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

This paper demonstrates that a one-dimensional Kerr parametric oscillator with parity-violating drives can still exhibit doubly-degenerate energy levels and protected qubit potential through an alternative antiunitary symmetry, challenging the conventional reliance on parity symmetry for such degeneracy.

The Gist

The Big Picture: Finding "Twins" in a Broken World

Imagine you are looking at a landscape with two identical valleys separated by a high mountain. In the world of quantum physics, a particle can exist in the left valley or the right valley. Usually, if the landscape is perfectly symmetrical (the left valley is a mirror image of the right), the particle can be in a "superposition" of both, creating a special kind of twin state. Physicists call this degeneracy, and it's often protected by a rule called parity symmetry (like a perfect mirror).

However, this paper asks a tricky question: What happens if we break the mirror? What if we tilt the landscape so the two valleys are no longer identical? Usually, in this "broken" world, the twin states disappear, and the energy levels separate.

The authors of this paper discovered something surprising: Even when you break the mirror, you can still find these "twin" states. They found a way to create a system where the energy levels remain almost identical (quasi-degenerate) even though the system is no longer symmetrical.

The Setup: The "Kerr" Oscillator

To test this, the researchers used a model called a Kerr Parametric Oscillator (KPO).

• The Analogy: Think of this as a very fancy, non-linear swing. Unlike a normal swing that moves back and forth in a simple arc, this swing changes its stiffness depending on how hard you push it.
• The Drives: They pushed this swing in two ways:
  1. Two-photon drive: This is like pushing the swing at a specific rhythm that keeps the landscape symmetrical (the two valleys are equal).
  2. One-photon drive: This is like adding a constant wind or a tilt to the landscape, breaking the symmetry so one valley is deeper than the other.

The Discovery: The "Time-Reverse" Secret

In the past, physicists thought that if you broke the symmetry (tilted the landscape), the twin energy states would vanish. But this paper shows that a different kind of "hidden symmetry" takes over.

• The Old Rule (Parity): If you flip the landscape left-to-right, it looks the same. This protects the twins.
• The New Rule (Time-Reversal): The authors found that even in the tilted, asymmetrical landscape, there is a rule related to time. If you were to play a movie of the particle's movement backward, the physics would still make sense.

The Metaphor: Imagine a dancer spinning on a stage.

• If the stage is perfectly round (symmetrical), the dancer looks the same spinning clockwise or counter-clockwise.
• If the stage is oval (asymmetrical), usually the spin looks different.
• However, the authors found that for this specific "Kerr" swing, even on the oval stage, there is a hidden rule: if you reverse the direction of time (play the movie backward), the dancer's path still fits perfectly. This "Time-Reversal" symmetry acts like a safety net, keeping the energy levels of the two states incredibly close together, even though the landscape is broken.

The Results: How Close are the Twins?

The researchers ran complex computer simulations to see how close these energy levels get.

1. The "Kissing" Effect: They found that as the system gets larger (approaching a "classical" limit where quantum effects are tiny), the energy gap between these two states shrinks exponentially.
2. The Analogy: Imagine two friends walking toward each other. In a normal broken system, they might stop a few feet apart. In this system, as they get closer to the "classical" limit, they get so close that they are practically touching, but they never quite merge. They are "quasi-degenerate."
3. The Math: They proved that the speed at which these levels get closer follows a specific mathematical pattern (an exponential decay), and this pattern is the same whether the system is symmetrical or broken.

Why Does This Matter? (According to the Paper)

The paper highlights two main reasons this is interesting, based strictly on their findings:

1. Protected Qubits: In the world of quantum computing, "qubits" are fragile. They need protection from noise. Usually, scientists use symmetrical systems to protect them. This paper suggests that even in asymmetrical systems (which are easier to build in some real-world circuits), you might still get this protection because of the "Time-Reversal" rule. This could help in building more robust quantum computers using superconducting circuits.
2. Adiabatic Calculations: When scientists try to solve problems by slowly changing a system (a method called the adiabatic approximation), they need to know if energy levels will cross or get stuck. This paper warns that even in broken systems, you might encounter these "kissing" levels, which could trip up calculations if you aren't careful.

Summary

In short, the paper shows that you don't need a perfect mirror (parity symmetry) to get "twin" energy states in a quantum system. Even if you break the symmetry, a different rule (time-reversal symmetry) can step in and keep those energy levels locked together, almost as if they were twins, provided the system has the right kind of "non-linear" behavior. This opens up new possibilities for designing quantum devices that don't rely on perfect symmetry.

Technical Summary

Technical Summary: Degeneracy beyond Parity-Symmetry Protection in 1D Spinless Models

Problem Statement
In quantum mechanics, the von Neumann-Wigner theorem establishes that degeneracy in one-dimensional (1D) systems typically arises from symmetries, most commonly parity ($P: \{q, p\} \to \{-q, -p\}$). In parity-conserving systems, spontaneous symmetry breaking leads to quasi-degenerate energy doublets in the broken-symmetry phase, where the energy gap decreases exponentially with system size due to tunneling between symmetric potential wells. However, a significant theoretical gap exists regarding systems that lack parity symmetry. Standard antiunitary symmetries (like time-reversal $\mathcal{T}$) usually do not guarantee degeneracy in spinless systems with integer total spin, as $\mathcal{T}$ acting on an eigenstate does not necessarily produce a linearly independent state. The central problem addressed is whether a 1D spinless quantum system, explicitly breaking parity symmetry, can still exhibit exponentially small energy gaps (quasi-degeneracy) and, if so, what symmetry mechanism protects these states.

Methodology
The authors investigate this phenomenon using a non-linear Kerr parametric oscillator (KPO) driven by both one- and two-photon terms. The system is described by the Hamiltonian:
$$\frac{\hat{H}(\xi_2, \xi_1)}{K\hbar} = \frac{\hat{a}^{\dagger 2}\hat{a}^2}{N_e} - \xi_2(\hat{a}^{\dagger 2} + \hat{a}^2) + \frac{\xi_1}{\sqrt{N_e}}(\hat{a}^\dagger + \hat{a})$$
where $\xi_2$ and $\xi_1$ are control parameters for two-photon and one-photon drives, respectively, and $N_e$ drives the system toward the classical limit.

The study employs a dual approach:

1. Quantum Numerical Simulation: The energy spectra are calculated using a truncated Fock basis. The authors analyze the energy levels as functions of $\xi_1$ and $\xi_2$, specifically comparing the parity-conserving case ($\xi_1 = 0$) with the parity-broken case ($\xi_1 \neq 0$). High-precision arithmetic is utilized to resolve exponentially small energy gaps that would otherwise suffer from numerical underflow.
2. Classical Limit Analysis: The classical limit ($N_e \to \infty$) is derived to visualize phase-space trajectories. The authors examine the energy contours and symmetries of the classical Hamiltonian, specifically looking for disconnected trajectories invariant under specific symmetry operations.
3. Semiclassical Estimation: An analytical estimation of the energy gap scaling exponent $\delta$ is performed using the overlap of Glauber coherent states localized in the potential wells.

Key Contributions and Results

• Existence of Degeneracy without Parity: The paper demonstrates that a 1D spinless system can exhibit quasi-degenerate energy doublets even when parity symmetry is explicitly broken by a one-photon drive ($\xi_1 \neq 0$).
• Role of Time-Reversal Symmetry: In the parity-broken regime, the system retains a specific antiunitary symmetry: time-reversal ($\mathcal{T}: (q, p) \to (q, -p)$). While $\mathcal{T}$ alone does not guarantee degeneracy in standard spinless potentials, the anharmonic nature of the Kerr interaction introduces a momentum dependence that allows for the existence of two equivalent, disconnected classical trajectories invariant under $\mathcal{T}$. This "spectral kissing" (quasi-degeneracy) is protected by this time-reversal symmetry rather than parity.
• Scaling of Energy Gaps: The authors verify that the energy gap $\Delta_j$ between the quasi-degenerate levels scales exponentially with the system size parameter $N_e$ ($\Delta_j \propto e^{-\delta N_e}$) in both parity-conserving and parity-broken cases.
  • Numerical fits yield exponents $\delta$ that are nearly identical for both cases.
  • An analytical approximation confirms $\delta \approx 2|\xi_2|$, derived from the overlap of coherent states in the potential wells.
• Phase Space Topology: The study maps the classical energy surfaces, showing that for negative $\xi_2$ (the symmetry-broken phase), the system possesses two symmetric minima. In the parity-conserving case, these are related by parity; in the parity-broken case, they are related by time-reversal symmetry. The expectation values of position ($\hat{Q}$) and momentum ($\hat{P}$) operators confirm the localization of wavefunctions in these distinct minima.

Significance and Claims
The authors claim that their work identifies a mechanism for protecting quantum states in systems where parity symmetry is absent. The primary significance lies in the demonstration that time-reversal symmetry, in conjunction with specific momentum-dependent anharmonicities, can induce quasi-degeneracy in 1D spinless systems.

The paper highlights the relevance of these findings for:

• Superconducting Circuits: The KPO model serves as an effective Hamiltonian for driven superconducting circuits, which are candidates for quantum computing and simulation. The existence of protected quasi-degenerate levels in parity-breaking setups could be practically useful for creating robust qubits.
• Adiabatic Quantum Computation: The results suggest that researchers relying on the quantum adiabatic theorem (e.g., in quantum annealing) must carefully consider all possible symmetries, including antiunitary ones, to avoid unexpected level crossings or degeneracies that could compromise the adiabatic approximation.

The authors modestly note that while the theoretical existence of these states is established, the specific "spectral kissing" protected by time-reversal in parity-violating systems is currently beyond experimental access, though recent experimental progress in asymmetric double-well potentials suggests this may change. The work also points to future research directions involving open KPO systems and local symmetries associated with level crossings.