Engineering integrated pure narrow-band photon sources

Engineering and controlling well defined states of light for quantum information applications is of increasing importance as the complexity of quantum systems grows. For example, in quantum networks high multi-photon interference visibility requires properly devised single mode sources. In this paper we propose a spontaneous parametric down conversion source based on an integrated cavity-waveguide, where single narrow-band, possibly distinct, spectral modes for the idler and the signal fields can be generated. This mode selection takes advantage of the clustering effect, due to the intrinsic dispersion of the nonlinear material. In combination with a CW laser and fast detection, our approach provides a means to engineer a source that can efficiently generate pure photons, without filtering, that is compatible with long distance quantum communication. Furthermore, it is extremely flexible and could easily be adapted to a wide variety of wavelengths and applications.


Introduction
Engineering pure quantum states of light is becoming increasingly important for a multitude of applications associated with quantum information technologies, such as the implementation of quantum repeaters [1][2][3], device-independent quantum key distribution (QKD) [4], entangled states for quantum metrology [5,6] or for optical quantum computation [7,8]. This is especially relevant for the scalability of resources in complex quantum systems: for instance, for the realization of a quantum communication network, for which multiple independent systems need to be interfaced with high interference visibility, engineering suitable photon pair sources is crucial.
In recent years, we have seen a concerted effort to develop pure photon sources via spontaneous parametric downconversion (SPDC) or four-wave mixing [9][10][11]. However, these approaches usually generate relatively broadband photons. Long-distance quantum communication [12,13], on the other hand, requires narrow-band photons, either for coupling to quantum memories [3] or simply for robustness against thermal dilation of optical fibers [14]. Moreover, we require these sources to be efficient, by which we mean that both the spatial and the spectral modes have to be efficiently selected. The former requires optimal coupling into a single-mode optical fiber, whereas the latter implies spectral purity, which is generally achieved via filtering.
The problem with filtering is that it is inherently lossy: a Gaussian-shaped filter, even if perfectly transparent at its central wavelength, will always introduce some loss in the wings, resulting in the transmission of only 1/ √ 2 ≈ 70% of the light inside its own spectrum. Emerging applications such as device-independent QKD [4], as well as more fundamental challenges such as all-optical loophole-free Bell tests [15], cannot be realized even with this trivial loss mechanism. Sources compatible with long-distance quantum communication have so far followed two paths: SPDC sources pumped in a continuous-wave (CW) mode and using strong filtering [16,17], or optical parametric oscillators (OPO) [18][19][20][21]. The latter provide quite a good solution, since they naturally produce narrow-band photons and avoid the Gaussian loss, but are generally complex and require complicated and time-consuming (off-line) stabilization routines.
In this paper, we introduce an integrated OPO approach, which exploits a spectral clustering effect, related to the intrinsic dispersion of the nonlinear crystal and observed in [22], to suppress undesired resonant modes and obviate the need for filtering. Avoiding the use of external filters allows one to increase the rates for long-distance quantum communication experiments. This integrated OPO requires only a simple (and continuous) temperature stabilization. One can obtain a simple, flexible and efficient source of narrow-band heralded pure single photons by detecting one of the down converted photons with a sufficiently fast detector [23,24]. Such a heralded pure single-photon source can be built up even with a CW pump.
First, we explain the concept of an integrated OPO photon pair source, what the clustering effect is and how it arises from the dispersion of the nonlinear crystal. We then test our theoretical model against some preliminary experimental results and provide easily adaptable guidelines on how to design an integrated cavity-waveguide for a wide variety of wavelengths and applications. An example of such a source is also given. Finally, we discuss the potential advantages of our approach in the context of engineering complex quantum communication architectures.

An integrated cavity-waveguide spontaneous parametric downconversion source
For practical applications in quantum information and communication, having an integrated device [25][26][27] offers a number of advantages. For example, stabilizing a high-finesse doubly resonant cavity can prove challenging. In a monolithic design, such as the one provided by an integrated cavity-waveguide, only the temperature of a single component needs to be stabilized. It is also by changing the temperature that the device is phase-matched for a variety of wavelengths. Another advantage of integration is that very short cavity lengths can be achieved, allowing one to increase the free spectral range (FSR), which will be important for what follows.
The generic structure of the proposed device is shown in figure 1 and consists of a periodically poled lithium niobate (PPLN) crystal, in which a waveguide has been created [28][29][30][31]. Other materials, such as periodically poled potassium titanyl phosphate (PPKTP) [32], can also be adopted. In an optimal device the entrance facet would be coated for high reflection at the signal and idler wavelengths, while the exit facet would have a reflectivity sufficient for achieving the desired finesse, but low enough to be the dominant loss mechanism.
We can treat this device as a Fabry-Perot cavity, whose finesse is completely determined by the losses and reflectivities: where ρ is related to the power in the cavity after a round trip and can be expressed as ρ = R 1 R 2 10 −2αL/10 , R 1 and R 2 are the reflectivities of the mirrors, α is the absorption coefficient expressed in dB cm −1 , L is the length of the crystal in cm and 2 accounts for the fact that the optical path per round trip is 2L. In practice, the coating reflectivities R 1 and R 2 can be estimated by coating an optical blank with the same coating as the integrated device. Once the finesse of the cavity is measured, one can solve equation (1) for the waveguide loss coefficient α. In our initial test device [22], α was found to be 0.06 dB cm −1 . Having low internal losses is important, as these will ultimately limit the achievable finesse for a given cavity length.

Clustering in doubly resonant spontaneous parametric downconversion
The spectrum of photons generated by SPDC is usually determined by the phase-matching conditions between the pump, signal and idler fields in a nonlinear crystal [32], resulting in a typical joint frequency spectrum like that shown in figure 2(a). When the nonlinear crystal is placed inside a cavity [18][19][20][21], an extra set of constraints must be satisfied: photons can only be emitted in modes which are supported by the resonator. For a Fabry-Perot cavity the resonant modes are given by the Airy function. This second constraint on the joint frequency spectrum is represented in figure 2(b). When pumping with a narrow CW laser, energy conservation ω i ω s Figure 3. In a dispersive medium the optical length of a cavity varies with wavelength. SPDC occurs in such a cavity only when the double resonance condition is satisfied, i.e. when both signal and idler frequencies (satisfying ω p = ω s + ω i ) are resonant.
(ω s + ω i = ω p ) will further restrict all photons to be emitted on the diagonal of the joint spectral function ( figure 2(b)). If the refractive index is equal for both signal and idler photons, then the optical path length of the cavity, and the Airy functions, will be the same for both photons such that the distance between the peaks will be defined by the FSR of the cavity. However, if the signal and idler photons have different dispersion characteristics, for example when they are non-degenerate in frequency or have distinct polarizations, the FSR at the signal and idler wavelengths will be different. In this situation, simultaneous resonance for both the signal and the idler arises only in regions of the spectrum where the peaks of the Fabry-Perot for the signal and the idler align (figures 2(c) and 3). These regions are called 'clusters' [33,34], as they usually contain a number of emission peaks. However, as we shall see below, it is possible to design the finesse of the cavity, so as to produce clusters containing only a single peak.

Calculated and observed clustering spectra
The state of the photon pairs emitted by a doubly resonant OPO is given by [35] where S(ω s , ω i ) is the joint spectral intensity, with the assumption of an infinite number of round trips of the photons inside the cavity. The quantity f (ω s , ω i ) represents the joint spectral amplitude in the absence of the cavity and, in the presence of a monochromatic pump, which is the case when the crystal is pumped in a CW mode, it reduces to the phase-matching function. A j (ω j ) are the Airy functions for the signal  (2) with the spectral resolution of the monochromator (Gaussian, 4 nm FWHM) with detector noise added (dotted green line). The residual background is due to the overlap of the tails of the Gaussian shape of the monochromator with the peaks. The detector noise is different in the two figures as the signal-to-noise ratio varied between the two measurements. See the text for details. and the idler photons, which, as functions of the wavelength, are expressed as where φ j = 2π Ln eff (λ j )/λ j . Here n eff is the wavelength-dependent effective refractive index inside the waveguide, taking into account the material and the waveguide dispersion.
To demonstrate that the clustering effect arises from the dispersion, we use the previous equations to reproduce the measured frequency spectra previously reported [22]. We consider a PPLN waveguide with length L = 3.6 cm, poling period = 16.6 µm, at a temperature T = 128.6 • C with mirror coating reflectivities of R 1 = R 2 = 0.85. For a pump at 780 nm, the phase matching is satisfied for the generation of photon pairs degenerate around 1560 nm. In this first instance, as we are interested only in the clustering effect; we can neglect the phase-matching condition, which is much larger than the effects we are studying. To model the emission spectrum we multiply the Airy function transmissions for the idler and the signal photons, whose frequencies are set by the energy conservation. The wavelength dependency of the extraordinary refractive index for the PPLN is given by the Sellmeier equation reported in [36], which have been adapted to take into account the waveguide dispersion. Figure 4 shows the SPDC spectra emitted by this waveguide resonator for two different temperatures, as measured in [22] (red dots) and calculated (black solid line). The measurements were carried out by sending the SPDC photons through a monochromator and by detecting them with an InGaAs APD. The low resolution of the monochromator (4 nm full width at half maximum (FWHM)) averages over the fine structure within the clusters. As one moves away from the degeneracy point (central cluster) the clusters become narrow with respect to the monochromator's resolution, which results in an apparent reduction in their height. This effect 7 is well reproduced experimentally. The detector efficiency gradually decreases above 1560 nm so that the experimental peaks at higher wavelengths appear smaller than predicted by theory.
Our model is in good agreement with the measured data, with the clusters found at the same frequencies and the temperature difference between the configuration with ( figure 4(a)) and without ( figure 4(b)) the central cluster at 1560 nm is ≈0.07 • C in the experiment and 0.08 • C in the calculations. Furthermore, the model confirms its dependence on the finesse of the cavity, as observed in [22].

Designing a source of pure narrow-band photons with an integrated cavity-waveguide
Engineering a single-mode SPDC source via the clustering effect requires the clusters to be sufficiently narrow such that they can contain only one spectral mode of the cavity and that the FSR is sufficiently large so as to allow for low-loss filtering of the desired mode.

Determining the finesse
A cluster contains M modes when the difference between the spectral range corresponding to these modes for the idler and the same range for the signal is equal to the single-mode bandwidth. In quantitative terms, the number of modes within the cluster bandwidth can be expressed as where FSR i,s is the FSR for the idler and signal and ν is the mode bandwidth. We consider for ν the mean value of the idler and signal bandwidths: where F is the finesse of the cavity. In equation (6), F is assumed for simplicity to be the same for the idler and signal photons, implying that the mirrors' reflectivities and the internal losses of the waveguide resonator are independent of the wavelength. However, when this is not the case, similar calculations can be perfomed. Since FSR i,s = c/(2N i,s L), where N i,s is the group index at the idler and signal wavelengths, we can solve for the number of modes inside the cluster and determine the constraint on the finesse to have just a single mode: Therefore, finesse values larger than F M=1 are good to select single spectral modes. In figure 5, we give examples of the finesse F M=1 , as a function of the signal wavelength, in the case of a 532 nm or a 780 nm pump. The different dispersions in the case of type 0 (eee) and type II (eoe) interactions for PPLN and PPKTP are considered. For the dispersion of the ordinary refractive index in PPLN and PPKTP, we use the Sellmeier equations reported in [37] and [38], respectively. For a certain signal wavelength, the idler is determined via energy conservation and then the dispersion at the two wavelengths is considered. The phase-matching condition for different wavelengths is not taken into account but can be defined for a variety of interactions with specific values of the poling period. It is evident from these examples that, in the telecom regime, Type II PPLN and PPKTP crystals require lower values of the finesse, making the realization of integrated cavitywaveguides easier. For the type 0 interactions the required finesse becomes very high when the signal wavelength is double the pump. In this configuration, the signal and idler have the same wavelength and undergo the same dispersion, increasing the finesse value according to equation (7).
Note that these considerations only provide a lower bound for the value of the finesse and, according to the needs, one has to define the length and the reflectivities of the mirrors of the cavity in order to have the desired mode bandwidth and internal losses.

Finding the optimal parameters of the integrated cavity-waveguide
In order to design a single-mode source with an integrated waveguide resonator with a precise value of the finesse, one has to know the absorption coefficient and optimize the values of R 2 and L to take into consideration the losses inside the crystal. Therefore, let us calculate the probability p out of emitting photon pairs. With a reasoning similar to that in [22], assuming the initial generation of the photon pairs in the center of the waveguide, we find that The probability p out depends on the length L of the cavity-waveguide. Clearly, for larger values of L the probability p out decreases, although, to keep the finesse fixed, the reflectivity R 2 needs to be increased. Therefore, shorter crystals guarantee a small amount of overall loss and a large FSR, which is necessary for the efficient selection of a single spectral mode in the clusters.

Purity of the heralded photon state
This engineered photon pair source, in conjunction with sufficiently fast detectors, can be shown to produce heralded pure single photons. The clustering effect allows one to isolate a single spectral mode for the idler and the signal photon. Then, in order to achieve the spectro-temporal purity of the heralded photon [16,39], one has to adopt a detector on the heralding side with a resolution time τ res smaller than the coherence time of the photon τ c = 1/(π ν) [18]. In [39], the condition ν · τ res < 4 has to be fulfilled in order to guarantee a single-mode measurement. Standard detectors, having a timing resolution of the order of hundreds of picoseconds, require ν 1 GHz, which is easily satisfied by this source. Furthermore, the spatial purity of the heralded photon is guaranteed by coupling it into a single-mode optical fibre at the cost of some losses that depend on the emission properties of the source. However, single-mode waveguides have been shown to have very efficient (∼90%) coupling to single-mode fibers [31].

An optimized integrated cavity-waveguide source
Let us now consider a specific device design. A CW laser at 780 nm pumps a type II PPLN cavity-waveguide. From equation (7), we know that, in order to obtain a single-mode source at 1560 nm, we need a finesse larger than 10. Now, to have a sufficiently large FSR, we consider a crystal of length L = 1 mm. By assuming R 1 ≈ 1 and R 2 = 0.95, which are attainable with current reflection coating technology, we obtain a finesse of 116.2 from equation (1). For the absorption coefficient we use the value of α = 0.06 dB cm −1 , indicated in [22]. Using equation (8), we find a value for p out of 95%. By adopting the model described in section 2.2, we calculate the resonances of the cavity affected by the clustering effect at the temperature of T = 80.14 • C. This is illustrated in figure 6, where the spectrum is reported with respect to the signal wavelength and the red (dashed) curve represents the phase-matching envelope, calculated using the model reported in [32]. The phase mismatch k can be expressed as k = k p − k i − k s − 2π/ , where k p,i,s are the projection of the pump, idler and signal wave numbers to the direction of the waveguide and is the effective poling period, which ensures phase matching.
As can be seen from figure 6, the spectrum is characterized by a series of resonances, which are inside distinct clusters. The phase-matching envelope contains only three single-mode clusters, which are present at a spectral distance of λ clust ≈ 15 nm: it is therefore possible to filter the central mode, for example, by simply using a prism or a notch filter, which have negligible transmission loss. This device represents an example of a single-mode SPDC cavitywaveguide which does not need additional, lossy, narrow-band filters and produces photons with a bandwidth of λ = 4.6 pm (right inset).
Let us now consider the purity of this system when used as a source of heralded single photons. A bandwidth of 4.6 pm ( ν = 0.57 GHz at 1560 nm) means that we need a detection timing resolution less than 7 ns to satisfy τ res · ν < 4 [39]. Standard InGaAs APDs (like the ID210, IDQuantique) typically have τ res ≈ 350 ps. Therefore, heralded photons (with a high degree of purity) are achievable using quite straightforward and practical detection techniques.
In order to evaluate the brightness of the photon pairs generated by such a system, we take into account the measured brightness value reported in [40] for a 3 cm type II PPLN waveguide pumped by a CW laser at 777 nm and generating photons in a degenerate configuration at 1554 nm. The brightness of such a source is 6 × 10 2 photon pairs generated per second, per MHz of bandwidth and per mW of pump power. The brightness of a similar waveguide of 1 mm in a spectral window of λ = 4.6 pm would be 1.1 × 10 4 (s mW) −1 . According to [18], double-resonant OPOs undergo an average enhancement factor per resonant mode equal to the square of the finesse F over π. Therefore, the optimized integrated cavity-waveguide reported in this subsection would have a brightness of 4.8 × 10 7 (s mW) −1 in the 4.6 pm single resonant mode shown in figure 6. In general, for applications in quantum communication, SPDC sources need to generate less than 0.1 photon pair per coherence time in order to limit the emission of double pairs. Since the coherence time of these photons is τ c = 1 π ν = 560 ps, the pump power necessary for emitting 0.1 photon pair per coherence time is about 4 mW. Therefore, the enhancement factor provided by the cavity allows one to use a low pump power.

Discussion
Engineering pure narrowband photon states that avoid the use of external filters is fundamental for performing a variety of quantum communication applications and complex multi-photon experiments. Because of the Gaussian loss induced by the filters, some protocols, such as device-independent QKD, simply cannot be implemented, since it becomes impossible to close the detection loophole. On the other hand, for the same Gaussian filtering factor, multiphoton experiments become extremely slow and practically unfeasible. Figure 7 represents one of these most fundamental and important measurement schemes in quantum information science: the interference between two independent quantum systems. This configuration is commonly used, for example, to measure the Hong-Ou-Mandel interference between the photons γ 2 and γ 3 , which are heralded by the detectors D 1 and D 4 , respectively. Achieving unit interference visibility at beamsplitter B requires that the photons are in pure and indistinguishable states. Alternatively, by using the detectors D 2 and D 3 one can use this as a Bell state measurement (BSM) to herald a successful teleportation, as required for deviceindependent QKD [4], or entanglement swapping, as shown in [16]. We have shown that with this integrated OPO source, in conjunction with fast detection, one can efficiently herald pure Figure 7. Photon interference scheme. The visibility of the Hong-Ou-Mandel interference of the photons γ 2 and γ 3 , which are heralded by the detectors D 1 and D 4 , respectively, is maximum when these two photons are in pure and indistinguishable states.
single photons. As such, the interference between the pure single photons produced by two such systems should be capable of a high degree of HOM interference. In [16], a Hong-Ou-Mandel interference with waveguide-based sources followed by 10 pm external filters has been reported. If we were to replace the sources used in this experiment with two cavity-waveguides and, hence, to remove the lossy filtering system, we estimate an increase in the rate of such an experiment by almost two orders of magnitude, even before considering the improvements in fiber coupling and detector efficiencies achieved over the last few years. The visibility of the Hong-Ou-Mandel experiment in [16] was limited by multi-pair emission and by polarization fluctuations due to long integration times. The increased efficiency of this new approach would allow one to reduce the pump power, and hence the multi-pair emission rate, as well as reducing the integration times, thus significantly reducing the constraints on stabilization of the system. It is clear that this integrated solution, which is capable of exploiting standard detection schemes, provides significant advantages with respect to current approaches and is an important resource for future quantum networks.

Conclusions
We have outlined a model for the clustering effect in an integrated cavity-waveguide SPDC photon pair source pumped in a CW mode. We have shown how this effect can be engineered in order to emit narrow-band photons and to avoid the use of external filters. In conjunction with the fast detection of one photon of the generated pairs, one can efficiently herald pure narrowband single photons. Due to the very narrow bandwidth of the photons, we expect that this will be of much interest, not only for quantum communication, but for a wide range of emerging quantum technologies and more fundamental investigations such as all-optical, loophole-free, Bell experiments.