The spontaneous parametric downconversion (SPDC) process is ubiquitous in quantum optics, as a source of correlated photon pairs, heralded single photons, entangled photons, or field-quadrature squeezing. Such non-classical states of light have applications in emerging quantum technologies including quantum communication [1], quantum metrology [2], and optical quantum information processing [3]. Bringing these technologies to a useful scale places stringent requirements on the performance of sources, which have driven improvements in the brightness and efficiency of SPDC sources [4,5]. Controlling the spectral and temporal properties of the generated light, which can be encapsulated by a joint spectral amplitude (JSA) between a signal and an idler beam, is critical for optimising the quantum features of the twin beams. The JSA can be engineered or affected by the shape of the pump laser pulse and by the properties of the nonlinear medium used: its dispersion and birefringence [6], by placing the medium inside a cavity or resonator [7,8], or even by using a photonic bandgap structure as the nonlinear medium to suppress emission at certain wavelengths [9]. Many applications require a factorable JSA (i.e. an absence of correlation or entanglement), such that each photon is emitted into an identical and well-defined spectral-temporal mode, and can undergo high-quality interference with every other photon [10]. The same concerns exist for spontaneous four-wave mixing (FWM) sources, which achieve similar effects as SPDC but using a $\chi ^{(3)}$ rather than a $\chi ^{(2)}$ nonlinearity, and hence can be implemented in a wider variety of materials [11–13]. Progress has also been made in utilising the spectral-temporal mode structure of single photons to encode quantum information [14], with recent demonstrations of heralded single photons in arbitrary temporal shapes and photon-pairs in engineered entangled states [15,16]. As a result, increasing attention has been given to methods of characterising the JSA for SPDC and FWM sources, to ensure that a source has the desired spectral-temporal properties, and to diagnose any short-fall in quantum interference visibility.

The most common methods of characterisation are stimulated emission tomography (SET) [17], where a bright seed laser is used to stimulate a classical analogue of SPDC or FWM, so that the emission can be measured by conventional spectral intensity measurements; or direct measurement at the single photon level, often using a highly dispersive medium as a time-of-flight spectrometer followed by a single photon detector [18]. In their basic forms, both methods provide the joint spectral intensity (JSI), excluding the phase-information of the JSA, and hence do not constitute a full spectral-temporal characterisation. SET can be made phase-sensitive with an additional phase-reference beam injected into the emission mode [19]; similarly, a quantum pulse gate or spectral-shear interferometry can make a direct measurement sensitive to spectral-temporal mode [20,21]. All of these techniques require considerable additional experimental complexity in order to retrieve the phase information of the JSA, whether in the form of extra phase-stabilized reference beams, a quantum pulse gate, or an interferometer implementing a spectral shear in one path. Hence for developing quantum photonic technologies with increasing numbers of sources, it is desirable to develop simple and scalable methods of characterisation which are able to diagnose phase-correlations in the JSA which would be missed by a JSI measurement.

In this work, we make spectrally-resolved measurements of photons generated in double-pair emission from a SPDC source. While the frequency-resolved detection of individual photon-pairs only provides information about the JSI, multi-pair emission is sensitive to phase-correlations. This can be understood as interference between different pathways to creating the multiphoton state [22]: the interfering paths correspond to different ways the photons could be paired with each other, in a manner closely related to Gaussian boson sampling [23]. We operate a type II SPDC source which produces signal and idler beams in horizontal and vertical polarizations. We introduce varying levels of phase-correlation to the JSA, by varying the chirp applied to the pump laser, and observe the resulting interference pattern appearing in the four photon events. We also show that the phase-correlation manifests as spectral intensity correlation between pairs of signal photons, even when the two corresponding idler photons are not detected - this is essentially the spectrally-resolved version of an unheralded $g^{(2)}(0)$ measurement, which is known to be directly related to the purity of the photons [24]. The spectrally-resolved version provides additional information about the density function of a single photon from a pair, and can provide a more robust estimate of its purity. We note that analogously it has been observed in SET of high-gain sources that the light generated by cascaded or higher-order nonlinear processes is sensitive to phase factors in the first-order SPDC process, though a method of retrieving the full JSA is still lacking [25].

An ideal two-mode squeezed vacuum (TMSV) state can be written as

We now consider the effect of applying a chirp to the pump pulse, on a JSA which is otherwise free of correlation. Assuming low parametric gain and low group-velocity dispersion, the JSA has the form

Figure 1(a) shows the experimental setup to measure this effect. A 10MHz mode-locked fiber-laser produces $\sim 100$fs pulses with a broad bandwidth around 1550nm, which are frequency-doubled to $\sim 777$nm with a periodically-poled lithium niobate crystal. A spectral pulse shaper (SPS) is used to set the bandwidth of the pulses and apply a variable chirp, and consists of a spatial light modulator inside a folded 4f line, with a 500mm focal length. The back-reflected pulses from the SPS are picked off and launched into a type II SPDC source based on a highly nonlinear waveguide in periodically-poled potassium titanyl phosphate (ppKTP) crystal, which generates telecom-wavelength near-degenerate photon pairs. The pump is filtered out, while the downconverted photons are collected into fiber. To measure the photons’ frequencies, a large group-velocity dispersion is applied using reflection from a chirped fiber Bragg grating and a circulator, which translates wavelength into arrival time with 2.3ns/nm relative delay per change in wavelength. The signal and idler photons have orthogonal polarizations and are separated by a fiber polarizing beamsplitter; then each output is split to four superconducting nanowire single photon detectors, to allow multiple photons to be detected within the detector deadtime. The detector outputs, along with a synchronization signal from the laser, are sent to time-tagging electronics, resulting in an overall $\sim 150$ps time resolution, implying $\sim 65$pm wavelength resolution for the measured photons. Figure 1(b) shows the JSI measured using coincidences between one signal and one idler photon, for a 2nm pump bandwidth, which generates a near-factorable JSI. The phase-matching of the SPDC process is expected to result in a sinc-squared profile containing side-lobes around the central peak: a clear side-lobe can be seen to the top-left of the JSI, but in the bottom-right it appears more smeared out, which is thought to be due to some non-uniformity along the length of the waveguide. Slight diagonal ripples can be seen across the central peak of the JSI, which are caused by ripples in the pump spectrum due to imperfect calibration of the pulse shaper.

Figure 2(a) shows the measured four photon counts, accumulated for 30 minutes, as a function of the difference between the two signal wavelengths, $|\lambda _s-\lambda _s'|$, and the difference between idler wavelengths, $|\lambda _i-\lambda _i'|$. The chirp applied to the pump pulse is varied from 0 to 20ps/nm in steps of 5ps/nm. The interference is clearly visible as curved fringes, as predicted by Eq. (9). Figure 2(b) shows the corresponding interference fringes predicted from a theoretical model of the JSA, where the pump is taken to have a Gaussian spectrum multiplied by a quadratic phase as described above, whereas the phase-matching is taken to follow a sinc function. The experimental data is in good agreement with the model, though finite count statistics make the fringes hard to observe away from the maximum. The fringes provide a clear signature of the presence of the chirp-induced spectral phase in the two photon joint spectrum, and their modulation period can be used to infer the value of $\beta$, though they are not dependent on its sign. For an unknown $\beta$, the sign ambiguity could be resolved by adding a controlled chirp to the pump with a known sign, and observing how the modulation period changes. The contrast of the fringes is limited by residual joint spectral intensity correlations (if the JSI is not separable the two interfering terms in Eq. (9) will be imbalanced and not cancel fully), background from higher-order events (where more than two pairs are generated), and the finite spectral resolution.

 

Fig. 1. (a) Experimental setup. MLL: mode-locked laser at 1550nm; SHG: second harmonic generation in periodically-poled lithium niobate; SP: short-pass filter; PBS: polarizing beamsplitter; DG: diffraction grating; F: 500mm focusing mirror; SLM: spatial light modulator; SPDC: downconversion in a periodically-poled potassium titanyl phosphate waveguide; LP: long-pass filter; CBG: chirped bragg grating; 1$\rightarrow$4: fiber splitter; SPDs: single photon detectors. (b) Measured JSI for a 2nm pump bandwidth centered at 777nm.

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Fig. 2. (a) Four photon counts as a function of the difference between the two signal wavelengths and between the two idler wavelengths, for different applied pump chirps between zero and 20 ps/nm. (b) Corresponding plots from the theoretical model.

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Next we consider the spectral correlations induced between two signal photons by phase correlation in the JSA. If we ignore, i.e. integrate out, the two idler frequencies, from Eq. (9), we obtain an expression for the joint probability of two signal photons with frequencies $\omega _s$, $\omega _s'$:

For a more general JSA function $\psi (\omega _s,\omega _i)$, i.e. no longer assuming the form given in Eq. (8), the probability of detecting two signal photons can still be written as the sum of a non-interfering and an interfering term:

$|\rho (\omega _s,\omega _s')|^2$ can be measured quickly because it is only necessary to detect two rather than four of the photons; it appears to be relatively sensitive to small amounts of phase correlation; and it is not influenced by loss, even for high levels of squeezing, because with the idler traced out the signal is essentially a multi-mode thermal state.Fig. 3(a) shows this function measured by detecting two signal photons in coincidence, with the non-interfering term subtracted. To ensure an accurate background subtraction, we measured the corresponding correlation function between adjacent pump pulses, which is equal to the non-interfering term. Statistical uncertainty still leads to imperfect subtraction, which can be seen as a region of noise surrounding the main function; here, when the resulting counts are negative, they are rounded up to zero. The chirp applied to the pump was varied from -5ps/nm to +5ps/nmm, in steps of 2ps/nm, and for each value the photon statistics were recorded for 10 minutes. For larger values of chirp, a thin line can be seen along the diagonal, which broadens as expected when the chirp is reduced. This width along the anti-diagonal reaches a maximum at an applied chirp of around -1ps/nm, rather than zero as one would expect - this is most likely due to the presence of residual chirp generated by the 4f line in the SPS. The measured phase correlation is minimised when the applied chirp compensates that of the delay line. Figure 3(b) shows the corresponding plots generated from the theoretical model, where this offset to the pump chirp has been incorporated. The model and experiment are generally in good agreement, though the experimental function does not become as close to circular as the model, suggesting some short-fall in purity. This deviation from the model could be due to the fact that the phase-matching is not an ideal sinc function - this could be seen in Fig. 1(b), in the smearing effect on the JSI likely from non-uniformity along the waveguide - and from imperfect calibration of the SPS. Additional artifacts in the experimental plots seen as sharp anti-diagonal lines result from a slight after-pulsing effect in the detection electronics.

 

Fig. 3. (a) The measured interference term for detection of two signal photons, corresponding to $|\rho _s(\omega _s,\omega _s')|^2$, with the chirp applied to the pump pulse varied from -5ps/nm to +5ps/nm. (b) The corresponding predicted plots from the theoretical model.

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The ratio between the total counts in the interfering term $|\rho _s(\omega _s,\omega _s')|^2$ and in the non-interfering term $\Psi _s(\omega _s)\Psi (\omega _s')$ should be equal to the purity of the photons. Since this involves integrating over the remaining frequencies, this does not require any spectral resolution, and is equivalent to an unheralded $g^{(2)}(0)$ measurement, which is commonly used as a test of purity [24]. Figure 4 shows the expected value of purity from a simulation, based on the shape of the JSI and the applied chirp (black line). The inferred purity from measurement of the unheralded $g^{(2)}(0)$ (red squares) follows broadly the predicted trend, but at larger chirps the value does not fall as far as expected. This measure is easily affected by uncorrelated background effects and imperfections such as idler photons leaking into the signal channels. Here, the coincidences due to after-pulsing seen in Fig. 3(a) form part of the background which flattens out the red squares’ dependence on pump chirp. Making use of the frequency-resolved measurements in Fig. 3(a) can result in an improved assessment of the purity - the blue circles in Fig. 4 show the purity inferred from the narrowed width of the peak along the anti-diagonal compared to the full width along the diagonal. These values are in much better agreement with the simulation, suggesting that observations from the frequency-resolved measurement can provide a more robust test of purity, avoiding the effects of background and leaked idler photons.

 

Fig. 4. The estimated purity as a function of pump chirp. black line: simulated value based on the applied chirp; blue circles: value inferred from the narrowing of $I_s(\omega _s,\omega _s')$; red squares: value inferred from the unheralded $g^{(2)}(0)$.

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In conclusion, we have shown that the frequency-resolved double-pair emission from a SPDC source is sensitive to the complex phase of the joint spectral amplitude, and demonstrated that this can be used to detect the presence of a bilinear form of correlated phase. Its effects can be seen as interference fringes in the four photon coincidence probabilities. Further, the coincidences between two signal photons can be used to infer the absolute value of the corresponding single photon density function. This quantifies the degree of coherence between separate frequencies, and can be used to make a robust measure of purity, which is not specific to the bilinear correlated phase. In future work, this could be extended to characterization of more general JSA functions and unknown spectral phase correlations. In analogy with Ref [28], where one and two photon interference are used to characterize a linear-optical interferometer, the two and four photon coincidence probabilities should be sufficient for a complete reconstruction of a general JSA, with the proviso that these probabilities are insensitive to non-correlated spectral phases functions affecting the signal or idler individually. Other than this ambiguity, a barrier to this approach is the large number of possible four photon outcomes - for 250 discrete points across the spectral width, around $10^8$ distinct outcomes - which may make it impractical to accumulate sufficient statistics to determine the probability of a particular outcome. A possible way forward is to assume the spectral phase can be modelled using a small number of parameters, e.g. $\beta$ as above and a few describing higher-order phase correlations, and then to retrieve these parameters using maximum likelihood optimization to the photon statistics. Such a method of characterizing the JSA of a photon pair source, including its phase correlations, would be relatively simple to implement, requiring only a time-of-flight spectrometer and the ability to detect at least two photons in each output, and would be valuable in benchmarking new sources and devices for quantum photonic technologies.