实验观察到两自旋系统量子控制景观中的鞍点

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本文是一篇研究论文，发表于《物理评论A》第91卷，第4期，2015年，题为"实验观察到量子控制景观中的鞍点：一个双量子系统的案例"。随着量子控制实验的日益成功，控制景观分析作为一种理论工具被开发出来，用于解析实验结果。在量子系统中，通过电磁场进行调控时，可观测量作为控制场的函数会形成一个控制景观。理论研究预测了这些景观上存在关键点，包括具有不确定Hessian（雅可比矩阵的二阶导数矩阵）的鞍点。论文的核心内容聚焦于对双量子系统（例如13C标记的氯仿13CHCl3中的两个核自旋）的核磁共振控制实验。作者Qiuyang Sun、István Pelczer、Gregory Riviello、Re-Bing Wu和Herschel Rabitz来自普林斯顿大学化学系和清华大学自动化系，以及量子信息科学和技术中心，他们共同探讨了如何在实际实验条件下观察并验证理论预测的鞍点现象。实验中，他们细致地设计了控制序列，通过对电磁场的精细调整，探索了系统响应的敏感性，并通过数值分析方法解析了控制景观的形状和鞍点特性。通过实验，研究人员观察到了量子控制系统中鞍点的存在，这些鞍点对应着控制参数空间中的局部极小值与极大值之间的临界点，其Hessian矩阵的某些特征值为零，表明局部稳定性和不稳定性同时存在。这种发现对于理解量子控制的复杂性、优化控制策略以及避免不必要的动力学陷阱具有重要意义。此外，这项工作还为未来的量子计算和量子信息处理提供了实证基础，促进了理论与实验的深度融合，推动了量子控制领域的前沿研究。

PHYSICAL REVIEW A 91, 043412 (2015)

Experimental observation of saddle points over the quantum control landscape of a two-spin system

Qiuyang Sun,

Istv

an Pelczer,

Gregory Riviello,

Re-Bing Wu,

and Herschel Rabitz

Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

Department of Automation, Tsinghua University and Center for Quantum Information Science and Technology,

TNList, Beijing 100084, China

(Received 21 August 2014; published 17 April 2015)

The growing successes in performing quantum control experiments motivated the development of control

landscape analysis as a basis to explain these ﬁndings. When a quantum system is controlled by an electromagnetic

ﬁeld, the observable as a functional of the control ﬁeld forms a landscape. Theoretical analyses have predicted

the existence of critical points over the landscapes, including saddle points with indeﬁnite Hessians. This paper

presents a systematic experimental study of quantum control landscape saddle points. Nuclear magnetic resonance

control experiments are performed on a coupled two-spin system in a

C-labeled chloroform (

CHCl

) sample.

We address the saddles with a combined theoretical and experimental approach, measure the Hessian at each

identiﬁed saddle point, and study how their presence can inﬂuence the search effort utilizing a gradient algorithm

to seek an optimal control outcome. The results have signiﬁcance beyond spin systems, as landscape saddles are

expected to be present for the control of broad classes of quantum systems.

DOI: 10.1103/PhysRevA.91.043412 PACS number(s): 32.80.Qk, 37.10.Jk

I. INTRODUCTION

The control of quantum phenomena is gathering increasing

interest for fundamental reasons and potential applications.

Quantum system optimal control is concerned with active

manipulation of physical and chemical processes at the atomic

and molecular scale, such as creation of particular molecular

vibrational excitations, selective breaking of chemical bonds,

and manipulation of electron transport in nanoscale devices

(see [1] for a review). The control objective is generally

addressed through the introduction of a semiclassical elec-

tromagnetic ﬁeld whose shape is honed for the particular

application. The generally successful outcomes of optimal

control experiments and the extensive simulations on model

systems suggest that while searching through the vast space

of possible control ﬁelds it is relatively easy to ﬁnd good

solutions. Seeking a fundamental explanation for this good for-

tune motivated the development of quantum control landscape

analysis [2–4], which provides quantitative predictions on the

landscape features that can be assessed in the laboratory. For

laser ﬁeld manipulation of atomic, molecular, or condensed

phase systems, various experimental complexities can make

quantitative testing of landscape principles challenging. The

advanced nature of nuclear magnetic resonance (NMR) instru-

mentation producing high signal-to-noise ratios (S/N) provides

ideal laboratory circumstances for testing the predictions from

control landscape analysis. In this work we will utilize these

capabilities for the study of a control landscape in a two-spin

system which possesses saddle points.

In quantum control applications, the system evolution

is governed by a time-dependent Hamiltonian including an

applied electromagnetic ﬁeld. The amplitudes, phases, and/or

frequencies of the ﬁeld can be modulated to meet the control

objective, i.e., maximizing or minimizing an experimental

measurable objective J at the target time T . The time interval

T is conventionally chosen to be sufﬁciently long to permit

unfettered control, while being short enough to consider the

dynamics to form a closed system. The functional dependence

of J upon the control ﬁeld forms a control landscape, whose

topology, especially the distribution of critical points, may

determine the effectiveness of a search for an optimal control.

A critical point is deﬁned as a location on the landscape

where the gradient (the ﬁrst functional derivative of J with

respect to the control) is zero for t ∈ [0,T ]. An analysis of

the Hessian matrix (second derivative of J ) can identify the

intrinsic topological character at a critical point [4]: a negative

(or positive) semideﬁnite Hessian indicates that the critical

point is locally maximal (or minimal) to second order, while

an indeﬁnite Hessian is associated with a saddle point.

Upon satisfaction of some underlying assumptions (see

Sec. II for details), control landscape analysis predicts that the

critical points only exist at particular values of the objective J

and that there are no local suboptimal maxima or minima

(traps) over the landscape; i.e., the critical points located

between the global maximum and minimum, if any, have

to be saddle points [3]. Another important conclusion is the

low rank of the Hessian at critical points; i.e., there exists a

speciﬁc maximum number of positive and negative eigenval-

ues dependent on the nature of the quantum system and the

control problem [4]. Many of the theoretical predictions have

been successfully tested in large numbers of simulations [5,6],

but experimental afﬁrmation in the laboratory is important

for fundamental and practical reasons. A laser experiment on

pure state transitions in atomic rubidium was found consistent

with expectations in terms of the Hessian structure at the

global minimum and maximum of the landscape [7], and as

a foundation for the present paper we explored the control

landscape for a proton spin-1/2 two-level system [8]. The

landscapes in both of the two latter works are devoid of saddle

points. Thus, in order to fully test the predictions provided

by landscape theoretical analysis we have to consider other

more complex quantum systems with the simple example in

this paper being a coupled two-spin system.

NMR provides a desirable domain for studying fundamen-

tal properties of quantum control due to the simple and well-

deﬁned Hamiltonians involved, slow relaxation, high signal-

to-noise ratio, etc. [8]. The dynamical process underlying

NMR can be viewed as the manipulation of magnetization

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