Coupled Photonic Quantum Memristors

• The paper demonstrates a coupled architecture where crossed-feedback between photonic quantum memristors enhances hysteresis behavior and non-Markovian dynamics.
• It employs integrated Mach–Zehnder interferometers on a silicon nitride chip, driven by room-temperature single photons from an SiV⁻ color center, to establish measurement-induced memristive states.
• The system achieves complex, history-dependent memory dynamics with self-intersecting hysteresis loops, paving the way for advanced quantum neuromorphic computing applications.

Coupled photonic quantum memristors are photonic quantum memristors arranged so that their internal memristive states are mutually updated through crossed feedback, producing a nonlinear, history-dependent, and effectively non-Markovian dynamics at the level of observable photon numbers. In the integrated realization reported in "Coupled integrated photonic quantum memristors using a single photon source made of a colour center" (Baldazzi et al., 16 Feb 2026), the platform consists of two Mach–Zehnder interferometers (MZIs) on a silicon nitride photonic integrated circuit, driven by single photons emitted at room temperature by a silicon-vacancy color center SiV$^-$ in a nanodiamond. Relative to isolated photonic quantum memristors, the crossed-feedback architecture yields enhanced memristive behavior, including larger hysteresis form factors, non-pinched inter-device loops, and self-intersecting hysteresis curves associated with topologically non-trivial memory dynamics (Baldazzi et al., 16 Feb 2026).

1. Formal definition and photonic realization

The underlying memristive formalism follows the generalized two-variable description

$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$

where $x$ is the input, $y$ the output, and $s$ an internal state variable. In the photonic setting of (Baldazzi et al., 16 Feb 2026), $x$ is identified with the average input photon number, $y$ with the average output photon number, and $s$ with the MZI reflectivity $R$. The basic input–output relation is

$$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$

Memristive behavior emerges once $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$0 becomes history-dependent through measurement-conditioned feedback (Baldazzi et al., 16 Feb 2026).

The quantum-optical implementation uses a dual-rail encoding in which a photonic qubit is represented by two path modes $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$1 and $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$2,

$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$3

with an ancillary vacuum mode $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$4. The memristive element is an MZI acting on modes $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$5 and $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$6 with tunable reflectivity $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$7. After tracing over the ancilla, the output density matrix yields

$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$8

so the observable response has the required state-dependent multiplicative structure (Baldazzi et al., 16 Feb 2026).

In an ideal MZI, the reflectivity is

$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$9

and, with finite visibility $x$0,

$x$1

This gives a direct physical interpretation of the internal memristive state: the state variable is implemented as a controllable interferometric phase converted into reflectivity by the MZI transfer function (Baldazzi et al., 16 Feb 2026).

The broader theoretical lineage begins with the proposal of photonic quantum memristors as tunable beam splitters equipped with weak measurements and classical feedback (Sanz et al., 2017), extends to frequency-bin implementations using frequency mixers and feedback-conditioned phase updates (Gonzalez-Raya et al., 2019), and reaches integrated single-photon experiments in femtosecond-laser-written photonic circuits (Spagnolo et al., 2021). The coupled Si$x$2N$x$3 realization of (Baldazzi et al., 16 Feb 2026) advances this trajectory from single-device and simple cascaded configurations to a crossed-feedback network of two photonic quantum memristors.

2. Integrated architecture and single-photon source

The integrated hardware in (Baldazzi et al., 16 Feb 2026) is a silicon nitride photonic integrated circuit fabricated by Ligentec. The waveguides are single-mode TE$x$4 Si$x$5N$x$6 waveguides with cross-section $x$7 and silica cladding. Beam splitters are multimode interferometers, and phase shifters are thermo-optic heaters placed on both arms of each MZI. The relevant MZI blocks are MZI-0, MZI-1, and MZI-2 as front-end modulators, together with MZI-M1 and MZI-M2 as the two photonic quantum memristors (Baldazzi et al., 16 Feb 2026).

Single photons are injected into one input waveguide of MZI-0. At the output, only two waveguides are fiber-coupled to SPADs, and the $x$8 phase relation between MZI outputs is used to infer both output ports with one physical detector per memristor by re-programming phases in two sub-runs. In the single-memristor configuration, only MZI-M1 is used as the memristor, while MZI-M2 is a fixed reference MZI used to collect total counts. In the coupled configuration, both MZI-M1 and MZI-M2 act as photonic quantum memristors in parallel, MZI-0 splits the input flux equally to its two outputs, and MZI-1 and MZI-2 modulate the photon fluxes entering MZI-M1 and MZI-M2 with sinusoidal intensities and controlled relative phase $x$9 (Baldazzi et al., 16 Feb 2026).

The source is a single silicon-vacancy color center SiV$y$0 in an HPHT-grown nanodiamond, spin-coated on a silicon substrate. The emission has a zero-phonon line at approximately $y$1, shifted from the nominal $y$2 due to strain. The measured lifetime is $y$3, the second-order correlation is $y$4, the antibunching dip has FWHM $y$5, and the brightness is approximately $y$6 at the output of a single-mode fiber (Baldazzi et al., 16 Feb 2026). These values establish single-photon emission at room temperature and distinguish the device from earlier photonic memristor experiments that relied on SPDC photons or cryogenic emitters.

Coupling into the chip is achieved with a tapered lensed fiber, while the output is collected by a fiber array and sent to two silicon SPADs. The total insertion loss of the PIC with the standard output fiber array is $y$7 dB, and typical output fluxes are $y$8 kHz (Baldazzi et al., 16 Feb 2026). This loss budget is central to the experimental timing and feedback protocol discussed below.

3. Crossed feedback, memory depth, and non-Markovian dynamics

For a single photonic quantum memristor, the feedback law is

$y$9

which, over a memory window $s$0, integrates to

$s$1

In discrete time, with time-bin duration $s$2, memory depth $s$3, and $s$4,

$s$5

For $s$6, the response is almost instantaneous; for large $s$7, the device integrates over many past steps (Baldazzi et al., 16 Feb 2026).

The defining feature of the coupled configuration is crossed feedback. The memristive reflectivity of one device is updated from the past input of the other:

$s$8

$s$9

This makes the evolution nonlocal both in time and in device index: each memristor stores the history of the other memristor’s input stream (Baldazzi et al., 16 Feb 2026).

The measurement-induced protocol operates in discrete time bins. For each bin, the input MZIs are programmed, the memristor phases are set according to $x$0 and $x$1, counts are collected, the memristor phases are shifted by $x$2 to probe the complementary output port, additional counts are collected, the input MZIs are shifted by $x$3 for normalization, and the normalized $x$4 and $x$5 are computed and stored before updating the internal states for the next bin (Baldazzi et al., 16 Feb 2026).

This architecture clarifies an important distinction. Each MZI, taken in isolation, is a linear, unitary interferometric component. The overall photonic quantum memristor is nonlinear because the phase, and hence the transfer matrix, is adaptively changed on the basis of measurement outcomes. The memory is not introduced through an explicit memory-kernel master equation; rather, it is realized directly by iterative buffer-based update rules over the previous $x$6 time bins (Baldazzi et al., 16 Feb 2026). A common misconception is that the coupled device demonstrates fully coherent inter-memristor quantum dynamics. The reported experiment instead realizes a measurement-based hybrid dynamics in which the input field is genuinely non-classical, but the state update is mediated by classical records of photon detections (Baldazzi et al., 16 Feb 2026).

4. Input–output hysteresis and loop topology

To reveal memristive behavior, the input photon flux is periodically modulated with time-bin duration $x$7, modulation period

$x$8

and normalized input photon number

$x$9

which sweeps from 0 to 1 and back over each cycle (Baldazzi et al., 16 Feb 2026).

For a single memristor, the input–output relation $y$0 versus $y$1 depends strongly on $y$2. At $y$3, corresponding to $y$4, the curve is close to parabolic and shows almost no hysteresis. For $y$5, the curves become pinched hysteresis loops, pinched at the origin and clearly dependent on the traversal direction. At $y$6, the relation becomes almost linear and single-valued because the internal state averages over a full cycle (Baldazzi et al., 16 Feb 2026). This reproduces the standard memristor phenomenology in a single-photon interferometric device.

The coupled system introduces two classes of relations. The intra-memristor relations are

$y$7

whereas the inter-memristor relations are

$y$8

With crossed feedback and $y$9, the inter-relations become nontrivial hysteresis curves (Baldazzi et al., 16 Feb 2026).

Three experimentally emphasized parameter sets are $s$0 with $s$1 rad, $s$2 with $s$3 rad, and $s$4 with $s$5 rad. In these regimes, the intra-relations remain qualitatively similar to single-memristor loops but are deformed by cross-coupling. The inter-relations display qualitatively new structures: some are large-area non-pinched loops that do not pass through the origin, while others are self-intersecting “∞-shaped” loops (Baldazzi et al., 16 Feb 2026).

The self-intersection condition is that there exist distinct times $s$6 for which

$s$7

while the local derivative or traversal direction differs. In the mapped parameter regions, self-intersections never occur in intra-relations; they occur only in inter-relations, and if one inter-relation has a self-intersection, the other does not (Baldazzi et al., 16 Feb 2026). The experiment interprets these curves as signatures of richer multistability and topologically non-trivial memory dynamics than those accessible in isolated devices.

5. Quantification, simulations, and parameter dependence

The hysteresis size and shape are quantified by the form factor

$s$8

where $s$9 is the loop area and $R$0 the loop perimeter. In this normalization, $R$1 for a circle, and smaller values correspond to more elongated or complex curves (Baldazzi et al., 16 Feb 2026). In the single-memristor case, the maximum form factor is $R$2 at $R$3. In the coupled system, the intra-relations reach $R$4 at very short memory $R$5 and $R$6 or $R$7 rad. The inter-relations reach form factors of approximately $R$8, near $R$9, with optimal phases $$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$0 rad for $$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$1 and $$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$2 rad for $$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$3 (Baldazzi et al., 16 Feb 2026).

These values show that coupling enhances memristive hysteresis beyond the single-device limit, particularly in inter-device mappings. An important numerical observation is that even when each device is individually almost memoryless, crossed feedback can induce strong effective memory and large intra-memristor form factors through inter-device correlations (Baldazzi et al., 16 Feb 2026).

The simulations use a semi-classical photon-number model with classical internal states $$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$4. Non-ideal visibility $$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$5, static phase offset $$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$6, detector efficiency $$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$7, and dark counts $$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$8 are included through an effective classical transfer function:

$$\langle N_{\rm out}(t)\rangle = [1-R(t)]\,\langle N_{\rm in}(t)\rangle.$$9

with

$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$00

and

$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$01

Using visibilities and phase offsets extracted from calibration and slightly optimized within measured confidence bounds, the model reproduces the transition from parabolic to pinched to nearly linear hysteresis in single devices, as well as the large-area and self-intersecting loops observed in the coupled experiment (Baldazzi et al., 16 Feb 2026).

The dependence on $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$02 and $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$03 defines a phase-diagram-like structure. Moderate memory depths $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$04–$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$05 maximize hysteresis, while $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$06 collapses it. Simulations identify phase “sweet spots” around $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$07–$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$08 rad and $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$09–$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$10 rad, and show that $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$11 effectively reduces the cross-feedback’s ability to generate complex dynamics (Baldazzi et al., 16 Feb 2026).

A related, although non-photonic, study used supervised regression to optimize the form factor of single and coupled quantum memristors and found that maximizing memristivity leads to large values in the degree of entanglement in a bosonic model directly mappable to coupled photonic quantum memristors (Hernani-Morales et al., 2023). For larger arrays, a tripartite study of three coupled quantum memristors reported that the relation between entanglement and memristive behavior depends on network geometry: in triangular coupling they follow the same behavior, whereas in linear coupling they follow opposite behavior (Kumar et al., 2022). These results suggest that topology is not ancillary but constitutive in memristive quantum networks.

6. Context, applications, distinctions, and limitations

The coupled Si$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$12N$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$13 experiment establishes what the data describe as the first implementation of coupled photonic quantum memristors with crossed feedback and the first combination of integrated memristive devices with a deterministic room-temperature single-photon source based on an SiV$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$14 center (Baldazzi et al., 16 Feb 2026). Earlier photonic work proposed MZI-based quantum memristors with weak measurements and feedback (Sanz et al., 2017), developed frequency-bin memristors in frequency-entangled optical fields (Gonzalez-Raya et al., 2019), and experimentally demonstrated a single integrated quantum-optical memristor acting on single photons (Spagnolo et al., 2021). That single-device experiment also numerically simulated a three-memristor quantum reservoir architecture and reported 95% classification accuracy on unseen MNIST images restricted to digits 0, 3, and 8, together with 98% accuracy on an entanglement-detection task (Spagnolo et al., 2021). The coupled-device results of (Baldazzi et al., 16 Feb 2026) provide a direct hardware basis for extending such reservoir and neuromorphic ideas from isolated elements to genuinely interacting photonic memristive subsystems.

In this context, photonic quantum memristors are motivated as nonlinear synapse-like elements for quantum neuromorphic computing. Standard MZI meshes implement linear weight matrices, whereas photonic quantum memristors add measurement-induced nonlinearity and tunable temporal memory. The coupled architecture is fully reconfigurable and uses only standard MZI and phase-shifter elements, which the experiment identifies as favorable for scaling to larger networks and for compact quantum reservoir computing and quantum neural-network architectures (Baldazzi et al., 16 Feb 2026).

A distinction is required between photonic quantum memristors of the measurement-induced type and optical memristors based on material non-volatility. For example, PZT optical memristors rely on ferroelectric-domain-controlled refractive index and provide non-volatile optical states, $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$15 loss, $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$16-bit precision, and 48 Gbps volatile modulation on the same platform (Li et al., 2024). Those devices are optical memristors, but they are not the measurement-based single-photon quantum memristors realized in (Baldazzi et al., 16 Feb 2026). This suggests a broader taxonomy in which “optical memristor” and “photonic quantum memristor” are overlapping but not identical categories.

The reported limitations are substantial and define the immediate research frontier. The total insertion loss of $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$17 dB forces long time bins of $y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$18 s to accumulate adequate statistics. Polarization drifts caused by non-PM fiber require recalibration after each alignment. The SiV$y = f(s,x,t)\,x,\qquad \dot{s}=g(s,x,t),$19 source is off-chip and fiber-coupled. Feedback is implemented with external electronics and software, so real-time adaptation is limited by heater bandwidth and classical control. On the quantum side, the dynamics is classical in the sense that single-photon probabilities are updated based on classical measurement records, and entanglement and full quantum coherence across memristors are not explored experimentally (Baldazzi et al., 16 Feb 2026).

The stated routes forward include on-chip single-photon sources, larger networks based on MZI meshes, and multiple spectrally matched color centers, possibly at low temperatures for more demanding multi-photon interference tasks (Baldazzi et al., 16 Feb 2026). A plausible implication is that coupled photonic quantum memristors are best understood not as isolated components but as network primitives whose computational value depends on how memory depth, relative phase, coupling topology, and source integration are co-designed.