The phase problem can be considered as one of the cornerstones of quantum mechanics intimately connected to the detection process and the uncertainty relation. The latter impose fundamental limits on the manifold phase reconstruction schemes invented to date, in particular, at small magnitudes of the quantum wave. Here, we show that a rigorous solution of the transport of intensity reconstruction (TIE) scheme in terms of a linear elliptic partial differential equation for the phase provides reconstructions even in the presence of wave zeros if particular boundary conditions are given. We furthermore discuss how partial coherence hampers phase reconstruction and show that a modified version of the TIE reconstructs the curl-free current density at arbitrary (in)coherence. Our results open the way for TIE-based phase retrieval of arbitrary wave fields, eventually containing zeros such as phase vortices.
The phase problem can be considered as one of the cornerstones of quantum mechanics intimately connected to the detection process and the uncertainty relation. The latter impose fundamental limits on the manifold phase reconstruction schemes invented to date, in particular, at small magnitudes of the quantum wave. Here, we show that a rigorous solution of the transport of intensity reconstruction (TIE) scheme in terms of a linear elliptic partial differential equation for the phase provides reconstructions even in the presence of wave zeros if particular boundary conditions are given. We furthermore discuss how partial coherence hampers phase reconstruction and show that a modified version of the TIE reconstructs the curl-free current density at arbitrary (in)coherence. Our results open the way for TIE-based phase retrieval of arbitrary wave fields, eventually containing zeros such as phase vortices.
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.
Quantum mechanics is conventionally formulated in terms of complex Hilbert spaces, linear operators, and the Schrödinger equation. Although geometric structures play an increasingly important role in modern formulations of quantum theory, the wavefunction itself is generally regarded as a fundamental object. In this work, we investigate an alternative formulation in which the fundamental object is a coherence coordinate , where Omega represents radial coherence and Theta represents phase coherence associated with geometric transport. Starting from three assumptions—a coherence coordinate, a probability density generated by radial coherence, and a geometric Hamiltonian—we reconstruct the continuity equation, the quantum Hamilton–Jacobi equation, and ultimately the Schrödinger equation. Within this framework, the wavefunction appears as a derived quantity, while Hilbert spaces emerge naturally as spaces of square-integrable functions on a coherence manifold. The resulting construction establishes a differential-geometric representation of quantum mechanics and provides a common geometric language connecting quantum states, probability flow, geometric phases, holonomies, and gauge transport.
We propose and demonstrate numerically a measurement scheme for complete reconstruction of the 2D quantum wave function of a Bose-Einstein condensate, amplitude and phase, from a time-of-flight measurement. We identify a fundamental ambiguity present in the measurement of phase structures of high-symmetry excitations (e.g., vortices) and show how to overcome it by allowing for different expansion durations in different directions. We demonstrate this approach with the reconstruction of matter-wave vortices and arrays of vortices.
Retrieving the phase of a complex-valued field from the measurements of its amplitude is a crucial problem with a wide range of applications in microscopy and ultracold atomic physics. In particular, obtaining an accurate and efficient solution to this problem is a key step in shaping laser beams for trapping atoms in optical tweezer arrays and applying high-fidelity entangling gates on a neutral atom quantum computer. Current approaches to this problem fail to converge on the optimal solution due to a phenomenon known as vortex formation. In this work, we present an efficient optimization algorithm using Optimal Transport. Our approach completely bypasses the creation of phase vortices and allows for a state-of-the-art solution both in terms of accuracy and efficiency. Furthermore, we show a deep theoretical connection between the Optimal Transport plan and the ray-optics limit of the Wigner distribution of the unknown complex-valued field, and show that our method can be used to retrieve the phase-space transformation of any unknown quadratic phase system. Finally, we reinterpret this problem in the modern quantum learning framework. The techniques we develop provide both useful intuition and practical tools for advancing the frontiers of phase retrieval and laser beam shaping.
This preprint presents a geometric reconstruction of Schrödinger dynamics based on coherent transport in an axial–phase configuration space. Within this framework, physical evolution is represented as a spiral process combining irreversible axial progression with transverse phase rotation. The approach seeks to provide a minimal geometric interpretation of quantum dynamics while preserving the established empirical structure of quantum mechanics. By imposing phase closure, preservation of superposition structure, and invariance of overlap relations, the admissible evolution law is constrained to a linear first-order transport equation generated by a Hermitian operator. When intrinsic evolution is projected onto an observational frame in which laboratory time is effectively defined, the resulting transport law reduces to the conventional Schrödinger equation. The framework suggests that laboratory time need not coincide with the fundamental evolution parameter but may emerge as an effective coordinate through projection. This perspective is consistent with ongoing discussions in both quantum theory and cosmology regarding the possibility that conventional notions of time are incomplete while leaving the successful predictive structure of quantum mechanics unchanged. Potential deviations from the standard description are expected only in regimes where intrinsic axial support becomes experimentally accessible or where the effective time-local approximation breaks down. The work therefore provides a geometric interpretation of Schrödinger dynamics that reproduces standard quantum mechanics within its established domain of validity while identifying possible directions for future theoretical and experimental investigation. This manuscript was originally prepared for submission to a peer-reviewed journal and is archived here as a preprint. Keywords: quantum mechanics, Schrödinger equation, geometric interpretation, time, phase space, Hermitian operators, quantum foundations, wave mechanics, coherent transport, emergent time.
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.