Jaynes-Cummings Hamiltonian

Updated 21 December 2025

Jaynes–Cummings Hamiltonian is a fundamental quantum-optical model describing a two-level system interacting with a single photon mode under the rotating-wave approximation.
Its exact solvability enables detailed analysis of phenomena such as vacuum Rabi oscillations, dressed states, and photon blockade in various experimental settings.
Extensions of the model incorporate non-RWA effects, ultrastrong coupling, and many-body generalizations, broadening its applications in modern quantum optics.

The Jaynes–Cummings Hamiltonian is the fundamental quantum-optical model describing the interaction between a two-level system—commonly referred to as an "atom" or "qubit"—and a single-mode quantized electromagnetic field (photon mode) under the rotating-wave approximation (RWA). This model forms the theoretical foundation of cavity quantum electrodynamics (cQED) and circuit QED, providing the canonical description of phenomena such as vacuum Rabi oscillations, dressed states, photon blockade, hybridized polaritons, and quantum phase transitions in light–matter systems. Its simplicity and exact solvability enable rigorous comparison between analytical theory and a variety of experiments, ranging from atomic cavity QED to superconducting circuits and engineered quantum photonic structures (Maisi, 10 Mar 2025).

1. Canonical Formulation and Diagonalization

In natural units ( $\hbar = 1$ ), the standard Jaynes–Cummings Hamiltonian is:

$\hat{H}_{JC} = \omega_c\, a^\dagger a + \frac{\omega_q}{2}\, \sigma_z + g\, (a\, \sigma_+ + a^\dagger\, \sigma_-)$

where $a$ ( $a^\dagger$ ) annihilates (creates) a photon in the mode with frequency $\omega_c$ , $\sigma_z$ is the Pauli- $z$ operator for the two-level system (transition frequency $\omega_q$ ), and $g$ is the vacuum Rabi coupling (Maisi, 10 Mar 2025, Bina, 2011).

The total excitation number $N = a^\dagger a + \sigma^+ \sigma^-$ is conserved. Diagonalization in each invariant subspace defined by $N$ gives, for the $n$ -th manifold (with $n$ photons or atomic excitation):

Dressed-state energies

$E_{n,\pm} = \omega_c (n+\frac{1}{2}) \pm \frac{\Omega_n}{2}$

where $\Omega_n = \sqrt{(\omega_q - \omega_c)^2 + 4g^2(n+1)}$ .

Dressed eigenstates: superpositions of $|e,n\rangle$ (atom excited, $n$ photons) and $|g,n+1\rangle$ (atom in ground, $n+1$ photons), with photon-number-dependent mixing angles (Bina, 2011, Frenzel, 2012).

On resonance ( $\omega_q = \omega_c$ ), the Rabi splitting grows as $2g\sqrt{n+1}$ , a direct signature of the field quantization.

2. Input–Output Theory and Photonic Mode Description

For practical quantum optics, the key measurable is the transmission response to a probing field. The input–output formulation yields, in the weak-drive limit, the transmission coefficient as

$T(\omega)=|\kappa_e A(\omega)|^2,$

with $A(\omega)$ incorporating both cavity and qubit losses and the cavity–qubit coupling (Maisi, 10 Mar 2025). In the strong-coupling regime, the spectral lines split into two well-separated Lorentzians at $\omega_\pm = (\omega_c+\omega_q)/2 \pm \frac{\Omega}{2}$ , each corresponding to a "hybrid photonic mode":

$T(\omega) \simeq \frac{\kappa_{e\pm}^2}{\left[\frac{\kappa_{\mathrm{tot}}^\pm}{2}\right]^2 + (\omega - \omega_\pm)^2}$

Here, the effective external coupling $\kappa_{e\pm}$ , internal loss $\kappa_{i\pm}$ , and total linewidth $\kappa_{\mathrm{tot}}^\pm$ are given by simple weighting of the bare cavity and qubit channels by the photonic and electronic admixtures in each dressed state. The "visibility" and linewidth of each resonance provide immediate spectroscopic access to the degrees of hybridization and the limiting decoherence rates:

For zero detuning, both dressed modes inherit half the cavity and qubit loss, and their visibility is maximal.
If the qubit is more coherent than the resonator ( $\Gamma \ll \kappa$ ), it is possible to realize increasing resonance sharpness while maintaining strong visibility as the hybridized mode approaches the qubit frequency (Maisi, 10 Mar 2025).

3. Generalizations, Extensions, and Infinite-Coupling Regimes

Non-RWA (Full Quantum Rabi Hamiltonian)

Without the rotating-wave approximation, the Hamiltonian acquires counter-rotating terms:

$H = a^\dagger a + \frac{\Delta}{2} \sigma_z + g(a + a^\dagger)\sigma_x$

The resulting operator maps to an infinite Jacobi matrix with purely discrete spectrum. Unlike the RWA case, the high-excitation spectrum is asymptotically linear, with only a constant $O(1)$ shift, and lacks the characteristic $\sqrt{n}$ splitting. The excitation number is no longer conserved, and the model displays fundamentally different spectral properties for large photon numbers (Monvel et al., 2015).

Ultrastrong Jaynes–Cummings and Quantum Phase Transitions

Via synchronous modulation of both the qubit and cavity, all counter-rotating terms in the quantum Rabi Hamiltonian can be dynamically suppressed, yielding an exact Jaynes–Cummings form even for $g$ approaching or exceeding the bare transition frequencies ("ultrastrong" regime). In this regime, ultrafast gate operations and new types of quantum phase transitions in the excitation-number-conserving JC model become accessible, including ground-state crossings at specific critical values of $g/\omega_0$ (Huang et al., 2019).

4. Many-Body and Multi-Mode Generalizations

The JC Hamiltonian generalizes naturally to arrays (Jaynes–Cummings–Hubbard models), optomechanical hybrids, and multimode extensions:

Two-site Jaynes–Cummings–Hubbard: Each site is a JC system, with photon hopping as the intersite coupling. Systematic factorization and Wei–Norman techniques enable analytic propagators in regimes where intersite and atom–photon couplings are noncommuting but hierarchically organized (Ramos-Prieto et al., 2019).
Many-body expansions: The operator diagonalization of the JC Hamiltonian reveals an infinite, but typically rapidly convergent, $k$ -body bosonic expansion in terms of normal-ordered powers of effective "dressed" operators. In the dispersive limit ( $|g/\Delta| \ll 1$ ), higher-order terms die off; at resonance, $k$ -body contributions remain significant, marking a breakdown of the naive Bose–Hubbard analogy (Smith et al., 2021).

Characteristic quantum phases in the two-site system (Mott insulator, superfluid, polaritonic analogs) can be precisely mapped to the effective nonlinearity and intersite hybridization calculated from the full JC structure.

5. Symmetry, Supersymmetry, and Hierarchies

The Jaynes–Cummings Hamiltonian’s algebraic structure is deeply connected to supersymmetric quantum mechanics (SUSY-QM). SUSY transformations relate JC Hamiltonians with different detunings, and the "anti–Jaynes–Cummings" model arises as the SUSY partner of the standard JC Hamiltonian. Iterating the SUSY construction organizes families of JC models into shape-invariant hierarchies, with explicit intertwining operators mapping eigenvectors between detunings and providing a unifying framework for selection rules, emergent symmetries, and effective classification of hybrid light–matter systems (Ateş et al., 28 Apr 2025).

6. Non-Hermitian Extensions and Open Quantum Systems

Incorporating decoherence and loss, the JC Hamiltonian can be promoted to a non-Hermitian operator with complex energies for the electronic states. This extension gives rise to $\mathcal{PT}$ -symmetric physics and exceptional points (EPs) in the spectrum. At these exceptional points, the eigenvalues and eigenvectors coalesce, leading to abrupt changes in the system’s optical response, enhanced parameter sensitivity, and new dynamical features relevant for open quantum optics and parameter estimation protocols (Bagarello et al., 2016).

7. Physical Interpretation and Limitations

The Jaynes–Cummings Hamiltonian provides a minimal yet comprehensive model for the quantum dynamics of light–matter interaction, fully capturing:

Photon-number-dependent Rabi oscillations, collapses and revivals, and entanglement generation.
Quantitative descriptions of linewidths, decoherence, and spectroscopic signatures in cavity QED and circuit QED.
The breakdown of the rotating-wave approximation at ultrastrong coupling or in the presence of squeezing, where more general models (e.g., Rabi or squeezed-coherent JC Hamiltonians) are necessary (Alexanian, 2022).

Its applicability is limited by the validity of the two-level approximation, the rotating-wave regime ( $g \ll \omega_c, \omega_q$ ), and neglect of irreversible atom and cavity losses, although generalizations incorporating these effects—by master-equation or non-Hermitian extensions—are well developed.