Excitations of strongly correlated lattice polaritons S. Schmidt and G. Blatter Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland (Dated: October 31, 2018) We present an analytic slave-boson approach to calculate the elementary excitations of the Jaynes- Cummings-Hubbard Model (JCHM) describing strongly correlated polaritons on a lattice in various quantum optical systems. In the superfluid phase near the Mott transition we find a gapless, linear Goldstone mode and a gapped Amplitude mode corresponding to phase and density fluctuations, respectively. The sound velocity of the Goldstone mode develops a peculiar anomaly as a function of detuning at low densities, which persists into the weakly interacting regime of a polariton BEC. The realization of a BEC of weakly interacting polari- tons in high-Q microcavities [1], i.e., quasiparticles which form when photons strongly interact with quantum well excitons, has triggered an immense interest in quantum condensed and coherent light-matter systems. The ad- dressibility of a cavity with external lasers, device inte- gration, and high tunability make these optical systems ideally suited for quantum information and simulation tasks. Furthermore, the inherently composite nature of polaritons promises rich and interesting new physics. One of the most exciting questions in this emerging field is whether one can realize a Mott insulator - su- perfluid (MI-SF) transition of strongly correlated polari- tons. The Jaynes-Cummings-Hubbard Model (JCHM) has been introduced to describe such a quantum phase transition of light in an array of coupled QED cavities, each containing a single photonic mode interacting with a two-level system (qubit) [2]. Meanwhile the JCHM has become a paradigm lattice model for polaritons describ- ing various systems at the interface of condensed matter and quantum optics [3]. Proposals based on cavity QED arrays can be realized using atoms [4], excitons [5], or superconducting qubits [6]. In trapped ion systems, e.g., in a linear Paul trap, a transverse phonon assumes the role of the photon and couples to hyperfine levels of the ion via an external laser [7]; tunneling of phonons is me- diated by Coulomb interactions between the ions. Ultra- cold atom systems with two optical lattices that couple to the photon mode of a single QED cavity may also realize a JCHM [8]. Theoretically, the phase diagram and the excitations in the Mott phase have been calculated accurately in D = 1 , 2, and 3 dimensions [9–15]. However, very lit- tle is known about the nature of the superfluid phase, except the existence of a Goldstone mode predicted by heavy Monte-Carlo simulations in D = 1 [14]. In this let- ter we present an analytic slave-boson approach valid in both phases. In the Mott phase we find gapped particle and hole modes in quantitative agreement with previous results obtained from a diagrammatic linked-cluster ex- pansion [12]. In the superfluid phase we find a gapless linear Goldstone mode and a gapped Amplitude mode corresponding to phase and density fluctuations (similar excitations were found in ultracold gases [16]). where h JC i denotes the local Jaynes-Cummings Hamilto- nian h JC i = ω$_{c}$ a † $_{i}$a$_{i}$ + ω$_{x}$σ + i σ − i + g ( σ + i a$_{i}$ + σ − i a † $_{i}$) with site index i , boson creation (annihilation) operators a ( † ) i and qubit raising (lowering) operators σ +( − ) i . The bosonic mode frequency is ω$_{c}$ , the two qubit levels are separated by the energy ω$_{x}$ and the coupling is given by g (we set ℏ = 1). We also assume that the total number of excita- tions, i.e., polaritons N = ∑ $_{i}$( a † $_{i}$a$_{i}$ + σ + i σ − i ), is conserved and fixed by the chemical potential µ . We note that even under more general conditions including the effects of an external drive and dissipation the underlying equi- librium model still captures the essential physics of the MI-SF transition as recently shown for the Bose-Hubbard model (BHM) [19]. The third term in (1) describes the delocalization of bosons over the whole lattice due to hop- ping between nearest neighbour sites with amplitude J . It competes with an effective on-site repulsion between bosons mediated by the coupling g . This competition The Hamiltonian of the JCHM is given by In addition, our analytic approach is ideally suited to discuss the behavior of excitations in the superfluid phase as a function of detuning, which provides an important experimentally accessible parameter. Detuning can be used to change (i) the nature of polaritonic excitations, i.e., the relative weight of its bosonic and atomic parts, and (ii) the strength of the effective repulsion between polaritons. We find that the phase diagram as well as the sound velocity of the Goldstone mode show a surprising anomaly when this parameter is varied: For high polari- ton densities, the size of the Mott lobes and the sound velocity decrease for any finite detuning, while at low densities they steadily increase when tuning through the resonance. Comparison to an effective Bogoliubov the- ory for the JCHM shows that this anomaly prevails into the regime of a weakly interacting polariton BEC [17]. In this regime, superfluidity has been demonstrated recently by measuring the sound velocity of exciton-polaritons in a single micro-cavity using angle-resolved photon spec- troscopy [18]. Similar experimental techniques should also be applicable to lattice models as discussed here. H = ∑ i h JC i − µN − J ∑ 〈 ij 〉 a † $_{i}$a$_{j}$ , (1) arXiv:1001.1677v2 [cond-mat.other] 31 May 2010