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Journal of Hebei North University

Fractal in Mixed State Geometric Phase

Yanxia Liu

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The Weierstrass function approach is applied to investigate the fractal of geometric phase.A noninteracting 4-spin-12 system is considered,of which each spin is driven by a local and rotating magnetic field.A mixed state of this system is constructed.The adiabatic geometric phase for the mixed state is calculated.The mixed state geometric phase is studied under two conditions:① adiabatic approximation;②letting the geometric phases in the mixed state small.The geometric phase for the mixed state becomes an average of the geometric phases of the pure states,weighed by their eigenvalues.The mixed state geometric phase is constructed into the form of Weierstrass function.A novel fractal evolution of the geometric phase is found.The evolution curve of the geometric phase is presented in a special example.The box-counting dimension of the evolution curve of the geometric phase is calculated.

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The Weierstrass function approach is applied to investigate the fractal of geometric phase.A noninteracting 4-spin-12 system is considered,of which each spin is driven by a local and rotating magnetic field.A mixed state of this system is constructed.The adiabatic geometric phase for the mixed state is calculated.The mixed state geometric phase is studied under two conditions:① adiabatic approximation;②letting the geometric phases in the mixed state small.The geometric phase for the mixed state becomes an average of the geometric phases of the pure states,weighed by their eigenvalues.The mixed state geometric phase is constructed into the form of Weierstrass function.A novel fractal evolution of the geometric phase is found.The evolution curve of the geometric phase is presented in a special example.The box-counting dimension of the evolution curve of the geometric phase is calculated.
The geometric phase for states of two spin-1/2 particles in rotating magnetic field is calculated, in particular, the noncyclic and cyclic non-adiabatic phases for the general case are explicitly derived and discussed. We find that the cyclic geometric phase for the entangled state can always be written as a sum of the phases of the two particles respectively; the same cannot be said for the noncyclic phase. We also investigate the geometric phase of mixed state of one particle in a biparticle system, and we find that the geometric phase for one subsystem of an entangled system is always affected by another subsystem of the entangled system.
Quantum mechanical methods for getting geometric phases for mixed states are analyzed. Parallel transport equations for pure states are generalized to mixed states by which dynamical phases are eliminated. The geometric phases are derived by SU(2) transformations of mixed states wgeometric phase,hich are different from those used in NMR and neutron interference experiments. The mixed states are obtained as two-dimensional thermal states. The SU(2) transformations include time dependent parameters which are function of the magnetic fields interactions in times which are short relative to relaxation times. These conditions lead to a special form of the unitary transformation of the mixed thermal states by which results for geometric phase and for interference intensity are derived
Quantum mechanical methods for getting geometric phases for mixed states are analyzed. Parallel transport equations for pure states are generalized to mixed states by which dynamical phases are eliminated. The geometric phases are derived by SU(2) transformations of mixed states wgeometric phase,hich are different from those used in NMR and neutron interference experiments. The mixed states are obtained as two-dimensional thermal states. The SU(2) transformations include time dependent parameters which are function of the magnetic fields interactions in times which are short relative to relaxation times. These conditions lead to a special form of the unitary transformation of the mixed thermal states by which results for geometric phase and for interference intensity are derived
Quantum mechanical methods for getting geometric phases for mixed states are analyzed. Parallel transport equations for pure states are generalized to mixed states by which dynamical phases are eliminated. The geometric phases of mixed states are obtained as Pancharatnam phases which are valid also for open cycles. The geometric phases are derived here by SU(2) transformations of mixed thermal states which are different from those used in NMR and neutron interference experiments. While the zeroth order Hamiltonian is given by the interaction of a magnetic moment and constant magnetic field in the z direction, the high order perturbations assumed in the present article are composed of two oscillating magnetic fields in the same z direction. These assumptions lead to a special form of the SU(2) unitary transformation of the mixed thermal states by which results for geometric phase and for interference intensity are derived.
The geometric phases of cyclic evolutions for mixed states are discussed in the framework of unitary evolution. A canonical 1-form is defined whose line integral gives the geometric phase, which is gauge invariant. It reduces to the Aharonov and Anandan phase in the pure state case. Our definition is consistent with the phase shift in the proposed experiment (Sjöqvist et al 2000 Phys. Rev. Lett. 85 2845) for a cyclic evolution if the unitary transformation satisfies the parallel transport condition. A comprehensive geometric interpretation is also given. It shows that the geometric phases for mixed states share the same geometric sense with the pure states.