Extended Nondegenerate Two-Photon Jaynes-Cummings Model

• The extended nondegenerate two-photon Jaynes-Cummings model is a framework where a two-level system interacts with two distinct bosonic modes via a joint two-photon transition.
• This model extends classical JC interactions by incorporating detuning, Kerr effects, and multimode generalizations to yield effective multiphoton couplings and detailed resonance conditions.
• Its dynamic behavior reveals number-dependent Rabi oscillations, quantum correlations, and analytical tractability through algebraic methods such as SU(1,1) tilting transformations.

Searching arXiv for recent and foundational papers related to the extended nondegenerate two-photon Jaynes-Cummings model. The extended nondegenerate two-photon Jaynes-Cummings model denotes a family of Jaynes-Cummings descendants in which an effective two-level system interacts with two distinct bosonic modes through a joint two-photon transition. In the notation used across the relevant literature, the characteristic interaction is of the form $$a_1 a_2 \sigma_+ + a_1^\dagger a_2^\dagger \sigma_-$$, so that one atomic excitation or de-excitation is accompanied by the annihilation or creation of one quantum in each mode. In this setting, the nondegenerate resonance condition is set by the sum of the mode frequencies, typically $\omega_0 \approx \omega_1+\omega_2$ or exactly $\omega_1+\omega_2=\omega_0$, while the adjective “extended” refers to the addition of detuning, Stark-shift, Kerr, intensity-dependent, drive, dephasing, dissipative, or broader multimode generalizations (Dhar et al., 2014, Larson et al., 2022, Jiang et al., 2022).

1. Conceptual definition and model class

The defining distinction of the nondegenerate two-photon model is that the two photons belong to different modes. This separates it from the standard degenerate two-photon Jaynes-Cummings model, where both photons occupy the same bosonic mode and the interaction is written with $a^2$ and $a^{\dagger 2}$. In the nondegenerate case, the elementary two-photon process instead involves two mode operators, such as $a_1 a_2 \sigma_+$ and $a_1^\dagger a_2^\dagger \sigma_-$, and the natural vacuum-to-pair transition is $|0,0\rangle \leftrightarrow |1,1\rangle$ rather than $|0\rangle \leftrightarrow |2\rangle$ (Jiang et al., 2022, Prieto et al., 2014).

Within the surveyed literature, the model does not appear as a single universally fixed Hamiltonian with one canonical extension. Rather, it appears through closely related constructions: as the $N=2$ specialization of a multimode nondegenerate multiphoton Jaynes-Cummings interaction, as an effective second-order two-mode coupling obtained by adiabatic elimination of an intermediate level, and as an algebraically related two-mode JC/AJC system whose decoupled bosonic sector contains $\omega_0 \approx \omega_1+\omega_2$0 and $\omega_0 \approx \omega_1+\omega_2$1. This suggests that the expression “extended nondegenerate two-photon Jaynes-Cummings model” is best understood as a model class rather than a single normal form (Dhar et al., 2014, Larson et al., 2022, Choreño et al., 2017).

2. Core Hamiltonians and resonance structure

A direct and explicit formulation is given by the generalized multimode nondegenerate Jaynes-Cummings Hamiltonian

$\omega_0 \approx \omega_1+\omega_2$2

Here $\omega_0 \approx \omega_1+\omega_2$3 is an effective multiphoton coupling, the interaction is written under the rotating-wave approximation, and the resonance condition is

$\omega_0 \approx \omega_1+\omega_2$4

For $\omega_0 \approx \omega_1+\omega_2$5, this reduces to

$\omega_0 \approx \omega_1+\omega_2$6

which is the conventional effective nondegenerate two-photon Jaynes-Cummings Hamiltonian up to notation and the prefactor convention in the $\omega_0 \approx \omega_1+\omega_2$7 term (Dhar et al., 2014).

A second route is provided by three-level two-mode models. In the $\omega_0 \approx \omega_1+\omega_2$8-configuration discussed in the monograph, the Hamiltonian

$\omega_0 \approx \omega_1+\omega_2$9

reduces, for large detunings $\omega_1+\omega_2=\omega_0$0 and $\omega_1+\omega_2=\omega_0$1, to the effective interaction

$\omega_1+\omega_2=\omega_0$2

This is not the canonical pair-creation form $\omega_1+\omega_2=\omega_0$3, but it is an effective two-mode second-order two-photon interaction generated by adiabatic elimination of a virtual intermediate level (Larson et al., 2022).

3. Effective origin and extended variants

The nondegenerate two-photon Hamiltonian is typically an effective description. In the generalized multimode treatment, the intermediate levels associated with the multiphoton process are assumed to be nonresonant and are adiabatically removed, leaving a direct effective coupling between $\omega_1+\omega_2=\omega_0$4 and $\omega_1+\omega_2=\omega_0$5. The resulting $\omega_1+\omega_2=\omega_0$6 is therefore an effective multiphoton coupling rather than a bare one-photon vacuum Rabi parameter. This effective viewpoint is central to the nondegenerate model because the joint mode product $\omega_1+\omega_2=\omega_0$7 already encodes the elimination of intermediate transitions (Dhar et al., 2014).

The surveyed literature uses “extended” in several precise senses. One extension is multimode generalization,

$\omega_1+\omega_2=\omega_0$8

of which the nondegenerate two-photon case is the $\omega_1+\omega_2=\omega_0$9 member. A second extension is nonlinear or Kerr-dressed Jaynes-Cummings structure,

$a^2$0

where $a^2$1 gives intensity-dependent coupling and $a^2$2 gives multiphoton coupling. A third extension is open or driven Jaynes-Cummings physics, where coherent driving and Lindblad dissipation are added to the basic exchange dynamics. In this broader sense, the extended nondegenerate two-photon model belongs to the large class of driven, nonlinear, multimode, and effective Jaynes-Cummings descendants (Larson et al., 2022, Jiang et al., 2022).

4. Algebraic structure and exactly solvable generalizations

A particularly important two-mode extension is the simultaneous JC/AJC Hamiltonian

$a^2$3

This is not itself the textbook nondegenerate two-photon Jaynes-Cummings interaction. However, after elimination of one spinor component, the effective bosonic sector contains

$a^2$4

and therefore exhibits exactly the pair-annihilation and pair-creation structure characteristic of nondegenerate two-photon processes (Choreño et al., 2017).

The associated algebra is $a^2$5, with generators

$a^2$6

and conserved mode-difference operator

$a^2$7

Diagonalization proceeds by a tilting transformation $a^2$8, and the eigenfunctions are written in terms of Perelomov number coherent states of the two-dimensional harmonic oscillator. In the nonrelativistic limit, the same construction reduces to the time-independent nondegenerate parametric amplifier Hamiltonian

$a^2$9

which makes the relation to nondegenerate pair physics explicit (Choreño et al., 2017).

The one-mode generalized JC/AJC model provides a degenerate precursor rather than a nondegenerate realization. There, the decoupled bosonic equations contain $a^{\dagger 2}$0 and $a^{\dagger 2}$1, are solved via the same $a^{\dagger 2}$2 tilting method, and are interpreted as a relativistic degenerate parametric amplifier. The authors explicitly remark that the one-oscillator formulation can be extended to a model that includes two oscillators, which places that work as a methodological antecedent rather than a direct treatment of the nondegenerate model (Ojeda-Guillén et al., 2014).

5. Dynamics, Rabi structure, and quantum correlations

For the generalized nondegenerate multiphoton model, the exact interaction-picture propagator is written with operator-valued Rabi frequencies

$a^{\dagger 2}$3

and for an initial atomic excited state the field amplitudes evolve with generalized trigonometric factors

$a^{\dagger 2}$4

In the nondegenerate two-photon specialization, the effective Rabi frequency becomes

$a^{\dagger 2}$5

which is the characteristic number-dependent two-mode coupling structure of the model. The same work relates these oscillations to collapse and revival, and notes that in the two-mode case the quantum correlations become oscillatory and discontinuous because collapse and revival happen rapidly in short intervals (Dhar et al., 2014).

A distinctive contribution of the generalized multimode treatment is the mapping of the pure-state atom-field dynamics into an effective $a^{\dagger 2}$6 bipartite system. The evolving state can be rewritten so that the atomic qubit $a^{\dagger 2}$7 is coupled only to a two-dimensional field subspace spanned by $a^{\dagger 2}$8. This permits direct evaluation of entanglement of formation for pure states. When atomic dephasing is added through

$a^{\dagger 2}$9

the exact $a_1 a_2 \sigma_+$0 compression no longer holds, and the analysis is transferred to a truncated $a_1 a_2 \sigma_+$1 space using optical truncation or quantum scissors. The reported truncations are $a_1 a_2 \sigma_+$2 for good reproduction of the undamped pure-state dynamics at $a_1 a_2 \sigma_+$3, and $a_1 a_2 \sigma_+$4 for dissipative calculations. In that dephased setting, logarithmic negativity decays faster than quantum discord, while discord remains nonzero for longer times (Dhar et al., 2014).

6. Driven-dissipative analogues, related descendants, and scope

The most detailed driven-dissipative template in the surveyed material is not nondegenerate but degenerate: the $a_1 a_2 \sigma_+$5-photon Jaynes-Cummings model with single-mode interaction $a_1 a_2 \sigma_+$6, specialized at $a_1 a_2 \sigma_+$7. In the rotating frame, the driven two-photon Hamiltonian is

$a_1 a_2 \sigma_+$8

and in the Mollow regime $a_1 a_2 \sigma_+$9 the strong drive dresses the two-level system and enables resonant super-Rabi oscillation between $a_1^\dagger a_2^\dagger \sigma_-$0 and $a_1^\dagger a_2^\dagger \sigma_-$1. Dissipation is introduced through

$a_1^\dagger a_2^\dagger \sigma_-$2

Although this is a single-mode degenerate theory, it supplies a precise dressed-state methodology for resonance engineering, emission diagnostics, and bundle correlations (Jiang et al., 2022).

A plausible translation of that driven-dissipative logic to the nondegenerate two-photon case is

$a_1^\dagger a_2^\dagger \sigma_-$3

Under this translation, the vacuum-to-two-photon resonance $a_1^\dagger a_2^\dagger \sigma_-$4 becomes $a_1^\dagger a_2^\dagger \sigma_-$5, and the natural bundle operator is $a_1^\dagger a_2^\dagger \sigma_-$6 rather than $a_1^\dagger a_2^\dagger \sigma_-$7. The same comparison suggests that bundle-emission observables in a nondegenerate model should be built from mode-resolved pair operators instead of single-mode powers of $a_1^\dagger a_2^\dagger \sigma_-$8 (Jiang et al., 2022).

Related descendants clarify the model boundaries. The single-mode two-photon Jaynes-Cummings model on resonance $a_1^\dagger a_2^\dagger \sigma_-$9 uses the Rabi factor $|0,0\rangle \leftrightarrow |1,1\rangle$0 and supports nonlinear coherent-state generation through Susskind-Glogower operators, but it remains degenerate rather than nondegenerate (Prieto et al., 2014). The two-mode polarization-resolved Jaynes-Cummings model generates quasi-periodic photonic entanglement because the characteristic frequencies depend nonlinearly on occupation numbers, but it is still a two-mode one-photon model rather than a joint two-photon interaction (Erementchouk et al., 2010). Taken together, these works show that the extended nondegenerate two-photon Jaynes-Cummings model sits at the intersection of three themes: two-mode coupling, multiphoton effective interaction, and model extensions such as driving, dephasing, Kerr structure, or algebraic generalization (Larson et al., 2022).

Experimentally, the neighboring literature identifies trapped ions, cavity QED, and superconducting circuits as plausible settings for the required ingredients. In the driven degenerate two-photon template, the reported regime uses weak multiphoton coupling $|0,0\rangle \leftrightarrow |1,1\rangle$1, strong drive $|0,0\rangle \leftrightarrow |1,1\rangle$2, and bad-cavity conditions $|0,0\rangle \leftrightarrow |1,1\rangle$3, with moderate pure dephasing not destroying antibunched bundle-emission signatures. This does not constitute a direct implementation of the nondegenerate model, but it indicates that strong driving and dressed-state engineering are regarded as experimentally accessible tools for multiphoton Jaynes-Cummings physics (Jiang et al., 2022).