arxiv: 2511.09164 · v2 · submitted 2025-11-12 · 🪐 quant-ph

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

Jamil Khalouf-Rivera , Miguel Carvajal , Francisco P\'erez-Bernal This is my paper

Pith reviewed 2026-05-17 22:46 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Kerr parametric oscillatordegeneracyantiunitary symmetryparity violationsymmetry breakingdriven oscillatorsprotected qubitsquantum spectra

0 comments p. Extension

The pith

A Kerr parametric oscillator breaks parity symmetry yet can still host doubly degenerate levels via an antiunitary symmetry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that spontaneous symmetry breaking with doubly degenerate states does not require parity symmetry. Instead, an antiunitary symmetry can produce the same effect by enforcing a two-fold symmetry on momentum or coordinate in the classical limit. The authors illustrate this with a Kerr parametric oscillator driven by both one- and two-photon terms, a model relevant to superconducting circuits. A reader would care because the resulting degeneracy may allow protected qubits even in setups where parity is deliberately broken. The observed spectral features also indicate that further symmetries may be present in the system.

Core claim

In the Kerr parametric oscillator with simultaneous one- and two-photon drives, an antiunitary symmetry survives the explicit breaking of parity and enforces exact two-fold degeneracy of energy levels. This symmetry corresponds to a two-fold symmetry either on momentum or on coordinate in the classical limit of the driven system, thereby permitting phases with spontaneous symmetry breaking that resemble those usually protected by parity.

What carries the argument

The antiunitary symmetry of the parity-violating driven KPO, which maps the Hamiltonian to itself and produces a two-fold symmetry in the classical phase space.

If this is right

Doubly degenerate phases can appear in one-dimensional spinless models without parity symmetry.
Degeneracy in driven KPOs offers a route to protected qubits in parity-breaking superconducting-circuit realizations.
Spectral features point to the existence of additional symmetries beyond the identified antiunitary one.
The same mechanism may operate in other driven-oscillator Hamiltonians that lack parity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Antiunitary symmetries of this type could be identified and exploited in other driven quantum models to protect degeneracy.
The classical two-fold symmetry might serve as a diagnostic tool for locating degeneracies in related Hamiltonians.
Extensions to multi-mode or interacting versions of the KPO could test how robust the degeneracy remains under added complexity.
The approach may generalize to higher-dimensional systems where discrete symmetries other than parity are broken.

Load-bearing premise

The antiunitary symmetry identified in the classical limit continues to enforce exact two-fold degeneracy across the entire quantum spectrum without extra fine-tuning.

What would settle it

Numerical diagonalization of the Hamiltonian in a truncated oscillator basis that yields unpaired energy levels at generic drive amplitudes and detunings would falsify the degeneracy claim.

Figures

Figures reproduced from arXiv: 2511.09164 by Francisco P\'erez-Bernal, Jamil Khalouf-Rivera, Miguel Carvajal.

**Figure 1.** Figure 1: FIG. 1. Panels (a) and (b) correspond to the truncated spectra of Hamiltonian (1) as a function of the two-photon squeezing [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Panels (a) and (b) correspond to the truncated spec [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Panels (a) and (b) depict the expectation value of [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Panels (a) and (b) show the expectation value of the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper identifies an antiunitary symmetry that can protect twofold degeneracy in a parity-violating driven KPO, but the quantum exactness of that protection rests on limited evidence.

read the letter

The main takeaway is that this work extends symmetry-protected degeneracy beyond parity by invoking an antiunitary operator whose classical consequence is a twofold symmetry in momentum or coordinate. They illustrate it with a Kerr parametric oscillator that includes both one- and two-photon drives, breaking parity explicitly yet still showing spectral features consistent with degeneracy. That move is new relative to the usual parity-protected models in the literature they cite, and it is directly motivated by the practical goal of protected qubits in driven superconducting circuits where parity may not be available or desirable to preserve. The paper does a reasonable job laying out the classical picture and noting that the reported spectra strongly suggest additional symmetries at work. The connection to circuit-QED applications is stated plainly without overclaiming immediate utility. The soft spot is that the abstract and title lean heavily on the classical implication, while the quantum claim is phrased cautiously as levels that “may have” degeneracy. Without an explicit check that the proposed antiunitary operator leaves the full quantum Hamiltonian invariant (or maps the spectrum onto itself without splitting), it is not yet clear whether the degeneracy is exact and symmetry-protected or only approximate and parameter-dependent. The stress-test concern lands here: if the drive or Kerr terms do not commute with the operator in the quantum case, the protection weakens. No machine-checked proof or detailed invariance calculation is referenced in the provided material. This is the sort of paper that belongs in a reading group for people working on symmetry-protected states in open quantum systems or circuit QED. A reader interested in new routes to degeneracy without parity will get something useful from it, even if the quantum rigor needs tightening. It is worth sending to peer review; the core idea is coherent and the application angle is timely, but referees will need to see the explicit symmetry check and any numerical confirmation that the degeneracy survives quantization.

Referee Report

3 major / 2 minor

Summary. The paper examines one-dimensional spinless quantum systems that exhibit spontaneous symmetry breaking without relying on parity symmetry. It introduces an antiunitary symmetry that enforces a two-fold symmetry on momentum or coordinate in the classical limit, and illustrates this with a Kerr parametric oscillator (KPO) subject to one- and two-photon drives. Despite explicit parity violation, the model is claimed to support doubly-degenerate levels, with spectral features suggesting additional symmetries. The work is motivated by potential applications to protected qubits in driven superconducting circuits.

Significance. If the antiunitary symmetry is shown to protect exact twofold degeneracy in the quantum spectrum, the result would broaden the class of systems supporting symmetry-protected degeneracies beyond parity-based mechanisms. This could be relevant for qubit designs in parity-breaking hardware. The manuscript correctly identifies the classical implication but leaves the quantum protection as an indication from spectra rather than a derived invariance.

major comments (3)

[§3] §3 (Quantum Hamiltonian): The antiunitary operator is defined and shown to imply twofold symmetry in the classical limit, but no explicit check is given that it commutes with or leaves invariant the full quantum Hamiltonian including the one-photon drive and Kerr terms. Without this, the degeneracy may be approximate rather than symmetry-protected.
[§4] §4 (Spectral analysis): The claim that spectral features 'strongly indicate' additional symmetries and exact double degeneracy relies on numerical diagonalization without error bounds or finite-size scaling. It is unclear whether level splittings remain zero in the thermodynamic or infinite-drive limit.
[Eq. (12)] Eq. (12) (Classical equations of motion): The two-fold symmetry is derived under the assumption that the antiunitary transformation maps the phase space identically, but the mapping to the quantum spectrum is not derived; the paper does not show that the quantum operator squares to a phase or identity in a way that forces Kramers-like degeneracy.

minor comments (2)

Notation for the antiunitary operator should be introduced earlier and used consistently when discussing both classical and quantum cases.
Figure captions for the energy spectra should include the specific parameter values and Hilbert-space truncation used for the numerics.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments have helped us clarify the quantum protection mechanism and strengthen the numerical evidence. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

Referee: [§3] §3 (Quantum Hamiltonian): The antiunitary operator is defined and shown to imply twofold symmetry in the classical limit, but no explicit check is given that it commutes with or leaves invariant the full quantum Hamiltonian including the one-photon drive and Kerr terms. Without this, the degeneracy may be approximate rather than symmetry-protected.

Authors: We agree that an explicit verification is required. In the revised manuscript we now demonstrate that the antiunitary operator commutes with the complete quantum Hamiltonian, including the one-photon drive, two-photon drive, and Kerr nonlinearity. The commutation relation is shown by direct substitution of the operator into each term of the Hamiltonian, confirming that the symmetry is preserved at the quantum level and protects the exact twofold degeneracy. revision: yes
Referee: [§4] §4 (Spectral analysis): The claim that spectral features 'strongly indicate' additional symmetries and exact double degeneracy relies on numerical diagonalization without error bounds or finite-size scaling. It is unclear whether level splittings remain zero in the thermodynamic or infinite-drive limit.

Authors: We have added finite-size scaling of the level splittings together with numerical error bounds obtained from the diagonalization routine. The revised figures show that the splittings extrapolate to zero with increasing system size, consistent with exact degeneracy protected by the antiunitary symmetry. We also include an analytic argument for the infinite-drive limit in which the symmetry remains unbroken. revision: yes
Referee: [Eq. (12)] Eq. (12) (Classical equations of motion): The two-fold symmetry is derived under the assumption that the antiunitary transformation maps the phase space identically, but the mapping to the quantum spectrum is not derived; the paper does not show that the quantum operator squares to a phase or identity in a way that forces Kramers-like degeneracy.

Authors: We have now derived the quantum consequence explicitly. The antiunitary operator satisfies A² = −1 and anticommutes with the Hamiltonian in the relevant sector, which is the precise condition that enforces Kramers-type degeneracy for this antiunitary symmetry. This derivation is added as a new subsection that bridges the classical two-fold symmetry to the quantum spectrum. revision: yes

Circularity Check

0 steps flagged

No circularity: degeneracy claim rests on independent antiunitary symmetry identification

full rationale

The abstract presents the doubly-degenerate levels as a consequence of an antiunitary symmetry whose two-fold implication on momentum or coordinate is identified in the classical limit of the driven KPO. This symmetry is invoked to explain degeneracy despite explicit parity breaking, without any quoted equations, fitted parameters, or self-citations that reduce the quantum degeneracy result to a definition or input by construction. The derivation chain therefore remains self-contained; the central claim is not forced by renaming, fitting, or load-bearing self-reference within the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The antiunitary symmetry is invoked but its precise definition and proof of degeneracy protection are not provided.

pith-pipeline@v0.9.0 · 5486 in / 1183 out tokens · 21056 ms · 2026-05-17T22:46:41.905889+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

for nonzero ξ₁ values, parity is broken, but Hamiltonian (1) still conserves the time-reversal symmetry, T̂ Ĥ(ξ₂,ξ₁) T̂⁻¹ = Ĥ(ξ₂,ξ₁). ... two equivalent closed trajectories that are invariant under the symmetry transformation T explains the existence of doubly-degenerate energy levels.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the anharmonic nature of the Kerr parametric resonator Hamiltonian results in a momentum dependence that allows for the existence of degenerate states associated with the time-reversal symmetry.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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