A staircase formula for a hidden quantum symmetry

A symmetry that survives asymmetry

The quantum Rabi model describes a two-level system coupled to a quantum oscillator, such as a mode of light. The two-level system has an internal energy splitting, while the oscillator can hold different numbers of excitation quanta. Their interaction is controlled by a coupling strength.

Without a bias, the model has a symmetry called parity. Its energy states fall into two classes, and levels belonging to different classes can cross as the coupling changes. Adding a bias breaks this familiar parity symmetry, generally removing those crossings.

But there is an exception: crossings can occur when the bias is an integer multiple of the oscillator frequency, in the paper’s convention. Earlier research established that hidden symmetry operators exist at these special biases. The new work does not discover those crossings or establish the symmetry’s existence for the first time. Its contribution is an explicit, closed formula for the operator at every integer bias.

Kollar and Dutta describe the result in an arXiv preprint dated November 9, 2026. The supplied source does not establish peer-reviewed publication.

Building the operator as a staircase

A symmetry operator is a mathematical transformation compatible with the system’s Hamiltonian, the operator describing its energy. Here, compatibility means that applying the symmetry and applying the Hamiltonian give the same result whichever comes first.

The authors construct this symmetry indirectly. They first build an operator connecting the Hamiltonian at positive bias to its counterpart at the opposite, negative bias. Ordinary parity, combined with an exchange of the two-level components in their chosen basis, then maps that mirror system back to the original one.

Their connecting operator is assembled from a staircase of simpler factors. At integer bias $N$, it links bias $N$ to bias $-N$ through $N$ steps, each changing the bias by two units.

Individual factors do not generally make this connection exactly: they leave residual terms. The solution combines their ordered product with correction terms that cancel those residuals. Mathematically, this combination is a continuant, a determinant-like construction associated with a tridiagonal matrix.

The order matters because these factors are operators, not ordinary numbers: swapping them can change the answer. The authors specify a recurrence that preserves their order and fixes the correction weights. An induction proof then shows that the completed construction has the required symmetry for every positive integer $N$. At zero bias, it reduces to ordinary parity.

The staircase is an algebraic connection between Hamiltonians, not a proposed experimental sequence for changing a device’s bias.

A simpler limit explains the matching

The physical picture becomes clearer when the internal splitting parameter, called delta, is set to zero. In the rotated basis used by the authors, the problem then separates into two displaced oscillators, each with an evenly spaced ladder of energy levels.

At integer bias $N$, the ladders line up after an offset of $N$ rungs. Every level above the lowest $N$ states belongs to a degenerate pair; those lowest states remain unpaired. The symmetry operator exchanges the members of each pair and annihilates the unpaired states.

For nonzero delta, this simple universal pairing no longer describes the spectrum. The full formula nevertheless supplies the hidden symmetry that permits crossings under the appropriate conditions. Integer bias does not mean that every energy level is doubly degenerate.

One construction, three models

The proof relies on a small set of algebraic identities. The authors show that the same identities hold in asymmetric two-photon and two-mode Rabi models, after expressing them through generators of an algebra called su(1,1). The same continuant therefore produces their symmetry operators at every order, extending previously available low-order formulas.

For these variants, the special bias is an even integer multiple of a coupling-dependent, renormalized frequency—not simply the original oscillator frequency.

Alongside the proof, the authors report exact-arithmetic computer checks: through order 30 for the ordinary asymmetric model, through order six for the two-photon model, and through order four in several representations relevant to the generalized construction. These finite checks support, but do not replace, the general proof.

What remains unresolved

This is a mathematical study, not an experimental demonstration. It provides an explicit symmetry generator rather than a complete new account of the spectrum.

In particular, the paper does not determine the general polynomial relating the square of the symmetry operator to the Hamiltonian. Its connection to earlier conjectures, the operator’s zero states, and detailed consequences for low-energy levels remain open. The result makes the hidden symmetry explicit; several implications still need to be worked out.