Non-Markovian Toy Models

This notebook guide specifies a reproducible toy-model calculation for non-Markovian dynamics. The baseline model is a qubit coupled coherently to a single environmental mode. The exact joint dynamics is finite-dimensional and unitary, while the reduced qubit dynamics shows revivals that a Markovian amplitude-damping equation cannot reproduce. Repeated-interaction variants belong with Collision Models.

As of this review, no executable notebook under notebooks/density-open-systems/non-markovian-toy-models/ is promoted as a reproduced artifact. This page is the admission contract for that notebook: it defines the model, exact solution checks, trace-distance diagnostics, Markovian comparison, convergence requirements, and acceptable outputs.

Purpose

The notebook should demonstrate how to:

• evolve a small system plus explicit environmental mode exactly;
• trace out the environmental mode to obtain reduced qubit states;
• compare exact reduced dynamics with a Markovian master equation;
• compute trace distance between two reduced states;
• identify distinguishability revivals as information-backflow diagnostics;
• test Hilbert-space truncation and solver convergence;
• avoid overinterpreting a toy model as a universal definition of non-Markovianity.

The first version should use the smallest nontrivial model: one qubit, one mode, one excitation.

Directory Plan

Use a dedicated directory:

notebooks/density-open-systems/non-markovian-toy-models/
  non-markovian-toy-models.ipynb
  README.md

The notebook or README.md should state:

• Python and package versions;
• basis ordering for qubit and mode;
• tensor-product ordering;
• oscillator truncation;
• Hamiltonian parameters and units;
• ODE solver method and tolerances;
• trace-distance norm convention;
• random seed, if any random state pairs are sampled;
• date and commit identifier when the notebook is promoted.

Baseline Hamiltonian

Use the resonant Jaynes–Cummings-type Hamiltonian

$$H = \hbar\omega_q\sigma_+\sigma_- + \hbar\omega_c a^\dagger a + \hbar g \left( \sigma_+ a + \sigma_- a^\dagger \right).$$

For the baseline benchmark set

$$\omega_q=\omega_c,$$

and work in the one-excitation sector. The initial environmental mode is vacuum:

$$\rho_E(0)=\lvert 0\rangle\langle 0\rvert.$$

If the qubit starts excited, the joint state evolves as

$$\lvert e,0\rangle \longmapsto \cos(gt)\lvert e,0\rangle - i\sin(gt)\lvert g,1\rangle$$

in the interaction picture.

Tracing out the mode gives the qubit excited-state probability

$$p_e(t) = \cos^2(gt).$$

This exact oscillation is the primary validation target.

Reduced Dynamics

The reduced qubit state is

$$\rho_S(t) = \operatorname{Tr}_E \rho_{SE}(t).$$

The notebook should implement the partial trace explicitly and test it on product states before applying it to evolved states.

For an arbitrary initial qubit state

$$\rho_S(0) = \begin{pmatrix} \rho_{ee}(0) & \rho_{eg}(0)\\ \rho_{ge}(0) & \rho_{gg}(0) \end{pmatrix},$$

with the mode initially in vacuum, the one-mode resonant model gives the amplitude-damping-like form

$$\begin{aligned} \rho_{ee}(t) &= |G(t)|^2\rho_{ee}(0), \\ \rho_{eg}(t) &= G(t)\rho_{eg}(0), \\ \rho_{gg}(t) &= 1-|G(t)|^2\rho_{ee}(0), \end{aligned}$$

where

$$G(t)=\cos(gt)$$

for the closed single-mode benchmark. Unlike a Markovian amplitude-damping channel, $|G(t)|^2$ is not monotone.

Markovian Comparison

Compare the exact reduced dynamics with the zero-temperature Markovian amplitude-damping equation

$$\dot\rho = \Gamma\mathcal D[\sigma_-]\rho.$$

For an initially excited qubit,

$$p_e^{\mathrm M}(t) = e^{-\Gamma t}.$$

Choose $\Gamma$ deliberately. For example, match the initial short-time behavior poorly but transparently, or match a chosen decay time:

Gamma = 2 g

The point is not to make the Markovian curve “fit” the oscillations. The point is to show that a memoryless exponential cannot reproduce excitation returning from an explicit environmental mode.

Trace Distance

For two reduced qubit states $\rho_1(t)$ and $\rho_2(t)$, define

$$D(t) = \frac12 \lVert \rho_1(t)-\rho_2(t) \rVert_1.$$

Quantum channels contract trace distance. Thus, if $D(t)$ increases over some interval for the reduced dynamics, the increase is interpreted as distinguishability returning from the environment to the system. See Information Backflow.

For the state pair

$$\rho_1(0)=\lvert e\rangle\langle e\rvert, \qquad \rho_2(0)=\lvert g\rangle\langle g\rvert,$$

the exact single-mode model gives

$$D(t) = \cos^2(gt).$$

This quantity decreases and then revives periodically. The Markovian amplitude-damping comparison gives

$$D_{\mathrm M}(t) = e^{-\Gamma t},$$

which is monotone.

Optional Damped-Mode Extension

After the closed single-mode benchmark passes, add damping of the environmental mode:

$$\dot\rho_{SE} = - \frac{i}{\hbar}[H,\rho_{SE}] + \kappa\mathcal D[a]\rho_{SE}.$$

This model is a simple reaction-coordinate or pseudomode example. The qubit exchanges excitation with a mode, and the mode leaks irreversibly into a Markovian background. For strong enough $g/\kappa$, the qubit can still show non-Markovian revivals; for weak coupling or large $\kappa$, the reduced dynamics approaches Markovian decay.

This extension connects the notebook to Pseudomode Methods and Reaction-Coordinate Mapping.

Numerical Workflow

The notebook should follow this order:

1. Define qubit and mode bases.
2. Declare tensor-product ordering.
3. Build $a$, $a^\dagger$, $\sigma_-$, and $\sigma_+$ in the joint Hilbert space.
4. Verify commutators within the chosen oscillator truncation.
5. Build the Hamiltonian.
6. Evolve the joint state exactly with a unitary or ODE solver.
7. Partial-trace the mode.
8. Validate $p_e(t)=\cos^2(gt)$ for the closed benchmark.
9. Compute trace distance for at least one state pair.
10. Compare with Markovian amplitude damping.
11. Repeat selected calculations for tighter time grids and larger truncations.

The baseline notebook should not start with a large bath. The point is clarity: exact joint dynamics, reduced memory, and a visible revival.

Convergence Checks

Oscillator truncation

For the one-excitation benchmark, a two-dimensional mode truncation is enough:

$$n_{\max}=1.$$

The notebook should still test a larger truncation, such as $n_{\max}=2$, and confirm that the baseline results do not change. For driven or finite-temperature extensions, truncation must be tested more carefully.

Solver tolerance

Check that reducing the ODE tolerances or time step does not change:

• total joint-state trace;
• joint-state purity for closed unitary evolution;
• $p_e(t)$;
• trace-distance revival times;
• maximum trace-distance error against the analytic benchmark.

Partial-trace convention

Test the partial trace on a product state:

$$\rho_{SE} = \rho_S\otimes\rho_E.$$

The result should be exactly $\rho_S$ up to roundoff. This catches tensor-ordering mistakes before they contaminate the non-Markovian diagnostics.

Validation Tests

A promoted notebook should pass:

Test	Expected result
closed joint trace	$\operatorname{Tr}\rho_{SE}(t)=1$
closed joint purity	constant for pure initial states
exact excited population	$p_e(t)=\cos^2(gt)$
reduced trace	$\operatorname{Tr}\rho_S(t)=1$
reduced positivity	eigenvalues nonnegative up to tolerance
trace distance	$D(t)=\cos^2(gt)$ for $\lvert e\rangle,\lvert g\rangle$ pair
Markovian comparison	$D_{\mathrm M}(t)=e^{-\Gamma t}$ monotone
truncation check	$n_{\max}=1$ and $n_{\max}=2$ agree for one-excitation benchmark

The notebook should report numerical tolerances explicitly.

What the Toy Model Does Not Prove

This model is intentionally small. It shows that reduced dynamics can have memory when information is stored in an environmental degree of freedom and later returned. It does not prove that every nonexponential decay is non-Markovian, that trace-distance backflow is the only diagnostic, or that finite environments are realistic reservoirs.

Use the toy model to build intuition, then connect it to the diagnostic pages:

• CP Divisibility
• Information Backflow
• Memory Kernels

Common Mistakes

• Calling any oscillation in the reduced state “information backflow” without comparing state distinguishability.
• Forgetting that a finite explicit mode is not a broadband reservoir.
• Using too small an oscillator truncation after adding drives or finite temperature.
• Comparing exact unitary dynamics with a Markovian curve whose rate was fitted after seeing the revival.
• Treating trace-distance backflow as the only definition of non-Markovianity.
• Losing tensor-product ordering in the partial trace.
• Interpreting the damped-mode extension without checking whether $\kappa$ makes the mode effectively Markovian.

Exercises

Single-Mode Excited Population

Starting from $\lvert e,0\rangle$, derive $p_e(t)=\cos^2(gt)$ for the resonant one-excitation model.

Solution

In the one-excitation subspace, the interaction-picture Hamiltonian couples only

$$\lvert e,0\rangle \quad\text{and}\quad \lvert g,1\rangle.$$

It acts like

$$H_I = \hbar g \left( \lvert e,0\rangle\langle g,1\rvert + \lvert g,1\rangle\langle e,0\rvert \right).$$

Therefore

$$\lvert e,0\rangle \longmapsto \cos(gt)\lvert e,0\rangle - i\sin(gt)\lvert g,1\rangle.$$

The excited-state probability after tracing out the mode is

$$p_e(t)=\cos^2(gt).$$

Trace-Distance Revival

For the initial pair $\lvert e\rangle$ and $\lvert g\rangle$, show that $D(t)=\cos^2(gt)$ in the closed single-mode model.

Solution

The initially ground qubit remains ground because the mode starts in vacuum and there is no excitation. The initially excited qubit reduces to

$$\rho_e(t) = \cos^2(gt) \lvert e\rangle\langle e\rvert + \sin^2(gt) \lvert g\rangle\langle g\rvert.$$

The ground reference state is

$$\rho_g(t)=\lvert g\rangle\langle g\rvert.$$

Their difference has eigenvalues $\cos^2(gt)$ and $-\cos^2(gt)$. Hence

$$D(t) = \frac12 \left( \cos^2(gt)+\cos^2(gt) \right) = \cos^2(gt).$$

Markovian Comparison

Show that the same initial pair under zero-temperature Markovian amplitude damping has monotone trace distance $D_{\mathrm M}(t)=e^{-\Gamma t}$.

Solution

The initially ground state remains ground. The initially excited state becomes

$$\rho_e^{\mathrm M}(t) = e^{-\Gamma t} \lvert e\rangle\langle e\rvert + \left( 1-e^{-\Gamma t} \right) \lvert g\rangle\langle g\rvert.$$

The difference from $\lvert g\rangle\langle g\rvert$ has eigenvalues $e^{-\Gamma t}$ and $-e^{-\Gamma t}$. Thus

$$D_{\mathrm M}(t) = e^{-\Gamma t},$$

which decreases monotonically.

Partial Trace Test

Why should the notebook test $\operatorname{Tr}_E(\rho_S\otimes\rho_E)=\rho_S$ before studying revivals?

Solution

Trace-distance revivals are computed from reduced states. If the partial-trace routine has the wrong tensor ordering, it can produce incorrect reduced states while still returning matrices of the right size. Testing

$$\operatorname{Tr}_E(\rho_S\otimes\rho_E)=\rho_S$$

with known product states verifies the convention before the diagnostic is used.

Cross-Links

• Non-Markovian Dynamics
• Information Backflow
• CP Divisibility
• Pseudomode Methods
• Reaction-Coordinate Mapping
• Collision Models
• Memory Kernels
• Amplitude Damping Master Equation
• Solving Lindblad Equations
• Formula Sheet
• Approximation Checklist

References

• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• H.-P. Breuer, E.-M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Physical Review Letters 103, 210401 (2009).
• A. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and non-Markovianity of quantum evolutions,” Physical Review Letters 105, 050403 (2010).
• B. M. Garraway, “Nonperturbative decay of an atomic system in a cavity,” Physical Review A 55, 2290-2303 (1997).
• I. de Vega and D. Alonso, “Dynamics of non-Markovian open quantum systems,” Reviews of Modern Physics 89, 015001 (2017).