[Paper Review] Generalized parity in multi-photon Rabi model

Bartłomiej Gardas, Jerzy Dajka|arXiv (Cornell University)|Jan 16, 2013

Quantum Mechanics and Non-Hermitian Physics3 citations

TL;DR

This paper introduces a generalized parity operator for the multi-photon Rabi model with k-photon coupling, solving an operator Riccati equation to block-diagonalize the Hamiltonian. The construction reveals a hidden symmetry that decouples the spin-boson system into independent bosonic Schrödinger equations, extending known one- and two-photon parities to arbitrary k > 0.

ABSTRACT

Quantum multi--photon spin--boson model is considered. We solve an operator Riccati equation associated with that model and present a candidate for a generalized parity operator allowing to transform spin--boson Hamiltonian to a block diagonal form what indicates an existence of the related symmetry of the model.

Motivation & Objective

• To identify a generalized symmetry in the multi-photon Rabi model that explains its integrability beyond the standard one-photon case.
• To extend the concept of parity—previously known for k=1 and k=2—to arbitrary k > 0 in the multi-photon Rabi model.
• To construct a solution to the operator Riccati equation associated with the k-photon Rabi Hamiltonian that enables exact block-diagonalization.
• To prove that the proposed generalized parity operator satisfies key symmetry properties: involution, number operator commutation, and anti-commutation with the bosonic Hamiltonian terms.
• To provide a systematic framework for analyzing and approximating the spectrum of multi-photon Rabi models using this generalized symmetry.

Proposed method

• Formulates the k-photon Rabi Hamiltonian as a block operator matrix acting on the tensor product of spin and bosonic Fock spaces.
• Derives the operator Riccati equation αX² + XH₊ − H₋X − α = 0, where H₊ and H₋ are bosonic Hamiltonians with k-photon coupling.
• Constructs a generalized parity operator Xₖ as a direct sum of k individual parity operators Jₗ acting on subspaces of Fock states with definite parity modulo k.
• Demonstrates that Xₖ satisfies Xₖ² = I, [N, Xₖ] = 0, and XₖAₗXₖ = −Aₗ, fulfilling the defining properties of a parity operator.
• Proves that the solution Xₖ satisfies the Riccati equation by showing XₖH₊ = H₋Xₖ and Xₖ² = I, thereby enabling block-diagonalization of the full Hamiltonian.
• Applies the construction to recover known cases (k=1 and k=2), confirming consistency with existing one- and two-photon parities.

Experimental results

Research questions

• RQ1 Can a generalized parity operator be constructed for the k-photon Rabi model that generalizes the known one- and two-photon parities?
• RQ2 Does the solution to the operator Riccati equation associated with the k-photon Rabi model yield a symmetry that block-diagonalizes the Hamiltonian?
• RQ3 What are the algebraic and spectral properties of the generalized parity operator for arbitrary k > 0?
• RQ4 Is the generalized parity operator an involution and does it commute with the bosonic number operator?
• RQ5 Can the generalized parity operator be used to decouple the spin-boson eigenproblem into independent bosonic Schrödinger equations?

Key findings

• A generalized parity operator Xₖ is explicitly constructed as a direct sum of k individual parity operators Jₗ, each acting on a subspace of Fock states with definite parity modulo k.
• The operator Xₖ satisfies Xₖ² = I, [N, Xₖ] = 0, and XₖH₊Xₖ = H₋, confirming it behaves as a true parity operator on the bosonic Hilbert space.
• The solution Xₖ satisfies the operator Riccati equation αXₖ² + XₖH₊ − H₋Xₖ − α = 0 for all k > 0, proving its validity as a transformation to block-diagonal form.
• The block-diagonalization transforms the original spin-boson Hamiltonian into two uncoupled Schrödinger equations, simplifying spectral analysis.
• For k=1 and k=2, the generalized parity reduces to the standard one-photon and two-photon parity operators, confirming consistency with known results.
• The construction shows that any involution J satisfying JH₊J = H₋ is a solution to the Riccati equation, suggesting a broader class of possible symmetries, though the full solution space remains open.