A problem in Quantum
Control:

Solving an Instance of the Schrödinger Equation

** **

**Brown University**

Under the Guidance of:

Professor Avy Soffer and
Pieter Blue,

Rutgers University
Department of Mathematics

The
physical question:

Given
a particular initial quantum state of a several particle system, can we
introduce perturbations into the system to bring the system into a quantum
state of our choosing?

-In practice, the perturbations take the form of laser pulses acting on the particles.

-This
problem has applications in quantum computing.

The
Schrödinger equation is used to model the behavior of a quantum system :

* *

*i(d***a***/dt)=H***a***(t)*

** **

*-H* is a (possibly time
dependent) linear operator whose specific form is a manifestation of the
particular physical situation we are modeling.

-**a***(t)*** **is a vector-valued function.
(It represents the quantum state of our system as a function of time)

** **

* *

My problem concerns the behavior of a quantum system
whose particles are acted on by lasers with intensities in time of form *ce ^{-}*

For large l, this function approximates a short laser pulse.

This system can be modeled
using the following instance of the Schrödinger equation:

_{}

*i(d***a***/dt) = (H _{0}+ ce^{-}*

** **

where

-**a(t)** is finite-dimensional
vector-valued function

-H_{0} is a
hermitian matrix

-M_{0} is a block
diagonal matrix whose blocks are rotation matrices

-**a***(0)***, **the
initial state of the system, is known.

My task for this summer has been to solve this
equation for the vector **a***(t)*.
Once the equation is solved we can specify a desired quantum state and
use the solution to algebraically determine physical parameters that will yield
this desired state over large time.

* *

*i(d***a***/dt) = (H _{0}+ ce^{-}*

-This matrix differential equation represents a
“non-autonomous” system of ordinary differential equations.

-it cannot be solved by the methods used to easily
solve non-linear problems (such as decoupling by diagonalization of the
operator)

NOTE:The imaginary constant can be absorbed into the
operator to make the computation simpler.
Henceforth, all H_{0}, M_{0} used here will be
understood to be the matrices specified above, each multiplied by the scalar
constant *–i*.

My
approach to solving the problem:

-I take the ** Laplace transform,** denoted

_{}

-I solve this difference equation explicitly for the
(vector valued) Laplace transform **F***(s)* of **a***(t)*. This solution takes the form of an infinite
sum.

-I use the theory of complex variables to:

1)Show that taking __inverse Laplace
transform,__ or Bromwich integral, of
**F***(s)* recovers **a***(t)*.

2)Demonstrate certain algebraic equalities that we would like to employ to bring the integrand in the Bromwich integral into a more easily calculable form.

3)Calculate the value of the
inverse Laplace transform of **F***(s)* using the theory of residues
from complex analysis.

The difference equation obtained by taking the
Laplace transform of both sides of the matrix equation is:

**F***(s)=(sI-H _{0})^{-1}*

_{}

_{}

_{}

In
the special case where H_{0} and M_{0} commute, this reduces to
the neater sum:

_{}

If
certain conditions on either **F***(s)* or the unknown function **a***(t)*
are satisfied, **a***(t)* is equal to the __inverse Laplace transform__*
*of **F***(s)*, denoted

*L ^{-1}{*

_{}

for
(t>0) and for some real g sufficiently large.

So,
in summary, to find **a***(t)* we must first show that *L ^{-1}{*

Results:

For
the case where M_{0 }and H_{0 }commute, I have a tentative
solution:

_{}

Work
in progress:

-meticulous
verification of this result

-solving
the general case

References/Recommended
Reading:

-Churchill**, Operational Mathematics**-Laplace

transforms

-Brown, Churchill, *Complex Variables and *

** Applications**-calculating residues

-** http://www.qubit.com** –web resource
for

information on applications of quantum control to quantum computing

-Schirmer,
Greentree, Ramakrishna, Rabitz, *Quantum *

*Control Using
Sequences of Simple Control Pulses*, arXiv:quant-ph/0105155 v1, May 31
2001. A paper that tackles the problem
of quantum control using sinusoidal laser pulses.

* *