Long-range interactions and entanglement of slow single-photon pulses

We show that very large nonlocal nonlinear interactions between pairs of colliding slow-light pulses can be realized in atomic vapors in the regime of electromagnetically induced transparency. These nonlinearities are mediated by strong, long-range dipole-dipole interactions between Rydberg states of the multilevel atoms in a ladder configuration. In contrast to previously studied schemes, this mechanism can yield a homogeneous conditional phase shift of $\pi$ even for weakly focused single-photon pulses, thereby allowing a deterministic realization of the photonic phase gate.

Figure 1

(Color online) (a) Level scheme of atoms interacting with weak (quantum) fields $E_{1,2}$ on the transitions $\mid g⟩\to \mid e_{1,2}⟩$ and strong driving fields of Rabi frequencies ${\Omega }_{1,2}$ on the transitions $\mid e_{1,2}⟩\to \mid d_{1,2}⟩$, respectively. $V_{\mathrm{dd}}$ denotes the dipole–dipole interaction between pairs of atoms in Rydberg states $\mid d⟩$. (b) Upon entering the medium, each field having Gaussian transverse intensity profile is converted into the corresponding polariton ${\Psi }_{1,2}$ representing a coupled excitation of the field and atomic coherence. These polaritons propagate in the opposite directions with slow group velocities $v_{1,2}$ and interact via the dipole–dipole interaction.

Figure 2

(a) The 1D dipole–dipole potential $\Delta \left(\zeta \right)$ of Eq. (9) as a function of dimensionless distance $\zeta =(z-z^{\prime })∕w$, in units of $2C∕w^3\mathrm{Hz}$. (b) The resulting phase shift $\varphi \left(\tau \right)\equiv \varphi (vt,L-vt,t)$ of Eq. (14) as a function of dimensionless time $\tau =vt∕w$, in units of $2C∕\left(v{w}^2\right)\mathrm{rad}$.