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Optical Trapping and Optical Micromanipulation

A water droplet as a toy atom

Javier Tello Marmolejo, Adriana Canales, Ricardo Méndez-Fragoso, D. Hanstorp

Published Oct 3, 2022• 1 citation
Publisher

Abstract

Common atomic physics courses jump from the square and harmonic well potentials straight to the hydrogen atom. However, there is a missing link in between, the spherical well potential. Although it is included in some textbooks, the lack of an experimental backing means the problem quickly becomes mathematically complex. Here we have built an optical toy atom using the scattering of an optically levitated, evaporating water droplet. We find a greatly simplified Mie scattering spectrum composed of a series of evolving Fano resonances organized in a set of combs. The whole spectrum can be intuitively explained through an analogy to a quantum spherical well potential. This produces a model of an atom including ground and excited states, quantized angular momentum, and tunneling.

DOI
10.1117/12.2631867

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Common atomic physics courses jump from the square and harmonic well potentials straight to the hydrogen atom. However, there is a missing link in between, the spherical well potential. Although it is included in some textbooks, the lack of an experimental backing means the problem quickly becomes mathematically complex. Here we have built an optical toy atom using the scattering of an optically levitated, evaporating water droplet. We find a greatly simplified Mie scattering spectrum composed of a series of evolving Fano resonances organized in a set of combs. The whole spectrum can be intuitively explained through an analogy to a quantum spherical well potential. This produces a model of an atom including ground and excited states, quantized angular momentum, and tunneling.
Abstract Modeling the hydrogen atom is often considered an essential milestone in an introductory quantum mechanics course. Mathematically, the instructional path is clear: from the square potential well, through the harmonic potential, the three-dimensional box, and the spherical well potential prior to the 1/r Coulomb potential. However, there are few experimental examples to help visualize and ground the key physical concepts. Here, we present an experiment where the light scattering of an evaporating, optically levitating water droplet can be used as an analogy to a quantum spherical well potential. We have distilled two important concepts in this analogy: angular momentum and tunneling. This approach is akin to the analogy between a square well potential and a Fabry–Pérot cavity. It can be used as a visual example on the path toward the hydrogen atom in undergraduate-level quantum mechanics courses.
Self-bound quantum droplets are a newly discovered phase in the context of ultracold atoms. In this Letter, we report their experimental realization following the original proposal by Petrov [Phys. Rev. Lett. 115, 155302 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.155302], using an attractive bosonic mixture. In this system, spherical droplets form due to the balance of competing attractive and repulsive forces, provided by the mean-field energy close to the collapse threshold and the first-order correction due to quantum fluctuations. Thanks to an optical levitating potential with negligible residual confinement, we observe self-bound droplets in free space, and we characterize the conditions for their formation as well as their size and composition. This work sets the stage for future studies on quantum droplets, from the measurement of their peculiar excitation spectrum to the exploration of their superfluid nature.
Our work utilizes the quantum model of the hydrogen atom which is based on the Schrödinger equation with Coulomb potential. Specifically, we concentrate on the angular components of the wave eigenfunctions derived from this model. We consider the quantum states with n ≤ 4. In order to visualize the orbital shapes of these states, we built in the spherical coordinates system their 3D geometric representations. Furthermore, we use the corresponding spherical harmonics, to calculate the θ nodal values that describe the configurations of these orbital states.
The quantization of orbital angular momentum can be introduced simply through the model of the two- and three-dimensional simple harmonic oscillator. Most of the analysis is carried out in the familiar Cartesian coordinate system in which the simple harmonic oscillator Schrödinger equation is separable. Angular momentum eigenfunctions are constructed by superposition, and the resulting functions easily transformed into spherical coordinates. The angle-dependent parts of the solutions are the spherical harmonics, which are the orbital angular momentum eigenfunctions for all central potentials including the Coulomb potential of the hydrogen atom. The method also introduces the separation of variables technique and provides practice in applying the concepts of completeness, orthogonality, superposition of states, and alternative representations.
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We investigate the rotational properties of quantum droplets, which form in a mixture of two Bose-Einstein condensates, in the presence of an anharmonic trapping potential. We identify various phases as the atom number and the angular momentum or angular velocity of the trap vary. These phases include center-of-mass–like excitation (without or with vortices), vortices of single and multiple quantization, etc. Finally, we compare our results with those of the single-component problem. Published by the American Physical Society 2024