Approximation Checklist

An open-system approximation is useful when it predicts the chosen observables over the chosen time interval. Born, Markov, and secular approximations answer different questions about coupling, memory, and frequency resolution. Checking them separately helps explain why an equation can be completely positive yet inaccurate, or accurate to a stated perturbative order without being completely positive on every input.

Required background. Reduced Dynamics distinguishes exact reduced maps from their approximations; The Lindblad–GKSL Equation supplies the structure of completely positive Markov evolution.

System, preparation, and observable

Start with the degrees of freedom you retain and the prediction you need: a population, a coherence, a spectrum, a stationary state, or a conditioned measurement record. A small error in a dominant population can coexist with a large relative error in a weak coherence.

For a dimensionless coupling parameter $\lambda$, write

$$\begin{gathered} H=H_S\otimes I_B+I_S\otimes H_B+H_I,\\ H_I=\lambda\sum_\alpha A_\alpha\otimes B_\alpha. \end{gathered}$$

If $\langle B_\alpha\rangle_B\ne0$, subtract this mean from $B_\alpha$ and add $\lambda\sum_\alpha A_\alpha\langle B_\alpha\rangle_B$ to $H_S$. Otherwise a first-order Hamiltonian contribution may be missed. A narrow, strongly coupled mode may be better retained in the system than treated perturbatively as part of the bath. See System–Bath Hamiltonians and Reaction-Coordinate Mapping.

A product preparation $\rho_S(0)\otimes\rho_B$ with the same $\rho_B$ for every input gives an exact channel from the initial time. Equilibrating the interacting system and bath generally prepares correlations instead. In that case, specify how inputs are prepared before asking for a reduced map on arbitrary states. See Initial Correlations.

Also distinguish the averaged state from the state conditioned on a detector record. Averaging a consistent trajectory model recovers the unconditional equation for that experiment; one trajectory is not a prediction for the whole ensemble.

Bath correlations and frequency conventions

For Hermitian coupling operators and a stationary bath, use

$$\begin{gathered} C_{\alpha\beta}(t)=\operatorname{Tr}_B [B_\alpha(t)B_\beta(0)\rho_B],\\ S_{\alpha\beta}(\omega)=\int_{-\infty}^{\infty} e^{i\omega t}C_{\alpha\beta}(t)\,dt. \end{gathered}$$

If $A_\alpha$ is dimensionless and $B_\alpha$ has energy units, the spectral rate matrix is $\gamma_{\alpha\beta}(\omega)=\lambda^2S_{\alpha\beta}(\omega)/\hbar^2$. Confusing $S$ with a rate matrix loses dimensions and changes numerical rates.

Estimate memory from the correlations that actually enter the reduced dynamics. Long tails, recurrences, and narrow resonances may prevent a useful single correlation time. For equilibrium scalar noise, the convention above gives $S(-\omega)=e^{-\beta\hbar\omega}S(+\omega)$. Matrix cross-spectra require the corresponding exchange of operator indices. See Spectral Densities and Common Noise Spectra.

Born, Markov, and secular approximations

Born approximation. Treat bath disturbance and higher-order correlations perturbatively. Check the coupling, retained order, and elapsed time; weak coupling alone does not guarantee accuracy indefinitely.

Markov approximation. Replace the past reduced state inside the memory integral by its current value. Compare bath memory with the slow reduced evolution in the relevant interaction picture.

Secular approximation. Average cross terms between distinct transition frequencies. Their beat periods must be short compared with the averaging window, which must itself remain short compared with relaxation.

Born factorization inside a perturbative equation does not assert that the exact joint state remains uncorrelated. Likewise, a time-local equation does not by itself establish short memory: time-convolutionless equations can describe non-Markovian evolution.

For a stationary weak-coupling problem, a common useful averaging window is

$$\tau_B\ll\Delta t\ll\tau_R.$$

Dropping a pair of frequency cross terms additionally requires $|\omega-\omega'|\Delta t\gg1$. A short bath memory can therefore justify Markov reasoning while full secularization still fails near a doublet. Fast free-system oscillations are usually handled explicitly through energy eigenoperators, rather than requiring $\tau_B$ shorter than every inverse Bohr frequency.

With $H_S=\sum_\epsilon\epsilon\Pi(\epsilon)$, define

$$\begin{gathered} A_\alpha(\omega)= \sum_{\epsilon'-\epsilon=\hbar\omega} \Pi(\epsilon)A_\alpha\Pi(\epsilon'),\\ [H_S,A_\alpha(\omega)]=-\hbar\omega A_\alpha(\omega). \end{gathered}$$

Keep coherent sums within an exactly degenerate frequency block. Treating each matrix element as an independent jump can remove interference and dark states even when all resulting rates are positive. Strong driving changes the appropriate frequency decomposition; transform the Hamiltonian and dissipators consistently.

Structural checks and physical accuracy

For $\dot\rho=\mathcal L\rho$, check trace preservation $\operatorname{Tr}\mathcal L(X)=0$ and Hermiticity preservation $\mathcal L(X^\dagger)=\mathcal L(X)^\dagger$. For a finite-time map, the Choi matrix tests complete positivity and its marginal tests trace preservation. Testing a few isolated input states cannot establish complete positivity.

A time-independent finite-dimensional GKLS generator produces a CPTP semigroup. A regular time-dependent generator with nonnegative canonical rates produces CP-divisible evolution. A negative coefficient in a redundant, noncanonical operator list does not alone establish loss of CP divisibility; inspect the canonical dissipative matrix or the intermediate maps.

The Redfield equation can violate positivity even immediately for some boundary states. Such a violation is a real limitation, but its size must be compared with neglected perturbative terms and the observable error; it does not automatically locate an algebra mistake. Clipping eigenvalues or adding dephasing changes the model and needs its own accuracy check.

Conversely, a completely positive approximation can omit a relevant near-degenerate coherence or use the wrong reservoir spectrum. Physical structure and approximation accuracy are separate tests.

Stationary states, limiting cases, and numerics

Solve $\mathcal L(\rho_{\rm ss})=0$, then distinguish existence, uniqueness, and attraction. A thermal weak-coupling generator may have the Gibbs state as a stationary state without driving every input to it. Pure dephasing, disconnected transition sectors, and dark states are simple counterexamples. See Steady States and Relaxation and Detailed Balance.

Useful comparisons include zero coupling, a solvable qubit, a passive bath at zero temperature, and a larger explicit environment. Exact survival is quadratic at sufficiently short times when the initial state’s energy variance is finite; a coarse-grained exponential need not reproduce that microscopic interval. Do not extrapolate an approximation beyond its intended time range solely because the numerical solver remains stable.

For numerical work, vary timestep and oscillator cutoff separately. Inspect trace, Hermiticity, positivity, and the observable of interest. Check population near the cutoff and compare stochastic averages with sampling uncertainty. Numerical convergence solves the chosen model accurately; it does not establish that the model approximates the experiment. See Solving Lindblad Equations.

Exercises

No secular window. Let $\tau_B=10^{-7}\,\mathrm s$, $\tau_R=10^{-4}\,\mathrm s$, and $|\omega-\omega'|=10^3\,\mathrm{s}^{-1}$. Can both short-memory coarse graining and full secular averaging of this pair be justified?

Solution

There is a window between bath memory and relaxation. But averaging the pair requires $\Delta t\gg10^{-3}\,\mathrm s$, incompatible with $\Delta t\ll10^{-4}\,\mathrm s$. The pair should not be discarded merely because its frequencies are unequal.

A negative damping rate. For $\dot\rho=\gamma(t)\mathcal D[|g\rangle\langle e|]\rho$ with no Hamiltonian, find the condition for the initial-to-time map to remain CPTP.

Solution

The excited-state survival factor is $s(t)=\exp[-\int_0^t\gamma(u)du]$, and coherence has multiplier $\sqrt{s(t)}$. The amplitude-damping channel is CPTP exactly when $0\le s(t)\le1$, hence $\int_0^t\gamma(u)du\ge0$ at every time considered. A negative rate can temporarily increase $s$ while it stays below one. For regular finite rates, the map is invertible and a negative rate prevents CP divisibility over that interval, without necessarily invalidating the initial-time map.

Stationary is not attracting. A qubit undergoes pure energy-basis dephasing in contact with an equilibrium noise source. Must it relax to Gibbs populations?

Solution

No. Every diagonal qubit state is stationary under pure dephasing with a diagonal Hamiltonian. A Gibbs state is among them, but no population-transfer process selects it from other diagonal states.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Weak-coupling approximations and their limitations.
• Davies, E. B. “Markovian master equations.” Communications in Mathematical Physics 39, 91–110 (1974). doi:10.1007/BF01608389. The controlled weak-coupling limit.
• Rivas, Á., and S. F. Huelga. Open Quantum Systems: An Introduction. Springer, 2012. Dynamical maps and Markovian generators.
• Wiseman, H. M., and G. J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2010. Conditional and averaged descriptions.