Designing High-Power, Octave Spanning Entangled Photon Sources for Quantum Spectroscopy

Prior to quantifying the experimental behavior of the SPDC sources, we lay out the theoretical and numerical details used in their design.

In SPDC, the creation of two entangled photons arises from the quantum mechanical process of a single pump photon mixing with the underlying vacuum state. The two down-converted daughter photons display strong correlations in time, energy, and momentum due to the parametric mixing process Burnham and Weinberg (1970); Klyshko, Penin, and Polkovnikov (1970). The time and energy correlations originate from the individual photons being generated simultaneously from the pump photon:

$$E_{p}\left(\omega_{p}\right)=E_{s}\left(\omega_{s}\right)+E_{i}\left(\omega_{i}\right)$$

(1)

$$\therefore\;\omega_{i}=\omega_{p}-\omega_{s}$$

(2)

where $\omega_{p}$, $\omega_{s}$, and $\omega_{i}$ are the frequencies of the pump, signal, and idler photon respectively. The momentum correlation is dictated by the phase matching condition, which is a consequence of the wave-like nature of the interacting fields. Each field has a corresponding wave vector defined via its respective phase velocity $\textit{v}_{p}=c/n\left(\omega,T\right)$ in the medium:

$$k\left(\omega,T\right)=\frac{\omega n\left(\omega,T\right)}{c}$$

(3)

with $n\left(\omega,T\right)$ being the frequency and temperature dependent refractive index of the material. The following vector sum, the longitudinal component of which is depicted in Fig.1, should be satisfied in order to ensure that a proper phase relationship between the interacting waves is maintained throughout the nonlinear crystal.

	$\displaystyle\Delta\textbf{{k}}\left(\omega_{p},\omega_{s},\omega_{i}\right)=\textbf{{k}}_{p}\left(\omega_{p},n_{p}\left(\omega_{p},T\right)\right)-\textbf{{k}}_{s}\left(\omega_{s},n_{s}\left(\omega_{s},T\right)\right)$		(4)
	$\displaystyle-\textbf{{k}}_{i}\left(\omega_{i},n_{i}\left(\omega_{i},T\right)\right)$

It is only when this relationship holds true $\left(\Delta k=0\right)$ that constructive interference between the propagating waves can occur and energy is transferred dominantly in one direction, rather than oscillating back and forth between the two optical modes Boyd and Prato (2008).

A natural extension of this phase-matching principle, termed quasi-phase-matching, is apparent if one considers the duality between position and momentum space. A periodic lattice in position space directly corresponds to a well-defined wave vector k in momentum space via a Fourier transform. Therefore, if the nonlinear dielectric tensor domains of ferroelectric nonlinear crystal are modulated with a specific periodicity $\Lambda$, a quasi-momentum term $\Gamma\left(\Lambda\right)$ arises, which can be used to satisfy a non-phase-matched nonlinear interaction.

$$\Delta\tilde{\textbf{{k}}}\left(\omega_{p},\omega_{s},\omega_{i}\right)=\Delta\textbf{{k}}\left(\omega_{p},\omega_{s},\omega_{i}\right)-\Gamma\left(\Lambda\right)$$

(5)

The quasi-phase-matching term $\Gamma\left(\Lambda\right)$ also has an additional advantage, in that it can be chosen to be higher-order. Here, the order of the configuration is defined by m.

$$\Gamma\left(\Lambda\right)=\frac{2\pi m}{\Lambda},\;m\in 2\mathbb{Z}+1\left(\neg 0\right)$$

(6)

This serves a particular experimental benefit as it allows for nonlinear interactions to be phase-matched (albeit with lower efficiency), even if the required $1^{st}$ order periodicity is too small to be fabricated Fejer et al. (1992).

We now describe the design of a grating in congruent lithium tantalate with the design goal of a 406 nm pump wavelength that creates a >400 nm bandwidth around the degenerate 812 nm SPDC point. The source must also be capable of creating near $\mu$W powers given a 1 W input power. The first step is to obtain the Sellmeier equations that describe the frequency and temperature dependent refractive indices of the material Moutzouris et al. (2011). The numeric values and a representative plot are found in the supplementary information. Using the Sellmeier equations, the phase mismatch (Eq.5) is then evaluated for a wide range of pump-signal/idler wavelength configurations while keeping one entangled photon frequency a constant. External conditions such as the desired temperature range and minimum achievable poling periodicity are also enforced such that the parameter space remains within experimentally realistic boundaries. The root of Eq.5 can then be found by a root-finding algorithm. For the congruent lithium tantalate and a 406-812 SPDC process, and imposing a realistic < 200°C temperature condition, solving for the root of Eq.5 yields a poling period of 9.5 $\mu$m at 133°C (see figure in SI for the graphical solution).

Quasi-phase-matching is a strongly temperature dependent technique. By solving Eq.5 for a range of temperatures near the calculated degeneracy point (133°C), the splitting and tuning of the SPDC spectrum can be calculated. Tuning the SPDC spectrum via the crystal temperature is useful for controlling what intermediate states are involved in, the magnitude of, and the temporal resolution of an entangled two photon process. The temperature dependent behavior is shown in Fig.2. Here, the transition between the degenerate and non-degenerate domains is clearly observable, with lower temperatures corresponding to a weaker degenerate emission profile. The lower plot of Fig.2 also clearly shows a very strong narrowing of the signal and idler bandwidths as one temperature tunes further away from degeneracy.

Additionally, both signal and idler have an inherent angular emission property determined by the momentum conservation condition in Eq.5. The strong frequency dependence of the outgoing angles can be numerically calculated from the momentum conservation equation if one decomposes the wave vectors into their longitudinal and transverse components, and solves the trigonometric equations for transverse phase-matching. Subsequently computing the emission angle for each frequency of interest, in conjunction with the frequency dependent refractive index of the nonlinear crystal, the SPDC cone’s spectral characteristic for various crystal temperatures can be plotted (Fig.3). As predicted, at the degenerate emission temperature of 133°C, the plot shows that the emission angle of the 812 nm signal and idler photons is 0°. More importantly however, comparing the results to those in Fig.2, the shaded non-degenerate region also corresponds to a collinear emission profile. Vice versa, temperatures below 133°C display a (albeit weak) degenerate, non-collinear character.

In a chirped crystal, the bandwidth of the down-converted entangled photons is increased by changing the poling period throughout the crystal. The increasing width elements correspond to different phase-matched wavelength configurations of Eq.5 (Fig.4). The photons still add up to the narrow linewidth of the pump laser but the temporal resolution is given by the temporal alignment and down-converted bandwidth of the entangled photons. This is why entangled photon experiments are said to have independent spectral and temporal resolutions. The importance of the pump laser’s linewidth in determining the entangled photon state’s purity, and thus the access to non-classical spectroscopic properties, is discussed in the SI.

Following a similar treatment to Di Giuseppe et al. (2002), the crystal can be decomposed into its individual domains, in each of which the downconversion amplitude follows the usual sinc function-like phase-matching dependence Burlakov et al. (1997). This is attributed to the Fourier transform of a rectangular function in real-space representing the probability of a downconversion event occurring inside the crystal element.

$$A_{n}\left(\omega_{s},\omega_{i}\right)=L_{n}sinc\left(\frac{L_{n}\Delta k}{2}\right)$$

(7)

With $L_{n}$ being the length of the $n^{th}$ crystal element. Similarly, a cumulative-phase term can be defined, which is related to a forward propagating wave with phase-mismatch $\Delta k$ traversing a crystal length $L_{prop}$ after it has been generated in the SPDC process at crystal element n.

$$\varphi=\Delta kL_{prop}=\Delta k\left(\frac{L_{n}}{2}+\sum_{m=n+1}^{N}L_{m}\right)$$

(8)

Carrying out a sum over all individual crystal elements then allows for the resulting SPDC spectrum to be calculated (Eq.9) and thereby observe the effects of different chirping parameters. For lithium tantalate, this is shown in Fig.5. In the case of CLT, a $\pm$10% chirp around the degenerate poling periodicity calculated above yields a full octave spanning SPDC bandwidth. Trivially, in the limit of no chirp, where all crystal elements are equal in size, the solution for a periodically poled crystal of length L is recovered.

	$\displaystyle A\left(\omega_{s},\omega_{i}\right)=\chi_{0}\sum_{n=1}^{N}\left(-1\right)^{n}L_{n}sinc\left(\frac{L_{n}\Delta k}{2}\right)$		(9)
	$\displaystyle\times e^{-i\Delta k\left(\frac{L_{n}}{2}+\sum_{m=n+1}^{N}L_{m}\right)}$

By plotting the solution of Eq.9 as a function of the position inside the crystal along the propagation direction, we can gain some intuition for how the constructive interference results in a broadened spectrum. In the case of CLT, a $\pm$10% chirp around the degenerate poling periodicity calculated above yields a full octave spanning SPDC bandwidth as shown in Fig.6. Here we used a total grating length of 18 mm equal to the chip used in subsequent experiments.

The 2D plot in Fig.6 provides some additional insight into how modulation of the poling period affects the build-up of the emitted spectrum. In particular, it’s important to note that while the first 8mm length of the crystal appears to show no downconversion taking place, this is indeed not the case. While all crystal elements in this region satisfy a particular phase-matching condition, it is only after a certain propagation length that the phases of each spectral mode coherently sum to a strong enough amplitude that would be perceivable on the plot. Furthermore, the broadening of the downconverted bandwidth does have a particular drawback, in that the frequencies close to degeneracy become weaker in intensity. This is best observed in the line cuts for positions L3 and L4, where instead of a constant intensity across all frequencies, the emission spectrum peaks at the lower and higher cut-offs but begins to droop more severely as the bandwidth increases. Thus, increasing the chirping of the poling period indefinitely does not necessarily yield a flat power spectral density. It should be pointed out that this issue can be addressed via chirp profiles which follow not a linear but some higher order functional form Brańczyk et al. (2011).

In contrast to narrowband SPDC where the emission angles for the photon pairs are well confined within the two narrow arcs satisfying momentum conservation, such extremely broadband downconversion poses inherent difficulties in the spatial degree of freedom due angular (chromatic) dispersion of the resulting beam. As Fig.3 shows, the emission angles increase quickly away from degeneracy. This presents several experimental issues, as the beam no longer represents an ideal point source, which will be discussed in the experimental section below. Taking into account the physical dimensions of the grating cross section (detailed in the experimental section below), it is easy to see that the output aperture of the grating can at most only accommodate $arctan(0.15mm/9mm)\approx 1\degree$ of emission in the horizontal axis, and $arctan(0.5mm/9mm)\approx 3.2\degree$ along the vertical axis relative to the center point of the crystal where we focus. Beyond this, large parts of the SPDC emission cone will begin to interact with the domain boundaries which form the grating channel. While in principle the refractive index is the same for both un-poled and poled crystal regions, the periodic poling process causes some stress at the domain walls. Therefore, the edge of the each pattern in the lateral direction acts as a kind of scattering center, possibly resulting in unwanted interference patterns. Therefore special attention should be paid to accommodate the spatial distribution of the cone as it diverges inside the channel.

A final point to consider is the total length of the grating.is to be chosen by considering two interlinked criteria. Firstly, the expected power spectral density of the entangled photons at the output scales as $L^{2}$. Emission intensity can therefore be significantly increased by utilizing longer structures. Conversely, the expected power spectral density also depends on a $sinc\left(\frac{L_{n}\Delta k}{2}\right)$ multiplicative factor. Hence, longer gratings result in narrower emission bandwidths. This ‘dual’-argument also applies to poling patterns which have a chirp, as while increasing the amount of variation in the poling periodicity serves to broaden the emission bandwidth, each spectral component will be less populated with photons and thus less brilliant.

The periodically poled congruent lithium tantalate gratings were manufactured by HC Photonics. The unchirped chip consists of 8 gratings with different poling periods (8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12 $\mu$m). It is 20 mm long and each channel has a 0.9 mm by 0.5 mm cross section. The chirped chip consists of 6 gratings with varying chirp parameters and entrance poling periods (Table 1). Each grating is 18 mm long and has a 1 mm by 0.3 mm cross section. The gratings are AR-coated on both input and output faces at 406nm(R<0.5%)/812nm(R<0.5%).

The experimental setup, as illustrated in Fig.8, consisted of a computer-tunable, CW Ti:Sapphire laser (M2 SolsTiS) and an external SHG cavity to act as the pump source. The maximal output power of the SHG at 406 nm was 1.15 W with a 0.89 nm linewidth. The pump beam is focused through the ppCLT grating using an aspheric lens with a 40 cm focal length. This focal length is chosen such that the size of the focused beam was smaller than the grating’s cross sectional dimension and further optimized for peak SPDC brightness. The position of the crystal can be finely adjusted in the XYZ directions using a piezoelectric stage, which is critical for maximizing the entangled photon flux. The angular deviation of the grating’s long axis relative to the input beam was not as crucial as the X-Y centering of the chip around the focused beam and the z-control over exact focal point positioning. The temperature of the crystal is controlled by a ceramic heater and PID loop (Covesion) to an accuracy of 10 mK.

To measure the SPDC spectrum as well as the power of the downconverted flux, the output of the grating is passed through a total of three OD-4 500 nm longpass filters (Edmund Optics) to eliminate the residual 406 nm pump. The SPDC cone is collimated with an off-axis parabolic mirror and then focused into a USB spectrometer (Thorlabs). For more detailed characterization, the SPDC cone is collimated with a telephoto lens and then focused into a spectrometer (Princeton Instruments IsoPlane) with a 15cm focal length lens. In this configuration, spectral and power measurements are performed with an electron multiplying intensified CCD camera (emICCD, Princeton Instruments MAX4). For the broadband spectral measurements, spectra are collected and stitched together as a 800 nm blazed grating with 150 grooves/mm is scanned across several center wavelengths. These spectra are then background corrected, quantum efficiency corrected according to the detection efficiency of the emICCD camera, and normalized.

Fig.7 (a) depicts the SPDC spectra for the 9.50 µm unchirped grating as the temperature of the crystal is varied from 60°C to 160°C. Above 150°C, the shift from degenerate to non-degenerate emission conditions is apparent from the decrease in intensity around the degenerate wavelength region of 812 nm (gray dashed line). In comparison, as shown in Fig.7 (b), at 145 °C, the chirped chip (blue curve) exhibits much broader and flatter emission than the unchirped counterpart (red curve).

The fourth-order interference between entangled photon pairs was quantified with a two-photon Michelson interferometer, depicted in Fig.8. The Michelson interferometer is chosen because the configuration allows a broader bandwidth of the SPDC cone to be easily utilized, in contrast to a Mach-Zehnder configuration. Following the end of the ppLT chip, the entangled photons are collimated with an off-axis parabolic mirror and a cylindrical lens. These photons are directed through a broadband nonpolarizing 50:50 beamsplitter (BS1, Layertec) which separates the incident photons into two paths. In the reflected path, the photons pass through an achromatic quarter-waveplate (Thorlabs) twice to rotate their polarizations. The optical path length difference between the two arms is adjusted by scanning a mirror mounted to a linear stage in the transmitted arm. Coincidence counts vs. optical path length difference are then collected at one of the output arms of BS1. The coincidence counting setup consists of a second broadband 50:50 beamsplitter (BS2, Layertec), and two free-space-to-fiber coupling setups connecting to single-photon avalanche diodes (SPADs, Laser Component COUNT) and time-tagging electronics (PicoHarp 300). Because of the high SPDC flux, the pump beam must first be dimmed by six orders of magnitude to avoid SPAD saturation (counting rate $10^{6}$ photons/s). Full counting rate experiments can be completed using the CCD alone Reichert, Defienne, and Fleischer (2018); Zhang et al. (2020) but were not implemented at the time of writing this paper.

The free-space-to-fiber coupling setups each consist of 2 mirrors to align the beamsplitter outputs through a 4.51 mm focal length asphere, which then focuses the beam into a multimode fiber (105 $\mu m$ core diameter, 0.22 NA) connected to the SPADs. For these measurements, the coincidence time-bin resolution is set to 8 ps and coincidences are summed within a 10 ns window. Fourth-order interferograms were measured for different bandwidths of entangled photon pairs. For the first experiment, bandpass filters centered at 810 nm with a 10 nm FWHM are mounted to the front of each of the fiber coupling units. Note that this filter is not at the 812nm degenerate wavelength and reduces the degree of entanglement. For the second experiment, the 810 nm bandpass filters are removed to couple the broadband, collinear SPDC into the SPADs. Finally, the oven is allowed to cool to the non-collinear but degenerate temperature of 150°C, and broadband two-photon interference was measured again. Note that due to the large bandwidth of the SPDC, the fiber coupling efficiency of the non-degenerate photons is also poor at less than 10$\%$ of total power. As discussed at the end of this section, optics development is still needed similar to using classical broadband ultrafast lasers.

Fig.9 shows the simulated fourth-order interferences using the produced bandwidth of the SPDC source and then the measured fourth-order interferences for two collinear bandwidths and a third non-collinear, full spectrum measurement. When the quarter-wave plate is set to 0°, the photons arriving at BS1 have the same polarization, and interfere coherently as correlated pairs. The interference pattern is robust against pump power fluctuations. When the quarter-wave plate is rotated to 45°, the reflected arm becomes perpendicularly polarized to the transmitted arm, the interference pattern disappears, and the coincidence counts follow the average pump power reading collected before and after the coincidence detection (SI Fig.8-9). The resulting coherence length of the total flux can be determined from the width of the interference peak. Table 2 summarizes the theoretical and measured coherence lengths.

For the measurement with 810-(10) nm bandpass filters mounted to the front of the fiber coupling units, shown in Fig.9(d), the coherence length is approximately 73 µm (or 245 fs). The simulated interference pattern as well as the coherence length of 60µm (200 fs), shown in Fig.9(a), agree fairly with the measurement given that the 10nm bandpass filter is offset from the degenerate wavelength of 812 nm. Note that the measured bandwidth after the fiber is roughly double what it actually is because the spectrometer slit had to be run wide to get sufficient flux for spectral analysis after the fiber coupling. The theoretical plot uses the correct bandwidth. Fig.9(e) shows the interference from collinear broadband SPDC flux without bandpass filters. The bandwidth is reduced and the SPDC spectrum is split (nondegenerate) because the phase matching temperature is maintained at the elevated collinear emission (Fig.2). From the width of the center peak, the coherence length is approximately 17.2 µm (or 57.4 fs). For reference, the simulated coherence length is approximately 13 µm (45 fs). Similarly, Fig.9(f) shows the lower temperature non-collinear broadband interference. Here, the full 200 nm bandwidth of the SPDC source is attempted to be used (expected temporal width 8fs). However the spectrum reaching the detector is limited to 125 nm because of collimation issues, upstream optics, and fiber coupling. The coherence length calculated from the measured width of the peak is 18.9 µm (or 62 fs). This indicates that the degree of entanglement has degraded due to poor fiber coupling at non-collinear temperatures and other upstream optics. For example, the spectrum after the multiple beamsplitters is not balanced (Fig.9(f), inset) as compared to the other cases. The increased temporal width is also indicative of the need for dispersion management upstream, just as is similar to a broadband ultrafast laser pulse of an expected  20 fs pulse. The same would be true for the chirped (>200 nm) SPDC source so an interferometer is not shown here.

The lack of fiber coupling efficiency in the broadband, non-collinear case is not simply one of input optics. Immediately following the chip, the spatial profile of the entangled photons is cone-like with a wavelength-dependent distribution (Fig.10 left). The measured angular deviation with wavelength matches the simulated entangled photon emission. Within a non-degenerate pair, the idler photon has a larger emission angle than the signal photon. The wavelength-dependent angular emission cone present a challenge in the implementation of free-space broadband entangled photon setups because the emission does not act like single point source. We attempted to collimate the emission cone using a variety of free-space optical configurations and found that a telephoto lens or an off-axis parabolic mirror (sometimes with an additional cylindrical lens for beam shaping) provides the best long range spatial profiles. After a propagation length of 45 cm into the far field, the entangled photon cone collapses (Fig.10 middle). This spatial profile collapse is most likely due to the interference of photon pairs created along the length of the crystal and scattering effects near the domain boundaries of the grating. The end result shows characteristics of somewhere between a Laguerre and a Hermite mode which would match the circular emission in a square profile created by the source. The beam profile will likely be improved by using adaptive optics. The far field effects could also be reduced by using a waveguide instead of a grating; however, nanophotonic implementations cannot handle the high input powers used in this paper and fiber coupling of the broadband emission spectrum from the waveguide is not well developed. Another obvious option is, instead of using fiber coupled SPADs for the detection, is to move to emICCD photon counting techniques that would not require focusing of the collapsed beam profile. However, this approach still needs exploration and requires an emICCD even more costly than SPADs. Further investigation into the creation and collimation of broadband entangled photons is critical for the implementation of short temporal length, high flux entangled photon spectroscopy.