  The orbit tracking of free-evolutionary target system in closed quantum
systems is studied in this paper. Based on the concept of system control
theory, the unitary transformation is applied to change the time-dependent
target function into a stationary target state so that the orbit tracking
problem is changed into the state transfer one. A Lyapunov function with
virtual mechanical quantity P is employed to design a control law for such a
state transferring. The target states in density matrix are grouped into two
classes: diagonal and non-diagonal. The specific convergent conditions for
target state of diagonal mixed-states are derived. In the case that the target
state is a non-diagonal superposition state, we propose a non-diagonal P
construction method; if the target state is a non-diagonal mixed-state we use a
unitary transformation to change it into a diagonal state and design a diagonal
P. In such a way, the orbit tracking problem with arbitrary initial state is
properly solved. The explicit expressions of P are derived to obtain a
convergent control law. At last, the system simulation experiments are
performed on a two-level quantum system and the tracking process is illustrated
on the Bloch sphere.
