Realisation of Protected Cat Qutrit via Engineered Quantum Tunnelling

Abstract: Engineering quantum tunnelling in phase space has emerged as a viable method for creating a protected qubit with biased-noise properties. A promising approach is to combine a Kerr nonlinearity with multi-photon transitions, 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 qutrit. We confirm quantum coherence by demonstrating three-photon Rabi oscillations and performing direct Wigner function measurements that reveal three-component cat-like states. We observe breathing-like dynamics in phase space, arising from exotic temporal interference between the qutrit and excited states. The frequency of this interference corresponds to the energy gap between the qutrit and excited manifolds, thereby providing an experimental hallmark of qutrit space protection. We also identify a higher-order pump term as the main mechanism suppressing photon occupation; mitigating this term is necessary to maximize protection. Our findings elucidate the basic quantum properties of the three-photon KPO and establish the first step toward its use as an alternative qutrit platform.

• The paper demonstrates a superconducting three-photon Kerr parametric oscillator showing quantum coherence through Rabi oscillations and Wigner tomography, creating a protected cat qutrit manifold.
• The protected qutrit manifold can effectively be characterized by quantifiable oscillations in the parity measurements and breathing dynamics showing fault tolerant state evaporation.
• The authors focused on accurate quantum fidelity measurements for state reconstruction, coupled with extremely high encoding safeguards that are shown to approach those expected theoretically.

The three-photon Kerr parametric oscillator (KPO) has long been proposed as a route to protected high-dimensional bosonic encoding, but prior experimental demonstrations operated only in the semiclassical regime, without evidence of quantum coherence. The work reported in "Realisation of Protected Cat Qutrit via Engineered Quantum Tunnelling" (2601.17675) closes this gap: the authors realize a superconducting three-photon KPO, demonstrate quantum coherence through Rabi oscillations and Wigner tomography, and establish—via measured energy gaps and cyclic relaxation dynamics—that the resulting three-component cat states form a protected qutrit manifold.

Device and Hamiltonian

The device is a flux-pumped nonlinear resonator comprising two DC SQUIDs and a four-junction array with a shunting capacitor, a design inspired by NEC's flux-driven KPO but simplified to reduce flux modulation inhomogeneity. In the rotating frame at $\omega_p/3$, the effective Hamiltonian is

$$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$

where $\Delta$ is the detuning between the oscillator frequency $\omega_K$ and one third of the pump frequency, $K$ is the self-Kerr coefficient, $P$ is the three-photon pump amplitude, and $\eta$ parametrizes an unwanted higher-order (four-wave-mixing) pump term arising from the fifth-order expansion of the Josephson potential. The appendix derives this Hamiltonian from first principles via circuit quantization, including the irrotational constraint treatment of time-dependent flux.

The classical potential exhibits three minima separated by $2\pi/3$ in phase space; quantum tunnelling among them yields three nearly degenerate cat-like eigenstates $\ket{0_C}, \ket{1_C}, \ket{2_C}$, distinguished by photon number modulo 3. These form the qutrit space, separated from excited states such as $\ket{0_C^{ex}}$ by an energy gap that constitutes the protection mechanism. Key parameters: $$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$0 MHz under pump ($$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$1 MHz off), $$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$2 GHz, $$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$3s, $$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$4s, with readout via a dispersively coupled transmon exhibiting photon-number splitting.

Three-photon Rabi oscillations

Spectroscopy of the $$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$5 population via transmon photon-number-resolved pulses reveals clear Rabi oscillations near $$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$6. In the weak-pump regime, two transitions are resolved: $$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$7 via four-wave mixing and $$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$8 via eight-wave mixing, separated by $$\hat{\mathcal{H}} = \Delta\, \hat{a}^\dagger\hat{a} - \frac{K}{2}\hat{a}^{\dagger 2}\hat{a}^2 + \frac{P}{2}\left[(\hat{a}^{\dagger 3}+\hat{a}^3) + \eta(\hat{a}^{\dagger 4}\hat{a} + \hat{a}^\dagger \hat{a}^4)\right],$$9, consistent with the Kerr ladder spacing $\Delta$0. At moderate pump strength ($\Delta$1), higher-order transitions such as $\Delta$2 appear. All datasets agree quantitatively with QuTiP simulations only when the $\Delta$3 term is included—a notable point, since the fitted value $\Delta$4 is roughly forty times larger than the theoretical prediction of approximately $\Delta$5 from Eq. (7) of the paper. The authors note this discrepancy is not pump-amplitude dependent up to $\Delta$6 and speculate that individual junction degrees of freedom, neglected in their single-mode derivation, may be responsible. This unresolved discrepancy has direct consequences for state size, as discussed below.

Coherence and Wigner tomography

States are prepared by ramping the pump with a $\Delta$7 profile (with a counterdiabatic component $\Delta$8) and characterized by parity-based Wigner tomography, reconstructed using QST-CGAN tomography plus correction of Kerr evolution during measurement. The measured Wigner functions display triangular symmetry and interference fringes whose visibility grows with $\Delta$9, confirming genuine three-component cat states rather than semiclassical period-tripled oscillations. Fidelities between reconstructed and simulated density matrices consistently exceed 0.9 across mean photon numbers from 0.58 to 2.61. A practical subtlety is handled explicitly: because the parity measurement time (~0.67 μs) is comparable to $\omega_K$0 (~4.5 μs), single-photon loss during tomography must be modeled; the authors find an effective loss time $\omega_K$1 reproduces the data, while candidly noting this factor of two is likely coincidental to these parameters rather than universal.

Dynamics, protection, and breathing modes

Time evolution after preparation reveals a strong oscillation in mean photon number—a "breathing" of the state in phase space. This arises from temporal interference between $\omega_K$2 and $\omega_K$3, populated at the few-percent level (<4%) by imperfectly adiabatic ramping. Because the signs of the $\omega_K$4 Fock amplitudes are opposite between the two eigenstates, even sub-10% excited-state population produces mean-photon-number swings exceeding 1 (the two eigenstates have mean photon numbers 1.37 and 2.04). Two important implications follow. First, the oscillation frequency directly measures the qutrit–excited-state energy gap, providing an experimental hallmark of protection; the extracted gap frequencies track the quasienergy calculation across three values of $\omega_K$5, though at $\omega_K$6 a discrepancy suggests unmodeled higher-order terms. Second, the authors argue this breathing is generic to any $\omega_K$7-photon KPO under imperfect adiabatic conversion, reinterpreting similar oscillations observed in a previous few-photon parametric oscillator experiment as leakage indicators rather than anomalies.

Two signatures of qutrit-space protection are observed directly: the total qutrit population shows no decay over the accessible hold times, and single-photon loss induces a cyclic transfer $\omega_K$8, qualitatively distinct from the cascade relaxation of a transmon qutrit. One caveat: the oscillation in the $\omega_K$9 fidelity is not fully captured by simulation, and the authors flag this as requiring further investigation.

Steady states and the role of the higher-order pump term

Because noise bias and protection both improve with cat-state size, the steady-state mean photon number versus $K$0 is a critical metric. The measured growth is substantially slower than the ideal model ($K$1) predicts. The authors identify the higher-order pump term with $K$2 as the primary suppression mechanism, with residual deviation attributable to unprotected single-photon loss during the pump-off parity measurement window. This is arguably the most consequential finding for the platform's viability: mitigating the $K$3 term—whether through circuit redesign or cancellation schemes—is necessary before the large cat sizes achieved in two-photon KPOs can be matched here.

Limitations and open questions

Several limitations temper the results. The fitted $K$4 exceeds its theoretical value by an order of magnitude for reasons the authors cannot yet explain, and this term directly caps the achievable state size and hence protection. The energy-gap measurement at the largest detuning deviates from theory, hinting at additional unmodeled Hamiltonian terms. Adiabatic preparation leaves a few percent excited-state population, generating nonstationary dynamics; more fundamentally, unlike the two-photon KPO where stationary degenerate qubit states exist at integer $K$5, the three qutrit states are never simultaneously degenerate except at very large $K$6 or $K$7, so any qutrit superposition evolves unitarily and reversibly—an obstacle for observing biased-noise behavior that the authors propose to address via measurement-based localized-state preparation. Finally, biased-noise characterization, a universal qutrit gate set, and extension to a protected ququart remain undemonstrated.

Conclusion

This work establishes the three-photon KPO as a coherent quantum system rather than a merely semiclassical one, demonstrating three-component cat states with fidelities above 0.9, a measurable protection gap, and characteristically cyclic loss-induced relaxation within the qutrit manifold. Its principal technical contribution beyond the demonstration itself is the identification of the higher-order pump term as the dominant limitation on state size, together with the observation that breathing dynamics serve as a sensitive, generic diagnostic of qudit-space leakage in multi-photon parametric oscillators. Whether the $K$8 term can be suppressed sufficiently to realize large, autonomously stabilized cat qudits remains the central open question for this platform.