Quantum simulation of the bosonic Kitaev chain

Analog quantum simulation (AQS) enables the study of quantum systems that are classically intractable, naturally representing bosonic degrees of freedom without large overhead. Lattice models with topological properties are especially compelling yet challenging for classical methods due to the sign problem. Recently, non-Hermitian topological physics has drawn significant interest for its distinctive phenomena (e.g., nonorthogonal eigenstates, complex spectra with exceptional points). In quantum settings, accessing non-Hermitian dynamics via post-selected trajectories is difficult due to the rarity of no-jump evolutions. An alternative uses Hermitian bosonic Hamiltonians with unitary squeezing/antisqueezing terms, yielding effectively non-Hermitian equations of motion without dissipation. Building on a prior programmable multimode superconducting parametric cavity platform that realized synthetic lattices and static gauge fields, this work asks whether one can implement the bosonic Kitaev chain (BKC) with both hopping and pairing to realize chiral transport, quadrature-localized wavefunctions, and boundary-condition-sensitive spectra—key signatures associated with non-Hermitian topology—under genuinely quantum conditions.

The study situates itself within AQS advances for bosonic and topological models and the growing field of non-Hermitian physics. Prior works addressed non-Hermitian topology, exceptional points, and the non-Hermitian skin effect (NHSE), as well as alternative realizations in optics and mechanics. Post-selected non-Hermitian quantum dynamics in open systems have been demonstrated but are experimentally demanding. Theoretical proposals showed that quadratic bosonic Hamiltonians with pairing can yield effective non-Hermitian dynamics without dissipation. The authors previously demonstrated synthetic lattices and gauge control in a parametric cavity (bosonic Creutz ladder). The BKC, a bosonic analog of the fermionic Kitaev chain, was introduced theoretically and predicted to support phase-dependent chiral transport and topological winding in the complex spectrum, along with NHSE under open boundaries. This work experimentally realizes and probes these predictions.

Platform and mapping: A multimode superconducting parametric cavity with a SQUID terminator provides a tunable boundary condition. Impedance engineering yields uneven mode spacings across a 4–12 GHz band (fundamental ~400 MHz), enabling selective parametric coupling between chosen mode pairs. Lattice sites are mapped to cavity frequency modes, and synthetic 1D BKCs are programmed via parametric pumps applied to the SQUID. Parametric couplings: Complex hopping and pairing between adjacent sites are engineered using pumps at mode-difference and mode-sum frequencies, respectively, with independently tunable amplitudes and phases controlling the effective couplings. The phases implement local synthetic gauge control. For a 3-site chain, two links realize an open chain; adding a third link closes the chain (periodic boundary conditions). Measurement protocols: Spectra are measured using a vector network analyzer by probing reflection around each site frequency. Phase-sensitive transport is characterized by injecting a coherent tone of fixed magnitude at a site while sweeping its phase from −180° to 180°, and detecting reflected and transported signals at all site frequencies. Setting zero probe detuning makes signal and idler frequencies coincide, producing interference that yields phase-dependent transport. Calibration: Hopping strengths are calibrated via mode splitting (2t greater than decay rates to resolve strong coupling). Pairing strengths are calibrated from phase-dependent transport contrast. Link phases are calibrated by comparing full transport maps to theory, using a gauge-invariant phase θ controlling tube alignment (twisted-tubes picture). Normalization yields reflection/transport coefficients with unit input. Theory and analysis: The target BKC Hamiltonian includes nearest-neighbor complex hopping and pairing. Heisenberg-Langevin equations incorporate uniform single-photon loss κ. A squeezing transformation reveals a topological transition at t|cosφ|=Δ. Open-chain spectrum E=√(t′2−Δ′2)cosk−iκ/2 is independent of φ; periodic spectrum Ep=t sinφ sink + i√(Δ2−t2 cos2φ) cosk − iκ/2 traces ellipses with nonzero winding for t|cosφ|<Δ, indicating boundary-condition sensitivity (NHSE). Input-output theory provides scattering matrices for signal/idler and closed-form transport coefficients for 2- and 3-mode cases, capturing favored/suppressed quadratures and gauge-invariant phase control. The twisted-tubes picture visualizes favored transport quadratures and their phase-controlled alignment/misalignment at shared sites, predicting trivial (phase-insensitive) and chiral (phase-selective) regimes. Closed-chain stability: By tuning link phases in the closed 3-site chain, the determinant of the dynamical matrix can vanish for sufficient pairing, signaling dynamical instability; reflection magnitudes exceeding unity indicate approaching this threshold. Analytical conditions tie instability to gauge-invariant combinations of link phases and to Δ relative to κ and t.

- Experimental realization of a 3-site bosonic Kitaev chain in synthetic frequency dimensions on a multimode superconducting parametric cavity, with independently tunable complex hopping and pairing couplings. - Phase-dependent chiral transport in the open chain: transport magnitudes between ends show strong dependence on the input phase in the chiral regime (θ=0°), with favored quadratures enhanced and orthogonal ones suppressed; phase-insensitive transport in the trivial regime (θ=90°). Transport phase exhibits staircase-like locking in the chiral case, contrasting with near-linear tracking in the trivial case. - Quadrature wavefunction localization (NHSE precursor): extracted x and p quadrature wavefunctions localize at opposite chain ends in the chiral regime, while both are delocalized in the trivial regime; minimal support on the center site due to odd-chain zero-mode structure. - Boundary-condition sensitivity of the spectrum: Open-chain spectra follow Eq. (4) and are φ-independent after appropriate transformations; closed-chain spectra (periodic boundary conditions) trace ellipses with nonzero winding and exhibit braiding as a function of link phases, with observed discontinuities (spectral jumps) at specific phase settings indicating approach to dynamical instability. - Quantitative estimates: From transport enhancement/suppression relative to a trivial baseline, Δ≈0.4 t; from closed-chain spectral fits, Δ≈0.41 κ. Line cuts at higher pairing show reflection magnitudes |S|>1 near alignment (Δ≈0.68 κ), consistent with imminent instability. - Theoretical phase diagram and stability: Topological regime for t|cosφ|<Δ with nonzero spectral winding under periodic boundary conditions; stability condition demonstrates regimes where open chains are stable while periodic chains are unstable, a hallmark of effective non-Hermiticity induced by pairing. - Methodological advance: Demonstration of programmable control of local coupling phases enabling gauge-invariant tuning between trivial and chiral regimes, validated by agreement between experiment and input-output theory.

The central goal was to realize the bosonic Kitaev chain on a quantum platform and probe non-Hermitian topological phenomena without relying on dissipation or post-selection. The experiments confirm the theoretically predicted phase-dependent chirality: alignment of favored quadratures along a chain (θ=0°) yields unidirectional-like transport in complementary quadratures, while misalignment (θ=90°) suppresses phase selectivity. The extracted localization of x and p quadrature wavefunctions at opposite ends under open boundaries, together with the marked difference between open- and closed-chain spectra (including braiding and phase-tunable spectral discontinuities), demonstrates sensitivity to boundary conditions, a defining feature of the non-Hermitian skin effect. These results validate that Hermitian bosonic Hamiltonians with pairing can emulate effective non-Hermitian dynamics in a controllable, quantum-coherent setting. The findings establish the parametric cavity platform as a versatile AQS tool for studying non-Hermitian topology and dynamics, with implications for directional amplification, quantum sensing, and entanglement generation in continuous-variable systems.

This work implements the bosonic Kitaev chain using a multimode superconducting parametric cavity and demonstrates key signatures of effective non-Hermitian topology: phase-dependent chiral transport, quadrature-resolved edge localization, and boundary-condition-sensitive spectra with phase-controlled approach to instability. The platform’s in situ tunable complex couplings and gauge control enable systematic exploration of non-Hermitian effects in genuinely quantum conditions. Future research directions include scaling to longer chains and cavity networks to probe bulk-boundary correspondence and many-body effects, driving deeper into the topological regime to study non-Hermitian nonlinear dynamics and parametric oscillations, leveraging vacuum squeezing for multimode entanglement resources, and realizing phase-programmable directional amplifiers and enhanced-sensitivity sensors.

- System size is limited to a 3-site chain, demonstrating precursors rather than full bulk topological behavior in the thermodynamic limit. - Accurate absolute calibration of pump phases at the sample is challenging; experiments rely on gauge-invariant phase control and fitting to theory for calibration. - Measurements of closed and open chains were performed on separate devices, potentially introducing device-to-device variations. - Near the instability threshold, system nonlinearity becomes significant and deviates from linear input-output theory fits. - The study focuses on single-particle (quadratic) physics; many-body interactions and genuine non-Hermitian many-body phenomena remain to be explored.