Time uncertainty of a photon pair creation in a bulk periodically poled potassium titanyl phosphate pumped by a femtosecond laser

Periodically poled crystals, such as periodically poled potassium titanyl phosphate (PPKTP) or lithium niobate, recently became more attractive because of their high efficiency for producing a photon pair in both cw and pulse cases. The time uncertainty of photon pair creation in a crystal is crucial for many important applications. In this paper, we experimentally demonstrate that the time uncertainty of the photon pair creation in a PPKTP crystal is much larger than the coherence time of a femtosecond pump laser by using phase-sensitive two-photon interference using a Michelson interferometer, which means the femtosecond laser cannot be considered as a clock that announces the creation of the photon pairs. The experimental results can be well explained by an effective Franson-type interferometer, if the PPKTP is regarded as a narrow band filter at the same time. Our experiment also shows a way by which the Franson-typetwo-photon interference phenomenon can be observed even if the coherence length of the pump laser is less than the path-length difference between the two interfering beams.

laser used is much larger than the path-length difference [28]- [30]. Although this kind of source is not very useful for the aforementioned applications, it may be used to get a heralded high efficient single photon source [23].
The experimental proofs consist of two parts. Before giving them, we briefly give some parameters of the PPKTP crystal used in this experiment. It was bought from Raicol Crystal Company. The size of the crystal is: 1.05 mm (z) × 2.1 mm (y) × 5 mm (x), where z, y and x are mean height, width and length respectively. The grating period is of 3.25 µm. It is cut for type-I phase matching and is antireflection coated on both faces at 400 and 800 nm, and placed in a holder with a built-in Peltier device with a stability of 0.01 • C. The first proof given here is based on the measured spectrum of the second harmonic generation (SHG) of a strong 800 nm femtosecond mode-locked Ti: sapphire laser. The pulse width of the laser is less than 100 fs and the repetition rate is 80 MHz. We check the SHG power dependence on the temperature of the PPKTP crystal. The optimal temperature for our crystal and our laser is 58 • C. The spectrum of the SHG wave with 148 mW pump power is shown in figure 1. The measured full width at half maximum (FWHM) of the SHG wave by an optical spectrum analyser (ANDO AQ-6315A with a resolution of 0.05 nm) is about 0.093 nm. This intrinsic bandwidth determines the coherence time of the PPKTP crystal, which is about 2.5 ps, equals to about 750 µm coherence lengths assuming that the pulse shape is Gaussian. The main reason why PPKTP has a very narrow intrinsic bandwidth is that it has a large group velocity mismatch in the ultrashort pulsed regime case. 3 In [24], the FWHM of the SHG wave with a 2.12 mm long PPKTP crystal is approximately 0.18 nm, compared with the 3.2 nm wide SHG wave spectrum when pumping a 1 mm BBO crystal with the same pump laser.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Although we can infer indirectly from the spectrum of the SHG that the PPKTP acts effectively as a narrow band filter at the same time, prolonging the coherence time of the pump laser, this cannot be seen as a direct proof of the temporal behaviour of the photon pairs. In the following, we give a clearer proof. The basic idea can be explained by a two-photon interference experiment using a Franson-type interferometer. When we consider a Franson-type interference experiment done with a femtosecond laser, two different cases occur: when the path-length difference between the two interfering beams is larger than the coherence length of the single photons, but smaller than that of the pump laser, the coincidence rate, but not the single rate, depends on the fine path-length difference of the order of the pump laser wavelength. The maximal 100% visibility can be obtained if the two interfering terms 'short-short' and 'longlong' can be distinguished from the other two non-interfering terms 'short-long' and 'long-short' [26]. Otherwise, the maximal visibility is limited to 50%. This is a straightened condition since the coherence length of a femtosecond laser is very small demanding very small path-length difference. If the path-length difference is larger than the coherence length of the pump laser, the coincidences are also independent on the fine path-length difference. We hope to roughly estimate the actual uncertainty of the photon pair creation times by observing the two-photon interference and checking the relation between the path-length difference and the coherence length of the pump laser. Based on this idea, we perform a phase-sensitive two-photon interference experiment using a Michelson interferometer to directly check the temporal behaviour of the photon pairs generated in the PPKTP. An interesting solution given by Brendel et al [32] to solve the problem with a femtosecond laser should be mentioned. In their scheme, a new interferometer on the path of the pump laser is added in order to create a coherent superposition state of the pump photons. By this method, a 'time-bin' entangled state can be created. One advantage of this method is that the coherence of the pump laser is of no importance, the uncertainty of the pump photon's arrival time at the crystal is replaced by two sharp values. So the path-length difference can be arbitrary chosen provided it is the same than that of the pump interferometer. The two-photon interference can be still observed even if the path-length difference is larger than the coherence length of the pump laser, and the maximal 100% visibility can also be expected using a post-selection on the two interfering terms.
The experimental setup is shown in figure 2. The output of the mode-locked Ti : sapphire laser is doubled in a 1 mm type-I phase matched BBO crystal to generate violet pulses with a centre wavelength of 400 nm. The half waveplate and the polarizer are used to control the intensity of the 800 nm laser. Before the 400 nm laser pumps the PPKTP crystal, the violet pulses first pass through two 1.5 mm pinholes, and are attenuated by a continuous variable attenuator. The distance between the two pinholes is about 23 cm. The FWHM of the spectral bandwidth of the violet pulses measured by the optical spectrum analyser before the PPKTP is about 3.14 nm, and the centre wavelength is 400.51 nm. A lens with 100 cm focal length weakly focuses the violet pulses. The temperature of the PPKTP crystal is 40 • C. Note that in this case the optimal temperature is different in contrast to the SHG process, because their centre wavelengths are different. 4 After passing through the PPKTP crystal, two red filters (RG715) are firstly used to cut the remaining violet pump pulses. Then the SPDC photon pairs are sent to a small Michelson interferometer. One of the reflective mirrors is mounted on a motorized micrometer stage (K101-20MS, Suruga). The other one is mounted on a piezo-electric actuator (PZT) (AE0505D08, Thorlabs, displacement of 6.1 ± 1.5 µm@100V). Using this micrometer stage (PZT), we can coarsely (finely) adjust the path-length difference between the two interfering beams. One of the outputs of the interferometer is coupled to a 2 m long single-mode fibre (P1-4224-FC-2, numerical aperture (NA) = 0.12, manufactured by 3M) by an objective lens (NA = 0.15). The output of the fibre is coupled to a 50/50 fibre beamsplitter with the operating wavelength of 798 nm (manufactured by OFR) by using a fibre connector (FC). Each output of the fibre beamsplitter is connected to a singlephoton detector (PerkinElmer SPCM-AQR-14-FC, showing a quantum efficiency of ∼55% at 800 nm). The outputs of the detectors are sent to a coincidence circuit for coincidence counting, which mainly consists of a logic level adapter module (KN200), a discriminator (KN1300), a variable delay (KN330), a coincidence gate (KN470), (all above from Kaizu Works) and a counter (SR400, Standford Research Systems). The coincidence window is about 5 ns, which is much larger than the photon travel time difference associated to the interferometric path-length difference in the following experiments.
In our experiment, we first balance the two interfering beams by using the motorized micrometer stage, and measure the coincidence counts and one of single counts as a function of the voltage of the PZT. Then we change the path-length difference between the two interfering beams and repeat the same measurements. The experimental results are shown in figure 3. When the two beams are coarsely balanced, both coincidence and single counts show clear oscillations. It can be calculated from standard quantum mechanics that the variation of the coincidence rate corresponds to the mathematical formula R c.c = 3 + 4 cos + cos (2 ) in the ideal case, where, = 2π L/λ 800 , and L is the path-length difference and λ 800 refers to the 800 nm wavelength.   The special formula comes from three possible processes of photon pair propagation inside the Michelson interferometer: both photons propagate along the same path, or along different paths.When the two photons propagate along the different paths, there is a photon bunching effect [33]. The solid line is the fitted line with the above formula. The fitted period of the oscillations is 1050 ± 261 nm. The small mismatch between the experimental data and the fitted curve is due to mechanical instabilities or misalignment of the interferometer, etc. The oscillations of the single counts correspond to the formula R s = 1 + cos in the ideal case. This is also called white light interference [34]. The solid line is the fitted curve using the latter formula. The fitted period is 1000 ± 246 nm. All experimental data are given with errors. A more interesting thing appears when the coarse path-length difference is much larger than the coherence length of the pump laser (which is in our experiment, the 400 nm laser with 3.14 nm bandwidth. The calculated coherence time is about 75 fs, which corresponds to about 23 µm coherence length assuming that the pulse shape is Gaussian), the coincidence counts still show clear oscillations. At the same time, we monitor the single counts and find the single counts remain constant (for example, the single counts of the detector D1 are 38 ks −1 with accidentals in all three cases, not shown in the related figure). The oscillations of the coincidence counts correspond to the formula R c.c = 1 + 1/2 cos(2 ) in the ideal case. Obviously, there is no photon bunching effect in these cases with respect to the previously discussed balanced case. The experimental data for different path-length differences and the related fitted curves are shown in figure 3(b). The oscillation periods are 500 ± 123, 532 ± 131 and 532 ± 131 nm respectively for the coarse path differences of 80, 240 and 400 µm. The visibilities are 42 ± 4.1%, 33.2 ± 2.8% and 23.9 ± 3% respectively calculated from the fitted curves including errors. 5 Correcting for errors results in a few percentage changes. As we mentioned before, if the maximal 100% visibility is expected, the two interfering terms and two non-interfering terms should be distinguished. In our experiment, the path-length difference is so small that there is no way to differentiate them because of lower temporal resolution of the analyser. So, the maximal visibility is limited to 50%. The two-photon interference oscillation period is almost half of the single-photon interference. In all figures, the periods of all fits are larger than the theoretical values of 400 and 800 nm, we think mainly from the PZT used which has a large uncertainty, and maybe also from instability of the experimental system. Now, we consider how to explain the results. Even if our experiment features collinear outputs and a single Michelson interferometer in contrast to other experiments [28,30], this phenomenon can be effectively explained by a two-photon interference in a Franson interferometer [26]. The coherence length of the pump laser is much less than the path-length difference between the 'long-short' paths in our experiment compared to other Franson-type experiments [28]- [30], in which the coherence length of the pump laser is much larger than the path-length difference. The only reason that can explain our experimental phenomenon is that the PPKTP acts as a narrow band filter at the same time, prolonging the coherence length of the effective pump laser. Another difference between our experimental result and the results of the other experiments is that the visibility is gradually decreased with the increase of the path-length difference in our experiment. In contrast, the visibility remains constant in other experiments 5 The error in the figure can be calculated by the following equation where c m is the mean of coincidence counts, σ is standard deviation, c i , is the coincidence counts of fitting, c i is the actual coincidence counts and n is the number of the data. The visibility uncertainty can be calculated by equation V = 2c max c min (c max +c min ) 2 E, where c max and c min are maximum and minimum values of the coincidence counts.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT [28]- [30]. The main reason is that the effective pump laser provides pulses with a finite bandwidth. The coherence becomes bad near the edge of the pulse. From another point of view, this is not surprising because the larger difference between the two interfering beams becomes comparable with the coherence length of the effective pump laser. This experiment clearly demonstrates that the femtosecond pump laser does not act as a clock as in other experiments by which the photon pair creation times can be known with a satisfying resolution. We also try to experimentally roughly evaluate the time uncertainty based on the fringe visibility as the function of path-length difference between two interfering beams [35], and find that when the path-length difference is larger than about 1 mm, there is almost no coincidence oscillation, therefore the whole interference range is about 2 mm. If the Gaussian pump pulse shape is assumed, then the coherence length is about 0.88 mm. We also calculate the coherence length from the group velocity mismatch. It is about 1.08 mm assuming the Gaussian pulse shape of the pump laser with 100 fs FWHM pulse width. Both are almost comparable to the value calculated from the measured SHG bandwidth, and all are much larger than the coherence length of the pump laser. Besides although our experiment could be explained by an effective Franson-type interferometer [26], we observe the Franson-type two-photon interference even if the coherence length of the pump laser is less than the path-length difference between the two interfering beams, in contrast to many previously performed experiments in which the coherent time of the pump laser used is much larger than path difference [28]- [30]. Obviously, our experiment is different from the scheme presented in [33] in which an artificial coherence of the pump laser is introduced by adding an interferometer on the pump path, in contrast to using the intrinsic bandwidth of the crystal to prolong the coherence time of the pump laser in our experiment.
In summary, although PPKTP can be used to highly efficiently generate photon pairs, the photon pairs occur within a larger time uncertainty compared to other crystal, such as BBO, LBO, even if an ultrashort pulsed laser is used as a pump. This large uncertainty in time is mainly determined by the intrinsic bandwidth of the PPKTP crystal, as this intrinsic narrow band filter prolongs the coherence time of the pump laser. This means a femtosecond laser cannot be considered as the clock that announces the creation of the photon pairs. We experimentally demonstrate this effect by using a phase-sensitive two-photon interference using a Michelson interferometer. The experimental phenomenon can effectively explained by a Fransontype interferometer. This experiment shows that the two-photon interference in a Franson-type interferometer can also be observed even if the coherence length of the pump laser is much less than the path-length difference between the two interfering paths.