Physical analogies are particularly useful to study not easily accessible systems. This is the case of the quantum pendulum, whose evolution in time is not easily observed. Fortunately, its mathematical description is shared by the Pendulum beams, which can be readily created in optical laboratories. We engineered mode superpositions that upon propagation mimic the time evolution of the quantum pendulum. We stablished experimental conditions in which the analogy is valid and present results for several relevant cases, finding excellent agreement between experiment and theoretical predictions. We expect to explore other superpositions to extend our analogy to pendulums with more complex dynamics.
Einstein beams are coherent optical beams generated by the conditions of gravitational lensing. In the ray picture, Einstein beams are formed by the intersection of light rays deflected by a lensing mass, similar to non-diffracting Bessel beams, but with the difference that adjacent rays diverge slightly. When accounting for the wave properties of light, they form beam-like diffraction patterns that preserve their shape but expand as the light propagates. The addition of a topological charge to the light, leads to more complex patterns carrying orbital angular momentum. For symmetric lensing conditions, Einstein beams carry modes described by confluent hypergeometric functions, which can be approximated by Bessel functions. A theoretical analysis of this is presented here. In astrophysical observations, many of these features can only be inferred because conditions of coherence and alignment make them difficult to observe. Studies of Einstein beams in the laboratory can be used to inform astrophysical observations and discover new non-astrophysical laboratory applications.
We used spatial light modulation to deflect coherent light beams into trajectories specified by gravitational lensing. We programmed symmetric and asymmetric lensing objects, including adding orbital angular momentum. In the far field we observed Einstein rings and arcs for symmetric lensing objects, and various types of Einstein arcs for asymmetric lensing objects, similar to what is seen in astronomical observations. In the near field we observed the caustics produced by gravitational lensing, not possible in astronomical observations, which consisted of Bessel patterns for symmetric objects, and Mathieu-like patterns and astroids for asymmetric objects.
The similarity between the 2D Helmholtz equation in elliptical coordinates and the Schr¨odinger equation for the simple mechanical pendulum inspires us to use light to mimic this quantum system. When optical beams are prepared in Mathieu modes, their intensity in the Fourier plane is proportional to the quantum mechanical probability for the pendulum. Previous works have produced a two-dimensional pendulum beam that oscillates as a function of time through the superpositions of Mathieu modes with phases proportional to pendulum energies. Here we create a three-dimensional pendulum wavepacket made of a superposition of Helical Mathieu-Gaussian modes, prepared in such a way that the components of the wave-vectors along the propagation direction are proportional to the pendulum energies. The resulting pattern oscillates or rotates as it propagates, in 3D, with the propagation coordinate playing the role of time. We obtained several different propagating beam patterns for the unbound-rotor and the bound-swinging pendulum cases. We measured the beam intensity as a function of the propagation distance. The integrated beam intensity along elliptical angles plays the role of quantum pendulum probabilities. Our measurements are in excellent agreement with numerical simulations.
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