Coulomb Bound States of Strongly Interacting Photons

We show that two photons coupled to Rydberg states via electromagnetically induced transparency can interact via an effective Coulomb potential. This interaction gives rise to a continuum of two-body bound states. Within the continuum, metastable bound states are distinguished in analogy with quasibound states tunneling through a potential barrier. We find multiple branches of metastable bound states whose energy spectrum is governed by the Coulomb potential, thus obtaining a photonic analogue of the hydrogen atom. Under certain conditions, the wave function resembles that of a diatomic molecule in which the two polaritons are separated by a finite “bond length.” These states propagate with a negative group velocity in the medium, allowing for a simple preparation and detection scheme, before they slowly decay to pairs of bound Rydberg atoms.

Figure 1

(a) The probe field couples the ground state $|g⟩$ to the excited state $|e⟩$ and is red detuned by $\mathrm{\Delta }$. A control field with Rabi frequency $\mathrm{\Omega }$ couples $|e⟩$ to the Rydberg state $|R⟩$ and is blue detuned by $\mathrm{\Delta }$, thus putting the probe on an EIT transmission resonance. The Rydberg state is thus shifted downward by ${\mathrm{\Omega }}^2/\mathrm{\Delta }$. The van der Waals interaction with another reference Rydberg excitation at $r=0$ can bring $|R⟩$ into an absorption resonance with the two-photon transition. (b) The effective potential of two Rydberg polaritons as a function of their separation $r$ exhibits a singularity at $|r|=r_b$ (the blockade radius) and behaves near this singularity as a Coulomb potential.

Figure 2

(a) Dispersion curves for the exact eigenstates underlying the metastable Coulomb bound states (only the first four branches $n=1–4$ are shown) with $g^2{r}_b/c\mathrm{\Delta }=40$ and $\mathrm{\Omega }/g=0.05$. The solid lines give the exact solution, while the dashed lines represent the WKB results. For $K\to 2g^2/c\mathrm{\Delta }$, the WKB results are almost exact. The dispersion curves converge to one point with a negative slope, or group velocity. (b) Decomposition of the metastable Coulomb bound states (${\mathrm{\Psi }}_n^c$, defined as ${\mathrm{\Psi }}_{{\omega }_n,K_n}$ with the $\delta$-function contribution removed) into the continuum of exact eigenstates. Here, we took $g^2{r}_b/c\mathrm{\Delta }=40$ and $\mathrm{\Omega }/g=0.01$ with a fixed center of mass momentum $cK\mathrm{\Delta }/2g^2=0.95$. The width of these distributions is much less than the energy spacing, indicating they are spectrally distinguishable. The inset shows the wave function components for the $n=1$ Coulomb state with the parameters in (a) and $cK\mathrm{\Delta }/2g^2=0.98$. (The $EE$ component is exaggerated by a factor of 1.5 for better visibility.)

Figure 3

(a) Level structure used to prepare the initial $SS$ distribution. (b)–(d) Time evolution of a wave packet with all components initially zero except $SS$, which is chosen to be a Gaussian wave packet of variational $n=1$ Coulomb state solutions (with the $\delta$ function removed) centered at $\omega =0$ and having width ${\mathrm{\Omega }}^2/2\mathrm{\Delta }$ [61]. Specifically, $|EE|$, initially zero, is shown after the initial transient evolution subsides at (b) $t=t_f/4$, and at (c) $t=t_f/2$ and (d) $t=t_f$, where $t_f=5.5\mathrm{\Delta }/{\mathrm{\Omega }}^2$. The wave packet within the blockade radius has the expected shape of the Coulomb state, propagates backward, and decays, while the wave packet outside the blockade radius propagates forward with $v_g$ and disperses. We took a medium of length $L=16r_b$, $g^2{r}_b/c\mathrm{\Delta }=5$, $g/2\pi =17\mathrm{GHz}$, $\mathrm{\Omega }/2\pi =1.5\mathrm{MHz}$, $r_b=25\mu \mathrm{m}$, $\mathrm{\Delta }/2\pi =30\mathrm{MHz}$, $\gamma /2\pi =3\mathrm{MHz}$, and ${\gamma }^{\prime }/2\pi =5\mathrm{kHz}$ [61].