Nonclassical correlated photon pairs are very important in quantum information field. Usually, spontaneous parametric down-conversion in a nonlinear crystal is used in the generation of correlated photon pairs [1]. However there is a drawback that the photons generated by using this method have very broad line width, therefore are very inefficient in coupling with atoms. Narrow band correlated photon pairs can be generated by using spontaneous Raman scattering or four-wave mixing [2, 3, 4, 5, 6, 7, 8] in a cold atomic ensemble [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], or a hot atomic ensemble [21, 22, 23]. The early works [9, 10,11, 12, 21, 22] employed co-propagating write and read laser fields and on-axis Raman-scattered light was collected. These on-axis setups require that the filters have an extinction ratio of larger than 10¹¹ : 1 at a very small frequency difference, for example, about 6.8 GHz in ⁸⁷Rb atomic ensemble, which is difficult to be realized in technique. In contrast to these works, Balić et al. pioneered the efficient non-degenerate photon-pair production [13] in a cold atomic ensemble via off-axis four-wave mixing, using counter-propagating write and read fields. This off-axis setup spatially separates the collected photons and the lasers, and make the requirement of the filter very simple. After this work, almost all the works in the cold atomic ensemble used the counter-propagating and off-axis setup for the generating correlated photons [14, 15, 16, 17, 18, 19, 20], while the works in the hot atomic ensemble used the counter-propagating but on-axis setup [23].

In this paper, we demonstrate the generation of nonclassical correlated photon pairs in a hot atomic ensemble by using a counter-propagating and off-axis setup via non-degenerate fourwave mixing. Following the work of Balić et al. [13], two lasers, which are resonant with the D1 and D2 transitions of ⁸⁷Rb respectively, are applied to a hot rubidium cell to generate non-degenerate photon pairs. The time-resolved second order coincidence between the Stokes photons and the anti-Stokes photons, and the auto-coincidence of these photons are measured, from which the violation of Cauchy-Schwarz inequality by a factor of 1.69±0.14 is obtained.

The schematic setup of the experiment is shown in Fig. 1. The working medium is a rubidium cell with a length of 5 cm, which is kept at a temperature of about 35 °C, corresponding to an atomic density of about 2.5×10¹⁰ cm^-3. A strong and linear polarized coupling laser, which is resonant with the |2〉→|3〉 transition, continuously drives the atoms into the level |1〉. A weak pump laser, which is orthogonally polarized with the coupling laser and resonant with the |1〉→ |4〉 transition, counter propagates with the coupling laser. These two lasers meet in the

 

Fig. 1. Schematic setup of this experiment. Left, Energy levels of ⁸⁷Rb, and the frequency arrangement of the lasers used in this experiment, where |1〉 and |2〉 are the hyperfine sublevels of the ground state, and |3〉 and |4〉 are excited states. Right, experimental setup. A strong coupling laser and a weak pump laser with orthogonal linear polarization counter propagate through a rubidium cell. Paired Stokes and anti-Stokes photons are generated in phase-matched directions. These photons are collected by single-mode fibers and detected by single photon counting modules. F1 and F2 are filters, each consists of a polarizer, an optical pumped Rb cell and a ruled diffraction grating.

Download Full Size | PDF

rubidium cell. In the ideal case, the atomic ensemble are well prepared in state |1〉. The weak pump field induces a |4〉→|2〉 transition occasionally, and a Stokes photon is generated. The coupling laser retrieves the atomic ensemble back to the initial state, and an anti-Stokes photon is generated. The Stokes photon and the anti-Stokes photon are in pair and have nonclassical correlation.

In the experiment, the pump laser and the coupling laser are generated by two external cavity diode lasers. After filtered by using single-mode fibers, the 1/e ² diameters of them are about 2 mm in the rubidium cell. These two lasers are aligned to each other. The directions at an angle of about 4° to the lasers are chosen to collect the generated photons. Because the wavelengths of the pump laser and the coupling laser are 780 nm and 795 nm respectively, the best phase-matched directions between the Stokes photons and the anti-Stokes photons are not collinear exactly with this off-axis setup. Therefore the single-mode fibers used to collect the photons are not align to each other, instead, an additional laser, which is resonant with the |1〉→|3〉 transition, is used to simulate the anti-Stokes photons and find out the phasematched direction of the Stokes photons. When the additional laser is applied to the rubidium cell, a stimulated four-wave mixing process happens[24, 25, 26], and the generated four-wave mixing signal propagates in the phase-matched direction. The two single-mode fibers used for collecting the Stokes photons and the anti-Stokes photons are aligned to the four-wave mixing signal and the additional laser, respectively. This additional laser is cut off in the experiment, and the single-mode fibers collected the Stokes photons and the anti-Stokes photons propagating in phase-matched directions. The collected photons are sent to photon-counting modules (Perkin-Elmer SPCM-AQR-15), and the time resolved coincidence are recorded by using a time digitizer (FAST ComTec P7888-1E) with 2 ns bin width and totally 160 bins.

First, the coincidence between the Stokes photons and the anti-Stokes photons are recorded, in which the Stokes photons are used as the START signals and the anti-Stokes photons are used as the STOP signals of the time digitizer. The result is shown in Fig. 2(a), which is measured when the powers of the coupling field and the pump field are about 7.0 mW and 40 µW respectively. The counting rates of the Stokes photons and the anti-Stokes photons are about 14000 and 40000 per second respectively. In the ideal case the counting rate of the anti-Stokes photons should less or equal to the counting rate of the Stokes photons. The larger counting rate of the anti-Stokes photons here is caused by the quick moving of atoms. In a hot rubidium cell, the quick moving out of and into the coupling beam of the atoms leads to a large effective decay rate between the ground states. The atoms in the state |2〉 moving into the coupling field contribute to the uncorrelated anti-Stokes photons. This can be proved by the fact that the counting rate of the anti-Stokes photons is larger than 20000/s even when the pump field is absent. These uncorrelated photons causes the large background in the coincidence, which is estimated to be about 0.7 per second per bin. Figure 2(a) shows that the correlated time between the Stokes photons and the anti-Stokes photons is about 20 ns, and the maximum coincidence is obtained at the relative delay of about 12 ns. This delay is caused by the time needed to generate the anti-Stokes photons. The fact shows that the photons are generated in pairs, but they are not simultaneous, in contrast to the photon pairs generated in spontaneous parametric down-conversion in a nonlinear crystal [1]. The auto-coincidences of the photons are also measured by using fiber beam splitters. The results are shown in Fig. 2(b) and Fig. 2(c).

 

Fig. 2. The time-resolved coincidences between the fields versus time delay τ with the powers of the coupling and pump of 7.0 mW and 40 µW, respectively. All the data are normalized in time. (a) the coincident statistics between the Stokes photons and the anti-Stokes photons is shown. Data are collected over 1000 seconds. (b) and (c) show the self-coincidence of the Stokes photons and the anti-Stokes photons. Data are collected over 15000 and 3000 seconds respectively. cell, the quick moving out of and into the coupling

Download Full Size | PDF

The violation of the Cauchy-Schwarz inequality can be demonstrated by adopting the treatment of Ref. [9]. The Cauchy-Schwarz inequality is

where g̃_s,as(τ)≡n_s,as(τ)/ms,as, g̃s,s≡n_s,s/m_s,s and g̃_as,as≡n_as,as/m_as,as are the cross-correlation and auto-correlations of the photons respectively, n _s,as(τ) is the coincidence counts between the Stokes photons and the anti-Stokes photons obtained at the bin with the delay τ, n_s,s and n_as,as are the auto-coincidence counts of the photons at the bin τ=0, and m_{i, j}s, where (i, j)=(s,as), are the average counts of the background at one bin. From the data shown in Fig. 2, we obtain g̃_s,as(12 ns)=1.87±0.04, g̃_s,s=1.48±0.04 and g̃_as,as=1.40±0.04.

 

Fig. 3. The two photon correlation g̃_s,as(τ) and the net total coincidence counts N_s,as versus coupling powers with a fixed pump power of 40 µW.

Download Full Size | PDF

 

Fig. 4. The two photon correlation g̃_s,as(τ) and the net total coincidence counts N_s,as versus pump power with a fixed coupling power of about 7.0 mW.

Download Full Size | PDF

These results show that the auto-correlation of the photons is larger than 1, which should be 2 in the ideal case [9, 4], and are degraded by diverse sources of background counts. The g̃ _{i, j}s given above show the inequality of Eq. (1) for classical fields is strongly violated, namely [g̃_s,as(τ)²=3.50±0.11]≰≤ [g̃_s,s g̃_as,as=2.07±0.08], where all errors indicate the statistical uncertainties. This violation of the Cauchy-Schwarz inequality clearly demonstrates the non-classical character of correlation between the photons in a pair.

The coincidences between the photons versus the powers of the lasers are measured to find out the best power working point. The ${\tilde{g}}_{s,as}(\tau )$ and the N_s,as versus the coupling powers with a fixed pump power of 40µW are shown in Fig. 3, where N_s,as is the total coincidence counts at a relative delay between 2 ns and 32 ns, with the background counts subtracted, which is the net correlated counts between the photons. This figure shows that the g̃_s,as(τ) and N_s,as increase with the coupling power until the maximum output of our laser. Figure 4 gives the g̃_s,as(τ) and the N_s,as versus the pump powers with a fixed coupling power of about 7.0 mW, which shows that as the decrement of the pump power, the g̃_s,as(τ) increases while the N_s,as decreases. These experimental results agree with the intuitions: As the increment of the coupling power, more atoms are prepared to be in the state |1〉, and the efficiency of retrieving the atoms in the state |2〉 increases too, all these will increase the g̃_s,as(τ) and N_s,as; As the decrement of the pump power, the probability of inducing the |4〉→|2〉 transition decreases, and the probability of exciting multiple photons decreases too. Therefore N_s,as decreases while the g̃_s,as(τ) increases as the decrement of pump power. It should be noted that, the cause lead to rapid decrement of g̃_s,as(τ) at the two most left points in Fig. 4 is very simple: The exciting rate of the Stokes photons is too small compared with the background counts.

In the above experiments, the frequencies of the Stokes photons and the anti-Stokes photons are resonant with the D2 and D1 transitions of the ⁸⁷Rb respectively, and the rubidium cell used is a normal one. As we know, the rubidium cells filled with buffer gas or coated with paraffin can greatly reduce the decay between the ground states, and are extensively used in the hot atomic experiments. Therefore we also try these cells, hopefully to improve the experimental results. However, surprising things happen: there is no correlation between the Stokes photons and the anti-Stokes photons at all when these special cells are used. Because this phenomenon is not very close to the topic of this work, we will show the possible reason and detail analysis elsewhere.

In conclusion, we have experimentally demonstrated the generation of non-degenerate photon pairs in a hot atomic ensemble using off-axis setup, via spontaneous four-wave mixing. This paper shows a clear time-resolved two-photon correlation function between the generated photon pairs, and gives a violation of Cauchy-Schwarz inequality of 1.69±0.14. The correlated time of about 20 ns between the photon pairs demonstrates the photon pairs are narrow band photons.