Ph.D. Thesis - Physics

A.2 Simulation with a linear force gradient in space
The following is a function that computes the average J value, as well as the simulation
error for a given gradient of F.
% File: ising_small_rs_fn.m
% Author: Rob Clark
% Simulates Ising model with and without micromotion,
% beginning with a random two-qubit state.
% A linear gradient in space is used.
% This is a function that returns the average J value when the ion undergoes
% micromotion and also the error, calculated as the deviation from the
% micromotion-free simulation.
% Basis generated by {spin space of ion 1} \otimes {spin space of ion 2}
% STAGE 1: simple calculation, ising, no vibration
function [Jm,err_result] = isrf(J0,Jav,A,grad)
% J = -10000;
% B = -J;
% J = 1.0306e3;
J = Jav;
B = -J;
% A = .1;
Omega = 1.0e6;
t = 0:1/(2*Omega):10/J;
sx = [0 1; 1 0];
sz = [1 0; 0 -1];
sztot1 = kron(eye([2 2]),sz) + kron(sz,eye([2 2]));
H1 = J*kron(sz,sz) + B*(kron(eye([2 2]), sx) + kron(sx,eye([2 2])));
% Randomize initial state
amps = [rand(1),rand(1),rand(1),rand(1)] + i*[rand(1),rand(1),rand(1),rand(1)];
init1 = amps/norm(amps); init1 = init1’;
expsz1 = [init1’*sztot1*init1];
for k = 2:length(t)
curr = expm(-i*H1*t(k))*init1;
expsz1 = [expsz1 curr’*sztot1*curr];
end
expsz2 = [init1’*sztot1*init1];
tstep=t(2)-t(1);
curr = init1;
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