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Geometric Phases, Coherent States and Resonant Hamiltonians

Masao Matsumoto, Hiroshi Kuratsuji

Published Dec 29, 1995• 0 citations
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Abstract

We study characteristic aspects of the geometric phase which is associated with the generalized coherent states. This is determined by special orbits in the parameter space defining the coherent state, which is obtained as a solution of the variational equation governed by a simple model Hamiltonian called the ”resonant Hamiltonian”. Three typical coherent states are considered: SU(2), SU(1,1) and Heisenberg-Weyl. A possible experimental detection of the phases is proposed in such a way that the geometric phases can be discriminated from the dynamical phase

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We study characteristic aspects of the geometric phase which is associated with the generalized coherent states. This is determined by special orbits in the parameter space defining the coherent state, which is obtained as a solution of the variational equation governed by a simple model Hamiltonian called the ”resonant Hamiltonian”. Three typical coherent states are considered: SU(2), SU(1,1) and Heisenberg-Weyl. A possible experimental detection of the phases is proposed in such a way that the geometric phases can be discriminated from the dynamical phase
The geometric phase is studied for quantum polarized light described by an SU(2) coherent state. This is revealed by using quantum interference. The SU(2) coherent state is constructed from the quantized Stokes parameters. The quantum interferometry is arranged by the evolution operator implemented by the Hamiltonian inspired from the birefringence of the Faraday and Kerr effect. The geometric phase is given by the coherent state path integral. In particular, by considering a resonant Hamiltonian, the geometric phase is extracted from the total factor that consists of both geometric and dynamical phases. This extraction forms the basis of an experimental way to detect the geometric phase.
The quantum dynamics of time-dependent Hamiltonians of the kind H(t)=U(t)H(0)U(t)° is studied in the framework of generalized coherent states. It is shown that the quantum and the classical dynamics are isomorphic and that the phase of ‖ψ(t)〉 is simply the classical action. In adiabatic approximation it is easy to extract Berry’s phase from this. A discussion of generalized coherent states (CS’s) necessary to deal with excited states is given; in particular for SU(2), SU(1,1) and E(2) their structure is completely characterized, showing that the deep connection with geometric quantization extends to these generalized CS’s. This technique may be employed to estimate higher-order correction to the adiabatic approximation.
Consider a Hamiltonian H which is an element of the enveloping algebra of a Lie algebra L. For simplicity the authors choose L as SU(2) or SU(1,1). They look for the coherent state mod sc) which solves the variational equation delta (sc mod H mod sc)=0. This condition in state space is transcribed into a condition in a classical phase space. Geometrical concepts are given for the state of lowest energy, and for a self-consistent linearised Hamiltonian. The interpretation yields solutions of the equation for the state of lowest energy by iteration procedures.
This paper is concerned with the construction of phase operators, phase states, vector phase states, and coherent states for a generalized Weyl-Heisenberg algebra. This polynomial algebra (that depends on real parameters) is briefly described. The various states are defined on a finite- or infinite-dimensional space depending on the parameters. This report constitutes an introduction to three papers published by the authors in J. Phys. A [43 (2010) 115303 and 45 (2012) 244036] and J. Math. Phys. [52 (2011) 082101]. See these three papers for the relevant references.