Non-Markovian Dynamics

Non-Markovian dynamics concerns memory in a reduced quantum description, but different tests ask different questions. A changing decay rate, a failure of CP divisibility, a return of state distinguishability, and memory revealed by interventions are not interchangeable observations. This chapter teaches how to name the property being tested and choose a model that can predict it.

Required background. Reduced Dynamics gives the system–environment starting point; the Markov Approximation explains the short-memory limit; Quantum Dynamical Semigroups supplies the time-homogeneous comparison.

Failure of an exponential law is not a complete memory test

A qubit dephasing map multiplies $\rho_{01}$ by $\eta(t)$ and leaves populations fixed. If $|\eta(t)|\le1$, each map from the initial time is a channel. Different choices illustrate three distinct possibilities:

Coherence factor	Property
$e^{-\Gamma t}$, $\Gamma\ge0$	A time-homogeneous dephasing semigroup
$e^{-(t/\tau)^2}$, $\tau>0$	Not a semigroup, but CP divisible for $t\ge0$
$1-a\sin^2(\omega t)$, $0<a<1$	Valid initial-time channels, with intervals of increasing coherence and failed CP divisibility

The second row is already enough to disprove the claim that every nonexponential decay fails CP divisibility. For the third row, the intermediate coherence multiplier is $\eta(t)/\eta(s)$, since $\eta$ never vanishes. Whenever the ratio exceeds one, the intermediate dephasing map cannot be a channel on arbitrary inputs.

Read What Non-Markovian Means before using a numerical measure. In particular, a family of reduced maps does not supply all the response statistics of an experiment with interventions at several times.

Divisibility and distinguishability test different information

CP Divisibility asks whether channels $V_{t,s}$ can be chosen so that $\Phi_t=V_{t,s}\Phi_s$ for every $t\ge s$. Invertible maps give a unique candidate $V_{t,s}=\Phi_t\Phi_s^{-1}$; noninvertible maps need a more careful existence test.

Information Backflow instead follows the trace distance

$$D(\rho_1,\rho_2)=\tfrac12\|\rho_1-\rho_2\|_1.$$

Channels cannot increase this distinguishability. Its increase between two evolved preparations therefore rules out a channel propagating both states over that interval. For the dephasing example and initial $|+\rangle,|-\rangle$, the distance is $|\eta(t)|$.

The absence of a revival for one chosen pair proves little. Even optimizing ordinary system-state pairs does not make every diagnostic equivalent to CP divisibility in general. Non-Markovianity Measures explains the optimization, ancilla, and preparation assumptions behind different measures.

A time-local equation can contain the same revivals

For real positive $\eta(t)$, the same dephasing family obeys

$$\begin{gathered} \dot\rho=\frac{\gamma(t)}2(Z\rho Z-\rho),\\ \gamma(t)=-\frac{d}{dt}\ln\eta(t). \end{gathered}$$

The rate is negative when coherence grows. This does not invalidate the initial-time channel: the integrated evolution still has $0<\eta(t)\le1$. It instead changes divisibility. In general, signs of coefficients are meaningful for this purpose only after identifying the canonical generator representation and its regularity assumptions.

For general models, compare Memory Kernels with Time-Convolutionless Master Equations. These are different ways to represent reduced evolution, not definitions of memory by themselves.

Choose a constructive model

Physical structure	Method and limitation
A few sharp environmental resonances	Pseudomode Methods represents suitable correlations through auxiliary damped modes. The mapping depends on the spectrum and preparation.
A strongly coupled collective bath coordinate	Reaction-Coordinate Mapping enlarges the system; the residual bath still needs a justified treatment.
Correlations represented by convergent exponential expansions	Hierarchical Equations of Motion introduces auxiliary operators, with hierarchy and correlation-expansion convergence checks.
Repeated environmental interactions	Collision Models exposes how ancilla reuse or correlations carry memory.
Interaction energy and dressing are substantial	Strong Coupling revisits the system boundary, preparation, and perturbative assumptions.

Finish with Pitfalls in Non-Markovian Modeling. Revivals can be numerical recurrence artifacts, an approximate kernel can violate positivity, and adding parameters can improve a fit without explaining a mechanism. Compare predictions across observables and refine the numerical representation before interpreting a memory measure.

Exercises

Gaussian decay. Show that $\eta(t)=e^{-(t/\tau)^2}$ is CP divisible but fails the semigroup law for positive $s,t$.

Solution

For $t\ge s\ge0$, $\eta(t)/\eta(s)=e^{-(t^2-s^2)/\tau^2}$ lies between zero and one, so it defines an intermediate dephasing channel. But $\eta(t+s)=\eta(t)\eta(s)e^{-2ts/\tau^2}$ differs from the product. Equivalently, its time-local rate is $2t/\tau^2\ge0$, which is time dependent.

A revival without an unphysical initial-time map. Let $a=3/4$ in the oscillatory example. Compare $\eta$ at $s=\pi/(2\omega)$ and $t=\pi/\omega$.

Solution

$\eta(s)=1/4$ and $\eta(t)=1$. Each initial-time map is completely positive, but their unique intermediate dephasing multiplier is four. It would take some valid Bloch vectors outside the ball, so it is not a channel on arbitrary system states. For the initial $X$ eigenstates, distinguishability returns from $1/4$ to one.

References

• Breuer, H.-P., and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, 2002. Reduced maps and memory equations.
• de Vega, I., and D. Alonso. “Dynamics of non-Markovian open quantum systems.” Reviews of Modern Physics 89, 015001 (2017). doi:10.1103/RevModPhys.89.015001. Comparison of constructive methods.
• Rivas, Á., S. F. Huelga, and M. B. Plenio. “Quantum non-Markovianity: characterization, quantification and detection.” Reports on Progress in Physics 77, 094001 (2014). Definitions and diagnostics.