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Automation and Remote Control

Optimization of processes in a spin chain

Гурман Владимир Иосифович, Irina Viktorovna Rasina

Published Dec 1, 2014• 3 citations
Publisher

Abstract

There are considered the problems of optimal excitation transfer in spin chains on the base of Shrödinger equation for different types of Hamiltonians containing linear control. Exact analytic solutions for the Landau-Zener model and like 2-spin systems are obtained via double transformation to derived system known from the theory of degenerate problems. An approach to application of these solutions for the investigation of multi-spin chains is proposed.

DOI
10.1134/s0005117914120108

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There are considered the problems of optimal excitation transfer in spin chains on the base of Shrödinger equation for different types of Hamiltonians containing linear control. Exact analytic solutions for the Landau-Zener model and like 2-spin systems are obtained via double transformation to derived system known from the theory of degenerate problems. An approach to application of these solutions for the investigation of multi-spin chains is proposed.
It is considered the problem of excitation transfer along a spin chain related to the applied problem of quantum computations.The model of a quantum system of interacting spins based on the Shrödinger equation with unbounded linear control is transformed to an equivalent derived system (known from the degenerate problems theory), and then approximately to derived systems of higher stages with reducing order.Their investigation performed analytically or via simple computations leads at least to approximate solutions and lower estimates of the transfer time, which can be used in subsequent improving procedures.
A revised formulation of the problem of fastest transfer of the excitation in a spin chain is considered on the base of Shrödinger equation which Hamiltonian depends linearly on control. It is taken into account that the excitation of the first or last spin means that it has greatest amplitude equal to the chain invariant whereas its phase is undefined and can be considered as an additional control variable. The role of this additional control is analyzed via transformation of the original problem with unbounded linear control to the regular derived problem known from the theory of degenerate problems [1, 2], in the same way as in [2]. The overall procedure is demonstrated in computational experiments with the use of visual examples.
The exact analytical solution of the degenerate Landau-Zener model, wherein two bands of degenerate energies cross in time, is presented. The solution is derived by using the Morris-Shore transformation, which reduces the fully coupled system to a set of independent nondegenerate two-state systems and a set of decoupled states. Due to the divergence of the phase of the off-diagonal element of the propagator in the original Landau-Zener model, not all transition probabilities exist for infinite time duration. In general, apart from some special cases, only the transition probabilities between states within the same degenerate set exist, but not between states of different sets. An illustration is presented for the transition between the magnetic sublevels of two atomic levels with total angular momenta J=2J=2J=2 and 1.
The author considers solutions of the Yang-Baxter equation such that the logarithmic derivative of the transfer matrix yields a quantum spin Hamiltonian which is isotropic in spin space, i.e. SU(2)-invariant. Four such solutions are known for each value of the spin S. (For S=1/2 they degenerate into the same solution, and for S=1 they only give three different solutions). For S<or=6 he shows that these are the only solutions which are SU(2)-invariant, except for S=3 when there is a fifth solution.