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
($\hbar=1$):
:$H = \sum_^\omega_c a_^a_ +\sum_^\omega_a \sigma_n^+\sigma_n^- + \kappa \sum_^ \left(a_^a_+a_^a_\right) + \eta \sum_^ \left(a_\sigma_^ + a_^\sigma_^\right)$
where $\sigma_^$ 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 $\hat_c \equiv \sum_^a_n^a_n$ and $\hat_a \equiv \sum_^ \sigma_^\sigma_^$, the total number of excitations is a conserved quantity,
i.e., $\lbrack H,\hat_c+\hat_a\rbrack=0$.

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.

Further reading

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

References

{{DEFAULTSORT:Jaynes-Cummings-Hubbard model
Quantum optics