Jaynes–Cummings Model in Quantum Optics

• The Jaynes–Cummings model is a solvable quantum framework describing coherent excitation exchange between a two-level system and a single quantized mode under the rotating-wave approximation.
• It explains key phenomena such as Rabi oscillations, collapse and revival of quantum states, and the formation of dressed states through conserved excitation manifolds.
• The model underpins diverse applications in cavity QED, circuit QED, and quantum information science, while inspiring engineered variants for exploring ultra-strong coupling and beyond.

The Jaynes–Cummings model (JCM) is the standard rotating-wave description of a single two-level system interacting with a single quantized bosonic mode. In its canonical form it combines a two-level truncation for matter, a single-mode approximation for the field, and the rotating-wave approximation (RWA), thereby reducing light–matter dynamics to an exactly solvable excitation-exchange problem. Its algebraic structure underlies dressed-state spectroscopy, vacuum and few-photon Rabi oscillations, collapse-and-revival phenomena, dispersive shifts, and a large family of descendants spanning cavity QED, circuit QED, trapped ions, waveguide QED, and quantum information science (Larson et al., 2022, Bina, 2011).

1. Canonical formulation and approximations

In the cavity-QED formulation, the model describes a two-level atom with states $|g\rangle, |e\rangle$ and transition frequency $\omega_a$ coupled to one field mode of frequency $\omega_f$, with annihilation and creation operators $a,a^\dagger$. After the dipole approximation, the single-mode approximation, the two-level truncation, and the RWA, the Hamiltonian takes the standard form

$$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$

with $$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$ and coupling $g$ (Bina, 2011). A common equivalent notation is

$$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$

in units $\hbar=1$ (Quesada et al., 2013).

The RWA removes the counter-rotating terms $\sigma_+a^\dagger$ and $\omega_a$0, which oscillate at frequencies $\omega_a$1 in the interaction picture. The underlying full light–matter Hamiltonian before the RWA contains both rotating and counter-rotating contributions,

$\omega_a$2

and the JCM is recovered when the anti-resonant terms are neglected near resonance and for $\omega_a$3 (Larson et al., 2022, Bina, 2011).

The model is routinely used in optical and microwave cavity QED, trapped ions, circuit QED, waveguide QED, quantum dots, NV centers, and related settings (Larson et al., 2022). This breadth reflects the fact that the JCM is a formal template for coherent excitation exchange rather than a platform-specific Hamiltonian.

2. Conserved excitation number and dressed-state spectrum

A defining structural property is the conserved excitation number

$\omega_a$4

which counts photons plus atomic excitations (Quesada et al., 2013). Because of this conservation law, the Hilbert space decomposes into invariant two-dimensional manifolds $\omega_a$5, and the Hamiltonian reduces in each manifold to a $\omega_a$6 block (Bina, 2011).

With detuning $\omega_a$7, the generalized Rabi frequency is

$\omega_a$8

and the dressed energies are

$\omega_a$9

The corresponding dressed states are superpositions of $\omega_f$0 and $\omega_f$1, with mixing angle defined by

$\omega_f$2

(Bina, 2011).

On resonance, $\omega_f$3, one has $\omega_f$4, so the dressed states become the symmetric and antisymmetric combinations

$\omega_f$5

In the large-detuning regime, $\omega_f$6, the model reduces to a dispersive Hamiltonian of the form

$\omega_f$7

which produces Stark shifts $\omega_f$8 (Larson et al., 2022).

This block structure is the basis for most of the model’s exact results. It also defines the common bare-state versus dressed-state distinction: bare states diagonalize the uncoupled atom and field, whereas dressed states diagonalize the interacting Hamiltonian within each fixed-excitation manifold.

3. Coherent dynamics, collapse and revival, and phase-space descriptions

For an initial Fock state $\omega_f$9, the JCM predicts coherent excitation exchange between $a,a^\dagger$0 and $a,a^\dagger$1. In resonance, the populations are

$a,a^\dagger$2

with $a,a^\dagger$3 (Bina, 2011). The corresponding inversion oscillates at frequency $a,a^\dagger$4, making the $a,a^\dagger$5 nonlinearity the central spectral signature of the model.

If the field is instead prepared in a coherent state $a,a^\dagger$6, the excited-state population becomes

$a,a^\dagger$7

so the superposition of many incommensurate Rabi frequencies produces collapse and revival (Bina, 2011). For $a,a^\dagger$8, the revival time is

$a,a^\dagger$9

and the monograph further notes cat-state formation at $$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$0, where the atom disentangles and the field ends in a Schrödinger-cat superposition of two coherent states (Bina, 2011, Larson et al., 2022).

A phase-space reformulation has recently been given in terms of a hybrid Stratonovich–Weyl correspondence on $$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$1. In that construction the full Wigner function

$$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$2

encodes Rabi oscillations, inversion, and entanglement generation, while the reduced field Wigner function yields the field purity

$$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$3

(Sanchez-Cordova et al., 29 Jun 2025). In this formulation, the Wigner representation is informationally complete for the hybrid qubit–boson system.

Time-dependent resonant couplings preserve exact solvability when the interaction-picture Hamiltonian is

$$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$4

because $$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$5 for all $$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$6 under resonance. The evolution then depends only on the pulse area

$$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$7

so the constant-coupling solution is recovered by the substitution $$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$8 (Tsutsui et al., 26 Aug 2025). Linear ramps compress collapse–revival patterns, hyperbolic-secant pulses freeze inversion and entanglement to asymptotic values, and sinusoidal modulations can induce periodic revivals and population trapping for thermal initial fields (Tsutsui et al., 26 Aug 2025).

4. Open-system formulations, coherent driving, and traveling pulses

In open-system treatments, the cavity and atom couple to separate reservoirs with leakage rates $$H_{\rm JC} = \frac{\hbar\omega_a}{2}\,\sigma_z +\hbar\omega_f\,a^\dagger a +\hbar g\,(\sigma_+ a+\sigma_- a^\dagger),$$9 and $$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$0. In the coherent-drive setting, the rotating-frame Hamiltonian can be written as

$$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$1

with cavity drive

$$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$2

or direct atomic drive

$$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$3

(Fischer et al., 2018).

A central result is that cavity drive and atom drive generate identical system dynamics except for a trivial coherent offset. Using the Mollow displacement transformation and the operator shift

$$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$4

with $$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$5, the atom dynamics under cavity drive are identical to those under direct atomic drive with

$$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$6

up to an overall coherent offset in the cavity output field (Fischer et al., 2018). The offset can be canceled interferometrically by mixing $$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$7 with a local oscillator on a $$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$8 beamsplitter and choosing

$$\sigma_\pm=|e\rangle\langle g|, |g\rangle\langle e|$$9

so that the remaining field contains only the quantum fluctuations generated by the JCM interaction (Fischer et al., 2018). The result holds for arbitrary pulsed multimode coherent inputs, not only for cw drives.

A distinct limitation appears for propagating quantum radiation. For a traveling pulse in free space or a waveguide, the standard single-mode JCM should not be expected to properly describe the interaction between an emitter and a traveling pulse because of the multimode continuum of eigenmodes (Christiansen et al., 2024). A cascaded-quantum-system formulation addresses this by introducing a “virtual” input cavity $g$0 that emits the target temporal mode $g$1, optionally together with an output cavity $g$2 for a chosen output mode $g$3. This yields modified JCM-like Hamiltonians with time-dependent couplings and Lindblad loss terms rather than a closed single-mode unitary model (Christiansen et al., 2024). In the long-pulse limit, $g$4 constant, one recovers a standard time-dependent JCM-like coupling; in the weak TLS–continuum limit, $g$5, the loss terms vanish and the dynamics reduce to a closed single-mode virtual-cavity problem (Christiansen et al., 2024).

A common misconception is therefore that the JCM directly describes all pulse-scattering scenarios. The cascaded treatment shows that this is generally false for traveling pulses, even though JCM-like structure can be recovered in controlled limits (Christiansen et al., 2024).

5. Beyond the RWA: parity, counter-rotating terms, and deep-strong coupling

When counter-rotating terms are retained, the light–matter Hamiltonian becomes

$g$6

and the RWA no longer applies in general (Casanova et al., 2010). In this regime the conserved quantity is not the excitation number but a parity operator, written for example as

$g$7

or, in an alternative representation,

$g$8

(Casanova et al., 2010, He et al., 2012). The Hilbert space then splits into two decoupled parity chains.

Using parity and an extended Swain’s ansatz, the full non-RWA problem can be solved exactly as a tridiagonal determinant problem (He et al., 2012). Systematic approximations emerge naturally from this structure. The first-order truncation reproduces the usual JCM spectrum under the RWA, while the second-order truncation yields the leading effects of counter-rotating terms, including explicit corrections to dressed energies, the ground-state energy, and the vacuum Rabi splitting (He et al., 2012). In that analysis, the second-order expressions remain accurate up to the ultra-strong coupling regime (He et al., 2012).

In the deep-strong coupling regime, defined by $g$9, the model admits a different physical picture. In a displaced parity basis, photon-number wavepackets propagate along parity chains, reach a turning point near $$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$0, and bounce back, yielding collapse and revival of the initial population even when the oscillator starts in vacuum (Casanova et al., 2010). For $$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$1, the revival probability is

$$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$2

so complete collapses and full revivals occur with period $$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$3 (Casanova et al., 2010).

A classical analogue of this dynamics has been proposed in engineered waveguide superlattices. There the full non-RWA model maps onto two uncoupled tight-binding chains with nonuniform couplings $$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$4, a linear index gradient $$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$5, and an alternating propagation-constant mismatch $$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$6. In the deep-strong-coupling limit, wavepacket bouncing in Fock space is reinterpreted as generalized Bloch oscillations in an inhomogeneous lattice (Longhi, 2011).

6. Descendants, engineered variants, and entanglement structure

Many descendants preserve the JCM excitation-exchange core while altering the oscillator algebra, the number of modes, the effective matter degree of freedom, or the route by which the Hamiltonian is engineered. The 2022 monograph groups such descendants across atomic physics, quantum optics, solid-state physics, and quantum information science (Larson et al., 2022).

One line of generalization replaces the bosonic mode by other effective oscillators. In the collective-mode construction, a single control atom and an atomic sample interact with a nonresonant cavity mode, and under appropriate dispersive conditions the coherent energy exchange between the control atom and the sample is described by an effective JCM for a collective atomic mode (Zheng, 2012). In the finite-Kerr-medium construction, an $$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$7 representation produces a bounded field excitation number and an intensity-dependent coupling

$$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$8

so the standard JCM is recovered only in the $$H_{\rm JC} = \omega\,a^\dagger a+\frac{\Omega}{2}\sigma_z +g(\sigma_+ a+\sigma_- a^\dagger),$$9 limit (Ruiz et al., 2013).

A second line enlarges the mode structure. Two-mode extensions include a simultaneous JC–AJC model with SU(1,1) symmetry and a two-JC model with SU(2) symmetry, both solved by tilting transformations and Perelomov number coherent states (Choreño et al., 2017). A single-mode generalized JC model with simultaneous JC and AJC couplings,

$\hbar=1$0

connects the theory to the relativistic parametric amplifier and to the quantum simulation of a single trapped ion (Ojeda-Guillén et al., 2014).

A third line concerns engineered strong-coupling Hamiltonians and quantum phase transitions. By synchronous modulation of both qubit and oscillator frequencies in the quantum Rabi model, counter-rotating terms can be suppressed without reducing the rotating-wave coupling strength, producing an ultrastrong JC Hamiltonian

$\hbar=1$1

in the interaction frame (Huang et al., 2019). A related qubit-frequency modulation scheme yields an effective deep-strong JC Hamiltonian with drive-tunable parameters and a finite-size quantum phase transition signaled by nonzero average cavity photons in the ground state (Liu et al., 2023). In the standard resonant JCM, the first level crossing between $\hbar=1$2 and $\hbar=1$3 occurs at $\hbar=1$4, above which the ground state changes and $\hbar=1$5 (Liu et al., 2023).

The entanglement structure of the model is richer than the common qubit–oscillator narratives based only on free entanglement or separability. Quesada and Sanpera analyzed $\hbar=1$6 states commuting with the conserved excitation number and showed that for $\hbar=1$7 the JCM admits PPT yet non-separable bound-entangled states; they also showed that Jaynes–Cummings interaction can dynamically generate bound entanglement (Quesada et al., 2013). This corrects the widespread expectation that the symmetries enforced by excitation-number conservation preclude such states.

Taken together, these descendants show that the JCM is less a single Hamiltonian than a solvable organizing principle. Its central invariants—fixed-excitation manifolds under the RWA, parity sectors beyond it, and analytically tractable dressed dynamics—continue to structure modern work on coherent control, engineered nonlinearities, multimode scattering, phase transitions, and hybrid-state entanglement (Larson et al., 2022).