Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2511.09164 · Physical Review A

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

Jamil Khalouf-Rivera, Miguel Carvajal, Francisco Pérez-Bernal

quant-phSubmitted Nov 12, 2025• Announced Nov 13, 2025• 0 citations• Revised Feb 18, 2026 (v2)
v1 · Nov 12, 2025v2 · Feb 18, 2026
arXivHTMLPDF

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.

DOI
10.1103/kmvc-88rh

Related papers

Degeneracy beyond the parity-symmetry protection in the Lipkin-Meshkov-Glick model

Jamil Khalouf-Rivera, Miguel Carvajal, Francisco Pérez-Bernal

Degeneracy patterns in quantum mechanics stem from the system symmetries. In particular, the broken-symmetry phase in the well-known Lipkin-Meshkov-Glick (LMG) model is composed of doubly-degenerate states of different parity. In this work, we show that such doublets can exist even if parity is not conserved. For this purpose, our starting point is an anharmonic LMG Hamiltonian with a second-order ground-state quantum phase transition (GSQPT) and a rich spectrum, with two different excited-state quantum phase transitions. The inclusion in the Hamiltonian of a term inducing a first-order GSQPT breaks the parity symmetry but conserving the exponential degeneracy in the energy doublets. We demonstrate that this phenomenon can be traced back to the existence of a $\mathbb{Z}_2$ symmetry (reflection symmetry) in the system's classical limit phase space that leads to an anti-unitary $\mathbb{Z}_2$ symmetry in the quantum system.

quant-pharXiv:2608.21010•Aug 24, 2026•0 citations

Parity-Time Symmetry Breaking beyond One Dimension: The Role of Degeneracy

Li Ge, A. Douglas Stone

We consider the role of degeneracy in Parity-Time (PT) symmetry breaking for non-hermitian wave equations beyond one dimension. We show that if the spectrum is degenerate in the absence of T-breaking, and T is broken in a generic manner (without preserving other discrete symmetries), then the standard PT-symmetry breaking transition does not occur, meaning that the spectrum is complex even for infinitesimal strength of gain and loss. However the realness of the entire spectrum can be preserved over a finite interval if additional discrete symmetries X are imposed when T is broken, if X decouple all degenerate modes. When this is true only for a subset of the degenerate spectrum, there can be a partial PT transition in which this subset remains real over a finite interval of T-breaking. If the spectrum has odd-degeneracy, a fraction of the degenerate spectrum can remain in the symmetric phase even without imposing additional discrete symmetries, and they are analogous to dark states in atomic physics. These results are illustrated by the example of different T-breaking perturbations of a uniform dielectric disk and sphere, and a group theoretical analysis is given in the disk case. Finally, we show that multimode coupling is capable of restoring the T-symmetric phase at finite T-breaking. We also analyze these questions when the parity operator is replaced by another spatial symmetry operator and find that the behavior can be qualitatively different.

arXiv:1402.0428•Feb 3, 2014•10 citations

Phase transitions, symmetries, and tunneling in Kerr parametric oscillators

Miguel A. Prado Reynoso, Edson M. Signor, Jamil Khalouf-Rivera +3 more

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.

arXiv:2504.15347•Physical Review A•Aug 7, 2025•2 citations

A driven Kerr oscillator with two-fold degeneracies for qubit protection

Jayameenakshi Venkatraman, Rodrigo G. Cortiñas, Nicholas Frattini +2 more

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.

Proceedings of the National Academy of Sciences•Jun 5, 2024•27 citations

Quantum theory of a three-photon Kerr parametric oscillator

Alessandro Bruno, Patrick P. Potts, Alexander Grimm +1 more

We investigate the quantum properties of a nonlinear Kerr oscillator driven by a three-photon pump. We derive both exact and approximate analytical expressions for the ground state of this interacting model. The exact solution arises at an exact spectral degeneracy, while the approximate solution describes regimes of quasi-degeneracy of the energy spectrum. In both cases, the threefold (quasi)degenerate ground-state manifold consists of quantum superpositions of three macroscopically distinct states. These states differ qualitatively from conventional three-component Schrödinger's cat states due to the presence of squeezing with a distinctive parametric dependence. By varying the detuning between the oscillator and the three-photon pump, we show that the squeezing can be enhanced, suppressed, or even reversed, leading to a squeezing-to-anti-squeezing transition. We analyze the generation and stabilization of these superposition states, their robustness against perturbations and analytically quantify the leakage to excited states. Our analysis elucidates how the three-photon Kerr parametric oscillator can be used to encode a Kerr-cat qutrit protected against phase-flip errors.

quant-pharXiv:2605.21036•May 21, 2026•0 citations

Inter-transition interference in spectrum of Kerr parametric oscillators

Shumpei Masuda

Abstract A Kerr parametric oscillator (KPO) is a Kerr-nonlinear resonator driven parametrically, which can serve as a qubit in quantum information processing. Spectroscopy using a weak probe field provides a practical method for extracting system parameters. In this work, we theoretically investigate reflection and transmission measurements of two-photon and four-photon KPOs, introducing interference effects between inter-level transitions. Because KPOs typically exhibit energy-level degeneracy, a probe field can simultaneously resonate with multiple inter-level transitions. We extend the previous theory of reflection measurements by incorporating the interaction between inter-level transitions and off-diagonal elements of the density matrix which had previously been neglected. We demonstrate that interference among these transitions substantially modifies the spectrum. We identify the conditions for the interference, as well as those under which the off-diagonal elements of the density matrix affect the spectrum. The theory is also generalized to transmission measurements, and is applicable to a broad class of systems beyond KPOs.

arXiv:2510.27122•New Journal of Physics•Jan 1, 2026•0 citations
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.
Degeneracy patterns in quantum mechanics stem from the system symmetries. In particular, the broken-symmetry phase in the well-known Lipkin-Meshkov-Glick (LMG) model is composed of doubly-degenerate states of different parity. In this work, we show that such doublets can exist even if parity is not conserved. For this purpose, our starting point is an anharmonic LMG Hamiltonian with a second-order ground-state quantum phase transition (GSQPT) and a rich spectrum, with two different excited-state quantum phase transitions. The inclusion in the Hamiltonian of a term inducing a first-order GSQPT breaks the parity symmetry but conserving the exponential degeneracy in the energy doublets. We demonstrate that this phenomenon can be traced back to the existence of a Z2\mathbb{Z}_2Z2​ symmetry (reflection symmetry) in the system's classical limit phase space that leads to an anti-unitary Z2\mathbb{Z}_2Z2​ symmetry in the quantum system.
We consider the role of degeneracy in Parity-Time (PT) symmetry breaking for non-hermitian wave equations beyond one dimension. We show that if the spectrum is degenerate in the absence of T-breaking, and T is broken in a generic manner (without preserving other discrete symmetries), then the standard PT-symmetry breaking transition does not occur, meaning that the spectrum is complex even for infinitesimal strength of gain and loss. However the realness of the entire spectrum can be preserved over a finite interval if additional discrete symmetries X are imposed when T is broken, if X decouple all degenerate modes. When this is true only for a subset of the degenerate spectrum, there can be a partial PT transition in which this subset remains real over a finite interval of T-breaking. If the spectrum has odd-degeneracy, a fraction of the degenerate spectrum can remain in the symmetric phase even without imposing additional discrete symmetries, and they are analogous to dark states in atomic physics. These results are illustrated by the example of different T-breaking perturbations of a uniform dielectric disk and sphere, and a group theoretical analysis is given in the disk case. Finally, we show that multimode coupling is capable of restoring the T-symmetric phase at finite T-breaking. We also analyze these questions when the parity operator is replaced by another spatial symmetry operator and find that the behavior can be qualitatively different.
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 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.
We investigate the quantum properties of a nonlinear Kerr oscillator driven by a three-photon pump. We derive both exact and approximate analytical expressions for the ground state of this interacting model. The exact solution arises at an exact spectral degeneracy, while the approximate solution describes regimes of quasi-degeneracy of the energy spectrum. In both cases, the threefold (quasi)degenerate ground-state manifold consists of quantum superpositions of three macroscopically distinct states. These states differ qualitatively from conventional three-component Schrödinger's cat states due to the presence of squeezing with a distinctive parametric dependence. By varying the detuning between the oscillator and the three-photon pump, we show that the squeezing can be enhanced, suppressed, or even reversed, leading to a squeezing-to-anti-squeezing transition. We analyze the generation and stabilization of these superposition states, their robustness against perturbations and analytically quantify the leakage to excited states. Our analysis elucidates how the three-photon Kerr parametric oscillator can be used to encode a Kerr-cat qutrit protected against phase-flip errors.
Abstract A Kerr parametric oscillator (KPO) is a Kerr-nonlinear resonator driven parametrically, which can serve as a qubit in quantum information processing. Spectroscopy using a weak probe field provides a practical method for extracting system parameters. In this work, we theoretically investigate reflection and transmission measurements of two-photon and four-photon KPOs, introducing interference effects between inter-level transitions. Because KPOs typically exhibit energy-level degeneracy, a probe field can simultaneously resonate with multiple inter-level transitions. We extend the previous theory of reflection measurements by incorporating the interaction between inter-level transitions and off-diagonal elements of the density matrix which had previously been neglected. We demonstrate that interference among these transitions substantially modifies the spectrum. We identify the conditions for the interference, as well as those under which the off-diagonal elements of the density matrix affect the spectrum. The theory is also generalized to transmission measurements, and is applicable to a broad class of systems beyond KPOs.