Dicke phase transition in a disordered emitter–graphene-plasmon system

We study the Dicke phase transition in a disordered system of emitters coupled to the plasmonic modes of a graphene monolayer. This system has unique properties associated with the tunable, dissipative, and broadband characters of the graphene surface plasmons, as well as the disorder due to the random spatial distribution and the inhomogeneous linewidth broadening of the emitters. We apply the Keldysh functional-integral approach and identify a normal phase, a superradiant phase, and a spin-glass phase of the system. The conditions for these phases and their experimental signatures are discussed.

Figure 1

Emitter-graphene system. An ensemble of $N$ emitters with spontaneous-emission rate ${\gamma }_0$ and transition frequency inhomogeneously broadened by $\mathrm{\Delta }$ around a central transition frequency ${\omega }_z$ is distributed in a layer with horizontal dimension $L$ at height $z$. The Fermi energy $E_F$ of the graphene electrons can be tuned by gate doping.

Figure 2

Results for systems with ${\omega }_z=0.5\mathrm{eV}$ and $E_F=0.1\mathrm{eV}$: (a) the $N-L$ phase diagram for ${\gamma }_0={10}^{-5}\mathrm{eV}$, with different normal-SG and SG-SR boundaries for $z=20\mathrm{nm}$ (red lower dashed curves) and $z=40\mathrm{nm}$ (upper dashed curves); (b) the value of $Re{\overline{h}}_{\left(1\right)}\left(\omega \right)$, the energy shift induced by graphene; (c) the value of $Im{\overline{h}}_{\left(1\right)}\left(\omega \right)$, the graphene-induced emitter damping, as a function of frequency and different heights for $z=20$ nm (red top curve), $z=30$ nm (orange curve), $z=40$ nm (blue curve), $z=50$ nm (green bottom curve); and (d) the $E_F-L$ phase diagram for $z=50\mathrm{nm}, {\gamma }_0={10}^{-8}\mathrm{eV}$, and different normal-SG and SG-SR boundaries for $N={10}^4$ (blue lower dashed curves) and $N={10}^5$ (red upper dashed curves).

Figure 3

The $E_F-{\omega }_z$ phase diagram for a system with $z=50\mathrm{nm}, L={10}^3\mathrm{nm}, N=2\times {10}^4$, and two different values of ${\gamma }_0/{\omega }_z^3$ so that when ${\omega }_z=0.5\mathrm{eV}, {\gamma }_0={10}^{-7}$ (blue lower dashed curve) or ${10}^{-6}\mathrm{eV}$ (red upper dashed curve). For the case of ${\omega }_z=1\mathrm{eV}$, the spectral densities $A^{\text{SR}}=-2Im{Q}_{cq}\left(\omega \right)$ (red shaded, darkest), $Im{\overline{h}}_{\left(1\right)}$ (blue shaded, peaks at the right-hand side), and $Im{\overline{h}}_{\left(2\right)}$ (gray shaded, lightest) are shown for different Fermi energies $E_F=0.1,0.032,0.004,0.001\mathrm{eV}$. The values of $Im{\overline{h}}_{(1,2)}$ are shown on the right-hand vertical axes.

Figure 4

Coefficients of the emitter–graphene-surface-plasmon coupling for systems with ${\omega }_z=0.5\mathrm{eV}, E_F=0.1\mathrm{eV}$, and $L={10}^3$ nm. From the top to the bottom in the figures we show results for the different heights $z=20$ (red top curve), $z=30$ (orange curve), $z=40$ (blue curve), and $z=50$ (green bottom curve) nm. The dimensionless values are normalized by the emitter spontaneous-emission rate ${\gamma }_0$.