Common Lindblad Operators

A Lindblad operator identifies which states a dissipative term connects or distinguishes. Its normalization determines the observable decay rate. This reference translates common operators into population and coherence dynamics; a detector interpretation additionally requires a monitoring scheme.

Required background. The Lindblad–GKLS Equation defines the generator, and Lindblad Operators explains its nonunique representations.

Dissipator normalization and units

Use

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

Rates may be separate, $\gamma\mathcal D[L]$, or absorbed, $\mathcal D[\sqrt\gamma L]$. When $L$ is dimensionless, $\gamma$ has units of inverse time; the absorbed operator has units of inverse square root of time. If $L$ itself is dimensional, the coefficient must compensate. For example, a coefficient of $\mathcal D[x]$ has units $1/(\text{length}^2\,\text{time})$.

The formulas below give only the dissipative contribution. A Hamiltonian can add rotations, change populations in the displayed basis, or move states out of a protected subspace.

Qubit relaxation and dephasing

Let $\sigma_-=|g\rangle\langle e|$, $\sigma_+=|e\rangle\langle g|$, and $Z=|g\rangle\langle g|-|e\rangle\langle e|$.

Term	Population change	Coherence decay
$\Gamma_\downarrow\mathcal D[\sigma_-]$	$\dot p_e=-\Gamma_\downarrow p_e$	$\dot\rho_{eg}=-\Gamma_\downarrow\rho_{eg}/2$
$\Gamma_\uparrow\mathcal D[\sigma_+]$	$\dot p_e=\Gamma_\uparrow(1-p_e)$	$\dot\rho_{eg}=-\Gamma_\uparrow\rho_{eg}/2$
$(\Gamma_\phi/2)\mathcal D[Z]$	Populations fixed	$\dot\rho_{eg}=-\Gamma_\phi\rho_{eg}$

With both transitions and pure dephasing,

$$\begin{gathered} \frac1{T_1}=\Gamma_\downarrow+\Gamma_\uparrow,\\ \frac1{T_2}=\frac1{2T_1}+\Gamma_\phi. \end{gathered}$$

These are the constant nonnegative-rate, two-level Markov formulas. They should not be applied indiscriminately to nonexponential decay, ensemble inhomogeneity, or echo times. An equilibrium bath at inverse temperature $\beta$ gives $\Gamma_\uparrow/\Gamma_\downarrow=e^{-\beta\hbar\omega_0}$ for gap $\hbar\omega_0$. Arbitrary pumping rates need not represent such a bath. See Amplitude Damping Master Equation and Thermal Master Equations.

For $P_e=|e\rangle\langle e|$ there is the exact two-level identity

$$2\Gamma_\phi\mathcal D[P_e] =\frac{\Gamma_\phi}{2}\mathcal D[Z].$$

It explains a common factor-of-four difference between projector and Pauli operator coefficients. The coherence-decay rate is the same. See Pure Dephasing Master Equation.

Oscillator loss, gain, and number monitoring

For a bosonic mode, $a|n\rangle=\sqrt n\,|n-1\rangle$ and $N=a^\dagger a$.

Term	Mean occupation	Mean field
$\kappa\mathcal D[a]$	$d\langle N\rangle/dt=-\kappa\langle N\rangle$	$d\langle a\rangle/dt=-\kappa\langle a\rangle/2$
$g\mathcal D[a^\dagger]$	$d\langle N\rangle/dt=g(\langle N\rangle+1)$	$d\langle a\rangle/dt=g\langle a\rangle/2$

The $+1$ in the gain law is spontaneous excitation. The ideal linear amplifier has no saturation and its occupation grows without bound at long times. A thermal reservoir combines loss and gain with coefficients $\kappa(\bar n+1)$ and $\kappa\bar n$; then $d\langle N\rangle/dt=-\kappa(\langle N\rangle-\bar n)$.

Number dephasing leaves all Fock populations fixed:

$$\bigl(\gamma_N\mathcal D[N]\rho\bigr)_{mn} =-\frac{\gamma_N}{2}(m-n)^2\rho_{mn}.$$

Large number separations lose coherence faster. This term describes phase diffusion or unread number monitoring, not photon loss. The identities assume the required moments exist; a finite Fock cutoff changes oscillator commutators at its upper boundary. Check occupation near that boundary as described in Solving Lindblad Equations.

Projector, Pauli, and collective operators

For any orthogonal projector $P$, $\gamma_P\mathcal D[P]$ damps coherences between the $P$ and $I-P$ sectors at rate $\gamma_P/2$, while preserving each diagonal block. Such a generator can represent continuous unread monitoring. One instantaneous nonselective projective measurement is the channel $\rho\mapsto P\rho P+(I-P)\rho(I-P)$, rather than this finite-rate generator. A POVM effect alone specifies neither process; see Quantum Instruments.

Independent Pauli terms $\sum_{j=x,y,z}\gamma_j\mathcal D[\sigma_j]$ give

$$\dot r_x=-2(\gamma_y+\gamma_z)r_x,$$

with cyclic versions for $r_y,r_z$. Equal coefficients $\gamma_j=\gamma$ therefore shrink the Bloch vector at rate $4\gamma$, not $\gamma$. See Pauli Channels for the corresponding finite-time map.

Collective emission uses $J_-=\sum_j\sigma_-^{(j)}$. Its dissipator contains cross terms between emitters and differs from $\sum_j\mathcal D[\sigma_-^{(j)}]$. For two identical qubits, the antisymmetric single-excitation state is dark under $J_-$ but decays under independent loss. Remaining dark during the full evolution also requires that the Hamiltonian preserve the dark sector; see Decoherence-Free Subspaces.

Equivalent operators and different records

For rate-absorbed operators, a unitary mixing $M_\alpha=\sum_k U_{\alpha k}L_k$ leaves $\sum_\alpha\mathcal D[M_\alpha]=\sum_k\mathcal D[L_k]$ exactly unchanged. Identity shifts can also be compensated by a Hamiltonian change. Thus an operator list does not uniquely identify microscopic couplings or individual clicks. Unravelings explains how different monitoring choices can produce the same averaged equation.

Exercises

Read a coefficient. A model contains $3\,\mathrm{ms}^{-1}\mathcal D[Z]$. What is the pure coherence-decay time?

Solution

The off-diagonal entry of $Z\rho Z-\rho$ is $-2\rho_{eg}$. Its decay rate is $6\,\mathrm{ms}^{-1}$, so $T_\phi=1/6\,\mathrm{ms}$. The dissipator coefficient is not itself the coherence-decay rate.

Number separation. Compare the decay rates of $\rho_{01}$ and $\rho_{03}$ under $\gamma_N\mathcal D[N]$.

Solution

They decay at $\gamma_N/2$ and $9\gamma_N/2$, respectively. Their populations remain unchanged. A single quoted dephasing time cannot describe all number coherences without specifying the pair.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Qubit and oscillator generators.
• Gardiner, C. W., and P. Zoller. Quantum Noise. 3rd edition, Springer, 2004. Loss, amplification, and phase diffusion.
• Lindblad, G. “On the generators of quantum dynamical semigroups.” Communications in Mathematical Physics 48, 119–130 (1976). doi:10.1007/BF01608499.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Monitoring and alternative unravelings.