Equivalent spin-orbit interaction in the two-polariton Jaynes-Cummings-Hubbard model

A cavity quantum electrodynamics (cavity-QED) system combines two or more distinct quantum components, exhibiting features not seen in the individual systems. In this work, we study the one-dimensional Jaynes-Cummings-Hubbard model in the two-excitation (two-polariton) subspace. We find that the centre momentum of two-excitation induces a magnetic flux piercing the equivalent Hamiltonian Hk in the invariant subspace with momentum k, which can be described as a 4-leg ladder in the auxiliary space. Furthermore, it is shown that the system in π-centre-momentum subspace is equivalent to a lattice system for spin-1 particle with spin-orbit coupling. On the basis of this concise description, a series of bound-pair eigenstates which display long-range polaritonic entanglement is presented as a simple application.

dynamics involving photonic and atomic degrees of freedom, which is in contrast to the widely studied Bose-Hubbard model. Such a cavity-QED system can be implemented with a defect array in a photonic crystal 13,14 or a Josephson junction array in a cavity 8,15,16 . The Hamiltonian of a cavity-QED system, or indeed a lattice atom-photon system  17    The expression of H k in Eq. (8) has a clear physical meaning: j m k , , denotes the site state for the j-th site on the m-leg of a 4-leg ladder system with the effective magnetic flux piercing the plaquette. The flux is proportional to the centre momentum of two excitations. The structure of H k is schematically illustrated in Fig. 1. We note that the matrix representation of H k in the basis of Eqs. (10) and (11), H k , breaks the time-reversal symmetry. Nevertheless, we still have Σ k H k = Σ k H k * , as H k * = H -k = H 4π-k . In essence, the nonzero plaquette flux arises from the relationship between the complex coupling con-

4-Leg Ladder with flux. The system is translational invariant
. In contrast, one can see from H k that the complex λ cannot induce a nonzero plaquette flux. We would like to stress that the effective magnetic field in the present model is intrinsic, not depending on an external control, but relying on the value of k. We note that there are two kinds of excitations, the spin-up (excited atom) state and a photon, which obey two different statistics (that for hardcore bosons and bosons). This may be the origin of the equivalent plaquette flux. Then, the underlying mechanisms for obtaining the equivalent plaquette flux in our work and that of Ref. 18,19 are different.
In order to understand the mechanism of the effective flux, we investigate the exchange process for photon and atomic excitations beginning in the state l l j ψ ( , + )  . This investigation implies that the origin of the effective magnetic field may be the special statistical properties of two quasi-particles in each invariant subspace. Equivalent Hamiltonian in π-momentum subspace. We focus on the case k = π and ω a = ω b , which leads to H a AP  ω =ˆ. It is a simple but non-trivial case, since the hopping along leg 2 is switched off but the plaquette flux still exists. We note that the on-site potentials μ l of different legs are identical, which allows us to ignore the diagonal terms in H π .
Introducing the three-dimensional vector bra and ket for the Hamiltonian H π in the π-momentum subspace can be expressed as (b) For k = π, H k is equivalent to a spin-1 chain with spin-orbit interaction. The graph of H π consists of two unconnected subgraphs, characterized by the parity Π = ± 1. H k indicates that H π can be further decomposed into two independent parts H o (dark) and H e (blue).
, which indicates that H π is block-diagonal. Here j S z , represents a spin-1 particle at the j-th site with spin polarization S z = 0, ± 1 defined as π π π π π π       Consequently, within a specific invariant subspace, a system made of N-cavity array with a single two-level atom embedded in each cavity appears to be equivalent to a tight-binding chain of spin-1 particle with spin-orbit interaction. The structure of H so is schematically illustrated in Fig. 1. Intuitively, the graph of H so consists of two unconnected subgraphs. This can be clarified by observing that the parity operator 1 23  i.e. j ϕ is an eigenstate of H so . This is a direct application of the bound state theorem given in 20 , which states that any eigenstate of a sub-graph is also an eigenstate of the whole, if the nodes cover all the joint points. We are interested in the expression of these states in the atom-photon basis, given by with excited atomic state. There should be a large probability for the transition between states with the same energy, including our target states. This scheme requires a temporal control of parameters λ and κ in experiment. At this stage, this is just a qualitative analysis, but will motivate further quantitative investigation for the procedure of entangled state preparation in a cavity-QED system. The key process referred to as mixed-type in Eq. (51) shows that the cancellation of the transitions requires an optimal ratio between the parameters λ and κ.