Temporal correlations of spectrally narrowband photon pair sources

We report on theoretical and experimental investigations of time-resolved cross- and auto-correlation measurements of spectrally narrowband photon pairs generated in sources based on parametric down conversion in resonant waveguide structures. We show that time-resolved measurements provide detailed and useful information about the spectral and modal structure of the bi-photon state. The shape of the cross-correlation function is asymmetric with exponential decays determined by the lifetimes of the signal and idler photons in the cavity. The time-resolved auto-correlation has Lorentzian shape. The measured g ( 2 ) ( 0 ) value convoluted with the detector windows and mode beating can be used to characterise the spectral longitudinal mode behaviour. The temporal width of the auto-correlation function is more than two times longer that the cross-correlation time. This reveals that the spectral bandwidth of the single-photon component is much broader than the spectral width of the two-photon component.

practical terms because to systematically investigate the temporal behaviour of single and multiple photon pairs generation, the practical implementation requires several NPP sources with high brightness, excellent stability, compact design and easy operation.
We investigated two different miniaturised photon pair sources based on doubly resonant waveguides exhibiting one or several longitudinal modes [22]. These investigations enabled us to resolve temporal structures of the cross-correlation between signal and idler and the auto-correlation of two-photon components, which are usually covered by the detector jitter when broadband photon pairs are investigated. Thus, the more information of cross and auto-correlation functions, such as temporal shape, correlation time and mode behaviour can be revealed. Figure 1 illustrates the basic scheme of our measurements. Figure 1 a ( ) shows the standard time-integrated correlation measurement. The spectrally broad photon pairs lead to a temporarily narrow correlation, which cannot be resolved due to jitter in the SPDs which limits the resolution of the SPD to an effective detector window T. However, if the photon pair is spectrally narrowband, the temporal structure of the correlation is broader than this detection window (as sketched in figure 1(b)); thus, temporal structures of the correlation functions can be investigated.
A widespread method to generate NPP is cavity-enhanced PDC. In such a nonlinear process, a single pump photon splits into two photons (signal and idler) inside a cavity, obeying energy conservation, phase matching and resonance conditions. For our experiments, we used miniaturised integrated photon pair sources that are based on doubly resonant waveguides exploiting type II PDC phase matching in a Ti-indiffused periodically poled LiNbO 3 (PPLN) waveguide. A schematic sketch of the integrated source is shown in figure 2. The endfaces of the waveguide are directly coated with high-reflective dielectric mirrors to form the cavity. In the dispersive waveguide, cavity resonances occur at distinct frequencies separated by the free spectral range (FSR) of the resonator. These resonances form Lorentzian frequency combs spaced with respective FSRs in the signal and idler wavelength range. Because of different FSRs in the resonant waveguide due to the different dispersions at the signal and idler wavelengths and polarisations, the resonances of signal and idler only overlap at certain frequencies, so called 'cluster'. As maximum enhancement is only obtained if both signal and idler are resonant simultaneously, PDC is generated only in such cluster regions of the spectrum. Including multiple photon pairs generation up to the second-order, the photon pair states generated within such a resonator can be expressed by )is the cavity-modified joint spectral function (JSF) determined by pump spectral distributed function, phase-matching function and field distributions inside the cavity. When signal and idler are both resonant simultaneously in a dispersive cavity, we can approximate the individual cavity resonance by a Lorentzian function i g describes the damping constants of cavity at signal and idler frequency, respectively. If there is only one pair of signal and idler cavity modes with narrowband range dw generated (or filtered) in the centre of phase matching, we obtain a complex JSF: The temporal signal-idler cross-correlation function g si 1,1 t ( ) ( ) is measured as the coincidence distribution of time differences between the signal and idler photons t t s i t = -. Using the inverse Fourier transform of the JSF, the signal-idler correlation can be simplified to g a t a t a t a t u e u e , 3 ) . The normalised second-order auto-correlation function is given by Similar to the derivation of the signal-idler correlation function, we consider a signal cavity mode with a Lorentzian spectral distribution and use the inverse Fourier transform of its intensity spectrum. Then, equation (4) can be approximately simplified to Obviously, this auto-correlation function is symmetric no matter which kind of PDC process it produced. This is different compared to cross-correlation function g si 1,1 t ( ) ( ) with its asymmetrical exponential decays out from cavity. In principle, this asymmetrical behaviour is from the cavity-modified JSF f , ), which has the chirped phase from signal and idler resonance. The shape of the auto-correlation function is of a type of a Cauchy-Lorentz distribution. The auto-correlation time of signal photon T au ss is approximately given by Together with the signal-idler correlation time c t , we have The absolute square interference term tells us that for the multi-mode case a beating under the exponentially decaying envelope occurs. In the same way, the multi-mode auto-correlation function is given as . In a realistic scenario, the detection window is in the range of 0.5 ns and F D is in the order of tens to hundreds of GHz. Thus, the time window of the detection system covers several beating periods, hence the measured g 0 Consequently, the number of cavity modes N can be directly estimated from the auto-correlation value. Please note that this conclusion is similar with pulsed time-integrated correlation in [23], although finite time-resolved correlation function is different with time-integrated one. Figure 3 provides the calculated coincidences between signal and idler photon generated from the double cavities. The simulated and convoluted auto-correlation results for signal photons from two resonant waveguides are theoretically predicted as well.
To study experimentally the correlation between photon pairs, two different resonant PDC waveguide sources given in table 1 were investigated. They are composed of a 12.3 mm (first source) and 14.5 mm (second source) long Ti-indiffused waveguide which is periodically poled with a poling period of 4.44 μm to provide type II phase matching for a PDC process pumped at 532 nm to generate nondegenerate photon pairs a around 890 nm (signal) and 1320 nm (idler). Both sources have a front mirror with a high reflectivity for signal and idler wavelengths. The rear mirror of the first source has also a high reflectivity (R 99% r1 » ), whereas the reflectivity of the rear mirror of the second source is only R 90% r2 » . The cavity finesse of the first source is 100 s1 ~and 80 i1 ~for the signal and idler wavelengths, respectively, while the second has a lower finesse of 22 s2 ~and 25 i2 ~. The experimental characterisation of such resonant waveguide device has already been discussed in detail in [22]. Here we only focus on revealing the correlation information from different samples which is not covered by the time resolution of detector system.
The PDC spectra of both sources consist of three clusters with a spectral separation of about 90 GHz and 75 GHz, individually. A volume Bragg grating is inserted into the signal beam to act as bandpass filter to select a single cluster. The modal structure within the selected cluster was investigated using a confocal scanning Fabry-Perot resonator with a free spectral range of 15 GHz (figures 4 (a) and (b)). These measurement revealed the modal structure of our sources: the first one operated only on a single longitudinal mode, whereas in the second one with the lower finesse three longitudinal modes with different strength could be observed. Please note that the periodical peaks are due to the scan of FP through three FSRs of FP etalon. With these measurements, the spectral bandwidth of the longitudinal modes could not be determined because the Fabry-Perot resonator has only a resolution of about 700 MHz, which is much larger than the expected bandwidth of the PDC modes.
Cross-and auto-correlation measurements were performed using these two sources. In figures 4 (c) and (d), signal-idler cross-correlation coincidence results are shown. The presence of the cavity implies that the shape of the coincidence curve should be determined by exponential functions. This is in good accordance with the theoretically prediction, as shown in equation , respectively, can be estimated. This is in reasonably good qualitative agreement with the theory and the measured spectra shown in figures 4(a) and (b). The signal auto-correlation time (T 12.5 au ss 1~ ns) is about 2.8 times broader than the signal-idler correlation time ( 4.8 c1 t~ns), which is matched well a theoretical analysis using equation (7). A Lorentzian fit to experimental curve also coincides with our theory. For the lower finesse sample, T au ss 2 (around 4 ns) is only two times longer than c2 t (∼2 ns). The reason why this measured value deviates more from the theoretically predicted than corresponding results for the high finesse source is that the correlation time is close to the time resolution of the detection system (∼0.5 ns). Thus, this measurement is partially but not completely time-resolved. This clearly reveals that interpreting such measurements requires a careful analysis of the timing resolution of the measurement system. The results can also be understood by using another but equivalent interpretation looking at the signal or idler multi-photon contributions. These arise from the second term in equation (1). Whereas the crosscorrelation coincidences mostly occur from single-photon pair generation events, i.e. the first term in equation (1), the auto-correlation is solely due to the multi-photon contributions. Thus, there are different parts  of the photon pair states which are probed by the cross-correlation and the auto-correlation measurements. The auto-correlation time T au is the correlation time between two (signal or idler) photon components, while the cross-correlation time c t is the correlation time between signal and idler photons. From the experimental investigations and also from our theoretical considerations, we showed that the auto-correlation time T au is larger than correlation time c t . Correspondingly, the two-photon components have a narrower frequency bandwidth (related to the product f f , , ) ( )) than then the one-pair component (related to f , R s i w w ( )) in the spectral domain. Intuitively, there is a higher probability to generate multiple photon pairs in the centre of phase-matching condition. Therefore, it is easily understandable that single-photon pairs and multiple photon pairs have different spectral and temporal properties.
In summary, we have theoretically and experimentally investigated correlation measurements and the relation to the temporal resolution of the detection system. For this purpose we investigated different narrowband photon pair sources based on PDC in doubly resonant PPLN waveguides. We showed that crossand auto-correlation functions have different temporal shapes: the cross-correlation has asymmetric exponential decays whereas the auto-correlation is of Lorentzian shape. The temporal width of the auto- . Correlation results from two narrowband sources. The first column corresponds the source with higher finesse and the second one with lower finesse. Panels (a) and (b): signal fine spectra at single-photon level in one cluster recorded at different temperatures. Panels (c) and (d): signal-idler coincidences of photon pairs as a function of arrival time difference between photon pair with exponential decay fits. Panels (e) and (f): measured signal-signal auto-correlation as function of arrival time difference between two signal photons with Lorentz fits. Please note that the measured temperatures are slightly different might due to the tiny shift of measurement conditions. correlation function is longer than the cross-correlation time. Thus, we could conclude that the two-photon pair part of the PDC state is spectrally narrower than the single pair contribution. Additionally, we analysed how the measured g 0 2 ( ) ( ) is influenced by the timing jitter of the detection system, e.g. we could relate the measured g 0 2 ( ) ( ) value with the number of longitudinal modes from our sources. From our investigations, we can conclude that a profound understanding of temporal and spectral correlations is of key importance. A detailed knowledge of the physics behind time-resolved measurements provides essential insight for analysing source properties. What we have learned from the time-resolved detection can generally be used to other narrowband sources as well, but the correlation shape is associated with the generation mechanism of narrowband sources. Thus, such time-resolved measurements are useful tool in particular for optimised engineering of all kinds of narrowband photon pair sources.