Phase Geometry

Jerrold Eldon Marsden

Thomas A. Reichardt, Robert P. Lucht, Paul M. Danehy +1 more

We examine theoretically the degenerate four-wave mixing (DFWM) signal intensities and line shapes obtained with the forward phase-matched geometry in which all beams propagate in the same direction and compare the results to those of the phase-conjugate geometry with counterpropagating pump beams. To examine the forward phase-matched geometry, we modify a theoretical approach used previously to calculate phase-conjugate DFWM signal intensities. This theoretical approach, which involves numerical integration of the time-dependent density-matrix equations, is validated for the forward phase-matched geometry by comparison of our calculated line shapes to both a perturbative solution and to experimental data. This methodology is then used to compare the signal intensities and line shapes obtained with the forward phase-matched geometry and the phase-conjugate geometry in the perturbative (low laser power) and saturated (high laser power) regimes. In the perturbative regime the forward phase-matched signal exhibits less sensitivity to the Doppler linewidth. At pump laser intensities approximately equal to the saturation intensity the signal for the forward phase-matched geometry is stronger than that for the phase-conjugate geometry for primarily Doppler-broadened resonances, assuming the same probe volume for both geometries. These advantages warrant further investigations employing the forward phase-matched configuration for DFWM measurements of gas-phase species.

Jincheng Zhang

The field of quantum geometry is experiencing a resurgence, driven by the desire to understand and manipulate the fundamental structure of quantum systems. This paper introduces the Phase-Field Quantum Geometry (PFQ) framework, a novel approach to representing and manipulating geometric configurations within quantum systems. PFQ leverages the concept of phase fields to dynamically construct complex geometric structures, offering a powerful tool for studying quantum phenomena and potentially enabling new quantum technologies. The core mechanism focuses on an automated "Phase Field Mapping" algorithm, which constructs intricate geometric representations from quantum data. This paper presents a detailed explanation of the PFQ framework, including its mathematical foundations, algorithm implementation, and potential applications across diverse quantum systems. The research explores the advantages of PFQ compared to existing geometric techniques and highlights its potential to unlock deeper insights into quantum mechanics.

Péter Lévay

Edward Meisner

Internal Phase, Amplitude, and Probability in UPG extends the Unified Phase Geometry (UPG) program by adding a mathematically precise amplitude and probability layer to the existing three‑paper framework. In UPG, quantum phase geometry and Lorentzian spacetime geometry are treated as distinct but related structures: the phase manifold carries a positive Fubini–Study–type Kähler metric, while spacetime carries a Lorentzian metric or causal cone, with the two connected by a projection map and horizontal lifts. This paper introduces an internal U(1) phase degree of freedom modeled as a Hermitian line bundle over the phase manifold. The internal U(1) action is generated by a Hamiltonian vector field whose moment map gives the conserved internal action or charge. A key clarification is that the complex amplitude is not itself the Noether charge; instead, the squared Hermitian norm ∣A∣2 is the natural local action or probability density. The paper distinguishes coherent projection, which sums fiber contributions as complex amplitudes before taking the norm, from incoherent pushforward, which sums intensities. Under coherent projection, and assuming Hermitian norm, linear superposition, positivity, normalization, and additivity over exclusive alternatives, the projected probability density takes the standard Born form p=∣ψ∣2. Projection alone does not enforce the Born rule; it only restricts observables to internal‑phase‑invariant quantities. The paper also shows that gauge‑invariant interference requires a U(1) connection so that relative phase between fibers is defined via holonomy. This construction integrates cleanly with the relational action of UPG Paper #3 and preserves the central two‑geometry principle: the Fubini–Study metric measures quantum phase separation, while Lorentzian geometry governs spacetime causality. This work completes the foundational kinematic layer of UPG by providing a rigorous treatment of internal phase, amplitude, and probability within the two‑geometry framework.