Markov Approximation

The Markov approximation replaces a short-memory integral by a local-in-time equation for the slowly changing reduced state. Its accuracy depends on bath correlations, coupling, the time interval, and the observables being predicted. A local equation by itself does not establish that every multi-time measurement process is memoryless.

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 the interaction-picture reduced state changes. Free Bohr-frequency oscillations are retained in the coupling operators, not discarded by this replacement.

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.

Helpful background. The Born approximation supplies the second-order kernel; bath correlation functions explain its time dependence.

Born closure and short bath memory

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

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

Here the coupling parameter $\lambda$ is dimensionless, the bath operators are centered and Hermitian, and $[H_E,\rho_E]=0$. 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 the interaction picture it still contains free system oscillations through $S_\alpha(t)$ and $S_\beta(t-\tau)$. Replacing the delayed state alone can also arise as a second-order time-convolutionless reorganization at fixed time, without extending the integral or assuming short memory. The physical Markov approximation needs the time-scale argument in addition to that algebraic step.

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

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

For several coupling operators, these are Hermitian and anti-Hermitian matrix parts, not entrywise real and imaginary parts:

$$\gamma_{\alpha\beta}=\mathcal G_{\alpha\beta}+\mathcal G_{\beta\alpha}^*, \qquad \Delta_{\alpha\beta}=\frac{\mathcal G_{\alpha\beta}-\mathcal G_{\beta\alpha}^*}{2i}.$$

The two-sided spectral matrix $\gamma(\omega)$ is positive semidefinite for stationary Hermitian bath operators under the usual integrability assumptions. Actual dissipative coefficients carry $\lambda^2/\hbar^2$; the corresponding secular Lamb-shift Hamiltonian carries $\lambda^2/\hbar$ multiplying $\sum_{\omega,\alpha,\beta}\Delta_{\alpha\beta}(\omega)S_\alpha^\dagger(\omega)S_\beta(\omega)$.

For one correlation $C(\tau)=C_0e^{-\tau/\tau_E}$ at $\tau\ge0$, with real $C_0>0$,

$$\mathcal G(\omega)=\frac{C_0\tau_E}{1-i\omega\tau_E},\quad \gamma(\omega)=\frac{2C_0\tau_E}{1+\omega^2\tau_E^2},\quad \Delta(\omega)=\frac{C_0\omega\tau_E^2}{1+\omega^2\tau_E^2}.$$

The frequency dependence survives even though the equation is local in the system state.

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.

Worked example: the initial transient in colored dephasing

Let a qubit experience a centered stationary classical Gaussian frequency fluctuation $\xi(t)$ through $H(t)=\hbar\xi(t)Z/2$, with

$$\mathbb E[\xi(t)\xi(0)]=\sigma^2e^{-|t|/\tau_E}.$$

The exact coherence factor is the Gaussian phase average. Integrating its variance gives

$$W(t)=\exp\{-\Gamma[t-\tau_E(1-e^{-t/\tau_E})]\}, \qquad \Gamma=\sigma^2\tau_E.$$

At short times, $W(t)=1-\sigma^2t^2/2+O(t^3)$, whereas the Markov result $W_M(t)=e^{-\Gamma t}$ starts with a nonzero linear slope. At late times,

$$\frac{W(t)}{W_M(t)}\longrightarrow e^{\Gamma\tau_E}.$$

Thus $\Gamma\tau_E=0.02$ limits the asymptotic relative envelope mismatch to about $2.02\%$; for $\Gamma\tau_E=0.5$ it is about $64.9\%$. Taking $t\gg\tau_E$ does not alone make the accumulated transient error small.

This Gaussian example has an exact time-local equation with nonnegative rate $\Gamma(1-e^{-t/\tau_E})$, despite the noise’s finite correlation time. It illustrates why time locality or CP divisibility is not interchangeable with absence of all physical noise memory. An initial-slip correction can improve a coarse description in some models, but must be derived consistently rather than used to hide nonpositive predictions.

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 gives a sequence of memoryless CPTP steps when the ancillas begin mutually independent and uncorrelated with the system, each interacts once, and there are no intervening ancilla–ancilla couplings or feedback. Identical steps give a discrete semigroup; a continuous GKSL limit requires a suitable scaling. If the same ancilla returns, or if ancillas are correlated with one another, the reduced system can remember its history.

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

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