Induced Cavities for Photonic Quantum Gates

Effective cavities can be optically induced in atomic media and employed to strengthen optical nonlinearities. Here we study the integration of induced cavities with a photonic quantum gate based on Rydberg blockade. Accounting for loss in the atomic medium, we calculate the corresponding finesse and gate infidelity. Our analysis shows that the conventional limits imposed by the blockade optical depth are mitigated by the induced cavity in long media, thus establishing the total optical depth of the medium as a complementary resource.

Figure 1

(a) When light traverses an atomic medium and acquires a nonzero phase, it always experiences some loss. Given the resonant optical depth $2d_B$, the blue circle traces the relation between phase and loss for two-level atoms or under the conditions of ideal EIT [Eq. (1), see the text]. In a cavity, the radius of the circle effectively grows with the cavity finesse $\mathcal{F}$ (red). (b) A phase gate based on Rydberg blockade by a stored photon.

Figure 2

(a) Optically induced grating in a Sagnac interferometer. (b) Atomic level scheme. A control field (green) couples the probe transition (red) to a Rydberg state, rendering EIT. (c) To induce a cavity, the EIT resonance frequency is longitudinally modulated by a far-detuned dressing standing wave (blue). For example, this scheme can be implemented with rubidium atoms using a probe, control, and dressing fields at 780, 479, and 475 nm, respectively. The angles between the optical axis and the dressing beams are tuned to form a standing wave with a period that is half the probe wavelength.

Figure 3

Cavities encircling atomic media. (a) Ring cavity. (b) The loss vs phase, defined in Eq. (2), for the ring cavity ($\mathcal{F}=20\pi$, $2d_B=4\pi$). The exact relation in Eq. (3) (green) is well approximated by a circle (dashed purple) with a radius $\pi \mathcal{F}/8$ larger than that obtained with no cavity (red). (c) A symmetric Fabry-Pérot cavity inside a Sagnac interferometer is equivalent to a ring cavity. (d) Fabry-Pérot cavity.

Figure 4

Transmission spectra of (a) bare Bragg grating, and [(b) and (c)] grating inside a Sagnac interferometer. We compare (solid blue) a purely dispersive Bragg grating ($\kappa L=5.24$, $\mathrm{Im}\sigma =0$) with (dashed red) a grating formed by dressing an atomic medium ($d={10}^4$, and the dressing parameters are chosen to minimize the gate loss when $2d_B=4\pi$). The resonances of the bare grating are $m=\pm (1,2,3,4,\dots )$, but only $m=1,-2,3,-4,\dots$ retain the steep phase slope in the Sagnac setup. Insets: longitudinal intensity profiles $\parallel \overrightarrow{E}(z){\parallel }^2$ at the corresponding resonances $\pm m=1–4$ (blue, red, orange, and purple).

Figure 5

Scaling of performance with optical depth. Green: numerical maximization of the induced cavity finesse (dots) compared to the analytic result $\mathcal{F}\approx 1.3\sqrt{d}$ (line). Red: numerical minimization of the loss for a phase gate with $2d_B=4\pi$ (dots) and a power-law fit (line). For example, for $d={10}^4$, the optimization finds $\epsilon =0.08$ at $x=6.0\times {10}^{-4}$, $y=5.2\times {10}^{-4}$. The dashed lines in Figs. 4 and 4 are plotted for $y=5.2\times {10}^{-4}$ and scanning $x$. Without the induced grating, the loss is very high ($\epsilon \approx 1$; horizontal orange line). See Ref. [59] for more details on the optimized parameters.