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Geometric phases for a thermal two-dimensional mixed spin 1/2 system

Y. Ben-Aryeh

Published Jul 7, 2023• 0 citations
Publisher

Abstract

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

DOI
10.1364/opticaopen.23641206.v1

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Geometric phases for a thermal two-dimensional mixed spin 1/2 system

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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

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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.
Quantum Carnot engine whose working medium is a two-dimensional spin 1/2 system, with a time-dependent magnetic field in the symmetric z direction is described. The dynamic of this engine is obtained by using four steps, where in two steps the system is coupled alternatively to hot and cold heat baths, and in the other two steps the time development is adiabatic and isentropic (with constant entropy). The conditions for getting a reversible Carnot cycle and the role of time duration for its irreversibility are discussed. Since the calculations are made for the expectation values of the Hamiltonian, only dynamical phases are obtained which cannot be used for interference effects. An alternative method is developed for getting geometric phases, which can be used in interferometry. Parallel transport equations for pure states are generalized to mixed states by which dynamical phases are eliminated. The geometric phases are derived by unitary SU(2) transformations, including time-dependent parameters which are a function of the magnetic fields interactions. A special form of the unitary transformation for the mixed thermal states is developed, by which geometric phases are obtained, which are different from those obtained in NMR and neutron interferometry.
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.
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.