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Time Optimal Control for Quantum Systems with SU(1,1) Symmetry

吴建武, 李春文, 张婧

Published Aug 25, 2009• 0 citations
Publisher

Abstract

Based on Lie group decomposition,the time optimal control problem is discussed for quantum systems with SU(1,1) symmetry.Minimal time required for realizing a specified evolution operator is calculated for two cases that the control Hamiltonian is elliptical and parabolical.Moreover,an upper bound is provided for the case that the control Hamiltonian is hyperbolical.

DOI
10.12341/jssms08378

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Based on Lie group decomposition,the time optimal control problem is discussed for quantum systems with SU(1,1) symmetry.Minimal time required for realizing a specified evolution operator is calculated for two cases that the control Hamiltonian is elliptical and parabolical.Moreover,an upper bound is provided for the case that the control Hamiltonian is hyperbolical.
This paper examines the controllability for quantum control systems with SU(1,1) dynamical symmetry, namely, the ability to use some electromagnetic field to redirect the quantum system toward a desired evolution. The problem is formalized as the control of a right invariant bilinear system evolving on the Lie group SU(1,1) of two dimensional special pseudo-unitary matrices. It is proved that the elliptic condition of the total Hamiltonian is both sufficient and necessary for the controllability. Conditions are also given for small time local controllability and strong controllability. The results obtained are also valid for the control systems on the Lie groups SO(2,1) and SL(2,R).
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 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 study the time optimal control problem for the evolution operator of an n-level quantum system from the identity to any desired final condition. For the considered class of quantum systems the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. From a mathematical perspective, such a problem is a sub-Riemannian K-P problem, whose underlying symmetric space is SU(n)/S(U(n-1) x U(1)). Following the method of symmetry reduction, we consider the action of S(U(n-1) xU(1)) on SU(n) as a conjugation X ---> AXA^{-1}. This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels (Lambda-systems). We solve this latter problem by reducing it to an integer quadratic optimization problem with linear constraints.
Abstract In this paper we study the relationship between factorization problems on SU(2n) or more generally on compact Lie groups G and time optimal control problems. Both types of problems naturally arise in physics, such as in quantum computing and in controlling coupled spin systems (NMR‐spectroscopy). In the first part we show that certain factorization problems can be reformulated as time optimal control problems on G. In the second part a necessary condition for the existence of finite optimal factorizations is discussed. At the end we illustrate our results by an example on Euler angle factorizations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We consider the problem of time-optimal control for a three-level quantum system where one level is coupled by the control field to the lowest two, which are not coupled to each other. A bound is assumed on the norm of the control at every time. Such a problem belongs to the class of KP sub-Riemannian problems for which we can perform a symmetry reduction and reduce to a Riemannian problem on the associated quotient space. We prove several properties of such a quotient space in our case, including the fact that it is an example of an almost-Kähler manifold that is not Kähler. We provide the explicit form of the optimal controls for any unitary transformation on the lowest two levels and discuss the geometric and practical implications of this result.