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Diffstat (limited to 'faraday/useful_functions.m')
| l---------[-rw-r--r--] | faraday/useful_functions.m | 338 |
1 files changed, 1 insertions, 337 deletions
diff --git a/faraday/useful_functions.m b/faraday/useful_functions.m index eb3e880..a2d3237 100644..120000 --- a/faraday/useful_functions.m +++ b/faraday/useful_functions.m @@ -1,337 +1 @@ -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); - % WARNING INTRODUCED HERE BUT MUST BE DEFINE IN ATOM PROPERTIES FILE - %n_atoms - proportional to a number of interacting atoms - n_atoms=500; - transmited_intensities_vector = ... - abs(E_field.linear + n_atoms*(1i)*dEdz_linear).^2 ... - +abs(E_field.right + n_atoms*(1i)*dEdz_right).^2 ... - +abs(E_field.left + n_atoms*(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 +../useful_functions.m
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