Observation of Pairwise Level Degeneracies and the Quantum Regime of the Arrhenius Law in a Double-Well Parametric Oscillator
Nicholas Frattini, Rodrigo G. Cortiñas, Jayameenakshi Venkatraman, Xu Xiao, Qile David Su, Chan U Lei, Benjamin J. Chapman, Vidul R. Joshi, S. M. Girvin, Robert J. Schoelkopf, Shruti Puri, Michel Devoret
By applying a microwave drive to a specially designed Josephson circuit, we have realized a double-well model system: a Kerr oscillator submitted to a squeezing force. We have observed, for the first time, the spectroscopic fingerprint of a quantum double-well Hamiltonian when its barrier height is increased: a pairwise level kissing (coalescence) corresponding to the exponential reduction of tunnel splitting in the excited states as they sink under the barrier. The discrete levels in the wells also manifest themselves in the activation time across the barrier which, instead of increasing smoothly as a function of the barrier height, presents steps each time a pair of excited states is captured by the wells. This experiment illustrates the quantum regime of Arrhenius’s law, whose observation is made possible here by the unprecedented combination of low dissipation, time-resolved state control, 98.5% quantum nondemolition single shot measurement fidelity, and complete microwave control over all Hamiltonian parameters in the quantum regime. Direct applications to quantum computation and simulation are discussed. Published by the American Physical Society 2024
By applying a microwave drive to a specially designed Josephson circuit, we have realized a double-well model system: a Kerr oscillator submitted to a squeezing force. We have observed, for the first time, the spectroscopic fingerprint of a quantum double-well Hamiltonian when its barrier height is increased: a pairwise level kissing (coalescence) corresponding to the exponential reduction of tunnel splitting in the excited states as they sink under the barrier. The discrete levels in the wells also manifest themselves in the activation time across the barrier which, instead of increasing smoothly as a function of the barrier height, presents steps each time a pair of excited states is captured by the wells. This experiment illustrates the quantum regime of Arrhenius’s law, whose observation is made possible here by the unprecedented combination of low dissipation, time-resolved state control, 98.5% quantum nondemolition single shot measurement fidelity, and complete microwave control over all Hamiltonian parameters in the quantum regime. Direct applications to quantum computation and simulation are discussed. Published by the American Physical Society 2024
By applying a microwave drive to a specially designed Josephson circuit, we have realized an elementary quantum optics model, the squeezed Kerr oscillator. This model displays, as the squeezing amplitude is increased, a cross-over from a single ground state regime to a doubly-degenerate ground state regime. In the latter case, the ground state manifold is spanned by Schrödinger-cat states, i.e. quantum superpositions of coherent states with opposite phases. For the first time, having resolved up to the tenth excited state in a spectroscopic experiment, we confirm that the proposed emergent static effective Hamiltonian correctly describes the system, despite its driven character. We also find that the lifetime of the coherent state components of the cat states increases in steps as a function of the squeezing amplitude. We interpret the staircase pattern as resulting from pairwise level kissing in the excited state spectrum. Considering the Kerr-cat qubit encoded in this ground state manifold, we achieve for the first time quantum nondemolition readout fidelities greater than 99%, and enhancement of the phase-flip lifetime by more than two orders of magnitude, while retaining universal quantum control. Our experiment illustrates the crucial role of parametric drive Hamiltonian engineering for hardware-efficient quantum computation.
The spontaneous switching of a quantum particle between the wells of a double-well potential is a phenomenon of general interest to physics and chemistry. It was broadly believed that the switching rate decreases steadily as a function of the size of the energy barrier. This view was challenged by a recent experiment on a driven superconducting Kerr nonlinear oscillator (often called the Kerr-cat qubit or the Kerr parametric oscillator), whose energy barrier can be continuously increased by ramping up the drive. Remarkably, as the drive amplitude increases, the switching rate exhibits a steplike decrease termed the ``staircase.'' The view challenged by the experimental staircase demands a deep review of our understanding of the role of quantum effects in double wells. In this work, we use a Lindbladian model of dissipation to derive a semianalytical formula for the switching rate, resolving a continuous transition between tunneling-dominated dynamics and dissipation-dominated dynamics. These two dynamics are observed respectively in the flat part and the steep part of each step in the staircase. Our formula exposes two distinct dissipative processes that limit tunneling: the dephasing of interwell superpositions freezes tunneling via the quantum Zeno effect, and the decay from one excited state to another limits the time during which tunneling can be attempted. This physical understanding allows us to pinpoint the critical drive amplitude that separates the flat and steep part of each step using an equation involving the rates of the two dissipative processes and the tunnel splitting. In addition, analyzing the transition matrix elements in the formula shows that in the regime of a few ($\ensuremath{\lesssim}10$) states in the well and under moderate to low temperatures, highly excited states are populated predominantly via cascaded and direct thermal heating rather than quantum heating. At very low temperatures, we find that the perturbation induced by the non-Hermitian Hamiltonian part of the Lindbladian model becomes increasingly important and facilitates a form of quantum heating that has not been identified before. We numerically map the activation mechanism as a function of drive amplitude, damping rate, and temperature. Our theory deepens the understanding of switching dynamics between metastable quantum states, highlights the importance of a general interplay between tunneling and dissipation, and identifies a novel quantum regime in activated transitions.
Transmon qubits are the predominant element in circuit-based quantum information processing, such as existing quantum computers, due to their controllability and ease of engineering implementation. But more than qubits, transmons are multilevel nonlinear oscillators that can be used to investigate fundamental physics questions. Here, they are explored as simulators of excited state quantum phase transitions (ESQPTs), which are generalizations of quantum phase transitions to excited states. We show that the spectral kissing (coalescence of pairs of energy levels) experimentally observed in the effective Hamiltonian of a driven SNAIL-transmon is an ESQPT precursor. We explore the dynamical consequences of the ESQPT, which include the exponential growth of out-of-time-ordered correlators, followed by periodic revivals, and the slow evolution of the survival probability due to localization. These signatures of ESQPT are within reach for current superconducting circuits platforms and are of interest to experiments with cold atoms and ion traps.
Multi-tone external driving offers a route to parametric physics without directly modulating the device. However, the validity of the parametric response in the few-photon regime remains underexplored. Here, we apply two coherent microwave tones to a Josephson-junction Kerr oscillator and stimulate degenerate parametric downconversion via four-wave mixing. Using transmission spectroscopy, we observe that the response retains the qualitative semiclassical Kerr parametric oscillator structure, including its instability lobe and bistable phase-space topology. Interestingly, we demonstrate that a conventional single-mode reduction fails to capture the system quantitatively: the predicted AC Stark shift is severely underestimated, and the reported distributions might not be fully physical when the single-photon Kerr shift K exceeds the cavity linewidth κ. Instead, we show that a full three-tone quantum description accurately reproduces the experimental observables. There, quantum fluctuations of the drive tones become dynamically dominant over dissipation, and all three interacting tones operate in a deep few-photon limit where the expected semiclassical macroscopic lobes undergo fundamental renormalization due to profound mixing with quantum variance. Our results establish two-tone-driven Kerr oscillators as potential parametric amplifiers and open new horizons to explore the quantum-to-classical crossover in driven-dissipative circuits.
Multi-tone external driving offers a route to parametric physics without directly modulating the device. However, the validity of the parametric response in the few-photon regime remains underexplored. Here, we apply two coherent microwave tones to a Josephson-junction Kerr oscillator and stimulate degenerate parametric downconversion via four-wave mixing. Using transmission spectroscopy, we observe that the response retains the qualitative semiclassical Kerr parametric oscillator structure, including its instability lobe and bistable phase-space topology. Interestingly, we demonstrate that a conventional single-mode reduction fails to capture the system quantitatively: the predicted AC Stark shift is severely underestimated, and the reported distributions might not be fully physical when the single-photon Kerr shift K exceeds the cavity linewidth κ. Instead, we show that a full three-tone quantum description accurately reproduces the experimental observables. There, quantum fluctuations of the drive tones become dynamically dominant over dissipation, and all three interacting tones operate in a deep few-photon limit where the expected semiclassical macroscopic lobes undergo fundamental renormalization due to profound mixing with quantum variance. Our results establish two-tone-driven Kerr oscillators as potential parametric amplifiers and open new horizons to explore the quantum-to-classical crossover in driven-dissipative circuits.
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.