Extended Jaynes-Cummings Model (eJCM) Overview

• eJCM is a family of generalized Jaynes-Cummings models that incorporate non-RWA terms, nonlinear couplings, and multimode extensions to capture complex light–matter interactions.
• The model employs analytic and numerical methods—such as a coherent-state ansatz and recurrence relations—to compute spectra with high accuracy even at ultra-strong coupling regimes.
• Generalized variants include lattice, continuum, and relativistic formulations, offering versatile applications in circuit-QED, quantum simulation, and critical phenomena studies.

Searching arXiv for the cited papers and related uses of “extended Jaynes-Cummings model.” arXiv search query: "extended Jaynes-Cummings model" The extended Jaynes–Cummings model (eJCM) denotes, across the cited literature, a family of non-equivalent extensions of the Jaynes–Cummings Hamiltonian used to describe light–matter coupling beyond the textbook single-mode rotating-wave setting. In this usage, “extended” may refer to inclusion of counter-rotating terms, nonlinear and multi-boson exchange, squeezed or displaced bosonic modes, coherent driving, collective atomic modes, traveling-wave pulses, continuum-coupled autoionizing states, or lattice generalizations of Jaynes–Cummings–Hubbard type. A unifying feature is that the two-level subsystem remains coupled to bosonic or bosonized degrees of freedom, while the spectral, dynamical, or many-body structure departs from the standard JCM in a controlled way (Chen et al., 2010, Maldonado-Villamizar et al., 2020, Gutiérrez-Jáuregui et al., 2018, Alexanian, 2024, Wei et al., 2020).

1. Exact non-RWA formulation

A canonical meaning of eJCM is the Jaynes–Cummings model without the rotating-wave approximation. In laboratory variables the Hamiltonian is

$$H_0=\frac{\Delta}{2}\sigma_z+\omega a^\dagger a+g(a+a^\dagger)\sigma_x,$$

and after a $90^\circ$ rotation about $y$, with energies rescaled by $\omega$, it becomes

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

This Hamiltonian commutes with the parity operator

$\Pi=\sigma_x e^{i\pi a^\dagger a},$

so the Hilbert space splits into even and odd parity sectors. The parity decomposition is central because counter-rotating terms mix all photon numbers, whereas parity remains conserved (Chen et al., 2010).

Chen et al. solve this non-RWA problem by an extended bosonic coherent-state ansatz built from two coherent states of opposite displacements $\pm\alpha$, each dressed by a polynomial in $a^\dagger$ of degree $M$. After projection onto $$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$, the stationary Schrödinger equation reduces to a three-term recurrence for coefficients $90^\circ$0, with $90^\circ$1 and $90^\circ$2. The truncation condition $90^\circ$3 yields a single-variable polynomial equation of degree $90^\circ$4 in $90^\circ$5, and each real root $90^\circ$6 gives an eigenenergy

$90^\circ$7

The associated recurrence determines all coefficients $90^\circ$8, and hence the full eigenfunction. In this construction, the infinite-dimensional bosonic problem is mapped exactly onto a one-variable secular equation whose real roots determine the complete spectrum for arbitrary coupling $90^\circ$9 and detuning $y$0 (Chen et al., 2010).

The method is both analytic and numerical. For given $y$1, one increases $y$2 until the real roots stabilize; a relative tolerance $y$3 is explicitly cited as a convergence criterion. Typical $y$4 suffices to converge the first $y$5 levels even at ultra-strong coupling $y$6, and in the ultra-strong regime the required truncation can fall to $y$7, whereas direct Fock-space diagonalization would require thousands of photon states for $y$8. The construction was motivated in part by circuit-QED systems reaching $y$9 and even $\omega$0, where Bloch–Siegert shifts, squeezing, and ground-state entanglement become nonperturbative (Chen et al., 2010).

2. Systematic approximations beyond the rotating-wave approximation

A complementary exact formulation uses an extended Swain ansatz truncated directly in the Fock basis,

$\omega$1

again exploiting the conserved parity $\omega$2. Insertion into the Schrödinger equation yields the recursion

$\omega$3

with $\omega$4, $\omega$5, and boundary condition $\omega$6. As $\omega$7, the roots of the resulting degree-$\omega$8 polynomial in $\omega$9 converge to the exact spectrum (He et al., 2012).

This formulation supports a hierarchy of analytic approximations. The zero-order approximation gives

$$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$0

with ground-state energy $$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$1, and contains no counter-rotating-term effects. The first-order $$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$2 block approximation reproduces exactly the standard RWA spectrum,

$$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$3

The second-order $$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$4 block approximation produces the first counter-rotating corrections: at resonance, the $$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$5 shifts and three-state admixtures are identified explicitly as the first signatures of counter-rotating terms. The second-order energies remain accurate with errors $$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$6 up to $$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$7, and therefore cover the $$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$8 regime cited for modern circuit QED (He et al., 2012).

The same second-order treatment modifies vacuum Rabi splitting. Under the RWA, the probe operator excites the ground state into an equal-weight superposition of $$H=-\frac{\Delta}{2}\sigma_x+a^\dagger a+g(a+a^\dagger)\sigma_z.$$9, producing two peaks of equal height separated by $\Pi=\sigma_x e^{i\pi a^\dagger a},$0. With counter-rotating terms, the dressed ground state acquires $\Pi=\sigma_x e^{i\pi a^\dagger a},$1 and $\Pi=\sigma_x e^{i\pi a^\dagger a},$2 admixtures, the peak frequencies shift to

$\Pi=\sigma_x e^{i\pi a^\dagger a},$3

and the total splitting becomes

$\Pi=\sigma_x e^{i\pi a^\dagger a},$4

The peak heights are unequal, with $\Pi=\sigma_x e^{i\pi a^\dagger a},$5; at $\Pi=\sigma_x e^{i\pi a^\dagger a},$6, the splitting shift is approximately $\Pi=\sigma_x e^{i\pi a^\dagger a},$7, whereas the height ratio $\Pi=\sigma_x e^{i\pi a^\dagger a},$8 is predicted to be experimentally visible (He et al., 2012).

3. Generalized rotating-wave extensions

Beyond the non-RWA single-mode problem, the label eJCM is also applied to several generalized Hamiltonians that retain the rotating-wave approximation but enlarge the interaction structure. One broad formulation includes Stark shifts, Kerr-type nonlinearities, multi-boson exchange, and intensity-dependent coupling: $\Pi=\sigma_x e^{i\pi a^\dagger a},$9 Here $\pm\alpha$0 generates qubit level shifts, $\pm\alpha$1 nonlinear boson dispersion, $\pm\alpha$2 is the number of exchanged bosons, and $\pm\alpha$3 makes the coupling intensity dependent. The model possesses a graded Lie algebra built from nilpotent supercharges $\pm\alpha$4, a SUSY Hamiltonian $\pm\alpha$5, and excitation operators $\pm\alpha$6. A unitary transformation $\pm\alpha$7 diagonalizes the effective Hamiltonian sector by sector, yielding a generalized Rabi frequency $\pm\alpha$8, closed-form eigenstates, and a closed-form propagator. Standard JC, Kerr, Buck–Sukumar, Stark-shifted two-photon, and algebraically deformed models appear as special cases (Maldonado-Villamizar et al., 2020).

A different generalized eJCM mixes Jaynes–Cummings and anti-Jaynes–Cummings terms linearly,

$\pm\alpha$9

Using the $a^\dagger$0 generators $a^\dagger$1, squaring the Hamiltonian leads to a Klein–Gordon-type operator that is diagonalized by a squeezing, or tilting, transformation $a^\dagger$2. The exact spectrum is

$a^\dagger$3

and the eigenfunctions are squeezed number states. The same model is connected in the source to the relativistic parametric amplifier and to a single trapped ion driven simultaneously on red and blue motional sidebands (Ojeda-Guillén et al., 2014).

Related modified models are obtained by replacing the cavity operators by squeezed or squeezed-coherent operators. One construction introduces

$a^\dagger$4

so that the transformed Hamiltonian contains two-photon field terms $a^\dagger$5, counter-rotating atom–field terms, and an atomic-exchange term proportional to $a^\dagger$6. Under the RWA and with $a^\dagger$7, the interaction reduces to an effective coupling $a^\dagger$8, with dressed-state blocks identical in form to the ordinary JCM but sampled over altered squeezed-coherent photon statistics. A closely related squeezed-photon model writes the interaction in terms of $a^\dagger$9, thereby interpolating between the JCM at $M$0 and the Rabi Hamiltonian in the $M$1 limit (Alexanian, 2022, Alexanian, 2024).

Variant	Defining modification	Source
Non-RWA single-mode eJCM	Counter-rotating term and parity-resolved exact polynomial	(Chen et al., 2010, He et al., 2012)
Nonlinear generalized eJCM	$M$2, $M$3, $M$4, $M$5-boson exchange	(Maldonado-Villamizar et al., 2020)
JC/anti-JC mixture	Simultaneous $M$6 and $M$7 couplings with $M$8	(Ojeda-Guillén et al., 2014)
Squeezed or displaced exchange	$M$9 or $$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$0	(Alexanian, 2022, Alexanian, 2024)

4. Relativistic correspondences and analog mappings

Several eJCMs admit exact reformulations in relativistic language. The driven Jaynes–Cummings–Rabi Hamiltonian,

$$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$1

is mapped to a massless charged Dirac particle in constant homogeneous electric and magnetic fields. In the Dirac picture, the field invariant $$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$2 separates two regimes. For $$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$3, corresponding to subcritical drive, the quasienergies are discrete,

$$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$4

with $$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$5, and the spectrum collapses as $$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$6. For $$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$7, corresponding to supercritical drive, the spectrum becomes continuous. The critical amplitude is therefore $$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$8. In the generalized case with counter-rotating coupling ratio $$(a^\dagger)^m e^{\pm \alpha a^\dagger}\lvert 0\rangle$$9, a squeezing transform leads to $90^\circ$00. Below threshold, the eigenstates are Hermite–Gaussian spinors; above threshold, Weber-function spinors appear (Gutiérrez-Jáuregui et al., 2018).

The generalized linear $90^\circ$01 Hamiltonian discussed above is also presented as a relativistic parametric amplifier. In its nonrelativistic limit, the model reproduces the degenerate parametric-amplifier Hamiltonian

$90^\circ$02

The same source gives an exact trapped-ion realization under Lamb–Dicke and rotating-wave approximations, where simultaneous red- and blue-sideband driving reproduces the full eJCM coupling structure (Ojeda-Guillén et al., 2014).

A further mapping relates the $90^\circ$03 Dirac–Moshinsky oscillator coupled to an external isospin field to a two-atom single-mode eJCM,

$90^\circ$04

The conserved quantity $90^\circ$05 partitions the Hilbert space into invariant manifolds. Closed-form time-dependent states, reduced density matrices, concurrence or von Neumann entropy, and population inversion are given for both initial number states and initial coherent states. In the abstracted conclusion of that study, the coherent state is reported to give a better description than the number state for entanglement and population inversion (Obada et al., 2018).

5. Collective, traveling-wave, continuum, and lattice realizations

The eJCM can emerge as an effective low-energy theory for collective matter degrees of freedom. In a system consisting of a single control atom, a cavity, and a sample of $90^\circ$06 identical two-level atoms, adiabatic elimination in the dispersive regime $90^\circ$07, $90^\circ$08 yields a collective bosonic mode

$90^\circ$09

The effective Hamiltonian takes the Jaynes–Cummings form

$90^\circ$10

with $90^\circ$11, $90^\circ$12, and $90^\circ$13. The resulting dynamics includes vacuum-Rabi oscillations, collapse and revival for an initial coherent state of the collective mode, stepwise Fock-state generation, Schrödinger-cat preparation, and Wigner-function reconstruction through selective displacements and atomic readout (Zheng, 2012).

For traveling quantum pulses, a single cavity mode is insufficient. The cascaded-systems construction introduces “virtual” cavity modes $90^\circ$14 and $90^\circ$15 whose leak rates are chosen to emit or absorb prescribed temporal profiles $90^\circ$16 and $90^\circ$17. In the emission-only model,

$90^\circ$18

with jump operator

$90^\circ$19

The input–output version adds a second virtual cavity and a cross term $90^\circ$20. In the interaction picture for $90^\circ$21, an auxiliary orthogonal mode survives explicitly, carrying non-Markovian memory. The source states directly that this differs from a “naïve” time-dependent JCM with coupling $90^\circ$22, because the correct couplings depend on temporal integrals of the pulse shape and the Lindblad operators collect radiation into all modes not explicitly retained (Christiansen et al., 2024).

Autoionizing polaritons furnish a continuum-coupled eJCM. The model contains a ground state $90^\circ$23, bright and dark autoionizing states $90^\circ$24, continua of opposite parity, and a quantized infrared mode $90^\circ$25. The interaction

$90^\circ$26

couples the two autoionizing states, while Fano configuration interaction supplies complex widths. Diagonalization of the effective non-Hermitian $90^\circ$27 polariton Hamiltonian yields complex eigenenergies $90^\circ$28. The widths $90^\circ$29 depend on mixing coefficients, photoionization rates, autoionization rates, and interference terms. At resonance, destructive interference can minimize $90^\circ$30, producing stabilization relative to the field-free width (Cariker et al., 2022).

The lattice extension most directly called the extended Jaynes–Cummings–Hubbard model adds photon hopping and nearest-neighbor repulsion between atomic excitations: $90^\circ$31 Large-scale worm quantum Monte Carlo on one-dimensional chains, square lattices, and triangular lattices shows that for $90^\circ$32, bipartite geometries exhibit a first-order solid–superfluid transition and no stable supersolid in the thermodynamic limit, whereas the triangular lattice supports a stable supersolid. At fixed $90^\circ$33, the supersolid satisfies $90^\circ$34 and $90^\circ$35 simultaneously for $90^\circ$36 as $90^\circ$37. The algorithm is reported to operate in continuous imaginary time, to measure stiffness from winding-number fluctuations, to reach $90^\circ$38, and to remain free of a sign problem in this bosonic setting (Wei et al., 2020).

6. Critical phenomena and computational complexity

The squeezed-photon extension provides an explicit interpolation between Jaynes–Cummings and Rabi limits. With

$90^\circ$39

the Hamiltonian in the original basis contains both rotating and counter-rotating couplings weighted by $90^\circ$40 and $90^\circ$41. In the limit $90^\circ$42, the ordinary JCM is recovered exactly; in the limit $90^\circ$43, one obtains the quantum Rabi Hamiltonian with effective coupling $90^\circ$44. After a unitary transformation and Bogoliubov diagonalization, the source identifies a soft-mode condition $90^\circ$45, giving the critical coupling

$90^\circ$46

At $90^\circ$47 this reduces to $90^\circ$48, while at $90^\circ$49 it yields $90^\circ$50. The associated phase structure is described as a second-order superradiant transition, with $90^\circ$51 serving as order parameter (Alexanian, 2024).

The driven Jaynes–Cummings–Rabi correspondence to the Dirac equation identifies a different critical phenomenon: quasienergy collapse at $90^\circ$52. Below critical drive, the spectrum is discrete and Landau-like; above critical drive, it is continuous. Taken together with the squeezed-photon model, these results show that the phrase “phase transition” in eJCM contexts can denote distinct mechanisms: a soft-mode superradiant transition in one construction and a discrete-to-continuous quasienergy collapse in another. This suggests that the term eJCM is structurally broad rather than tied to a single universal critical theory (Gutiérrez-Jáuregui et al., 2018, Alexanian, 2024).

The same breadth appears in quantum-simulation complexity analysis for multimode eJCMs under the rotating-wave approximation. For $90^\circ$53 truncated bosonic modes,

$90^\circ$54

with

$90^\circ$55

After Pauli decomposition, first- and second-order product formulas yield explicit operator-norm error bounds. The first-order scheme requires

$90^\circ$56

while the second-order scheme satisfies $90^\circ$57 and therefore $90^\circ$58. The paper also treats pure-state binary encoding, mixed-state vectorization through the Choi–Jamiolkowski isomorphism, and fault-tolerant compilation on a rotated surface-code architecture. For the parameter set $90^\circ$59, $90^\circ$60 ($90^\circ$61), $90^\circ$62, the reported estimate is approximately $90^\circ$63 $90^\circ$64 gates, about $90^\circ$65 physical qubits, and execution in $90^\circ$66 surface-code cycles (Nguyen et al., 16 Sep 2025).

In aggregate, the cited literature presents the eJCM less as a single model than as a technical umbrella for Jaynes–Cummings-type Hamiltonians enlarged by non-RWA physics, generalized couplings, additional modes, lattice structure, or effective reductions from more complex systems. The common payoff is exact or controlled access to regimes in which parity, squeezing, coherent displacement, collective bosonization, continuum interference, many-body order, or relativistic analogies materially change the spectrum and dynamics.