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 $\rho_S(s)$.

With an explicit instantaneous term, the general form is

$$\frac{d}{dt}\rho_S(t) = \mathcal L_0(t)\rho_S(t)+ \int_{t_0}^t ds\, \mathcal K(t,s)\rho_S(s) + \mathcal J(t).$$

Here $\mathcal L_0$ has rate units, $\mathcal K(t,s)$ has units inverse time squared, and the kernel is a superoperator 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.

For an autonomous total Hamiltonian and a fixed projection, the Schrödinger-picture kernel can depend only on the time difference. For simplicity, the convolution examples below have no separate instantaneous or source term:

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

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

Helpful background. Reduced dynamics defines the maps being modeled, and the Born approximation supplies a perturbative example. Non-Markovianity has several distinct operational definitions.

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

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

Stationary bath correlations alone do not make this interaction-picture kernel a function of $t-s$: $H_I(t)$ also contains the rotating system operators. A convolution form may instead be obtained in a suitable Schrödinger-picture autonomous formulation. A time-dependent drive generally leaves a two-time kernel.

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,

$$\mathcal K(\tau) = \mathcal L\,\delta_+(\tau),$$

uses a causal delta distribution of unit mass on the half-line, so its convolution returns $\mathcal L\rho(t)$ for $t>0$. For example, $\Lambda e^{-\Lambda\tau}$ at $\tau\ge0$ has this limit. This convention avoids a spurious factor of one-half at the integration endpoint. A physical memoryless quantum generator must also satisfy the appropriate GKSL conditions.

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 and the transform is taken in a half-plane where it converges and the inverse exists. Isolated poles produce exponential or oscillatory terms; branch cuts and other nonanalytic structure can produce algebraic long-time tails. With an instantaneous $\mathcal L_0$, the resolvent instead contains $z\mathcal I-\mathcal L_0-\widetilde{\mathcal K}(z)$.

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

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

Use $y(0)=0$ for the stated history starting at zero. The auxiliary variable is not automatically a physical state. This construction gives a larger time-local differential equation; a physical Markovian quantum embedding additionally needs a valid Hilbert space, state, and CP dynamics. Pseudomodes and reaction coordinates can provide such embeddings in suitable models, while auxiliary operators in hierarchical equations need not themselves be density matrices.

Eliminating $y$ gives $\ddot x+\Lambda\dot x+\gamma\Lambda x=0$, with $x(0)=1$ and $\dot x(0)=0$. For $\gamma>\Lambda/4$,

$$x(t)=e^{-\Lambda t/2}\left[\cos(\Omega t)+\frac\Lambda{2\Omega}\sin(\Omega t)\right], \qquad \Omega=\sqrt{\gamma\Lambda-\Lambda^2/4}.$$

The damped-oscillator energy $\dot x^2+\gamma\Lambda x^2$ is nonincreasing for $\gamma,\Lambda>0$, so $|x(t)|\le1$. This makes $x(t)$ a valid qubit dephasing factor when the diagonal entries remain unchanged, even when it becomes negative: negativity of a coherence factor is not negativity of a population.

The same function need not define a valid depolarizing channel. If all Bloch components are multiplied by $x(t)$, CP requires $x\ge-1/3$. At $\gamma=4\Lambda$, the first minimum is

$$t_{\min}=\frac{2\pi}{\sqrt{15}\,\Lambda}, \qquad x(t_{\min})=-e^{-\pi/\sqrt{15}}\simeq-0.444.$$

It violates that bound and gives a negative Bell-pair output eigenvalue $(1+3x)/4$. The scalar curve and trace preservation alone therefore cannot establish physicality of a quantum kernel.

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.

A standard sufficient structural condition for trace preservation is

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

for all allowed operators $X$, together with a trace-annihilating instantaneous term and $\operatorname{Tr}\mathcal J(t)=0$. These are satisfied by the usual exact projection construction. 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.

Exercises

Markov limit of a narrow kernel

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

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

For times larger than that width, show that $\mathcal K_\epsilon(\tau)=\mathcal L f_\epsilon(\tau)$ approaches the local equation $\dot\rho=\mathcal L\rho$ when $\rho(t-\tau)$ varies slowly across it. Assume separately that $\mathcal L$ is a valid GKSL generator if the limit is meant to describe a quantum dynamical semigroup.

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

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002). Publisher.
• I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Reviews of Modern Physics 89, 015001 (2017). Author manuscript.
• S. Nakajima, “On quantum theory of transport phenomena,” Progress of Theoretical Physics 20, 948–959 (1958). DOI.
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012). Author manuscript.
• R. Zwanzig, “Ensemble method in the theory of irreversibility,” Journal of Chemical Physics 33, 1338–1341 (1960). DOI.