---
title: Nonlinear Jaynes-Cummings Model
url: https://www.emergentmind.com/topics/nonlinear-jaynes-cummings-model
type: topic
---

# Nonlinear Jaynes-Cummings Model

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The nonlinear Jaynes–Cummings model designates a family of extensions of the Jaynes–Cummings (JC) Hamiltonian in which the light–matter interaction, the field sector, or both acquire explicit excitation-dependent structure. In the narrow sense, it includes Hamiltonians with Kerr-like field nonlinearities, intensity-dependent couplings, multiphoton exchange, or deformed bosonic operators; in a broader spectroscopic sense, it also includes the intrinsic anharmonicity of the JC dressed-state ladder, whose normal-mode splitting scales as \(\sqrt{n}\) [2202.00330]. Standard realizations begin from the rotating-wave Hamiltonian
\[
H_{\mathrm{JC}}=\omega_c a^\dagger a+\frac{\omega_a}{2}\sigma_z+g(a\sigma_+ + a^\dagger \sigma_-),
\]
for which the total excitation number is conserved, but nonlinear descendants replace the constant coupling and harmonic field dispersion by excitation-dependent functions that bend the spectrum, alter collapse–revival structure, and change the entanglement dynamics [1303.5892].

## 1. Canonical structure and the meaning of nonlinearity

The canonical JC model is block diagonal in the bare basis \(\{|e,n\rangle,|g,n+1\rangle\}\), with dressed energies
\[
E_{n,\pm}=\hbar\omega_c n+\frac{\hbar\Delta}{2}\pm \hbar\sqrt{g^2(n+1)+(\Delta/2)^2},
\]
where \(\Delta=\omega_0-\omega_c\). On resonance, the normal-mode splitting is \(2g\sqrt{n+1}\), and this \(\sqrt{n}\) dependence is already an intrinsic nonlinearity of the JC ladder [1609.03940]. In “Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms,” this intrinsic ladder nonlinearity was measured directly: on resonance, the ratio of the two-atom to single-atom splittings was \(1.43(0.03)\approx \sqrt{2}\) [1609.03940].

In the literature on nonlinear JC models, however, “nonlinearity” usually denotes more than the dressed-state anharmonicity of the linear JC Hamiltonian. The principal routes summarized in the monograph “The Jaynes-Cummings model and its descendants” are field nonlinearities, matter nonlinearities, coupling nonlinearities, and beyond-RWA extensions [2202.00330]. Field nonlinearities add terms such as \(\hbar\chi a^{\dagger 2}a^2\); matter nonlinearities include multiphoton transitions and multilevel atoms; coupling nonlinearities replace the constant \(g\) by an operator function \(f(\hat n)\); and beyond-RWA models retain counter-rotating terms, leading to the quantum Rabi class [2202.00330].

A common misconception is therefore to identify nonlinear JC physics exclusively with Kerr media or intensity-dependent couplings. The data support a broader distinction: the standard JC model is already anharmonic through its \(\sqrt{n}\) ladder, while nonlinear JC models in the stricter sense introduce explicit nonlinear functions of the photon number into the Hamiltonian [1609.03940].

## 2. Hamiltonian forms of nonlinear Jaynes–Cummings models

A compact general form, used for parity-conserving nonlinear JC and Rabi models, is
\[
H=h(\hat n)+\frac{\omega_0}{2}\sigma_z
+g_-\!\left[a\frac{f(\hat n)}{\sqrt{\hat n}}\sigma_+ + a^\dagger \frac{f(\hat n)}{\sqrt{\hat n}}\sigma_-\right]
+g_+\!\left[a\frac{f(\hat n)}{\sqrt{\hat n}}\sigma_- + a^\dagger \frac{f(\hat n)}{\sqrt{\hat n}}\sigma_+\right].
\]
Here \(h(\hat n)\) is a real, well-behaved function of \(\hat n\), and \(f(\hat n)\) specifies the intensity-dependent coupling. The choice \(h(\hat n)=\omega_f \hat n+\kappa \hat n^2\) incorporates a Kerr-like field nonlinearity, while \(f(\hat n)=\hat n\) produces the Buck–Sukumar model [1303.5892]. Setting \(g_+=0\) suppresses counter-rotating processes and yields a nonlinear JC Hamiltonian.

This generic structure subsumes several well-known descendants. The Kerr–JC model adds \(\hbar\chi a^{\dagger 2}a^2\) to the field sector. The two-photon JC model replaces single-boson exchange by
\[
H_{2\mathrm{ph}}=\hbar\omega \hat n+\frac{\hbar\omega_q}{2}\sigma_z+\hbar g_2(a^2\sigma_+ + a^{\dagger 2}\sigma_-),
\]
which enforces parity selection rules and supports squeezing and Schrödinger cat states [2202.00330]. Intensity-dependent models take the form
\[
H_{\mathrm{BS}}=\hbar\omega \hat n+\frac{\hbar\omega_q}{2}\sigma_z+\hbar g\big[a f(\hat n)\sigma_+ + a^\dagger f(\hat n)\sigma_-\big],
\]
with common choices \(f(\hat n)=\sqrt{\hat n+1}\) or \(f(\hat n)=\hat n\) [2202.00330].

A broader generalized JC family introduces simultaneously nonlinear bosonic terms, nonlinear dispersive shifts, multiphoton exchange, and algebraic deformations:
\[
\hat H=\omega\hat n+\frac{\omega_0}{2}\hat\sigma_z+\hat\sigma_z F(\hat n)+G(\hat n)
+g\big[\hat\sigma_+ \hat a^k f(\hat n)+\hat\sigma_- f(\hat n)\hat a^{\dagger k}\big].
\]
In this formulation, \(F(\hat n)\) is a Stark-like nonlinear dispersive term, \(G(\hat n)\) may encode Kerr-like bosonic processes, \(f(\hat n)\) specifies intensity dependence, and \(k\in\mathbb N\) controls multi-boson exchange [2010.13867].

## 3. Symmetry, invariant subspaces, and exact solution strategies

For the parity-conserving nonlinear Hamiltonian above, the conserved parity operator is
\[
\hat\Pi=-(-1)^{\hat n}\sigma_z,\qquad [H,\hat\Pi]=0.
\]
This decomposes the Hilbert space into two orthogonal parity subspaces spanned by
\[
|+,j\rangle=\{ |0,g\rangle,|1,e\rangle,|2,g\rangle,\ldots\},\qquad
|-,j\rangle=\{ |0,e\rangle,|1,g\rangle,|2,e\rangle,\ldots\},
\]
which remain dynamically independent [1303.5892]. When \(g_+=0\), parity is supplemented by conservation of the total excitation number \(\hat N=\hat n+\sigma_z/2\), restoring the block-diagonal JC structure.

In the nonlinear JC case, an exact solution can be written with Susskind–Glogower operators. Defining
\[
\Gamma(\hat n)=h(\hat n-1)-h(\hat n)+\omega_0,\qquad
\Omega(\hat n)=\sqrt{\Gamma^2(\hat n)+4g_-^2 f^2(\hat n)},
\]
the dressed “eigenfrequencies” become
\[
\lambda_\pm(\hat n)=\frac{h(\hat n-1)+h(\hat n)\pm \Omega(\hat n)}{2}.
\]
The spectrum is therefore no longer governed by a constant coupling and linear field dispersion; instead, both level spacings and effective Rabi frequencies become \(n\)-dependent functions determined by \(h\) and \(f\) [1303.5892].

The broader solution toolkit includes rotating frames, Schrieffer–Wolff transformations in the dispersive regime, exact diagonalization for deformed algebras, Bargmann methods, \(su(1,1)\) techniques for multiphoton models, parity block diagonalization, Born–Oppenheimer approximations in deep-strong coupling, and master-equation approaches for driven and open systems [2202.00330]. A distinct algebraic route uses an underlying graded Lie algebra symmetry reminiscent of supersymmetric quantum mechanics, which yields closed forms for eigenstates, eigenvalues, and time evolution in a unified generalized JC model [2010.13867].

## 4. Spectra, collapse–revival dynamics, and entanglement structure

Nonlinear JC spectra differ from the standard ladder because \(h(\hat n)\) modifies the bare field dispersion and \(f(\hat n)\) modifies the coupling matrix elements. In the formulation above, the relevant spectral quantities are \(\lambda_\pm(\hat n)\) and \(\Omega(\hat n)\), so level spacings are explicitly excitation dependent [1303.5892]. A direct consequence is altered collapse–revival structure, since the dephasing and rephasing of different \(n\)-components are controlled by a nonlinear distribution of effective Rabi frequencies rather than by the standard \(\sqrt{n+1}\) law [2202.00330].

The Buck–Sukumar model is the standard example. In the notation of the parity-conserving nonlinear Hamiltonian,
\[
H_{\mathrm{BS}}=\omega_f \hat n+\frac{\omega_0}{2}\sigma_z+g\big[a\sqrt{\hat n}\,\sigma_+ + \sqrt{\hat n}\,a^\dagger \sigma_-\big],
\]
corresponding to \(g_+=0\), \(g_-=g\), \(f(\hat n)=\hat n\), and \(h(\hat n)=\omega_f \hat n\) [1303.5892]. In the photonic-lattice simulation of this model, the reconstructed mean photon number, atomic inversion, and von Neumann entropy displayed characteristic nonlinear JC dynamics, while the fidelity exhibited periodic returns to the initial state, in line with the known exact periodicity of the atomic inversion in the Buck–Sukumar model [1303.5892].

Recent spectroscopy sharpened the distinction between linear and explicit nonlinear models. For an \(f\)-deformed nonlinear JC model, the long-time spectral response is intrinsically asymmetric with the nonlinear coupling, and this asymmetry is identified as “a signature of the impossibility of getting resonant conditions for finite field excitations” [2408.09061]. In the bare nonlinear field sector with \(f(n)=1+\chi n\), the long-time spectrum of a Fock initial state \(|n\rangle\) consists of Lorentzians centered at
\[
\omega=\omega_c+2\omega_c\chi n,
\]
showing directly that the field becomes spectrally anharmonic [2408.09061].

In driven dispersive JC physics, a different form of nonlinearity emerges from saturation. In the strong-dispersive bad-cavity regime, the effective cavity pull decreases with photon number,
\[
\chi(n)\simeq \sigma_z\,\frac{g^2}{\sqrt{\Delta^2+4g^2 n}},
\]
and this produces a finite bistable region bounded by two critical points, unlike the usual dispersive bistability from a Kerr nonlinearity [1005.0377]. This suggests that nonlinear JC behavior is not exhausted by explicit Hamiltonian deformations; saturation of the JC interaction itself can generate qualitatively different nonlinear oscillator response.

## 5. Realizations, simulators, and observable reconstruction

One implementation route replaces the quantum system by a classical analog. A pair of one-dimensional photonic lattices can simulate parity-conserving nonlinear JC and Rabi dynamics by mapping the evolution variable \(t\) to the propagation distance \(z\), the Fock index to the waveguide index, and the parity sectors to two parallel arrays [1303.5892]. In this architecture, the onsite refractive-index shift of waveguide \(j\) encodes
\[
d^{(\pm)}(j)=h(j)\mp (-1)^j \frac{\omega_0}{2},
\]
and nearest-neighbor couplings are engineered to reproduce the functions \(f(j)\) and the alternating weights \(g_\pm\) [1303.5892]. Output intensities reconstruct \(\langle \hat n(t)\rangle\) and \(\langle \sigma_z(t)\rangle\), while phase-resolved measurements also recover \(\langle \sigma_x(t)\rangle\), fidelity, and the reduced-atom von Neumann entropy [1303.5892].

Quantum platforms realize different sectors of the nonlinear JC landscape. Rydberg-dressed atoms implement the intrinsic nonlinear JC ladder rather than a deformed Hamiltonian: under perfect blockade, the bosonic excitation number is the number of symmetric spin flips, the coupling in the \(n\)-th manifold is \(\sqrt{n}\Omega_r/2\), and the Autler–Townes splitting follows
\[
\Omega_{\mathrm{eff}}(n)=\sqrt{n\Omega_r^2+\Delta_r^2},
\]
with resonant \(\sqrt{n}\) scaling [1609.03940]. Circuit quantum electrodynamics can realize a nonlinear JC model by coupling a transmon to a Kerr nonlinear resonator; in that setting the pumped resonator displays bistability, parametric amplification, and squeezing, and the interplay with strong coupling yields a nonlinear JC Hamiltonian of direct experimental relevance [1111.0501].

Trapped ions provide another route. A detuned nonlinear JC model for the quantized motion of a trapped ion, driven on a sideband with a small frequency mismatch, has an explicitly time-dependent interaction Hamiltonian, and exact solutions can be obtained by quantizing the pump field [1809.04391]. The same system can also be solved with a classical driving laser field, making it possible to study time-ordering effects and the nonclassicality of the motional state under nonlinear JC dynamics [1809.04391].

## 6. Open-system, many-body, and extended descendants

The nonlinear JC model admits systematic many-body and driven–dissipative extensions. In open cavity arrays with three-level atoms, adiabatic elimination produces a JC-like Hamiltonian with an additional nonlinear term. In the single-cavity case, the system features a bistable region; the extra nonlinear term gives rise to limit cycles through Hopf bifurcations; and in the limit of large nonlinearity the model exhibits an Ising-like phase transition as the coupling between light and matter is varied [1602.03491]. Within the mean-field treatment used there, the two-dimensional square geometry reduces to uniform single-cavity-like behavior, which indicates that beyond-mean-field correlations or different geometries may be needed for spatially ordered phases [1602.03491].

Collective generalizations introduce another notion of nonlinearity. In the homogeneous Tavis–Cummings model, the Hilbert space decomposes into independent higher-pseudospin JC ladders, and the effective doublet coupling becomes
\[
g_{\mathrm{eff}}(n,S,m)=g\sqrt{n+1}\sqrt{(S-m)(S+m+1)}.
\]
This produces a “square-root-\(n\)-type” nonlinearity inside each pseudospin ladder and leads to multi-frequency beating in observables [1301.4857]. The effect is distinct from Kerr or phenomenological intensity-dependent coupling because it arises microscopically from Dicke matrix elements [1301.4857].

Two-atom nonlinear JC models add still more structure. For two identical two-level atoms coupled to one cavity mode, the Hamiltonian may include a general intensity-dependent coupling \(f(\hat n)\), a Kerr-like nonlinear medium through \(h(\hat n)\), and interatomic Ising-like and dipole–dipole couplings. Under the rotating-wave approximation, a generalized excitation number remains conserved, the Hamiltonian block-diagonalizes into finite manifolds, and explicit diagonalization yields the evolution of atomic excitation, purity, concurrence, the entropy of the field, and the field phase-space distribution [1607.03216]. In the Buck–Sukumar plus Kerr case, the reported effects include beat structures in the inversion, modified purity recoherence windows, concurrence spikes, and cat-like splitting in the Husimi \(Q\)-function [1607.03216].

Taken together, these developments define the nonlinear Jaynes–Cummings model not as a single Hamiltonian but as a structured class of excitation-dependent light–matter theories. Their common feature is the replacement of constant ladder spacing and constant coupling by number-dependent functions. That replacement bends spectra, reshapes collapse–revival physics, modifies entanglement generation, and opens direct connections to photonic simulation, circuit QED, trapped ions, neutral atoms, and driven many-body quantum optics [2202.00330].

Source: https://www.emergentmind.com/topics/nonlinear-jaynes-cummings-model