Quantum Perfect State Transfer in a 2D Lattice
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Quantum Perfect State Transfer in a 2D Lattice Sarah Post
Received: 4 November 2013 / Accepted: 23 April 2014 © Springer Science+Business Media Dordrecht 2014
Abstract A finite oscillator model based on two-variable Krawtchouk polynomials is presented and its application to spin dynamics is discussed. The model is defined on a triangular lattice. The conditions that the system admit a form of perfect state transfer where the excitation spectrum is isolated to the boundary of the domain is investigated. We give the necessary bounds on the parameters of the model and a sufficient condition on the ratios of the frequencies of the Hamiltonian. The stronger case where the excitation is isolated at a point is also investigated and shown to exist only in degenerate cases. We then focus on systems with rational frequencies, namely the superintegrable cases and their perfect state transfer properties. We see that these systems interpolate between two, one-dimensional spin chains. By using a parameter in the model as a control parameter, we show that it is possible to steer the excitation spectrum to be isolated at either of the two vertices of the triangle with perfect fidelity. Keywords Discrete models · Spin systems · Harmonic oscillator · Orthogonal polynomials 1 Introduction Recently a two dimensional finite oscillator model was discovered that admits SU(2) symmetry, analogous with the continuous case [13]. The goal of this paper is to investigate a physical model for this system and its symmetries. Namely, to construct a quantum spin system on a triangular lattice and identify the conditions under which it will admit perfect state transfer. Oscillators are one of the most important and well understood models in physics; in classical and quantum physics, harmonic oscillators are paradigmatic examples of integrability and superintegrability. Consider the two-dimensional harmonic oscillator. The classical trajectories and quantum wave functions can be computed explicitly by separating variables in Cartesian coordinates. Indeed, the Hamiltonian is a linear superposition of two,
B
S. Post ( ) Department of Mathematics, University of Hawai‘i, Honolulu, HI 96822, USA e-mail: [email protected]
S. Post
one-dimensional Hamiltonians and as a result the system is integrable. That is, the system admits two conserved quantities in 2D and n quantities when the system is extended to nD. In addition, the most common form of a harmonic oscillator is an isotropic oscillator. This is one in which the frequencies of the one-dimensional Hamiltonians are equal. In this case there is an obvious group symmetry, namely rotational symmetry, and the trajectories or wavefunctions separate in polar coordinates as well. This additional symmetry and corresponding conserved quantity makes the isotropic harmonic oscillator superintegrable. A superintegrable system in nD is one that admits more than n algebrically independent conserved quantities. In fact, even when the frequencies are not equal but have rational ratios it is possible to construct
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