Decay Operators

For a two-level atom or qubit with excited state $\lvert e\rangle$ and ground state $\lvert g\rangle$,

$$L=\sigma_-=\lvert g\rangle\langle e\rvert$$

gives amplitude damping:

$$\frac{d\rho}{dt} = \Gamma\,\mathcal D[\sigma_-]\rho.$$

The populations obey

$$\dot\rho_{ee} = -\Gamma\rho_{ee}, \qquad \dot\rho_{gg} = \Gamma\rho_{ee}.$$

The coherence decays at half the population-decay rate, apart from Hamiltonian phase rotation:

$$\dot\rho_{eg} = -\frac{\Gamma}{2}\rho_{eg}.$$

This is the generator version of the Amplitude-Damping Channel.

For rate conventions, finite-time solutions, and thermal variants, see Amplitude Damping Master Equation.

For a harmonic oscillator, $L=a$ describes loss of one quantum. At finite temperature, both loss and gain appear:

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

The $a$ term removes quanta; the $a^\dagger$ term adds quanta from a thermally occupied bath.

Dephasing Operators

A Hermitian Lindblad operator often represents phase noise or continuous unread monitoring of an observable. For qubit pure dephasing, one common convention is

$$L=\sigma_z, \qquad \frac{d\rho}{dt} = \frac{\Gamma_\phi}{2}\mathcal D[\sigma_z]\rho.$$

In the $\sigma_z$ basis,

$$\rho = \begin{pmatrix} \rho_{00}&\rho_{01}\\ \rho_{10}&\rho_{11} \end{pmatrix},$$

the populations are unchanged and

$$\dot\rho_{01} = -\Gamma_\phi\rho_{01}.$$

More generally, suppose $L=L^\dagger$ and

$$L\lvert m\rangle = \ell_m\lvert m\rangle.$$

Then

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

Coherences between states with different eigenvalues decay; coherences inside degenerate eigenspaces are protected from that dissipator. This is the operator-level reason dephasing selects a preferred basis.

For the corresponding generator as a named model, see Pure Dephasing Master Equation.

Microscopic Origin

In weak-coupling derivations, Lindblad operators are often system coupling operators resolved into Bohr-frequency components. Start with

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

In the energy basis of $H_S$, each $A_\alpha$ is decomposed as

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

After the Markov and secular approximation, the dissipator has the schematic form

$$\sum_{\omega,\alpha,\beta} \Gamma_{\alpha\beta}(\omega) \left( A_\beta(\omega)\rho A_\alpha^\dagger(\omega) - \frac12 \{A_\alpha^\dagger(\omega)A_\beta(\omega),\rho\} \right).$$

The positive rate matrix $\Gamma_{\alpha\beta}(\omega)$ can be diagonalized. The resulting eigenvectors define Lindblad operators as linear combinations of the $A_\alpha(\omega)$ within a frequency block.

Thus a Lindblad operator is rarely arbitrary in a microscopic model. It reflects the system operator coupled to the bath, the bath spectrum at a transition frequency, and the approximations used to remove memory.

Jump Interpretation

When an environment is monitored in a photon-counting-like way, an operator

$$J_k=\sqrt{\gamma_k}\,L_k$$

can be interpreted as a jump operator. During a short interval $dt$, the probability of a jump of type $k$ is

$$p_k = dt\, \operatorname{Tr} \left( J_k^\dagger J_k\rho \right),$$

and the conditioned state after observing that jump is

$$\rho \longmapsto \frac{ J_k\rho J_k^\dagger }{ \operatorname{Tr}(J_k^\dagger J_k\rho) }.$$

If no jump is observed, the conditional state evolves with an effective non-Hermitian Hamiltonian

$$H_{\mathrm{eff}} = H - \frac{i\hbar}{2} \sum_k J_k^\dagger J_k,$$

followed by normalization. Averaging the conditioned jump and no-jump updates recovers the unconditional Lindblad equation.

The warning is important: a Lindblad operator in an equation is not automatically an actually observed jump. The jump interpretation depends on the monitoring scheme. A different measurement of the same environment can give a diffusive trajectory rather than jumps, while the unconditional master equation remains the same.

For the operational distinction between conditioned and unconditioned updates, see Quantum Instruments.

Nonuniqueness

Lindblad operators are not unique labels of physical mechanisms.

First, rates can be absorbed:

$$\gamma\,\mathcal D[L] = \mathcal D[\sqrt{\gamma}\,L].$$

Second, operators inside a set can be unitary-mixed. If

$$\widetilde L_a = \sum_k u_{ak}L_k$$

with $u$ unitary, then

$$\sum_a\mathcal D[\widetilde L_a] = \sum_k\mathcal D[L_k].$$

Third, shifting an operator by a scalar multiple of the identity can be compensated by changing the Hamiltonian. For one channel,

$$L\mapsto L+\alpha I$$

changes the dissipator by a commutator term, which can be absorbed into $H$. This is sometimes called a gauge freedom of the Lindblad representation.

The invariant object is the full generator, not a particular list of $L_k$ written on the page.

Dark States and Pointer States

A state $\lvert\psi\rangle$ is dark with respect to a jump operator $J$ if

$$J\lvert\psi\rangle=0.$$

For amplitude damping, $\sigma_-\lvert g\rangle=0$, so the ground state is dark with respect to emission. If the Hamiltonian and all dissipators leave a dark subspace invariant, it can become a steady or Decoherence-Free Subspace. Designing dissipators so that useful states are dark and attracting is the subject of Reservoir Engineering.

For Hermitian dephasing operators, eigenstates are not dark in the same jump sense, but they are stable against coherence loss from that dissipator. Superpositions of different eigenvalues lose phase coherence; mixtures within the eigenbasis are unchanged.

This connects Lindblad operators to pointer states and to the distinction between dephasing and dissipation.

How to Read a Proposed Operator

When someone writes a Lindblad term, ask:

• Is the rate separated from the operator or absorbed into it?
• What physical bath, detector, or noise source justifies this operator?
• Is the operator diagonal, lowering, raising, collective, local, or nonlocal?
• Does it conserve energy, exchange energy, or only dephase?
• Does the same bath require several operators, such as both $\sigma_-$ and $\sigma_+$?
• Is the representation unique enough to support the claimed physical interpretation?
• Is a jump trajectory actually monitored, or is the equation unconditional?
• Does the steady state match the intended environment?

These questions prevent a common mistake: reading too much microscopic detail into a convenient generator.

Common Mistakes

Treating Lindblad operators as unique observables

The same generator can be written with different operator sets. Individual operators are not unique physical observables unless an experimental monitoring scheme or microscopic derivation fixes the representation.

Calling every dissipator energy dissipation

The notation $\mathcal D[L]$ does not imply energy relaxation. A Hermitian $L$ commuting with $H$ can describe pure dephasing without population transfer.

Forgetting the anticommutator

The term $L\rho L^\dagger$ alone is not trace preserving. The anticommutator supplies the normalization back-action required for a legitimate generator.

Assuming jumps exist without a record

Jumps are conditioned descriptions associated with a particular unraveling. If the environment is not monitored, the master equation describes the averaged state.

Ignoring collective structure

Replacing a collective operator such as $J_-=\sum_j\sigma_-^{(j)}$ by independent local operators $\sigma_-^{(j)}$ changes the physics. Collective and independent dissipation have different dark states, rates, and correlations.

Exercises

Dephasing in an eigenbasis

Let $L=L^\dagger$ and $L\lvert m\rangle=\ell_m\lvert m\rangle$. Show that

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

Solution

The first term gives

$$(L\rho L)_{mn} = \ell_m\ell_n\rho_{mn}.$$

The anticommutator gives

$$\frac12(L^2\rho+\rho L^2)_{mn} = \frac12(\ell_m^2+\ell_n^2)\rho_{mn}.$$

Therefore

$$\left(\mathcal D[L]\rho\right)_{mn} = \left[ \ell_m\ell_n - \frac12(\ell_m^2+\ell_n^2) \right]\rho_{mn} = - \frac12(\ell_m-\ell_n)^2\rho_{mn}.$$

Amplitude damping populations

For $L=\sigma_-=\lvert g\rangle\langle e\rvert$, compute $\dot\rho_{ee}$ and $\dot\rho_{gg}$ in

$$\dot\rho=\Gamma\mathcal D[\sigma_-]\rho.$$

Solution

Using $\sigma_+\sigma_-=\lvert e\rangle\langle e\rvert$ and $\sigma_-\rho\sigma_+=\rho_{ee}\lvert g\rangle\langle g\rvert$,

$$\dot\rho_{ee} = \Gamma \langle e|\mathcal D[\sigma_-]\rho|e\rangle = -\Gamma\rho_{ee},$$

and

$$\dot\rho_{gg} = \Gamma \langle g|\mathcal D[\sigma_-]\rho|g\rangle = \Gamma\rho_{ee}.$$

Population flows from $\lvert e\rangle$ to $\lvert g\rangle$.

Jump probability

Let $J=\sqrt{\Gamma}\sigma_-$. What is the probability of an emission jump in a short interval $dt$ for a state $\rho$?

Solution

The jump probability is

$$p = dt\,\operatorname{Tr}(J^\dagger J\rho) = \Gamma dt\, \operatorname{Tr}(\sigma_+\sigma_-\rho).$$

Since $\sigma_+\sigma_-=\lvert e\rangle\langle e\rvert$,

$$p = \Gamma dt\,\rho_{ee}.$$

Unitary mixing

Suppose $\widetilde L_a=\sum_k u_{ak}L_k$ with $u$ unitary. Show that $\sum_a\mathcal D[\widetilde L_a]\rho=\sum_k\mathcal D[L_k]\rho$.

Solution

For the positive term,

$$\sum_a \widetilde L_a\rho\widetilde L_a^\dagger = \sum_{a,k,l} u_{ak}u_{al}^* L_k\rho L_l^\dagger.$$

Unitarity gives $\sum_a u_{ak}u_{al}^*=\delta_{kl}$, so this becomes

$$\sum_k L_k\rho L_k^\dagger.$$

Similarly,

$$\sum_a \widetilde L_a^\dagger\widetilde L_a = \sum_k L_k^\dagger L_k.$$

The anticommutator parts are therefore also equal.

References

• V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,” Journal of Mathematical Physics 17, 821-825 (1976).
• G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications in Mathematical Physics 48, 119-130 (1976).
• E. B. Davies, Quantum Theory of Open Systems, Academic Press (1976).
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer (1993).
• 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).