Figure 1

Schematic of Hamiltonian, where the cavity field and walls are omitted for clarity. The system consists of bosonic atoms with two internal levels (red and green). The cavity field allows transitions (blue arrows) between the two internal levels due to atom-photon coupling $g$. $J$ denotes the atomic tunneling between nearest neighbor sites. For simplicity, we consider only hardcore intraspecies interactions and zero interspecies interaction.Reuse & Permissions

Figure 2

The phase diagram shows four distinct phases in the ${\mu }_2/g-J/g$ plane. Here ${\mu }_2$ controls the number of excitations, $J$ is the intersite tunneling rate, and $g$ is the atom-photon coupling. The lower half of the phase diagram has no photons or atomic excitations, leading to a typical MI-SF transition of unispecies hardcore bosons. For larger ${\mu }_2$, we predict a field-matter entangled system. As one tunes $J/g$ in this region, there is a MI-SF transition of both atomic species. The double arrow highlights the phase transition which we explore.Reuse & Permissions

Figure 3

(a) Dependence of mean field order parameters of the “a atoms” and “b atoms” indicating a first order transition when the system is light entangled; (b) photon number density $⟨n_{ph}⟩$; and (c) photon absorption $|⟨{\psi }_i{b}_i^{†}{a}_i⟩|/\sqrt{n_{ph}{n}_b{n}_a}$, as a function of hopping strength $zJ/g$. Plots (a)–(c) indicate that the Mott-insulating to superfluid phase transition is accompanied by a change in both the intracavity photon number and the photon absorption strength.Reuse & Permissions

Figure 4

(a) Quench dynamics of the intracavity photon number density, found from time evolving an initial superfluid state. The red curve assumes zero dissipation. The blue curve accounts for photon leakage by averaging over 10 000 wave function Monte Carlo realizations ($(\kappa /g)=0.05$). The dissipation modifies the photon dynamics by causing an overall decay, but the oscillation is preserved for sufficiently small $\kappa$. The Monte Carlo error bars are smaller than the plot line width. (b) Fourier transform of the zero-dissipation dynamics indicates two dominant frequencies with corresponding amplitudes of photon oscillation $A_1$ and $A_2$. (c) The amplitudes $A_1$ and $A_2$ in the absence of dissipation, upon varying initial hopping closely correspond to the superfluid order parameters. This trend offers a novel experimental method to deduce the order parameters from photon number dynamics alone. In the presence of dissipation, the value of $A_1$ is offset by roughly a constant value from zero, even when quenching a MI state.Reuse & Permissions