dorsal/arxiv
View SchemaNon-Abelian Geometrical Phase for General Three-Dimensional Quantum Systems
| Authors | Ali Mostafazadeh |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9608031 |
| URL | https://arxiv.org/abs/quant-ph/9608031 |
| DOI | 10.1088/0305-4470/30/21/023 |
| Journal | J.Phys.A30:7525-7535,1997 |
Abstract
Adiabatic $U(2)$ geometric phases are studied for arbitrary quantum systems with a three-dimensional Hilbert space. Necessary and sufficient conditions for the occurrence of the non-Abelian geometrical phases are obtained without actually solving the full eigenvalue problem for the instantaneous Hamiltonian. The parameter space of such systems which has the structure of $\xC P^2$ is explicitly constructed. The results of this article are applicable for arbitrary multipole interaction Hamiltonians $H=Q^{i_1,\cdots i_n}J_{i_1}\cdots J_{i_n}$ and their linear combinations for spin $j=1$ systems. In particular it is shown that the nuclear quadrupole Hamiltonian $H=Q^{ij}J_iJ_j$ does actually lead to non-Abelian geometric phases for $j=1$. This system, being bosonic, is time-reversal-invariant. Therefore it cannot support Abelian adiabatic geometrical phases.
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"abstract": "Adiabatic $U(2)$ geometric phases are studied for arbitrary quantum systems\nwith a three-dimensional Hilbert space. Necessary and sufficient conditions for\nthe occurrence of the non-Abelian geometrical phases are obtained without\nactually solving the full eigenvalue problem for the instantaneous Hamiltonian.\nThe parameter space of such systems which has the structure of $\\xC P^2$ is\nexplicitly constructed. The results of this article are applicable for\narbitrary multipole interaction Hamiltonians $H=Q^{i_1,\\cdots i_n}J_{i_1}\\cdots\nJ_{i_n}$ and their linear combinations for spin $j=1$ systems. In particular it\nis shown that the nuclear quadrupole Hamiltonian $H=Q^{ij}J_iJ_j$ does actually\nlead to non-Abelian geometric phases for $j=1$. This system, being bosonic, is\ntime-reversal-invariant. Therefore it cannot support Abelian adiabatic\ngeometrical phases.",
"arxiv_id": "quant-ph/9608031",
"authors": [
"Ali Mostafazadeh"
],
"categories": [
"quant-ph",
"hep-th"
],
"doi": "10.1088/0305-4470/30/21/023",
"journal_ref": "J.Phys.A30:7525-7535,1997",
"title": "Non-Abelian Geometrical Phase for General Three-Dimensional Quantum Systems",
"url": "https://arxiv.org/abs/quant-ph/9608031"
},
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