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arXiv:0902.3521 · Annals of Physics

Unitary transformations purely based on geometric phases of two spins with Ising interaction in a rotating magnetic field

Yu Shi

Published Apr 14, 2010• 6 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2010.02.014

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We investigate how to obtain a non-trivial geometric phase gate for a two-qubit spin chain, with Ising interaction in different magnetic fields. Indeed, one of the spins is driven by a time-varying rotating magnetic field, and the other is coupled with a static magnetic field in the direction of the rotation axis. This is an interesting problem both for the purpose of measuring the geometric phases and in quantum computing applications. It is shown that the static magnetic field does not change the adiabatic states of the system, and it does not affect the geometric phases, whereas it may be used to control the dynamic phases. In addition, by considering the exact two-spin adiabatic geometric phases, we find that a non-trivial two-spin unitary transformation, purely based on Berry phases, can be obtained by using two consecutive cycles with opposite directions of the magnetic fields, opposite signs of the interaction constant and the phase shift of the rotating magnetic field. In addition, in the non-adiabatic case, starting with a certain initial state, a cycle can be achieved and thus the Aharonov–Anandan phase is calculated.

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Geometric Phase of a Heisenberg-chain

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Objective The algebriaic dynamics is applied to investigate the geometric phase of a Heisenberg-chain,in order to find out how the coulping strengths between spins influence the geometric phase of this system.Method An interacting anisotropic 3-spin 1/2 system,which is driven by a rotating magnetic field,is considered.The system has a su(2) algebraic structure.By an unitary gauge transformation,this nonautonomous system becomes an autonomous system.The Schrodinger equation of the autonomous system is solved.By virtue of the inverse transformation of the original gauge transformation,the exact analytical solution to the time dependent Schrodinger equation is obtained in total spin 1/2 subspace.Resutlt The geometric phases in this subspace are calculated.Conclusion The result shows that(i)Coulping strengths don't influence the geometric phases of this system;(ii) The geometric phases of the system is the same as that of one spin 1/2 system in a rotating magnetic field.

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Quantum thermal engine with spin 1/2 system and geometric phases and interference obtained by unitary transformations of mixed states

Y. Ben-Aryeh

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.

International Journal of Thermodynamics•Jun 1, 2025•0 citations
We investigate how to obtain a non-trivial geometric phase gate for a two-qubit spin chain, with Ising interaction in different magnetic fields. Indeed, one of the spins is driven by a time-varying rotating magnetic field, and the other is coupled with a static magnetic field in the direction of the rotation axis. This is an interesting problem both for the purpose of measuring the geometric phases and in quantum computing applications. It is shown that the static magnetic field does not change the adiabatic states of the system, and it does not affect the geometric phases, whereas it may be used to control the dynamic phases. In addition, by considering the exact two-spin adiabatic geometric phases, we find that a non-trivial two-spin unitary transformation, purely based on Berry phases, can be obtained by using two consecutive cycles with opposite directions of the magnetic fields, opposite signs of the interaction constant and the phase shift of the rotating magnetic field. In addition, in the non-adiabatic case, starting with a certain initial state, a cycle can be achieved and thus the Aharonov–Anandan phase is calculated.
Objective The algebriaic dynamics is applied to investigate the geometric phase of a Heisenberg-chain,in order to find out how the coulping strengths between spins influence the geometric phase of this system.Method An interacting anisotropic 3-spin 1/2 system,which is driven by a rotating magnetic field,is considered.The system has a su(2) algebraic structure.By an unitary gauge transformation,this nonautonomous system becomes an autonomous system.The Schrodinger equation of the autonomous system is solved.By virtue of the inverse transformation of the original gauge transformation,the exact analytical solution to the time dependent Schrodinger equation is obtained in total spin 1/2 subspace.Resutlt The geometric phases in this subspace are calculated.Conclusion The result shows that(i)Coulping strengths don't influence the geometric phases of this system;(ii) The geometric phases of the system is the same as that of one spin 1/2 system in a rotating magnetic field.
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.