Band structure, phase transitions, and semiconductor analogs in one-dimensional solid light systems

The conjunction of atom-cavity physics and photonic structures (“solid light” systems) offers new opportunities in terms of more device functionality and the probing of designed emulators of condensed-matter systems. By analogy to the canonical one-electron approximation of solid-state physics, we propose a one-polariton approximation to study these systems. Using this approximation, we apply Bloch states to the uniformly tuned Jaynes-Cummings-Hubbard model to analytically determine the energy-band structure. By analyzing the response of the band structure to local atom-cavity control, we explore its application as a quantum simulator and show phase-transition features absent in mean-field theory. Using this approach for solid light systems, we extend the analysis to include detuning impurities to show the solid light analogy of the semiconductor. This investigation also shows features with no semiconductor analog.

Figure 1

(Color online) A possible realization of a one-dimensional Jaynes-Cummings-Hubbard model. Here, holes are drilled into a thin membrane and lattice defects serve as the optical cavities housing two-level atoms. The cavities are coupled evanescently with single-photon hopping rate $\kappa$ and the atom is coupled to the local cavity mode with coupling strength $\beta$.

Figure 2

The solid (dashed) lines are the one-polariton (hole) band structures for $\Delta =0$, $\left(\mu -\omega \right)/\beta =-0.5$, and $\kappa /\beta =0.01$. The numbers indicate the $n$ value.

Figure 3

(Color online) (a) Dashed (solid) lines plot the one-polariton-hole band structure at point $A\left(A^{\prime }\right)$. At $A\left(A^{\prime }\right)$, the energy cost of adding a hole is less (more) than the polariton counterpart. (b) Dashed (solid) lines plot ${\mu }_c^{p,h}$ at point $B\left(B^{\prime }\right)$. The gap at $k=0$ gives the width of the Mott lobes. (c) Plot of the energy gap indicating the Mott lobes. Overlayed is the phase-transition boundary. Indicated also are the Mott lobe states.

Figure 4

(Color online) Comparison of one-polariton approximation (solid line), MF (dotted line), and finite cavity simulations (colored dashed lines). One mean excitation plateau boundary is calculated for 2–5 cavities; two mean excitation plateau boundaries are calculated for 2–4 cavities. As the number of cavities increases, it appears that finite cavity simulations tend to one-polariton analysis and not MF.

Figure 5

(Color online) Every second atomic site is detuned from, but not far from, cavity resonance $\left({\Delta }_0=0,{\Delta }_1=\beta \right)$. The unit cell (dotted box) constitutes two adjacent sites. This introduces extra energy bands within the energy gap of the tuned system, in analogy to the semiconductor doping process.

Figure 6

(Color online) (a) The solid (dashed) lines are the polariton (hole) band structure at point I. The left, middle, and right panel is the band structure for the tuned, doped, and detuned system, respectively. The impurity has introduced extra energy bands in the doped system reducing the energy band gap. The lowest-energy band in the doped system is that of an extra polariton. This is analogous to an $n$-type semiconductor. (b) Solid (dashed) lines are the polariton (hole) band structure at point II. The lowest-energy band in the doped system is that of a hole. This is analogous to a $p$-type semiconductor. (c) Solid (dashed) lines are the polariton (hole) band structure at point III. Interestingly, the impurity increases the band gap: this has no semiconductor analogy. (d) The filled regions are the ${|n⟩}_{{\Delta }_0}\otimes {|n⟩}_{{\Delta }_1}$ (gray) and ${|n+1⟩}_{{\Delta }_0}\otimes {|n⟩}_{{\Delta }_1}$ (light gray) Mott lobes of the doped system. (e) The filled regions are the ${|n⟩}_{{\Delta }_0}\otimes {|n⟩}_{{\Delta }_1}$ (gray) and ${|n-1⟩}_{{\Delta }_0}\otimes {|n⟩}_{{\Delta }_1}$ (dark gray) Mott lobes of the doped system. Overlayed in (d) and (e) in solid (dashed) lines is the phase-transition border for the tuned (detuned) system. Note the different scales of (d) and (e). For notational convenience, ${|n⟩}_{{\Delta }_0}\otimes {|m⟩}_{{\Delta }_1}$ is written as $|n⟩|m⟩$.