The wave function is the cornerstone of quantum theory, and various techniques have been proposed for its direct measurement. The reconstruction of the wave function can be achieved by utilizing the real and imaginary components of momentum weak value, which correspond to amplitude and phase differentiation, respectively. However, these methods are limited in their applicability to wave functions with large phase gradients due to the weak coupling condition, which is a fundamental constraint. In this paper, we propose a method for reconstructing wave functions with large phase gradients based on optical differentiation through weak measurement. We have developed a more generalized theory of phase differentiation and implemented a common optical path and fast measurement for wave functions with large phase gradients. The experimental detection of wave functions with varying phase gradients has validated the reliability of our method. This work expands the range of applications for weak measurement while retaining its high resolution and sensitivity advantages, particularly in the field of wavefront sensing.
The wave function is the cornerstone of quantum theory, and various techniques have been proposed for its direct measurement. The reconstruction of the wave function can be achieved by utilizing the real and imaginary components of momentum weak value, which correspond to amplitude and phase differentiation, respectively. However, these methods are limited in their applicability to wave functions with large phase gradients due to the weak coupling condition, which is a fundamental constraint. In this paper, we propose a method for reconstructing wave functions with large phase gradients based on optical differentiation through weak measurement. We have developed a more generalized theory of phase differentiation and implemented a common optical path and fast measurement for wave functions with large phase gradients. The experimental detection of wave functions with varying phase gradients has validated the reliability of our method. This work expands the range of applications for weak measurement while retaining its high resolution and sensitivity advantages, particularly in the field of wavefront sensing.
The wave function is the cornerstone of quantum mechanics, although it is still open to different interpretations. It has been shown that the amplitude and the phase of a wave function are manifested in the imaginary and real parts of the momentum weak value, respectively. Based on this, we develop a weak measurement scheme to reconstruct the wave function through measuring the momentum weak value. A proof-of-principle experiment is made to confirm the feasibility of our measurement scheme, in which the polarization states and the momentum of light are coupled. We believe the experimental protocol has the potential to be applied in diverse fields such as wave-front sensing, edge detection, and optical communication, and measurement of the momentum weak value may offer valuable insight to better understand quantum mechanics.
The measurement of a wave function plays a pivotal role in quantum physics and presents a distinctive challenge in experiment. Recent works have shown that both the real and imaginary components of the wave function can be extracted by employing weak or strong measurements, thereby enabling the determination of its amplitude and phase. Here, we propose a simple approach for reconstructing the wave function utilizing the spin-orbit interaction of light at the air-glass interface. By directly measuring the amplitude and employing spatial differentiation to capture the phase gradient, it becomes possible to successfully reconstruct an unknown wave function. To demonstrate its feasibility, we experimentally measure the pure wave function of photons with a Gaussian state. Furthermore, we conduct measurements on a custom state featuring a targeted phase jump to examine the accuracy of our methodology. The measured results show a distribution with high contrast and considerable accuracy, with a fidelity that can exceed 85%. We believe that this work contributes valuable insights into the practical applications of spin-orbit interaction, including optical image processing, wave-front sensing, and quantitative phase imaging.
The wave function is important for understanding and predicting the behavior of photons. Recent studies have shown that wave function can be reconstructed through weak measurement. But currently, the relationships of wave function, weak value, and probe are single and not complete. Here, we propose a wave function reconstruction method based on complex weak values and complex shift probes. Four relationships of them are built and the wave function can be reconstructed by position and momentum shift probes. The complex weak values can be obtained by adjusting the pre- and post-selection in weak measurement, while the complex shift probes originate from the complex nature of the Fresnel coefficient in total internal reflection. To demonstrate the feasibility of the proposed scheme, an experiment of phase distribution reconstruction is achieved by a real weak value and a momentum shift probe; the experimental results agree well with the theory and show a high contrast. Therefore, the wave function can be conveniently reconstructed by our scheme and the relationships between the wave function, weak values, and probes become more complete. This work provides a valuable reference for the application of weak measurement in image processing, quantitative phase imaging, and label-free biological imaging.
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.
With an extremely high dimensionality, the spatial degree of freedom of entangled photons is a key tool for quantum foundation and applied quantum techniques. To fully utilize the feature, the essential task is to experimentally characterize the multiphoton spatial wave function including the entangled amplitude and phase information at different evolutionary stages. However, there is no effective method to measure it. Quantum state tomography is costly, and quantum holography requires additional references. Here, we introduce quantum Shack-Hartmann wavefront sensing to perform efficient and reference-free measurement of the biphoton spatial wave function. The joint probability distribution of photon pairs at the back focal plane of a microlens array is measured and used for amplitude extraction and phase reconstruction. In the experiment, we observe that the biphoton amplitude correlation becomes weak while phase correlation shows up during free-space propagation. Our work is a crucial step in quantum physical and adaptive optics and paves the way for characterizing quantum optical fields with high-order correlations or topological patterns.
Weak measurements have thus far been considered instrumental in the so-called direct measurement of the quantum wave function [4J. S. Lundeen, Nature (London) 474, 188 (2011).]. Here we show that a direct measurement of the wave function can be obtained by using measurements of arbitrary strength. In particular, in the case of strong measurements, i.e., those in which the coupling between the system and the measuring apparatus is maximum, we compared the precision and the accuracy of the two methods, by showing that strong measurements outperform weak measurements in both for arbitrary quantum states in most cases. We also give the exact expression of the difference between the original and reconstructed wave function obtained by the weak measurement approach; this will allow one to define the range of applicability of such a method.