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arXiv:quant-ph/9912045 · Journal of Experimental and Theoretical Physics Letters

Quantum particle trajectories and geometric phase

Mihai Octavian Dima

Published Dec 1, 2000• 3 citations
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Abstract

Particle trajectories are defined as integrable dx μ dp μ =0 paths in projective space. Quantum states evolving on such trajectories, open or closed, do not delocalize in ( x, p ) projection, the phase associated with the trajectories related to the geometric (Berry) phase and the classical mechanics action. Properties at high energies of the states evolving on particle trajectories are discussed.

DOI
10.1134/1.1348476

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Particle trajectories are defined as integrable dx μ dp μ =0 paths in projective space. Quantum states evolving on such trajectories, open or closed, do not delocalize in ( x, p ) projection, the phase associated with the trajectories related to the geometric (Berry) phase and the classical mechanics action. Properties at high energies of the states evolving on particle trajectories are discussed.
Abstract When subject to measurements, quantum systems evolve along stochastic quantum trajectories that can be naturally equipped with a geometric phase observable via a post-selection in a final projective measurement. When post-selecting the trajectories to form a close loop, the geometric phase undergoes a topological transition driven by the measurement strength. Here, we study the geometric phase of a subset of self-closing trajectories induced by a continuous Gaussian measurement of a single qubit system. We utilize a stochastic path integral that enables the analysis of rare self-closing events using action methods and develop the formalism to incorporate the measurement-induced geometric phase therein. We show that the geometric phase of the most likely trajectories undergoes a topological transition for self-closing trajectories as a function of the measurement strength parameter. Moreover, the inclusion of Gaussian corrections in the vicinity of the most probable self-closing trajectory quantitatively changes the transition point in agreement with results from numerical simulations of the full set of quantum trajectories.
This paper presents an alternative approach to geometric phases from the observable point of view. Precisely, we introduce the notion of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. The observable-geometric phases are shown to be connected with the quantum geometry of the observable space evolving according to the Heisenberg equation. They are indeed distinct from Berry's phase \cite{Berry1984, Simon1983} as the system evolves adiabatically. It is shown that the observable-geometric phases can be used to realize a universal set of quantum gates in quantum computation. This scheme leads to the same gates as the Abelian geometric gates of Zhu and Wang \cite{ZW2002,ZW2003}, but based on the quantum geometry of the observable space beyond the state space.
An analysis of classical mechanics in a complex extension of phase space shows that a particle in such a space can behave in a way redolent of quantum mechanics; additional dimensions permit ‘tunnelling’ without recourse to instantons and time/energy uncertainties exist. In practice, ‘classical’ particle trajectories with additional degrees of freedom arise in several different formulations of quantum mechanics. In this talk we compare the extended phase space of the closed time-path formalism with that of complex classical mechanics, to suggest that ℏ has a role in our understanding of the latter. However, differences in the way that trajectories are used make a deeper comparison problematical. We conclude with some thoughts on quantisation as dimensional reduction.