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Abstract
Single-photon avalanche diode arrays can provide both the spatial and temporal information of each detected photon. We present here the characterization of spatially entangled photons with a 32 × 32 pixel sensor, specifically designed for quantum imaging applications. The sensor is time-tagging each detection event at pixel level with sub-nanosecond accuracy within frames of 50 ns. The spatial correlations between any number of detections in a defined temporal window can thus be directly extracted from the data.The space-momentum entanglement of photon pairs is demonstrated by violating an EPR-type inequality directly from the measured near-field correlations and far-field anti-correlations.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum states of light are a fundamental tool to implement quantum information processing protocols and quantum metrology methods in the optical domain, such as quantum imaging [1]. They can be described either by discrete variables, the photons, or by continuous fields. Correspondingly measurements of light can be mainly split into two classes: measurements of fields, by homodyne detection for example, and intensity measurements, that correspond fundamentally to photon counting. These are the tools of choice to characterize discrete quantum optical states; for example to measure correlations in quantum states with few photons. An ideal detector should be able to detect the position and time of arrival of all impinging photons. Such universal detectors do not yet exist, but there are various types of single photon sensitive detectors that are each optimized with respect to specific characteristics. Most of the experiments based on correlations between photons are realized with single photon detectors with high temporal resolution but no spatial resolution [2], including photomultipliers, single photon avalanche diodes (SPAD) or superconducting detectors. On the other side, imaging experiments are usually implemented, even at the single photon level, with low noise cameras as EMCCD or scientific CMOS cameras [3,4], that are able to localize single photons among many pixels, but have low temporal resolution. Spatial entanglement in photon pairs has been demonstrated with scanning single pixel detectors [5], EMCCD [6] and commercial gated single photon avalanche array sensors [7]. Correlations between photons from a pair, one detected by a single pixel detector and the other one by an ICCD camera, have also been measured [8,9].
A way to combine high temporal and spatial resolution is fast gating of imaging sensors. However, while gating gives access to short times, the full spatio-temporal correlations of light across a wide temporal range cannot be measured. In addition, gating is well suited for pulsed operations, but a continuous source of correlated photons would ideally be measured with a non-synchronous detector. This is easily implemented with single pixel detectors, that are always ready to trigger, up to their dead-time. However when operating many detectors in parallel, as in detector arrays, the readout mechanism usually requires a frame-based operation. In that case, one of the criterion to be optimized is the sensor duty cycle defined by the product of the frame repetition rate of the detector with the duration of each frame. For gated detectors, the requirement to have short gates in order to achieve a good temporal discrimination of correlated photons contradicts the need of long frames to improve the photon collection efficiency. The solution is the ability to time-stamp with high temporal resolution each detection event within a long frame. Combining single photon detectors and external time-stamping electronics can be realized only for a small number of detectors [10]. This cannot, however, be extended to imaging with thousands of pixels without the integration of light sensitive devices with digital electronics, as offered by CMOS technology. Event based high-speed optical cameras have the ability to time-stamp many detection events at a very high rate. Single photon sensitivity can be achieved in combination with an image intensifier, however at the expense of the temporal and spatial resolutions. Correlation between polarization-entangled photon pairs have been demonstrated with such devices [11]. In this work, we demonstrate the capability of a recently developed SPAD array sensor [12–14] to characterize spatial entanglement from spontaneous parametric down-conversion (SPDC) by measuring both, near- and far-field correlations [5,6]. We demonstrate entanglement of the state by testing an Einstein-Podolsky-Rosen (EPR)-type inequality. The direct access to each individual coincidence event allows to estimate the violation of the inequality directly from the measured joint-probabilities, without a priori assumption on the quantum state. In addition, it can be estimated from fitted data, assuming a Gaussian model for the SPDC emission.
In section 2. we recall the quantum state of the photon pairs emitted by SPDC and the expected joint-probability distribution that have to be measured in order to verify their spatial entanglement. Section 3. describes the experimental setup and the processing of the data acquired with the SPAD array, in particular the corrections of the accidental events and crosstalk. The results are presented in section 4., with the different estimations of the violation of the inequality. Finally, in section 5., we introduce figures of merit for the comparison of various sensor technologies in the case of the detection of $n$-photon correlations in continuous sources of light.
2. Theory of spatial entanglement in SPDC
SPDC is nowadays a common source of entangled photon pairs, where a pump photon of frequency $\omega _p$ is annihilated inside a non-linear crystal (NLC) with a non-vanishing second-order susceptibility $\chi ^{(2)}$, and two photons with frequencies $\omega _1$ and $\omega _2$ and wave-vectors $\mathbf {k}_1$ and $\mathbf {k}_2$ (historically called signal and idler) are simultaneously created [15]. Here we consider the case of type-0 quasi-phase-matching in a periodically poled crystal of length $L$ with poling period $G$ and refractive index $n(\omega )$, where all fields have the same polarization [16].
Under the approximation of a monochromatic pump with a frequency $\omega _{cp}$, the envelope of the pump field is given by
The position $\boldsymbol {\rho }_i=(x_i,y_i)$ and momentum $\hbar \boldsymbol {q}_i=(\hbar k_{x_i}, \hbar k_{y_i})$ operators do not commute and therefore a criterion for EPR-type correlations between two systems $1$ and $2$ (that correspond in our case to the signal and idler photons) can be established [18]. In the SPDC emission, the $x$ and $y$ components are decoupled. We thus consider the one dimensional case with $x_i$ and $p_i=\hbar k_{x_i}$. The results of correlation measurements are therefore described by the joint-probabilities $\mathcal {P}(\boldsymbol {\rho }_1,\boldsymbol {\rho }_2)$ and $\mathcal {P}(\boldsymbol {q}_1,\boldsymbol {q}_2)$, or $\mathcal {P}(x_1,x_2)$ and $\mathcal {P}(p_1,p_2)$ in the 1D case. The minimal inferred variance of $(x_1,x_2)$ is defined by
where $\Delta ^{2}(x_1|x_2)$ is the variance of the conditional probability $\mathcal {P}(x_1|x_2)$ and $\mathcal {P}(x_2)$ is the marginal probability of system $2$. Analogue definitions apply for the variables $(p_1,p_2)$. According to [18], the fulfillment of the following inequality on the product of minimum inferred variances indicates EPR-type correlations, or entanglement or expressed as a dimensionless inequality Experimentally, the joint-probabilities are derived from correlation measurements that correspond to second-order coherence functions of the electric field. They are given at transverse positions $\boldsymbol {\rho }_1$ and $\boldsymbol {\rho }_2$ and times $t_1$ and $t_2$ by2.1 Second-order near-field correlations
When imaging the crystal with a magnification $M$, the correlation function at the image plane using Fourier optics can be expressed as
2.2 Second-order far-field correlations
In a far-field imaging setup at wavelength $\lambda =2\pi / k$, using a lens of focal length $f$, the relation between the transverse position at the imaging plane $\boldsymbol {\rho }$ and the transverse momentum $\mathbf {q}$ at the object is
Hence, the second-order correlation function in the far-field reads3. Experimental evaluation of correlations
The joint-probability distribution that leads to the evaluation of Eq. (10) has to be estimated from the raw correlations acquired with the SPAD array sensor. Ideally it should be estimated directly from the non-processed data, in order to achieve a non-conditional violation of the inequality. However due to the sensor imperfections, such as reduced efficiency, dark counts and crosstalk between pixels, some assumptions have to be introduced to process and correct the raw data. In a second step, the corrected data can be either directly used to numerically evaluate Eq. (10) or, by introducing additional assumptions, fitted with a model of the expected correlations as it is done for instance in [6].
3.1 Experimental setup
The experimental setup is shown in Fig. 1. A continuous wave (CW) laser (Toptica DL PRO HP 405) with a maximal power output of 30 mW at a wavelength of 405 nm and a spectral bandwidth of 80 MHz serves as the pump. It is slightly focused onto the crystal’s center plane $\Sigma _0$ by a two-lens system ($f_1=f_2=200$ mm). The beam waist of the pump at this plane is $w_{0x}=250$ µm in the $x$ direction and and $w_{0y}=300$ µm in the $y$ direction. Quarter and half-wave plates are used to achieve the desired horizontal polarization in the crystal. A $1\times 2\times 12$ mm3 periodically poled KTiOPO$_4$ (PPKTP) NLC with poling period $G_0=3.51043$ µm is embedded into a temperature controlled oven and provides the source for the down-converted photons. The oven is maintained at a temperature of 26.0 °C for an almost collinear phase-matching, that maximizes the photon flux onto the sensor in the far field. The down-converted photons are separated from the pump by a bandpass filter (BP) centered at 810 nm with FWHM 10 nm. About 2 nW of entangled photons are produced, that corresponds to about $4\times 10^{9}$ photon pairs, or 200 photon per frames of 50 ns. Given the source parameters, waists of the pump beam and the length of the SPDC crystal, we can estimate the minimal inferred standard deviations to be
Two lenses ($f_3=f_4=50$ mm) form a $4f$ imaging system such that the electric field at the plane $\Sigma _{NF}$ is an exact replica of the field in the crystal at $\Sigma _{0}$. Either the near- or far-field of this plane is then imaged onto the sensor, depending on the selected lens. For near-field, a lens $f_{NF}=25.4$ mm images the plane $\Sigma _{NF}$ onto the sensor with a magnification factor $M=9$. For far-field, a lens $f_{FF}=150$ mm images the far-field such that the area covered by the sensor corresponds in $k$-space to $\pm 36.2$ mm−1, as $q = x \frac {k}{f_{FF}}$.
Fig. 1. Experimental setup for spatial correlation measurements. A CW laser at 405 nm is slightly focused by a telescope ($f_1$ and $f_2$) and pumps the crystal. The bandpass filter (BP) only transmits the down-converted photons at a central frequency of 810 nm. The 4-f lens configuration ($f_3$ and $f_4$) one to one images the crystal center $\Sigma _0$ onto the near field plane $\Sigma _{NF}$. Further imaging onto the detector array occurs by either a one lens configuration $f_{NF}$ for the near field or a Fourier lens $f_{FF}$ for the far field.
The coincidence detection is performed with a single photon avalanche detector (SPAD) array recently developed [12,13]. It is a fully digital $32\times 32$ pixel sensor array based on CMOS technology. The total sensitive area is $1.4\times 1.4 \textrm{mm}^{2}$ and the pixel pitch $\Delta L$ is $44.64$ µm with an overall fill-factor of $19.48\;{\%}$. A time-to-digital converter (TDC) integrated in each pixel allows for pixel-wise time stamping the first detection event within a frame. The time resolution (time-bin length) of the TDC is $210$ ps with a depth of 8 bits, corresponding to an observation time of 255 time-bins or about $50$ ns. The mean frame observation rate is about $850$ kHz, with peaks up to $1$ MHz, which corresponds to a duty cycle of $4.5\;{\%}$. The total expected photon detection efficiency, when the sensor is active, is $5\;{\%}$ at $400$ nm and $0.8\;{\%}$ at $810$ nm. The dark count rate per pixel is below 1 kHz at room temperature on average, that correspond to $5\times 10^{-5}$ events/frame. The raw data from the sensor’s FPGA are transfered to a computer through a USB 3.0 interface and are then post-processed.
3.2 Data processing
The coordinate and time of all detection events within one frame, together with the frame identification number, constitute the acquired raw data. In the following we associate the spatial coordinate $\boldsymbol {\rho }_i=(x_i,y_i)$ with the pixel coordinate $\mathbf {p}_{i}=(\mathrm {p}_{x_{i}},\mathrm {p}_{y_{i}})$ using $\boldsymbol {\rho }_{i}=\Delta L\mathbf {p}_{i}$ . An estimation of the raw correlation $G_{r a w}^{(2)}(\boldsymbol {\rho }_1,\boldsymbol {\rho }_2)$ between a pair of pixels, or equivalently of $G_{r a w}^{(2)}(\mathbf {p}_1,\mathbf {p}_2)$, is obtained by counting all coincidence events between those pixels that occur within a defined temporal window. The histogram of the absolute time differences between all pairs of detection events across the sensor is shown on Fig. 2. While the temporal jitter of each pixel is of the order of 200 ps, in practice temporal shifts across the sensor widen the coincidence peak. In principle, a pixel-wise temporal calibration would allow to narrow the window [13]. Here we define the coincidence window to be $\pm 10$ TDC steps ($\sim \pm 2$ ns) in order to catch all coincidence events. The obtained histogram is further normalized to a number of coincidences per frame (or alternatively per million of frame, MFrame) [17]. Hence for the data of Fig. 2, $5\times 10^{-3}$ true coincidences events per frame are detected within the coincidence window for $3.7\times 10^{-2}$ accidental coincidences. The processing of those raw data includes first the subtraction of accidental coincidences, and then a correction step for crosstalk.
Fig. 2. The temporal second-order correlation $G^{(2)}(|\Delta t|)$ accumulated over all pixels. The raw detection event counts are shown and after removal of uncorrelated accidentals.
Before describing in detail the processing steps, we introduce various representations of the second-order correlation. In a transverse plane it is a function of four variables ${G^{(2)}}(\mathbf {p}_1,\mathbf {p}_2)={G^{(2)}}(\mathrm {p}_{x_1},\mathrm {p}_{y_1}, \mathrm {p}_{x_2}, \mathrm {p}_{y_2})$. In order to be able to plot a 2D representation of the full correlations, we introduce the pixel indices numerating each pixel from $1\dots 1024$
Fig. 3 shows the full correlation matrix between every pair of pixels after subtraction of accidental events (see section 3.2.1) in the case of far-field measurement. The photon pairs exhibit anti-correlation in their detection positions, that reflects into the $\approx 8$ anti-diagonal lines. The nearest neighbors correlations (separated by pixel indexes $\pm 1$ and $\pm 32$) are affected from crosstalk, forming the four diagonal lines. The exact diagonal cancels, as the sensor cannot measure self-correlations on one pixel.Fig. 3. Full second-order correlations between every pixel pair in the far-field with corrected accidentals. Each pixel is addressed with a linear pixel index $\tilde {\mathrm {p}}_{i}=1,\dots ,1024$. The inset shows a close up of the crosstalk correlations appearing along the diagonals.
In order to apply the inequality (10) for only one dimension, the correlation functions are projected onto either the $x$ or $y$ dimension by
Alternatively, one can use the centroid and difference coordinates to project the 4D correlation function on 2D planes defined by fixed values of $\boldsymbol {\rho }_1+\boldsymbol {\rho }_2$ or $\boldsymbol {\rho }_1-\boldsymbol {\rho }_2$, respectively. The projected correlation functions read3.2.1 Removing accidentals
Accidental coincidences are coincidence events that occur in the defined coincidence window but are neither temporally nor spatially correlated. They stem from detections triggered by background light, dark counts, and mostly from SPDC photons that do not belong to the same photon pair. The total measured raw correlation signal is thus given by the sum of the accidental coincidences and the true temporal coincidences (as due to photons from a pair)
Fig. 4. Second-order far-field correlations. Measured raw data (a), after removing accidentals (b) and after crosstalk correction (c)
3.2.2 Crosstalk corrections
A single detection event, triggered by a dark count or a photon, can trigger nearby pixels. This leads to undesirable detection events that are an artifact intrinsic to the detector. The physical process behind optical crosstalk are photons, created by the charge avalanche of the first triggering SPAD. These photons may reach and trigger neighboring pixels in a very short time-scale. Crosstalk is especially undesired in the near-field correlations, as it overlays the real signal. The crosstalk probability should be estimated for every pixel pair. However, as a first approximation, we assume it to only depend on the distance between two pixels. Thus an average crosstalk value over all pixels can be extracted from the projected correlation function in the far-field $G_{FF}^{(2)}(\Delta {x},\Delta {y})$, as in that case only anti-correlations are expected. Hence, the mean probability that a pixel at distance $(\Delta {x},\Delta {y})$ is triggered from crosstalk is given by
Fig. 5. (a) Correlation peak due to crosstalk of the second-order correlation measurement in the far-field. (b) Crosstalk probability extracted from the correlation peak. The pixel at $(\Delta x, \Delta y)=0$ is the emitter of the crosstalk.
4. Measurement results
The full correlation map for far-field measurements is shown on Fig. 4. The same data can then be represented in the other coordinates previously introduced. Figure 6 shows the anti-correlation (a) and correlation (b) peaks in the far-field after crosstalk removal. As far-field measurements map the momentum space onto the sensor, the plots are labeled with the coordinates $\mathbf {q}_+$ and $\mathbf {q}_-$, but still in units of pixel. We clearly observe the presence of an anti-correlation peak and the almost disappearance of the correlation peak.
Fig. 6. Anti-correlation (a) and correlation (b) peaks of the second-order far-field correlation after removing crosstalk.
The same processing can be applied for the near-field measurements. The full correlation matrix is shown in Fig. 7 without (a) and with (b) crosstalk correction. We observe that the crosstalk along the diagonal overlays the actual signal, even after correction. As the crosstalk contribution is of the same order as the signal, its correction is very sensitive to the estimation of the crosstalk probabilities. A calibration at the pixel level would allow for a full correction in future experiments. This is why in the present work, the next neighbor correlations will not be taken into account for the quantitative estimation of the correlations. The partial suppression of the crosstalk can also be seen in the correlation peaks of Fig. 7(c) and (d), where the bright central peak is due to crosstalk, while the broad peak indicates correlation between photons. The magnification factor of the optical system has been chosen in order to obtain a correlation peak broader than the crosstalk.
Fig. 7. Second-order near-field correlation with accidentals removed, (a) full correlation matrix and (c) correlation peak. After crosstalk correction, (b) full correlation matrix and correlation peak (d).
Figure 8 shows the second-order correlations projected onto the $x$ and $y$ coordinates after removing crosstalk in the far- (a) and (c) and near-field (b) and (d). The correlations affected by crosstalk are set to zero on those plots. The anti-correlations in momentum and correlations in position space are clearly visible. Those data are further used to evaluate the EPR-type inequality.
Fig. 8. Second-order correlations in the far-field (a,c) and near-field (b, d) for the $x$ (a,b) and $y$ (c,d) coordinates.
While the identification of EPR-type correlations should be ideally done on the raw correlations ${G_{raw}^{(2)}}(\mathbf {p}_1,\mathbf {p}_2)$, usually some additional assumptions are applied to correct for the imperfections of the sensor. Because the detector sensitivity is limited at the SPDC photons wavelength, the accidental and crosstalk are not negligible and have to be corrected as described. In a second step, the corrected data from the second-order correlations in the near-field ${G_{NF,corr}^{(2)}}(\mathbf {p}_1,\mathbf {p}_2)$ and far-field ${G_{FF, corr}^{(2)}}(\mathbf {p}_1,\mathbf {p}_2)$ can be either directly numerically evaluated, or, introducing further assumption, fitted with a model of the SPDC emission. The results of the various evaluations are summarized in Table 1. The uncertainties are derived from the confidence intervals of the fits from which probability distributions were obtained.
Table 1. Results of the evaluation of the conditional variances
4.1 Numerical evaluation of the EPR inequality
We present in detail the numerical evaluation for the $x$ coordinates, i.e. ${G^{(2)}}(x_1,x_2)$ and ${G^{(2)}}(q_{x_1},q_{x_2})$. The treatment for the $y$ coordinate is identical.
The conditional and unconditional probability density functions are estimated from the corrected experimental data
Fig. 9. Example of measured conditional probabilities for the $x$ variable in the far-field $\mathcal {P}(q_{x_1}|q_{x_2})$ (a) and near-field $\mathcal {P}(x_{1}|x_{2})$ (b). The variance is either extracted numerically or using a Gaussian fit (continuous red line). The pixels affected from crosstalk are skipped (indicated by the red asterisks)
The EPR-type correlations are identified by violating the inequality
The same procedure is applied for the $y$ coordinate and the result is denoted as4.2 1D Gaussian fitting
An alternative approach to direct numerical evaluation is to constrain the data into a set of "reasonable" data, compatible with the expected characteristics of the light source. The shape of the JMA from SPDC emission can be, in some conditions, approximated by Gaussian functions. Therefore $\mathcal {P}(q_{x_2})$, $\mathcal {P}(x_{2})$, $\mathcal {P}(q_{x_1}|q_{x_2})$ and $\mathcal {P}(x_{1}|x_{2})$ can be estimated by fitting the data with Gaussian functions. Figures 9 and 10 show examples of fitted data. One can then extract the variances directly from these fitted probability density functions and calculate the EPR-criterion according to the definition of Eqs. (8) and (10). The results are shown in the second line of Table 1 and are in agreement with the expected values derived from the experimental parameters.
Fig. 10. Example of measured unconditional probabilities for the $x$ variable in the far-field (a) and near-field (b).
4.3 2D Gaussian fitting
Instead of fitting multiple Gaussians by slicing the data sets ${G^{(2)}}$, one can also directly fit ${G_{NF,corr}^{(2)}}(x_1,x_2)$ and ${G_{FF,corr}^{(2)}}(q_{x_1},q_{x_2})$ with 2D Gaussian functions (and similarly for $y$), as shown on Fig. 11. It is to be observed that we only assume the correlations to be well-approximated by 2D Gaussians with variances $\Delta ^{2}_{x+,x-}$ and $\Delta ^{2}_{qy+,qx-}$ as fitting parameters, but we don’t impose the quantum state to follow the double Gaussian model [20]. The conditional variances are then related to the fitting parameters by [21]
Fig. 11. The second-order correlations in the far-field (a) and near-field (b) are directly fitted with 2D Gaussian distributions and the variances are extracted from the fits.
4.4 (Anti-) Correlation peaks
Finally, a quick way to estimate the strength of the (anti-) correlations is to consider the projections onto $x$ and $y$ of the (anti-) correlation peaks ${G_{FF}^{(2)}}(\mathbf {q}_+)$ and ${G_{NF}^{(2)}}(\boldsymbol {\rho }_-)$ that are shown on Fig. 12. The width of the fitted Gaussian are respectively $\sigma _{x+},\sigma _{y+}$ and $\sigma _{qx+},\sigma _{qy+}$. They can be related to the conditional variances under the assumption that the correlations, respectively anti-correlations, appear only along the $-$ and $+$ coordinates. Equivalently, the correlations in the near-field are assumed to be expanded very far along the $+$ coordinate (i.e. $\sigma _{x+},\sigma _{y+} \to \infty$) and similarly in the far-field for the $-$ coordinate ($\sigma _{qx+},\sigma _{qy+} \to \infty$). Graphically in Fig. 8, the diagonals (b and d) and anti-diagonals (b and d) would thus extend to infinity. The conditional variances are then related to the width of the peaks ${G^{(2)}}(x_-)$ and ${G^{(2)}}(q_{x+})$ by
and similarly for the $y$ coordinate.Fig. 12. Anti-correlations peaks (a) of the second-order correlations in the far-field and correlation peaks (b) from near-field measurements alon $x$ and $y$ directions.
5. Sensors comparison
The previous results show the capacity of SPAD array sensors to detect spatio-temporal coincidences between correlated photon pairs. Hence they provide an alternative technology to ICCD and EMCCD cameras. Comparing different technologies is strongly dependent on the targeted applications. This is why we introduce here figures of merit for selected applications, in order to infer the potential of various sensors. For this purpose we estimate the relative effective efficiencies for detecting coincidences, including PDE and duty cycle, for the following detectors: the SPAD array used in this work, a commercial gated SPAD array (MPD SPC$^{3}$), a state-of-the-art ICCD (Andor iStar sCMOS 18$^{*}$ -A3) in full frame or binning modes and high efficiency scanning superconductor single photon detectors (Single Quantum EOS).
We specifically consider the detection of spatio-temporal correlations between $n$ photons emitted from a continuous source of light and we assume a regime of sparse photon detections. The first relevant criteria are the spatial and temporal resolutions. The spatial resolution is determined by the number of pixel $N_p$. The size of the pixels, or pixel pitch $\Delta p$, is here of secondary importance, as the pixel can be fit to arbitrary dimensions by optical magnification (within the restrictions of the optical setup). The temporal resolution is determined by the smallest accessible time-bin $\Delta t$, usually given by the detector jitter or the gating time. The number of accessible time-bins is $N_t$. In contrary to the spatial resolution, the temporal resolution cannot be changed by a stationary linear optical setup. For single (or few pixel detectors), spatial scanning is required, while for gated detectors, temporal scanning is needed. In general, sensors with both spatial and temporal resolution have access to a pixelized volume of space-time of size $N_{st}=N_p \times N_t$, the spatio-temporal resolution. The measurements are repeated at a frame rate $\nu$ and each frame gives access to the volume $N_{st}$. An important additional parameter is the photon detection efficiency $PDE$ including fill factor. The PDE is wavelength dependent, but we only consider the peak PDE across the spectral range. In order to concentrate on the fundamental limitations of different technologies, other characteristics, as noise or crosstalk will not be taken into account in the following estimations. Also we assume a negligible time for the scanning, in order to investigate the intrinsic limitation of the sensors only.
The accessible volume $N_{st}$ is a subset of the total space-time volume of interest that has to be measured, and thus both, spatial and temporal scanning may be required in general. The acquisition time is hence inversely proportional to $N_{st}$ or to an equivalent quantity for high-order correlations.
5.1 Single photon detection
At first we address the problem of building the spatio-temporal histogram of single photon detection $N(\vec {r},t)$ that corresponds to the intensity distribution $I(\vec {r},t)$, or in other terms the first order correlation function of the field $\langle{\hat {E}^{-}(\vec {r},t)\hat {E}^{+}(\vec {r},t)}\rangle$. This is the case for wide-field LIDAR or fluorescence lifetime imaging microscopy for instance.
In order to compare various detectors we estimate the time needed to scan through a targeted space-time volume $N^{target}_{st}=N^{target}_pN_t^{target}$, defined by a spatial resolution of $N^{target}_p$ and a temporal resolution, in the nanosecond range, of $N_t^{target}$ time-bins. The time required to scan this volume is hence proportional to $N^{target}_{st}/N_{st}/\nu$, or furthermore, by taking into account the PDE to $N^{target}_{st}/N_{st}/\nu /PDE$. Thus, two figures of merit are relevant for spatio-temporal single photon detection, a quantity proportional to the acquisition time
that should be minimal and $\Delta t$ that defines the smallest detectable time interval.5.2 $n$ photons in coincidence
We extend the previous reasoning to the case where $n$ photons are expected to reach the sensor simultaneously (within a time window smaller than the sensor’s temporal resolution). The relevant events are therefore given by $n$ spatial positions and one time coordinate only. This reflects a $n$-order correlation function $\langle{\hat {E}^{-}(\vec {r_1},t)\cdots \hat {E}^{-}(\vec {r_n},t)\hat {E}^{+}(\vec {r_n},t)\cdots \hat {E}^{+}(\vec {r_1},t)}\rangle$. Here, the targeted volume is given by $N^{target}_{st}=(N^{target}_p)^{n} N_t^{target}$ and the accessible volume is $N_{st}=(N_p)^{n} N_t$. Hence the figures of merit are
and $\Delta t$.5.3 $n$ photons at arbitrary times
In the most general case, any possible $n$-fold coincidence between any position and time, should be registered. This is formaly given by the full $n$th order correlation function
Table 2 shows the characteristics of the selected sensors together with the estimated figures of merit for one-photon detection ($M_1$), two-photon detection at arbitrary times ($M_2$), two photons in coincidence ($M^{*}_2$) and three photons at arbitrary times ($M_3$). The values are normalized to the values for the SPAD used in the present work that are $M^{0}_1=5.7\times 10^{-11}, M^{0}_2=2.8\times 10^{-15}, M^{0*}_2=7.3\times 10^{-13}$ and $M_3^0=1.4\times 10^{-19}$. Hence they are proportional to the ratio of acquisition times and therefore smaller is better.
Table 2. Figures of merit of selected single photon sensors.
We can observe that SPAD arrays and single pixel detector trade the high number of pixels of ICCD for high acquisition rates. In all the cases, the present SPAD array is much faster than a gated SPAD, thanks to its higher speed and high number of time-bins, while offering a higher temporal resolution. It also competes quite favorably with scanning single photon detectors, that are of advantage only when very high temporal resolution is required. The comparison with ICCD is less clear. For high spatial resolution, the high number of pixels of the ICCD gives it a clear advantage to measure higher order correlation measurements. For moderate spatial resolution however, the present sensor competes favorably, thanks to its high space-time volume $N_{st}$.
6. Conclusion
In this work we exploit the capacity of a recently developed 32x32 pixel CMOS SPAD array sensor to time-tag each individual photon with high temporal resolution. This allows us to measure the emission of spatially entangled photon pairs from SPDC in the near- and far-field. Anti-correlations in the far-field and correlations in the near-field are observed. From the experimentally measured probability distributions, the violation of an EPR-type inequality is demonstrated. Interestingly the inequality is violated even without fitting the data with a specific model. When assuming Gaussian shape for the probability distributions, the obtained value for the variances are in good agreement with the expected ones derived from the parameters of the photon source. Those results demonstrate the ability of CMOS SPAD arrays to effectively measure simultaneously spatial and temporal correlations between photons. Estimated figures of merit show that SPAD arrays compare favorably with other technologies. Future developments will aim at increasing the number of pixels on the one side, and at achieving higher acquisition speed, for instance by event-based technology. SPAD arrays then could become an effective tool for practical implementations of quantum imaging schemes.
Funding
Horizon 2020 Framework Programme (project SUPERTWIN 686731 and the Research and Innovation Programme 852045).
Disclosures
The authors declare no conflicts of interest.
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