[Paper Review] Absence of Energy Level Crossing for the Ground State Energy of the Rabi Model

Masao Hirokawa, Fumio Hiroshima|arXiv (Cornell University)|Jul 17, 2012

Quantum Information and Cryptography14 references3 citations

TL;DR

This paper proves that the ground state energy of the Rabi model is simple (i.e., non-degenerate) for all coupling strengths, implying no energy level crossings occur. Using a path integral representation and functional analysis, the authors establish that the ground state remains unique across all parameter regimes, contrasting sharply with the Jaynes-Cummings model which exhibits multiple level crossings in the ultra-strong coupling regime of circuit QED.

ABSTRACT

The Hamiltonian of the Rabi model is considered. It is shown that the ground state energy of the Rabi Hamiltonian is simple for all values of the coupling strength, which implies the ground state energy does not cross other energy

Motivation & Objective

• To resolve the open question of whether energy level crossings occur in the ground state energy of the Rabi model across all coupling strengths.
• To clarify the fundamental difference between the Rabi model and the Jaynes-Cummings model in the ultra-strong coupling regime of circuit QED.
• To establish the simplicity of the ground state energy as a mathematical foundation for understanding spectral behavior in the Rabi model.
• To provide a rigorous functional-integral proof of ground state uniqueness using path measure techniques.
• To confirm numerically observed behavior—absence of level crossings in the Rabi model—through analytical means.

Proposed method

• Constructs a path integral representation of the time-evolution operator $ e^{-tH_{\text{Rabi}}} $ using the Feynman-Kac formula with spin degrees of freedom.
• Adapts recent results on the spin-boson model to the single-mode Rabi model, treating it as a special case of the spin-boson Hamiltonian.
• Derives a functional integral formula for the matrix element $ (f, e^{-tH}g)_{\mathscr{H}} $ involving stochastic processes on position and spin paths.
• Applies the positivity-improving property of the semigroup $ e^{-tH} $ to prove the ground state is unique via the Perron-Frobenius theorem.
• Uses the fact that $ \Delta > 0 $ and the positivity of the kernel to show that the ground state eigenspace is one-dimensional.
• Establishes that the absence of level crossings follows directly from the simplicity of the ground state energy.

Experimental results

Research questions

• RQ1 Does the ground state energy of the Rabi model exhibit energy level crossings as a function of coupling strength $ g $?
• RQ2 How does the spectral behavior of the Rabi model differ from that of the Jaynes-Cummings model in the ultra-strong coupling regime?
• RQ3 Can the absence of level crossings in the Rabi model be rigorously proven using functional-integral methods?
• RQ4 Is the ground state of the Rabi Hamiltonian unique (i.e., non-degenerate) for all values of $ g $, $ \Delta $, and $ \omega $?
• RQ5 What is the mathematical mechanism ensuring that the ground state energy remains non-degenerate across all coupling regimes?

Key findings

• The ground state energy of the Rabi Hamiltonian is simple (i.e., non-degenerate) for all values of the coupling strength $ g $, $ \Delta $, and $ \omega $.
• The absence of energy level crossings in the ground state energy is rigorously proven as a direct consequence of the ground state's simplicity.
• The path integral representation of $ e^{-tH_{\text{Rabi}}} $ is positivity improving, which implies the ground state is unique.
• The proof relies on the positivity of the kernel in the functional integral and the application of the Perron-Frobenius theorem to the semigroup.
• The result confirms numerical observations that the Rabi model does not exhibit level crossings in contrast to the Jaynes-Cummings model.
• This distinction highlights a fundamental qualitative difference between the Rabi and Jaynes-Cummings models in the ultra-strong coupling regime of circuit QED.