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Physical Review A

Geometry of quantum evolution and the coherent state

Subir K. Bose, Binayak Dutta-Roy

Published Apr 1, 1991• 16 citations
Publisher

Abstract

The geometric approach to quantum mechanics initiated by Berry's remarkable discovery [Proc. R. Soc. London Ser. A 392, 45 (1984)] of the anholonomy of the phase of the wave function and subsequent developments leading to a recent reformulation of the geometry of quantum evolution by Anandan and Aharonov [Phys. Rev. Lett. 65, 1697 (1990)] is shown to find an explicit and suggestive realization through the coherent-state representation.

DOI
10.1103/physreva.43.3217

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The geometric approach to quantum mechanics initiated by Berry's remarkable discovery [Proc. R. Soc. London Ser. A 392, 45 (1984)] of the anholonomy of the phase of the wave function and subsequent developments leading to a recent reformulation of the geometry of quantum evolution by Anandan and Aharonov [Phys. Rev. Lett. 65, 1697 (1990)] is shown to find an explicit and suggestive realization through the coherent-state representation.
One milestone in quantum physics is Berry's seminal work [Proc.~R.~Soc.~Lond.~A \textbf{392}, 45 (1984)], in which a quantal phase factor known as geometric phase was discovered to solely depend on the evolution path in state space. Here, we unveil that even an infinitesimal deviation of the initial state from the eigenstate of the initial Hamiltonian can yield a significant change of the geometric phase accompanying an adiabatic evolution. This leads to the surprising observation that two geometric phases can dramatically differ from each other even if their evolution paths are sufficiently close in a pointwise manner.
The geometric phase in quantum mechanics is formulated for charged particles in a gauge-invariant, geometric manner. It is then extended to an evolution resulting from a sequence of measurements as in the work of Pancharatnam and Aharonov and Vardi. Its close connection to the Feynman formulation of quantum mechanics is pointed out. The geometric angles, which are generalizations of the classical, adiabatic angles introduced by Hannay and the quantum, adiabatic angles introduced by Anandan and Stodolsky in their group-theoretic treatment of Berry's phase, are studied in quantum and classical physics. The geometric phase for a quantum spin in a magnetic field due to a second particle is obtained using the quantum reference frame defined by the latter. The question of whether the geometric phase and angles are local or nonlocal and their relationship to the electromagnetic and gravitational phases are also discussed.
The concept of Geometric Phase in Quantum Mechanics is generally formulated en- tirely in terms of geometric structure of complex Hilbert Space. This tutorial article gives a general mathematical overview of the Geometric phase introduced by Berry with the conditions over the system’s evolution embedded through adiabaticity and cyclicity, which were subsequently relaxed sequentially by Aharonov-Anandan, Samuel-Bhandari, and later by Mukunda and Simon by using the idea of Bargmann Invariants. The arti- cle presents a thorough and illustrative overview regarding the mathematical derivation behind the upliftment of the conditions, which results in the generalized definition of the Geometric phase for quantum systems. In addition to that, the article also gives an overall idea about the general analytical aspects of finding the Geometric phase for three level open quantum systems undergoing decoherence and dephasing due to interaction with the surroundings in the weak system reservoir coupling limit described by quantum master equations.