Degeneracy beyond the parity-symmetry protection in one-dimensional spinless models: The parity-violating Kerr parametric oscillator

TL;DR

The paper investigates whether degeneracy beyond parity protection can occur in one-dimensional spinless quantum systems when parity is broken. It analyzes a squeeze-driven Kerr oscillator with one- and two-photon drives, studying its quantum spectra, classical limit, and symmetry properties to identify the origin of degeneracy. The main finding is that, in parity-broken cases, doubly-degenerate levels arise due to a time-reversal-like antiunitary symmetry, and the energy gaps decay exponentially as the classical-limit parameter Ne grows, with a semiclassical estimate delta_app approx 2|xi2| explaining the scaling. This demonstrates degeneracy protected by antiunitary symmetry in parity-violating 1D systems, with potential implications for protected qubits in superconducting circuits and for adiabatic quantum computation; future work includes open-system extensions and experimental exploration.

Abstract

One-dimensional quantum systems that undergo spontaneous symmetry-breaking, having a symmetric (non-degenerate) and a broken-symmetry (doubly-degenerate) phase, have been intensely studied in different branches of physics. In most cases, the spontaneously-broken symmetry is parity. However, it is possible to obtain similar phases in systems without parity symmetry, through an antiunitary symmetry that implies a two-fold symmetry either on momentum or coordinate in the system's classical limit. To illustrate this phenomenon, we use a Kerr parametric oscillator (KPO) with one- and two-photon drives that, despite the breaking of parity symmetry, may have doubly-degenerate levels. Different realizations of squeezed KPOs convey a great deal of attention, as effective Hamiltonians for driven superconducting circuits and the occurrence of degeneracy in such systems could be of practical interest in their application to obtain protected qubits in parity-breaking setups. In addition to this, the reported spectral features strongly indicate the existence of additional symmetries in the system.

Figures (4)

• Figure 1: Panels (a) and (b) correspond to the truncated spectra of Hamiltonian \ref{['eq:kerrH']} as a function of the two-photon squeezing amplitude $\xi_2$ with $\xi_1=0$ (parity-conserving case) and $\xi_1=-30/\sqrt{2}$. The inset panels (I-IV) correspond to the energy contours of Hamiltonian \ref{['eq:classH']} for the cases marked by dashed vertical lines with parameters $(\xi_2,\xi_1)$: I (-20,0), II (20,0), III (-20,-30$/\sqrt{2}$), and IV (20,-30$/\sqrt{2}$). Panels (c) and (d) show the expectation value of coordinate and momentum operators, $\hat{Q}$ and $\hat{P}$, for the two lowest energy eigenstates, $\ket{\phi_0}$ and $\ket{\phi_1}$, versus the control parameter $\xi_2$ for $\xi_1 = 0$ in panel (c) and $-30/\sqrt{2}$ in panel (d). Quantum calculations have been carried out with $N_e=1$.
• Figure 2: Panels (a) and (b) correspond to the truncated spectra of Hamiltonian \ref{['eq:kerrH']} as a function of the one-photon squeezing amplitude $\xi_1$ with $\xi_2=-30$ and $\xi_2=30$, respectively. The inset panels (I-IV) contain the energy contours of Hamiltonian \ref{['eq:classH']} for the cases marked by dashed vertical lines in panels (a) and (b), with parameters $(\xi_2,\xi_1)$: I ($-30$, $-30/\sqrt{2}$), II ($-30$, $30/\sqrt{2}$), III ($30$, $-30/\sqrt{2}$), and IV ($30$, $30/\sqrt{2}$). Quantum calculations have been carried out with $N_e=1$.
• Figure 3: Panels (a) and (b) depict the expectation value of $\hat{Q}$ and $\hat{P}$, while panel (c) shows the energy gap of adjacent levels, $\Delta_j = E_{j+1} - E_j$, versus the system energy for $(\xi_2=40,\xi_1=-30/\sqrt{2})$ (blue points) and $(\xi_2=-40,\xi_1=-30/\sqrt{2})$ (orange crosses), and $N_e=1$. The inset in panel (c) is a zoom of the main plot using logarithmic scale. In panel (d), we show the energy gap $\Delta_2$ of different parity-symmetric systems ( $\xi_1 =0$, filled markers) and $\Delta_0$ of different parity-deformed systems ($\xi_1=-30/\sqrt{2}$, empty markers) using log-lin scales. The values of $\xi_2$ are $-30$ (red circles), $-40$ (blue crosses), and $-50$ (green triangles). The dashed lines correspond to the exponential fitting (see the main text).
• Figure 4: Panels (a) and (b) show the expectation value of the coordinate and momentum operators, $\hat{Q}$ and $\hat{P}$, for the two lowest energy eigenstates, $\ket{\phi_0}$ and $\ket{\phi_1}$, versus the control parameter $\xi_1$ for $\xi_2 = -30$ in panel (a) and $30$ in panel (b). Quantum calculations have been carried out with $N_e=1$.