Nonlocal Rydberg enhancement for four-wave-mixing biphoton generation

Tunable Rydberg interactions in cold atoms can be used to generate strongly correlated photon pairs through a spontaneous four-wave-mixing process in which all driving fields are allowed to work on resonance. The scheme relies on electromagnetically induced transparency and an imbalanced dark state with most atoms trapped in a ground state and a weakly populated Rydberg state. We show in particular that the combination of these two effects results in negligible linear absorption and Raman gain at the Stokes and anti-Stokes frequencies of generated photon pairs, while the nonlocal (Rydberg) nonlinearities contribute significantly to the enhancement of photon-pair generation rates and cross correlations at very low pump strengths and optical depths.

Figure 1

(a) Two five-level cold atoms interacting with pump (${\omega }_p$), coupling (${\omega }_c$), and driving (${\omega }_d$) fields while in the presence (left) or absence (right) of Stokes ${\omega }_{\mathrm{s}}$ and anti-Stokes ${\omega }_{\mathrm{as}}$ biphotons generated via a resonant SFWM process. The vdW interactions between some atoms (left) in a coherent superposition of the ground state $|1\rangle$ and Rydberg state $|5\rangle$ and other atoms (right) in the Rydberg state will result in the two-body correlator ${\mathcal{V}}_{55,51}^{\text{int}}$, which then yields an extra nonlocal nonlinearity facilitating biphoton generation. (b) Schematic of the nonlinear generation of Stokes and anti-Stokes biphotons in a sample of cold atoms illuminated by the copropagating pump and coupling fields as well as a transverse driving field. A specific case has been considered where at most two atoms (large yellow balls) can be excited to the Rydberg state while other atoms (small orange balls) are in the ground state within each sphere (dashed circles) of the triple blockade radius $3R_b$ characterizing the range of vdW interactions.

Figure 2

(a)–(c) Zeroth-order atomic populations and (d) dark-state population ratios plotted against Rabi frequency ${\mathrm{\Omega }}_d$ with ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_p={\mathrm{\Delta }}_d=0$, ${\mathrm{\Omega }}_c=5\gamma$, and ${\mathrm{\Omega }}_p=0.13\gamma$. Other parameters for ${}^{87}\mathrm{Rb}$ atoms [61] are ${\mathrm{\Gamma }}_{54}\simeq {\mathrm{\Gamma }}_{53}\simeq {\gamma }^{\prime }=2\pi \times 2.0$ kHz, ${\mathrm{\Gamma }}_{42}={\mathrm{\Gamma }}_{41}=\gamma =2\pi \times 2.873$ MHz, ${\mathrm{\Gamma }}_{32}=5\gamma /3$, ${\mathrm{\Gamma }}_{31}=\gamma /3$, ${\gamma }_{54,53,42,41,32,31}=\gamma$, ${\gamma }_{52,51,21}={\gamma }^{\prime }$, $C_6=2\pi \times 140$ GHz $\mu {\text{m}}^6$, $N_{\mathrm{a}}=1.0\times {10}^{10}{\mathrm{cm}}^{-3}$, and $L=1.0$ mm. Dashed lines are obtained in the absence of vdW interactions.

Figure 3

(${\mathrm{a}}_1$) and (${\mathrm{b}}_1$) Linear susceptibilities together with their (${\mathrm{a}}_2$) and (${\mathrm{b}}_2$) local and (${\mathrm{a}}_3$) and (${\mathrm{b}}_3$) nonlocal components plotted against two-photon detuning $\delta$ for anti-Stokes (left) and Stokes (right) fields. Red solid and blue dashed lines refer, respectively, to the real and imaginary parts of corresponding susceptibilities. Parameters are the same as in Fig. 2, except ${\mathrm{\Omega }}_d=1.3\gamma$.

Figure 4

(${\mathrm{a}}_1$) and (${\mathrm{b}}_1$) Nonlinear susceptibilities together with (${\mathrm{a}}_2$) and (${\mathrm{b}}_2$) local and (${\mathrm{a}}_3$) and (${\mathrm{b}}_3$) nonlocal components plotted against two-photon detuning $\delta$ arising from atomic polarizations $P_{31}$ (left) and $P_{42}$ (right). Red solid and blue dashed lines refer, respectively, to the real and imaginary parts of corresponding susceptibilities. Parameters are the same as in Fig. 3.

Figure 5

Absolute values of (a) the wave-number mismatch, (b) phase-mismatch factor, (c) wave-mixing function, and (d) joint spectral amplitude plotted against two-photon detuning $\delta$. Blue dashed lines correspond to the case with only local (linear and nonlinear) susceptibilities considered. Parameters are the same as in Fig. 3, except ${\lambda }_p\simeq {\lambda }_c\simeq {\lambda }_{\mathrm{s}}\simeq {\lambda }_{\mathrm{as}}\simeq 795.0$ nm.

Figure 6

(a) Biphoton generation rate plotted against ${\mathrm{\Omega }}_d/{\mathrm{\Omega }}_p$ with ${\mathrm{\Omega }}_p=0.13\gamma$, as well as (b) cross-correlation and (c), (d) autocorrelation functions plotted against time delay $\tau$. The blue dashed line in (a) refers to the case with only local (linear and nonlinear) susceptibilities considered. Parameters are the same as in Fig. 3, except ${\lambda }_p\simeq {\lambda }_c\simeq {\lambda }_{\mathrm{s}}\simeq {\lambda }_{\mathrm{as}}\simeq 795.0$ nm.

Figure 7

Zeroth-order (${\mathrm{a}}_1$) and (${\mathrm{b}}_1$) one-body and (${\mathrm{a}}_2$) and (${\mathrm{b}}_2$) two-body density-matrix elements plotted against the interatomic distance $|{\mathbf{r}}^{\prime }-\mathbf{r}|$ on which the latter depend. Here, ${\mathrm{\Omega }}_d=1.3\gamma$, while other parameters are the same as in Fig. 2. Red solid and blue dashed lines refer, respectively, to the real and imaginary parts of corresponding density-matrix elements.