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Physical Review A

Quantum and classical dynamics correspondence and the brachistochrone problem

Huner Fanchiotti A, Carlos A. García Canal, Miguel Angel Mayosky, Armando Pérez, A. Veiga

Published Oct 22, 2024• 2 citations
Publisher

Abstract

The decomplexification procedure, which allows showing mathematically the isomorphism between classical and quantum dynamics of systems with a finite number of basis states, is exploited to propose resonant electric circuits with gyrator-based couplings and to experimentally study the quantum brachistochrone problem, particularly the passage time in Hermitian and parity-time-symmetric cases.

DOI
10.1103/physreva.110.042219

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The decomplexification procedure, which allows showing mathematically the isomorphism between classical and quantum dynamics of systems with a finite number of basis states, is exploited to propose resonant electric circuits with gyrator-based couplings and to experimentally study the quantum brachistochrone problem, particularly the passage time in Hermitian and parity-time-symmetric cases.
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In this thesis I take a look at various aspects of the brachistochrone problem. In the first half I consider the classical brachistochrone problem, without friction, after which I generalize to also consider drag; friction proportional to the squared speed. I do this numerically, with the shooting method, and following some papers, which Euler wrote. Then I will look at how Euler's results compare to modern results and discuss how they are different. In the second half I consider the quantum brachistochrone problem, following a paper by A. Carlini, and afterwards I look at a related problem, where there is not enough time to change an electron spin between two predetermined states. This problem has been numerically solved in an example.
The quantum brachistochrone problem -- to find the time-optimal transition between given initial and final quantum states -- is investigated in this bachelor's thesis. First the quantum equivalent of distance (the Fubini-Study metric) is formulated, using geometry of spheres. Together with constraints, it is used to create the quantum action. Functional derivatives are then taken of said action to find equations of motion (eoms). For the unconstrained case (excluding finite energy), these eoms are solved in closed form, whilst general to be solved formulae are obtained for the constrained case. The latter is used to explicitly solve an example quantum brachistochrone problem: a spin-1/2 particle in a controllable magnetic field constrained to an x-y plane. Finally, the link with quantum computing is illustrated through the time-optimization of unitary transformations that the quantum brachistochrone yields us.
Abstract The decomplexification procedure allows one to show mathematically (stricto sensu) the equivalence (isomorphism) between the quantum dynamics of a system with a finite number of basis states and a classical dynamics system. This unique way of connecting different dynamics was used in the past to analyze the relationship between the well-known geometric phase present in the quantum evolution discovered by Berry and its generalizations, with their analogs, the Hannay phases, in the classical domain. In here, this analysis is carried out for several quantum hermitian and non-hermitian PT-symmetric Hamiltonians and compared with the Hannay phase analysis in their classical isomorphic equivalent systems. As the equivalence ends in the classical domain with oscillator dynamics, we exploit the analogy to propose resonant electric circuits coupled with a gyrator, to reproduce the geometric phase coming from the theoretical solutions, in simulated laboratory experiments.