⚛️ quantum physics

Entangling Power and Symmetries in the Quantum Rabi Model

This paper demonstrates that time-averaged entangling power serves as an effective operator-level diagnostic for distinguishing between manifest $U(1)$ symmetry and parameter-dependent hidden symmetries in the Quantum Rabi model family, revealing distinct spectral responses such as peaks at integer-bias points for the asymmetric model and dips for the Jaynes-Cummings limit.

Original authors: Ian Low, Jens Koch, Sahel Ashhab

Published 2026-07-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ian Low, Jens Koch, Sahel Ashhab

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the quantum world as a giant, invisible dance floor where two very different partners are trying to move together. One partner is a qubit (a tiny, two-state quantum bit, like a coin that can be heads or tails), and the other is a harmonic oscillator (a vibrating field of light, like a spring that can wiggle with any amount of energy). The "Quantum Rabi Model" is the rulebook for how these two dance.

Usually, this dance is chaotic and hard to predict. But sometimes, the music changes in a very specific way, and the dancers suddenly fall into a secret, hidden rhythm. This paper is about finding a way to spot that secret rhythm, even when you can't see the sheet music.

The Secret Rhythm (The Hidden Symmetry)

In some versions of this dance, the rules are obvious. For example, in the famous Jaynes–Cummings (JC) model, the dancers are strictly forbidden from changing their total number of "excitations" (like a rule that says you can't add or remove steps). This is a "manifest" symmetry—it's written right on the wall.

But in the Asymmetric Quantum Rabi Model (AQRM), things look messy. There is a "bias" (a tilt in the dance floor) that seems to break all the obvious rules. However, the paper reveals that if you tilt the floor by just the right amount—specifically when the bias $\varepsilon$ is an integer multiple of the oscillator's frequency $\omega$ (so $\varepsilon/\omega$ is 1, 2, 3, etc.)—a hidden symmetry wakes up.

This isn't a symmetry you can see in the basic rules. It's like a secret handshake that only the dancers know when the music hits a specific note. The paper shows that at these exact integer points, the dancers organize themselves into special pairs, but this organization doesn't show up as a simple "tie" in their scores (energy levels). In fact, if you just look at the energy scores, you might miss it entirely.

The Detective Tool: "Entangling Power"

So, how do you catch a hidden symmetry that doesn't show up in the scores? The authors use a tool called time-averaged entangling power.

Think of "entanglement" as the dancers getting so mixed up that you can't tell where one ends and the other begins. The "entangling power" measures how good the dance is at mixing them up, on average, over a long time.

The authors ran simulations (computer experiments) to see how this "mixing score" changes as they slowly adjust the tilt of the floor (the bias $\varepsilon$ ). They used two different ways to pick the starting positions for the dancers:

Finite-Haar: Picking random starting spots within a limited box of energy.
Coherent-state: Picking starting spots that look like a smooth, classical wave (like a real laser beam).

The Big Discovery:
In these simulations, whenever the bias hit an integer value ( $\varepsilon/\omega = 1, 2, 3...$ ), the "mixing score" didn't drop; it spiked. It shot up to a peak.

This is surprising! Usually, when a system has a strong symmetry (like the obvious JC model), the mixing score tends to dip or stay low because the rules are too rigid. But here, the hidden symmetry at the integer points causes the dancers to mix more efficiently. The paper suggests this happens because the hidden symmetry forces the dancers into specific "displaced doublets" (pairs of states) that get perfectly mixed together at these integer points, creating a burst of entanglement.

What It Is NOT (And What It Rules Out)

It's important to know what this isn't. The paper explicitly rules out a few ideas:

It's not just a tie in the scores: The authors checked carefully. At these integer points, the energy levels of the dancers do not necessarily cross or become exactly the same (degenerate) in a generic scan. If they were just looking for a "tie" in the energy scores, they wouldn't have found this peak. The peak comes from how the dancers' movements (eigenvectors) reorganize, not just their scores.
It's not a universal rule for all tilts: The hidden symmetry only appears at those specific integer values. If you tilt the floor by a non-integer amount (like 1.5), the secret handshake disappears, and the mixing score drops back down.
It's not the same as the JC model: In the JC model (where the rules are obvious), the symmetry causes a dip in the mixing score. The paper shows that the AQRM hidden symmetry does the exact opposite, creating a peak. This proves the two symmetries work in fundamentally different ways.

How Sure Are We?

The authors are very confident in their simulations. They ran the numbers on computers for different strengths of the dance coupling (from weak to "deep-strong" coupling, where the dancers are glued together). In every single case, the peaks appeared exactly at the integer values.

They also derived a mathematical explanation using a "displaced-doublet" picture. They showed that at the integer points, the two dancers in a pair get perfectly balanced, which mathematically forces the mixing to be maximized. While they haven't built a physical machine to prove this in a lab yet, the computer evidence and the math behind it are very strong.

The Takeaway

This paper suggests that entanglement (the mixing of quantum partners) can act like a super-sensitive detector for hidden symmetries. Even when a symmetry is invisible in the basic rules and doesn't create obvious "ties" in the energy levels, it leaves a fingerprint: a sharp peak in how much the system entangles.

So, if you want to find a hidden secret in a quantum system, don't just look at the scores. Watch how much the dancers mix. If you see a sudden spike in the mixing at an integer setting, you've found the hidden symmetry.

Technical Summary: Entangling Power and Symmetries in the Quantum Rabi Model

Problem Statement
The Quantum Rabi Model (QRM) serves as the standard effective Hamiltonian for light–matter interaction, describing a qubit coupled to a single harmonic oscillator. While the symmetric QRM ( $\epsilon=0$ ) possesses a manifest discrete $\mathbb{Z}_2$ parity symmetry, the Asymmetric Quantum Rabi Model (AQRM), characterized by a transverse bias $\epsilon$ , breaks this parity. However, the AQRM exhibits a "hidden" symmetry at integer bias values ( $\epsilon/\omega \in \mathbb{Z}$ ), where the spectrum contains exceptional crossings and the Hilbert space can be organized into symmetry sectors. Unlike the Jaynes–Cummings (JC) model, where symmetry is manifest via a conserved excitation number ( $U(1)$ symmetry), the AQRM's hidden symmetry is not visible in the Hamiltonian's structure and does not generically produce spectral degeneracies at fixed coupling parameters $(g, \Delta)$ .

The central problem addressed is whether an operator-level diagnostic, specifically the time-averaged entangling power, can detect such a hidden symmetry that leaves no generic spectral degeneracy signature. Furthermore, the infinite-dimensional nature of the oscillator Hilbert space complicates standard entanglement diagnostics, which typically rely on finite-dimensional Haar measures.

Methodology
The authors employ the time-averaged entangling power ( $e_p$ ) as a diagnostic tool. This quantity measures the average entanglement generated by the time-evolution operator $U(t) = e^{-iHt}$ from product inputs, averaged over both time and an ensemble of input states.

To handle the infinite-dimensional oscillator, the paper compares two distinct finite input ensembles:

Finite-Haar Average: The oscillator Hilbert space is truncated to $N_{tr}$ Fock states. A Haar average is performed over this finite subspace. The mean occupation number of this ensemble scales with the cutoff ( $\bar{n}_H \approx N_{tr}/2$ ).
Coherent-State Average: The input ensemble consists of coherent states $|\alpha\rangle$ with a fixed mean occupation number $\bar{n}$ . This approach separates the physical input scale ( $\bar{n}$ ) from the numerical truncation cutoff ( $N_{tr}$ ), which is chosen sufficiently large to capture the coherent state's support.

The entangling power is calculated using the linear entropy of the reduced density matrix of the qubit subsystem. The time average is computed by grouping eigenstate pairs according to their energy differences, a method that isolates contributions from exact degeneracies and repeated energy gaps.

Key Contributions and Results

Detection of Hidden Symmetry:
Numerical scans of the time-averaged entangling power across the bias parameter $\epsilon/\omega$ reveal sharp peaks at integer values ( $\epsilon/\omega \in \mathbb{Z}$ ). This feature is robust across weak, ultrastrong, and deep-strong coupling regimes and is observed consistently in both the finite-Haar and coherent-state diagnostics. This confirms that the entangling power responds to the hidden symmetry of the AQRM even in the absence of generic spectral degeneracies.
Mechanism of the Peak (Eigenvector Reorganization):
The paper demonstrates that the integer-bias peaks are not caused by the spectral mechanism typical of symmetry-induced dips (i.e., the enlargement of energy-difference groups due to degeneracies). Instead, the peaks arise from eigenvector reorganization.
- Using a displaced-doublet picture, the authors show that at $\epsilon = k\omega$ , the mixing between displaced oscillator ladders is maximized.
- This maximal mixing causes the reduced qubit Bloch vectors of the eigenstates to contract (shorten) toward the origin, signaling stronger qubit-oscillator entanglement.
- Since the time-averaged purity $\langle \text{Tr}\rho_A^2 \rangle_t$ is reduced by this contraction, the entangling power $e_p = 2(1 - \langle \text{Tr}\rho_A^2 \rangle_t)$ increases, resulting in a peak.
Contrast with the Jaynes–Cummings Limit:
In contrast to the AQRM peaks, the JC point (where $U(1)$ symmetry is manifest) exhibits a weak dip in entangling power.
- The JC symmetry decomposes the Hilbert space into fixed-charge sectors.
- While the diagonal channel (Bloch vector length) is minimized (Bloch vectors vanish), the sparse structure of the $U(1)$ sectors concentrates off-diagonal contributions in the spectral expansion, leading to a net reduction in entangling power.
- This highlights that the sign of the response (peak vs. dip) depends on how the symmetry reorganizes the spectral expansion (eigenvector structure vs. degeneracy grouping).
Diagnostic Robustness:
The location of the integer-bias peaks is determined by the AQRM eigenstructure rather than the specific choice of input ensemble. While the two diagnostics (finite-Haar and coherent) differ in amplitude and require different scaling of parameters to resolve the peaks, they agree on the peak locations, validating the result as a property of the Hamiltonian dynamics.

Significance
The paper claims that the time-averaged entangling power serves as a viable operator-level diagnostic for symmetries in light-matter systems, even when those symmetries are "hidden" and do not manifest as generic spectral degeneracies.

Beyond Degeneracy: The work extends the utility of entanglement diagnostics beyond systems with obvious degeneracies (like the JC model or spin chains with SU(2) symmetry) to systems where symmetry is parameter-dependent and non-manifest.
Eigenvector Sensitivity: It establishes that entangling power is sensitive to the reorganization of eigenvectors (specifically the contraction of Bloch vectors in displaced doublets), providing a dynamical signature of the hidden symmetry operator $J_k$ .
Methodological Utility: The comparison between finite-Haar and coherent-state sampling demonstrates that coherent-state averages provide a physically motivated, finite-energy probe for entanglement in infinite-dimensional bosonic systems, separating physical scales from numerical artifacts.

The results support the broader hypothesis that quantum resources, such as entanglement, may be extremized at points of enhanced symmetry, offering a new perspective on organizing fundamental parameters in quantum systems.