Markov Approximation

The Markov approximation is the short-memory step in an open-system derivation. It replaces history-dependent reduced dynamics by an equation in which the future state depends, to the desired accuracy, only on the present reduced state.

Schematically,

$$\int_0^t ds\, C(t-s)\rho_S(s) \quad \longrightarrow \quad \rho_S(t) \int_0^\infty d\tau\, C(\tau),$$

when the bath correlation function $C(\tau)$ decays on a time $\tau_E$ much shorter than the time scale on which $\rho_S$ changes.

This does not mean there is no environment. It does not mean there is no dissipation. It means that the environment does not retain dynamically relevant memory over the time resolution of the reduced description.

The terminology behind “environment,” “bath,” and “reservoir” is fixed in Baths, Reservoirs, and Environments.

Relation to Born Approximation

The Born Approximation and Markov approximation are often used together, but they answer different questions.

Born asks whether the environment can be kept near a reference state:

$$\rho_{SE}^{I}(s) \approx \rho_S^{I}(s)\otimes\rho_E.$$

Markov asks whether the delayed system state can be replaced by the present one inside a memory integral:

$$\rho_S^{I}(t-\tau) \approx \rho_S^{I}(t)$$

over the range of $\tau$ where bath correlations are appreciable.

A Born equation can still be non-Markovian. A time-local Markov equation can still fail complete positivity if additional assumptions, such as secular averaging or suitable coarse graining, are not made.

Starting Point

After Born closure, a stationary bath with centered operators gives an interaction-picture equation of the form

$$\begin{aligned} \frac{d}{dt}\rho_S^{I}(t) = -\frac{\lambda^2}{\hbar^2} \sum_{\alpha,\beta} \int_0^t d\tau\, \big( & C_{\alpha\beta}(\tau) \left[ S_\alpha(t), S_\beta(t-\tau)\rho_S^{I}(t-\tau) \right] \\ & + C_{\beta\alpha}(-\tau) \left[ \rho_S^{I}(t-\tau)S_\beta(t-\tau), S_\alpha(t) \right] \big), \end{aligned}$$

where $\tau=t-s$ and

$$C_{\alpha\beta}(\tau) = \operatorname{Tr}_E \left[ B_\alpha(\tau)B_\beta(0)\rho_E \right].$$

The equation is time nonlocal: the derivative at $t$ depends on $\rho_S^{I}(t-\tau)$ over the bath memory interval.

The Markov Step

The Markov approximation has two common ingredients.

First, replace the delayed reduced state by the present reduced state:

$$\rho_S^{I}(t-\tau) \approx \rho_S^{I}(t).$$

This is justified when $\rho_S^{I}$ varies little over the support of the bath correlation functions. If $\Gamma$ is a characteristic relaxation or dephasing rate,

$$\Gamma\tau_E\ll1$$

is the standard schematic condition.

Second, if one is interested in times $t\gg\tau_E$, extend the upper limit of the memory integral:

$$\int_0^t d\tau\, C_{\alpha\beta}(\tau)(\cdots) \approx \int_0^\infty d\tau\, C_{\alpha\beta}(\tau)(\cdots).$$

The resulting equation is local in the reduced state. In interaction picture it still contains the free system oscillations through $S_\alpha(t)$ and $S_\beta(t-\tau)$.

One-Sided Bath Transforms

For weak-coupling derivations it is useful to decompose system operators into Bohr-frequency components:

$$S_\alpha(t) = \sum_\omega e^{-i\omega t} S_\alpha(\omega).$$

The Markov integral then produces one-sided bath transforms

$$\mathcal G_{\alpha\beta}(\omega) = \int_0^\infty d\tau\, e^{i\omega\tau} C_{\alpha\beta}(\tau).$$

Under suitable regularity assumptions this can be separated into dissipative and Hamiltonian parts:

$$\mathcal G_{\alpha\beta}(\omega) = \frac12\gamma_{\alpha\beta}(\omega) + i\Delta_{\alpha\beta}(\omega),$$

where the two-sided spectrum is

$$\gamma_{\alpha\beta}(\omega) = \int_{-\infty}^{\infty}d\tau\, e^{i\omega\tau} C_{\alpha\beta}(\tau).$$

The real part controls dissipative rates. The imaginary principal-value part contributes to a Lamb-shift Hamiltonian. Different books distribute factors of $2\pi$ differently, so definitions should be compared before comparing numerical rates.

Time Scales

The Markov approximation is controlled by a hierarchy, not by a magic word.

Common scales include:

• $\tau_E$: bath correlation time;
• $\Gamma^{-1}$: relaxation or dephasing time generated by the coupling;
• $\tau_{\text{obs}}$: time resolution of the experiment or model;
• $|\omega-\omega'|^{-1}$: beat time between Bohr-frequency sectors;
• $\Omega_{\text{drive}}^{-1}$: drive or rotating-frame time scale, when present;
• recurrence time of a finite environment.

A common working hierarchy is

$$\tau_E \ll \Gamma^{-1}, \qquad \tau_E \ll \tau_{\text{obs}}.$$

The Secular Approximation adds a different comparison involving Bohr-frequency splittings and dissipative rates. Markovianity alone does not perform that averaging.

Initial Transients and Slippage

The replacement

$$\int_0^t d\tau \longrightarrow \int_0^\infty d\tau$$

cannot be trusted at very early times $t\lesssim\tau_E$. The bath has not yet forgotten the initial preparation, and the exact reduced dynamics can have short-time curvature that a time-local Markov equation misses.

For this reason a Markov master equation is usually interpreted as valid after an initial transient. In refined treatments one may introduce an initial slip or restrict the equation to coarse-grained times. The need for such a correction is not a pathology; it is a reminder that a memoryless description has a finite resolution.

Markovian Does Not Always Mean GKSL

The word “Markovian” has several meanings in quantum theory.

In microscopic derivations, the Markov approximation usually means the short-memory replacement described above. In quantum information, Markovianity may refer to completely positive divisibility, absence of information backflow, or semigroup structure. These notions are related but not identical.

For a chapter-level map of those competing meanings, see Non-Markovian Dynamics.

After Born and Markov approximations, one often obtains a Redfield Equation. It may be accurate in its perturbative regime but need not generate a completely positive map for arbitrary initial states and times. The Lindblad–GKSL Equation requires additional structure. See Completely Positive Maps for the channel-level condition.

Examples

Broadband vacuum

A weakly coupled atom in free-space electromagnetic vacuum is often well described by a Markov approximation. The vacuum correlation time at optical scales is short compared with the spontaneous-emission lifetime, and the emitted field does not return coherently to the atom in ordinary free space.

Damped cavity as an effective bath

A single cavity mode is not automatically Markovian. If the cavity itself is strongly damped by a broad external continuum, then the combined effect of the damped mode can sometimes be approximated by a short-memory reservoir. Without that damping, coherent exchange with the mode creates memory.

Photonic band edge

Near a photonic band edge the density of states can vary sharply and bath correlations can have long tails. Extending the memory integral to infinity while treating the spectrum as flat may miss bound-state formation, fractional decay, or nonexponential dynamics.

Low-frequency noise

Noise with strong low-frequency weight, such as idealized $1/f$ noise over a wide range, can have long memory. It may be better modeled as quasistatic disorder, colored classical noise, or a non-Markovian quantum bath rather than as a simple Lindblad dephasing rate.

Fresh collision models

A repeated-interaction model is Markovian when each ancilla interacts once and is discarded. If the same ancilla returns, or if ancillas are correlated with one another, the reduced system can remember its history.

Common Mistakes

Equating Markovian with closed-system evolution

Closed-system evolution is unitary. Markovian open-system evolution can include relaxation, dephasing, thermalization, pumping, loss, and measurement-induced decoherence.

Equating Markovian with no bath

The bath is precisely what produces the rates in a Markov master equation. The approximation says that the bath memory is short, not that the bath is absent.

Applying the approximation before the transient is over

For $t$ comparable to $\tau_E$, the upper limit $t$ matters. Short-time exact dynamics often differs from an exponential Markov law.

Assuming complete positivity follows automatically

Born–Markov equations can be time local without being GKSL. Complete positivity usually requires a Lindblad structure, a valid coarse graining, or an exact channel construction.

Ignoring frames and drives

A Markov approximation made in one frame may not be equivalent to one made in another if a drive introduces additional time scales. Floquet or rotating-frame derivations need their own hierarchy checks.

Treating structured reservoirs as flat continua

Band gaps, sharp resonances, finite environments, critical baths, and algebraic correlation tails are warning signs. The approximation may still work in a restricted regime, but it should not be assumed.

Exercises

Exponential bath correlation

Let

$$C(\tau) = C_0 e^{-\tau/\tau_E}$$

for $\tau\ge0$. Compute

$$\int_0^\infty d\tau\, C(\tau)$$

and state the Markov condition in terms of $\tau_E$ and a system rate $\Gamma$.

Solution

The integral is

$$\int_0^\infty d\tau\, C_0e^{-\tau/\tau_E} = C_0\tau_E.$$

The Markov replacement is reliable when the reduced state changes little over the support of the correlation function:

$$\Gamma\tau_E\ll1.$$

First correction to the delayed state

Expand $\rho_S^{I}(t-\tau)$ to first order in $\tau$ and estimate why the Markov replacement fails when $\Gamma\tau_E$ is not small.

Solution

Taylor expand:

$$\rho_S^{I}(t-\tau) = \rho_S^{I}(t) - \tau\frac{d}{dt}\rho_S^{I}(t) + \cdots.$$

If $\|\dot\rho_S^{I}\|\sim\Gamma\|\rho_S^{I}\|$, the first correction is of relative size roughly $\Gamma\tau$ over the part of the integral that contributes. Averaging over a correlation function supported on $\tau_E$ gives a relative correction of order $\Gamma\tau_E$. The Markov approximation requires this to be small.

Markovian but dissipative

Give an example of a Markovian equation that is not closed-system unitary evolution.

Solution

The amplitude-damping master equation for a two-level atom,

$$\frac{d\rho}{dt} = \gamma \left( \sigma_-\rho\sigma_+ - \frac12\{\sigma_+\sigma_-,\rho\} \right),$$

is Markovian in the usual Lindblad sense and is dissipative. It relaxes population from the excited state to the ground state.

Fresh and recycled ancillas

A system interacts for a short time with one ancilla, then with a new uncorrelated ancilla in the same state, and so on. Compare this with a model in which the same ancilla returns repeatedly.

Solution

Fresh, uncorrelated ancillas can produce a memoryless reduced evolution because each interaction samples a new environment degree of freedom. If the same ancilla returns, it can carry information from earlier interactions back to the system. The reduced dynamics then has memory and is generally not Markovian.

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer, 2012.
• C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer, 2004.
• H. J. Carmichael, An Open Systems Approach to Quantum Optics, Springer, 1993.
• R. Alicki and K. Lendi, Quantum Dynamical Semigroups and Applications, 2nd ed., Springer, 2007.
• E. B. Davies, “Markovian master equations,” Communications in Mathematical Physics 39, 91–110 (1974).