A problem in Quantum Control:

Solving an Instance of the Schrödinger Equation

 

 

 

 

 

 

 

 

Michael Lesnick

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(da/dt)=Ha(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-lt, where c is a constant and l is a positive valued constant.

 

 

 

 

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

 

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

i(da/dt) = (H0+ ce-ltM0) a(t)

 

 

where

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

-H0 is a hermitian matrix

-M0 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(da/dt) = (H0+ ce-ltM0) a(t)

 

 

-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 H0, M0 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 L{f(t)} for a function f(t), of both sides of the equation to turn the differential equation into a difference equation of a complex variable s. 

 

-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-H0)-1a(0)+M0F(s+l)

provided that s is not equal to any eigenvalue of H0.

 

 

Using this equation to write F(s+l) in terms of F(s+2l), F(s+2l) in terms of F(s+3l), etc, we can find F(s) explicitly as an infinite sum:

 

 

In the special case where H0 and M0 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{F(s)}:

 

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

 

So, in summary, to find a(t) we must first show that    L-1{F(s)}=a(t), and then calculate  L-1{F(s)} using residue theory.

 

Results:

 

For the case where M0 and H0 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.