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

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.

💡 Research Summary

The manuscript investigates the emergence of double‑degenerate energy levels in a one‑dimensional, spin‑less quantum system that does not conserve the usual spatial parity symmetry. The authors demonstrate that an anti‑unitary symmetry—specifically time‑reversal (T) or the combined PT operation—can protect a two‑fold degeneracy even when parity is explicitly broken. To illustrate this, they study a Kerr parametric oscillator (KPO) driven simultaneously by one‑photon (linear) and two‑photon (squeezing) fields. The Hamiltonian reads

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