[Paper Review] Analytic solutions of 2-photon and two-mode Rabi models

Yao-Zhong Zhang|arXiv (Cornell University)|Apr 30, 2013

Quantum Information and Cryptography26 references3 citations

TL;DR

This paper presents analytic solutions for the 2-photon and two-mode Rabi models using Bogoliubov transformations and Bargmann-Hilbert spaces, deriving a transcendental function whose zeros yield the exact energy spectrum. The method also extends to the driven Rabi model with broken ${\cal Z}_2$ symmetry, enabling numerical computation of energy levels via standard root-finding techniques.

ABSTRACT

Applying Bogoliubov transformations and Bargmann-Hilbert spaces, we obtain analytic representations of solutions to the 2-photon and two-mode quantum Rabi models. In each case, a transcendental function is analytically derived whose zeros give the energy spectrum of the model. The zeros can be numerically found by standard root-search techniques. We also present analytic solution to the driven Rabi model with broken ${\cal Z}_2$ symmetry.

Motivation & Objective

• To derive analytic solutions for the 2-photon and two-mode quantum Rabi models using advanced mathematical techniques.
• To identify a transcendental function whose zeros correspond to the energy levels of these models.
• To extend the analytic framework to the driven Rabi model with broken ${\cal Z}_2$ symmetry.
• To provide a systematic method for numerically computing energy spectra using standard root-finding algorithms.

Proposed method

• Employing Bogoliubov transformations to map the Hamiltonian into a solvable form in Fock space.
• Utilizing Bargmann-Hilbert spaces to represent wavefunctions as analytic functions.
• Deriving a transcendental equation whose roots determine the energy spectrum of the 2-photon and two-mode Rabi models.
• Applying the same formalism to the driven Rabi model with explicit breaking of ${\cal Z}_2$ symmetry.
• Expressing the energy spectrum as zeros of a derived analytic function, enabling numerical evaluation.
• Using standard root-finding techniques to compute energy levels from the transcendental function.

Experimental results

Research questions

• RQ1 How can analytic solutions be systematically derived for the 2-photon Rabi model using symmetry and Hilbert space techniques?
• RQ2 What transcendental function emerges from the solution of the two-mode Rabi model, and how does it encode the energy spectrum?
• RQ3 Can the analytic framework be extended to the driven Rabi model when ${\cal Z}_2$ symmetry is broken?
• RQ4 What role do Bogoliubov transformations play in simplifying the Hamiltonian structure of these models?
• RQ5 How can the derived transcendental function be efficiently solved numerically to obtain energy levels?

Key findings

• A transcendental function is analytically derived whose zeros correspond exactly to the energy levels of the 2-photon Rabi model.
• The energy spectrum of the two-mode Rabi model is fully determined by the zeros of a derived analytic function.
• The method successfully extends to the driven Rabi model with broken ${\cal Z}_2$ symmetry, providing a consistent analytic framework.
• The use of Bargmann-Hilbert spaces enables a rigorous representation of wavefunctions as entire functions.
• The energy levels can be computed numerically with standard root-finding algorithms applied to the transcendental function.
• The approach provides a unified analytic treatment for multiple variants of the Rabi model, including symmetric and broken-symmetry cases.