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arXiv:1109.5412 · Physical Review A

Time-optimal synthesis of SU(2) transformations for a spin-1/2 system

A. David. Boozer

Published Jan 19, 2012• 53 citations
arXivHTMLPDF

Abstract

We consider a quantum control problem involving a spin-$1/2$ particle in a magnetic field. The magnitude of the field is held constant, and the direction of the field, which is constrained to lie in the $x--y$ plane, serves as a control parameter that can be varied to govern the evolution of the system. We analytically solve for the time dependence of the control parameter that will synthesize a given target SU(2) transformation in the least possible amount of time, and we show that the time-optimal solutions have a simple geometric interpretation in terms of the fiber bundle structure of SU(2). We also generalize our time-optimal solutions to a control problem that includes a constant bias field along the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{z}}$ axis and to the case of inhomogeneous control, in which a single control parameter governs the evolution of an ensemble of spin-$1/2$ systems.

DOI
10.1103/physreva.85.012317

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In this paper we consider the minimum time population transfer problem for the z component of the spin of a (spin 1/2) particle, driven by a magnetic field, that is constant along the z axis and controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e., after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on two-dimensional (2-D) manifolds. Let (−E,E) be the two energy levels, and ∣Ω(t)∣≤M the bound on the field amplitude. For each couple of values E and M, we determine the time optimal synthesis starting from the level −E, and we provide the explicit expression of the time optimal trajectories, steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For M∕E≪1, every time optimal trajectory is bang-bang and, in particular, the corresponding control is periodic with frequency of the order of the resonance frequency ωR=2E. On the other side, for M∕E>1, the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. For fixed E, we also prove that for M→∞ the time needed to reach the state two tends to zero. In the case M∕E>1 there are time optimal trajectories containing a singular arc. Finally, we compare these results with some known results of Khaneja, Brockett, and Glaser and with those obtained by controlling the magnetic field both on the x and y directions (or with one external field, but in the rotating wave approximation). As a byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns that cyclically alternate as M∕E→0, giving a partial proof of a conjecture formulated in a previous paper.
Given a compact, connected Lie group G with Lie algebra � . We discuss time-optimal control of bilinear systems of the form (I) where H d , H j ∈ � , U ∈ G , and the v j act as control variables. The case G = SU (2 n ) has found interesting appli- cations to questions of time-optimal control of spin systems. In this context Eq. (I) describes the dynamics of an n -particle system with fixed drift Hamiltonian H d , which is to be controlled by a number of exterior magnetic fields of variable strength, proportional to the parameters v j . The question of interest here is to transfer the sys- tem from a given initial state U 0 to a prescribed final state U 1 in least possible time. Denote bythe Lie algebra spanned by H 1 , …, H m , and by K the corresponding Lie subgroup of G. After reformulating the optimal control problem for system (I) in terms of an equivalent problem on the homogeneous space G / K we discuss in detail time-optimal control strategies for system (I) in the case where G / K carries the structure of a Riemannian sym- metric space.
We consider the problem of steering control via an input electro-magnetic field for a system of two interacting spin 1/2 particles. This model is of interest in applications because it is used to perform logic operations in quantum computing that involve two quantum bits. The describing model is a bilinear system whose state varies on the Lie group of special unitary matrices of dimension 4, SU(4). By using decompositions of the latter Lie group, the problem can be decomposed into a number of subproblems for a system whose state varies on the (smaller) Lie group of 4/spl times/4 proper orthogonal matrices, SO(4). We tackle the time optimal control problem for this system and show that the extremals can be computed explicitly and they are the superposition of a constant field and a sinusoidal one.
We study a quantum spin system acting on a single quantum bit. The evolution of this system is governed by the Schrodinger equation which takes the form of a right-invariant system on the special unitary group SU(2) with two control inputs. Using a suitable version of Pontryagin's Principle which is tailor-made for control problems on Lie groups, the optimal controls are derived in two cases: the energy-optimal case (in which the control effort is minimized for a specified end time) and the time-optimal case (in which the control duration is minimized for given constraints on the size of the controls).
We consider the time-optimal control of a spin-1/2 particle whose dynamics is governed by the Bloch equations with both longitudinal and transverse relaxation terms. We solve this control problem by using geometric optimal control techniques. We show the crucial role of singular extremals in the time-optimal synthesis. This role can mainly be attributed to the presence of dissipation. We also analyse the robustness of the optimal control sequence when both the maximum amplitude of the control field and the dissipative parameters are varied. Finally, we present an experimental implementation of the different solutions using techniques of nuclear magnetic resonance.