Strongly Correlated States of Light and Repulsive Photons in Chiral Chains of Three-Level Quantum Emitters

We study the correlated transport of photons through a chain of three-level emitters that are coupled chirally to a photonic mode of a waveguide. It is found that this system can transfer a weak classical input into a strongly correlated state of light in a unitary manner. Our analysis reveals two-photon scattering eigenstates, that are akin to Fano resonances or shape resonances in particle collisions and facilitate the emergence of antibunched light with long-range correlations upon crossing a critical length of the chain. By operating close to conditions of electromagnetically induced transparency of the three-level medium, a high degree of antibunching and photon transmission can be maintained in the presence of moderate losses. These features suggest a promising mechanism for single-photon generation and may open the door to exploring correlated quantum many-body states of light with repulsively interacting photons.

Figure 1

A chain of three-level emitters is chirally coupled to a photonic mode of a waveguide. The guided photons drive a transition between a stable state $|g⟩$ and an excited state $|e⟩$, which in turn is coupled to another stable state $|b⟩$. The corresponding coupling strengths, level energies, and frequency detunings are indicated in the depicted level diagram. The collective coupling to a chain of such three-level emitters can give rise to extended-range correlation, as indicated by the two-photon correlation function $g^{(2)}(r)$ of transmitted photons in panel (a). Strongly antibunched photons with $g^{(2)}(0)\approx 0$ can emerge in a broad parameter region upon crossing a critical chain length $N$, as shown in panel (b) for different control field Rabi frequencies $\mathrm{\Omega }$. The photon output is determined by the two-photon eigenstates ${\mathrm{\Psi }}_{E,\nu }(x,x^{\prime })$ of the underlying scattering matrix. We find three types of such states, which are illustrated in panel (c) and discussed in more detail in the main text. All calculations are performed for $\gamma =0$ and $\mathrm{\Delta }=\mathrm{\Gamma }/4=-\overline{\mathrm{\Delta }}$, and panels (a) and (c) have been obtained for $\mathrm{\Omega }=\mathrm{\Gamma }/2$ and 10 emitters.

Figure 2

Wave function decomposition of incident state ${\psi }_{\mathrm{in}}(x,x^{\prime })=⟨x,x^{\prime }|\mathrm{in}⟩$ [(a) and (e)] and transmitted photon states ${\psi }_{\mathrm{out}}(x,x^{\prime })=⟨x,x^{\prime }|\mathrm{out}⟩$ [(b)–(d) and (f)–(h)] into the three types of eigenstates of the $S$-matrix, whose color coding is the same as in Fig. 1. Results are shown for a chain of three-level emitters in panels (a)–(d) and for a chain of two-level emitters in panels (e)–(h). The total wave function is indicated by the black line and grey shaded area. The two-photon scattering-resonance states (red line) do not exist for two-level emitters, where coherent antibunching is not possible. In panels (i)–(k) we show the output for a dissipative ($\gamma =9\mathrm{\Gamma }$) two-level chain where the input field is on resonance. In panels (a)–(h) we use single-photon detuning $\mathrm{\Delta }=\mathrm{\Gamma }/4$, and we use $\mathrm{\Omega }=\mathrm{\Gamma }/2$ and $\overline{\mathrm{\Delta }}=-\mathrm{\Gamma }/4$ for the three-level emitter [(a)–(d)]. Panel (e) shows the decomposition of the input for all panels (f)–(k).

Figure 3

(a) The two-photon correlation function $g^{(2)}(r)$ for $\mathrm{\Omega }=\mathrm{\Gamma }/2$, $\mathrm{\Delta }=\mathrm{\Gamma }/4$, $\overline{\mathrm{\Delta }}=-\mathrm{\Gamma }/4$, $N=10$, and different decay rates $\gamma$ into other modes than the guided. (b) Two-photon transmission through a chain of 10 emitters with the parameters of panel (a) as a function of $\gamma$. The points mark the transmission for the values of $\gamma$ used in panel (a). (c) Minimum number of emitters that yields antibunching with $g^{(2)}(0)\le 0.1$ as a function of $\mathrm{\Omega }$ for $\mathrm{\Delta }=0.4\mathrm{\Gamma }$ and $\overline{\mathrm{\Delta }}=-0.2\mathrm{\Gamma }$. The corresponding optimal two-photon transmission $|T_{\mathrm{opt}}{|}^2$ is shown in panel (d).