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arXiv:quant-ph/0407236 · Physical Review A

Model for entangled states with spin-spin interaction

Yakir Aharonov, J. Anandan, G. Jordan Maclay, Jun Suzuki

Published Nov 24, 2004• 4 citations

Abstract

A system consisting of two neutral spin-1/2 particles is analyzed for two magnetic field perturbations: (1) an inhomogeneous magnetic field over all space, and (2) external fields over a half space containing only one of the particles. The field is chosen to point from one particle to the other, which results in essentially a one-dimensional problem. A number of interesting features are revealed for the first case: the singlet, which has zero potential energy in the unperturbed case, remains unstable in the perturbing field. The spin-zero component of the triplet evolves into a bound state with a double well potential, with the possibility of tunneling. Superposition states can be constructed which oscillate between entangled and unentangled states. For the second case, we show that changes in the magnetic field around one particle affect measurements of the spin of the entangled particle not in the magnetic field nonlocally. By using protective measurements, we show it is possible in principle to establish a nonlocal interaction using the two particles, provided the dipole-dipole potential energy does not vanish and is comparable to the potential energy of the particle in the external field.

DOI: 10.1103/physreva.70.052114

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Entanglement is the superposition of two or more particles in multiple states. An example is given for the double-well potential; another involves particles with spin 1/2. Measurement of spin is carried out by a Stern–Gerlach apparatus, using the deflection of a spin by a magnet. Entangled spin-1/2 states, namely triplet states having total spin 1, and the singlet state, with spin 0, are also examined; the latter state appears in same form when expressed in spin up/down along any Cartesian coordinate. The determination of one of the spins in an entangled singlet state immediately determines the other, resulting in a “spooky action at a distance.” Einstein, Podolsky, and Rosen (EPR) claimed to show, on the basis of local realism, that quantum mechanics must be incomplete because of this feature. Their logic suggests that one might explain the instantaneous results by invoking “hidden variables” not contained in a quantum treatment.

We examine spinor Bose-Einstein condensates in optical superlattices theoretically using a Bose-Hubbard Hamiltonian that takes spin effects into account. Assuming that a small number of spin-1 bosons is loaded in an optical potential, we study single-particle tunneling that occurs when one lattice site is ramped up relative to a neighboring site. Spin-dependent effects modify the tunneling events in a qualitative and quantitative way. Depending on the asymmetry of the double well, different types of magnetic order occur, making the system of spin-1 bosons in an optical superlattice a model for mesoscopic magnetism. We use a double-well potential as a unit cell for a one-dimensional superlattice. Homogeneous and inhomogeneous magnetic fields are applied, and the effects of the linear and the quadratic Zeeman shifts are examined. We also investigate the bipartite entanglement between the sites and construct states of maximal entanglement. The entanglement in our system is due to both orbital and spin degrees of freedom. We calculate the contribution of orbital and spin entanglements and show that the sum of these two terms gives a lower bound for the total entanglement.

The entanglement dependence of Berry's geometric phase is studied in the case of two spin-1/2 particles with a spin-spin interaction.Berry's phases for two spin-1/2 entanglement particles in the presence of spin-spin interaction and noninteraction are calculated,respectively.The expression of calculating geometric phase,when entanglement vanishes,is obtained.Moreover,the geometric phase is discussed when only one of the two particles is affected by the external magnetic field.It is shown that its geometric phase is not equal to that of single particle which is acted by the external magnetic field,and the result is related toαthat affect the strength of entanglement. A general entanglement-dependence geometric phase is formulated.

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We discuss the thermal and magnetic entanglement in the one-dimensional Kondo necklace model. First, we show how the entanglement naturally present at zero temperature is distributed among pairs of spins according to the strength of the two couplings of the chain, namely, the Kondo exchange interaction and the hopping energy. The effect of temperature and the presence of an external magnetic field are then investigated. We also discuss the adjustment of these variables in order to control the entanglement available in the system. In particular, it is indicated that there exists a critical magnetic field above which the entanglement undergoes a sharp variation, leading the ground state to a completely unentangled phase.