Generation, Characterization and Use of Atom-Resonant Indistinguishable Photon Pairs

Abstract

We describe the generation of atom-resonant indistinguishable photon pairs using nonlinear optical techniques, their spectral purification using atomic filters, characterization using multi-photon interference, and application to quantum-enhanced sensing with atoms. Using either type-I or type-II cavity-enhanced spontaneous parametric down-conversion, we generate pairs of photons in the resonant modes of optical cavities with linewidths comparable to the natural linewidths of strong atomic transitions. The cavities and pump lasers are tuned so that emission occurs in a mode or a pair of orthogonally-polarized modes that are resonant to the $\text {D}_1$ line, at 794.7 nm. The emission from these frequency-degenerate modes is separated from other cavity emission using ultra-narrow atomic frequency filters, either a Faraday anomalous dispersion optical filter (FADOF) with a 445 MHz linewidth and 57 dB of out-of-band rejection or an induced dichroism filter with an 80 MHz linewidth and $\ge$ 35 dB out-of-band rejection. Using the type-I source, we demonstrate interference of photon pair amplitudes against a coherent state and a new method for full characterization of the temporal wave-function of narrow-band photon pairs. With the type-II source we demonstrate high-visibility super-resolving interference, a high-fidelity atom-tuned NooN state, and quantum enhanced sensing of atoms using indistinguishable photon pairs.

Appendix: Second-Order Correlation Functions of Filtered Output

In this section we consider the second order correlation function of the field operators $a_\mathrm{out}$ in a form:

$$\begin{aligned} G^{(2)}(T) \propto \langle a_\mathrm{out}^\dagger (t) a_\mathrm{out}^\dagger (t+T) a_\mathrm{out}(t+T) a_\mathrm{out}(t) \rangle \end{aligned}$$

(7.11)

for multimode (unfiltered) and single-mode (filtered) output of the OPO.

As shown by Lu et al. [69], $G^{(2)}(T)$ describing the output of a single-mode, far-below-threshold OPO has the form of double exponential decay

$$\begin{aligned} G_\mathrm{single}^{(2)}(T)\propto e^{-|T|(\gamma _1+\gamma _2)}, \end{aligned}$$

(7.12)

where the reflectivity of the output coupler is $r_1=\exp [{-\gamma _1 \tau }]$, the effective reflectivity resulting from intracavity losses is $r_2=\exp [{-\gamma _2 \tau }]$ and $\tau$ is the cavity round-trip time. An ideal narrowband filter would remove all the nondegenerate cavity-enhanced spontaneous down-conversion CESPDC modes, reducing the $G^{(2)}(T)$ to $G_\mathrm{single}^{(2)}(T)$. This filtering effect was demonstrated in [38] for a type-II OPO and an induced dichroism atomic filter.

In [69] it is also predicted that when the filter is off, so that the output consists of N cavity modes, $G_{}^{(2)}(T)$ takes the form

$$\begin{aligned} G_\mathrm{multi}^{(2)}(T)\propto & {} G_\mathrm{single}^{(2)}(T) \frac{\sin ^2[(2N+1)\pi T/ \tau ]}{(2N+1)\sin ^2[\pi T/\tau ]} \end{aligned}$$

(7.13)

$$\begin{aligned}\approx & {} G_\mathrm{single}^{(2)}(T)\sum _{n=-\infty }^{\infty }\delta (T-n\tau ), \end{aligned}$$

(7.14)

i.e., with the same double exponential decay but modulated by a comb with a period equal to the cavity round-trip time $\tau$. In our case the bandwidth of the output contains more than 200 cavity modes, and the fraction in (7.13) is well approximated by a comb of Dirac delta functions.

The comb period of $\tau = 1.99$ ns is comparable to the $t_\mathrm{bin} = 1$ ns resolution of our counting electronics, a digital time-of-flight counter (Fast ComTec P7888). This counter assigns arrival times to the signal and idler arrivals relative to an internal clock. We take the “window function” for the ith bin, i.e., the probability of an arrival at time T being assigned to that bin, to be

$$\begin{aligned} f^{(i)}(T)={\left\{ \begin{array}{ll} 1,&{} \text {if } T\in [i t_{bin}, (i+1) t_{bin}] \,, \\ 0, &{} \text {otherwise}\,. \end{array}\right. } \end{aligned}$$

(7.15)

Without loss of generality we assign the signal photon’s bin as $i=0$, and we include an unknown relative delay $T_0$ between signal and idler due to path length, electronics, cabling, and so forth. For a given signal arrival time $t_s$, the rate of idler arrivals in the ith bin is $\int dt_i \, f^{(i)}(t_i) G_\mathrm{multi}^{(2)}(t_i - t_s - T_0)$ ($t_i$ is the idler arrival time). This expression must be averaged over the possible $t_s$ within bin $i=0$. We also include the “accidental” coincidence rate $G_\mathrm{acc}^{(2)}= t_\mathrm{bin} R_1 R_2$, where $R_1, R_2$ are the singles detection rates at detectors 1, 2, respectively. The rate at which coincidence events are registered with i bins of separation is then

$$\begin{aligned} G_\mathrm{multi,det}^{(2)}(i)= & {} \frac{1}{t_\mathrm{bin}} \int dt_s \, f^{(0)}(t_s) \int dt_i \, f^{(i)}(t_i) G_\mathrm{multi}^{(2)}(t_i - t_s - T_0) + G_\mathrm{acc}^{(2)} \end{aligned}$$

(7.16)

$$\begin{aligned}= & {} \sum _{n=-\infty }^{\infty } G_\mathrm{single}^{(2)}(n \tau ) \frac{1}{t_\mathrm{bin}} \int _0^{t_\mathrm{bin}} dt_s \, f^{(i)}(t_s + T_0 + n \tau ) + G_\mathrm{acc}^{(2)}. \end{aligned}$$

(7.17)

We take $T_0$ is a free parameter in fitting to the data. Note that if we write $T_0=k t_\mathrm{bin}+\delta$ then the simultaneous events fall into kth bin and $\delta \in [-t_\mathrm{bin}/2,t_\mathrm{bin}/2]$ determines where the histogram has the maximum visibility due to the beating between the 1 ns sampling frequency of the detection system and the 1.99 ns comb period. APD time resolution is estimated to be 350 ps FWHM (manufacturer’s specification), i.e. significantly less than the TOF uncertainty, and is not included here.