报告摘要：Quantum mechanical models for systems with spin are among the simpler
Quantum systems to study, if they are in isolation and at zero
temperature. In this case the systems can be described in terms of a
density matrix $\rho $ which is Hermitean and whose evolution in time is
governed by a linear differential equation of the form
$$\dot{\rho }=[A+uB,\rho ]$$
where $u$ is a scalar control function representing the effect of a
externally applied radio frequency field. However, this degree of
simplicity is rarely realized; in most situations the system of interest
is in contact with a ``heat bath" whose dynamics are complex and
imperfectly known. If the dynamics of the entire system were to be
modeled then then the model would be impossibly large; an effective
approximation, widely used in NMR work, is to replace the heat bath
dynamics by a stochastic term, yielding a stochastic equation of the form
$$ \dot{\rho } =- [A+uB,\rho ] + [R\eta (t), \rho] $$

The next step is to decide on the nature of the stochastic process $\eta
$. Intuitively speaking, the autocorrelation function of $\eta $ should
reflect the dynamics of the heat bath, (also called lattice the lattice
dynamics in this context). The entries in $R$ then represent coupling
strengths. On the other hand, if $\eta $ is approximated by white noise
of variance $k$ then the analysis of the expected value of the evolution
of $\rho $ is greatly simplified. In fact its description is given in
terms of a liner differential equation
$$\mathcal E \dot{\rho } = [A+ +uB, \mathcal E \rho ] +k [R,[R,\rho ]]$$
This analysis is part of a more general theory which allows one to
reduce the computation of any particular set of moments of this equation
to the solution of a linear constant coefficient differential equation.
Moreover, the correlation function of $\rho $ is also easily computed in
terms of $A,B,u,R$.
These considerations help shape a relevant control problem in the
following way. When an NMR experiment is designed for spectroscopic
purposes then one is interested in the eigenvalues of $A=-A^*$. When $u$
is set to zero the response with $k=0$ is purely oscillatory and the
Fourier transform of the signal it radiates serves to identify its
eigenvalues. On the other hand, the effect of $R$ is to introduce
damping, in this context called decoherance, making the spectrum of the
observed signal harder to measure accurately. (This decoherance is also
a significant probem in most types of quantum computing.)

In this talk we will attempt to explain this background and present some
newer results on the control theoretic description of decoherance free
trajectories.
报告人简历：哈佛大学的Roger W. Brockett教授，主要研究兴趣量子控制，非线性系统控制，几何控制理论，最优控制。Brockett教授是几何控制理论的奠基人之一，在非线性系统控制，最优控制等方面曾做出过重要贡献。在量子控制领域，他与合作者在量子系统时间最优控制以及量子系统可观性均做出过重要工作，其中关于量子系统时间最优控制的工作引用已经接近100。