Polarization entanglement and quantum beats of photon pairs from four-wave mixing in a cold Rb-87 ensemble

We characterize correlations in polarization and time of photon pairs generated from a cold cloud of Rb-87 atoms via a four-wave mixing process in a cascade level scheme. Quantum state tomography reveals entangled polarization states of high purity for each of the decay paths through two different intermediate hyperfine levels. When allowing both decay paths, we observe quantum beats in time-resolved correlation measurements.


Introduction
Time-correlated and entangled photon pairs have been an important resource for a wide range of quantum optics experiments, ranging from fundamental tests [1] to applications in quantum communication, cryptography, teleportation and computation [2].
The first sources of correlated photon pairs were based on a cascade decay in neutral atoms [3,4]. The cascade imposes a time correlation, and with an appropriate choice of the geometry and intermediate states it is possible to observe a strong non-classical correlation in the polarization of the photons. These sources are rarely used today in quantum optics experiments for their limited generation rates, and have been superseded by schemes based on three-wave mixing in non-linear optical crystals, or four-wave mixing (FWM) in optical fibers [5]. To obtain a narrow optical bandwidth, it is possible to use near-resonant transitions in atoms to provide the large third order nonlinear susceptibility for efficient FWM, leading to photon pairs with a central wavelength matching those of the transitions involved [6,7,8]. These sources are a hybrid between the traditional atomic cascade approaches, and those based on three-or four-wave mixing in solids in the sense that they deliver both a useful pair rate collected into single mode optical fibers, and exhibit interesting temporal correlations.
In this paper, we present the characterization of a source of time-correlated and polarization-entangled photon pairs based on four-wave mixing in a cold cloud of 87 Rb atoms. The involved atomic levels, selected by the choice of pump and target wavelengths, form a cascade decay scheme, providing an asymmetrical time correlation similar to the one from the cascade decay of single atoms.

Polarization entanglement
Entanglement between photons can be established in several degrees of freedom like time bins, polarization, and orbital angular momentum [9], with polarization entanglement having been extensively studied due to its robustness and the availability of very stable optical elements for manipulating and detecting the polarization of single photons [10].
Observation of photon polarization entanglement begin with early photon pair sources based on cascade decays in atomic beams [11], followed by spontaneous parametric down conversion in nonlinear optical crystals [12], cold [8] and hot [13] atomic vapors, and recently also in cascade emission from quantum dots [14]. In this paper, we characterize the correlation polarization properties of nearly Fourier-limited photon pairs generated from the cold cloud of 87 Rb [15,16].

Experimental setup
The experimental setup is similar to our earlier work [15,16], but uses a collinear beam geometry (see figure 1). An ensemble of cold 87 Rb atoms is prepared with a magnetooptical trap (MOT) of optical density OD ≈ 32 for light resonant to the 5S 1/2 , F = 2 → 5P 3/2 , F = 3 transition. The atoms are excited from 5S 1/2 , F = 2 to 5D 3/2 , F = 3 via a two-photon transition, with a two-photon detuning of ≈ 5 MHz. Pump beams of wavelength 795 nm and 762 nm overlap in a co-propagating geometry inside the cloud. The 795 nm pump is red detuned by 30 MHz from the intermediate level 5P 1/2 , F = 2 to reduce the incoherent scattering rate. From the 5D 3/2 , F = 3 excited level, atoms can decay through several paths. We select "signal" photons around 776 nm, and "idler" photons around 780 nm with interference filters of 3 nm FWHM bandwidth. Within this bandwidth, two decays can be observed (figure 1, top right): decay X through the hyperfine level 5P 3/2 , F = 3, and Y going through 5P 3/2 , F = 2.
Energy conservation and phase matching results in the generation of signal and idler photon pairs from both decay paths with a frequency difference of δ=266 MHz corresponding to the hyperfine splitting of the intermediate level. The generated photons are collected into single-mode fibers and detected by avalanche photodetectors (quantum efficiency ≈ 40%, jitter time ≈ 1 ns). In the experiment, we cycle between a 150 µs long cooling period with the MOT turned on, and a 10 µs long period for pair generation.

Polarization state tomography
We investigate the polarization state of the photon pairs for decay paths X and Y independently. We select light in the signal mode (see figure 1) using a 2 cm long solid fused silica etalon with a transmission bandwidth of 52 MHz (FWHM). The etalon is temperature-tuned to match to the resonance frequency of either the 5P 3/2 , F = 3 → 5D 3/2 , F = 3 or 5P 3/2 , F = 2 → 5D 3/2 , F = 3 transition; its temperature is stabilized to within 1 mK to minimize frequency drifts. We completely characterize the polarization state of photon pairs via quantum state tomography [17] by projective detections of individual photons in a combination of linear and circular polarizations. For this, we insert quarter and half wave plates (q, h) followed by beam splitter cubes (PBS) in the signal and idler modes. Figure 2 shows the real part of the reconstructed biphoton states ρ X and ρ Y corresponding to decay paths X and Y . The imaginary parts of all elements are smaller than ±0.09i. The strong off-diagonal elements (LR, RL) signify the polarization entanglement. From the reconstructed matrices, we can extract typical entanglement measures for the state; we evaluate the concurrence C [18], and the entanglement of formation E [19]. Furthermore, we can also determine the purity P = Tr[ρ 2 X,Y ] of the biphoton state for each decay path. The values of these indicators are given in table 1.
Despite the fact that the atomic ensemble is not prepared in a particular Zeeman sublevel, the polarization states for photon pairs from both decay paths show a remarkably high purity. This is compatible with theoretical models presented in [20,21], which we briefly summarize here.
In a cascade decay, polarization entanglement arises from indistinguishable decay paths, in our case provided by sufficiently degenerate Zeeman states of each hyperfine level. With the quantization axis along the beam propagation direction of all modes, we only drive transitions with ∆m F = ±1 with orthogonally circularly polarized pump beams. In parametric processes [22], the quantum state of the medium remains unchanged through the interaction [23]. Further, rotational symmetry of the atomic cloud in beam propagation direction implies angular momentum conservation. Along with the angular momentum selection rules, this limits the possible polarizations of the generated signal-idler photon pairs to |LR and |RL . Since the process is coherent and |LR and |RL are indistinguishable otherwise, the resulting state |ψ of a target mode photon pair can be written as The probability amplitudes a 0,1 and the phase φ 0 can be derived using a model based on the relative transition strength between different Zeeman sublevels [20,21] as where α S,I are the helicities of the signal and idler photons, and x α S ,α I is the product of relevant Clebsh-Gordan coefficients [24] that couple the individual |m F states of the different hyperfine levels involved in the four-wave mixing process, and where F g,b,e,d = 2, 2, 3, 3 corresponding to the respective total angular momentum F of the participating levels. From (2), we obtain the expected state |ψ X ≈ 0.55|LR − 0.83|RL for the decay path X. The reconstructed state ρ X matches the expected one with a fidelity of 94±1%. For the decay path Y , the model predicts state |ψ Y ≈ 0.92|LR − 0.39|RL , which agrees with ρ Y with a fidelity of 93±1%.

Transition strength of different decay paths
Both decay paths exhibit different decay time constants due to different transition strengths for the decays. The transition 5P 3/2 , F = 3 to 5S 1/2 , F = 2 is 2.8 times  Figure 3. Coincidence events as a function of the detection time difference ∆t between signal and idler photon detection for the decay path X and Y , selected with a temperature tuned etalon. Integration times were 7 and 14 minutes, respectively. stronger than from 5P 3/2 , F = 2 [25]. This results in a higher optical density (OD) for the F = 3 transition [26], and should lead to a faster decay via path X [27]. To experimentally investigate this, we perform separate time correlation measurements between the detection of signal and idler photons for each decay path. The histogram of coincidence events as a function of time delay ∆t between the detection of signal and idler photons sampled into 1 ns wide time bins is shown in figure 3. The solid line in both cases shows a fit to a heuristic model, inspired by the joint effect of a decay time corresponding to the two-photon transition [15,28], and the effect of a finite ring-down time of the filter etalon: For path X, we obtain a decay constant τ X =5.6±0.1 ns for an idler photon heralded by a signal photon. In the same way, with the etalon tuned to transmit the resonance frequency of the 5P 3/2 , F = 2→ 5D 3/2 , F = 3 transition for path Y , we find τ Y = 13.1±0.2 ns. Both decay constants τ X,Y are shorter than the spontaneous decay time τ sp = 27 ns of the 5P 3/2 level of a single Rubidium atom in free space due to the collective enhancement effects observed in an optically thick atomic ensemble . The rise time τ r = 3.1 ± 0.3 ns for decay path X and τ r = 3.3 ± 0.4 ns for Y is a consequence of the finite response time of the etalon. Both values are compatible with the value of 3.0 ± 0.1 ns obtained in an independent characterization of the etalon.

Quantum beats
Without the etalon (see figure 1), the decay paths X and Y cannot be distinguished by wideband photodetectors. Consequently, the energy difference between photons from the two paths leads to a modulation of the time correlation function between signal  and idler photodetection, as shown in Figure 4. This, and other similar phenomena, is known as quantum beats: it was predicted at an early stage of quantum physics [29], and first experimentally observed in pulsed optical excitation of atoms with two upper states decaying to the same ground state [30,31]. Quantum beats have also been observed in cascade decay systems of dilute atomic vapours [32], dense thermal atomic vapours [33], and for single ions [34]. In our case, the beat frequency δ/2π=266 MHz is equal to the energy difference between the hyperfine levels 5P 3/2 , F = 3 and F = 2. The measurements shown in Figure 4 are performed with polarization of pump 1 and 2 set to H and V , respectively, and polarizations of signal and idler modes are set to observe a large interference contrast.
To model the interference between the different decay paths, we take into account the different coherence time of the emitted photon pairs, characterized by the time constants τ X and τ Y , and the relative difference in the generation amplitude R, and phase φ. These last two terms can be calculated via (2). We express the probability amplitudes c X,Y for paths X and Y as function of the detection time difference ∆t between signal and idler photons, which interfere to a joint detection probability Using the measured coherence times τ X and τ Y and the values of R and φ calculated from the interaction strengths of the transitions, we fit the experimental data in figure 4 using (6) with only G 0 and an accidental count rate as free parameters, and find a good agreement with this relatively simple model. Different interaction strengths for the different polarizations in the participating levels allow control of the relative amplitude and phase of the possible decay paths. We can observe the dependence of the amplitude of the oscillation on the polarization settings and compare it with the expected values calculated from the interaction strengths of the transitions. In figure 5 we present three different cases, all fitted in a similar way as for figure 4. Of particular interest is the case where the beats are almost entirely suppressed [ figure 5(a)], an indication that most of the photon pairs observed are generated by the X decay. Figure 5(b) and (c) show the situation for polarization selections that lead to quantum beats with a high contrast, but opposite phases. This demonstrates that it is possible to select one frequency component only by an appropriate choice of polarizations, without using an etalon. However, it is not possible to select only photon pairs from the Y decay in a similar manner due to the relative weakness of the transitions involving the 5P 3/2 , F = 2 level.

Conclusion
In summary, we have characterized the polarization entangled state of photon pairs from a cold cloud of atoms by performing quantum state tomography, individually for two decay paths of the cascade. We find that the resulting polarization-entangled states for both decay paths are not maximally entangled, but reasonably close to it. This is compatible with a model combining the transition strengths between different participating intermediate states in the four-wave mixing process. We observe highcontrast quantum beats in a time correlation measurement between the generated photon pairs. The contrast and the initial phase of beats can be controlled with the choice of polarization of pumps and projective measurements on the generated photons.