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arXiv:1712.02746 · Physical Review A

Compressive direct imaging of a billion-dimensional optical phase space

Samuel H. Knarr, Daniel J. Lum, James Schneeloch, John C. Howell

Published Aug 28, 2018• 12 citations
arXivHTMLPDF

Abstract

Optical phase spaces represent fields of any spatial coherence and are typically measured through phase-retrieval methods involving a computational inversion, optical interference, or a resolution-limiting lenslet array. Recently, a weak-values technique demonstrated that a beam's Dirac phase space is proportional to the measurable complex weak value, regardless of coherence. These direct measurements require raster scanning through all position-polarization couplings, limiting their dimensionality to less than 100 000 [C. Bamber and J. S. Lundeen, Phys. Rev. Lett. 112, 070405 (2014)]. We circumvent these limitations using compressive sensing, a numerical protocol that allows us to undersample, yet efficiently measure, high-dimensional phase spaces. We also propose an improved technique that allows us to directly measure phase spaces with high spatial resolution with scalable frequency resolution. With this method, we are able to easily and rapidly measure a 1.07-billion-dimensional phase space. The distributions are numerically propagated to an object in the beam path, with excellent agreement for coherent and partially coherent sources. This protocol has broad implications in quantum research, signal processing, and imaging, including the recovery of Fourier amplitudes in any dimension with linear algorithmic solutions and ultra-high-dimensional phase-space imaging.

DOI
10.1103/physreva.98.023854

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Optical phase spaces represent fields of any spatial coherence and are typically measured through phase-retrieval methods involving a computational inversion, optical interference, or a resolution-limiting lenslet array. Recently, a weak-values technique demonstrated that a beam's Dirac phase space is proportional to the measurable complex weak value, regardless of coherence. These direct measurements require raster scanning through all position-polarization couplings, limiting their dimensionality to less than 100 000 [C. Bamber and J. S. Lundeen, Phys. Rev. Lett. 112, 070405 (2014)]. We circumvent these limitations using compressive sensing, a numerical protocol that allows us to undersample, yet efficiently measure, high-dimensional phase spaces. We also propose an improved technique that allows us to directly measure phase spaces with high spatial resolution with scalable frequency resolution. With this method, we are able to easily and rapidly measure a 1.07-billion-dimensional phase space. The distributions are numerically propagated to an object in the beam path, with excellent agreement for coherent and partially coherent sources. This protocol has broad implications in quantum research, signal processing, and imaging, including the recovery of Fourier amplitudes in any dimension with linear algorithmic solutions and ultra-high-dimensional phase-space imaging.
Compressive sensing utilizes sparsity to realize efficient image reconstruction. It is a valuable processing technique when cost, power, technology or computational overhead are limited or high. In the quantum domain technology usually limits efficient acquisition of weak or fragile signals. I will discuss the basics of information theory, compression, and compressive sensing. I will then discuss our recent work in compressive sensing. The topics of discussion include low-flux laser Radar, photonic phase transitions, high resolution biphoton ghost imaging, Ghost object tracking, 3D object tracking and high dimensional entanglement characterization. I will touch lightly on our current work of rapid wavefunction reconstruction and wavefront sensing. As an example (shown below), we were able efficiently and rapidly reconstruct high dimensional joint probability functions of biphotons in momentum and position. With conventional raster scanning this process would take approximately a year, but using double-pixel compressive sensing, the pictures were acquired in a few hours with modest flux.
Phase space tomography estimates correlation functions entirely from multiple snapshots in the evolution of the system, rather than traditional interferometric methods requiring measurement of multiple two-point correlations. However, as in every tomographic formulation, undersampling poses a severe limitation. In the context of quantum correlation function estimation, a new theory utilizing compressed sensing was recently established [D. Gross et al. Phys. Rev. Lett. 105, 150401 (2010)] whereby both measurement and post-processing dimensionality are reduced without giving up reconstruction fidelity. Here we present the first, to our knowledge, experimental demonstration of compressive reconstruction of the classical optical correlation function, i.e. the mutual intensity function which is of course the analogue to a conservative quantum state. Our compressive algorithm makes explicit use of the physically justifiable assumption of a low-entropy source (or state.) Since the source was directly accessible in our classical experiment, we were able to compare the compressive estimate of the mutual intensity to an independent ground-truth estimate from the van Cittert-Zernike theorem and verify substantial quantitative improvements in the reconstruction.
It is of great importance to determine an unknown quantum state for fundamental studies of quantum mechanics, yet it is still difficult to characterize systems of large dimensions in practice. Although the scan-free direct measurement approach based on a weak measurement scheme was proposed to measure a high-dimensional photonic state, how weak the interaction should be to give a correct estimation remains unclear. Here we propose and experimentally demonstrate a technique that measures a high-dimensional quantum state with the combination of scan-free measurement and direct strong measurement. The procedure involves sequential strong measurement, in which case no approximation is made similarly to the conventional direct weak measurement. We use this method to measure a transverse state of a photon with effective dimensionality of 65000 without the time-consumed scanning process. Furthermore, the high fidelity of the result and the simplicity of the experimental apparatus show that our approach can be readily used to measure the complex field of a beam in diverse applications such as wavefront sensing and quantitative phase imaging.
Optical imaging of quantum emitters is essential for a wide range of quantum applications. Conventional confocal imaging relies on point-by-point raster scanning, which is inherently time-consuming and photon-inefficient, particularly for sparse emitter distributions and photon-limited samples. Here, we demonstrate a compressive sensing-based imaging approach, where spatially structured wide-field excitation replaces raster scanning, enabling reconstruction of sparse emitters. In our implementation, random binary patterns are used to acquire compressive measurements, from which the spatial fluorescence distribution is reconstructed using a GPSR-BB algorithm. We experimentally demonstrate this approach using nitrogen-vacancy (NV) centers in diamond as a representative platform, with high-fidelity image reconstruction achieved using only approximately 20%20\%20% of the measurements required for conventional raster scanning. In addition to intensity reconstruction, we extend this framework to reconstruct spatial maps of the second-order correlation function g(2)(0)g^{(2)}(0)g(2)(0) from compressive measurements. This enables identification of single-photon emitters through antibunching signatures using significantly reduced data.
Retrieving the vast amount of information carried by a photon is an enduring challenge in quantum metrology science and quantum photonics research. The transverse spatial state of a photon is a convenient high-dimensional quantum system for study, as it has a well-understood classical analogue as the transverse complex field profile of an optical beam. One severe drawback of all currently available quantum metrology techniques is the need for a time-consuming characterization process, which scales very unfavorably with the dimensionality of the quantum system. Here we demonstrate a technique that directly measures a million-dimensional photonic spatial state in a single setting. Through the arrangement of a weak measurement of momentum and parallel strong measurements of position, the complex values of the entire photon state vector become measurable directly. The dimension of our measured state is approximately four orders of magnitude larger than previously measured. Our work opens up a practical route for characterizing high-dimensional quantum systems in real time. Furthermore, our demonstration also serve as a high-speed, extremely-high-resolution unambiguous complex field measurement technique for diverse classical applications.
Resolving sources beyond the diffraction limit is important in imaging, communications, and metrology. Current image-based methods of super-resolution require phase information (either of the source points or an added filter) and perfect alignment with the centroid of the object. Both inhibit the practical application of these methods, as uniform motion and/or relative jitter destroy their assumptions. Here, we show that measuring intensity in the Fourier plane enables super-resolution without any of the issues of image-based methods. We start with the shift-invariance of the Fourier transform and the observation that the two-point position problem x1,x2{x_1,x_2}x1​,x2​ in the near field corresponds to the single-point wavenumber problem k =2π/(x2−x1)k\ =2π/(x_2-x_1)k =2π/(x2​−x1​) in the far field. We consider the full range of mutual coherence and show that for fully coherent sources, the Fourier method saturates the quantum limit, i.e. it gives the best possible measurement. Similar results hold for sub-Rayleigh constellations of NNN sources, which can act collectively as a spatially averaged metasurface and/or individually as elements of a phased-array antenna. The theory paves the way to merge Fourier optics with super-resolution techniques, enabling experimental devices that are both simpler and more robust than previous designs.