path: root/useful_functions.m

1;

function ret=decay_total(g_decay,i)
% calculate total decay for particular level taking in account all branches
	ret=sum(g_decay(i,:));
endfunction

function ret=kron_delta(i,j)
% kroneker delta symbol
	if ((i==j))
		ret=1;
	else
		ret=0;
	endif
endfunction

function rho=rhoOfFreq(rhoLiouville, freqIndex, Nlevels)
% this function create from Liouville density vector 
% the density matrix with given modulation frequency
	rho=zeros(Nlevels);
	rho(:)=rhoLiouville((freqIndex-1)*Nlevels^2+1:(freqIndex)*Nlevels^2);
	rho=rho.';
endfunction

function [N, rhoLiouville_w, rhoLiouville_r, rhoLiouville_c]=unfold_density_matrix(Nlevels,Nfreq)
% unwrap density matrix to Liouville density vector and assign all possible
% modulation frequencies as well
% resulting vector should be Nlevels x Nlevels x length(modulation_freq)
	N  = Nfreq*Nlevels*Nlevels;
	rho_size = Nlevels*Nlevels;
	rhoLiouville_w=zeros(N,1);
	rhoLiouville_r=zeros(N,1);
	rhoLiouville_c=zeros(N,1);
	
	w=1:Nfreq;
	w_tmplate(:)=repmat(w,rho_size,1);
	rhoLiouville_w=w_tmplate';
	r=1:Nlevels;
	r_tmplate(:)=repmat(r,Nlevels,1);
	rhoLiouville_r(:)=repmat(r_tmplate',Nfreq,1);
	c=(1:Nlevels)';% hold column value of rho_rc
	rhoLiouville_c=repmat(c,Nfreq*Nlevels,1);
endfunction	


function [L0m, polarizability_m]=L0_and_polarization_submatrices( ...
		Nlevels, ...
		H0, g_decay, g_dephasing, dipole_elements ...
		)
% create (Nlevels*Nlevels)x*(Nlevels*Nlevels)
% sub matrices of Liouville operator 
% which repeat themselves for each modulation frequency
% based on recipe from Eugeniy Mikhailov thesis
	%-------------------------
	useful_constants;
	rho_size=Nlevels*Nlevels;

	% now we create Liouville indexes list
	[Ndummy, rhoLiouville_w_notused, rhoLiouville_r, rhoLiouville_c]=unfold_density_matrix(Nlevels,1);

	kron_delta_m=eye(Nlevels);
	% note that L0 and decay parts depend only on combination of indexes
	% jk,mn but repeats itself for every frequency
	L0m=zeros(rho_size); % (Nlevels^2)x(Nlevels^2) matrix
	decay_part_m=zeros(rho_size); % (NxN)x(NxN) matrix
	% polarization matrix will be multiplied by field amplitude letter
	% polarization is part of perturbation part of Hamiltonian
	polarizability_m.linear = zeros(rho_size); % (NxN)x(NxN) matrix
	polarizability_m.left   = zeros(rho_size); % (NxN)x(NxN) matrix
	polarizability_m.right  = zeros(rho_size); % (NxN)x(NxN) matrix
	for p=1:rho_size
		% p= j*Nlevels+k 
		% this might speed up stuff since less matrix passed back and force
		j=rhoLiouville_r(p);
		k=rhoLiouville_c(p);
		for s=1:rho_size
			% s= m*Nlevels+n 
			m=rhoLiouville_r(s);
			n=rhoLiouville_c(s);

			% calculate unperturbed part (Hamiltonian without EM field)
			L0m(p,s)=H0(j,m)*kron_delta_m(k,n)-H0(n,k)*kron_delta_m(j,m);
			decay_part_m(p,s)= ...
				( ...
					  decay_total(g_decay,k)/2 ...
					+ decay_total(g_decay,j)/2 ...
					+ g_dephasing(j,k) ...
				)* kron_delta_m(j,m)*kron_delta_m(k,n) ...
				- kron_delta_m(m,n)*kron_delta_m(j,k)*g_decay(m,j) ;
			polarizability_m.linear(p,s)= ( dipole_elements.linear(j,m)*kron_delta_m(k,n)-dipole_elements.linear(n,k)*kron_delta_m(j,m) );
			polarizability_m.left(p,s)= ( dipole_elements.left(j,m)*kron_delta_m(k,n)-dipole_elements.left(n,k)*kron_delta_m(j,m) );
			polarizability_m.right(p,s)= ( dipole_elements.right(j,m)*kron_delta_m(k,n)-dipole_elements.right(n,k)*kron_delta_m(j,m) );
		endfor
	endfor
	L0m=-im_one/hbar*L0m - decay_part_m; 
endfunction

function L=Liouville_operator_matrix( ...
		N, ... 
		L0m, polarizability_m, ...
		E_field, ...
		modulation_freq, rhoLiouville_w, rhoLiouville_r, rhoLiouville_c ...
		)
% Liouville operator matrix construction
% based on recipe from Eugeniy Mikhailov thesis
	%-------------------------
	useful_constants;
	L=zeros(N); % NxN matrix
	Nfreq=length(modulation_freq);

	% Lets be supper smart and speed up L matrix construction
	% since it has a lot of voids.
	% By creation of rhoLiouville we know that there are
	% consequent chunks of rho_ij modulated with same frequency
	% this means that rhoLiouville is split in to Nfreq chunks 
	% with length Nlevels*Nlevels=N/Nfreq
	rho_size=N/Nfreq;
	
	% creating building blocks of L by rho_size * rho_size
	for w3i=1:Nfreq
		w_iner=modulation_freq(w3i);
		if ((w_iner == 0))
			% calculate unperturbed part (Hamiltonian without EM field)
			L_sub{w3i}=L0m;
		else
			% calculate perturbed part (Hamiltonian with EM field)
			% in other word interactive part of Hamiltonian 
			L_sub{w3i} = ...
				-im_one/hbar*polarizability_m.linear * E_field.linear(w3i) ...
				-im_one/hbar*polarizability_m.left   * E_field.left(w3i)   ...
				-im_one/hbar*polarizability_m.right  * E_field.right(w3i)  ...
				;
		endif
	endfor

	% Liouville matrix operator has Nlevels*Nlevels blocks 
	% which governed by the same modulation frequency
	for p_freq_cntr=1:Nfreq
		p0=1+(p_freq_cntr-1)*rho_size;
		% we guaranteed to know frequency of final and initial rhoLiouville
		w1i=rhoLiouville_w(p0); % final
		w_jk=modulation_freq(w1i);
		for s_freq_cntr=1:Nfreq
			s0=1+(s_freq_cntr-1)*rho_size;
			w2i=rhoLiouville_w(s0); % initial
			w_mn=modulation_freq(w2i);
			% thus we know L matrix element frequency which we need to match
			w_l=w_jk-w_mn;
			% lets search this frequency in the list of available frequencies
			% but since we not guaranteed to find it lets assign temporary 0 to Liouville matrix element
			w3i=(w_l == modulation_freq);
			if (any(w3i))	
				% yey, requested modulation frequency exist
				% lets do  L sub matrix filling
				% at most we should have only one matching frequency
				w_iner=modulation_freq(w3i);
				L(p0:p0+rho_size-1,s0:s0+rho_size-1) = L_sub{w3i};

			endif
		endfor
		% diagonal elements are self modulated
		% due to rotating wave approximation
		L(p0:p0+rho_size-1,p0:p0+rho_size-1)+= -im_one*w_jk*eye(rho_size);
	endfor
endfunction


function [rhoLiouville_dot, L]=constrain_rho_and_match_L( ...
		N, L, ...
		modulation_freq, rhoLiouville_w, rhoLiouville_r, rhoLiouville_c)
% now generally rhoL_dot=0=L*rhoL has infinite number of solutions
% since we always can resclale rho vector with arbitrary constant
% lets constrain our density matrix with some physical meaning
% sum(rho_ii)=1 (sum of all populations (with zero modulation frequency) scales to 1
% we will replace first row of Liouville operator with this condition
% thus rhoLiouville_dot(1)=1
	for i=1:N
		w2i=rhoLiouville_w(i);
		m=rhoLiouville_r(i);
		n=rhoLiouville_c(i);
		w=modulation_freq(w2i);
		if ((w==0) & (m==n))
			L(1,i)=1;
		else
			L(1,i)=0;
		endif
	endfor
	rhoLiouville_dot= zeros(N,1);
	% sum(rho_ii)=1 (sum of all populations (with zero modulation frequency) scales to 1
	% we will replace first row of Liouville operator with this condition
	% thus rhoLiouville_dot(1)=1
	rhoLiouville_dot(1)=1;
endfunction

function kappa=susceptibility(wi, rhoLiouville, dipole_elements, E_field)
% calculate susceptibility for the field at given frequency index
	Nlevels=( size(dipole_elements.linear)(1) );
	rho=rhoOfFreq(rhoLiouville, wi, Nlevels);
	kappa.linear=0;
	kappa.left=0;
	kappa.right=0;

	kappa.linear = sum( sum( transpose(dipole_elements.linear) .* rho ) );
	kappa.left   = sum( sum( transpose(dipole_elements.left)   .* rho ) );
	kappa.right  = sum( sum( transpose(dipole_elements.right)  .* rho ) );

	kappa.linear /= E_field.linear( wi ); 
	kappa.left   /= E_field.left( wi );
	kappa.right  /= E_field.right( wi ); 
endfunction

function index=freq2index(freq, modulation_freq) 
% convert modulation freq to its index in the modulation_freq vector
	index=[1:length(modulation_freq)](modulation_freq==freq);
endfunction

function rhoLiouville=rhoLiouville_steady_state(L0m, polarizability_m, E_field, modulation_freq)
% calculates rhoLiouville vector assuming steady state situation and normalization of rho_ii to 1
	Nlevels=sqrt( size(L0m)(1) );
	Nfreq=length(modulation_freq);

	% now we create Liouville indexes list
	[N, rhoLiouville_w, rhoLiouville_r, rhoLiouville_c]=unfold_density_matrix(Nlevels,Nfreq);

	% Liouville operator matrix construction
	L=Liouville_operator_matrix( 
			N, 
			L0m, polarizability_m,
			E_field, 
			modulation_freq, rhoLiouville_w, rhoLiouville_r, rhoLiouville_c
			);

	%use the fact that sum(rho_ii)=1 to constrain solution
	[rhoLiouville_dot, L]=constrain_rho_and_match_L(
			N, L,
			modulation_freq, rhoLiouville_w, rhoLiouville_r, rhoLiouville_c);

	%solving for density matrix vector
	rhoLiouville=L\rhoLiouville_dot;
endfunction

function [xi_linear, xi_left, xi_right]=susceptibility_steady_state_at_freq( atom_field_problem)
	% find steady state susceptibility at particular modulation frequency element
	% at given E_field 
	global atom_properties;
	L0m                = atom_properties.L0m              ; 
	polarizability_m   = atom_properties.polarizability_m ; 
	dipole_elements    = atom_properties.dipole_elements  ; 

	E_field            = atom_field_problem.E_field          ; 
	modulation_freq    = atom_field_problem.modulation_freq  ; 
	freq_index         = atom_field_problem.freq_index       ; 

	rhoLiouville=rhoLiouville_steady_state(L0m, polarizability_m, E_field, modulation_freq);
	xi=susceptibility(freq_index, rhoLiouville, dipole_elements, E_field);
	xi_linear = xi.linear;
	xi_right  = xi.right;
	xi_left   = xi.left;
endfunction		

function [dEdz_coef_linear, dEdz_coef_left, dEdz_coef_right] =dEdz_at_freq(wi, rhoLiouville, dipole_elements, E_field)
% complex absorption coefficient 
% at given E_field frequency component
	Nlevels=( size(dipole_elements.linear)(1) );
	rho=rhoOfFreq(rhoLiouville, wi, Nlevels);

	dEdz_coef_linear = sum( sum( transpose(dipole_elements.linear) .* rho ) );
	dEdz_coef_left   = sum( sum( transpose(dipole_elements.left)   .* rho ) );
	dEdz_coef_right  = sum( sum( transpose(dipole_elements.right)  .* rho ) );

endfunction 

function [dEdz_linear, dEdz_left, dEdz_right]=dEdz( atom_field_problem)
% complex absorption coefficient for each field
	global atom_properties;
	L0m                = atom_properties.L0m              ; 
	polarizability_m   = atom_properties.polarizability_m ; 
	dipole_elements    = atom_properties.dipole_elements  ; 

	E_field            = atom_field_problem.E_field          ; 
	modulation_freq    = atom_field_problem.modulation_freq  ; 
	Nfreq=length(modulation_freq);

	rhoLiouville=rhoLiouville_steady_state(L0m, polarizability_m, E_field, modulation_freq);
	for wi=1:Nfreq
	 [dEdz_linear(wi), dEdz_left(wi), dEdz_right(wi)] = dEdz_at_freq(wi, rhoLiouville, dipole_elements, E_field);
	endfor
endfunction		

function relative_transmission=total_relative_transmission(atom_field_problem)
% summed across all frequencies field absorption coefficient
	global atom_properties;
	[dEdz_linear, dEdz_left, dEdz_right]=dEdz( atom_field_problem);
	%dEdz_total= dEdz_linear .* conj(dEdz_linear) ...
									 %+dEdz_left   .* conj(dEdz_left) ...
									 %+dEdz_right  .* conj(dEdz_right);
	%total_absorption = sum(dEdz_total);

	%total_absorption = |E|^2 - | E - dEdz | ^2		
	E_field.linear = atom_field_problem.E_field.linear;
	E_field.right  = atom_field_problem.E_field.right;
	E_field.left   = atom_field_problem.E_field.left;
	
	modulation_freq    = atom_field_problem.modulation_freq  ; 
	pos_freq_indexes=(modulation_freq>0);
	transmited_intensities_vector = ...	
		 abs(E_field.linear + (1i)*dEdz_linear).^2 ...
		+abs(E_field.right  + (1i)*dEdz_right).^2 ...
		+abs(E_field.left   + (1i)*dEdz_left).^2;
	transmited_intensities_vector = transmited_intensities_vector(pos_freq_indexes);
	input_intensities_vector =  ...	
		 abs(E_field.linear).^2   ...
		+abs(E_field.right).^2    ...
		+abs(E_field.left).^2;
	input_intensities_vector = input_intensities_vector(pos_freq_indexes);
		
	relative_transmission = sum(transmited_intensities_vector) / sum(input_intensities_vector);
endfunction

% create full list of atom modulation frequencies from positive light frequency amplitudes
function [modulation_freq, E_field_amplitudes] = ...
light_positive_frequencies_and_amplitudes2full_set_of_modulation_frequencies_and_amlitudes(...
	positive_light_frequencies, positive_light_field_amplitudes )
	% we should add 0 frequency as first element of our frequencies list
	modulation_freq    = cat(2, 0, positive_light_frequencies, -positive_light_frequencies);
	% negative frequencies have complex conjugated light fields amplitudes
	E_field_amplitudes.left = cat(2, 0, positive_light_field_amplitudes.left, conj(positive_light_field_amplitudes.left) );
	E_field_amplitudes.right = cat(2, 0, positive_light_field_amplitudes.right, conj(positive_light_field_amplitudes.right) );
	E_field_amplitudes.linear = cat(2, 0, positive_light_field_amplitudes.linear, conj(positive_light_field_amplitudes.linear) );
endfunction	



% vim: ts=2:sw=2:fdm=indent