Ginzburg-Landau theory for the Jaynes-Cummings-Hubbard model

We develop a Ginzburg-Landau theory for the Jaynes-Cummings-Hubbard model which effectively describes both static and dynamic properties of photons evolving in a cubic lattice of cavities, each filled with a two-level atom. To this end we calculate the effective action to first order in the hopping parameter. Within a Landau description of a spatially and temporally constant order parameter we calculate the finite-temperature mean-field quantum phase boundary between a Mott insulating and a superfluid phase of polaritons. Furthermore, within the Ginzburg-Landau description of a spatiotemporal varying order parameter we determine the excitation spectra in both phases and, in particular, the sound velocity of light in the superfluid phase.

Figure 1

Quantum phase boundary for zero and finite temperatures at resonance $\Delta =0$. Interior of the lobes for $T=0$ K corresponds to the Mott insulator phase whereas the exterior corresponds to the superfluid phase. For finite temperatures the Mott insulator vanishes and becomes a mixture of normal and Mott phase. Note that the drop of the phase boundary at ${\mu }_{\mathrm{eff}}\approx -0.1g$ is a numerical remnant resulting from a summation cutoff in Eq. (29). A full summation would lead to an infinite sequence of Mott lobes approaching ${\mu }_{\mathrm{eff}}=0$.

Figure 2

The pair-excitation energy gap (left) and the effective mass (right) as a function of temperature for the first lobe $n=1$ at ${\mu }_{\mathrm{eff}}={\mu }_{\mathrm{crit}}=0.78g$ with zero detuning $\Delta =0$ and $\kappa ={\kappa }_{\mathrm{crit}}=0.16g$ (solid), $\kappa =0.1g$ (dashed), and $\kappa ={10}^{-4}g$ (dotted). The right picture represents the effective mass of both the hole (lower branch in light shade) and particle (upper branch in dark shade) excitations.

Figure 3

Various dynamic results for a mean particle density $n=2$ at zero temperature and vanishing detuning $\Delta =0$. Left column at ${\mu }_{\mathrm{eff}}=-0.4g$, middle column for tip of the lobe at ${\mu }_{\mathrm{eff}}=-0.37g$, and right column at ${\mu }_{\mathrm{eff}}=-0.34g$. (a) Particle (dotted) and hole (dashed) dispersion relations in the Mott phase (gray), on the phase boundary (black) and in the superfluid phase (solid) in the direction $\mathbf{k}=k(1,1,1)$. (b) Energy gap for particle (dotted) and hole (dashed) excitations in the Mott phase and for the massive mode in the superfluid phase (solid). (c) Effective mass for particle (dotted) and hole (dashed) excitations in the Mott phase and for the massive mode in the superfluid phase (solid). (d) Sound velocity for the massive mode in the superfluid phase.

Figure 4

Sound velocity $c$ normalized by the sound velocity at zero detuning $c_0$ as a function of the detuning parameter $\Delta$ for $n=2$ with $\kappa =0.001{\kappa }_{\mathrm{crit}}$ (solid) and $\kappa =0.15{\kappa }_{\mathrm{crit}}$ (dashed).