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Complex wave-field reconstruction by using phase-space tomography

M. G. Raymer, M. Beck, D. F. McAlister

Published May 8, 1994• 3 citations
OpenAlex

Abstract

A new class of phase-retrieval methods for 2-D fields is introduced. Phase-space tomography can be used for the experimental determination of the amplitude and phase structure of a quasi-monochromatic wave field in a plane normal to its propagation direction. The wave field ψ(r) may represent either a scalar electromagnetic (EM) field or the quantum-mechanical (QM) wave function of a matter wave. The complex wave field may be coherent or partially coherent, in which case the method reconstructs the two-point spatial correlation function, Γ(r, r′) = ⟨ψ(r)ψ*(r′)⟩. (In the QM case, the analogous quantity is the density matrix.) The experiment uses only intensity measurements and refractive optics (lenses), and the data-analysis algorithm is noniterative and requires no deconvolution.

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A new class of phase-retrieval methods for 2-D fields is introduced. Phase-space tomography can be used for the experimental determination of the amplitude and phase structure of a quasi-monochromatic wave field in a plane normal to its propagation direction. The wave field ψ(r) may represent either a scalar electromagnetic (EM) field or the quantum-mechanical (QM) wave function of a matter wave. The complex wave field may be coherent or partially coherent, in which case the method reconstructs the two-point spatial correlation function, Γ(r, r′) = ⟨ψ(r)ψ*(r′)⟩. (In the QM case, the analogous quantity is the density matrix.) The experiment uses only intensity measurements and refractive optics (lenses), and the data-analysis algorithm is noniterative and requires no deconvolution.
A method is proposed for the experimental determination of the amplitude and phase structure of a quasimonochromatic wave field in a plane normal to its propagation direction. The wave field may represent either a scalar electromagnetic (EM) field or the quantum mechanical (QM) wave function of a matter wave. For coherent EM fields or pure QM states, the method uniquely reconstructs the complex wave fields. For partially coherent EM fields or mixed QM states, it reconstructs the two-point correlation function or density matrix. The experiment uses only intensity measurements and refractive optics (lenses), and the data analysis algorithm is noniterative and requires no deconvolution.
The two-point coherence of an electromagnetic field is represented completely by a 4×4 coherency matrix G that encodes the joint polarization-spatial-field correlations. Here, we describe a systematic sequence of cascaded spatial and polarization projective measurements that are sufficient to tomographically reconstruct G--a task that, to the best of our knowledge, has not yet been realized. Our approach benefits from the correspondence between this reconstruction problem in classical optics and that of quantum state tomography for two-photon states in quantum optics. Identifying G uniquely determines all the measurable correlation characteristics of the field and, thus, lifts ambiguities that arise from reliance on traditional scalar descriptors, especially when the field's degrees of freedom are correlated or classically entangled.
A method of reconstructing a wave function from part of the Q function in a pure state of light is proposed. This method involves the use of a noniterative phase-retrieval method based on Gaussian filtering, which allows one to determine a convolution of the wave function with a known Gaussian function for the vacuum state from measured data of the Q function along only two parallel lines in phase space. By using a deconvolution process, the wave function is reconstructed from the convolution, provided that the resolution of the reconstructed wave function is limited by the extent of the Fourier transform of the Gaussian function. Numerical simulations demonstrate the applicability of this method to the reconstruction of the wave function from the Q function obtained from the photoelectric counting statistics in unbalanced homodyne detection.
We demonstrate a technique of quantum-state tomography for measuring the complex temporal wave function of narrowband biphotons with polarization-dependent and time-resolved two-photon interference. While the amplitude function of the biphoton waveform is directly related to the second-order correlation function which is determined by the two-photon coincidence measurement, the phase function is retrieved from six sets of time-resolved two-photon interference measurements projected onto different polarization subspaces. We apply this technique to experimentally reconstruct the temporal quantum states of the narrow-band biphotons generated from the spontaneous four-wave mixing in cold atoms. As compared with the homodyne detection, our method doesn't require any external phase reference.
We introduce the concept of quantum field tomography, the efficient and reliable reconstruction of unknown quantum fields based on data of correlation functions. At the basis of the analysis is the concept of continuous matrix product states (cMPS), a complete set of variational states grasping states in one-dimensional quantum field theory. We innovate a practical method, making use of and developing tools in estimation theory used in the context of compressed sensing such as Prony methods and matrix pencils, allowing us to faithfully reconstruct quantum field states based on low-order correlation functions. In the absence of a phase reference, we highlight how specific higher order correlation functions can still be predicted. We exemplify the functioning of the approach by reconstructing randomized cMPS from their correlation data and study the robustness of the reconstruction for different noise models. Furthermore, we apply the method to data generated by simulations based on cMPS and using the time-dependent variational principle. The presented approach is expected to open up a new window into experimentally studying continuous quantum systems, such as those encountered in experiments with ultra-cold atoms on top of atom chips. By virtue of the analogy with the input–output formalism in quantum optics, it also allows for studying open quantum systems.