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arXiv:0810.0309

Considerations about the Aharonov-Anandan Phase for Time Independent Hamiltonians

Pierre-Louis Giscard

Published Oct 1, 2008• 0 citations
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Abstract

We present a method for calculating the Aharonov-Anandan phase for time-independent Hamiltonians that avoids the calculation of evolution operators. We compare the generic method used to calculate the Aharonov-Anandan phase with the method proposed here through four examples; a spin-1/2 particle in a constant magnetic field, an arbitrary infinite-sized Hamiltonian with two known eigenvalues, a Fabry-Perot cavity with one movable mirror and a three mirrors cavity with a slightly transmissive movable middle mirror.

DOI
10.48550/arxiv.0810.0309

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We present a method for calculating the Aharonov-Anandan phase for time-independent Hamiltonians that avoids the calculation of evolution operators. We compare the generic method used to calculate the Aharonov-Anandan phase with the method proposed here through four examples; a spin-1/2 particle in a constant magnetic field, an arbitrary infinite-sized Hamiltonian with two known eigenvalues, a Fabry-Perot cavity with one movable mirror and a three mirrors cavity with a slightly transmissive movable middle mirror.
We construct an operator for the Aharonov-Anandan phase for time independent Hamiltonians. This operator is shown to generate the motion of cyclic quantum systems through an equation of evolution involving only geometric quantities, i.e. the distance between quantum states, the geometric phase and the total length of evolution. From this equation, we derive an operator for the Samuel and Bhandari phase (SB-phase) for non cyclic evolutions. Finally we show how the SB-phase can be used to construct an operator corresponding to a quantum clock which commutator with the Hamiltonian has a canonical expectation value.
Berry's (1984) formulation of the topological phase for periodic Hamiltonians is extended to the case of non-adiabatic evolution by proper choice of initial states. This restores a full parallel with the non-adiabatic formalism of Aharonov and Anandan (1987) for periodic orbits in a projective Hilbert space. The Berry phase for the novel example of a single electron in a rigidly rotating octahedral environment is calculated using both this formalism and that used previously by Aharonov and Anandan and by Page (1987).
In this paper, the exact solution and time-evolution operator for the system with a su(1,1)?s{\mathrm{?}}_{\mathit{s}}?s​h(4) Hamiltonian are found by making use of Lewis-Riesenfeld quantum theory. The time-evolution operator is then used to derive the Aharonov-Anandan phase. Finally, the noncyclic evolution is discussed with the help of the Aharonov-Anandan phase and the Berry phase is calculated by taking the adiabatic limit.
We evaluate the exact (Pancharatnam) phase differences between the final state \ensuremath{\Vert}\ensuremath{\psi}(t)〉 and various initial states for a spin-1/2 particle in a rotating magnetic field B(t). For initial states \ensuremath{\Vert}n;Bef{\mathbf{B}}_{\mathrm{ef}}Bef​(0)〉, which are eigenstates of the spin component along the direction of the initial effective field Bef{\mathbf{B}}_{\mathrm{ef}}Bef​(0), the exact phase has an energy-dependent part and an energy-independent part. It is shown that these states \ensuremath{\Vert}n;Bef{\mathbf{B}}_{\mathrm{ef}}Bef​(0)〉 are cyclic and their corresponding Aharonov-Anandan phases are evaluated. In the adiabatic limit we discuss different choices of time-dependent bases and the relationship between the exact phase, the Born-Fock-Schiff phase, and Berry's phase. We propose neutron interference experiments to test separately the exact and the adiabatic evolution laws, as well as to measure exact and adiabatic time-dependent phases.