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# Quantum phase-sensitive diffraction and imaging using entangled photons

Contributed by Shaul Mukamel, April 18, 2019 (sent for review March 21, 2019; reviewed by Sharon Shwartz and Ivan A. Vartanyants)

## Significance

A quantum diffraction imaging technique is proposed, whereby one photon of an entangled pair is diffracted off a sample and detected in coincidence with its twin. Scanning the photon that did not interact with matter, we show that the phase information imprinted in the state of the field is detectable. We discuss several experimental applications: (*i*) Obtaining real-space images in diffraction imaging avoids the “phase problem.” (*ii*) The image scales as *iii*) A Schmidt decomposition of the field can be used for image enhancement by reweighting the Schmidt modes contributions.

## Abstract

We propose a quantum diffraction imaging technique whereby one photon of an entangled pair is diffracted off a sample and detected in coincidence with its twin. The image is obtained by scanning the photon that did not interact with matter. We show that when a dynamical quantum system interacts with an external field, the phase information is imprinted in the state of the field in a detectable way. The contribution to the signal from photons that interact with the sample scales as

Rapid advances in short-wavelength ultrafast light sources have revolutionized our ability to observe the microscopic world. With bright free-electron lasers and high-harmonics tabletop sources, short time (femtosecond) and length (subnanometer) scales become accessible experimentally. These offer new exciting possibilities to study spatio-spectral properties of quantum systems driven out of equilibrium and monitor dynamical relaxation processes and chemical reactions. The spatial features of small-scale charge distributions can be recorded in time. Far-field off-resonant X-ray diffraction measurements provide useful information on the charge density

In this paper we consider the setup shown in Fig. 1. We focus on off-resonant scattering of entangled photons in which only one photon, denoted as the “signal,” interacts with a sample. Its entangled counterpart, the “idler,” is spatially scanned and measured in coincidence with the arrival of the signal photon. The idler reveals the image and also uncovers phase information, as was recently shown in ref. 13 for linear diffraction where heterodyne-like detection has been achieved due to vacuum fluctuation of the detector.

Our first main result is that for small diffraction angles, using Schmidt decomposition of the two-photon amplitude

Our second main result tackles the spatial resolution enhancement. In entanglement-based imaging, the resolution is limited by the degree of correlation of the two beams. Schmidt decomposition of the image allows us to enhance desired spatial features of the charge density. High-order Schmidt modes (which correspond to angular momentum transverse modes with high topological charge) offer more detailed matter information. Reweighting of Schmidt modes maximizes modal entropy which yields matter information gain and reveals fine details of the charge density. Moreover, **1** has no classical analog; the contribution to the overall image from the signal photons scales as

## Spatial Entanglement

Various sources of entangled photons are available, from quantum dots (14) to cold atomic gases (15) and nonlinear crystals, and are reviewed in ref. 4. A general two-photon state can be written in the form

### Schmidt Decomposition of Entangled Two-Photon States.

The hallmark of entangled photon pairs is that they cannot be considered as two separate entities. This is expressed by the inseparability of the field amplitude Φ into a product of single-photon amplitude; all of the interesting quantum optical effects discussed below are derivatives of this feature. Φ can be represented as a superposition of separable states using the Schmidt decomposition (19⇓–21)

The spatial profile of the photons in the transverse plane (perpendicular to the propagation direction) can be expanded and measured using a variety of basis functions; e.g., Laguerre–Gauss (LG) or Hermite–Gauss (HG) has been demonstrated experimentally (22, 23). These sets satisfy orthonormality **3** is approximated by a Gaussian, the Schmidt number is given in a closed form (24),

## The Reduced Idler Density Matrix in the Schmidt Basis

The reduced density matrix of the idler reveals the role of quantum correlations in the proposed detection measurement scheme (Fig. 1). The joint light–matter density matrix in the interaction picture is given by*SI Appendix*, section 1)

Fig. 3 displays the induced Schmidt-space coherence of the reduced density matrix of idler (the noninteracting photon) due to the interaction of its twin (signal) with an object. We have chosen the Hermite–Gauss basis, depicted in Fig. 4 for this visualization. Each mode is labeled by two indexes, one for each spatial dimension of the image. In Fig. 3 *C* and *D*, we have traced over the corresponding index, resulting in a 1D dataset. Each coherence corresponds to a projection of the object between two modes. Eq. **1** can be derived as the intensity expectation value calculated from the idler’s reduced density matrix given in Eq. **10**.

## Far-Field Diffraction

We next turn to far-field diffraction with arbitrary scattering directions. While the incoming field is understood to be paraxial, the scattered field is not. The coincidence image in the far field yields a similar expression to the one calculated from the reduced density matrix in Eq. **10** with an additional spatial phase factor characteristic to far-field diffraction. Using Eq. **4** for the setup described in Fig. 1, the coincidence image is given by the intensity–intensity correlation function (*SI Appendix*),**12** can be calculated straight from the reduced density matrix of the idler, despite the fact that it includes the signal’s intensity operator. The reason stems from the fact that the intensity operator expectation value monitors the single-photon space. The partial trace over a singly occupied signal state results in the same conclusion. Estimating this expression includes a 10-field operators correlation function which is shown explicitly in *SI Appendix*, section 2, Eq. **S4**. In the far field, after rotational averaging we obtain (*SI Appendix*, section 2)**5**) into Eq. **13** gives**11**. From the definition of

It is also possible to obtain the real-space image of the charge density when the signal is frequency dispersed. Assuming for simplicity perfect quantum correlations between the signal and idler, we obtain*SI Appendix*).

## Reweighted Modal Contributions

The apparent classical-like form of the coherent superposition in the Schmidt representation, where each mode carries distinct spatial matter information, suggests experiments in which a single Schmidt mode is measured at a time (23). This bears some resemblence to the coherent mode representation of partially coherent sources studied in refs. 27 and 28. Moreover, it allows the reweighting of high angular momentum modes available experimentally (29) and known to have a decreasing effect on the image upon naive summation. Reweighting of truncated sums is extensively used as a sharpening tool in digital signal processing, especially in medical image enhancement (30). This approach raises questions regarding the analysis of optimal Schmidt weights, error minimization, and engineered functional decrease of weights as done in theory for sampled signals. The structure of the spatial information mapping from the signal to the idler takes a simpler form for small scattering angles. When we examine the first- and second-order contributions due to a single charge distribution, the resulting image of a truncated sum composed of the first N modes is given by**11**, holds phase information of the studied object and has no classical counterpart. Its momentum space representation reads,

Fig. 5*A* presents the Schmidt spectrum for a beam characterized by *B* illustrates the improvement of the acquired image due to resummation of the Hermite–Gauss modes of the object decomposed in Fig. 3. By using Eq. **18** with a flattened Schmidt spectrum we demonstrate the enhancement of fine features of the diffracted image. Phase measurement is demonstrated in Fig. 5 *C* and *D*.

## Discussion

The scattered quantum light from matter carries phase information at odd orders in the charge distribution

We have demonstrated that coincidence diffraction measurements of entangled photons with quantum detection can also achieve enhanced imaging resolution. Eq. **18** provides an intuitive picture for the information transfer from the signal to the idler beams. By reweighting the spatial modes that span the measured image, one can refine the matter information. High angular momentum states of light have been recently demonstrated experimentally with quantum numbers above

The imaging of single localized biological molecules has been a major driving force for building free-electron X-ray lasers (31). Such molecules are complex, are fragile, and typically have multiple-timescale dynamics. One strategy is to use a fresh sample in each iteration, assuming a destructive measurement. Ultrashort X-ray pulses have been proposed to reduce damage (32). Entangled hard X-ray photons have been generated by parametric down conversion, using a diamond crystal (33). Avoiding damage of such complexes by using weak fields allows us to follow the evolution of initially perturbed charge densities. Linear diffraction scales as

## Acknowledgments

The support of the Chemical Sciences, Geosciences, and Biosciences Division, Office of Basic Energy Sciences, Office of Science, US Department of Energy is gratefully acknowledged. Collaborative visits of K.E.D. to the University of California, Irvine were supported by Award DEFG02-04ER15571, and. S.M. was supported by Award DESC0019484. S.A.’s fellowship was supported by the National Science Foundation (Grant CHE-1663822). K.E.D. acknowledges the support from Zijiang Endowed Young Scholar Fund, East China Normal University; Overseas Expertise Introduction Project for Discipline Innovation (111 Project, B12024). We also thank Noa Asban for the graphical illustrations.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: smukamel{at}uci.edu, sasban{at}uci.edu, or dorfmank{at}lps.ecnu.edu.cn.

Author contributions: S.A., K.E.D., and S.M. designed research; S.A. performed research; S.A. analyzed data; and S.A., K.E.D., and S.M. wrote the paper.

Reviewers: S.S., Bar Ilan University; and I.A.V., Deutsches Elektronen-Synchrotron (DESY).

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1904839116/-/DCSupplemental.

Published under the PNAS license.

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