Jaynes–Cummings–Hubbard model

The Jaynes–Cummings–Hubbard (JCH) model is a many-body quantum system modeling the quantum phase transition of light. As the name suggests, the Jaynes–Cummings–Hubbard model is a variant on the Jaynes–Cummings model; a one-dimensional JCH model consists of a chain of N coupled single-mode cavities, each with a two-level atom. Unlike in the competing Bose–Hubbard model, Jaynes–Cummings–Hubbard dynamics depend on photonic and atomic degrees of freedom and hence require strong-coupling theory for treatment. One method for realizing an experimental model of the system uses circularly-linked superconducting qubits.

History
The JCH model was originally proposed in June 2006 in the context of Mott transitions for strongly interacting photons in coupled cavity arrays. A different interaction scheme was synchronically suggested, wherein four level atoms interacted with external fields, leading to polaritons with strongly correlated dynamics.

Properties
Using mean-field theory to predict the phase diagram of the JCH model, the JCH model should exhibit Mott insulator and superfluid phases.

Hamiltonian
The Hamiltonian of the JCH model is ($$\hbar=1$$):
 * $$H = \sum_{n=1}^{N}\omega_c a_{n}^{\dagger}a_{n}

+\sum_{n=1}^{N}\omega_a \sigma_n^+\sigma_n^- + \kappa \sum_{n=1}^{N} \left(a_{n+1}^{\dagger}a_{n}+a_{n}^{\dagger}a_{n+1}\right) + \eta \sum_{n=1}^{N} \left(a_{n}\sigma_{n}^{+}        + a_{n}^{\dagger}\sigma_{n}^{-}\right) $$ where $$\sigma_{n}^{\pm}$$ are Pauli operators for the two-level atom at the n-th cavity. The $$\kappa$$ is the tunneling rate between neighboring cavities, and $$\eta$$ is the vacuum Rabi frequency which characterizes to the photon-atom interaction strength. The cavity frequency is $$\omega_c$$ and atomic transition frequency is $$\omega_a$$. The cavities are treated as periodic, so that the cavity labelled by n = N+1 corresponds to the cavity n = 1. Note that the model exhibits quantum tunneling; this process is similar to the Josephson effect.

Defining the photonic and atomic excitation number operators as $$\hat{N}_c \equiv \sum_{n=1}^{N}a_n^{\dagger}a_n$$ and $$\hat{N}_a \equiv \sum_{n=1}^{N} \sigma_{n}^{+}\sigma_{n}^{-}$$, the total number of excitations is a conserved quantity, i.e., $$\lbrack H,\hat{N}_c+\hat{N}_a\rbrack=0$$.

Two-polariton bound states
The JCH Hamiltonian supports two-polariton bound states when the photon-atom interaction is sufficiently strong. In particular, the two polaritons associated with the bound states exhibit a strong correlation such that they stay close to each other in position space. This process is similar to the formation of a bound pair of repulsive bosonic atoms in an optical lattice.