Realisation of Protected Cat Qutrit via Engineered Quantum Tunnelling

TL;DR

The paper tackles the challenge of fault-tolerant quantum computation by engineering a three-photon Kerr parametric oscillator (KPO) to realize a protected qutrit. It combines a Kerr nonlinearity with a three-photon pump to create a threefold-symmetric energy landscape whose low-lying eigenstates form a protected qutrit manifold and an adjacent excited space. Through three-photon Rabi oscillations and Wigner-function tomography, the authors demonstrate quantum coherence and three-component cat-like states, while observing breathing-like phase-space dynamics that reveal the presence of an energy gap safeguarding the qutrit. They identify a higher-order pump term, characterized by $\eta$, as the primary factor limiting protection, emphasizing the need to mitigate this term for optimal performance and outlining paths toward universal qutrit control and higher-dimensional protected qudits.

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.

Figures (7)

• Figure 1: Chip and qutrit states.a Photograph of the chip containing the KPO. The state of the KPO is measured by a nearby transmon and its readout resonator. b Circuit diagram of the Kerr parametric oscillator (KPO), which consists of two direct-current superconducting quantum interference devices (DC SQUIDs) and four junctions with a shunting capacitor. c The classical energy landscape corresponding to the Hamiltonian in Eq. \ref{['eq:3PhKPO_cl']}. d,e Wigner function and the occupation probability in the Fock basis of the eigenstates of the Hamiltonian in Eq. \ref{['eq:3PhKPO']}. The states $\ket{0_\textrm{C}}$, $\ket{1_\textrm{C}}$, and $\ket{2_\textrm{C}}$ form the qutrit space; $\ket{0_\textrm{C}^\textrm{ex}}$ is an excited state that is energetically closest to the qutrit space. For (c)--(e), the parameters are $P/K = 0.822$, $\Delta/K = 0.66$, and $\eta = -0.04$.
• Figure 2: Three-photon Rabi oscillations.a Pulse sequence used for measuring Rabi oscillations. The control parameters are shown in magenta, and "$\pi$" denotes the transmon $\pi$-pulses. b,c Rabi oscillations observed in the weak ($P \ll K$) and moderate ($P \sim K$) pump regimes, respectively. The simulation results in (c) clearly show that the contribution of the higher-order pump term ($\eta$ term) must be included to quantitatively understand the Rabi oscillations. Without $\eta$, the period of the higher-order signal (e.g., near $-10$ MHz) does not match well with the experimental data. The colour in both panels represents the population of the KPO's $\ket{0}$ state as detected by the transmon. d Energy level diagram of the KPO. The symbols indicate the correspondence between the transitions and the observed signals.
• Figure 3: Wigner functions of three-photon KPO states.a Pulse sequence for measuring the Wigner function. The pump-pulse conditions are as follows: $P/K = 0.822$, $P_\textrm{CD}/P = 0.3$, where $P_\textrm{CD}$ is the amplitude of the counterdiabatic component of the pump, $\tau_\textrm{ramp} = 0.4$, and $\tau_\textrm{hold} = 0.1$. b Wigner functions of the KPO states for various $\Delta/K$ values. The measured raw data are shown in the top row, the corresponding reconstructed Wigner functions in the middle row, and the Wigner functions of the simulated states in the bottom row. For the simulations, $\eta = -0.04$ was used and $T_1$ was assumed to be 4 . From left to right, the mean photon number and fidelity values are: (reconstructed density matrix, simulated density matrix, fidelity) = (0.58, 0.52, 0.98), (1.22, 1.06, 0.92), (2.20, 1.99, 0.88), and (2.61, 2.60, 0.93).
• Figure 4: Dynamics and relaxation.a Time evolution of the mean photon number, showing oscillations caused by imperfect adiabatic conversion from the $\ket{0}$ state to the $\ket{0_\textrm{C}}$ state. The dashed line is a fit to a simple decaying cosine function, with fitted parameters $f_\text{fit}$ and $\tau_\text{fit}$ shown. b Time evolution of the fidelity for the qutrit states and $\ket{0_\textrm{C}^\textrm{ex}}$. Within the label "Qutrit + excited space," the excited space refers to the subspace spanned by the states $\ket{0_\textrm{C}^\textrm{ex}}$, $\ket{1_\textrm{C}^\textrm{ex}}$, and $\ket{2_\textrm{C}^\textrm{ex}}$. In both (a) and (b), the simulation results (solid lines) account for single-photon loss in the KPO during parity measurement using an effective time $T_{\text{1eff}}=2T_\textrm{1sim}$. The parameters used for the simulation, $\Delta_\text{sim}$ and $T_\text{1sim}$, are provided in panel (a). c Wigner functions of the KPO state at the times indicated by arrows in (a). d The amplitudes of the Fock basis for the $\ket{0_\textrm{C}}$ and $\ket{0_\textrm{C}^\textrm{ex}}$ states. e Superpositions of the states $\ket{0_\textrm{C}}$ and $\ket{0_\textrm{C}^\textrm{ex}}$ with two opposing relative phases. The mean photon number is $0.75$ for the state $\ket{0_\textrm{C}}+0.2\ket{0_\textrm{C}^\textrm{ex}}$ and $2.03$ for $\ket{0_\textrm{C}}-0.2\ket{0_\textrm{C}^\textrm{ex}}$. For (d) and (e), $\Delta/K = 0.40$. f The first five quasienergy levels $E$ calculated using Eq. \ref{['eq:3PhKPO']}. The solid lines indicate the quasienergy levels for the qutrit states: black ($\ket{0_\textrm{C}}$, $E_\textrm{0c}$), red ($\ket{1_\textrm{C}}$), and green ($\ket{2_\textrm{C}}$). The dashed lines show the quasienergy levels for the excited states: black ($\ket{0_\textrm{C}^\textrm{ex}}$) and red ($\ket{1_\textrm{C}^\textrm{ex}}$). The stars represent the oscillation frequencies extracted from the mean photon number data; the black solid star specifically corresponds to the frequency determined from the data presented in (a). Here, the energy levels of the excited states are lower than those of the qutrit states, a consequence of the negative sign of $K$ in Eq. \ref{['eq:3PhKPO']}. For all simulations in this figure, $P/K = 0.822$ and $\eta = -0.04$.
• Figure 5: Steady states.a The mean photon number of the steady states as a function of the detuning, $\Delta/K$. The symbols represent the values extracted from the density matrix reconstructed via Wigner function measurement. The lines are results from the steady-state solution of the Lindblad master equation derived from Eq. \ref{['eq:3PhKPO']}. The effect of single-photon loss during parity measurement is included in the solid line by using a relaxation time of $T_1 = 4$. The pump-pulse conditions are as follows: $P/K = 0.822$, $P_\textrm{CD}/P = 0$, $\tau_\textrm{ramp} = 15$, and $\tau_\textrm{hold} = 0.2$. A long $\tau_\text{ramp}$ was chosen instead of a long $\tau_\text{hold}$ to eliminate the occupation of excited states and, consequently, to avoid the complex dynamics shown in Fig. \ref{['fig:relax']}. b Wigner functions of the steady state at the $\Delta/K$ values indicated by the arrows in (a).