Chapter 16: Design of Femtosecond Pulse Sequences to Control Reactions
Consider a nonlinear quantum control problem of the form:
where 
is a functional of the control, 
This is a called nonlinear control since the control, is multiplied by the state vector,
Complete controllability is defined as the ability to synthesize any unitary transformation in the
space of operators that operate in the N dimensional Hilbert space; this is a sufficient (but not necessary) condition to get from any initial state to any final state in the system. A test for complete controllability is whether the Lie algebra spanned by the commutators,
, 
etc. has dimension 
(for traceless 
).
In many situations there are multiple 's
and then the space spanned is evaluated using the commutators of all of {}
with 
and with each other. The program below computes the dimension of the Lie algebra, given a matrix
and a set of matrices {}.
is a functional of the control,
This is a called nonlinear control since the control, is multiplied by the state vector,
Complete controllability is defined as the ability to synthesize any unitary transformation in the
space of operators that operate in the N dimensional Hilbert space; this is a sufficient (but not necessary) condition to get from any initial state to any final state in the system. A test for complete controllability is whether the Lie algebra spanned by the commutators,
,
etc. has dimension (for traceless ).
In many situations there are multiple 's
and then the space spanned is evaluated using the commutators of all of {}
with and with each other. The program below computes the dimension of the Lie algebra, given a matrix
and a set of matrices {}.