Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

The constraint polynomials of the two-photon Rabi model at and beyond the critical coupling

Abstract

The constraint polynomials of the two-photon asymmetric quantum Rabi model factor into linear terms y + 2n(2n + 2rho - 1) at the critical coupling x = 1, and for x > 1 and nonnegative bias all their coefficients are positive, because the tridiagonal matrix behind them is positive definite.

Definition 1.1 (The constraint polynomials).

Formalization. D5/S3/Quantum/Dynamics/TwoPhotonRabiConstraintPolynomials.constraintPoly (✓ std3).

Citation. Cid Reyes-Bustos, Masato Wakayama (2026). Two-photon quantum Rabi models – Spectral degeneracy and symmetries. DOI: 10.48550/arXiv.2609.00750. URL: https://arxiv.org/abs/2609.00750v1.

Commentary.

Definition 4.1 of the source, read as polynomials in y with real parameters x, bias eps and parity rho, and the index N of the constraint polynomial P_N.

Definition 1.2 (The conjecture).

Formalization. D5/S3/Quantum/Dynamics/TwoPhotonRabiConstraintPolynomials.claim (✓ std3).

Citation. Cid Reyes-Bustos, Masato Wakayama (2026). Two-photon quantum Rabi models – Spectral degeneracy and symmetries. DOI: 10.48550/arXiv.2609.00750. URL: https://arxiv.org/abs/2609.00750v1.

Commentary.

The first statement is taken for every real bias. The second statement, positive coefficients and therefore no positive roots in y, is taken for bias eps >= 0, the sign the source uses when it derives the constraint condition; for eps = -2, N = 1 and rho = 0 one has P_1 = y - 2x + 4, which is negative at y = 0 once x > 2.

Theorem 1.3 (Proof of the conjecture for nonnegative bias).

Proof. Machine-checked in Lean as D5/S3/Quantum/Dynamics/TwoPhotonRabiConstraintPolynomials.result (✓ std3). ∎

Resolves. Problems/reyes-bustos-wakayama-2026-two-photon-rabi-constraint-polynomials (proved) by D5/S3/Quantum/Dynamics/TwoPhotonRabiConstraintPolynomials.result.

Source. Repository-derived.

Acknowledgement. Cid Reyes-Bustos, Masato Wakayama (2026). Two-photon quantum Rabi models – Spectral degeneracy and symmetries. DOI: 10.48550/arXiv.2609.00750. URL: https://arxiv.org/abs/2609.00750v1.

Commentary.

Part (a). At x = 1 the bias cancels from the recursion. In the basis of partial products Q_i = (y + lambda_1)…(y + lambda_i), with lambda_n = 2n(2n + 2rho - 1), the polynomial P_k(1, y) has coefficient C(k, i) 4^(k-i) (N-i-1)(N-i-2)…(N-k) (k+1)k…(i+2) at Q_i. Since y Q_i = Q_(i+1) - lambda_(i+1) Q_i, the recursion for these coefficients reduces to a polynomial identity that holds exactly when rho^2 = rho. At k = N every coefficient with i < N contains the factor N - N = 0, so P_N(1, y) = Q_N. Part (b). P_k(x, y) is det(y + J) for the symmetric tridiagonal matrix J(x) with diagonal entries d_k(x) = 2xk(4N + 2rho - 2k + 2eps + 1) - 4k(k + eps) and off-diagonal entries the square roots of the nonnegative couplings b_k x; expanding the determinant along the first row gives the recursion. By part (a) the eigenvalues of J(1) are the numbers lambda_n > 0, so J(1) is positive definite. For x >= 1 and eps >= 0, J(x) = sqrt(x) J(1) + D with D diagonal and D_k = 2k(sqrt(x) - 1)(sqrt(x)(4N + 2rho - 2k + 2eps + 1) + 2(k + eps)) >= 0, so J(x) is positive definite. Its eigenvalues mu_i are positive, P_N(x, y) is the product of the factors y + mu_i, and every coefficient is a sum of products of the mu_i, hence positive; for y > 0 each factor y + mu_i is positive, so y is not a root.

References