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The quantum Rabi model: solution and dynamics
Qiongtao Xie, Honghua Zhong, Murray T. Batchelor, Chaohong Lee
TL;DR
The review addresses how to understand the quantum Rabi model and related generalisations, including regimes where the rotating wave approximation is no longer valid. It surveys analytic solutions, spectrum and dynamics, and experiments, concluding that these solutions span the models’ full parameter spaces while important applications remain open.
Problem
The usual rotating wave approximation fails in ultrastrong and deep strong coupling regimes, motivating study of the quantum Rabi model and its generalisations beyond that approximation.
Source: §1 · p. 2Method
The article reviews analytic solutions for the quantum Rabi model and generalisations, along with dynamics, level statistics, and experimental realisations.
Source: §6 · p. 36Results
The reviewed analytic solutions apply across the models’ full parameter spaces, including regimes that have traditionally challenged numerical approaches.
Source: §6 · p. 36Takeaways & Limitations
A central challenge is identifying new phenomena that experimental and quantum-simulation advances can test in ultrastrong and deep strong coupling regimes.
Source: §6 · p. 36How the analytic solutions apply to dynamics, geometric phases, fidelity, and multipartite entanglement remains an open question.
Source: §6 · p. 36
Abstract
from arXiv · showhide
This article presents a review of recent developments on various aspects of the quantum Rabi model. Particular emphasis is given on the exact analytic solution obtained in terms of confluent Heun functions. The analytic solutions for various generalisations of the quantum Rabi model are also discussed. Results are also reviewed on the level statistics and the dynamics of the quantum Rabi model. The article concludes with an introductory overview of several experimental realisations of the quantum Rabi model. An outlook towards future developments is also given.
1. Introduction
The quantum Rabi model extends early atom–field models beyond the Jaynes–Cummings approximation, which neglects the counter-rotating term. The review emphasizes exact analytic solutions and their generalisations alongside coupling-regime developments and experimental realisations.
1. Introduction
- Rabi’s model addressed a rapidly varying weak magnetic field acting on an oriented atom, while the Jaynes–Cummings model described a two-level atom coupled to a quantised cavity mode.The Jaynes–Cummings model was treated using the rotating wave approximation (RWA), which neglects the counter-rotating term.Source: §1 · p. 1
- The RWA is valid near resonance and at weak coupling, and the resulting Jaynes–Cummings model has explained phenomena including vacuum Rabi mode splitting and quantum Rabi oscillation.Bloch and Siegert’s predicted resonance shift has also been observed in a driven superconducting qubit.Source: §1 · pp. 1–2
- Engineered systems with tunable parameters reach ultrastrong coupling, where the RWA fails and the counter-rotating term cannot be neglected.A classical simulator has also realised the deep strong coupling regime; generalized RWA methods have been proposed to approximate the eigenspectrum.Source: §1 · p. 2
- Following Braak’s exact solution, analytic solutions for the quantum Rabi model and its generalisations have been developed, including solutions in terms of confluent Heun functions.The isolated exact Judd solutions appear naturally in this analytic framework.Source: §1 · pp. 2–3
- The review covers analytic solutions, spectrum and dynamics, level statistics, experimental realisations in several platforms, and future directions.Its experimental overview includes single atoms in cavities, superconducting circuits, and hybrid mechanical systems.Source: §1 · p. 3
2. Eigenvalue problem and analytic solutions
This section reviews analytic approaches to the quantum Rabi model, from Bogoliubov transformations and Bargmann–Fock methods to confluent Heun functions. It also discusses the model’s symmetries and the unsettled question of quantum integrability.
2. Eigenvalue problem and analytic solutions
- The quantum Rabi Hamiltonian couples a two-level system to a bosonic mode, and its rotating and counter-rotating terms distinguish the full model from the Jaynes–Cummings approximation.The reviewed analytic methods map the model to coupled differential equations and exploit its Z2 symmetry to determine the spectrum.Source: §2 · pp. 3–4
2.4. Integrability
- The model has a conserved Z2 parity, while the Jaynes–Cummings model’s continuous U(1) symmetry is broken by the counter-rotating term.Quantum integrability has multiple definitions: Braak’s level-labelling criterion is satisfied, but the model does not appear Yang–Baxter integrable in the usual form and full parameter space.Source: §2 · p. 3, §2.4 · p. 12
2.1. Solution via Bogoliubov transformation
- The Bogoliubov transformation removes linear bosonic terms; matching its two solution forms yields the G±(x) = 0 spectral condition.The two forms should represent the same state when the eigenvalue is nondegenerate.Source: §2.1 · pp. 5–6
2.2. Braak’s solution in Bargmann-Fock space
- In the Bargmann–Fock space, Braak’s analytic solution uses Z2 symmetry to obtain conditions for the full energy spectrum, with the spectrum characterized by analyticity of the wave-function components.The conditions G±(x, z) = 0 hold throughout the complex plane exactly when x = E + g2 is an energy-spectrum point.Source: §2 · p. 4, §2.2 · p. 8
2.3. Solution in terms of confluent Heun functions
- Variable transformations reduce the wave-function equations to confluent Heun functions, and the resulting spectral conditions agree with Braak’s G± conditions where the Heun functions converge.The G-function approach joins local series solutions into an entire global solution, expressing the eigenspectrum in terms of known functions.Source: §2.3 · pp. 10, 12
3. Energy spectrum and dynamics
This section reviews analytic G-function approaches to the quantum Rabi spectrum, including regular and exceptional levels, and surveys level statistics, dynamics, and Berry phases.
3. Energy spectrum and dynamics
- The spectrum is reviewed through analytic G-functions, alongside computational approaches including matrix truncation, series expansions, and continued fractions.The section emphasizes the G-function analytic approach.Source: §3 · p. 13
3.1. Energy spectrum
- Regular levels are the nondegenerate spectrum from G-function zeros; absent same-parity crossings, states are labeled by parity and bosonic index n.Source: §3.1 · p. 13
3.2. Exceptional points
- Exceptional levels include Juddian parity crossings, while G-function zero conditions apply only to the regular spectrum.Juddian solutions arise when confluent Heun functions terminate as polynomials under suitable parameter conditions.Source: §3.2 · pp. 16–17
3.3. Level spacing statistics
- Nearest-neighbor spacings show two peaks, which narrow and approach as g increases and widen and separate as ∆ increases.Source: §3.3 · p. 18
- For nonexceptional parameters, each energy interval contains zero, one, or two G-function roots per parity, and empty intervals cannot be adjacent.Source: §3.3 · p. 18
3.4. Dynamics
- Population-inversion collapse and revival occurs in weak coupling, disappears as g/ω increases, and reappears in the deep strong coupling regime.Source: §3.4 · pp. 19–20
- For initial state |+, 0b⟩ and ∆=0, the model has an exact evolving coherent-state solution exhibiting periodic collapse and revival.Source: §3.4 · p. 21
3.5. Berry phase
- A nonvanishing Berry phase appears in both the Jaynes–Cummings and Rabi models, with phases calculated numerically as a function of g/ω.Source: §3.5 · p. 21
4. Generalised quantum Rabi models
The review surveys analytic solutions for several generalized quantum Rabi models, emphasizing the asymmetric, anisotropic, and two-photon variants. Their spectra include parity-related exceptional-point structure, while the two-photon G-function method has a stated justification caveat.
4. Generalised quantum Rabi models
- The review also identifies solved generalizations including two-mode and two-qubit Rabi models, the Dicke model, and the N-state Rabi model.The two-qubit and Dicke models are noted as practically important, while work remains to develop the general Dicke case.Source: §4 · pp. 22–23
4.1. Asymmetric quantum Rabi model
- The asymmetric model’s bias breaks Z2 symmetry, yet its regular spectrum is determined by G-function zeros and its exceptional spectrum by confluent-Heun truncation conditions.The model can also be solved in the extended Hilbert space or by Bogoliubov transformation.Source: §4.1 · pp. 23–24
- Bias ǫ separates exceptional points with the same integer n; when ǫ is an integer multiple of 1/2ω, exceptional points with different n can form a two-fold degenerate energy point.The bias also induces conical intersections at two-fold degenerate Judd points at ǫ = 0, and intersections occur for ǫ an integer multiple of 1/2ω.Source: §4.1 · pp. 24–25
4.2. Anisotropic quantum Rabi model
- The anisotropic model assigns different couplings to rotating and counter-rotating terms: λ = 0 gives the JC model, while λ = 1 recovers the quantum Rabi model.Its regular spectrum is given by zeros of the G-functions.Source: §4.2 · pp. 25, 27
4.3. Two-photon quantum Rabi model
- The two-photon model’s regular spectrum is obtained from G_C(z, E), but those functions vanish identically, so the method is not well justified despite working with mixed entire and truncated functions.The same analytic approach is applicable to a model with an additional nonlinear coupling ±ωσ_z a†a.Source: §4.3 · p. 29
- Two-photon exceptional energies occur at level crossings, but do not cover every crossing under some parameter conditions; the remaining crossings are separately described, with the cited exceptional energies occurring for ∆/ω > 1/2.The spectrum is also shown using the G_C-function approach in figure 10.Source: §4.3 · pp. 29–30, Figure 10 · p. 30
5. Experimental realisation
The review outlines experimental realizations across cavity, superconducting-circuit, hybrid mechanical, and artificial-atom platforms. These examples range from JC-regime photon observations to ultrastrong and deep-strong coupling simulations.
5. Experimental realisation
- The review describes realizations of the quantum Rabi model across cavity systems, superconducting circuits, trapped ions, quantum dots, and hybrid systems.It gives an introductory survey of selected experimental settings.Source: §5 · p. 31
5.1. Single atom in a cavity
- A single-atom cavity experiment observed photon-number quantum jumps through nondestructive measurements, while its small g/ω places the interaction in the JC-model regime.The setup uses circular Rydberg states interacting dispersively with trapped intracavity photons.Source: §5.1 · pp. 31–32, Figure 11 · p. 32
5.2. Superconductor circuits
- A flux qubit coupled to an LC resonator reached ultrastrong coupling and exhibited a measured 50 MHz Bloch-Siegert shift.The qubit is galvanically attached to the resonator.Source: §5.2 · pp. 32–33, Figure 12 · p. 34
5.3. Hybrid quantum systems
- A Cooper Pair Box coupled to a silicon-nitride nanoresonator showed a dispersive shift of the nanomechanical frequency.The experiment measures the shift as a function of CPB parameters and compares selected traces with a model.Source: §5.3 · pp. 33–34, Figure 13 · p. 35
5.4. Artificial atoms in the ultrastrong and deep strong coupling regimes
- Circuit-QED artificial atoms and photonic-crystal setups are presented as platforms for simulating the quantum Rabi model, including its asymmetric version, in all coupling regimes.The review points to ultrastrong and deep-strong coupling simulations using flux qubits.Source: §5.4 · pp. 35–36
6. Concluding remarks
The article reviews analytic methods and full-eigenspectrum solutions for the quantum Rabi model and three generalizations, alongside dynamics and experimental realizations. It closes by identifying open questions in applying these solutions and extending them to more complex systems.
6. Concluding remarks
- The review covers analytic methods and full-eigenspectrum solutions for the quantum Rabi model and its asymmetric, anisotropic, and two-photon generalizations.It also reviews quantum Rabi model dynamics and experimental realizations.Source: §6 · p. 36
- The analytic solutions apply across the models’ full parameter spaces, including regimes that have traditionally challenged numerical approaches.Open questions include applications to dynamics, geometric phases, fidelity, and multipartite entanglement.Source: §6 · p. 36
- Further development is needed for analytic solutions of parafermion, N-state, and multi-qubit extensions, while uncovering phenomena testable by experiments and quantum simulators remains an overarching challenge.The review cites robust GHZ-like dark states built from isolated exact solutions as a promising theoretical development.Source: §6 · p. 36
Appendix A. Confluent Heun equation and confluent Heun function
The appendix derives the quantum Rabi model’s analytic solutions by reducing its wave-function equations to confluent Heun equations. It constructs two solution forms and relates their constants through the coupled equations and normalization.
Appendix A. Confluent Heun equation and confluent Heun function
- The confluent Heun function is represented by an infinite series regular around z = 0, whose coefficients obey a three-term recurrence relation.The appendix introduces the equation and its series framework.Source: Appendix A · p. 37
- Substituting φ1(z) = e^-gz f1(x1), with x1 = (g − z)/2g, reduces the wave-function equation to a confluent Heun equation; a corresponding transformation gives an equation for f2.The Heun parameters are specified in the derivation.Source: Appendix A · pp. 37–38
- The second local Frobenius solution represents a physical solution only when −β1 is a positive integer, because it contains a factor x1^-β1.The first solution is formally entire as a series in x1.Source: Appendix A · pp. 37–38
- The two constructed solution types must satisfy either coupled equation, fixing constant ratios, while wave-function normalization determines the remaining constants.The resulting forms are the analytic solutions listed in equations (57)–(60).Source: Appendix A · pp. 38–39