Phase transitions, symmetries, and tunneling in Kerr parametric oscillators

TL;DR

This work analyzes driven Kerr parametric oscillators (KPOs) under multi-photon drives with $\mu \in \{1,2,3,4\}$ to map ground-state and excited-state quantum phase transitions (QPTs and ESQPTs) and to understand tunneling dynamics. By deriving effective time-independent Hamiltonians in the rotating frame and exploring their classical limits, the authors identify stationary points, phase boundaries, and DOS features that reveal how discrete $\mathbb{Z}_\mu$ symmetries shape spectral structure via real and avoided crossings. They show that avoided crossings within the same symmetry sector enhance tunneling, while real crossings between sectors can suppress it, offering a route to spectral engineering for dynamical control and critical sensing. The study also highlights the unique unbounded regime as $|\xi_4|\to 1/2$, where mean-field descriptions fail and Husimi analyses indicate beyond-mean-field spectral organization with potential applications in superconducting circuit and photonic platforms for robust quantum state manipulation and sensing.

Abstract

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 $μ$ times the oscillator's natural frequency, specifically for $μ=1,2,3,4$. 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 critical 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.

Figures (9)

• Figure 1: Non-driven KPO. Panel (a) illustrates the energy landscape (top) of the classical Hamiltonian $H_{\delta}^c$ in Eq. (\ref{['Eq:Hdeltac']}) with respect to the lowest energy $E_0$ and the projection of the landscape in the two-dimensional phase space (bottom). Panel (b) represents the quantum DOS. The blue circle indicates the energy of the local maximum, where the DOS presents a discontinuous step.
• Figure 2: One-photon KPO. Panel (a) shows the phase diagram for the detuning parameter $\delta$ and the one-photon drive amplitude $\xi_1$. The QPT is represented by a solid line, while the dotted lines indicate ESQPTs. Panels (b) and (c) illustrate the energy landscape and phase-space structure (top) and the DOS (bottom) for region I and region II, respectively.
• Figure 3: Two-photon KPO. Panel (a) shows the phase diagram for the detuning parameter $\delta$ and the two-photon drive amplitude $\xi_2$. QPTs (ESQPTs) are represented by solid (dotted) lines. Panels (b)-(d) illustrate the energy landscape and phase-space structure (top) and the DOS (bottom) for three regions appearing in the phase diagram: region I (b), II (c), and III (d).
• Figure 4: Three-photon KPO. Panel (a): Phase diagram for the detuning parameter $\delta$ and the three-photon drive amplitude $\xi_3$. Panels (b)-(g): energy landscape and phase-space structure (top), and sketches of the density of states (bottom) for the selected cases I-VI, respectively. A ground-state quantum phase transition is represented by a solid line, while excited-state quantum phase transitions are indicated by dotted lines.
• Figure 5: Four-photon KPO. Panel (a): Phase diagram for the detuning parameter $\delta$ and the four-photon drive amplitude $\xi_4$. Panels (b)-(c): energy landscape and phase-space structure (top), and sketches of the density of states (bottom) for phases I-II, respectively. A ground-state quantum phase transition is represented by a solid line.