3.1 Heralding effect

For quantum information applications, it is important to know when a single photon is likely to be detected. Photons emitted in cascade facilitate this due to the intrinsic delay between photons of each pair. This can be evaluated from their second order correlation function which can be calculated by solving the evolution equation of the atom, idler and signal electromagnetic field state,

Instead of solving the Maxwell-Bloch equations, we assume a simplified scheme where just the atomic states participating in the cascade transitions are involved [²¹]. This scheme considers a single atom, it does not describe the pumping procedure, and incorporates the electromagnetic environment effects on the atomic states through the decaying rates. The relevant atom- electromagnetic field state is expressed as

For the FWM configuration in Fig 1, the first term of this equation corresponds to the atomic second-excited state |α〉=5D3/2 with zero signal or idler photons, the second term is linked to the atomic first excited state |β〉=5P1/2 with one signal photon occupying one vacuum mode k and zero idler photons, and the last term stands for the ground state |γ〉=5S1/2 with the presence of one signal and one idler photon in the mode q. The effective Hamiltonian in the interaction picture under the rotating-wave approximation is

where ωif is the frequency corresponding to the energy separation between states for each |i〉→|j〉 decay and, the raising operators are σ+(1)=|α〉〈β| and σ+(2)=|β〉〈γ|. In Equation (12) the coupling constants,

are the analogous to Rabi frequencies for single photons; they are defined in terms of the dipole transition matrix element dij.

One can solve for the coefficients c(t) of the state |ψ(t)〉 by inserting Eqs. (11-12) into Eq. (10). Introducing the Γα and Γβ the decay rates of the states |α〉 and |β〉, the system of differential equations

is obtained. The decay rate Γβ=1/(2πτc) determines a coherence time for the heralded idler photons. The coefficients c(t) are constrained to the initial conditions cα(0)=1 and cβk(0)=cγkq(0)=0 in order to resemble a cascade decay.

From the first equation

is obtained trivially. The second coefficient

is readily derived by inserting this solution into the corresponding equation and integrating; plugging this solution of cβk(t) into the third differential equation yields a state |ψ(t)〉 which allows an approximate description of a transient regime studied in Ref. [²²]. There, fast decay times and oscillations that depend on the detunings and intensities of the pump fields occur. Here, we are interested in a steady state that results from the long time limit

the other two coefficients vanish, cα(∞)=cβk(∞)=0. Note that cγkq(∞) determines the conditional probability of finding the atom at the lowest level and the two photons derived from a previous decay. Thus, its squared value does not represent an atomic population. As expected, these equations yield Lorentzian profiles for the two cascade decays.

The probability amplitude given by Eq. (9) can now be evaluated for the asymptotic steady state

Assuming a continuous vacuum spectrum, the correlation amplitude Ψsi(2) can be written as an integral over the wave vectors k and q. After solving its angular part in spherical coordinates one arrives to

with Ψ0 defined as

where k0=(ωαγ-iΓα/2)/c-q0 and q0=(Γβ/2+ωβγ)/c; Δr1,2 correspond to the distances between the position of the atom, where each photon was emitted, and the position where each photon is detected. The integrand of Eq. (18)corresponds to probabilities given by the product of two Lorentzians; looking at the integrand structure it is observed that the extension of the lower limit of the integrals over the k and q to -∞ is a reliable approximation. Then, the residue theorem of Cauchy leads to

where

and

In Eq. (20),ts and ti incorporate directly the retardation effects for the signal and idler photons. Finally, the second-order correlation function can be written as

with amplitude

and Δt=ti-ts. This equation shows that ts is a trigger time for a sudden rise of the probability of detecting an idler photon to its maximum value followed by an exponential decay with a rate Γβ. The same result can be found with a treatment based on Green functions [²³]. In our case, Γβ=36 MHz and Γα=0.6 MHz.

This treatment involves a single atom, it does not account for collective effects. However, corrections to this scheme can be estimated from effective decay rates [²⁴] as described in 4.2.1.

4.1 Laser system

As shown in Fig. 3, the laser system is built in a modular fashion. Each gray box represents an optical arrangement that has been setup on a breadboard made of granite with the optical posts attached to it with instant glue. As reported in [²⁵], we found this to be a convenient solution because it allows building compact optical setups that remain stable for years. ii The boxes with solid borders represent setups of spectroscopy; boxes with dashed borders are optical arrangements that distribute light. The five frequencies required for inducing FWM in the MOT are generated by three commercial lasers (Moglabs: LDL and CEL002) and one home-built extended cavity diode laser (ECDL).

Figure 3 Block diagram of our experiment. The master laser is located at the left, top corner. It delivers frequency-stabilized light to seed our tapered amplifier (TA) for laser cooling, a small portion of its light is further tuned to prepare the p1 beam and the third output provides the first photon in the two step spectroscopy setup for the p2 laser that is described in Sec.4.1.1. In addition to the spectroscopy arrangements (gray boxes with solid borders), the light is fine tuned and prepared in polarization on the distribution boards (gray boxes with dashed borders).  

The master laser is located at the left, top corner of Fig. 3. It emits light locked to the crossover between the 5S 1/2, F = 2 → 5P 3/2, F’ = 3 and the 5S 1/2, F = 2 → 5P 3/2, F’ = 2 transitions to three different points around the setup. About one third of it (20 mW) seeds a tapered amplifier (TA) that generates up to 2 Watts of light that are further tuned with AOMs for laser cooling and for measuring the optical density OD. On the AOM board, the amplified light is offset by + 60 MHz with a single-pass OAM. This leaves it -151.6 MHz from the cooling transition. Thus we can optimize the laser cooling through its detuning ε, and implement the OD measurements described in Sec. 4.2.1 with an appropriate control of the detuning Δ of its probe beam with an AOM double passes of + 76 MHz in each case. To close the two-level system required for laser cooling the re-pump beam is combined with the cooling light at the polarizing beam splitter right before its output port as depicted in Fig. 3. The re-pump laser is home-made, its design was inspired on the Littrow version reported on [²⁵]. iii Clockwise, around the master laser in Fig. 3, its second output supplies p1 light to the AOM board. In order to avoid incoherent scattering from the first step of the FWM it is convenient to set a large detuning Δ (Fig. 1). Its optimal value was found around -60 MHz in our experiments. This is achieved by bypassing the offset AOM with the p1 beam, which is delivered to the FWM board after double passing its AOM. To complete the FWM frequencies we have the seed and the p2 lasers. The former is stabilized to the 5S 1/2, F = 2 → 5P 1/2, F’ = 2 transition with standard saturated absorption spectroscopy. As described in Sec.4.4, it is used for alignment purposes without further frequency tuning.

4.1.1 Two-photon spectroscopy

The p2 laser, depicted in 3, is a source of light at 776 nm intended to stimulate the 5P 3/2→ 5D 3/2 transition in the MOT. This is the second step of a double atomic transition thus, it has a relatively low probability to happen. A laser stabilization scheme, based on fluorescence lock-in detection, is suggested in Ref. [²⁶]. However, the saturated absorption method used in [²⁷] to probe the 5D 5/2 state in room-temperature atoms works well also for the 5D 3/2 state if their partial pressure is increased by heating up the spectroscopy cell. Implementing polarization spectroscopy [²⁸] yields similar results. However, temperature drifts due to the cell heating makes stabilization hard to establish in the long term.

Figure 4a) shows a schematics of our two-photon spectroscopy setup. About 3 mW of the master laser light is delivered to it through the optical fiber IF. The frequency of this beam is shifted by 2×70 MHz. This compensates the -60 MHz detuning on the p1 beam from resonance and enables a convenient scanning of δ with the p2 laser. The same AOM modulates the frequency of the master laser by 250 kHz

Figure 4 Schematics of the two-photon spectroscopy setup a) and error signal obtained with it b). The red beam depicts light at 780 nm for exciting the first step of the ladder; the orange beam represents the second photon, at 776 nm. IF indicates the input fiber of the 780 nm beam and OF indicates the output fiber of p2 going to the experiment. Optical elements are labeled following the same code as in Fig. 3.  

as required for saturated absorption spectroscopy. This beam is then sent through the cell as the pump with horizontal polarization.

The p2 light is generated by a cateye ECDL [²⁹] (MOGLABS, CEL002). Its frequency is scanned across all the hyperfine levels of the 5D 3/2 state. Approximately 3 mW from this laser enters the spectroscopy cell as the probe beam with linear polarization. Both beams counter-propagate and fully overlap to cancel Doppler broadening and to increase the interaction volume. The later is further incremented by expanding their 1/e2 width from 1 to 4 mm, see Fig. 4a).

Modulating the frequency of the pump beam also modulates its interaction with atoms inside the spectroscopy cell. This facilitates to derive the saturation signal assisted with a servo-loop circuit and a low-pass filter [³⁰], yielding a 20 mV peak-to-peak error signal in this case. This is a factor of 15 smaller than typical traces probing D lines with saturation absorption spectroscopy. Heating the spectroscopy cell enhances the obtained signal, making it suitable for locking a laser with standard electronics. Two power resistors of 10 Ω carrying 2 A are attached to the base of the cell. Thus, a stable temperature close to 70 ∘C is reached after one hour of heating. Figure 4b) shows a typical error trace that serves for locking the p2 laser. The blue lines cross the curve at the three points in which we can lock it. For the experiments reported here we used the F'=3 to F''=3 of the 5P 3/2,F'→ 5D 3/2,F'' transition. If one desires locking the laser to F''=1 or 2, it would be necessary to further increase the temperature of the cell. This laser produces 25 mW of stabilized light. Its detuning δ (Fig. 1) is controlled with the AOM in the range - 12 to + 12 MHz with respect to the 5P 3/2,F'=3→ 5D 3/2,F''=3 transition.

4.2 Vacuum system and MOT

As depicted in Fig. 5, the vacuum system has six elements only. Its simplicity grants full optical access and enables to reach ultra-high vacuum (UHV). The science cell (1) is an octagon made of quartz with windows anti-reflection coated for 780 nm light. The top and bottom windows have a 1.375 in diameter whereas the diameter of the lateral windows is 1 in. This cell is connected to a CF30 cube that holds all the vacuum system (2). Its vertical branch runs upwards towards a hybrid ion-sublimation pump [NEXTorr D 100-5, (3)]. Lastly, (Fig. 4) is a tee connecting a four-conductor feedthrough (5) and a UHV valve (6). The feedthrough connects independently two Rubidium dispensers and the valve is used for plugging the system to a pumping station for building vacuum from atmospheric pressure. With this system it is possible to reach pressures below 10 -10 Torr without baking. Ultra-high vacuum is not a necessary requirement for creating a MOT. However, in it one can transfer atoms to a dipole or a magnetic trap. With those traps it is possible to change the shape of the atom cloud. This would open the possibility of modulating the superradiance generated by collective effects throughout the FWM process [³¹].

Figure 5 Render of our vacuum system. The components are: (1) science cell, (2) CF30 cube, (3) vacuum pump (NEXTorr D 100-5), (4) tee, (5) feedthrough for Rubidium dispensers, (6) UHV valve and (7) MOT coils.  

Figure 5 also shows the MOT coils (7). They have 13 layers, with 9 windings each, of 13 AWG wire. The coil formers were 3D printed and designed to preserve full optical access to the science cell. Once arranged in an anti-Helmholtz configuration this coils generate a magnetic quadrupole field with axial gradients up to 30 Gcm -1, driving 8 A, with a stable temperature under 60. We found easier to align the MOT and FWM beams for experiments without compensation coils. The three MOT beams (not shown) are delivered from their distribution board, depicted in Fig.3. They form a 3D retro-reflection laser cooling arrangement. Each beam is about 20 mm in diameter and carries 13 mW of light that is -24 MHz detuned from the cooling transition. As explained below, less than 10 mW of evenly distributed re-pump light are enough to saturate the number of trapped atoms.

4.2.1. Optical density

Collective quantum-optical phenomena depend upon the amount of emitting atoms. For the light generated via FWM this is determined by the number of atoms simultaneously interacting with both pump beams. A direct way to find out this number is by measuring OD in the atomic cloud. Together with its shape, this parameter governs the influence that superradiance may have on the relaxation of atoms from their excited states and, consequently, on the bi-photons coherence properties [³¹].

The absorption of a near-resonance laser beam traveling through an atomic sample is described by the Beer’s Law,

dIdz=-ℏωγpn, (22)

where I is its intensity, z is the propagation distance through the atomic sample, n is the density of atoms, ω is the angular frequency of the transition being probed and γp is the total scattering rate given by [³²]

γp=s0Γ/21+s0+(2Δ/Γ)2. (23)

In Eq. (23), Γ is the excited state natural decay rate; Δ=ωl-ω is the laser detuning from resonance; s0=I/Isat and Isat are the on-resonance saturation parameter and intensity, respectively. For a low saturation parameter (I≪Isat) Eq. (23) simplifies to

γp=IΓ2IsatΓ2Γ2+4Δ2. (24)

Substituting Eq. (24) into Eq. (22) and integrating along z gives rise to the ratio of intensity absorption

II0=exp-ODΓ2Γ2+4Δ2, (25)

where I0 is the intensity of the probe beam before been scattered by atoms. The optical density OD is defined in terms of the on-resonance cross section σ0, the atom number N, and the light mode transverse area A as OD=Nσ0/A.

To measure OD we target the center of the atomic cloud with a low intensity probe beam, scan its frequency from -18 MHz to +25 MHz around the cooling transition and monitor its absorption with a photodiode. We acquire three series of data: (i) with the MOT and probe beam on, (ii) without the MOT and the probe on, and (iii) with the MOT on and probe switched off. Figure 6 displays a typical absorption data set (black dots) and curve (red) fitted to it with (25). There, Γ is set to 6.065 MHz [³³] and OD is left as the only free parameter. The experimental points displayed on Fig. 6 were obtained with a I0=0.5 mW cm -2, horizontally polarized probe beam. The asymmetry around resonance is originated by the Zeeman sub-levels that are involved in the probed transition. From the fit we can estimate the atom number. For this experiments the probe beam is around 0.008 cm 2 and σ0=2.907×10-9 cm 2 for vertically polarized light. Thus, in this case OD=20±0.2, which is equivalent to 5×107 absorbing atoms.

Figure 6 Normalized intensity I/I0 as a function of the probe beam detuning. In black is the experimental data and the red curve is a fit using Eq. (25).  

The optical density of a MOT mostly depends on: (i) the detuning ε of its cooling light and its power, (ii) partial pressure of Rubidium inside the vacuum chamber, (iii) the power of the re-pump beam and, (iv) the gradient of magnetic field given by its coils. Figure 7a) displays the OD of our MOT as a function of the gradient (black dots). To get these data the other parameters were fixed to values established during a previous optimization: ε= - 23 MHz, a pressure of 10 -9 torr and (iii) 8 mW of re-pump power evenly distributed among the MOT beams. The blue dots in Fig. 7a) were taken to characterize OD as a function of the re-pump power. All parameters were kept at the mentioned values except the magnetic gradient which was left saturated at 22 Gauss cm -1. In this case OD increases approximately linearly up to about 8 mW, which was the power left on for the next data set.

Figure 7 Dependency of OD on a) the power of the re-pump beam (blue) and on the field gradient produced by the coils (black) and; b) on the detuning ε (blue) and on the partial pressure of Rubidium inside the vacuum chamber (black).  

The blue dots in Fig. 7b) illustrate OD as a function of ε. Note that, as well as with the re-pump power, this dependency is approximately linear down to about - 23 MHz. Finally, the black dots in Fig. 7b) were taken with this detuning. They show the OD behaviour as a function of the partial pressure of Rubidium inside the vacuum chamber. The first experimental data depicts a measurement of OD=6 for a partial pressure of Rubidium below 0.1×10-11 torr; these experiments were taken with a 1.5×10-11 torr base pressure. Every reading above this value is a direct measure of the Rubidium atoms available to be trapped, which is controlled by the current driven through the dispenser. The experimental data display a clear non-linear behaviour that saturates until the MOT cannot longer increase the number of confined atoms.

From the data shown in Fig. 7 one can conclude that OD can be varied from 9 to 22 in our setup given that the MOT is fully loaded. Most of this range is conveniently covered by the detuning ε as the knob. The rest of it may be completed by using the dependency of OD on the re-pump power. All measurements depicted in Fig. 7 were carried out with a total laser cooling power of 60 mW evenly distributed among the trapping beams with one inch e-1 beam waist.

4.3 Control and DAQ

Many atomic physics laboratories around the world have developed their own control system according to their experimental needs. One can find online examples ranging from completely open source [³⁴] to semi-open source [³⁵] and closed systems [³⁶]. They all have two common fundamental characteristics: parallelism in the input/output signals and the ability to transport high speed 50 Ω signals. The former is achieved with a FPGA or a DSP core (or DAQ card) and the latter with a high-speed buffer.

In order to design our control system we thought of two generically different experimental sequences: (1) The most relevant parameter of the MOT to our current research interests is OD. We measure it as described in Sec.4.2. This method requires two analog input/output channels with no major speed demand. (2) To generate photon pairs we need a duty cycle alternating MOT loading periods and FWM pulses, both in the order of hundreds of microseconds. As explained in Sec. 4.4 this can be achieved with 3 or 4 digital channels.

To achieve parallelism the core must have the complete sequence information before outputting any signal. For this to be possible it requires to access a physical memory device (RAM or ROM), storing the control sequence previously sent by the control host, a PC in our case, see Fig. 8. The time needed to deliver information from the host to the core, usually referred to as communication time, must be considered if multiple sequences are to be exported. It is typically in the order of hundreds of milliseconds; normally the core takes a few nanoseconds to access the device memory. This time should also be taken into account for the duty cycle repetition accuracy. Based on the number of digital and analog channels, and speed demands to carry out experiments, we selected the LabJack T7 [³⁷] as the core of our control system.

Figure 8 Block diagram of our control system. The experimental sequences are programmed in a PC. This information is sent to the chosen core, a LabJack T7 card, which in turn sends the coordinating signals to the experimental devices and the data acquisition system, see Sec.4.3.2.  

4.4 Pumping and detection

Four-wave mixing happens inside the science chamber that is located at the center of the schematics on Fig. 9, where the arrangements for the final preparation of the pump light and for the collection of the photon pairs are also depicted.

The three FWM beams, p1, p2 and the seed, are delivered to their distribution board, Fig. 9a). There, the three of them are collimated to a diameter of 1.1 mm with aspheric lenses (C230TMD-B) and are combined on the narrow-band interference filter IF1 (LL01-780-25) that transmits the 776 nm beam whilst reflecting light at the other two wavelengths. For the experiments reported here all polarizations were set linear. The three beams are overlapped in an optical path targeting the center of the atomic cloud. With this propagation geometry the mode-matching condition - (5) - is satisfied if the signal and the idler photons are generated in the same direction.

Figure 9 Schematics of the experimental setup to measure the time statistics of the photon pairs and their correlations in polarization: a) is the optics arrangement to collimate and combine the pump beams toward the center of the MOT and; b) is the optics for photon collection. The interference filter IF2 just transmits light with the signal frequency whereas IF3 transmits just idler photons and reflects all the pump beams remainders. Both output ports A1 and A2 couple each polarization component of the generated photons to an APD.  

IF2 (LL01-780-25) allows just photons with the signal wavelength to pass through; IF3 (LL01-808-25) transmits the idler photon wavelength only. This array also filters the remainders of pump light in the photon flux that would appear as noise in our data. As depicted by Fig. 9b), both the signal and idler channels (A1 and A2) consist on a λ/2 and PBS arrangement enabling controllably spliting each photon flux in two inputs of fiber optics leading to the corresponding pair of avalanche detectors.

Light sources at 762 nm and 795 nm are required to align the signal and idler photons because both channels are filtered in frequency. The two outputs of the signal channel A1 are aligned by enhancing the FWM process with the seed beam [⁴¹]. This enables detecting 762 nm photons with a simple CMOS camera, which serves as the reference to couple these photons to the fibers of APD1 and APD2. Fibers leading the idler photons to APD3 and APD4 are aligned with the seed laser beam itself as the reference. Thus we typically reach a 70% fiber coupling efficiency using aspherical lenses (C230TMD-B). Our APDs have a nominal quantum efficiency close to 80 % around 780 nm.

The optical setup illustrated in Fig. 9b) has been designed for measuring the signal and iddler crossed correlations and the auto-correlations of each channel. Bi-photon correlations are measured by adjusting the λ/2 wave-plate for delivering all the light of each channel to a single APD. For auto-correlation experiments one can set each channel to split evenly its photon flux among the two detectors. Measurements of polarization correlations can be carried out as well since each APD is setup to measure just one projection of the generated photons.

A number of figures of merit have been designed to evaluate the generation of photon pairs. For this work we chose the coincidence rate since it is important for increasing the correlation histogram and the spectral brightness of the photon source (see Sec. 5). This parameter is optimized by a detuning Δ=-70 MHz and a power of 500 μW for p1 and, a two-photon detuning δ=-3 MHz and a 7 mW power for p2 in our setup; the FWM process is enhanced by setting the polarization of both pump beams linear and orthogonal to each other [⁴²]. Figure 10 depicts the experimental duty cycle maximising the bi-photon coincidence rate. It consists on a loading time of 500 μs followed by a 200 μs pulse of FWM, when p1 and the AND circuits are switched on. Data acquisition is carried out during this period, without cooling light. The re-pump laser is left on at all times to maximize the number of atoms taking part in the double excitation scheme. The p2 beam is also switched on throughout the full duty cycle. This might be limiting our photon generation by expelling a few atoms from the MOT. We also leave the MOT coils switched on during the integration time. They generate a field around one Gauss at the atom-light interaction volume, close to the center of the MOT. This means that atoms are subject to a Zeeman shift in the order of 1.4 MHz during experiments. Our pump lasers have a bandwidth smaller than 250 kHz. We did not notice any effect of this magnetic field over the coherence results presented in Sec. 5. However, we might observe some of its influence over quantum polarization correlations during studies in the near future.

Figure 10 Experimental duty cycle optimizing the coincidence rate of photon pairs in our setup. It consists on a 500 μs MOT loading time (green) followed by a 200 μs FWM pulse (red). During the FWM stage the cooling light is switched off at the same time of switching on the pump beams and the AND gate. The re-pump and p2 beams are always switched on as well as the MOT coils.  

5.1 Crossed- correlation function: coherence of the heralded photons

From Eq. (21) is reasonable to assume that the measured crossed-correlation function should fit

where G0 is the maximum coincidence rate and

is the number of accidental coincidences. It is given by the coincidence rate expected from two uncorrelated random sources; R1 and R2 are the individual count rates measured at each APD; Δtbin is the temporal bin width of the cross-correlation histogram and T the total integration time.

We fit the experimental data to the degree of second-order coherence, which is obtained by normalizing Eq. (26)

To make a more realistic model, one needs to introduce the noise probability distribution f(Δt) of each APD. Assuming that Eq. (28) and f(Δt) are two independent stochastic distributions, this measurement is expected to be a convolution of both of them. It is in general non-trivial to determine the function f(Δt) of a detector. However, we found reasonable results by assuming that it is a Gaussian,

with a width τD that is related to the response time of the detectors. Carrying out the convolution results in the fitting function

Coincidence histograms are obtained as explained in Sec. 4. Since FWM is pumped with two beams with orthogonal, linear polarization the signal beam is measured on transmission and the idler beam is measured on reflection from their corresponding PBS (Fig. 9). In this way, the accidental coincidences are attenuated by approximately one-third. For the experimental results here presented the parameters of the pump beams are those given in Sec. 4.4 except that, in this case, the two-photon detuning δ=+6 MHz. The optical density is set to approximately 20, T=17 s and Δtbins=1.4 ns.

Figure 11 shows the experimental data (black dots) for typical cross-correlation experiments in our setup. The dotted red curve is a fitting of Eq. (31) to these results leaving G0, τD and τc as free parameters. One can immediately observe that the rise to its maximum value is smoother than the Heaviside function expected for ideal detection. The time dependency of these measurements is attenuated by the Gaussian noise distribution with bandwidth τD. The fitted curve is optimum for τD∼0.61 ns, which is in the order of our detectors nominal response time (400 ps). From the exponential decay of the fitted curve we measured a heralded coherence time of τc=4.41±0.09 ns for the idler photons. In a purely atomic cascade decay this would be expected to be equal to the lifetime of the intermediate state 5P 1/2 divided by 2π, i.e. 4.40 ns. Note that a reduction with respect to this value would have been the result of collective effects in atomic ensembles [⁴³].

Figure 11 Cross-correlation histogram (black dots) as a function of the time delay between the signal and the idler photons. The right axis scales for the raw data whereas the left displays values for the normalized histogram. The red curve is a fit with (31). The best fitting parameters are G0=1654±48, τD=0.61±0.04 ns and τc=4.4±0.1 ns.  

We use two separate estimates of the accidental coincidences Gacc: either by their direct calculation, with Eq. (27), or by fitting the experimental data with Eq. (26). Calculation from the measured rates, R1=16,295 s -1 and R2=15,860 s -1, and the time settings for these experiments yields Gacc∼5.6 s -1. In order to obtain a reliable estimate from Eq. (27), we consider the longer coincidence time interval of Δt=350 ns, resulting in Gacc=5.8±1.1, compatible with the statistically inferred value.

The measured coherence time of the heralded idler photons corresponds to a bandwidth of 36.2±0.8 MHz, which is a factor of ten smaller than the value obtained in experiments without cooling the atoms [⁴¹]. It in principle can be reduced to 20 MHz by optimizing the pumping parameters [¹⁵], and below 8 MHz by removing the time-dependent phases with a Fabry-Perot cavity and a electro-optic modulator [¹⁶]. The 5S  1/2→ 5P  1/2 transition of  87Rb has a natural linewidth of 5.75 MHz [³³]. Thus heralded photons sourced in this way are suitable for interacting with atoms.

The spectral brightness of a quantum light source is likely to become a determining parameter to achieve control over the interaction of its photons with atoms. It is given by B=2πτcrc, where 2πτc is the inverse of its single-photon bandwidth, 2π (4.4 ± 0.1 ns) in our case. The coincidence rate rc is obtained from the un-normalized histogram GSI2 plotted in Fig. 11. It is given by the number of coincidences detected during an integration window that we chose to be 40 ns. These yield a coincidence rate up to 104 s -1 under experimental conditions similar to the reported above but with δ=0.5 MHz. Thus we report here a spectral brightness around 280 coincidences s -1 MHz. This value is consistent with observations in similar sources [⁴²] and two orders of magnitude larger than the value reported for hot atomic ensembles [⁴¹].

5.2 Auto-correlation functions: coherence properties of each channel.

Since the FWM process is induced on many atoms, each decay channel is a source of photons with chaotic origin. One can find their coherence properties through the treatment shown in Sec. 3.1, or by recalling that the first order correlation function of each atomic relaxation is given by the Fourier transform of its Lorentzian spectrum Si(ω)=Γi/2[(ω-ωi)2-(Γi/2)2] [⁴⁴];

where G0≃Gacc and τc is the coherence time of the chaotic photons, signal or idler. Due to their nature, in both cases, the second order of coherence is related to normalized first order correlation

by

indicating an expected sharp time symmetry due to the absolute value in the exponent with an exponential decay with rate Γ=1/(2πτc) towards negative and positive times.

To fit this data, the detectors noise was also assumed Gaussian. Its convolution with Eq. (34) is

These experiments were performed by implementing a Hanbury Brown-Twiss interferometer at each output channel, A1 and A2, depicted on 9b). The photon flux of each one was evenly split at the PBS. Both reflecting and transmitting outputs were coupled into single-mode fibers and sent to their own APD.

The black dots in Fig. 12 display typical coincidence histograms for the signal (a) and idler (b) photons; the dashed, red curves are fitted to these data with Eq. (35). All the experimental parameters were set the same as for the cross-correlation plots except that the two photon detunings were δ=4 MHz and δ=7 MHz for (a) and (b) respectively and, the integration time was 20 minutes in both cases.

Figure 12 The black dots are the normalized autocorrelation histograms for a) signal and b) the idler photons. As expected, the auto-correlation gii(2)(Δt) approaches 1 for long integration times; an interval from -100 ns to 100 ns is enough to see this in our data. Both red curves represent the corresponding fits according to (35). In a) the best fitting parameters are g0=0.22±0.03, τD=9.77±2.12 ns and τc=18.92±2.65 ns; in b) g0=0.12±0.03, τD=0.17±0.04 ns and τc=12.80±0.90 ns.  

It is illustrative to point out the differences between these observations and the behaviour predicted for ideal detection. According to Eq. (34) the auto-correlation should present a maximum equal to 2 at the time zero. Neither the signal nor the idler maximum reach this value because the APDs quantum efficiencies are lower than 100%. Furthermore, it was found that gss(2)(0)>gii(2)(0), which is consistent with the greater quantum efficiency of our detectors at 762 nm than at 795 nm. Note that the experimental maxima have been smoothed by the noise bandwidth τD, which was left as a free parameter during the fittings. Their values, displayed on Table I, are different because the noise of the detectors is a function of the optical wavelength. In this case τD762>τD795 translates to a softer maximum for the signal auto-correlation.

5.3 Non-classicality

A vector analysis of the classical electromagnetic field, based on the Cauchy-Schwartz inequality, predicts that [⁴⁵]

where gsi(2)(Δtmax)2, gss(2)(0) and gii(2)(0) are the maximum values of the second order of coherence measured from cross-correlations and autocorrelations. Their values, extracted from curves fitted to the experimental data shown in Figs. 11 and 12, are: gsi(2)(Δtmax)=1270±48, gss(2)(Δtmax)=1.77±0.15 and gii(2)(Δtmax)=1.63±0.14. By plugging them into Eq. (36) one finds that R=(5.62±1.38)×105, which is evidence of a strong non-classical behavior in the time statistics of the photon pairs generated by our source.