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arXiv:2305.10483 · Physical Review A

Quantum tunneling and level crossings in the squeeze-driven Kerr oscillator

Miguel A. Prado Reynoso, Daniel Julian Nader, Jorge Chávez-Carlos, Belter E. Ordaz-Mendoza, Rodrigo G. Cortiñas, Víctor S. Batista, Sergio Lerma-Hernández, Francisco Pérez-Bernal, Léa Ferreira dos Santos

Published Sep 13, 2023• 19 citations
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Abstract

The quasienergy spectrum recently measured in experiments with a squeeze-driven superconducting Kerr oscillator showed good agreement with the energy spectrum of its corresponding static effective Hamiltonian. The experiments also demonstrated that the dynamics of low-energy states can be explained with the same emergent static effective model. The spectrum exhibits real (avoided) level crossings for specific values of the Hamiltonian parameters, which can then be chosen to suppress (enhance) quantum tunneling. Here we analyze the spectrum and the dynamics of the effective model up to high energies, which should soon be within experimental reach. We show that the parameter values for the crossings, which can be obtained from a semiclassical approach, can also be identified directly from the dynamics. Our analysis of quantum tunneling is done with the effective flux of the Husimi volume of the evolved states between different regions of the phase space. Both initial coherent states and quench dynamics are considered. We argue that the level crossings and their consequences on the dynamics are typical to any quantum system with one degree of freedom, whose density of states presents a local logarithmic divergence and a local step discontinuity.

DOI
10.1103/physreva.108.033709

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The quasienergy spectrum recently measured in experiments with a squeeze-driven superconducting Kerr oscillator showed good agreement with the energy spectrum of its corresponding static effective Hamiltonian. The experiments also demonstrated that the dynamics of low-energy states can be explained with the same emergent static effective model. The spectrum exhibits real (avoided) level crossings for specific values of the Hamiltonian parameters, which can then be chosen to suppress (enhance) quantum tunneling. Here we analyze the spectrum and the dynamics of the effective model up to high energies, which should soon be within experimental reach. We show that the parameter values for the crossings, which can be obtained from a semiclassical approach, can also be identified directly from the dynamics. Our analysis of quantum tunneling is done with the effective flux of the Husimi volume of the evolved states between different regions of the phase space. Both initial coherent states and quench dynamics are considered. We argue that the level crossings and their consequences on the dynamics are typical to any quantum system with one degree of freedom, whose density of states presents a local logarithmic divergence and a local step discontinuity.
Quantum Kerr parametric oscillators (KPOs) are systems out of equilibrium with a wide range of applications in quantum computing, quantum sensing, and fundamental research. They have been realized in superconducting circuits and photonic platforms. In this work, we explore the onset of ground-state and excited-state quantum phase transitions in KPOs, focusing on the role of the phase-space rotational symmetry when the driving frequency is $\ensuremath{\mu}$ times the oscillator's natural frequency, specifically for $\ensuremath{\mu}=1,2,3,4$. These cases are experimentally accessible in superconducting circuits, where the Floquet quasienergy spectrum can also be studied as a function of tunable control parameters. Using the classical Hamiltonian of the system, we identify the stationary points associated with quantum phase transitions and analyze the emergence of both real and avoided level crossings, examining their influence on the energy spectrum and tunneling dynamics. Our findings provide insights into the engineering of robust quantum states, quantum dynamics control, and onset of quantum phase transitions with implications for critical quantum sensing.
We analyze the driven resonantly coupled Jaynes-Cummings model in terms of a quasienergy approach by switching to a frame rotating with the external modulation frequency and by using the dressed atom picture. A quasienergy surface in phase space emerges whose level spacing is governed by a rescaled effective Planck constant. Moreover, the well-known multiphoton transitions can be reinterpreted as resonant tunneling transitions from the local maximum of the quasienergy surface. Most importantly, the driving defines a quasienergy well which is nonperturbative in nature. The quantum-mechanical quasienergy state localized at its bottom is squeezed. In the Purcell-limited regime, the potential well is metastable and the effective local temperature close to its minimum is uniquely determined by the squeezing factor. The activation occurs in this case via dressed spin-flip transitions rather than via quantum activation as in other driven nonlinear quantum systems such as the quantum Duffing oscillator. The local maximum is, in general, stable. However, in presence of resonant coherent or dissipative tunneling transitions the system can escape from it and a stationary state arises as a statistical mixture of quasienergy states being localized in the two basins of attraction. This gives rise to a resonant or an antiresonant nonlinear response of the cavity at multiphoton transitions. The model finds direct application in recent experiments with a driven superconducting circuit quantum electrodynamics setup.
Abstract We study the symmetries of the static effective Hamiltonian of a driven superconducting nonlinear oscillator, the so-called squeeze-driven Kerr Hamiltonian, and discover a remarkable quasi-spin symmetry su(2) at integer values of the ratio η = Δ / K of the detuning parameter Δ to the Kerr coefficient K. We investigate the stability of this newly discovered symmetry to high-order perturbations arising from the static effective expansion of the driven Hamiltonian. Our finding may find applications in the generation and stabilization of states useful for quantum computing. Finally, we discuss other Hamiltonians with similar properties and within reach of current technologies.
We considered the multiphoton resonance in the periodically driven quantum oscillator with Kerr nonlinearity in the presence of weak high-order nonlinearities. Multiphoton resonance leads to the emergence of peaks and dips in the dependence of the stationary occupations of the stable states on detuning. We demonstrated that due to high-order nonlinearities, these peaks and dips acquire additional fine structure and split into several closely spaced ones. Quasiclassically, multiphoton resonance is treated as tunneling between the regions of the oscillator phase portrait, and the fine structure of the multiphoton resonance is a consequence of a special quasienergy dependence of the tunneling rate between different regions of the classical phase portrait. For different values of damping and high-order nonlinearity coefficients, we identified the domain of quasienergies where tunneling strongly influences the system kinetics. The corresponding tunneling term in the Fokker-Planck equation in quasienergy space was derived directly from the quantum master equation.
Coherent tunneling can be suppressed or enhanced by exciting a vibrational mode coupled to the tunneling coordinate. The localization in a two-level system (TLS) driven by a periodic force is obtained as a semiclassical limit for such a mode. We derive and test a semiclassical quantization rule that relates the quasienergies of a classically driven TLS with the energy splittings in the model of a TLS coupled to a quantum oscillator. The dissipative stabilization of a quasilocalized state and the possibility of localization by a quasiperiodic force are analyzed from this perspective.
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.