Jaynes–Cummings–Hubbard model
The Jaynes–Cummings–Hubbard 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 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:
where are Pauli operators for the two-level atom at the
n-th cavity. The is the tunneling rate between neighboring cavities, and is the vacuum Rabi frequency which characterizes to the photon-atom interaction strength. The cavity frequency is and atomic transition frequency is. 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 and, the total number of excitations is a conserved quantity,
i.e.,.