Manipulation and Certification of High-Dimensional Entanglement through a Scattering Medium

High-dimensional entangled quantum states improve the performance of quantum technologies compared to qubit-based approaches. In particular, they enable quantum communications with higher information capacities or enhanced imaging protocols. However, the presence of optical disorder such as atmospheric turbulence or biological tissue perturbs quantum state propagation and hinders their practical use. Here, we demonstrate a wavefront shaping approach to transmit high-dimensional spatially entangled photon pairs through scattering media. Using a transmission matrix approach, we perform wave-front correction in the classical domain using an intense classical beam as a beacon to compensate for the disturbances suffered by a copropagating beam of entangled photons. Through violation of an Einstein-Podolski-Rosen criterion by 988$\sigma$, we show the presence of entanglement after the medium. Furthermore, we certify an entanglement dimensionality of 17. This work paves the way toward manipulation and transport of entanglement through scattering media, with potential applications in quantum microscopy and quantum key distribution.

Popular Summary

When produced in a particular way, light can have surprising properties called ”quantum properties.” Quantum entanglement is one of the most important. It is the basis of many future technologies, such as ultrasecure communication links and high-performance microscopes. However, it is very fragile. As soon as there are inhomogeneities in the optical path, such as atmospheric turbulence or biological tissues, it mixes and quickly becomes unusable. In this work, we develop an approach called entanglement shaping that compensates for this mixing to keep the entanglement intact and usable after propagation through the disordered medium.

To demonstrate the method, we built a proof-of-principle experiment in which pairs of entangled photons were sent through a scattering layer. For the demonstration, we used an opaque layer of parafilm. Usually this causes the photons to be randomly scattered in all directions and entanglement becomes undetectable. Surprisingly, by acting on the particles before they enter the scattering layer, we were able to compensate for the disturbance they experience during their propagation and restore entanglement at the output.

The approach we’ve developed has the potential to make quantum entanglement more robust in real-world environments by preserving entanglement during propagation. This could help advance new applications in quantum microscopy, in which entangled photons could provide higher-resolution images of tissue samples, or in communications, in which messages could be encrypted and transmitted more reliably.

Figure 1

The experimental setup. (a) Spatially entangled photon pairs are produced by type-I spontaneous parametric down-conversion (SPDC) by illuminating a $\beta$-barium borate (BBO) crystal (0.5-mm thickness) with a vertically polarized collimated laser diode (405 nm). Simultaneously, horizontally polarized and collimated light emitted by a superluminescent diode (SLED) is aligned along the pump beam using a dichroic mirror (DM). Long-pass and band-pass filters at $810\pm 5\mathrm{nm}$ (LPF) remove pump photons and filter the classical light. A two-lens system $f_1$-$f_2$ images the crystal surface onto a spatial light modulator (SLM), which is itself imaged by $f_3$-$f_4$ onto a scattering medium (a layer of parafilm on a microscope slide). In the momentum-basis configuration, a single-lens Fourier imaging system ($f_6$ or $f_7$) is used to image output light either onto a single-photon avalanche diode (SPAD) camera (with the movable mirror) or onto a charge-coupled device (CCD) camera (without the movable mirror). In the position-basis configuration, a movable lens $f_5$ is inserted to image the scattering layer onto the cameras. (b) An intensity image acquired by the CCD camera using classical light and no SLM correction. (c) The correction SLM pattern calculated using the transmission matrix. Note that the SLM is represented in transmission while it operates in reflection. The spatial units are in pixels.

Figure 2

Violation of the EPR criterion. Images of JPD minus-coordinate projections acquired using the position-basis configuration (a) without the scattering medium, (c) with the medium and no SLM correction, and (e) with the medium and SLM correction. Images of JPD sum-coordinate projections acquired using the momentum-basis configuration (b) without the scattering medium, (d) with the medium and no SLM correction, and (f) with the medium and SLM correction. Analysis is performed on a total of $1.5\times {10}^9$ images. The central pixel in the minus-coordinate projections is set to zero because the SPAD camera does not resolve the number of photons and therefore cannot measure photon coincidences at the same pixel. The spatial units are in pixels.

Figure 3

Entanglement certification. (a) An intensity image acquired in the position-basis configuration, showing the grid of $45$ pixels. (b)–(d) Correlation matrices between all the pixels in the grid of the position basis (b) without the medium, (c) with the medium and a flat SLM, and (d) with the medium and a correction SLM. Each pixel is labeling the spatial coordinates of the photon: $|m⟩$ in the position basis. (e) An intensity image acquired in the momentum-basis configuration, showing the grid of 45 pixels. (f)–(h) Correlation matrices between all the pixels in the grid in the momentum basis (f) without the medium, (g) with the medium and a flat SLM, and (h) with the medium and a correction SLM. Each pixel is labeling the spatial coordinates of the photon: $|\tilde{p}⟩$ in the momentum basis. Analysis is performed on a total of $1.5\times {10}^9$ images. The black lines and columns are associated with hot pixels of the sensor being set to zero.

Figure 4

The certified-dimension analysis. The number of certified dimensions for different total numbers of frames without the medium (blue curve), with the medium and SLM correction (red curve), and with the medium and no correction (green curve).

Figure 5

The crosstalk intensity pattern. The JPD minus-coordinate projection measured with no light falling on the sensor. A total of $7\times {10}^8$ frames are acquired.

Figure 6

Conditional probability images before and after crosstalk removal. (a) ${\mathrm{\Gamma }}_{ijkl}^{(\mathrm{raw})}$ and (b) ${\mathrm{\Gamma }}_{ijkl}$ acquired in the position-basis configuration without the medium. The reference pixel $(k,l)$ is the central pixel $(0,0)$. (c) ${\mathrm{\Gamma }}_{ijkl}^{(\mathrm{raw})}$ and (d) ${\mathrm{\Gamma }}_{ijkl}$ acquired in the position momentum-basis configuration without the medium. The reference pixel $(k,l)$ is the central pixel $(0,0)$. A total of $7\times {10}^8$ frames are acquired.

Figure 7

The identification of hot pixels. (a) An intensity image obtained by summing $1.2\times {10}^9$ frames acquired with no light on the sensor. (b) The same image with the 32 identified hot pixels appearing in white.

Figure 8

The measurement of unbiasedness. (a) An experimental configuration representing the optical transformation that links the position and momentum bases in our experiment. (b) The spatial arrangement of the $45$ selected pixels in the position and momentum optical planes. (c) The sinc-shaped intensity distribution produced by a pixel located in one optical plane into the reciprocal optical plane.

Figure 9

Simulated two-photon output fields. Sum-coordinate projections of $|{\mathrm{\Psi }}_{\mathrm{out}}{|}^2$ obtained with (a) a flat phase pattern and the (b) correction phase pattern. The spatial coordinates are in pixels.

Figure 10

The simulation of entangled photons propagating through a thick scattering medium. (a) The experimental setup considered in our simulation. The pump beam has a wavelength of $405\mathrm{nm}$ and a beam diameter of $5\mathrm{mm}$. The focal length is $f=500\mathrm{mm}$ and the propagation distance $d=200\mathrm{mm}$. The nonlinear crystal is considered to be infinitely thin. (b) The minus-coordinate projection measured in the position-basis configuration and (c) the sum-coordinate projection measured in the momentum-basis configuration after optimization using SLM1. (d) The minus-coordinate projection measured in the position-basis configuration and (e) the sum-coordinate projection measured in the momentum-basis configuration after optimization using SLM1 and SLM2. SM, scattering medium; SLM, spatial light modulator; BBO, $\beta$-barium borate.

Figure 11

The simulation of a correlation matrix. In the case of the maximally entangled state, a (a) correlation matrix is obtained by adding a (b) $45\times 45$ matrix made of random values with mean parameter $0$ and standard deviation $\sigma =1/\sqrt{N}$, where $N$ is the number of frames, to a (c) $45\times 45$ diagonal matrix with diagonal values $\alpha$. In this example, $N={10}^6$ and $\alpha =1/45$.

Figure 12

The simulated fidelity and number of certified dimensions as a function of the number of frames. (a) The fidelity as a function of the number of frames $N$ in the case of a maximally entangled state (blue) and a nonmaximally entangled state (red). (b) The number of certified dimensions as a function of the number of frames $N$ in the case of a maximally entangled state (blue) and a nonmaximally entangled state (red). The simulations parameters are $\alpha =1/45$, ${\alpha }^{\prime }=0.0001$, and $K=1$.