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Physical Review A

Entanglement dynamics of spin systems in pure states

G. B. Furman, V. Meerovich, Vladimir Sokolovsky

Published Sep 16, 2009• 5 citations
Publisher

Abstract

We investigate numerically the appearance and evolution of entanglement in spin systems prepared initially in a pure state. We consider the dipolar coupling spin systems of different molecular structures: benzene ${\text{C}}_{6}{\text{H}}_{6}$, cyclopentane ${\text{C}}_{5}{\text{H}}_{10}$, sodium butyrate ${\text{CH}}_{3}{({\text{CH}}_{2})}_{2}{\text{CO}}_{2}\text{Na}$, and calcium hydroxyapatite ${\text{Ca}}_{5}(\text{OH}){({\text{PO}}_{4})}_{3}$. Numerical simulations show that the close relationship exists between the intensity of second order (2Q) coherences and concurrences of nearest spins in a cyclopentane molecule.

DOI
10.1103/physreva.80.032316

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The dynamics of entanglement is investigated on the basis of exactly solvable models of multiple quantum (MQ) NMR spin dynamics. It is shown that the time evolution of MQ coherences of systems of coupled nuclear spins in solids is directly connected with dynamics of the quantum entanglement. We studied analytically the dynamics of entangled states for two- and three-spin systems coupled by the dipole-dipole interaction. In this case the dynamics of the quantum entanglement is uniquely determined by the time evolution of MQ coherences of the second order. The real part of the density matrix describing MQ dynamics in solids is responsible for MQ coherences of the zeroth order while its imaginary part is responsible for the second order. Thus, one can conclude that the dynamics of the entanglement is connected with transitions from the real part of the density matrix to the imaginary one, and vice versa. A pure state which generalizes the Greenberger-Horne-Zeilinger (GHZ) and W states is found. Different measures of the entanglement of this state are analyzed for tripartite systems.
We investigate numerically the time dependence of the multiple quantum coherences and entanglement in linear chains up to nine nuclear spins of 1∕21∕21∕2 coupled by the dipole-dipole interactions. Two models are considered: (1) a spin chain with nearest-neighbor dipole-dipole interactions and (2) a more realistic model with interactions between all spins. It is shown that the entangled states appear between remote particles which do not interact directly (model 1), while the interaction between all spins (model 2) not always results in entanglement between remote spins.
The evolution of entanglement in a 3-spin chain with nearest-neighbor Heisenberg-XY interactions for different initial states is investigated here. In an NMR experimental implementation, we generate multipartite entangled states starting from initial separable pseudo-pure states by simulating nearest-neighbor XY interactions in a 3-spin linear chain of nuclear spin qubits. For simulating XY interactions, we follow algebraic method of Zhang et al. [Phys. Rev. A72 (2005) 012331]. Bell state between end qubits has been generated by using only the unitary evolution of the XY Hamiltonian. For generating W-state and GHZ-state a single qubit rotation is applied on second and all the three qubits, respectively after the unitary evolution of the XY Hamiltonian.
Dynamics of zeroth order quantum coherences and preparation of the pseudopure states in homonuclear systems of dipolar coupling spins is closely examined. It has been shown an extreme important role of the non-diagonal part of zeroth order coherence in construction of the pseudopure state. Simulations of the preparation process of pseudopure states with the real molecular structures (a rectangular (-chloro- -nitrobenzene molecule), a chain (hydroxyapatite molecule), a ring (benzene molecule), and a double ring (cyclopentane molecule)) open the way to experimental testing of the obtained results.
We study the dynamics of entanglement in spin gases. A spin gas consists of a (large) number of interacting particles whose random motion is described classically while their internal degrees of freedom are described quantum-mechanically. We determine the entanglement that occurs naturally in such systems for specific types of quantum interactions. At the same time, these systems provide microscopic models for non–Markovian decoherence: the interaction of a group of particles with other particles belonging to a background gas are treated exactly, and differences between collective and non–collective decoherence processes are studied. We give quantitative results for the Boltzmann gas and also for a lattice gas, which could be realized by neutral atoms hopping in an optical lattice. These models can be simulated efficiently for systems of mesoscopic sizes (N ~ 105).