Memory Kernels

A memory-kernel master equation is a time-nonlocal equation for a reduced density operator. Instead of depending only on the present state $\rho_S(t)$, the derivative at time $t$ depends on earlier states $\rho_S(s)$.

The schematic form is

$$\frac{d}{dt}\rho_S(t) = \int_{t_0}^t ds\, \mathcal K(t,s)\rho_S(s) + \mathcal J(t).$$

Here $\mathcal K(t,s)$ is a superoperator-valued kernel and $\mathcal J(t)$ is an inhomogeneous term that can appear when the initial system-environment state is correlated or when the chosen projection does not remove all irrelevant variables.

In a stationary setting, the kernel often depends only on the time difference:

$$\frac{d}{dt}\rho_S(t) = \int_0^t d\tau\, \mathcal K(\tau)\rho_S(t-\tau).$$

This is a convolution memory equation. It is the natural language for delayed environmental feedback, finite reservoirs, structured spectra, and non-Markovian reduced dynamics.

For the broader comparison of non-Markovian diagnostics and terminology, see What Non-Markovian Means.

Time Local Versus Time Nonlocal

A time-convolutionless master equation is time local and has the form

$$\frac{d}{dt}\rho_S(t) = \mathcal L(t)\rho_S(t).$$

All memory is compressed into the time-dependent generator $\mathcal L(t)$, if such a generator exists on the relevant domain.

A time-nonlocal equation has the form

$$\frac{d}{dt}\rho_S(t) = \int_{t_0}^t ds\, \mathcal K(t,s)\rho_S(s).$$

The earlier reduced states remain explicit. This is often closer to the microscopic picture: information and excitation can leave the system, evolve in the environment, and later affect the system again.

The two descriptions are not mutually exclusive. If the reduced map is invertible, a time-local generator may exist even for dynamics that is physically memoryful. Conversely, a poorly chosen kernel can describe an unphysical map. The representation is a tool, not a definition of physical validity by itself.

Where Kernels Come From

Start from the exact reduced state

$$\rho_S(t) = \operatorname{Tr}_E \left[ U(t,t_0)\rho_{SE}(t_0)U^\dagger(t,t_0) \right].$$

This formula is exact but not closed as a differential equation for $\rho_S(t)$. Nakajima–Zwanzig projection methods split the total state into a relevant part, usually associated with $\rho_S$, and an irrelevant part containing environment variables and correlations. Eliminating the irrelevant part produces a memory integral.

At second order in a weak system–bath coupling, the Born Approximation already gives a time-nonlocal structure:

$$\frac{d}{dt}\rho_S^{I}(t) = -\frac{\lambda^2}{\hbar^2} \int_0^t ds\, \operatorname{Tr}_E \left[ H_I(t), \left[ H_I(s), \rho_S^{I}(s)\otimes\rho_E \right] \right].$$

This equation has a memory kernel because $\rho_S^{I}(s)$ remains inside the integral. The Markov Approximation is the later step that replaces this delayed state by the present one when bath correlations are short lived.

Kernel as a Superoperator

A memory kernel is not just a scalar function. It is a map acting on operators.

For example, a second-order Born kernel in the interaction picture may be written as

$$\mathcal K(t,s)\rho = -\frac{\lambda^2}{\hbar^2} \operatorname{Tr}_E \left[ H_I(t), \left[ H_I(s), \rho\otimes\rho_E \right] \right].$$

Then

$$\frac{d}{dt}\rho_S^{I}(t) = \int_0^t ds\, \mathcal K(t,s)\rho_S^{I}(s).$$

In stationary problems one often changes variables to $\tau=t-s$ and writes $\mathcal K(\tau)$.

Markov Limit

The Markov limit is a short-memory limit. If $\mathcal K(\tau)$ decays on a bath memory time $\tau_E$ and $\rho_S(t-\tau)$ changes slowly over that interval, then

$$\rho_S(t-\tau) \approx \rho_S(t)$$

inside the integral. This gives

$$\frac{d}{dt}\rho_S(t) \approx \left[ \int_0^\infty d\tau\, \mathcal K(\tau) \right]\rho_S(t).$$

The time-local generator is then

$$\mathcal L = \int_0^\infty d\tau\, \mathcal K(\tau),$$

provided the integral and approximation are meaningful.

A formal delta-function kernel,

$$\mathcal K(\tau) = \mathcal L\,\delta(\tau),$$

is the ideal memoryless limit. Real baths usually have finite correlation times, so the question is whether those times are short compared with the system evolution being resolved.

Laplace-Transform Solution

For a time-translation-invariant kernel, a convolution equation can be solved formally by a Laplace transform.

Let

$$\widetilde\rho(z) = \int_0^\infty dt\, e^{-zt}\rho(t), \qquad \widetilde{\mathcal K}(z) = \int_0^\infty dt\, e^{-zt}\mathcal K(t).$$

For

$$\frac{d}{dt}\rho(t) = \int_0^t d\tau\, \mathcal K(\tau)\rho(t-\tau),$$

the transform gives

$$z\widetilde\rho(z)-\rho(0) = \widetilde{\mathcal K}(z)\widetilde\rho(z).$$

Thus

$$\widetilde\rho(z) = \left[ z\mathcal I-\widetilde{\mathcal K}(z) \right]^{-1} \rho(0),$$

where $\mathcal I$ is the identity superoperator. Poles of the resolvent determine decay rates, oscillations, bound-state contributions, and long-time tails.

Exponential Memory and Auxiliary Variables

A simple scalar analogy is

$$\dot x(t) = -\int_0^t d\tau\, k(\tau)x(t-\tau), \qquad k(\tau)=\gamma\Lambda e^{-\Lambda\tau}.$$

Define an auxiliary variable

$$y(t) = \int_0^t d\tau\, \gamma\Lambda e^{-\Lambda\tau}x(t-\tau).$$

Then

$$\dot x(t)=-y(t), \qquad \dot y(t)=\gamma\Lambda x(t)-\Lambda y(t).$$

The memory equation has been embedded into a larger time-local system. In quantum models, pseudomodes, reaction coordinates, auxiliary density operators, and chain mappings use the same broad idea: a non-Markovian reduced problem can sometimes be made Markovian by enlarging the system.

Physical Sources of Memory

Memory kernels appear when environmental degrees of freedom do not immediately lose information about the system.

Common sources include:

• finite environments with recurrences;
• structured reservoirs, such as cavities or band edges;
• low-dimensional continua with slow correlation decay;
• strong coupling to one or a few modes;
• feedback loops and delay lines;
• spin baths with slow internal dynamics;
• initial system-environment correlations;
• coarse graining that is too fine compared with bath memory.

The same microscopic Hamiltonian can look Markovian or non-Markovian depending on the chosen system-environment boundary. Moving a strongly coupled mode into the system can turn a long-memory kernel into a larger time-local model.

Positivity Caveats

An arbitrary memory kernel does not automatically generate physical quantum dynamics. The resulting map must still preserve trace, Hermiticity, and positivity on the intended domain, and often complete positivity is required.

Trace preservation imposes

$$\operatorname{Tr}\mathcal K(t,s)X=0$$

for all allowed operators $X$, together with any contribution from $\mathcal J(t)$. Hermiticity preservation requires the kernel to map Hermitian operators to Hermitian operators in the integrated equation.

Complete positivity is more subtle. Unlike the Lindblad–GKSL Equation, there is no simple universal sign condition on $\mathcal K(t,s)$ that makes every memory-kernel equation completely positive. One usually checks the finite-time map, constructs a microscopic dilation, or tests the Choi matrix when the map is known. See Completely Positive Maps.

Common Mistakes

Replacing history by the present too early

The step $\rho(t-\tau)\approx\rho(t)$ is a Markov approximation. It should be justified by a time-scale hierarchy, not made just to simplify an integral.

Treating any fitted kernel as physical

A fitted scalar decay curve may reproduce one observable while failing positivity or trace preservation for other states. A quantum memory kernel must be checked as a superoperator.

Ignoring inhomogeneous terms

Initial correlations can add a driving term $\mathcal J(t)$. Dropping it may give the wrong short-time behavior even if the kernel is otherwise reasonable.

Equating time nonlocal with every definition of non-Markovian

Quantum non-Markovianity has several inequivalent definitions, including CP-divisibility failure and information backflow. A time-nonlocal representation is related to memory, but it is not by itself a complete classification.

Forgetting boundary choices

A mode treated as an environment can generate memory. The same mode included in the system can produce a larger time-local problem.

Exercises

Markov limit of a narrow kernel

Let $f_\epsilon(\tau)$ be a family of nonnegative functions supported near $\tau=0$ with

$$\int_0^\infty d\tau\, f_\epsilon(\tau)=1.$$

Show that the kernel $\mathcal K_\epsilon(\tau)=\mathcal L f_\epsilon(\tau)$ approaches a Markov generator when $\rho(t-\tau)$ varies slowly over the width of $f_\epsilon$.

Solution

If $\rho(t-\tau)\approx\rho(t)$ over the support of $f_\epsilon$, then

$$\int_0^t d\tau\, \mathcal K_\epsilon(\tau)\rho(t-\tau) \approx \mathcal L\rho(t) \int_0^\infty d\tau\, f_\epsilon(\tau) = \mathcal L\rho(t).$$

Thus the memory equation approaches $\dot\rho(t)=\mathcal L\rho(t)$.

Laplace transform of a scalar kernel equation

For

$$\dot x(t) = -\int_0^t d\tau\, k(\tau)x(t-\tau),$$

derive the transformed solution $\widetilde x(z)$.

Solution

The Laplace transform of the left side is

$$z\widetilde x(z)-x(0).$$

The convolution transforms into a product:

$$\mathcal L \left[ \int_0^t d\tau\, k(\tau)x(t-\tau) \right] = \widetilde k(z)\widetilde x(z).$$

Therefore

$$z\widetilde x(z)-x(0) = -\widetilde k(z)\widetilde x(z),$$

so

$$\widetilde x(z) = \frac{x(0)}{z+\widetilde k(z)}.$$

Embedding exponential memory

For $k(\tau)=\gamma\Lambda e^{-\Lambda\tau}$, verify that

$$y(t) = \int_0^t d\tau\, k(\tau)x(t-\tau)$$

obeys $\dot y(t)=\gamma\Lambda x(t)-\Lambda y(t)$.

Solution

Write

$$y(t) = \int_0^t ds\, \gamma\Lambda e^{-\Lambda(t-s)}x(s).$$

Differentiate:

$$\dot y(t) = \gamma\Lambda x(t) - \Lambda \int_0^t ds\, \gamma\Lambda e^{-\Lambda(t-s)}x(s).$$

The integral is $y(t)$, so

$$\dot y(t) = \gamma\Lambda x(t)-\Lambda y(t).$$

Memory and complete positivity

Why is the condition $\operatorname{Tr}\mathcal K(t,s)X=0$ not enough to make a memory-kernel equation physical?

Solution

That condition only addresses trace preservation. A physical density operator must also remain Hermitian and positive, and for a subsystem map one usually requires complete positivity. A kernel can preserve trace while driving some positive states to operators with negative eigenvalues. Positivity or complete positivity must be checked separately, often through the finite-time map or a microscopic dilation.

References

• S. Nakajima, “On quantum theory of transport phenomena,” Progress of Theoretical Physics 20, 948–959 (1958).
• R. Zwanzig, “Ensemble method in the theory of irreversibility,” Journal of Chemical Physics 33, 1338–1341 (1960).
• 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.
• I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Reviews of Modern Physics 89, 015001 (2017).