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.
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References
Albanese, C., Christandl, M., Datta, N., Ekert, A.: Mirror inversion of quantum states in linear registers. Phys. Rev. Lett. 93(23), 230502 (2004)
Atakishiyev, N., Pogosyan, G., Wolf, K.: Finite models of the oscillator. Phys. Part. Nucl. 36(3), 247–265 (2005)
Atakishiyev, N.M., Pogosyan, G.S., Vicent, L.E., Wolf, K.B.: Finite two-dimensional oscillator: I. The Cartesian model. J. Phys. A 34, 9381–9398 (2001)
Atakishiyev, N.M., Pogosyan, G.S., Vinet, L.E., Wolf, K.B.: Finite two-dimensional oscillator: II. The radial model. J. Phys. A 34, 9399–9415 (2001)
Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91(20), 207901 (2003)
Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92(18), 187902 (2004)
Genest, V.X., Vinet, L., Zhedanov, A.: The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states. J. Phys. A, Math. Theor. 46(50), 505,203 (2013)
Grünbaum, F., Rahman, M.: On a family of 2-variable orthogonal Krawtchouk polynomials. SIGMA 6, 090 (2010)
Iliev, P., Terwilliger, P.: The Rahman polynomials and the lie algebra sl 3(c). Trans. Am. Math. Soc. 364(8), 4225–4238 (2012)
Kay, A.: Perfect, efficient, state transfer and its application as a constructive tool. Int. J. Quantum Inf. 8(04), 641–676 (2010)
Levi, D., Olver, P., Thomova, Z., Winternitz, P. (eds.): Symmetries and Integrability of Difference Equations. London Mathematical Society Lecture Note Series, vol. 381. Cambridge University Press, Cambridge (2011)
Louck, J., Moshinsky, M., Wolf, K.: Canonical transformations and accidental degeneracy. I. The anisotropic oscillator. J. Math. Phys. 14(6), 692–695 (2003)
Miki, H., Post, S., Vinet, L., Zhedanov, A.: A finite model of the oscillator in two-dimensions with SU(2) symmetry. J. Phys. A, Math. Theor. 46, 125,207 (2013)
Miki, H., Tsujimoto, S., Vinet, L., Zhedanov, A.: Quantum state transfer in a two-dimensional regular spin lattice of triangular shape. Phys. Rev. A 85, 062306 (2012)
Miller, W. Jr, Post, S., Winternitz, P.: Classical and quantum superintegrability with applications. J. Phys. A, Math. Theor. 46(42), 423,001 (2013)
Subrahmanyam, V.: Entanglement dynamics and quantum-state transport in spin chains. Phys. Rev. A 69(3), 034304 (2004)
Vinet, L., Zhedanov, A.: How to construct spin chains with perfect state transfer. Phys. Rev. A 85(1), 012323 (2012)
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Post, S. Quantum Perfect State Transfer in a 2D Lattice. Acta Appl Math 135, 209–224 (2015). https://doi.org/10.1007/s10440-014-9953-5
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DOI: https://doi.org/10.1007/s10440-014-9953-5