Jaynes–Cummings–Hubbard model

Tunnelling of photons between coupled cavities. The $\kappa$ is the tunnelling rate of photons.

Illustration of the Jaynes–Cummings model. In the circle, photon emission and absorption are shown.

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 combination of Hubbard-type models with Jaynes-Cummings (atom-photon) interactions near the photon blockade regime originally appeared in three, roughly simultaneous papers in 2006.

All three papers explored systems of interacting atom-cavity systems, and shared much of the essential underlying physics. Nevertheless, the term Jaynes–Cummings–Hubbard was not coined until 2008.

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 ($\hslash =1$):

$H=\sum _{n=1}^N{\omega }_c{a}_n^{†}{a}_n+\sum _{n=1}^N{\omega }_a{\sigma }_n^{+}{\sigma }_n^{-}+\kappa \sum _{n=1}^N(a_{n+1}^{†}{a}_n+a_n^{†}{a}_{n+1})+\eta \sum _{n=1}^N(a_n{\sigma }_n^{+}+a_n^{†}{\sigma }_n^{-})$

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 ${\hat{N}}_c\equiv \sum _{n=1}^N{a}_n^{†}{a}_n$ and ${\hat{N}}_a\equiv \sum _{n=1}^N{\sigma }_n^{+}{\sigma }_n^{-}$, the total number of excitations is a conserved quantity, i.e., $[H,{\hat{N}}_c+{\hat{N}}_a]=0$.Template:Citation needed

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.

Further reading

• D. F. Walls and G. J. Milburn (1995), Quantum Optics, Springer-Verlag.