A quantum environment that remembers—and carries control between spins

Memory with somewhere to go

The study, posted as an arXiv preprint on November 10, examines an environment with two abilities: it remembers earlier interactions, and it carries their influence elsewhere.

Each site contains a quantum spin coupled to a cavity that holds light. Photons can hop between neighbouring cavities, while also leaking out. In the first examples, there is no direct interaction between spins. Their route to one another runs through the connected cavity environment.

The distinction between system and environment matters. If spins and cavities are described together, their evolution follows a memoryless, or Markovian, master equation. But when the cavities are eliminated from the description, the spins retain the influence of earlier exchanges with them. Their reduced dynamics are non-Markovian: the surroundings have memory.

Entanglement through the surroundings

Flannigan first simulates three spins initially in their ground states, with empty cavities. Switching on the model’s spin–cavity interaction allows its counter-rotating terms to create spin–photon excitations. Photon hopping then connects the subsequent dynamics across sites.

With hopping, both neighbouring spins and the two endpoints develop entanglement. Without hopping, the local spin–cavity pairs remain independent.

The calculations also compare two ways of distributing the same total photon loss: equal leakage everywhere, or weaker leakage at the endpoints and stronger leakage in the middle. The weak-end arrangement produces greater endpoint entanglement in the displayed comparison. Longer-lived environmental memory at the edges helps, and an antisymmetric cavity mode with no middle-cavity component offers another route for retaining excitations.

This is not a clean experiment that changes transport alone. The paper explicitly notes that changing cavity hopping changes both spatial propagation and the local memory experienced by a spin.

Control in the middle, consequences at the ends

The next step uses leaking light as a measurement signal. Continuous homodyne detection measures a field quadrature of an intermediate cavity’s output. A classical controller feeds that signal into a drive acting on the spin at the same site. The emitted photons themselves are not returned.

In the three-spin simulations, this local feedback enhances endpoint correlations and entanglement, particularly with weak endpoint loss. Crucially, the entanglement is calculated from the endpoint state averaged over measurement records—not from selected records or an average of separately calculated trajectory entanglements.

Controls help identify what matters. With no photon hopping, feedback does not create correlations between the separated spins. Independent random driving at the same gains does not reproduce the measured-signal feedback response. The connection between the detected field and the actuator matters, rather than simply adding drive noise.

Four- and five-spin calculations show feedback-enhanced endpoint correlations across longer networks. Those correlation results should not be confused with a demonstration of endpoint entanglement in every longer chain.

After the controller switches off

Switching feedback on and off exposes the role of history. Starting feedback produces a delayed endpoint response. After feedback stops, endpoint entanglement can continue increasing before it decays.

There is no instantaneous remote action here. A local, trace-preserving operation on the middle spin cannot immediately change the endpoint state. Instead, feedback changes the joint spin–cavity state; later photon hopping and spin–cavity exchange carry its influence onwards. Stopping the controller does not erase the state already prepared.

A method with numerical limits

To calculate these dynamics, Flannigan extends the hierarchy of pure states, or HOPS. Stochastic spin wavefunctions are accompanied by auxiliary states that track environmental memory, including its transfer between sites. These auxiliaries are bookkeeping coordinates, not physical photon states that can simply be traced out.

The formal hierarchy is exact before numerical approximations. Practical calculations truncate it and use finite timesteps and finite samples. Small-network results are checked against explicit-cavity master equations or independent stochastic calculations.

Combining HOPS with tensor networks also enables twenty-spin simulations with direct spin interactions. Environmental hopping changes spatial correlation patterns, with its sign affecting some responses even when the local free-environment memory kernels remain unchanged. These many-body examples do not include feedback.

The findings remain numerical preprint results, not an experimental demonstration. The feedback assumes zero controller delay, the environment starts in vacuum, and some distant many-body correlations remain sampling-limited. Nor does HOPS deliver a universal computational speedup: its cost advantage depends on the regime and observable. Within those boundaries, the work presents an environment as an active carrier of quantum control—not merely something to isolate a system from.