Controlling correlations of a polaritonic Luttinger liquid by engineered cross-Kerr nonlinearity

TL;DR

The paper analyzes polaritons in a one-dimensional multiconnected Jaynes--Cummings lattice with engineered cross-Kerr nonlinearity that induces an attractive nearest-neighbor interaction. By projecting to the lower polariton manifold and mapping onto a bipartite extended Bose--Hubbard model, the authors use bosonization to show that the antisymmetric sector becomes gapped and the long-wavelength physics is captured by a single gapless symmetric LL with an enhanced Luttinger parameter $K_+$. The cross-Kerr coupling increases $K_+$ (provided $U-2\chi>0$), leading to slower algebraic decay of the single-particle correlator $G^{(1)}(x) \propto |x|^{-1/(4K_+)}$ and stronger phase coherence, while density correlations remain short-ranged. Importantly, both photonic and qubit observables inherit the same LL scaling through the gapless $+$ mode, illustrating a unified low-energy description of the polaritonic fluid. The results establish the MCJC platform with cross-Kerr engineering as a tunable, 1D quantum fluid with potential applications in controllable coherence and quantum simulation of extended Hubbard physics.

Abstract

We study correlation properties of polaritons at zero temperature in a multiconnected Jaynes--Cummings (MCJC) lattice on a superconducting circuit quantum electrodynamics platform with engineered cross-Kerr nonlinearity that mimics attractive nearest-neighbour interaction. A multi-connected Jaynes--Cummings lattice is a one-dimensional lattice constructed from alternating qubits and resonators with different left and right couplings. The nearest-neighbour interaction or cross-Kerr coupling is implemented dispersively through ladder-type qutrits between each nearest neighboring pair of resonator modes. Projecting onto the lower-polaritonic manifold, we derive an extended two-mode (bipartite) Bose--Hubbard-like model featuring on-site and attractive nearest-neighbor interactions. Employing a continuum bosonization approach, we express the Hamiltonian in terms of symmetric ($+$) and antisymmetric ($-$) collective modes. In the regime where the ($-$) sector acquires a finite gap, one can reduce the system to an effective single-component Luttinger liquid model for the $+$ sector. The cross-Kerr term reduces the compressibility of the ($+$) mode, thereby enhancing the corresponding Luttinger parameter $K_{+}$, resulting in the slower algebraic decay of single-particle correlations, $G(x)\propto|x|^{-1/(4K_{+})}$.

Figures (7)

• Figure 1: Schematic of a multiconnected Jaynes--Cummings (MCJC) lattice. Each cavity mode is locally coupled to a qubit, while adjacent cavities are indirectly connected via qubit-mediated couplings with $g_l$ and $g_r$.
• Figure 2: MCJC lattice after the installation of qutrit-mediated cross-Kerr interactions. Auxiliary three-level systems couple neighboring cavities and generate effective nearest-neighbor cross-Kerr terms -$\chi\, n_i n_{i+1}$. This interaction alternates between adjacent links, rendering the lattice into a bipartite system with two inequivalent sublattices $A$ and $B$.
• Figure 3: First-order coherence $g^{(1)}(x)$ as a function of distance $x/a$ for different values of the cross-Kerr interaction $\chi/U$. The algebraic decay becomes progressively slower with increasing $\chi/U$, reflecting an increase of the Luttinger parameter $K_+$ and enhanced quasi-long-range phase coherence.
• Figure 4: Spatial density-density correlations $g^{(2)}(x)$ versus $x/a$ for different $\chi/U$. All curves rapidly approach the uncorrelated value $g^{(2)}(x)=1$, with weak oscillatory corrections. The inset highlights the asymptotic convergence at large distances, consistent with the Luttinger-liquid prediction.
• Figure 5: Connected qubit magnetization correlator $C_{zz}(x)$ as a function of distance $x/a$. The correlator decays algebraically to zero for all $\chi/U$. Variations with $\chi/U$ reflect nonuniversal amplitudes, while the asymptotic scaling is governed by the same Luttinger parameter $K_+$ as the photonic correlators.