Photonic qubits can be easily created in entangled states, communicated over many-kilometer distances, and efficiently measured [1]. Tremendous effort has been devoted to improving the success rates of quantum enhanced protocols and multimode solutions, often accompanied with active multiplexing, and are one of the most promising branches of this development [2–11], enabling both faster transfer and generation of photonic quantum states. In particular, systems harnessing several degrees of freedom (DoFs) offer superior performance [12–14], especially in selected protocols such as superdense coding [15], quantum teleportation [16], or complete Bell-state analysis [17]. Utilization of several DoFs brings a qualitatively new possibility to create hyperentangled states exhibiting nonclassical correlations in several DoFs simultaneously with a greatly expanded Hilbert space and informational capacity. Generation of entangled pairs of photons in spectral, temporal, transverse wave vector, spatial, and orbital angular momentum (OAM) and with multiple DoFs has been demonstrated. In particular, spontaneous parametric downconversion (SPDC) can be used to generate hyperentangled states in four DoFs simultaneously [18]. Nonetheless, experimental characterization of multidimensional states remains challenging. Single-pixel detectors such as superconducting nanowires offer excellent timing resolution [19], as well as spectral resolution when combined with dispersive elements such as chirped fiber gratings [20] or detector-integrated diffraction gratings [21]. Such setups provide a way to implement high-dimensional quantum communication [22] and temporal super-resolved imaging [23] or to observe quantum interference in time or frequency space [24,25]—a promising approach for quantum fingerprinting [26,27]. Single-photon-resolving cameras on the other hand naturally offer spatial or angular resolution, which can be exploited in super-resolution imaging [28–32], interferometry [33], characterization [34,35], or, similar to the previous case, observation of quantum interference effects such as in Hong–Ou–Mandel-type experiments [36]. Recently, however, the capability of cameras has been expanded by invoking a well-known mode conversion technique, in which Sun et al. simply observed spectral correlation with the help of a diffraction grating [37]. It is thus a promising approach to use a camera to observe many DoFs simultaneously.

Here, we experimentally demonstrate a measurement of full four-dimensional correlations between the transverse and spectral DoFs of a twin-photon state, generated in a noncollinear type I SPDC. An ultrafast single-photon-sensitive camera, yielding ${10^4}$ frames per second with $100 \times 1952$ pixels per frame, allows to quickly gather enormous statistic size while maintaining high resolution due to a large number of pixels. In conjunction with recent development in high-dimensional entanglement detection [38], our single-photon detection system would enable rapid characterization of such hyperentangled states. Furthermore, precise correlation measurements are vital to fully utilize the quantum advantage of entangled states, e.g., via non-local dispersion compensation recently demonstrated to improve quantum key distribution rates [39]. Higher-order correlation measurements also enable novel super-resolution imaging techniques [29,30] that particularly benefit from fast acquisition rates and high spatial resolution of employed single-photon detectors.

The employed camera prototype is an order of magnitude frame rate improvement over the off-the-shelf devices, necessary for a direct high-resolution measurement of four-dimensional correlations. Prior approaches involved scanning the wave vector space with point detectors and used time-of-flight spectrometers for spectral resolution, applicable at telecom wavelengths and requiring compressed sensing techniques [40]. We note that a measurement in two mutually unbiased bases characterizes entanglement of pure bipartite, high-dimensional states without a state tomography [38]. In this context, our method would require extension to a position–time measurement basis to fully measure quantum correlations.

To generate twin-photon states, we have employed a beta barium borate (BBO) nonlinear crystal in the type I SPDC process with noncollinear geometry, as depicted in Fig. 1. For the SPDC pumping, we first produce a second harmonic of 70 fs, 800 nm pulses from a Ti-sapphire laser (Spectra Physics Mai Tai, 80 MHz repetition rate) in a second, similar BBO crystal with length $L = 0.5\;{\rm mm}$. The 800 nm red pump is filtered out with dichroic mirrors and a bandpass filter (400 nm, 10 nm bandwidth). A blue ${\lambda _p} = 400\;{\rm nm}$ pump with an average power of 70 mW is focused in an $L = 2\;{\rm mm}$ BBO with a Gaussian beam width of ${w_0} = 70\;\unicode{x00B5}{\rm m}$ and finally filtered out with a dichroic mirror. The SPDC emission is far-field imaged with a lens (${f_1} = 60\;{\rm mm}$) on an adjustable rectangular slit that selects a range of wave vectors $[- \Delta {k_y}/2,\Delta {k_y}/2]$ around ${k_y} = 0$. A second lens (${f_2} = 300\;{\rm mm}$) images the BBO onto a ruled diffraction grating (${N_{{\rm lines}}} = 1200\;{{\rm lines}/{\rm mm}}$, resolution of $\delta \lambda = 2{\lambda _p}/{N_{{\rm lines}}} = 0.66\;{\rm nm}$) mounted vertically in the Littrow configuration and at a small horizontal angle. The grating adds a wavelength-dependent wave vector in the $y$ direction. A third lens (${f_3} = 100\;{\rm mm})$ far-field images the grating onto a single-photon camera. The effective focal size of the setup from BBO to the camera is ${f_{{\rm eff}}} = 30\;{\rm mm}$. A finite slit width $\Delta {k_y}$ corresponds to a resolution comparable to that of the diffraction grating $\delta \lambda$ when the image of the slit and a spectral point of $\delta \lambda$ size are compared in the camera image.

 

Fig. 1. (a) Annular ring of twin-photon emission in the far field of a BBO crystal mediating noncollinear type-I SPDC. Histogram of single photon positions is registered over $2 \times {10^5}$ camera frames. (b) Twin-photon emission in the far field after passing through the rectangular slit selecting ${k_y} \approx 0$ and after diffraction on the grating acting as wavelength-dependent wave vector shift ${k_y} \to {k_y} + {K_{{\rm grating}}}(\lambda)$. Distinct fragments correspond to signal and idler photons. Camera frame coordinates correspond to transverse wave vectors and wavelength of photons. (c), (d) Camera regions corresponding to (c) signal and (d) idler photons. The division into sub-regions with limited wave vectors $\{k_x^{(s)}\} ,\{k_x^{(i)}\}$ and wavelengths $\{{\lambda _s}\} ,\{{\lambda _i}\}$ is depicted with rectangles. For clarity, only (c) wave vector sub-regions or (d) wavelength sub-regions are shown. Signal and idler regions are divided equally. (e) Experimental setup for fast transverse–spectral correlation imaging of twin photons.

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While BBO is cut for type I SPDC, by slightly adjusting the angle of the crystal axis with respect to the pump beam, we can alter the diameter and width of the far-field annular SPDC emission ring. Compared with theory, the crystal-axis–$z$-axis angle (including cutting 29.2°) is ${31.95^ \circ} \pm {0.025^ \circ}$. We estimate the overall efficiency of our setup at 4% roughly corresponding to the twin-photon generation (50%), diffraction grating (50%), and detection (20%) efficiencies combined.

The single-photon camera consists of a two-stage image intensifier (Hamamatsu V7090-D) with high-voltage supply (Photek FP630) and a 10 kHz gating module (Photek GM10-50B) connected with a custom-built I-sCMOS camera based on a fast CMOS sensor (LUX2100, pixel pitch 10 µm). Communication with the camera sensor and low-level image processing are performed with a programmable logic field-programmable gate array (FPGA) module (Xilinx Zynq-7020). The image processing consists of background subtraction and single-photon localization. The FPGA module is bundled with an ARM-family processor, providing Ethernet data transfer to the PC. The CMOS sensor is set for the lowest time-dependent noise at the cost of a lower dynamic range. With a faster gating module, the camera could operate at ${10^5}$ frames per second with a frame of $10 \times 1952\;{\rm px}$. Single-photon sensitivity is achieved by operating the image intensifier (II) in the Geiger mode (on–off) [41] (see Supplement 1).

The gating time is 1.2 µs with an average of ${\bar n_{{\rm tot}}} = 0.12$ photons per frame. Signal and idler photons are observed in $40 \times 70\;{\rm px}$ regions, each corresponding to $416\;{{\rm rad}/{\rm mm}} \times 5.1\;{\rm nm}$ [with $5.95\;{\rm rad}/({\rm mm} \times {\rm px})$ and $0.127\;{\rm nm}/{\rm px}$]. The Gaussian mode size ${\sigma _{k {\text -} {\rm mode}}}$ was predicted to be $6.2\;{{\rm rad}/{\rm mm}}$ and measured as $7.1 \pm 0.3\;{\rm rad}/{\rm mm}$. The spectral mode size was measured to be ${\sigma _{\lambda {\text -} {\rm mode}}} = 4.20 \pm 0.06\;{\rm nm}$. We define the mode sizes as the Gaussian widths of a two-dimensional second-order photon number correlation in the sum coordinates $(k_x^{(s)} + k_x^{(i)},{\lambda _s} + {\lambda _i})$ (see Supplement 1). Using the theoretical prediction of the joint wave function, we numerically get $M = 1/\sum\nolimits_{j = 0}^\infty \lambda _j^2 \approx 4.7$ accessible entangled modes, where ${\lambda _j}$ are the Schmidt coefficients (see Supplement 1). Note that for our considerations regarding the mode size and the number of modes, we implicitly assumed a Gaussian two-photon wave function (leading to Gaussian second-order correlations), as well as purity of the generated state.

While with a spectrally broad, focused pump beam and a short crystal, the SPDC emission is highly multimode in the spectral and transverse DoFs, we begin with a single pair of signal ($s$)-idler ($i$) modes. A two-mode squeezed state $|\psi \rangle = \sum\nolimits_{j = 0} {\chi ^{j/2}}|j{\rangle _s}|j{\rangle _i}$, generated in SPDC, can be approximated to the first order in $\sqrt \chi$ as a pair of photons $|1{\rangle _s}|1{\rangle _i}$. Consider the joint wave function in transverse-wave-vector and spectral coordinates:

 

Fig. 2. Correlation in joint (a) transverse wave vectors and (b) spectrum of signal–idler photon pairs. (a) Color map or (b) left column represents the experimental data (photon number covariance summed over (a) spectral or (b) wave vector regions; see main text). (a) White contours or (b) right column represents theoretical prediction $|{\Psi _{\{{\lambda _s}\} ,\{{\lambda _i}\}}}(k_x^{(s)},k_x^{(i)}{)|^2}$ of a two-photon wave function modulus squared, (a) summed over spectral ranges $\{{\lambda _s}\} ,\{{\lambda _i}\}$ or (b) transverse-wave-vector ranges $\{k_x^{(s)}\} ,\{k_x^{(i)}\}$ and normalized to a unity maximum.

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We have demonstrated a capability to measure four-dimensional transverse-wave-vector–spectral correlations between pairs of photons generated in noncollinear SPDC. Due to a custom single-photon camera with very fast acquisition rates (an order of magnitude improvement), we were able to gather statistics of ${10^9}$ camera frames (experiment repetitions) in roughly one day. Large statistics enabled faithful reconstruction of a bi-photon wave function in spectral and transverse-wave-vector coordinates. For this demonstration, we selected a single component of the transverse wave vector that is far-field imaged (mapped) onto positions on the camera frame; similarly, the spectral part is mapped onto positions with a diffraction grating. Importantly, our system is inherently multimode and can be adapted for measurements in different mode bases, e.g., OAM and for different DoFs.