Exact polariton-condensate states via nonlinear Stark coupling

The gist: Nonlinear Stark coupling enables an RSGA-protected tower of exact states in hybrid photon–atom lattices, which exhibit dual physical natures as quantum scars embedded in chaotic spectra and polariton condensates with off-diagonal long-range order.

Model and Hamiltonian Construction

The study considers a hybrid system of coupled cavities, each potentially containing a two-level atom or remaining empty The resonant Hamiltonian is assembled from overlapping dimers, Hres = X⟨i,j⟩Hij. (1) The building blocks comprise three types of dimers: AA dimers, AC dimers, and CC dimers The parameters entering h a l and h c l specify the local contributions assigned to an individual dimer Under compatible resonance conditions, the three types of dimers can be assembled into lattices with different geometries and atom distributions The resonance conditions imply g 2 l = −omega lUl When Ul < 0 at every atom-doped site, the constructed exact states are ground states within their respective fixed-total-excitation sectors

Quantum Scar Tower and Off-Diagonal Long-Range Order Ground State

The exact eigenstates ψ nE⟩ possess unique spectral and coherence properties that distinguish them from typical many-body states A distinctive feature of the exact eigenstates ψ nE⟩ is their identification as manybody quantum scars embedded within a chaotic spectrum The spectral statistics P(s) in the middle of the spectrum in the even-reflection-parity sector with N = 12, compared with GOE (red solid) and Poisson (blue dashed) predictions; ⟨r⟩ = 0.529224 In contrast, the exact states have anomalously low bipartite entropy SA relative to surrounding eigenstates At fixed positive filling and fixed nonzero couplings, both correlation functions remain finite and nonzero at arbitrarily large separations in the thermodynamic limit, establishing ODLRO in both the photon and atomic sectors.

Mott-insulator to Superfluid Transition

The numerical results for N = 4 illustrate this redistribution in the κ–U plane [Fig. 4(a)] The resonance line U = UR, indicated by the white dashed curve, provides an analytical reference along which the exact ground states possess ODLRO Negative Stark coupling enhances local fluctuations and shifts their rapid increase toward weaker hopping The resulting fluctuation-based estimates are κest ≃ 0.17 for U = 0 and κest ≃ 0.14 for U = −0.5, as shown in the insets. These results are consistent with a Stark-induced shift of the MI–SF transition toward weaker inter-cavity hopping.

Polariton Condensates Exhibit ODLRO

For the uniformly atom-doped ring, both correlation functions remain finite and nonzero at arbitrarily large separations in the thermodynamic limit, establishing ODLRO in both the photon and atomic sectors The lower bound for the photon correlation is Ar = g squared D(N) n−1 D(N) n ≥ ρg squared g squared + U 2R > 0 Similarly, the spin-correlation lower bound Sr ≥ ρg2U2R (g2 + ρU2) > 0. Both lower bounds are independent of the system size and the separation between distinct sites.

Conclusion

We have developed an RSGA-based building-block construction of exact many-body states in photon–atom systems with nonlinear Stark coupling, applicable to arbitrary lattice geometries and atom distributions These states realize quantum scars with coherent revivals in a chaotic spectral background and polariton condensates with ODLRO on the ground-state branch Our fluctuation-based numerical analysis further suggests that negative Stark coupling shifts the MI–SF transition toward weaker inter-cavity hopping. The work provides a versatile framework for engineering both coherent quantum phases and non-thermalizing states in diverse light-matter architectures.

Supplemental Material Details

The Supplemental Material presents the derivation of the restricted spectrum generating algebra (RSGA) in the single-cavity Jaynes–Cummings (JC) model with nonlinear Stark coupling It also provides a proof that the constructed exact states are ground states within their respective fixed-total-excitation sectors under the specified resonance conditions and on the negative-Stark-coupling branch. The derivation of the normalization factor and photon and atomic correlation functions used to characterize off-diagonal long-range order (ODLRO) is also provided in the Supplemental Material. The construction of lattice Hamiltonians with arbitrary lattice structures is detailed using the building-block method. The analysis of the periodic ring shows that at U = UR < 0, it is a ground state in each fixed-total-excitation sector, without requiring the ground-state energies of different sectors to coincide. The calculation of Ar and Sr reduces to Ar = n/N in the limit U/g → 0, representing the correlation function of pure bosonic condensation The derivation of the spin-correlation lower bound Sr ≥ ρg2U2R (g2 + ρU2) > 0 is established. The two coefficient sequences also satisfy the exact identity D(N) n = De(N) n + 2U 2RDe(N) n−1 + U 4RDe(N) n−2. This follows by expanding the two additional atomic factors in their generating functions. The final derivation of the spin-correlation lower bound Sr = U 2R De(N) n−1 D(N) n ≥ ρg2U2R (g2 + ρU2) > 0 is established. This establishes ODLRO in both the photon and atomic sectors. The data that support the findings of this article are openly available [45].

1. Bold header: Quantum Scars for Nonthermal AI States

This system can engineer quantum scars embedded within a chaotic many-body spectrum, which violate the eigenstate thermalization hypothesis (ETH) and undergo weak ergodicity breaking. This allows for the creation of non-thermalizing AI states that exhibit persistent coherence or structure in otherwise thermalizing computational regimes.

2. Bold header: Polariton Condensate for Coherent Processing

The system can form polariton condensates exhibiting off-diagonal long-range order (ODLRO) within fixed-excitation sectors. This capability enables the construction of coherent quantum phases suitable for implementing robust, collective quantum computation or highly correlated machine learning models.

3. Bold header: Stark Coupling for Phase Transition Control

The system allows researchers to investigate how negative Stark coupling shifts the Mott-insulator to superfluid transition toward weaker inter-cavity hopping. This provides a mechanism to tune the fundamental phase of a hybrid light-matter architecture, enabling precise control over switching between localized (Mott) and delocalized (superfluid) computational regimes.

4. Bold header: Arbitrary Lattice Engineering

The building-block method enables the construction of lattice Hamiltonians with exact many-body eigenstates across arbitrary lattice geometries and atom distributions. This means the AI system architecture can be tailored precisely to specific hardware layouts without being restricted by translation or reflection symmetry, maximizing hardware efficiency.

5. Bold header: Ground State Protection

The constructed exact states are proven to be ground states within their respective fixed-total-excitation sectors under the specified resonance conditions and on the negative-Stark-coupling branch. This ensures that the engineered quantum states possess a robust, stable minimum energy configuration in specific excitation manifolds.

6. Bold header: Correlation Function Characterization

The system provides analytical expressions for correlation functions, such as Ar = g squared D(N)n−1 D(N)n and Sr = U squared R De(N)n−1 D(N)n, which show that correlations remain finite and nonzero at arbitrarily large separations in the thermodynamic limit. This confirms the presence of simultaneous photon and atomic ODLRO, essential for verifying complex correlation properties in large-scale systems.