Question

Exercise 1. [30 points.] A cloud of ultracold atoms trapped in an optical lattice is described by the Hamiltonian [4.N2-a 4.22 -a -a 4.12-a Ho = fr -a -- 0 - 4.12 -a 4.22 4.N2

Answer 1

CODE:

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HAMILTON FUNCTION:

% Hamiltonian function to generate Hamiltonian Matrix

% Input: fR -> recoil frecuency; a1-> depth, N-> momentum states

% Output: H -> Hamiltonian Matrix

function H = hamiltonian(fR,a1,N)

% defining size of H

n = 2*N + 1;

% generating identity matrix of size n using diag

H = diag(n);

% defining diagonal values of H

diagH = 4*(-N:N).^2;

% looping through matrix to set values

for i=1:n

% seting diagonal value

H(i,i) = diagH(i);

% seting lower to diagonal and upper to diagonal value = -a1

if i<n

% lower to diagonal

H(i+1,i) = -a1;

% upper to diagonal

H(i,i+1) = -a1;

end

end

% Multiplying matrix with fR

H = fR .* H;

end

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MAIN SCRIPT:

% MATLAB Script

% a. description of hamiltonian

help hamiltonian

% b. call hamiltonian

fR = 1;

a = 2.5;

N = 10;

% calling function

H0 = hamiltonian(fR,a,N);

% c. eig to compute eigenvalues and vectors

[V, D] = eig(H0);

% d. ploting first eigenvector of H0

% defining p

p = -N:N;

bar(p,V(:,1));

xlabel("X");

ylabel("Eigen Vector");

title("First Eigen vector of H0");

% e. ploting second eigenvector of H0

% new figure

figure();

bar(p,V(:,2));

xlabel("X");

ylabel("Eigen Vector");

title("Second Eigen vector of H0");

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OUTPUT:

First Eigen vector of HO 0.9 0.8F 0.7 0.6 0.5 Eigen Vector 0.4 0.3 0.2 0.1 0 -10 -5 5 10 Xo

Second Eigen vector of HO 0.8 0.6 0.4 0.2 Eigen Vector 0 -0.2 -0.4 -0.6 -0.8 -10 -5 5 10 xo