Markovian Master Equations

Markovian master equations describe relaxation, dephasing, and driven dissipative motion through a generator of reduced evolution. The GKLS form makes a finite-dimensional model mathematically consistent, while its Hamiltonian and dissipative terms determine what it predicts. Establishing that those terms accurately describe an experiment is a separate physical task.

Required background. Open Quantum Systems, the Markov Approximation, and the Secular Approximation explain a common microscopic derivation. Completely Positive Maps explains the consistency condition on finite-time evolution.

A generator balances coherent motion, gain, and loss

In the convention used here,

$$\dot\rho=-\frac{i}{\hbar}[H,\rho] +\sum_k\gamma_k\mathcal D[L_k]\rho,$$ $$\mathcal D[L]\rho=L\rho L^\dagger -\frac12\{L^\dagger L,\rho\}, \qquad \gamma_k\ge0.$$

The first term rotates the state coherently. In each dissipator the anticommutator balances the trace of the gain term. Rates can instead be absorbed into $\sqrt{\gamma_k}L_k$; comparing coefficients requires checking that convention.

For time-independent coefficients in finite dimension, this equation generates channels $\Phi_t=e^{t\mathcal L}$ satisfying $\Phi_{t+s}=\Phi_t\Phi_s$. Lindblad–GKSL Equation introduces the working formula, Lindblad Operators explains its representations, and Lindblad Theorem with Quantum Dynamical Semigroups states the structural result and assumptions.

Time-dependent nonnegative-rate GKLS equations generally produce CP-divisible evolution under appropriate regularity assumptions, but not a time-homogeneous semigroup. Neither a semigroup of reduced maps nor CP divisibility alone specifies all statistics of interventions at multiple times; the chosen meaning of “Markovian” matters.

Learn the rates through a two-level system

Let $|g\rangle,|e\rangle$ have energies $0,\hbar\omega_0$ and take $\sigma_-=|g\rangle\langle e|$. With downward rate $\Gamma_\downarrow$ and upward rate $\Gamma_\uparrow$,

$$\dot p_e=\Gamma_\uparrow(1-p_e) -\Gamma_\downarrow p_e.$$

If their sum is positive, the stationary excited population is

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

The population approaches this value at the sum of the rates. The same two dissipators reduce $\rho_{eg}$ at half that sum. Independent pure dephasing adds its own coherence-decay rate but does not change these populations.

Work through Amplitude Damping Master Equation and Pure Dephasing Master Equation before combining terms. This is more reliable than inferring a rate from the letter multiplying a dissipator.

Choose the additional physics

Question	Pages
Which rate ratios describe equilibrium?	Detailed Balance and Thermal Master Equations connect bath spectra to Gibbs populations.
How do atoms, cavities, and coherent driving combine?	Quantum Optical Master Equation and Optical Bloch Equations develop damping, excitation, saturation, and coherent transients.
When is a classical population equation enough?	Pauli Rate Equations identifies decoupled population dynamics and the information lost by discarding coherences.
Which state is reached at long times?	Steady States and Relaxation separates stationarity, attraction, conserved sectors, and slow modes.

For the qubit above, equilibrium at inverse temperature $\beta$ requires $\Gamma_\uparrow/\Gamma_\downarrow=e^{-\beta\hbar\omega_0}$. A stationary state by itself does not establish detailed balance: driven systems and multiple reservoirs can sustain nonequilibrium currents.

What the generator leaves unspecified

Different operator representations can give the same generator. A Lindblad operator is therefore not automatically a unique detector event. A monitoring arrangement is needed to obtain a physical record and its conditional trajectory; see Continuous Measurement and Quantum Trajectories.

A dark state is annihilated by the relevant jump operators, but it must also remain invariant under the Hamiltonian to be stationary in the simplest construction. Even a stationary dark state need not attract every input. Symmetries, other invariant sectors, and slow modes determine reachability and convergence.

For numerical work, Solving Lindblad Equations checks integration error and Hilbert-space truncation separately. A trace-one matrix can still have negative eigenvalues, and a small steady-state residual in a truncated oscillator model does not establish cutoff convergence.

Exercises

Extract two decay times. Let $\Gamma_\downarrow=3\,\mathrm{ms}^{-1}$, $\Gamma_\uparrow=1\,\mathrm{ms}^{-1}$, and pure-dephasing rate $\gamma_\phi=2\,\mathrm{ms}^{-1}$. Find $p_e^{\rm ss}$, the population relaxation time, and the coherence-decay time.

Solution

The stationary excited population is $1/4$. The population rate is $4\,\mathrm{ms}^{-1}$, giving $T_1=0.25\,\mathrm{ms}$. The coherence rate is $(3+1)/2+2=4\,\mathrm{ms}^{-1}$, so $T_2=0.25\,\mathrm{ms}$. Without pure dephasing it would instead be $0.5\,\mathrm{ms}$.

A dark state that moves. Suppose $L=|g\rangle\langle e|$ and $H=\hbar\Omega X/2$ with $\Omega\ne0$. Is $|g\rangle\langle g|$ stationary merely because $L|g\rangle=0$?

Solution

No. Its dissipator vanishes, but its Hamiltonian commutator does not. The drive creates coherence and transfers population out of the ground state. Darkness with respect to the dissipator alone is insufficient when the Hamiltonian does not preserve the state.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Generators, thermal rates, and weak-coupling models.
• Carmichael, H. J. An Open Systems Approach to Quantum Optics. Springer, 1993. Optical master equations and trajectories.
• Rivas, Á., and S. F. Huelga. Open Quantum Systems: An Introduction. Springer, 2012. Semigroups, divisibility, and physical approximations.