Engineering quantum tunnelling in phase space has emerged as a viable method for creating a protected logical qubit manifold with biased-noise properties. A promising approach is to combine a Kerr nonlinearity with a multi-photon drive, resulting in a system known as a Kerr parametric oscillator (KPO). In this work, we implement a three-photon KPO and explore its potential as a protected bosonic qutrit. We confirm quantum coherence by demonstrating three-photon Rabi oscillations and performing direct Wigner function measurements that reveal the formation of three-component cat-like states. Crucially, we observe a breathing-like dynamic in phase space, a characteristic feature of driven quantum systems. This dynamic arises from macroscopic temporal interference between the cat-qutrit manifold and the excited states. The frequency of resulting oscillations in the mean photon number provides a direct, time-domain measurement of the energy gap separating the qutrit from the excited states, thereby establishing an experimental hallmark of qutrit manifold protection. Furthermore, we identify a parasitic higher-order pump term as the primary mechanism constraining the mean photon number, highlighting its mitigation as a requisite for maximising protection. Our findings elucidate the basic quantum properties of the three-photon KPO and establish the first step towards its use as an alternative qutrit platform.
Engineering quantum tunnelling in phase space has emerged as a viable method for creating a protected logical qubit manifold with biased-noise properties. A promising approach is to combine a Kerr nonlinearity with a multi-photon drive, resulting in a system known as a Kerr parametric oscillator (KPO). In this work, we implement a three-photon KPO and explore its potential as a protected bosonic qutrit. We confirm quantum coherence by demonstrating three-photon Rabi oscillations and performing direct Wigner function measurements that reveal the formation of three-component cat-like states. Crucially, we observe a breathing-like dynamic in phase space, a characteristic feature of driven quantum systems. This dynamic arises from macroscopic temporal interference between the cat-qutrit manifold and the excited states. The frequency of resulting oscillations in the mean photon number provides a direct, time-domain measurement of the energy gap separating the qutrit from the excited states, thereby establishing an experimental hallmark of qutrit manifold protection. Furthermore, we identify a parasitic higher-order pump term as the primary mechanism constraining the mean photon number, highlighting its mitigation as a requisite for maximising protection. Our findings elucidate the basic quantum properties of the three-photon KPO and establish the first step towards its use as an alternative qutrit platform.
We investigate the quantum properties of a nonlinear Kerr oscillator driven by a three-photon pump. We derive both exact and approximate analytical expressions for the ground state of this interacting model. The exact solution arises at an exact spectral degeneracy, while the approximate solution describes regimes of quasi-degeneracy of the energy spectrum. In both cases, the threefold (quasi)degenerate ground-state manifold consists of quantum superpositions of three macroscopically distinct states. These states differ qualitatively from conventional three-component Schrödinger's cat states due to the presence of squeezing with a distinctive parametric dependence. By varying the detuning between the oscillator and the three-photon pump, we show that the squeezing can be enhanced, suppressed, or even reversed, leading to a squeezing-to-anti-squeezing transition. We analyze the generation and stabilization of these superposition states, their robustness against perturbations and analytically quantify the leakage to excited states. Our analysis elucidates how the three-photon Kerr parametric oscillator can be used to encode a Kerr-cat qutrit protected against phase-flip errors.
We present the experimental finding of multiple simultaneous two-fold degeneracies in the spectrum of a Kerr oscillator subjected to a squeezing drive. This squeezing drive resulting from a three-wave mixing process, in combination with the Kerr interaction, creates an effective static two-well potential in the phase space rotating at half the frequency of the sinusoidal drive generating the squeezing. Remarkably, these degeneracies can be turned on-and-off on demand, as well as their number by simply adjusting the frequency of the squeezing drive. We find that when the detuning Δ between the frequency of the oscillator and the second subharmonic of the drive equals an even multiple of the Kerr coefficient K , Δ / K = 2 m , the oscillator displays m + 1 exact, parity-protected, spectral degeneracies, insensitive to the drive amplitude. These degeneracies can be explained by the unusual destructive interference of tunnel paths in the classically forbidden region of the double well static effective potential that models our experiment. Exploiting this interference, we measure a peaked enhancement of the incoherent well-switching lifetime, thus creating a protected cat qubit in the ground state manifold of our oscillator. Our results illustrate the relationship between degeneracies and noise protection in a driven quantum system.
Quantum computing crucially relies on maintaining quantum coherence for the duration of a calculation. Bosonic quantum error correction protects this coherence by encoding qubits into superpositions of noise-resilient oscillator states. In the case of the Kerr-cat qubit (KCQ), these states derive their stability from being the quasi-degenerate ground states of an engineered Hamiltonian in a driven nonlinear oscillator. KCQs are experimentally compatible with on-chip architectures and high-fidelity operations, making them promising candidates for a scalable bosonic quantum processor. However, their bit-flip time must increase further to fully leverage these advantages. Here, we present direct evidence that the bit-flip time in a KCQ is limited by leakage out of the qubit manifold and experimentally mitigate this process. We coherently control the leakage population and measure it to be > 9%, twelve times higher than in the undriven system. We then cool this population back into the KCQ manifold with engineered dissipation, identify conditions under which this suppresses bit-flips, and demonstrate increased bit-flip times up to 3.6 milliseconds. By employing both Hamiltonian confinement and engineered dissipation, our experiment combines two paradigms for Schrödinger-cat qubit stabilization. Our results elucidate the interplay between these stabilization processes and indicate a path towards fully realizing the potential of these qubits for quantum error correction.
Driven-dissipative Kerr cavities serve as a versatile platform for generating nonclassical states, such as quantum cat states and high-photon-number Fock states, which are essential for quantum information processing. Building on the quantized Lugiato-Lefever equation (QLLE), which predicts steady-state coherent state superpositions in multi-mode Kerr resonators, and the framework for engineering Kerr nonlinearity ratios to mitigate spectral crowding in coupled oscillators, we propose an integrated approach for precise quantum control. By engineering the Kerr ratio K1/K2 to approximate an incommensurate value using complex rational approximations, we eliminate parasitic degeneracies and enable selective addressing of transitions in the QLLE steady state. Employing a Magnus expansion, we derive an effective Hamiltonian that incorporates Stark-shift corrections, facilitating deterministic synthesis of entangled cat states and Fock states with fidelities exceeding 99.9%. Numerical simulations using QuTiP validate robustness against dissipation and thermal noise, demonstrating quantum hysteresis and Wigner negativity in multi-mode configurations. This synthesis bridges dissipative phase transitions with architectural control, offering a blueprint for scalable bosonic quantum processors in circuit QED. The Wigner function plays a pivotal role in characterizing these nonclassical states, providing a phase-space representation that reveals quantum interference effects. In our framework, the emergence of negative values in the Wigner function for the central mode confirms the non-Gaussian nature of the cat states, as dictated by Hudson’s theorem, which states that only Gaussian pure states have non-negative Wigner functions everywhere [21]. This negativity arises from the interference term in the cat-state Wigner expression, quantifying macroscopic quantum coherence and serving as a witness for entanglement in multi-mode systems.For instance, our simulations at F = 2.0 yield a minimum Wigner value of approximately -0.006, highlighting the system’s departure from classical behavior and its utility for bosonic error correction, where cat codes leverage phase-space separation to suppress bit-flip errors exponentially with cat size. Furthermore, the relation to Wigner functions extends to dynamical properties: the steady-state Wigner negativity is robust against thermal noise due to the dissipative stabilization mechanism, aligning with results from driven Bose-Hubbard models where two-mode cat states maintain negativity under loss [46]. In the QLLE context, the multi-mode extension introduces spatial localization, where the Wigner function’s negative regions correspond to soliton-like structures, as evidenced by photon population distributions [0.33, 0.90, 0.33]. This negativity is not merely a feature but a consequence of fundamental limits on classical simulability, relating to the Gottesman-Knill theorem’s extension to continuous variables, where non-Gaussian operations are necessary for universal quantum computation. Quantum hysteresis, observed in our hysteresis plots, is intimately linked to the Wigner representation through the bifurcation dynamics. Near the codimension-2 point, the system’s bistability manifests as distinct paths in phase space, with Wigner functions switching between Gaussian-like (low-photon) and cat-like (high-photon) forms, supported by spectral theory of Liouvillians that guarantees metastable states in dissipative phase transitions [30]. The forward and backward sweeps in our simulations demonstrate this path dependence, with the Wigner negativity peaking in the high-drive regime, underscoring the role of Kerr nonlinearity in engineering quantum resources. Overall, the Wigner function’s negativity serves as a diagnostic tool for the efficacy of our engineered ratios, directly tying to experimental verifiability in circuit QED setups, where homodyne measurements can reconstruct Wigner distributions to confirm cat-state fidelity. This work not only advances control protocols but also highlights the profound interplay between phase-space quasi-probabilities and dissipative quantum engineering, paving the way for fault-tolerant bosonic qubits.
We present a systematic study of Schrödinger cat codes constructed from Kerr-type coherent states, including displaced Kerr coherent states and Barut--Girardello Kerr coherent states, each admitting two distinct families determined by the sign of the Kerr nonlinearity. By tuning the Kerr parameter and coherent-state amplitude, these states interpolate between su(2), su(1,1) coherent states, providing a unified and versatile foundation for this type of bosonic quantum error correction. Unlike standard two-component Schrödinger cat codes, where a single photon-loss event induces an uncorrectable bit-flip, the nonlinear phase-space structure of Kerr cat states enables simultaneous detection and correction of both photon-loss and dephasing errors within a unified recovery framework, with optimal recovery operations determined via convex optimization. We demonstrate that Kerr cat encodings significantly outperform conventional cat codes under combined loss and dephasing noise, and that judicious parameter optimization can suppress both error channels to a level that reduces the overhead of additional error correction layers. We further show that Kerr-deformed coherent-state manifolds under engineered two-photon driving emerge as effective steady states of driven-dissipative dynamics, with single-photon decoherence strongly suppressed and leakage outside the protected manifold appearing only as higher-order corrections in the deformation strength. Our extended formalism identifies generalized Kerr Schrödinger cat codes as promising candidates for fault-tolerant bosonic quantum computation in experimental platforms such as nonlinear photonics.
The rich dynamics and large Hilbert space of quantum harmonic oscillators make them natural candidates for hardware-efficient and error-correctable quantum information processing. However, implementing direct entangling operations between oscillators remains an outstanding challenge. Existing strategies typically rely on parametrically activating interactions that populate the excited states of a nonlinear element, which introduces additional dissipation channels and potential leakage from the encoded manifold. Here, we engineer a Raman-assisted cross-Kerr interaction between microwave photons hosted in two superconducting cavities. Crucially, this dynamics does not excite the mediating nonlinear coupler, thereby suppressing coupler induced decoherence and leakage out of the bosonic code space. We use this direct nonlinear coupling to implement a controlled-phase gate within the single- and two-photon subspaces of two oscillators, deterministically generating entanglement between them. Finally, we use these engineered dynamics to implement a photon-number parity check on a storage cavity via purely bosonic interactions with an ancillary cavity, demonstrating an enhancement in the storage lifetime. Our work provides a promising pathway toward engineering robust operations that act entirely within a protected bosonic code space and realizing fault-tolerant quantum information processing with bosonic elements.