Quantum phase-sensitive diffraction and imaging using entangled photons Shahaf Asbana,b,1, Konstantin E. Dorfmanc,1, and Shaul Mukamela,b,1 aDepartment of Chemistry, University of California, Irvine, CA 92697-2025; bDepartment of Physics and Astronomy, University of California, Irvine, CA 92697-2025; and cState Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China Contributed by Shaul Mukamel, April 18, 2019 (sent for review March 21, 2019; reviewed by Sharon Shwartz and Ivan A. Vartanyants) (1) R ∗ (2) R 2 We propose a quantum diffraction imaging technique whereby Here βnm = dr un (r)σ(r)um (r), βnm = dr un (r) jσ(r)j ∗ one photon of an entangled pair is diffracted off a sample and um (r), and ρ¯i is a two-dimensional vector in the transverse detected in coincidence with its twin. The image is obtained by detection plane. σ (r) is the charge density of the target object scanning the photon that did not interact with matter. We show prepared by an actinic pulse and p = (1, 2) represents the that when a dynamical quantum system interacts with an external order in σ (r). For large diffraction angles and frequency- field, the phase information is imprinted in the state of the field resolved signal, thep phase-dependent image is modified to in a detectable way. The contribution to the signal from photons P1 ∗ S [ρ¯i ]/ Re nm γnm λn λm vn (ρ¯i )vm (ρ¯i ), where γnm has a 1=2 (1) that interact with the sample scales as / Ip , where Ip is the source similar structure to βnm modulated by the Fourier decomposi- intensity, compared with / Ip of classical diffraction. This makes tion of the Schmidt basis. γnm is phase dependent, contrast to imaging with weak fields possible, providing high signal-to-noise diffraction with classical sources. ratio, avoiding damage to delicate samples. A Schmidt decomposi- Our second main result tackles the spatial resolution enhance- tion of the state of the field can be used for image enhancement by ment. In entanglement-based imaging, the resolution is lim- reweighting the Schmidt modes contributions. ited by the degree of correlation of the two beams. Schmidt decomposition of the image allows us to enhance desired spa- quantum imaging j entangled photons j quantum diffraction j tial features of the charge density. High-order Schmidt modes phase-sensitive imaging (which correspond to angular momentum transverse modes with PHYSICS high topological charge) offer more detailed matter information. apid advances in short-wavelength ultrafast light sources Reweighting of Schmidt modes maximizes modal entropy which Rhave revolutionized our ability to observe the microscopic yields matter information gain and reveals fine details of the world. With bright free-electron lasers and high-harmonics table- charge density. Moreover, S(1) in Eq. 1 has no classical analog; top sources, short time (femtosecond) and length (subnanome- the contribution to the overall image from the signal photons ter) scales become accessible experimentally. These offer new 1=2 scales as Ip , where Ip is the intensity of the source. This is exciting possibilities to study spatio-spectral properties of quan- a unique signature of the linear quantum diffraction (13). The tum systems driven out of equilibrium and monitor dynamical overall detected signal is obtained in coincidence and scales as relaxation processes and chemical reactions. The spatial fea- 3=2 tures of small-scale charge distributions can be recorded in time. / Ip . Classical diffraction in contrast requires two interactions Far-field off-resonant X-ray diffraction measurements provide with the incoming field and therefore scales as Ip , and the corre- 2 (2) useful information on the charge density σ (Q), where Q is the sponding coincidence scales as / Ip , which also applies for S . diffraction wavevector. The observed diffraction pattern S (Q) 2 is given by the modulus square S (Q)/ jσ (Q)j . Inverting these Significance signals to real-space σ (r) requires a Fourier transform. Since σ ( ) the phase of Q is not available, the inversion requires phase A quantum diffraction imaging technique is proposed, retrieval which can be done using either algorithmic solutions whereby one photon of an entangled pair is diffracted off (1, 2) or more sophisticated and costly experimental setups such a sample and detected in coincidence with its twin. Scan- as heterodyne measurements (3). Correlated beam techniques ning the photon that did not interact with matter, we show (4–10) in the visible regime have been shown to circumvent that the phase information imprinted in the state of the this problem by producing the real-space image of mesoscopic field is detectable. We discuss several experimental applica- objects. Such techniques have classical analogs using correlated tions: (i) Obtaining real-space images in diffraction imaging light. They reveal the modulus square of the studied object 1=2 2 avoids the “phase problem.” (ii) The image scales as / I jσ (r)j (11, 12). p I In this paper we consider the setup shown in Fig. 1. We focus with the interacting photons, where p is the source inten- / on off-resonant scattering of entangled photons in which only sity, compared with Ip of classical diffraction. This makes one photon, denoted as the “signal,” interacts with a sample. weak-field imaging possible, avoiding damage to delicate Its entangled counterpart, the “idler,” is spatially scanned and samples. (iii) A Schmidt decomposition of the field can be used measured in coincidence with the arrival of the signal photon. for image enhancement by reweighting the Schmidt modes The idler reveals the image and also uncovers phase informa- contributions. tion, as was recently shown in ref. 13 for linear diffraction where Author contributions: S.A., K.E.D., and S.M. designed research; S.A. performed research; heterodyne-like detection has been achieved due to vacuum S.A. analyzed data; and S.A., K.E.D., and S.M. wrote the paper.y fluctuation of the detector. Reviewers: S.S., Bar Ilan University; and I.A.V., Deutsches Elektronen-Synchrotron (DESY).y Our first main result is that for small diffraction angles, The authors declare no conflict of interest.y using Schmidt decomposition of the two-photon amplitude p Published under the PNAS license.y Φ(q , q ) = P1 λ u (q )v (q ) λ s i n n n s n i , where n is the respective 1 To whom correspondence may be addressed. Email: [email protected], sasban@ mode weight, reads uci.edu, or [email protected] This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. (p) X p (p) ∗ 1073/pnas.1904839116/-/DCSupplemental.y S [ρ¯i ] / Re λn λm βnm vn (ρ¯i )vm (ρ¯i ): [1] Published online May 23, 2019. nm www.pnas.org/cgi/doi/10.1073/pnas.1904839116 PNAS j June 11, 2019 j vol. 116 j no. 24 j 11673–11678 Downloaded by guest on September 26, 2021 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 super- position of separable states using the Schmidt decomposition (19–21) 1 X p Φ(qs , qi ) = λn un (qs )vn (qi ), [5] n where the Schmidt modes un (qs ) and vn (qi ) are the eigenvec- tors of the signal and the idler reduced density matrices, and P the eigenvalues λn satisfy the normalization n λn = 1 (20). The number of relevant modes serves as an indicator for the degree of inseparability of the amplitude, i.e., photon entangle- ment. Common measures for entanglement include the entropy P −1 P 2 Sent = − n λn log2 λn or the Schmidt number κ ≡ n λn . The latter is also known as the inverse participation ratio as it quantifies the number of important Schmidt modes or the effec- Fig. 1. Sketch of the proposed quantum imaging setup. A broadband (2) tive joint Hilbert space size of the two photons. In a maximally pump pulse with the wavevector kp propagates through a χ crystal, gen- erating an entangled photon pair denoted as signal and idler. The photons entangled wavefunction, all modes contribute equally. are distinguished either by polarization (type II) or by frequency (type I) and The spatial profile of the photons in the transverse plane are separated by a beam splitter (BS). The signal photon interacts with the (perpendicular to the propagation direction) can be expanded sample and can be further frequency dispersed and collected by a “bucket” and measured using a variety of basis functions; e.g., Laguerre– detector Ds with no spatial resolution. The idler is spatially resolved in Gauss (LG) or Hermite–Gauss (HG) has been demonstrated the transverse plane by the detector Di. The two photons are detected in experimentally (22, 23). These sets satisfy orthonormality coincidence (Eq. 12). R 2 P 0 d q un (q)vk (q)= δnk and closure relations n un (q)vn (q )= (2) 0 δ (q − q ). The deviation of λn from a uniform (flat) dis- tribution reflects the degree of entanglement. Perfect quan- Thanks to favorable scaling, weak fields can be used to study tum correlations correspond to maximal entanglement entropy fragile samples to avoid damage. and thus a flat distribution of modes. This is further clarified by the closure relations, which demonstrate the convergence Spatial Entanglement into a point-to-point mapping in the limit of perfect trans- Various sources of entangled photons are available, from quan- verse entanglement. The biphoton amplitude exhibits two lim- tum dots (14) to cold atomic gases (15) and nonlinear crystals, iting cases for the infinite inverse participation ratio which are and are reviewed in ref.