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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

An investigation of the antiferromagnet-based NMR quantum register in inhomogeneous magnetic field

Alexander A. Kokin, V. A. Kokin

Published Mar 3, 2008• 5 citations
Publisher

Abstract

It is given the refinements and further development of NMR model of quantum register based on easy-axis nuclear spin-free antiferromagnet, which was presented previously in paper. The model suggests that external magnetic field is directed along the easy axis, normally to the plate surface and has a weak constant gradient along the nuclear spin chain of quantum register. In present paper it is evaluated the general expression for indirect coupling between nuclear spins due to hyperfine nuclear-electron coupling in substituted atoms and spin-wave propagation in antiferromagnet close to critical point of quantum phase transition in antiferromagnet of spin-flop type. It is given also an estimation of quantum states decoherence rate, which is caused by the interaction of nuclear spins with spin waves in antiferromagnet.

DOI
10.1117/12.801902

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It is given the refinements and further development of NMR model of quantum register based on easy-axis nuclear spin-free antiferromagnet, which was presented previously in paper. The model suggests that external magnetic field is directed along the easy axis, normally to the plate surface and has a weak constant gradient along the nuclear spin chain of quantum register. In present paper it is evaluated the general expression for indirect coupling between nuclear spins due to hyperfine nuclear-electron coupling in substituted atoms and spin-wave propagation in antiferromagnet close to critical point of quantum phase transition in antiferromagnet of spin-flop type. It is given also an estimation of quantum states decoherence rate, which is caused by the interaction of nuclear spins with spin waves in antiferromagnet.
As a nuclear spin model of scalable quantum register, the one-dimensional chain of the magnetic atoms with nuclear spins 1/2 substituting the basic atoms in the plate of nuclear spin free easy-axis 3D antiferromagnet is considered. It is formulated the generalized antiferromagnet Hamiltonian in spin-wave approximation (low temperatures) considering the inhomogeneous external magnetic field, which is directed along the easy axis normally to plane of the plate and has a constant gradient along the nuclear spin chain. Assuming a weak gradient, the asymptotic expression for coefficients of unitary transformations to the diagonal form of antiferromagnet Hamiltonian is found. With this result the expression for indirect interspin coupling, which is due to hyperfine nuclear electron coupling in atoms and the virtual spin wave propagation in antiferromagnet ground state, was evaluated. It is shown that the inhomogeneous magnetic field essentially modifies the characteristics of indirect interspin coupling. The indirect interaction essentially grows and even oscillates in relation to the interspin distance when the local field value in the middle point of two considered nuclear spin is close to the critical field for quantum phase transition of spin-flop type in bulk antiferromagnet or close to antiferromagnetic resonance. Thus, the external magnetic field, its gradient, microwave frequency and power can play the role of control parameters for qubit states. Finally, the one and two qubit states decoherence and longitudinal relaxation rate are caused by the interaction of nuclear spins with virtual spin waves in antiferromagnet ground state are calculated.
It is considered the indirect inter-qubit coupling in 1D chain of atoms with nuclear spins 1/2 which plays role of qubits fin the quantum register. This chain of the atoms is placed by regular way in easy-axis 3D antiferromagnetic thin plate substrate which is cleaned from the other nuclear spin containing isotopes. It is shown that the range of indirect inter-spin coupling may run to a great number of lattice constants both near critical point of quantum phase transition in antiferromagnet of spin-flop type (control parameter is external magnetic field) and/or near homogeneous antiferromagnetic resonance (control parameter is microwave frequency).
This study deals with the further development of nuclear spin model of scalable quantum register, which presents the one-dimensional chain of the magnetic atoms with nuclear spins 1/2, substituting the basic atoms in the plate of nuclear spin-free easy-axis 3D antiferromagnet. The decoherence rates of one qubit state and entanglement state of two removed qubits and longitudinal relaxation rates are caused by the interaction of nuclear spins-qubits with virtual spin waves in antiferromagnet ground state were calculated. It was considered also one qubit adiabatic decoherence, is caused by the interaction of nuclear spin of quantum register with nuclear spins of randomly distributed isotopes, substituting the basic nuclear spin-free isotopes of antiferromagnet. We have considered finally encoded DFS (Decoherence-Free Subspaces) logical qubits are constructed on clusters of the four-physical qubits, given by the two states with zero total angular momentum.
This article presents the results of studies of the linear antiferromagnetic spin chain with uniaxial anisotropy (XXZ– model) placed in a transverse magnetic field directed along the anisotropy axis of the chain with a constant gradient along the chain. A spin Hamiltonian for the states close to the ground was written in the spin-wave approximation. It was diagonalized by an asymptotic expansion in the small parameter describing the magnitude of the field gradient. Local energy of the ground state and the reduced density matrix of the quantum states for a selected pair of spins in the chain have been considered in the gapped and gapless cases. Local spin polarization and the local toroidal magnetic moment have also been found. Local pair entanglement of quantum states as a characteristic of quantum correlations in the spin chain was studied.
Using the spin wave approximation, we study the decoherence dynamics of a central spin coupled to an antiferromagnetic environment under the application of an external global magnetic field. The external magnetic field affects the decoherence process through its effect on the antiferromagnetic environment. It is shown explicitly that the decoherence factor which displays a Gaussian decay with time depends on the strength of the external magnetic field and the crystal anisotropy field in the antiferromagnetic environment. When the strength of the external magnetic field is increased to the critical field point at which the spin–flop transition (a first-order quantum phase transition) happens in the antiferromagnetic environment, the decoherence of the central spin reaches its highest point. This result is consistent with several recent quantum phase transition witness studies. The influences of the environmental temperature on the decoherence behaviour of the central spin are also investigated.
The one-dimensional homonuclear periodic array of nuclear spins I = 1/2, owing to hyperfine interaction of nuclear spins with electronic magnetic moments in antiferromagnetic structure, is considered. The neighbor nuclear spins in such array are opposite oriented and have resonant frequencies determined by hyperfine interaction constant, applied magnetic field value and interaction with the left and right nuclear neighbor spins. The resonant frequencies difference of nuclear spins, when the neighbor spins have different and the same states, is used to control the spin dynamics by means of selective resonant RF-pulses both for single nuclear spins and for ensemble of nuclear spins with the same resonant frequency. A model for the NMR quantum computer of cellular-automata type based on an one-dimensional homonuclear periodic array of spins is proposed. This model may be generalized to a large ensemble of parallel working one-dimensional arrays and to two-dimensional and three-dimensional structures.