## Quantum mechanics allows a new kind of imaging.

The goal of quantum imaging schemes is harnessing the properties of quantum states to reconstruct an image. Unlike classical imaging, the quantum counterpart requires the means to (i) generate, (ii) manipulate, and (iii) measure individual quantum states.

In state-of-the-art research, the illumination of choice is a twin beam of photon pairs. This can now ubiquitously be produced through a process called spontaneous parametric down-conversion (SPDC).

### Momentum conservation

In this process, a bright pump beam is shone upon a nonlinear crystal where, with a given probability, a pump photon is converted into a pair of photons. The key aspect of this conversion is that it conserves momentum; the momentum of the down-converted photons adds to the momentum of the pump photon.

Longitudinally, this conservation implies a wavelength conversion, e.g., one photon at 405 nm is converted into a pair of photons at 810 nm. Transversally, the momentum conservation leads to spatial correlations between the down-converted photons, as shown in the top figure.

### Coincidence measurement

When one down-converted photon in a pair is measured at position A, the twin photon from the same pair will be localised at position A’ – similarly for the pair of positions B and B’. This joint measurement of two photons is called a coincidence measurement and is a fundamental step in harnessing spatial correlations in photon pairs.

In coincidence-based quantum imaging, it is necessary to obtain the spatial distribution of these coincidence events. This has now been enabled by single-photon cameras.

### Single-photon sensitivity

Development in imaging devices capable of single-photon sensitivity is one of the main drivers in quantum imaging. Electron-multiplying charge-coupled device (EMCCD) and intensified charge-coupled device (ICCD) cameras are common tools in state-of-the-art laboratories focused on quantum imaging, owing to their high quantum efficiency, low noise levels, and gating capabilities (triggered acquisition with external signal).

More recently, single-photon avalanche diode (SPAD) cameras have shown great potential to enable faster acquisition time. Indeed, SPAD cameras have efficiencies much lower than an EMCCD camera, for example.

### Faster acquisition

However, SPAD cameras can achieve much faster acquisition, with frame rates on the order of 105 frames per second, a thousand times faster than EMCCD cameras. This feature is particularly important considering the method employed to reconstruct spatial correlations in two-photon pairs.

In practice, measuring coincidence events across multiple spatial positions is not a trivial task. Defienne et al. have developed an elegant mathematical model to reconstruct the two-photon spatial distribution 𝛤(𝑟𝑖, 𝑟𝑗) from N recorded intensity images; for two photons of the same pair, the probability of measuring one photon at pixel 𝑟𝑖 and the other at pixel 𝑟𝑗 on the camera sensor, can be reconstructed as follows:

### The appeal of SPAD cameras

In the above model, 𝐼𝑙(𝑟𝑖) is the intensity measured in frame 𝑙 at pixel 𝑟𝑖. Interestingly, this model becomes more accurate the larger the number of recorded intensity images N, hence the appeal of SPAD cameras.

Figure 1(a) shows the intensity of a collinear twin-beam produced through SPDC and recorded by a SPAD camera with a nominal frame rate of 96 thousand frames per second. Using a total of 100 million frames, one can reconstruct the conditional distributions of two-photon pairs.

##### Figure 1: (a) shows the intensity of SPDC twin-beam. From the two-photon spatial distribution, reconstructed using 100 million images, one can extract the conditional spatial distribution of a photon, given that his partner was measured at (b) A or (c) B. Credit: Dr. Bienvenu Ndagano

Suppose for example that one photon is measured at pixel A or B, the spatial distributions of the partner photon from the same pair are shown in Figures 1(b) and 1(c), respectively. Observe that each of these distributions is peaked at a pixel that is diametrically opposed around the center of the SPDC beam; this is a consequence of conservation of momentum. Using this approach, one can reconstruct the coincidence image of an object placed in the path of one (or both) of the twin beams.

#### Bienvenu Ndagano, Researcher, Fraunhofer Centre for Applied Photonics, United Kingdom

This article has first been published by the African Physics Newsletter - © American Physical Society, 2021