Microwave quantum memcapacitor effect

The study addresses how to realize scalable quantum memory devices suitable for neuromorphic quantum computing. Classical memory elements (memristors, memcapacitors, meminductors) exhibit nonlinear input-output relations with memory, typically evidenced by pinched hysteresis loops. While memristors were theoretically introduced by Chua and later realized experimentally, the ideal memristor’s existence remains debated. Extending memory concepts to quantum technologies motivates quantum memdevices that operate with quantum degrees of freedom but need not satisfy classical passivity constraints. Recent theory and experiments have explored quantum memory behavior in superconducting circuits and photonics, aligning with neuromorphic quantum computing strategies and reservoir computing. The paper proposes a practical superconducting circuit that functions as a microwave quantum memcapacitor, detailing how weak-measurement-based feedback via a SQUID imparts memory and how such devices can be coupled into arrays. The research questions include: Can a superconducting circuit with feedback implement a robust memcapacitive response (pinched hysteresis) in the microwave quantum regime across a variety of initial states? Does this behavior persist when devices are coupled, and what quantum correlations emerge during operation?

Prior work on memory circuit elements established memristors, memcapacitors, and meminductors as nonlinear devices with memory, with characteristic pinched hysteresis curves, and extensive developments for neuromorphic computing. Quantum extensions include proposals for quantum memdevices in superconducting circuits and photonics, and an experimental photonic quantum memristor demonstrating quantum memory behavior. Neuromorphic quantum computing leverages such devices to build reservoirs and non-von Neumann architectures, with studies showing nontrivial quantum correlations in coupled quantum memory dynamics that can be advantageous for reservoir computing. Despite robust methods for quantizing standard circuit elements (capacitors, inductors, Josephson junctions, nonreciprocal elements), a general quantization framework for memory devices is lacking, motivating explicit designs and characterizations such as the present work.

Device architecture: Two capacitively coupled LC resonators; resonator 2 is galvanically coupled to a SQUID with tunable effective Josephson energy 2E_J cos(2πΦ_α/Φ_0). An external gate voltage V_g drives the system, and weak measurements on the auxiliary resonator (resonator 1) feed back into the device by updating the external flux Φ_α, which modulates the resonator frequencies and couplings.

Classical model and quantization: Starting from the circuit Lagrangian with node fluxes Φ_i and capacitances/inductances, the authors apply Legendre transformation to obtain a classical Hamiltonian. Using high-plasma-frequency and low-impedance approximations for the SQUID allows elimination of the SQUID degrees of freedom in favor of the resonator variables. The Hamiltonian contains effective, flux-dependent resonator frequencies, drive terms from V_g, and flux-dependent capacitive and inductive couplings. Quantization promotes charges and fluxes to operators with canonical commutation relations and expresses them via bosonic ladder operators to obtain a quantum Hamiltonian of two coupled harmonic oscillators with flux-tunable parameters: Ĥ(Φ_α) = Σ_i[ω_i(Φ_α)a_i†a_i + G_i(Φ_α,V)a_i† + h.c.] + λ*(Φ_α)(a_1†a_2 + a_2†a_1) + additional counter-rotating terms under the adopted approximations.

Feedback protocol: The external magnetic flux is updated based on weak measurements of the photon number in resonator 1 according to Φ_Ω(t_j) = c_1 + c_2 ⟨n̂_1(t_j)⟩^2, where c_1 = 1.84 and c_2 = 0.08 are optimized constants and updates occur on intervals Δt satisfying ωΔt ≪ 1 for near-continuous feedback. The input is V_d(t) = V_d cos(ω t). The output variable is the expectation value of the charge-related operator in resonator 2, ⟨n̂_2(t)⟩.

Simulation setup: The system is simulated in closed dynamics over time windows much shorter than the relaxation time τ_s ≈ 10^3/ω_1, so environmental coupling is neglected. Circuit/system parameters are chosen via constrained optimization (e.g., ω_1 ≈ ω_2 ≈ 5 GHz, ω_s ≈ 50 GHz; example couplings G_12/ω_1 ≈ 5.294×10^-3, G_11/ω_1 ≈ 2.00×10^-4; V_0 = 0.01 μV). The input frequency is varied (e.g., ω ≈ π/5.92 ω_1; a higher-frequency case 2ω) to probe memory signatures (pinched hysteresis vs. linear/oscillatory response).

Initial states: Both non-correlated and correlated quantum states are considered: vacuum; single-photon superpositions in one resonator with phase control; coherent states |α⟩ with α = r e^{iφ}; squeezed states S(ξ)|α⟩ with ξ = R e^{iθ}; Bell-like superpositions cosθ|0,0⟩ + sinθ|1,1⟩; NOON state (|2,0⟩ + |0,2⟩)/√2; and entangled coherent (cat) states (|α,0⟩ + |0,α⟩)/√2 with specified r, φ.

Coupled devices: The architecture is extended to two microwave quantum memcapacitors coupled via a capacitor (with one device inverted to reduce SQUID crosstalk). The coupled Hamiltonian comprises two single-device Hamiltonians plus inter-device coupling terms with effective flux-dependent capacitive and inductive couplings. Parameters (e.g., ω_1 ≈ 5 GHz, ω_2 ≈ 4.97 GHz, Ω_1 ≈ 5.579 GHz, Ω_2 ≈ 5.034 GHz; example γ_ij/ω_1 spanning ~10^-4 to 10^-2) are listed, and the same categories of initial states are tested.

Quantum correlations: For coupled devices, bipartite quantum discord is computed among resonator pairs ρ_ij by minimizing the conditional entropy over projective measurements. The optimization over unitary parameters is performed via basin-hopping (scipy.optimize.basinhopping), with U parameterized by products of two-level rotations each defined by three angles, yielding 3(d-1)d/2 parameters for a d-dimensional space.

Assessment of memory: Memory behavior is identified through pinched hysteresis in the input-output relation between V(t) and the normalized observable ⟨n̂_2(t)⟩ (or ⟨n̂⟩ of the readout resonator). High-frequency limits are examined for convergence to linear or oscillator-like loops, consistent with memcapacitive fingerprint behavior.

Single-device memcapacitive behavior: Across diverse initial states, the system exhibits pinched hysteresis loops at suitably chosen drive frequencies and tends toward linear or oscillator-like loops at higher frequencies, consistent with memcapacitive fingerprints.

• Vacuum initialization |0⟩|0⟩: At ω ≈ π/5.92 ω_1, the V–⟨n̂_2⟩ curve is pinched at the origin; at 2ω the response tends toward a line (quasi-linear behavior).
• Single-photon superposition in resonator 2 (|ψ_n,x⟩ = (|0⟩ + e^{ix}|1⟩)/√2): Phase x controls memory. At ω ≈ π/5.92–5.94 ω_1, x = 0 shows no hysteresis; x = π/2 yields a stable pinched loop. At high frequency, x = 0 yields near-linear response; x = π/2 yields a circle-like loop.
• Coherent states |α⟩ (r = π/4; φ = π/4 or π/8): Both phases produce pinched hysteresis; at high frequency, loops show oscillatory features.
• Squeezed states S(ξ)|α⟩ (examples R = 0.1 or 1, θ = π/4 or π/2): Pinched hysteresis is observed; high-frequency response is elliptical/oscillator-like.

Entangled inputs (single device): Bell-like states (θ = π/4, π/16), NOON (|2,0⟩ + |0,2⟩)/√2, and cat states (|α,0⟩ + |0,α⟩)/√2 with r = π/2 and φ = π/4 or π/2 maintain memory behavior similar to the vacuum/coherent cases. The degree of initial entanglement does not significantly alter the hysteresis signature at the chosen parameters; high-frequency responses trend to lines or circles.

Coupled devices: When two microwave quantum memcapacitors are capacitively coupled, both devices retain memcapacitive behavior for non-correlated inputs (superposition, coherent, squeezed). Often, the device farther from the input exhibits more stable pinched hysteresis curves. At higher frequencies (doubling), responses become elliptical/circular, indicating oscillator-like dynamics. For correlated initializations (Bell, NOON, cat), pinched hysteresis is obtained at ω_r ≈ 0.5 ω_0, while at high frequency the memcapacitive signature diminishes into circular/oscillatory loops.

Quantum correlations: Starting from product states (coherent, squeezed, superposition), quantum discord between resonators emerges dynamically, oscillating between intra-device and inter-device pairs. With entangled initializations (Bell, NOON, cat), discord shows transfer among resonator pairs over time, evidencing redistribution of quantum correlations across the coupled array. For NOON states (n=2), discord can exceed unity due to higher photon number. These behaviors indicate nontrivial interplay between memory dynamics and quantum correlations.

Representative parameters: Example drive amplitude V_0 = 0.01 μV; optimized single-device couplings G_12/ω_1 ≈ 5.294×10^-3, G_11/ω_1 ≈ 2.00×10^-4; drive frequencies near ω ≈ π/5.92 ω_1 and high-frequency at 2ω. Coupled system uses ω_1 ≈ 5 GHz, ω_2 ≈ 4.97 GHz, Ω_1 ≈ 5.579 GHz, Ω_2 ≈ 5.034 GHz, with inter-device couplings γ_ij/ω_1 on the order of 10^-4 to 10^-2. Feedback constants c_1 = 1.84, c_2 = 0.08.

The feedback-controlled, SQUID-tunable superconducting circuit realizes a microwave quantum memcapacitor by rendering the resonator parameters dependent on a dynamically updated external flux tied to weak measurement outcomes. This induces history dependence in the input-output relation, producing pinched hysteresis curves characteristic of memcapacitors, and linear/oscillatory responses at high drive frequencies. The robustness of hysteresis across a wide variety of initial states—classical (vacuum, coherent, squeezed) and quantum-entangled (Bell-like, NOON, cat)—demonstrates that the memory effect does not require specific quantum preparation and can be engineered via parameter tuning. Phase control in superposition states provides an additional handle to gate memory strength. Coupling two devices preserves memcapacitive behavior under appropriate parameters, with the downstream device often exhibiting a more stable loop, suggesting improved stability deeper in arrays. The observed generation and redistribution of quantum correlations (quantum discord) during operation underscores the quantum nature of the device and points to potential advantages in neuromorphic quantum reservoir computing, where temporal correlations and nonlinearity are key computational resources. Overall, the results address the research questions by confirming memcapacitive behavior in a feasible microwave architecture, its persistence in coupled configurations, and the emergence of useful quantum correlations.

The work introduces an experimentally feasible superconducting microwave quantum memcapacitor comprising two coupled resonators with a SQUID-based, weak-measurement feedback loop that modulates device properties via an external flux. Simulations show pinched hysteresis (memcapacitive) behavior for a broad set of initial states and operating frequencies, with high-frequency responses trending to linear/oscillator-like loops. Extending to coupled devices preserves memory properties and enables arrays, while quantum correlations develop and redistribute among resonators, as quantified by quantum discord. The architecture is promising for scalable neuromorphic quantum computing and reservoir computing platforms due to its connectivity, tunability, and ability to sustain quantum correlations. Future directions include experimental realization in superconducting cQED platforms (e.g., coplanar waveguide resonators), incorporation of realistic dissipation and noise, optimization of feedback protocols and measurement strength, scaling to larger arrays, and leveraging the memcapacitive dynamics and quantum correlations for specific reservoir computing and machine learning tasks.

Analyses neglect environmental dissipation and decoherence by restricting to timescales much shorter than relaxation times; behavior under realistic noise and loss remains to be quantified. The feedback is modeled via discrete-time updates of a flux function based on weak measurements, without an explicit Kraus map derivation; obtaining an analytical input-output (Kraus) representation is nontrivial and left open. The device does not realize an ideal memcapacitor (perfect closed loops) but shows memory signatures consistent with Kubo response considerations. Approximations used (high-plasma frequency, low-impedance SQUID regime, linearization of Josephson terms) may limit quantitative accuracy outside their validity. Parameters (e.g., feedback constants, frequencies) are numerically optimized for memory signatures; tolerance to fabrication variations and crosstalk requires experimental validation. Readout backaction and feedback latency constraints are not fully modeled. Results are numerical; no experimental demonstration is presented.