Efficient generation of twin photons at telecom wavelengths with 10 GHz repetition-rate tunable comb laser

Efficient generation and detection of indistinguishable twin photons are at the core of quantum information and communications technology (Q-ICT). These photons are conventionally generated by spontaneous parametric down conversion (SPDC), which is a probabilistic process, and hence occurs at a limited rate, which restricts wider applications of Q-ICT. To increase the rate, one had to excite SPDC by higher pump power, while it inevitably produced more unwanted multi-photon components, harmfully degrading quantum interference visibility.Here we solve this problem by using recently developed 10 GHz repetition-rate-tunable comb laser, combined with a group-velocity-matched nonlinear crystal, and superconducting nanowire single photon detectors. They operate at telecom wavelengths more efficiently with less noises than conventional schemes, those typically operate at visible and near infrared wavelengths generated by a 76 MHz Ti Sapphire laser and detected by Si detectors. We could show high interference visibilities, which are free from the pump-power induced degradation. Our laser, nonlinear crystal, and detectors constitute a powerful tool box, which will pave a way to implementing quantum photonics circuits with variety of good and low-cost telecom components, and will eventually realize scalable Q-ICT in optical infra-structures.


I n t e n s i t y ( V )
T i m e ( n s ) FIG. 1: (a-f) The spectra, autocorrelation, and temporal sequences of the comb laser at 10 GHz and 2.5 GHz repetition rates. The full-width-at-half-maximum (FWHM) of the autocorrelation data are round 3.6 ps, corresponding to FWHM of 2.6 ps for the fundamental lasers.

Supplementary-II
In this part we investigate the relationship between signal to noise ratio (SNR) and the main/side peaks in Time of Arrival (ToA) data. The output state from the spontaneous parametric down conversion (SPDC) can be expressed as |ψ = 1 − λ 2 ∞ n=0 λ n |n, n = 1 − λ 2 (|0, 0 + λ |1, 1 + λ 2 |2, 2 + ...), where λ is the squeezing parameter. The probability for the n-pair photons per pulse is The average number of photon pairs per pulse is The 1-pair photons (|1, 1 components in Eq. (1)) are the "signal", e.g., for constituting a single-photon source. All the other photons are contamination, and should be viewed as the "noise". Therefore, the signal to noise ratio (SNR) can be naturally defined as the ratio of single pair emission rate over all the other n-pair emission rates, In Eq. (4), it can be noticed that the SNR is the inverse of the averaged photon pair number per pulse, p. For a low pump power, where λ 2 1, In the realistic situation, it may be difficult to estimate the SNR by using the experimental data. In contrast, here we provide a new method that enables us to simply evaluate the SNR from the main/side peaks in the experimental ToA data. In the following calculation, we assume the pump power is very small so that λ 2 1 is satisfied. We can omit all the higher order terms in Eq. (1), and only consider the 0-, 1-, and 2-pair emissions. In the ToA data, the probability of the main peak (P main ) is proportional to the 1,1 click probability from total emission.
where we only consider the 1-and 2-pair emission. η is the overall detection efficiency for the signal and idler photons. Assuming the dead time of the detector (∼40 ns in our experiment) is longer than the peak-to-peak interval of the pump laser (∼13 ns in our experiment), the probability of the side peak (P side ) is proportional to the 1, 0 click probability at the main peak position (P 1,0m , 1-click in start channel, and 0-click in stop channel), multiplied by the 1 click probability at the side peak position (P 1s , 1-click in stop channel).
Therefore, we can evaluate [(main peak − side peak)/side peak] as main peak−side peak side peak In Eq. (10), the approximations were achieved by assuming λ 2 1. By comparing Eq. (5) and Eq. (10), we can learn that Eq.(5) can be used to approximate the SNR. Therefore, it is reasonable to calculate the SNR in a logarithmic scale as The last approximation is valid if η is sufficiently low.
In the SNR test in this experiment, at 30 mW pump power, the overall detection efficiencies (η) were estimated as 0.31 for 76 MHz laser and 0.30 for 2.5 GHz laser; the average photon numbers per pulse (p) were estimated as 0.0079 for 76 MHz laser and 0.00021 for 2.5 GHz laser; λ 2 = p/(1 + p) were estimated as 0.0078 for 76 MHz laser and 0.00021 for 2.5 GHz laser. Therefore, the condition of λ 2 1 is fully satisfied in the experiment. In Eq.(11), with η = 0.31, .08 for λ 2 = 0.0078.

Supplementary-III
In this part, we numerically analyze the relationship between photon-pair generation rate (i.e., average photon pair per pulse) and HOM interference visibility.

The model
Here, we describe a numerical model of the HOM experiment. The model is described in Fig. 3(a) (without delay) and (b) (with delay) where η A,B represent transmittances of mode A and B (losses are effectively described by beam splitters), respectively, and η D1 and η D2 are the detector efficiencies. The mode mismatch between the signal and idler pulses is directly reflected to the HOM interference visibility. In general, the signal and idler pulses occupy slightly different modes in frequency, time, or spatial degrees of freedom. This is phenomenologically modeled by introducing two virtual beam splitters with transmittance η M (which directly corresponds to the mode matching efficiency) that split the signal and idler into three modes, overlapped part (A and B) and unoverlapped parts occupied by the signal (E) and the idler (F) (see Fig. 3(a)). The HOM interference visibility is defined as where CC min and CC mean are the coincidence count rates with zero-delay and large delay (i.e., with and without interference between the signal and idler), respectively. In the following we derive CC min and CC mean separately from our model. The initial state from the SPDC source is given by a two-mode squeezed-vacuum state where λ is the squeezing parameter, and λ 2 /(1 − λ 2 ) = p is the average photon pairs per pulse. LetV η A(B) be a beam splitting operator on mode A(B) with transmittance η which transforms the photon number states |n 1 |n 2 aŝ Applying the beam splittersV AB onto the two-mode squeezed vacuum (state at X in Fig. 3(a)), we obtain where l = k 5 + k 6 and we have used the relation Note thatV 1/2 should be applied to mode E and F , which will be discussed later. From Eq. (15) we find the joint probability of having l, k 3 + k 4 − l, n − k 1 , n − k 2 , k 1 − k 3 , k 2 − k 4 photons in mode A-F at X: The 50/50 beam splitting of mode E (F ) into E A and E B (F A and F B ) adds extra binomial distribution terms k1−k3 k7 k2−k4 k8 1 2 k1+k2−k3−k4 to Eq. (17). The joint probability distribution for the state at the detectors is thus given by The coincidence rate CC min is then obtained by the sum of the joint probability: The derivation of CC mean is rather simple since there is no interference at the 50/50 beam splitter due to the delay. This is illustrated in Fig. 3(b). Note that we do not need η M . The two-mode squeezed vacuum from the SPDC source has a joint photon distribution: The beam splitting operation simply spread this distribution in a binomial manner. For example, after the beam splitter η A , the joint distribution is given by Applying the η B and 50/50 beam splitters in a similar way, we have before the detectors. The coincidence count CC mean is then given by The HOM visibility in Eq. (12) is thus calculable from Eqs. (19) and (23).

Numerical result
The transmittances (efficiencies) of each components in the experiment are summarized in Table I (see Fig. 3 for the theoretical model and the corresponding experimental setup in Main text. In fact, the HOM visibility is extremely sensitive to the mode matching factor η M . It is however not easy to estimate the mode matching factor η M experimentally with enough accuracy. In Fig. 4, we plot the numerical results with various η M , and the experimental data with the 76 MHz laser. The experimental average photon-pair p is estimated from the experimental count rates. The experimental data fit the theoretical lines well within 0.9848 ≤ η M ≤ 0.9888. With the parameters in Table I, we also calculated the performance of our scheme at high photon-pair generation rate, as shown in Fig. 5 and Table II. From this simulation, we find several interesting relationship. (1). The visibility is directly determined by the average photon-pairs. (2). The slope of this line is very sensitive to the unbalanced loss in the delay arm and non-delay arm. (3). The Y-intercept of this line very sensitive to the mode matching efficiency.