Rabi Lattice: Quantum and Condensate Models

• Rabi lattice is a many-body system defined by full Rabi coupling rather than a scalar periodic potential, enabling Z2 symmetry and unique phase behavior.
• In cavity-QED formulations, the system exhibits quantum phase transitions between incoherent compressible and coherent superradiant phases with emergent Majorana-like edge modes.
• Spatially modulated Rabi coupling in condensates and optical lattices creates effective potentials that support gap solitons, topological states, and photon pairing.

Rabi lattice denotes a class of lattice many-body systems in which the defining local or intercomponent coupling is of Rabi type. In the canonical cavity-QED usage, each site is a Rabi system—a two-level atom or qubit locally coupled to a single quantized bosonic mode—and the sites are connected by photon tunnelling, giving the Rabi-Hubbard model or closely related Rabi-lattice models (Flottat et al., 2016). In other usages, the term denotes lattices generated by spatially periodic Rabi coupling in multicomponent Bose–Einstein condensates, or related lattice settings in which coherent internal-state Rabi driving and spatial periodicity are inseparable (Chen et al., 2017, Li et al., 2023). Across these realizations, the common feature is that the lattice physics is controlled by full Rabi coupling or by a spatially structured Rabi-conversion field rather than by a purely scalar periodic potential.

1. Canonical cavity-QED formulations

In the standard quantum-optical sense, a Rabi lattice is an array of coupled quantum cavities, each containing a single two-level system and a single bosonic mode. A one-dimensional periodic-chain formulation studied in quantum Monte Carlo is the Rabi-Hubbard model

$$H = -J \sum_i(a_i + a^\dagger_i)(a_{i+1} + a^\dagger_{i+1}) + \sum_i \left(\omega_s \sigma^+_i \sigma^-_i + \omega_l a^\dagger_i a_i \right) + g \sum_i (\sigma^-_i + \sigma^+_i)(a_i + a^\dagger_i),$$

with photon frequency $\omega_l$, atomic transition energy $\omega_s$, local light-matter coupling $g$, and inter-cavity hopping $J$; the detuning is $\delta=\omega_s-\omega_l$ (Flottat et al., 2016). Expanding the hopping and on-site coupling makes explicit the counter-rotating contributions $a_i a_{i+1}+\mathrm{h.c.}$, $a_i^\dagger a_{i+1}^\dagger$, $\sigma_i^+ a_i^\dagger$, and $\sigma_i^- a_i$. This is the decisive structural distinction from the Jaynes–Cummings–Hubbard model obtained under the rotating-wave approximation.

A closely related formulation, used in strong-coupling Ising analyses, writes the lattice Hamiltonian as

$\omega_l$0

where $\omega_l$1 is nearest-neighbor photon hopping and $\omega_l$2 is the full dipole coupling (Kumar et al., 2012, Jalal, 2016). For a single site, the local parity $\omega_l$3 with $\omega_l$4 is conserved, whereas on the lattice only the global parity $\omega_l$5 remains conserved. The resulting symmetry is therefore global $\omega_l$6, not $\omega_l$7. This symmetry reduction is the reason the cavity-QED Rabi lattice does not generically reproduce the excitation-number-conserving Mott physics of the Jaynes–Cummings lattice.

2. Rabi-Hubbard phase structure and quantum criticality

Exact stochastic Green Function quantum Monte Carlo on one-dimensional periodic chains of size up to $\omega_l$8, with inverse temperatures $\omega_l$9, finds that the generic zero-detuning Rabi-Hubbard model has only two phases: an incoherent compressible phase at small hopping and a coherent phase at larger hopping, separated by a quantum phase transition (Flottat et al., 2016). In the incoherent phase, equal-time photon and spin Green functions decay exponentially with distance, the photon density $\omega_s$0 is small, and the absence of density plateaus signals compressibility rather than a Mott state. In the coherent phase, photon correlations, spin correlations, and mixed spin-photon correlators approach nonzero plateaus at large separation, indicating true long-range order permitted by the discrete $\omega_s$1 symmetry. The ordered spin component is ferromagnetic along the $\omega_s$2-axis, and the photon condensate occupies the $\omega_s$3 mode macroscopically, so the phase is described as superradiant or Dicke-like rather than as a $\omega_s$4 superfluid.

The principal photon order parameter is the condensate fraction

$\omega_s$5

while spin order is tracked through long-distance $\omega_s$6-correlations such as

$\omega_s$7

The rise of $\omega_s$8 and $\omega_s$9 is simultaneous; no intermediate regime is found in which only photons or only spins order. Finite-size scaling identifies the transition as belonging to the classical two-dimensional Ising universality class. With $g$0 and Ising exponents $g$1, $g$2, the scaled condensate fraction $g$3 exhibits size crossings at the critical hopping, and data collapse versus $g$4 confirms Ising criticality.

A central negative result is the disappearance of the Jaynes–Cummings–Hubbard Mott lobes when counter-rotating terms are appreciable. The paper attributes this to the loss of excitation-number conservation and the resulting destruction of photon blockade. The Rabi-Hubbard and Jaynes–Cummings–Hubbard phase diagrams become similar only for sufficiently large negative detuning or sufficiently small counter-rotating coupling. The sensitivity is strong: at zero detuning, $g$5 already makes the two models look similar, while $g$6 destroys the $g$7 Mott plateau. Deep in the coherent phase, the model develops a photon-number divergence; in the symmetric case $g$8, the instability occurs at $g$9, i.e. $J$0, and reflects the chemical-potential-like role of $J$1 once excitation number is no longer conserved.

3. Strong-coupling Ising limit and Majorana-like edge modes

In the strong-coupling regime $J$2, the one-dimensional Rabi lattice admits a controlled mapping to a transverse-field quantum Ising model (Kumar et al., 2012, Jalal, 2016). The derivation uses a unitary transformation

$J$3

followed by a displacement

$J$4

Neglecting photon fluctuations about the large static displacement and projecting onto the lowest photon sector $J$5 yields

$J$6

with

$J$7

The Ising coupling is ferro-electric for positive $J$8, while the transverse field is exponentially suppressed by strong atom-photon coupling. The control parameter is

$J$9

and in one dimension the critical point is at $\delta=\omega_s-\omega_l$0, corresponding to

$\delta=\omega_s-\omega_l$1

The two phases are para-electric and ferro-electric. The order parameter is the spontaneous polarization

$\delta=\omega_s-\omega_l$2

with $\delta=\omega_s-\omega_l$3 in the para-electric phase and $\delta=\omega_s-\omega_l$4 in the ferro-electric phase. In one dimension the exact thermodynamic-limit result is

$\delta=\omega_s-\omega_l$5

The associated photon expectation satisfies $\delta=\omega_s-\omega_l$6, so ferro-electric order corresponds to a coherent photon displacement. The ordered phase is nevertheless a $\delta=\omega_s-\omega_l$7-ordered phase with gapped excitations away from criticality, not a conventional gapless $\delta=\omega_s-\omega_l$8 superfluid.

Density-matrix renormalization group calculations on chains up to $\delta=\omega_s-\omega_l$9, keeping four photon states per cavity, support the Ising mapping. The polarization $a_i a_{i+1}+\mathrm{h.c.}$0 agrees well with the strong-coupling prediction, the many-body gap closes at the transition and reopens on either side, and the phase boundary approaches the strong-coupling universal curve for $a_i a_{i+1}+\mathrm{h.c.}$1. For open chains, the effective Ising description implies emergent Majorana-like edge modes in the ordered phase. Their proposed observable signature is the end-to-end correlation

$a_i a_{i+1}+\mathrm{h.c.}$2

in sharp contrast with bulk long-range correlations, which scale as $a_i a_{i+1}+\mathrm{h.c.}$3. The modes are exponentially localized, with finite-size splitting $a_i a_{i+1}+\mathrm{h.c.}$4 and localization length $a_i a_{i+1}+\mathrm{h.c.}$5. They are explicitly described as Majorana-like rather than topological Majorana zero modes because they are not protected against generic longitudinal perturbations $a_i a_{i+1}+\mathrm{h.c.}$6. The same work also reports that inter-cavity entanglement, quantified by the linear entropy of the reduced density matrix in a two-site problem, exhibits peaked singularities at the quantum phase transition.

4. Dispersive regime and effective low-energy interactions

When the detuning is large compared with the relevant couplings, the Rabi lattice enters the dispersive regime, in which qubit-photon conversion is only virtual and a Schrieffer–Wolff transformation yields an effective lattice Hamiltonian (Zhu et al., 2013). For the Rabi lattice

$a_i a_{i+1}+\mathrm{h.c.}$7

the photon band is $a_i a_{i+1}+\mathrm{h.c.}$8, so the relevant denominators are momentum dependent: $a_i a_{i+1}+\mathrm{h.c.}$9 To second order in $a_i^\dagger a_{i+1}^\dagger$0, the effective couplings are controlled by

$a_i^\dagger a_{i+1}^\dagger$1

which decay exponentially with separation $a_i^\dagger a_{i+1}^\dagger$2 in the weak-hopping limit.

The full dispersive Rabi-lattice Hamiltonian contains several qualitatively distinct interactions. First, there is an induced qubit-qubit term $a_i^\dagger a_{i+1}^\dagger$3, so the effective spin model is transverse-Ising-like rather than XY. Second, there are AC-Stark and conditional-hopping terms $a_i^\dagger a_{i+1}^\dagger$4. Third, and specific to the full Rabi case, there are anomalous pair-creation terms $a_i^\dagger a_{i+1}^\dagger$5, which generate photon pairing and squeezing. These pairing terms are absent in the Jaynes–Cummings lattice and arise from the counter-rotating denominators $a_i^\dagger a_{i+1}^\dagger$6.

In negative detuning, the low-energy sector is spin dominated and the effective model reduces to a transverse-field Ising model with exponentially decaying couplings. In positive detuning, the low-energy sector is photonic and becomes a bosonic pairing Hamiltonian whose Bogoliubov vacuum is a product of one-mode and two-mode squeezed states. The paper identifies a positive-detuning breakdown scale

$a_i^\dagger a_{i+1}^\dagger$7

beyond which the perturbative Bogoliubov spectrum ceases to be stable. Exact diagonalization of the Rabi dimer validates both the negative-detuning Ising description and the positive-detuning squeezed-photon description, and directly shows the enhancement of even-photon sectors and squeezed single-site Wigner functions expected from the effective theory.

5. Ising–Rabi lattice with local $a_i^\dagger a_{i+1}^\dagger$8 gauge symmetry

A distinct Rabi-lattice variant is the Ising–Rabi lattice,

$a_i^\dagger a_{i+1}^\dagger$9

in which the local Rabi coupling is combined with intersite Ising exchange rather than boson hopping (Nevado et al., 2015). This model carries a local discrete gauge symmetry generated by

$\sigma_i^+ a_i^\dagger$0

so $\sigma_i^+ a_i^\dagger$1 and $\sigma_i^+ a_i^\dagger$2 locally while $\sigma_i^+ a_i^\dagger$3 is unchanged. A direct consequence is that the gauge-odd observables $\sigma_i^+ a_i^\dagger$4 and $\sigma_i^+ a_i^\dagger$5 vanish throughout the phase diagram. The relevant gauge-invariant diagnostic is the mean boson number

$\sigma_i^+ a_i^\dagger$6

The analysis identifies two phases. The ferromagnetic phase is approximately boson vacuum times a ferromagnetic spin configuration for $\sigma_i^+ a_i^\dagger$7. The dressed-ferromagnetic phase appears for $\sigma_i^+ a_i^\dagger$8 and consists of local cat-like superpositions $\sigma_i^+ a_i^\dagger$9 with $\sigma_i^- a_i$0. In the slow-boson regime $\sigma_i^- a_i$1, perturbation theory, Born–Oppenheimer analysis, and DMRG indicate a first-order quantum phase transition between the two phases. In the fast-boson regime $\sigma_i^- a_i$2, the change from ferromagnetic to dressed-ferromagnetic behavior is continuous. DMRG on chains up to $\sigma_i^- a_i$3 with a bosonic cutoff $\sigma_i^- a_i$4 shows that the derivative $\sigma_i^- a_i$5 sharpens strongly as $\sigma_i^- a_i$6 decreases, and the extracted critical line for $\sigma_i^- a_i$7 follows $\sigma_i^- a_i$8 with $\sigma_i^- a_i$9. The same work proposes a trapped-ion implementation in which local transverse phonons provide the bosons, an effective Ising coupling is mediated by axial modes, and sideband drives generate the local spin-boson coupling.

6. Spatially modulated Rabi coupling in condensates and nonlinear media

In nonlinear-wave and cold-atom settings, “Rabi lattice” often refers not to a lattice of local cavity Rabi systems but to a spatially periodic linear interconversion between two components. The basic one-dimensional model is a pair of coupled Gross–Pitaevskii or nonlinear Schrödinger equations with periodically modulated off-diagonal coupling $\omega_l$00, no explicit scalar lattice potential, and self-repulsive nonlinearity (Chen et al., 2017). In that setting, the lattice is embedded in the conversion channel itself. The periodic off-diagonal coupling generates Bloch bands and finite gaps, and supports on-site-centered and off-site-centered gap solitons that may be symmetric or antisymmetric between components. Stable families are found chiefly in the first finite bandgap, while the second bandgap is predominantly unstable and exhibits alternating stable and unstable windows. In the strong-asymmetry limit, the two-component system reduces to a single effective equation with

$\omega_l$01

showing how a coupling lattice can induce an ordinary effective lattice potential.

The same idea has been extended to topological coupling-engineered lattices in binary Bose–Einstein condensates (Li et al., 2023). There, the spatially dependent Rabi coupling $\omega_l$02 is arranged in one- or two-dimensional SSH-type patterns, combined with Zeeman splitting $\omega_l$03, so that the coupling landscape itself opens topological gaps. In the one-dimensional Rabi SSH lattice, the topological phase occurs for positive shift $\omega_l$04, with winding numbers $\omega_l$05 and $\omega_l$06 for the two lowest bands; finite systems then support edge states inside the gap. In two dimensions, the nontrivial phase is characterized by polarization $\omega_l$07 and supports higher-order corner states, which in the finite $\omega_l$08 lattice appear only for sufficiently large positive shift $\omega_l$09. Nonlinearity produces thresholdless topological edge and corner solitons that bifurcate from the linear boundary states and can remain stable for both attractive and repulsive interactions, including in two dimensions.

A related one-dimensional spin-orbit-coupled condensate model combines an optical lattice $\omega_l$10 with a Rabi-coupling lattice

$\omega_l$11

so that the scalar trap and the spatially modulated spin-flip field act simultaneously (Sultana et al., 20 Sep 2025). Variational reduction shows that the Rabi-lattice term enters the effective separation potential of a two-soliton composite primarily through the overlap region. The resulting effective potential develops additional local minima near small separation, and numerical simulations show localization, oscillation, transport, and splitting of the total density profile as $\omega_l$12, $\omega_l$13, and the initial overlap are varied. In another closely related continuum setting, a uniform Rabi coupling in a spin-dependent optical lattice can fundamentally alter band geometry and superfluid stability. For the Rabi-coupled Zeeman lattice with $\omega_l$14, the linear Hamiltonian

$\omega_l$15

has the half-period spin-translation symmetry $\omega_l$16, which extends the Brillouin zone from $\omega_l$17 to $\omega_l$18 and produces a significant separation between Landau and dynamical instability thresholds (He et al., 2021).

7. Related lattice usages: vortex-molecule crystals and optical-lattice spectroscopy

A further usage of the term occurs in rotating two-component condensates with coherent Rabi coupling, where the elementary lattice object is a Rabi-bound vortex molecule rather than a local cavity unit (Uranga et al., 2018). In the infinite two-dimensional system, Rabi coupling binds a vortex in component $\omega_l$19 to a vortex in component $\omega_l$20, and the ground state is an infinite periodic lattice of such molecules. The geometry depends on the inter-component interaction ratio $\omega_l$21 and the Rabi strength $\omega_l$22. For $\omega_l$23, interlaced component-vortex lattices are recovered; at intermediate $\omega_l$24, bound molecules form with $\omega_l$25-dependent orientational order; and for $\omega_l$26, the two vortices within each molecule overlap and the system approaches the triangular scalar-condensate lattice. In the tightly bound limit, the intermolecular alignment energy is described by a quadrupolar lattice interaction written in terms of an elliptic function $\omega_l$27.

The phrase also appears in optical-lattice clock physics, although explicitly not in the standard condensed-matter sense of a cavity-based Rabi lattice. One line of work studies a periodically shaken shallow $\omega_l$28 optical lattice clock as a lattice of Rabi-driven two-level systems with motional Bloch bands (Yin et al., 2021). Triangular Floquet shaking simultaneously renormalizes the effective Rabi frequencies,

$\omega_l$29

and the tunnelling amplitudes through

$\omega_l$30

so that near $\omega_l$31 the Bloch bands are flattened and a broad shallow-lattice line is narrowed to $\omega_l$32. A different optical-clock setting analyzes Rabi spectroscopy in a three-dimensional optical lattice at unity filling, where the clock shift factorizes as

$\omega_l$33

with $\omega_l$34 determined by the Rabi pulse and $\omega_l$35 by the dipolar lattice geometry (Liu et al., 2019). These usages preserve the core combination of coherent Rabi driving and lattice structure, but they should be distinguished from the cavity-QED Rabi lattice proper.

Taken together, these developments show that “Rabi lattice” is not a single Hamiltonian but a family of lattice constructions organized around Rabi coupling. In the canonical cavity-QED setting, the essential consequences are loss of excitation-number conservation, $\omega_l$36 rather than $\omega_l$37 symmetry, Ising criticality, and counter-rotating-term-induced phenomena such as photon-number instability, photon pairing, and Majorana-like edge modes (Flottat et al., 2016, Kumar et al., 2012, Zhu et al., 2013). In condensate, nonlinear-wave, and metrological settings, the term is extended to systems where a spatially patterned or dynamically engineered Rabi field itself generates the effective lattice or reshapes the lattice response (Li et al., 2023, Yin et al., 2021). This multiplicity of usage is itself a salient feature of the subject.