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Jaynes–Cummings–Hubbard model

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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 Jayne-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, Jayne-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.

Contents

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 ( ℏ = 1 ):

H = ∑ n = 1 N ω c a n † a n + ∑ n = 1 N ω a σ n + σ n − + κ ∑ n = 1 N ( a n + 1 † a n + a n † a n + 1 ) + η ∑ n = 1 N ( a n σ n + + a n † σ n − )

where σ n ± are Pauli operators for the two-level atom at the n-th cavity. The κ is the tunneling rate between neighboring cavities, and η is the vacuum Rabi frequency which characterizes to the photon-atom interaction strength. The cavity frequency is ω c and atomic transition frequency is ω 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 is process is similar to the Josephson effect.

Defining the photonic and atomic excitation number operators as N ^ c ≡ ∑ n = 1 N a n † a n and N ^ a ≡ ∑ n = 1 N σ n + σ n − , the total number of excitations a conserved quantity, i.e., [ H , N ^ c + N ^ a ] = 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.

References

Jaynes–Cummings–Hubbard model Wikipedia
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