Gauge-Invariant Phase Mapping to Intensity Lobes of Structured Light via Closed-Loop Atomic Dark States

TL;DR

The paper addresses how a gauge-invariant loop phase in a closed-loop three-level atomic system maps to observable intensity patterns of a structured Laguerre-Gaussian probe. It develops an analytical model in the weak-probe limit, yielding an output field with three contributions: Beer-Lambert absorption, scattering, and a loop-phase–dependent interference term governed by $\Phi = \phi_{12} + \phi_{23} - \phi_{13}$, with optical depth $\alpha L$ controlling visibility. A key finding is that unknown phases can be read out by rotating the LG lobes in azimuth, and the authors introduce a Berry-phase mapping through dark-state holonomy on a torus, deriving $\gamma_B = -2\pi \frac{\Omega_{23}^2 + \Omega_{13}^2}{\Omega_{12}^2 + \Omega_{23}^2 + \Omega_{13}^2}$ and highlighting the role of $\Omega_{12}$ in generating nontrivial holonomy. The work proposes experimental routes in cold-atom or solid-state platforms and positions structured light in closed-loop quantum-optics as a versatile testbed for geometric phases and phase metrology.

Abstract

We present an analytical model showing how the gauge-invariant loop phase in a three-level closed-loop atomic system imprints as bright-dark lobes in Laguerre Gaussian probe beam intensity patterns. In the weak probe limit, the output intensity in such systems include Beer-Lambert absorption, a scattering term and loop phase dependent interference term with optical depth controlling visibility. These systems enable mapping of arbitrary phases via interference rotation and offer a platform to measure Berry phase. Berry phase emerge as a geometric holonomy acquired by the dark states during adiabatic traversal of LG phase defined in a toroidal parameter space. Manifesting as fringe shifts which are absent in open systems, experimental realization using cold atoms or solid state platforms appears feasible, positioning structured light in closed-loop systems as ideal testbeds for geometric phases in quantum optics.

Figures (8)

• Figure 1: Schematic of the closed-loop three-level system. The probe beam with Rabi frequency $\Omega_{13}$ drives the $\ket{1} \to \ket{3}$ transition resonantly, while the pump beam with Rabi frequency $\Omega_{23}$ couples $\ket{2} \to \ket{3}$. The loop is closed by the third control field with Rabi frequency $\Omega_{12}$ connecting $\ket{1} \to \ket{2}$. All fields carry phases $\phi_{ij}$, enabling mapping of the gauge-invariant loop phase $\Phi = \phi_{12} + \phi_{23} - \phi_{13}$ onto the output intensity pattern.
• Figure 2: Azimuthal intensity and phase profiles of LG$^1_0$ probe beam before and after interaction with the closed-loop system. (a) Input LG$^1_0$ probe beam intensity with beam waist $w_0=100 \mu$m with characteristic donut-shaped profile. (b) Input phase structure showing the helical ($l=1$) azimuthal phase. (c) Output probe beam intensity for optical depth $\alpha L =1$, Rabi frequencies $\Omega_{13}=\Omega_{12}=0.1 \gamma_{13}$, $\Omega_{23}=5\gamma_{13}$ and $\phi_{12}+\phi_{23}=0$ revealing modulation of the dark bright lobes due to interference. (d) Output phase profile of the probe beam.
• Figure 3: Azimuthal intensity and phase profiles of LG$^2_0$ probe beam before and after interaction with the closed-loop system. (a) Input LG$^2_0$ probe beam intensity with beam waist $w_0=100 \mu$m with characteristic donut-shaped profile. (b) Input phase structure showing the double helical ($l=2$) azimuthal phase. (c) Output probe beam intensity for optical depth $\alpha L =5$, Rabi frequencies $\Omega_{13}=\Omega_{12}=0.1 \gamma_{13}$, $\Omega_{23}=5\gamma_{13}$ and $\phi_{12}+\phi_{23}=0$ revealing modulation of the dark bright lobes due to interference. (d) Output phase profile of the probe beam.
• Figure 4: Output intensity of the probe beam, at different values of optical depth(OD). Panels (a) and (b) correspond to LG$^1_0$ probe beam while (c) and (d) show results for the LG$^2_0$ probe beam.
• Figure 5: Output intensity of the probe beam at two different values of control field phases $\phi_{12}$ demonstrating sensitivity to unkown phases. Both cases show the rotation of the dark-bright lobes in the intensity pattern.