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{{Quantum book backlink|Open quantum systems}}
'''Non-Markovian quantum dynamics''' describe the evolution of open quantum systems in the presence of memory effects. In this regime, the future evolution depends not only on the present state but also on the system’s history.<ref name="Breuer2016">{{cite journal |last=Breuer |first=H.-P. |last2=Laine |first2=E.-M. |last3=Piilo |first3=J. |last4=Vacchini |first4=B. |title=Colloquium: Non-Markovian dynamics in open quantum systems |journal=Reviews of Modern Physics |volume=88 |issue=2 |pages=021002 |year=2016 |url=https://link.aps.org/doi/10.1103/RevModPhys.88.021002 |doi=10.1103/RevModPhys.88.021002}}</ref>
Non-Markovian effects are important in strongly coupled systems, structured environments, and low-temperature physics.

[[File:Quantum_nonmarkovian_dynamics_fixed.svg|thumb|400px|Non-Markovian dynamics: memory effects allow information backflow between system and environment.]]

=Non-Markovian quantum dynamics=
== Definition ==

A quantum process is non-Markovian if its evolution cannot be described by a memoryless (time-local) generator.

=== Memory dependence ===

The evolution of the density operator may depend on earlier states:

<math>
\frac{d\rho(t)}{dt}
=
\int_0^t K(t-s)\,\rho(s)\,ds,
</math>

where <math>K(t)</math> is a memory kernel.<ref name="Breuer2016" />

This explicitly introduces dependence on the past history of the system.

== Breakdown of Markovian approximation ==

Non-Markovian behavior arises when the assumptions of the Markovian approximation fail.

=== Strong coupling ===

When the interaction between system and environment is strong, correlations persist and memory effects become significant.

=== Structured environments ===

Environments with non-flat spectral densities (e.g. photonic crystals) can store and return information to the system.

=== Finite environments ===

Small environments cannot act as perfect reservoirs and may feed information back into the system.

== Information backflow ==

A defining feature of non-Markovian dynamics is the possibility of '''information backflow'''.

=== Physical meaning ===

* information lost to the environment can return
* coherence may temporarily increase
* distinguishability between states can grow

This contrasts with Markovian evolution, where information is lost irreversibly.

=== Trace distance criterion ===

One way to detect non-Markovianity is through the trace distance:

<math>
D(\rho_1,\rho_2) = \frac{1}{2}\mathrm{Tr}|\rho_1 - \rho_2|.
</math>

If <math>D</math> increases at some time, this indicates information backflow.<ref name="Breuer2016" />

== Dynamical behavior ==

Non-Markovian systems exhibit richer time evolution than Markovian systems.

=== Non-exponential decay ===

Decay processes may deviate from simple exponential laws:

<math>
\rho_{ij}(t) \not\sim e^{-\gamma t}.
</math>

=== Coherence revival ===

Quantum coherence can partially recover after decay:

<math>
\rho_{ij}(t) \uparrow
</math>

over certain time intervals.

=== Oscillatory dynamics ===

Systems may show oscillations due to feedback from the environment.

== Time-local formulation ==

Even non-Markovian dynamics can sometimes be written in a time-local form:

<math>
\frac{d\rho(t)}{dt} = \mathcal{L}(t)[\rho(t)],
</math>

where the generator <math>\mathcal{L}(t)</math> is time-dependent.

In this case, non-Markovianity is associated with the breakdown of divisibility of the dynamical map.<ref name="Breuer2016" />

== Relation to decoherence ==

Decoherence in realistic systems often includes non-Markovian corrections.

=== Non-Markovian decoherence ===

Leads to:

* temporary recoherence
* slower decay of interference
* environment-induced memory effects

=== Physical relevance ===

These effects are especially important in solid-state qubits and nanoscale systems.

== Applications ==

Non-Markovian dynamics are relevant in many areas.

=== Quantum information ===

Can be exploited to:

* preserve coherence
* improve control protocols
* enhance quantum memory

=== Quantum optics ===

Structured reservoirs produce non-Markovian emission and absorption behavior.

=== Condensed matter ===

Strong coupling and low temperatures naturally lead to memory effects.

== Physical significance ==

Non-Markovian quantum dynamics provide a more complete description of open quantum systems beyond the Lindblad approximation. They reveal the role of memory, correlations, and feedback in quantum evolution.<ref name="Breuer2016" />

They are essential for understanding realistic quantum systems and advanced quantum technologies.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Non-Markovian quantum dynamics|1}}

Physics:Quantum Non-Markovian dynamics - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template