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A correspondence between the Rabi model and an Ising model with long-range interactions

Bruno Scheihing-Hitschfeld, Néstor Sepúlveda

TL;DR

The paper establishes a bidirectional mapping between the Rabi model and a one-dimensional Ising model with long-range interactions using Trotterization, linking real-time transition amplitudes to Ising partition functions. In the real-time setting, the Rabi dynamics are expressed as a sum over Ising configurations, with a continuum limit yielding a domain-wall expansion that mirrors perturbation theory in the two-level splitting $\omega_0$. In imaginary time, the mapping exposes a Yukawa-type long-range kernel and a finite-temperature partition function for the Rabi model, enabling cross-utilization of statistical-mechanical techniques. The results illuminate how prominent dynamical regimes of light-matter coupling correspond to specific Ising-like spin configurations, offering practical computational advantages and guiding future explorations in cavity QED and beyond.

Abstract

By means of Trotter's formula, we show that transition amplitudes between a class of generalized coherent states in the Rabi model can be understood in terms of a certain Ising model featuring long-range interactions beyond nearest neighbors in its thermodynamic limit. Specifically, we relate the transition amplitudes in the Rabi model to a sum over binary variables of the form of a partition function of an Ising model with a number of spin sites equal to the number of steps in Trotter's formula applied to the real-time evolution of the Rabi model. From this, we show that a perturbative expansion in the energy splitting of the two-level subsystem in the Rabi model is equivalent to an expansion in the number of spin domains in the Ising model. We conclude by discussing how calculations in one model give nontrivial information about the other model, and vice versa, as well as applications and generalizations this correspondence may find.

A correspondence between the Rabi model and an Ising model with long-range interactions

TL;DR

The paper establishes a bidirectional mapping between the Rabi model and a one-dimensional Ising model with long-range interactions using Trotterization, linking real-time transition amplitudes to Ising partition functions. In the real-time setting, the Rabi dynamics are expressed as a sum over Ising configurations, with a continuum limit yielding a domain-wall expansion that mirrors perturbation theory in the two-level splitting . In imaginary time, the mapping exposes a Yukawa-type long-range kernel and a finite-temperature partition function for the Rabi model, enabling cross-utilization of statistical-mechanical techniques. The results illuminate how prominent dynamical regimes of light-matter coupling correspond to specific Ising-like spin configurations, offering practical computational advantages and guiding future explorations in cavity QED and beyond.

Abstract

By means of Trotter's formula, we show that transition amplitudes between a class of generalized coherent states in the Rabi model can be understood in terms of a certain Ising model featuring long-range interactions beyond nearest neighbors in its thermodynamic limit. Specifically, we relate the transition amplitudes in the Rabi model to a sum over binary variables of the form of a partition function of an Ising model with a number of spin sites equal to the number of steps in Trotter's formula applied to the real-time evolution of the Rabi model. From this, we show that a perturbative expansion in the energy splitting of the two-level subsystem in the Rabi model is equivalent to an expansion in the number of spin domains in the Ising model. We conclude by discussing how calculations in one model give nontrivial information about the other model, and vice versa, as well as applications and generalizations this correspondence may find.
Paper Structure (15 sections, 124 equations, 1 figure)

This paper contains 15 sections, 124 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Trotter's formula and the real-time correspondence
  3. The continuum limit of the Ising model as a perturbative expansion in $\omega_0$
  4. Applications in imaginary time
  5. What this Ising model can do for the Rabi model
  6. What the Rabi model can do for this Ising model
  7. Conclusions and Outlook
  8. Hamiltonian Model
  9. Trotterized time evolution of a coherent state
  10. Trotter's Formula
  11. Scaling of Trotter discretization and comparison with Hilbert-space truncation
  12. Derivation of dot product formula
  13. Recurrence in the $a$ vector components
  14. Recurrence in the $b$ vector components
  15. Coherent to coherent state amplitudes

Figures (1)

  • Figure 1: Transition amplitudes between two pairs of generic coherent states: $|\alpha\rangle=|-0.2+0.5i\rangle$ and $|\beta\rangle=|0.1+0.3i\rangle$ (first column), and $|\alpha\rangle=|-2+5i\rangle$ and $|\beta\rangle=|1+3i\rangle$ (second column), considering parity equal to $+1$. The black curves represent the numerical solution of the Schrödinger equation for the Rabi Hamiltonian, while the white dotted curves correspond to Eq. \ref{['eq:w0-expansion-2']} with $m_{\text{m}ax}=10$ and $10^4$ sampling steps for each value of $m$ from 1 to $m_{\text{m}ax}$ to compute the mean value and obtain $F_m$ using Eq. \ref{['eq:Fm-explicit']}. For a given value of $m$, which represents the number of domain walls (or spin flips) in the spin chain Hamiltonian, a sampling step corresponds to the random assignment of spin flip positions along the chain, independently distributed within the interval $(0,1)$. In all plots we used $\omega_0/\omega = 0.3$. We compare both results across three coupling regimes: strong coupling ($g/\omega = 0.05$, first row), ultra-strong coupling ($g/\omega = 0.5$, second row), and deep strong coupling ($g/\omega = 5\ $, third row).