Compressive direct imaging of a billion-dimensional optical phase space

Abstract

Optical phase-spaces represent fields of any spatial coherence [1][2][3], and are typically measured through phase-retrieval methods involving a computational inversion, interference, or a resolutionlimiting lenslet array. Recently, a weak-values technique [4][5][6] demonstrated that a beam's Dirac phase-space [7-9] is proportional to the measurable complex weak-value, regardless of coherence. These direct measurements require scanning through all position-polarization couplings, limiting their dimensionality to less than 100,000 [6]. 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 measure a 1.07-billion-dimensional phase-space. The distributions are numerically propagated to an object in the beam path, with excellent agreement. This protocol has broad implications in signal processing and imaging, including recovery of Fourier amplitudes in any dimension with linear algorithmic solutions [10] and ultra-high dimensional phase-space imaging.

Direct characterization of coherence of quantum detectors by sequential measurements

“…This method, known as direct quantum state tomography, opens up a new avenue for quantum tomography technique. The direct tomography (DT) protocol has been extensively studied and the scope of its application is expanded to high-dimensional states, [43][44][45][46][47][48][49][50][51] mixed states [52][53][54][55][56] and entangled states, 57,58 quantum processes, 59 and quantum measurements. 60 The development of the DT theory from the original weak-coupling approximation to the rigorous validation with the arbitrary coupling strength ensures the accuracy and simultaneously improves the precision.…”

Compressive characterization of telecom photon pairs in the spatial and spectral degrees of freedom

“…The slight displacement of the reconstructed modes along the vertical axis is due to the imperfect polishing on the face of the waveguide. As a result, the signal and idler photons experience different effective refractive indices that cause a small spatial displacement between them [see Recently, there has been a strong impetus to employ discrete pixel basis in high-dimensional quantum protocols [2,[7][8][9][10]. This photonic degree of freedom has been utilized in the context of quantum state engineering, quantum state tomography, and information processing [2,7,8].…”

“…As a result, the signal and idler photons experience different effective refractive indices that cause a small spatial displacement between them [see Recently, there has been a strong impetus to employ discrete pixel basis in high-dimensional quantum protocols [2,[7][8][9][10]. This photonic degree of freedom has been utilized in the context of quantum state engineering, quantum state tomography, and information processing [2,7,8]. For example, pixel entanglement was utilized to generate qudits [9]; this idea was further extended by Dixon and co-workers, who demonstrated information encoding in high-dimensional entangled states defined in the pixel basis [2].…”

“…Thus, the spatial characterization of telecom photons shown in Fig. 2 can have important implications for quantum information protocols that utilize the spatial degree of freedom of photons [2,7,8,10].…”

“…Interestingly, the transverse spatial degree of freedom has offered a flexible platform to test complex quantum information protocols in a relatively simple fashion. For example, quantum high-dimensional protocols have been implemented by defining Hilbert spaces in the pixel basis [2,[7][8][9][10]. In general, photons entangled in these representative degrees of freedom have been utilized to perform quantum metrology, quantum imaging, quantum communication, and quantum simulations [1,2,5,6,11].…”