1. Introduction

Controlled generation of single photons is one of the key elements in quantum optics and in quantum information science because of its highly non-classical photon statistics and its applicability as an ideal carrier of quantum bits for quantum cryptography[1] and quantum networking[2]. Over the past years, the controlled generation of single photons has been realized in many different systems, including single ions[3], single molecules[4], single atoms[5, 6, 7, 8], single quantum dots[9] and nitrogen-vacancy color centers in diamond[10].

Among them, the coupled system of a single atom and a single cavity mode has drawn much attention for its high collection efficiency and its well-defined spatial profile of the radiated field[6, 7, 8]. Moreover, this system can generates single photons with identical properties, e.g., frequency, coherence time, and temporal/spatial wave packet. It is thus suited for experiments requiring indistinguishable photons such as two-photon interferometry[11] and quantum computing with photons[12]. In addition, the coupled atom-cavity system is well described in the theoretical framework of cavity quantum electrodynamics(c-QED)[13]. Many interesting subjects have been studied with this system such as single-atom lasers[14, 15, 16], two-photon gateway[17], photon-photon entanglement[18], singular topology in an atom-cavity quantum composite[19, 20], etc.

So far, the controlled generation of single photons in the atom-cavity system has usually been demonstrated in the strong coupling regime, and thus its implementation has been quite challenging, requiring the state-of-the-art techniques. Moreover, its consequent long coherence time – the cavity decay rate had to be much smaller than the atom-cavity coupling constant –restricted the repetition rate of single-photon generation in the sub-MHz level.

In this paper, we demonstrate that the atom-cavity system in a moderate coupling regime is more efficient for rapid single-photon generation. We designed the system in the moderate coupling regime and then pumped the system with a nanosecond-pulse laser. Under this condition, we could generate single photons at a repetition rate of 10 MHz with photon-generation efficiency up to 17%. Our figure of merit, defined as the product of the single-photon generation efficiency and its repetition rate, is about 30 times larger than those of any previous experiments [8] in the atom-cavity system. We also investigated the efficiency by varying the pump power and found an optimum pump power for the most efficient single photon generation. Finally, the single-photon-emission nature of the system was confirmed by the strong suppression of the two-photon coincidence.

2. Experimental Apparatus

2.1. Optimum condition for rapid single-photon generation

For a two-level atom, positioned at an antinode of a cavity mode, the temporal width of the generated single photon wave packet is inversely proportional to (κ + γ)[21], where κ and γ are the cavity decay rate and the atomic spontaneous emission rate, respectively (all half widths), while the efficiency of generating a single photon for each pump pulse is given by[21]

With these considerations our cavity was designed to have (g₀, κ, γ)=2π×(16, 19, 3)MHz with two mirrors having intensity transmittance of 0.5 ppm and 200 ppm, respectively. The finesse, mode waist and cavity length were 25,000, 25 μm and 155 μm, respectively. The asymmetric transmittance of the cavity mirrors ensures that the cavity emission is uni-directional in practice, and thus the generated photons can be efficiently collected. In this configuration, the maximum value of P_s is expected to be larger than 0.7 from Eq. (1) with a repetition rate larger than 10 MHz.

2.2. Experimental setup and procedure

Our experimental setup is shown in Fig. 1(a). The cavity was placed 6 mm below a magneto-optical trap(MOT) of ⁸⁵Rb atoms. A probe laser, mode-matched to the cavity, was used to keep the cavity to be resonant with ²S_1/2, F = 3 ↔² P_3/2, F′ = 4 transition of ⁸⁵Rb. The transmitted probe laser and ultimately the generated single photons were spatially filtered with a 50-μm pinhole [not shown in Fig. 1(a)] and then detected by two photon-counting avalanche-photodiode(APD) modules in a Hanbury-Brown-Twiss-type setup[22]. The probe laser power was maintained at about 0.5 nW (corresponding to cavity mean photon number ∼ 0.05). The overall photon-detection efficiency was about 0.35, mainly due to the pinhole spatial filter (0.8), long-wave-pass filters used in front of APDs (0.9), and the quantum efficiency of APDs (0.5). A re-pump laser perpendicular to the cavity was used to recycle the atom from the unwanted F = 2 ground state.

 

Fig. 1 (a) Schematic diagram of the experimental setup. Only major parts are drawn. (b) Time trace of a typical cavity transmission upon single-atom transit events with a trigger level of 0.35 illustrated. (c) Timing sequence of the experiment. It takes 450 ms for one cycle of experiment. During a period of 25 μs single photons are generated at a repetition rate of 9.8 MHz.

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Pump pulses were obtained by modulating the intensity of a cw laser with a fiber-based electro-optic intensity modulator combined with a nano-second pulse generator. In this configuration, 1∼10 ns laser pulses were obtained with 0∼25 MHz repetition rates. The on/off intensity ratio of the resulting laser pulses was about 500:1. The pump frequency was detuned by 10 MHz from the atomic transition in order to suppress the atomic excitation due to the remaining intensity of the pump laser during the off phase. The pump beam was then focused into the gap of the cavity mirrors with a 42-μm waist and with circular polarization. All data reported in this paper were taken with 6 ns pulse width and 9.8 MHz repetition rate for the pump.

The experimental procedure can be divided into two parts. One is loading a single atom into the cavity mode. The other is pumping the atom with the pump pulse and measuring the cavity output photons. In the loading procedure, about 10^{4 85}Rb atoms are trapped in the MOT for 400 ms, and then released toward the cavity mode by turning off the MOT fields. After 25∼30 ms, individual atoms pass through the cavity mode with an average transit time of 150 μs, about 1.8 × 10⁴ times longer than the cavity decay time. When an atom passes through the cavity mode, the probe transmission on resonance is reduced, as shown in Fig. 1(b), due to the atom-cavity interaction [19]. Single-atom-transit events with the probe transmission falling below a certain trigger level were then considered having optimal atom-cavity coupling constants and used for single-photon generation. For example, for the data presented below, we set the trigger level at 35% of the maximum transmission and thus only the single-atom transit events with 0.4g₀ ≤ g ≤ g₀ were selected. This particular trigger level was chosen since it gave a modest number of optimal-coupled single-atom transit events for each release of atoms while not degrading the coupling constant much. If we lower the trigger level too much in order to increase the coupling constant, the number of optimal-coupled single-atom transit events becomes too small to be any practical.

Once an optimal-coupled single-atom-transit event is detected, a pump trigger signal is generated, which triggers turning off the probe laser and then turning on the pump-pulse train for a period of 25 μs. For each pump pulse, the atom is excited and generates a photon into the cavity mode. Because the cavity decay rate is larger than the coupling constant, the generated photon is then quickly emitted out the cavity. About 50 ms after the release of atoms, we turn on the MOT field and execute the entire cycle again. The overall timing sequence is summarized in Fig. 1(c). Typically, about 10⁴ cycles were performed during a period of 1.25 hours for a single data point in Fig. 2(b).

 

Fig. 2 (a) Measured arrival-time distribution (filled circles) with respect to the pump pulse (shaded area). The pump profile is scaled down for better comparison with the arrival-time distribution and its QTS fit (solid curve). Spontaneous emission decay (dashed curve) is also shown for comparison. (b) Single-photon-generation efficiency P_s as a function of the square root of the pump pulse energy.

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3. Results and Discussions

3.1. Arrival time distribution

Figure 2(a) shows the measured single-photon wave packets or the arrival time distribution of single photons with respect to the pump pulses. For this measurement, we removed BS in Fig. 1(a) and the pump trigger was used as a start signal of a digital time analyzer while the output from APD1 was used as a stop signal in a 2.5-ns resolution. Red dots indicate the experimental result while the blue solid curve represents the theoretical expectation obtained by quantum trajectory simulation(QTS) with an average coupling constant ḡ of 2π × 6.8 MHz (see below). The full width at half maximum(FWHM) of the single-photon wave packet was 28 ns, well matching the result by the QTS. For this measurement, the energy of the individual pump pulse was about 2.0 × 10⁻¹³ J, corresponding to a pulse area of 1.7π radian for the cyclic transition (²S_1/2, F = m_F = 3 ↔² P_3/2, F′ = m′_F = 4) of ⁸⁵Rb.

3.2. Pump power vs. efficiency

We have also measured the efficiency P_s of single-photon generation as a function of the square root of the pump pulse energy, which is proportional to the pulse area of the pump pulse, as shown in Fig. 2(b). Ideally it should exhibit sinusoidal Rabi oscillations[5] with a period of $5.2\times {10}^{-7}\sqrt{\text{J}}$ corresponding to a pulse area of 2π radian. However, the visibility of oscillation in the observed P_s is much reduced with an increased period as shown and the maximum of P_s is only 0.17, much smaller than the estimate by Eq. (1).

There are two main reasons for these discrepancies. One is the reduced average coupling constant ḡ of 2π × 6.8 MHz, much smaller than g₀, This value, determined by the trigger-level set for the optimal-coupled single-atom events, leads to P_c ≃ 0.39 in Eq. (1). The other is the nonideal pumping caused by the multi-sublevel structure of ⁸⁵Rb atom. Because the pump beam direction is orthogonal to the cavity axis, optical pumping to a cycling transition is not possible. Therefore, the pulse area should be averaged over all possible inter-magnetic-sublevel-transitions. Nonideal pumping is also caused by the spatial Gaussian-beam profile of the pump laser. Both result in an effective Rabi angle smaller than its apparent value.

We have performed QTS to verify our explanation and its result is shown as a solid curve in Fig. 2(b). In the simulation, which is applicable to both Figs. 2 (a) and (b), we first randomly choose an atomic position which makes the coupling constant compatible with the given trigger level. We then calculate the quantum trajectory or the time evolution of the system, excited by periodic 6-ns pump pulses at that position, with quantum jumps (cavity decay and atom decay) for 25μs. We considered 7–9 ground-excited levels of ⁸⁵Rb atom and two orthogonal cavity modes. After calculating the quantum trajectory for the 25-μs duration, we choose another atomic position and repeat the same procedure. For the plots in Fig. 2, we calculated the quantum trajectories for 1000 atomic positions and then evaluated the ensemble average of the cavity field. The simulation curve in Fig. 2(b) agrees well with the experimental data, confirming the validity of our explanation.

3.3. Coincidence measurement

Finally, we measured the coincidence rate of the generated photons in order to verify their single-photon nature. For this measurement, the output from APD2(APD1) was used as a start(stop) signal for the time analyzer. Time resolution was 10 ns. Strong suppression of coincidence at τ = 0 was observed as shown in Fig. 3, confirming that the system emits single photons mostly.

 

Fig. 3 Two-photon coincidence rates (open circles) as a function of delay time τ. Solid curve represents a fit based on a two-level theory formulated in Ref. [21]. Dashed curve is a fit for a two-photon emission peak.

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To evaluated the two-photon-emission probability, we fit the data with the two-time average of the intra-cavity field as formulated in Ref. [21]. The two-time average fit is based on the observed single-photon wave packet curve of Fig. 2(a) and denoted by a solid curve in a Fig. 3. It alone cannot fit the background near τ = 0. We have to add an additional two-photon-emission peak (dashed curve) in order to fit that region, obtaining a two-photon emission probability of 0.12±0.02. This value is consistent with our statistical estimate of the probability of having simultaneous transit of two atoms, which could not be discriminated in the present trigger scheme. The rate of multi-atom events can be suppressed by reducing initial number of atoms in MOT.

Our fit indicates that the background offset of about 0.1 in Fig. 3 is mostly due to the overlap between adjacent peaks. We have confirmed that this background is almost gone when the adjacent peaks are well separated. When this main contribution is subtracted, the remaining background is negligibly small, only 0.027±0.011, the most of which comes from pump pulse scattering (0.015) and continuous excitation by a small (1/500 of the peak intensity) dc component of the pump beam (0.01).

As a final remark, the averaged coupling constant ḡ of 2π × 6.8 MHz in fact corresponds a weak coupling regime, contrary to our design goal of realizing a moderate coupling regime. As a result, the observed single-photon generation efficiency was much smaller than our initial estimate. However, this is due to technical imperfection in the pumping process: one can recover a moderate coupling regime and thus increase the efficiency further by employing an intra-cavity dipole trap[23, 24] for better localizing single atoms at the anti-node of cavity mode. Nonetheless, our figure of merit, defined as the product of the single-photon generation efficiency and its repetition rate, is about 30 times larger than those of any previous experiments[8] in the atom-cavity system.

4. Conclusion

We have demonstrated the controlled generation of single photons based on the coupled atom-cavity system. By designing the cavity in the moderate coupling regime, we could generate single photons with 17% efficiency at 10 MHz repetition rate, the fastest result in the atom-cavity systems so far. We studied the characteristics of the system by measuring the arrival time distribution of the generated photons and the single-photon-generation efficiency. The strong suppression of two-photon coincidence was also observed, confirming the single-photon emission of the system.