Direct generation of genuine single-longitudinal-mode narrowband photon pairs

PDC is an important process in quantum optics, used especially as a source of entangled photon pairs and of single photons. In a nonlinear ${\chi }^{(2)}$ medium pump, photons (at frequency ${\omega }_{p}$) are split into pairs of photons (signal ${\omega }_{s}$ and idler ${\omega }_{i}$, respectively), obeying energy and momentum conservation. As for most cases the latter cannot be obtained in homogeneous crystals, quasi-phase matching is exploited in a periodically inverted medium. Thus, the phase-matching condition becomes:

$$\begin{eqnarray}&&\Delta k=k({\omega }_{p})-k({\omega }_{s})-k({\omega }_{i})-\displaystyle \frac{2\pi }{\Lambda }\approx 0\end{eqnarray} \tag{ 1 }$$

where ${k}_{j}(j=p,s,i)$ are the propagation constants of the pump, signal, and idler fields, and Λ is the poling period. We express the state of the photon pairs generated by PDC up to the second-order expansion,

$$\begin{eqnarray}{| \Psi \rangle }_{\mathrm{PDC}} & \propto & \displaystyle \int {\rm{d}}{\omega }_{s}{\rm{d}}{\omega }_{i}f({\omega }_{s},{\omega }_{i}){\hat{a}}_{s}^{\dagger }({\omega }_{s}){\hat{a}}_{i}^{\dagger }({\omega }_{i})| 0\rangle \\ & & +\displaystyle \int {\rm{d}}{\omega }_{s}{\rm{d}}{\omega }_{s}^{\prime }{\rm{d}}{\omega }_{i}{\rm{d}}{\omega }_{i}^{\prime }f({\omega }_{s},{\omega }_{i})f({\omega }_{s}^{\prime },{\omega }_{i}^{\prime }){\hat{a}}_{s}^{\dagger }({\omega }_{s}){a}_{s}^{\dagger }({\omega }_{s}^{\prime }){\hat{a}}_{i}^{\dagger }({\omega }_{i}){\hat{a}}_{i}^{\dagger }({\omega }_{i}^{\prime })| 0\rangle ,\end{eqnarray} \tag{ 2 }$$

where ${{\hat{a}}^{\dagger }}_{s,i}({\omega }_{s,i})$ describes the photon creation operator at frequency ${\omega }_{s,i}$, and $f({\omega }_{s},{\omega }_{i})$ is the joint spectral function (JSF) of the nonlinear waveguide:

$$\begin{eqnarray}&&f({\omega }_{s},{\omega }_{i})=\alpha ({\omega }_{s}+{\omega }_{i})\times \varphi ({\omega }_{s},{\omega }_{i}).\end{eqnarray} \tag{ 3 }$$

The $\alpha ({\omega }_{s}+{\omega }_{i})$ describes the spectrum and the amplitude of the pump field, $\varphi ({\omega }_{s},{\omega }_{i})$, is a phase-matching function $\mathrm{sinc}[\Delta k({\omega }_{s},{\omega }_{i})\frac{L}{2}]\mathrm{exp}[{\rm{i}}\Delta k({\omega }_{s},{\omega }_{i})\frac{L}{2}]$.

The full-width at half-maximum (FWHM) of the PDC process in the waveguide is determined by the phase-matching condition. Assuming a monochromatic pump, i.e., $\alpha ({\omega }_{s}+{\omega }_{i})\propto \delta ({\omega }_{s}+{\omega }_{i})$, $\Delta k$ can be linearized

$$\begin{eqnarray}&&\Delta k({\omega }_{s,i})\approx \Delta {k}_{0}+\displaystyle \frac{1}{c}[{n}_{{gs}}-{n}_{{gi}}]\Delta {\omega }_{s,i},\end{eqnarray} \tag{ 4 }$$

where ${n}_{{gs}}$ and ${n}_{{gi}}$ are the group index of signal and idler, respectively. The spectral characteristics of the PDC process are given by a ${\mathrm{sinc}}^{2}[\Delta {\rm{k}}({\omega }_{s},{\omega }_{i})\frac{{\rm{L}}}{2}]$ function. Thus, the FWHM of the signal photons generated in a waveguide of length L is given by:

$$\begin{eqnarray}&&\Delta {{\nu }_{s}}^{\mathrm{FWHM}}\approx 5.56\displaystyle \frac{c}{2\pi L}\displaystyle \frac{1}{| {n}_{{gs}}-{n}_{{gi}}| };\end{eqnarray} \tag{ 5 }$$

i.e., the spectral bandwidth is inversely proportional to the difference of the group velocities of signal and idler. Thus, close to degeneracy the bandwidth is very large, whereas group velocity dispersion in most cases leads to a much narrower bandwidth far away from degeneracy.

Our source exploits photon pair generation by PDC within a cavity. The schematic of the integrated source is shown in figure 1(a). The resonant waveguide is a Fabry–Pérot cavity with asymmetric reflectivities (${R}_{1s,i}$ and ${R}_{2s,i}$), internal loss (${\alpha }_{s,i}$), and effective indices $n({\omega }_{s,i})$ for the fundamental modes at signal and idler wavelength, respectively. The photon-pair state generated within such a resonator can be expressed by:

$$\begin{eqnarray}{| \Psi \rangle }_{\mathrm{RPDC}} & \propto & \displaystyle \int {\rm{d}}{\omega }_{s}{\rm{d}}{\omega }_{i}{f}_{R}({\omega }_{s},{\omega }_{i}){\hat{a}}_{s}^{\dagger }({\omega }_{s}){\hat{a}}_{i}^{\dagger }({\omega }_{i})| 0\rangle \\ & & +\displaystyle \int {\rm{d}}{\omega }_{s}{\rm{d}}{\omega }_{s}^{\prime }{\rm{d}}{\omega }_{i}{\rm{d}}{\omega }_{i}^{\prime }{f}_{R}({\omega }_{s},{\omega }_{i}){f}_{R}({\omega }_{s}^{\prime },{\omega }_{i}^{\prime }){\hat{a}}_{s}^{\dagger }({\omega }_{s}){a}_{s}^{\dagger }({\omega }_{s}^{\prime }){\hat{a}}_{i}^{\dagger }({\omega }_{i}){\hat{a}}_{i}^{\dagger }({\omega }_{i}^{\prime })| 0\rangle ,\end{eqnarray} \tag{ 6 }$$

where ${f}_{R}({\omega }_{s},{\omega }_{i})$ is the cavity-modified JSF: ${f}_{R}({\omega }_{s},{\omega }_{i})=f({\omega }_{s},{\omega }_{i}){A}_{s}({\omega }_{s}){A}_{i}({\omega }_{i})$, with $f({\omega }_{s},{\omega }_{i})$ being the conventional JSF for the nonresonant PDC. ${A}_{s,i}({\omega }_{s,i})$ are proportional to the field distributions inside the cavity. They can easily be derived, treating the device as a classical Fabry–Pérot resonator with internal loss:

$$\begin{eqnarray}&&{A}_{s,i}({\omega }_{s,i})=\displaystyle \frac{\sqrt{(1-{R}_{1s,i})(1-{R}_{2s,i})}{{\rm{e}}}^{-{\alpha }_{s,i}L/2}}{1-\sqrt{{R}_{1s,i}{R}_{2s,i}}{{\rm{e}}}^{-{\alpha }_{s,i}L}{{\rm{e}}}^{i2{\omega }_{s,i}n({\omega }_{s,i})L/c}}.\end{eqnarray} \tag{ 7 }$$

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Therefore, we obtain an expression for the resulting joint spectral intensity,

$$\begin{eqnarray}&&{S}_{R}({\omega }_{s},{\omega }_{i})={| {f}_{R}({\omega }_{s},{\omega }_{i})| }^{2}={| f({\omega }_{s},{\omega }_{i})| }^{2}{{\mathcal{A}}}_{s}({\omega }_{s}){{\mathcal{A}}}_{i}({\omega }_{i}),\end{eqnarray} \tag{ 8 }$$

where ${{\mathcal{A}}}_{s,i}({\omega }_{s,i})={| {A}_{s,i}({\omega }_{s,i})| }^{2}$ are Airy functions of signal and idler photons, respectively. The bandwidth of each cavity mode is related to the finesse; more details in appendix A. Therefore the spectral characteristics are a product of the phase-matching sinc function and two combs of Lorentzian lines, spaced by the resonator's FSRs.

The explicit form of the signal spectrum can be determined completely as

$$\begin{eqnarray}{S}_{R}({\omega }_{s}) & \propto & {| \displaystyle \int {\rm{d}}{\omega }_{i}{f}_{R}({\omega }_{s},{\omega }_{i})| }^{2}\\ & \propto & {\mathrm{sinc}}^{2}[\Delta {\rm{k}}({\omega }_{s})\displaystyle \frac{{\rm{L}}}{2}]{{\mathcal{A}}}_{s}({\omega }_{s}){{\mathcal{A}}}_{i}({\omega }_{p}-{\omega }_{s}).\end{eqnarray} \tag{ 9 }$$

In the dispersive waveguide, cavity resonances occur at distinct frequencies separated by the FSR of the resonator. These resonances form frequency combs spaced with respective FSRs in the signal and idler wavelength range (as shown in the supplementary movie).

$$\begin{eqnarray}&&{\mathrm{FSR}}_{s,i}\approx \displaystyle \frac{c}{2{n}_{{gs},i}L}\end{eqnarray} \tag{ 10 }$$

The FSR is the inverse of the round-trip time of a photon in the cavity.

The FWHM of each resonance $\Delta \nu$ is related with the finesse ${\mathcal{F}}$ of the cavity:

$$\begin{eqnarray}&&{{\mathcal{F}}}_{s,i}=\displaystyle \frac{{\mathrm{FSR}}_{s,i}}{\Delta {\nu }_{s,i}}=\displaystyle \frac{\pi \sqrt{\sqrt{{R}_{{1}_{s,i}}{R}_{{2}_{s,i}}}{{\rm{e}}}^{-{\alpha }_{s,i}L}}}{(1-\sqrt{{R}_{{1}_{s,i}}{R}_{{2}_{s,i}}}{{\rm{e}}}^{-{\alpha }_{s,i}L})}.\end{eqnarray} \tag{ 11 }$$

For the type I degenerate case, the frequency combs of signal and idler photons are identical. Thus, PDC generation still happens at all comb frequencies within the phase-matching bandwidth. However, for type II and/or a degenerate case, there are different FSRs in the resonant waveguide due to the different group dispersions at the signal and idler wavelengths, as shown in the upper two curves in figure 1(b). Thus, corresponding resonances of signal and idler only overlap at a certain frequency, but adjacent modes do not coincide, as shown in the bottom curve in figure 1(b). For instance, if a pair is simultaneously resonant for, e.g., ${\omega }_{s0}$ and ${\omega }_{i0}$, the adjacent signal resonance lies at ${\omega }_{s0}+2\pi {\mathrm{FSR}}_{s}$, and the corresponding idler frequency at ${\omega }_{i0}-2\pi {\mathrm{FSR}}_{s}$ is not in the resonance peak due to the different FSRs. The couple of simultaneous resonances for signal and idler is called a cluster [37, 45]. However, after a certain number of FSRs, another cluster occurs again, if the phase-matching bandwidth is broad enough. This happens when N₀ (where N₀ is the number of FSRs in one cluster) times the signal FSR equals ${N}_{0}-1$ times the idler FSR. Thus, the cluster separation $\Delta {\nu }_{{\rm{cluster}}}$, i.e., the frequency spacing between clusters, is given by:

$$\begin{eqnarray}&&\Delta {\nu }_{{\rm{cluster}}}={N}_{0}\cdot {\mathrm{FSR}}_{s}=\displaystyle \frac{1}{| {n}_{{gi}}-{n}_{{gs}}| }\displaystyle \frac{c}{2L},\end{eqnarray} \tag{ 12 }$$

which is associated with the difference of the group indices. By comparing equations (5) and (12), we can see that the cluster separation $\Delta {\nu }_{{\rm{cluster}}}$ is slightly larger than half of the bandwidth of the PDC phase-matching envelope $\Delta {{\nu }_{s}}^{\mathrm{FWHM}}$. Therefore, the maximum number of clusters within the phase-matching range is three. Theoretically, if there is a dominant cluster in the center of the normalized phase-matching curve, two symmetric clusters will occur at the side wings of the envelope with about 41% weight. However, the fine-mode structures inside clusters are determined by signal and idler resonances. As shown in figure 1(b), the intensity of PDC emission in the two side clusters is weaker than the central cluster. Within a single cluster, the number of resonances contributing to the PDC generation depends on the bandwidth of the resonances and the difference of the FSRs.

Due to the spectral density redistribution in the cavity, PDC in the resonant waveguide is enhanced due to cavity narrowing and the cluster effect. The enhancement factor M is given by

$$\begin{eqnarray}&&M=\displaystyle \frac{\int {\rm{d}}{\omega }_{s}\int {\rm{d}}{\omega }_{i}{| f({\omega }_{s},{\omega }_{i})| }^{2}}{\int {\rm{d}}{\omega }_{s}\int {\rm{d}}{\omega }_{i}{| f({\omega }_{s},{\omega }_{i})| }^{2}{{\mathcal{A}}}_{s}({\omega }_{s}){{\mathcal{A}}}_{i}({\omega }_{i})}\sim {\eta }_{\mathrm{pp}}{{\mathcal{F}}}_{s}{{\mathcal{F}}}_{i}{N}_{0},\end{eqnarray} \tag{ 13 }$$

where ${\eta }_{\mathrm{pp}}$ is the photon pair escape probability defined by the ratio of the number of photon pairs, which leaves the cavity to the number of pairs created inside the cavity. More details are discussed in appendix B. Equation (13) tells us that the higher the finesse of the cavity is, the more enhancement can be achieved [27]. The enhancement factor is roughly proportional to the square of the finesse of the cavity.

However, for a waveguide-based resonant PDC source, the internal losses must be considered. (In bulk-type sources, these might be neglected due to the much smaller losses in bulk crystals compared to waveguides.) The higher the finesse is, the larger the number of round-trips is. Thus, the probability that a created photon pair is lost within the cavity increases with increasing the finesse. Hence, ${\eta }_{\mathrm{pp}}$ decreases with increasing the finesse by higher mirror reflectivities, keeping waveguide length and losses constant. On the other hand, the spectral redistribution due to the clustering results in a N₀ times increased spectral density at the synchronous signal and idler resonance.

To verify the photon pairs generation and to evaluate its efficiency, the standard way is to measure the cross-correlation between two generated photons. The two-photon cross-correlation function ${G}_{{si}}^{(1,1)}$ between signal and idler photons defines the shape of the coincidence events. Each resonance of the frequency comb of signal and idler can reasonably well be approximated by a Lorentzian function ${A}_{s,i}({\omega }_{s,i})\propto {({\gamma }_{s,i}+{\rm{i}}{\omega }_{s,i})}^{-1}$, where ${\gamma }_{s,i}$ describes the damping constants of the cavity. Assuming only a single pair of signal ${\omega }_{s}$ and idler modes ${\omega }_{i}$ are synchronously resonant with narrowband range $\delta \omega$, we get the complex JSF

$$\begin{eqnarray}&&{f}_{R}(\delta \omega )=\displaystyle \frac{{\gamma }_{s}}{{\gamma }_{s}+{\rm{i}}\delta \omega }\displaystyle \frac{{\gamma }_{i}}{{\gamma }_{i}-{\rm{i}}\delta \omega }.\end{eqnarray} \tag{ 14 }$$

The temporal ${G}_{{si}}^{(1,1)}(\tau )$ is measured as the coincidence distribution of detection time differences $\tau ={t}_{s}-{t}_{i}$ between the signal and idler photons,

$$\begin{eqnarray}&&{G}_{{si}}^{(1,1)}(\tau )=\langle {\hat{a}}_{s}^{\dagger }(t){\hat{a}}_{i}^{\dagger }(t+\tau ){\hat{a}}_{i}(t+\tau ){\hat{a}}_{s}(t)\rangle ={| {\tilde{f}}_{R}(\tau )| }^{2},\end{eqnarray} \tag{ 15 }$$

where ${\tilde{f}}_{R}(\tau )$ is the joint temporal function (JTF) defined as the inverse Fourier transform of the JSF. Thus, the signal–idler correlation can be simplified as

$$\begin{eqnarray}&&{G}_{{si}}^{(1,1)}(\tau )\propto u(\tau ){{\rm{e}}}^{-2{\gamma }_{s}\tau }+u(-\tau ){{\rm{e}}}^{2{\gamma }_{i}\tau },\end{eqnarray} \tag{ 16 }$$

where $u(\tau )$ is the step function. This shows that the coincidence funtion is determined by two different exponentially decaying functions, which are related to the damping of signal and idler photons, respectively. Thus, in most cases ${G}_{{si}}^{(1,1)}(\tau )$ is asymmetric due to ${\gamma }_{s}\ne {\gamma }_{i}$. The signal–idler correlation time is defined by ${\tau }_{c}\sim 2({\tau }_{s}+{\tau }_{i})/{\rm{e}}$. This means we can identify the lifetime of signal/idler ${\tau }_{s,i}={(2{\gamma }_{s,i})}^{-1}$ from the signal–idler coincidence measurements. Only if ${\tau }_{s}={\tau }_{i}$, which, for instance, is always the case if type I degenerate PDC processes are exploited [36], one gets a symmetric cross-correlation with correlation time ${\tau }_{c}\sim 4{\tau }_{s}/{\rm{e}}$.

Besides the one-photon pairs generated in the PDC process, it is still possible to generate double or higher photon pairs, as shown in equation (6). The temporal second-order auto-correlation function ${G}^{(2)}(\tau )$ of signal and idler photons, respectively, is related to the temporal coherence of the two-photon wave-packet component. The measurement of the auto-correlation is usually performed by inserting a 50:50 beam splitter into the signal (idler) arm and detecting the photons at the two output ports of the beam splitter with single-photon detectors. If coincidence clicks are registered, a second- (or higher) order photon pair generation must have happened. ${G}^{(2)}(\tau )$ is defined by

$$\begin{eqnarray}{G}^{(2)}(\tau ) & = & \langle {\hat{a}}^{\dagger }(t){\hat{a}}^{\dagger }(t+\tau )\hat{a}(t+\tau )\hat{a}(t)\rangle \\ & = & \langle {\hat{a}}^{\dagger }(t)\hat{a}(t)\rangle \langle {\hat{a}}^{\dagger }(t+\tau )\hat{a}(t+\tau )\rangle \\ & & +\langle {\hat{a}}^{\dagger }(t)\hat{a}(t+\tau )\rangle \langle {\hat{a}}^{\dagger }(t+\tau )\hat{a}(t)\rangle .\end{eqnarray} \tag{ 17 }$$

Thus, the normalized second-order auto-correlation function ${g}^{(2)}(\tau )$ is given by

$$\begin{eqnarray}&&{g}^{(2)}(\tau )=1+{| \displaystyle \frac{\langle {\hat{a}}^{\dagger }(t)\hat{a}(t+\tau )\rangle }{\langle {\hat{a}}^{\dagger }(t)\hat{a}(t)\rangle }| }^{2}=1+{| {g}^{(1)}(\tau )| }^{2}.\end{eqnarray} \tag{ 18 }$$

Consider a signal-cavity mode with a Lorentzian spectral distribution,

$$\begin{eqnarray}&&{f}_{s}(\delta \omega )=\displaystyle \frac{{\gamma }_{s}}{{\gamma }_{s}+{\rm{i}}\delta \omega }\displaystyle \frac{{\gamma }_{i}}{{\gamma }_{i}-{\rm{i}}\delta \omega };\end{eqnarray} \tag{ 19 }$$

the time-dependent correlation function is given by the inverse Fourier transform of its intensity spectrum,

$$\begin{eqnarray}&&\langle {\hat{a}}^{\dagger }(t)\hat{a}(t+\tau )\rangle ={\mathcal{F}}{{\mathcal{T}}}^{-1}[{f}_{s}(\delta \omega ){f}_{s}^{*}(\delta \omega )]=\displaystyle \frac{1}{4}{\gamma }_{s}{\gamma }_{i}{{\rm{e}}}^{-{\gamma }_{s}| \tau | }*{{\rm{e}}}^{-{\gamma }_{i}| \tau | }\\ \qquad =\;\{\begin{array}{ll}\frac{{\gamma }_{s}{\gamma }_{i}}{{\gamma }_{i}^{2}-{\gamma }_{s}^{2}}[\frac{1}{2}{\gamma }_{s}{{\rm{e}}}^{-{\gamma }_{s}| \tau | }-\frac{1}{2}{\gamma }_{i}{{\rm{e}}}^{-{\gamma }_{i}| \tau | }], & {\gamma }_{s}\ne {\gamma }_{i}\\ \frac{1}{4}{\gamma }_{s}(1+{\gamma }_{s}| \tau | ){{\rm{e}}}^{-{\gamma }_{s}| \tau | }, & {\gamma }_{s}={\gamma }_{i}\end{array}.\end{eqnarray} \tag{ 20 }$$

Correspondingly, equation (18) can be deduced as

$$\begin{eqnarray}&&{g}^{(2)}(\tau ) =1+{| {g}_{0}^{(1)}(\tau )| }^{2}\\ \qquad =\{\begin{array}{ll}1+{| \frac{1}{{\gamma }_{i}-{\gamma }_{s}}{{\rm{e}}}^{-\displaystyle \frac{{\gamma }_{s}+{\gamma }_{i}}{2}| \tau | }({\gamma }_{i}{{\rm{e}}}^{\displaystyle \frac{{\gamma }_{i}-{\gamma }_{s}}{2}| \tau | }-{\gamma }_{s}{{\rm{e}}}^{-\displaystyle \frac{{\gamma }_{i}-{\gamma }_{s}}{2}| \tau | })| }^{2}, & {\gamma }_{s}\ne {\gamma }_{i}\\ 1+{| {{\rm{e}}}^{-{\gamma }_{s}| \tau | }(1+{\gamma }_{s}| \tau | )| }^{2}, & {\gamma }_{s}={\gamma }_{i}\end{array}.\end{eqnarray} \tag{ 21 }$$

The above equation shows that the decay of the auto-correlation is not an exponential decay anymore. For our further analysis, it is useful to estimate the shape and half-width of the auto-correlation peak. By using a Taylor expansion, the above complicated function can be approximately simplified to

$$\begin{eqnarray}{g}^{(2)}(\tau ) & \simeq & 1+{| {{\rm{e}}}^{-\displaystyle \frac{1}{2}({\gamma }_{s}+{\gamma }_{i})| \tau | }(1+\displaystyle \frac{1}{2}({\gamma }_{s}+{\gamma }_{i})| \tau | )| }^{2}\\ & \approx & 1+{(1+{[\displaystyle \frac{1}{2}({\gamma }_{s}+{\gamma }_{i})\tau ]}^{2})}^{-1}.\end{eqnarray} \tag{ 22 }$$

From equation (22), we can conclude that the auto-correlation peak is symmetric, no matter which kind of PDC process it produced. This shows an intrinsic difference to the behavior of the cross-correlation function with its asymmetrically exponential decays. The decay of auto-correlation can be approximated as a Cauchy–Lorentz distribution. We can define the auto-correlation time ${T}_{{au}}$ as the FWHM of the auto-correlation function and obtain

$$\begin{eqnarray}&&{T}_{{au}}\approx \displaystyle \frac{4}{{\gamma }_{s}+{\gamma }_{i}}\approx \displaystyle \frac{2}{\mathrm{ln}2}{\tau }_{c}.\end{eqnarray} \tag{ 23 }$$

We find that the auto-correlation time exceeds the cross-correlation time by a factor of around 2.8. Our theoretical analysis is in good agreement with previous experimental observations [32, 40].

Another way of interpreting this auto-correlation is to consider the measurement from the multiple-photon pair-generation point of view. If two single-photon detectors in the signal (or idler) arm register a coincidence click, the second- (or higher) order part of the PDC state (second term in equation (6)) is probed. In contrast, the cross-correlation measurement provides mainly information about the first-order part. The auto-correlation time ${T}_{{au}}$ is the correlation time between the two-photon components of signal (or idler) photon components, while the cross-correlation time ${\tau }_{c}$ is the correlation time between signal and idler photons. In the temporal domain, the auto-correlation time ${T}_{{au}}$ is larger than correlation time ${\tau }_{c}$. Correspondingly, two photon components have narrower frequency bandwidth (related to $f({\omega }_{s},{\omega }_{i})f({\omega }_{s}^{\prime },{\omega }_{i}^{\prime })$) than the one-photon pair component (related to $f({\omega }_{s},{\omega }_{i})$).

To confirm the nonclassical property of generated narrowband photons, it is important to characterize heralded photons. The conditional auto-correlation function is defined by [48]

$$\begin{eqnarray}&&{g}_{\mathrm{heralded}}^{(2)}({t}_{s1},{t}_{s2}| {t}_{i})=\displaystyle \frac{{P}_{{ssi}}({t}_{s1},{t}_{s2},{t}_{i})}{{R}^{3}{g}_{{si}}^{(1,1)}({t}_{s1}-{t}_{i}){g}_{{si}}^{(1,1)}({t}_{s2}-{t}_{i})},\end{eqnarray} \tag{ 24 }$$

where ${P}_{{ssi}}({t}_{s1},{t}_{s2},{t}_{i})$ is the triple coincidence rate and R is the pair-generation rate. After simplification, the function of interest, ${g}_{\mathrm{heralded}}^{(2)}(\tau )={g}_{\mathrm{heralded}}^{(2)}({t}_{i},{t}_{i}+\tau | {t}_{i})$, can be represented as

$$\begin{eqnarray}&&{g}_{\mathrm{heralded}}^{(2)}(\tau )\approx 1-\mathrm{exp}[-2{(\displaystyle \frac{{\gamma }_{s}{\gamma }_{i}}{{\gamma }_{s}+{\gamma }_{i}}\tau )}^{2}],\end{eqnarray} \tag{ 25 }$$

which means the heralded correlation can be approximated by a Gaussian distribution. ${T}_{{he}}$ is the heralded time between one (signal or idler) photon component,

$$\begin{eqnarray}&&{T}_{{he}}\approx \displaystyle \frac{2}{{\gamma }_{s}+{\gamma }_{i}}=\displaystyle \frac{1}{2}{T}_{{au}}.\end{eqnarray} \tag{ 26 }$$

The waveguide was fabricated by an indiffusion of a lithographically patterned, 81 nm-thick, 5 μm-wide Ti-stripe into the Z-cut LiNbO₃ substrate. The diffusion parameters were designed to provide single-mode guiding at the TM-polarized idler mode. At signal and pump wavelength, however, the waveguide supports several guided modes.

Subsequent to the Ti-indiffusion, the periodic poling was performed. To obtain first-order phase matching for the selected wavelength combination, a short poling period of only $\Lambda \approx 4.44\mu {\rm{m}}$ is required. This could be achieved using a field-assisted domain inversion of the lithographically defined poling pattern. In the final fabrication process the end faces were polished. Special care was taken to ensure flatness and parallelism of the waveguide end faces to avoid excess loss of the resonating waves at the multiple reflections at these interfaces.

Based on the theoretical design rules discussed in the previous section and appendix A, the resonator finesse should exceed 20 to provide the spectral narrowing. The detailed structure of the integrated waveguide chip and experimental setup to study this resonant source are shown in figure 2. The source consists of a 12.3 mm-long Ti-indiffused waveguide in Z-cut PPLN and dielectric mirrors deposited on the waveguide end faces.

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To implement the resonant source, dielectric layers composed of alternating layers of SiO₂ and TiO₂ were deposited on the end faces of the waveguide. In practice, a stack with 17 layers deposited as a front mirror provides a reflectivity of ${R}_{s,i}\approx 99\%$ for signal and idler wavelengths, and the rear mirror, consisting of 17 layers has the targeted ${R}_{s,i}\approx 98\%$ for both. The reflectivities of front and rear mirrors at 532 nm are around 15% and 13%, respectively, to enable efficient incoupling of the pump and to prevent triple resonance effects.

After mirror deposition, the resonator was characterized by measuring signal and idler decay times. As we discuss in appendix B, we obtained a finesse of ${\mathcal{F}}\sim 100$ and ${\mathcal{F}}\sim 80$ for the signal and idler wavelengths, respectively.

With the asymmetry of the mirror reflectivities, the ratio of outcoupled signal (idler) photons to lost photons before escaping this cavity is about ${\eta }_{s}\approx 0.28$ (${\eta }_{i}\approx 0.18$), respectively. Thus, the overall photon-pair escape probability is given as ${\eta }_{\mathrm{pp}}={\eta }_{s}{\eta }_{i}$, which means about 5% of the generated photon pairs leave the cavity as couples at the desired output mirror. Therefore, the cavity-enhanced source provides high heralding efficiency.

The spatial profiles of signal and idler are determined by the waveguide properties. Both are the fundamental modes of the waveguide. Thus, the profiles are well matched for fiber network applications.

PDC spectra of one waveguide recorded prior and after mirror deposition are shown in figure 4. The spectrum of the uncoated waveguide shows a main peak with a bandwidth (FWHM) of 0.4 nm ($\approx$ 157 GHz), as predicted for the interaction length. The spectra of the same waveguide with dielectric mirrors, however, shows a pronounced sub-structure. Although the resolution of the spectrometer ($\approx \;0.15$ nm) is too coarse to reveal details of the spectra, one can already derive that the pair generation occurs within clusters with a cluster separation of about 0.24 nm ($\approx$ 90 GHz). This separation is determined by the beating of the FSRs of signal and idler, which theoretically are ${\mathrm{FSR}}_{s}\sim 5.2$ GHz and ${\mathrm{FSR}}_{i}\sim 5.5$ GHz. Generally, there is one dominant cluster in the center of the normalized phase-matching curve and another two symmetric clusters at the side wings of the envelope, as simulated in figure 1(b).

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A comparison of the spectral measurements of the coated and uncoated sample proves already the enhancement of PDC generation in the cavity: If the cavity would only act as a comb filter, only a small fraction of the generated PDC spectrum would 'transmit' through this filter. In this case we would have seen only a weak signal for the resonant PDC, because the measured spectra are always convolutions of the real spectra with the spectral resolution, i.e., a window function of about 60 GHz ($\approx \;0.15$ nm) width. In the experiments, however, we observed in both cases spectra with similar intensity levels. For the measurements shown in figure 4(a), all of the parameters, apart from adapting the temperature, have been kept constant. In particular, the pump power of about 1 mW in front of the incoupling lens has not been changed. Thus, we can conclude that due to the cavity there is a spectral density redistribution, resulting in an enhanced PDC generation at the resonances.

To study the spectral structure within a single cluster we performed measurements with the scanning Fabry–Pérot resonator. Measured single-photon-level PDC spectra with ∼700 MHz resolution are shown in figure 4(b). They reveal the modal structure within a single cluster. The spectrum consists of only one longitudinal mode. The scanning range of 15 GHz covers around 3 FSRs of the signal resonator. Please note that the measured linewidth of each longitudinal mode of about 700 MHz is mainly determined by the resolution of the scanning Fabry–Pérot, but it is not the natural bandwidth of the generated PDC photons.

The single-photon level PDC spectra with GHz resolution indicates that we are able to suppress the adjacent 'satellite' modes to single-mode operation by finely tuning the temperature—as shown in figure 5. A slight shift of the temperature leads to the spectrum composed of always one predominant mode. Only at a certain temperature does a weak two-mode operation happen.

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The temporal second-order signal–idler cross-correlation function [49] is measured as the coincidence counts between the signal and idler photons as function of the arrival time difference (more details were already discussed in 2.4).

To compare the experimental result with the theoretically expected behavior we first simulated the cross-correlation function for the used resonator parameters, assuming a truly single-mode PDC generation, as shown in figure 6(a). The presence of the cavity implies that signal and idler photons may be emitted at distinct times, corresponding to a different number of round trips within the cavity. Thus, the shape of the coincidence curve should be determined by exponential functions. For the multi-mode case the distribution of time of emission differences between the signal and idler modes should show well-defined peaks, with the peak separation corresponding to the cavity round-trip time. However, in our experiment we could not resolve such oscillations due to the finite temporal resolution of our detection system of about 500 ps. The convoluted results concerning the resolution of the detector system is a smooth exponential decay curve as well. Taking into account this limitation again by a convolution of the simulated curves, we find that the bandwidth of the coincidence peak is independent of the number of modes and depends only on the bandwidth of each mode.

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In the left diagram of figure 6(b), a result of such a coincidence measurement is shown. It reveals a correlation time (FWHM of the coincidence peak) of about 4.8 ns. This is significantly broader than the corresponding results obtained with non-resonant samples showing a width of about 0.5 ns, which is determined by the finite resolution of our measurement system. The correlation time is inversely proportional to the bandwidth of the down-conversion fields. From the measured correlation time ${\tau }_{c}$ of 4.8 ns, according to ${\tau }_{c}=1/\pi \Delta \nu$, where $\Delta \nu$ is the bandwidth of the down-converted photons, a spectral bandwidth of about 66 MHz can be deduced. This is in good accordance with the theoretically predicted width calculated for the given cavity parameters, as shown in figure 6(a). In the right diagram of figure 6(b) the measurement result is redrawn, using a logarithmic scaling together with exponential fits for the rising and falling parts. The slight asymmetry reflects the different finesses of signal and idler, resulting in slightly different leakage times out of the resonator. The different slopes of decay lines in logarithmic scaling give us the decay times of the resonators.

Based on the coincidence data and appendix B, the different lifetimes of signal and idler leaving from the resonant waveguide are ${\tau }_{s}\sim 3$ ns and ${\tau }_{i}\sim 2.4$ ns, respectively. Thus, the loss inside the waveguide cavity can be determined to be as low as ${\alpha }_{s}\approx 0.016$ dB/cm and ${\alpha }_{i}\approx 0.022$ dB/cm for signal and idler, respectively, as well as cavity finesses of ${{\mathcal{F}}}_{s}\simeq 100$ and ${{\mathcal{F}}}_{i}\simeq 80$.

The coincidence measurements can also determine the efficiency of the PDC generation process. From the ratio of the coincidence to single counts, the generated photon pair rate can be determined. We found that this rate is almost equal for non-resonant and resonant waveguides. For our source we determined a normalized generation rate (inside the resonator) of about $50\times {10}^{6}$ pairs/(s mW). Assuming these are distributed over three inequivalent clusters with different weights, we can estimate that about 90% of the generated photon pairs are within the most dominant longitudinal mode, with 66 MHz bandwidth. Taking into account the photon-pair escape probability of 5%, the spectral brightness can be estimated to be $B=3\times {10}^{4}$ pairs/(s mW MHz).

An alternative method to characterize the spectral properties of the source is the analysis of the auto-correlation Glauber function ${g}^{(2)}(\tau )$. The number of cavity modes N can be estimated from the measured normalized auto-correlation function at zero time delay, according to ${g}_{m}^{(2)}(0)=1+1/N$ (more details are given in appendix C).

Figure 7(a) provides the simulated and convoluted auto-correlation results for signal photons from the resonant waveguide. In the ideal case of a pure single-cavity mode field, the second-order auto-correlation is close to 2 at zero time delay. For a cluster obtained from this resonant waveguide, which consists of several inequivalent modes (as shown in figures 1(b) and 4(b)), the temporal auto-correlation behavior should reveal multi-mode interference as well. However, we cannot resolve the interference fringes due to the resolution of the detection system. As a result, only one single convoluted peak with lower ${g}_{m}^{(2)}(0)$ could be observed in the experiment.

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From the measured value ${g}_{m}^{(2)}(0)\approx 1.85$, shown in figure 7(b), an effective cavity mode number of $N\simeq 1.2$ can be estimated. This is in reasonably good qualitative agreement with the simulated auto-correlation function shown in figure 7(a) and the measured spectra shown in figure 4, where we observed only one mode within a single cluster. The signal auto-correlation time (∼12.6 ns) is about 2.8 times broader than the signal–idler correlation time (∼4.8 ns), which is in good agreement with the theoretical analysis (equation (23)). The Lorentz fit to the experimental curve also coincides with our theory model. Moreover, it indicates that shaping single-photon wavepackets by using amplitude and phase modulators is possible [50], because the signal auto-correlation time is much longer than the time jitters from detectors and electro-optic modulators.

To understand the intrinsic nature of the source, it is important to investigate the heralding performance for this resonant waveguide. The heralding efficiency would suffer from the asymmetric reflectivities, because the photon pairs escaping probability are impacted by the reflectivities of both dielectrics. However, the signal and idler long wave-packets generated from the resonant waveguide could help to resolve the wave-packet of single-signal photons due to the broadened temporal correlation length. The result of the normalized heralded second-order correlation function is shown in figure 8. The ${g}_{\mathrm{heralded}}^{(2)}(0)\lt 0.3$ at high pump power (10 mW) confirms a genuine sign of non-classicality. By using low-power pumping (∼1 mW), the minimum dropped to ${g}_{\mathrm{heralded}}^{(2)}(0)\lt 0.02$.

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Comparing heralded correlation with auto-correlation, it is found that they have different distributions and bandwidths. The heralded one (∼6.0 ns) is roughly half of the two-photon auto-correlated one (∼12.6 ns). The higher-photon components' contributions to heralded correlation measurements mainly influence the minimum value, since the probability to generate single-photon pairs is absolutely higher than the probability to generate double- or even triple-photon pairs.