Superfluid-Mott-Insulator Quantum Phase
Transition of Light in Jaynes-Cummings
Lattice
H. Zheng and Y. Takada
Quantum phase transitions and related phenomena
constitute a field of great interest in the physics ofstrong correlation. Recently, the Jaynes-Cummings
lattice (JCL) attracts much attention in this field,
partly because it is a good model system for studying
strongly correlated polariton physics and partly
because it offers potential use of a quantum simulator
for solid-state Hamiltonians, just like the cold-atomsystems.
The JCL model is composed of a lattice ofelectromagnetic micro-cavities with frequency , eachcoupled to a two-level atom with level spacing , andthe intrasite atom-photon coupling g is competed withthe intersite photon hopping J. A mean-field treatmentof the model reveals a phase transition from Mott
insulator (MI) to superfluid phase, resembling in large
parts the phase diagram of the Bose-Hubbard modelwith an eminent feature of the Mott lobes at theboundary between the insulating phase and thesuperfluid one (See Fig. 1). Both numerical andanalytical methods have been employed to study thephase diagram as well as the spectra of low-lying
excitations beyond the mean-field approximation andconfirmed the existence of the Mott lobes.
All those theoretical studies, however, have notbeen done on the physically correct original JC model
but under the rotating-wave approximation (RWA), in
which the Hamiltonian HRWA commutes with thepolariton number operator Np at each lattice site,leading to the conservation of the local polariton
number as specified by the chemical potential . Thisconservation law is, in fact, a key feature to provide
the Mott lobes in the phase diagram.
The counter-rotating (CR) term HCR, included in theoriginal JCL model but neglected so far, will beirrelevant in resonance experiments to detect realtransitions, but this will be physically relevant in
virtual transitions in the formation of the ground state,making the RWA for the JCL model less reliable indescribing the ground-state properties, especially for gof the order ofand/or . Besides, HCR breaks theconservation of the local polariton number, i.e.,[HCR,Np] ≠ 0, implying a radical change of the phase
diagram, once the CR is included. Moreover, HCR
induces an additional long-range interaction between
cavities, enhancing a long-range ordering in the JCLmodel.
Here we have studied the effects of HCR on thequantum phase transition of light in thetwo-dimensional (2D) square JCL with HCR faithfullyincluded. The ground state as well as spectra oflow-lying excitations in both the insulating and thesuperfluid phases is investigated with use of a
sophisticated unitary transformation, which has beendeveloped in the polaron physics. The obtained phase
diagram is shown in Fig. 1, indicating that the Mott
lobes are absent in sharp contrast with the case ofRWA. Thus we may conclude that the physics of theJCL is basically different from that of theBose-Hubbard model as opposed to the conclusiondrawn in the preceding works.

Fig. 1. Phase diagram in the [( - )/g, zJ/g] space, or theparticle average number vs intersite coupling strength J in unitsof the atom-photon coupling g, for the two-dimensional square
JCL at the resonant condition of= , where z is thecoordination number. The inclusion of the usually neglected
counter-rotating term makes such a radical change in the phase
diagram; in particular, the eminent Mott lobes are eliminated.

Finally we find that gapped excitations appear evenin the superfluid phase in our calculation. Note that for
the JCL in the RWA as well as the Bose-Hubbard
model, the particle number conservation is broken by
the superfluid transition, leading to the gapless bosons
according to the Goldstone theorem, but if HCR isincluded, the number conservation is broken from theoutset, making our system stay outside of theGoldstone theorem. In our superfluid transition, thediscrete parity conservation law is broken, instead.
References[1] H. Zheng and Y. Takada, Phys. Rev. A 84, 043819(2011).
Authors
H. Zhenga and Y. Takada
aShanghai Jiao Tong University