Latest - complete script for extraction of edd, n, k_roton for different tilts of the dipole moment and different trapping frequencies and minor modifications to other scripts.
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@ -77,7 +77,7 @@ MeanWidth = sigma * lz;
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VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth);
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VDk_fftshifted = fftshift(VDk);
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figure(11)
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figure(8)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*1e-6, fftshift(Transf.ky)*1e-6, VDk_fftshifted); % Specify x and y data for axes
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@ -106,7 +106,7 @@ end
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EpsilonK = double(imag(EpsilonK) ~= 0); % 'isreal' returns 0 for complex numbers and 1 for real numbers, so we negate it
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figure(12)
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figure(9)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*1e-6, fftshift(Transf.ky)*1e-6, EpsilonK); % Specify x and y data for axes
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@ -159,7 +159,7 @@ for theta = theta_values
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EpsilonK = double(imag(EpsilonK) ~= 0); % 'isreal' returns 0 for complex numbers and 1 for real numbers, so we negate it
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% Plot the result
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figure(13)
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figure(10)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*1e-6, fftshift(Transf.ky)*1e-6, EpsilonK); % Specify x and y data for axes
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@ -28,123 +28,114 @@ Dy164Mass = 163.929174751*AtomicMassUnit;
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Dy164IsotopicAbundance = 0.2826;
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DyMagneticMoment = 9.93*BohrMagneton;
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%% Roton instability boundary for tilted dipoles
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%% Extracting values from the roton instability boundary for tilted dipoles
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wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
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theta = 0; % Polar angle of dipole moment
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%-------TEST-------%
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% nadd2s = 0.05:0.005:0.25;
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% as_to_add = 0.76:0.0001:0.81;
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lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
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add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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%-------DEPLOY-------%
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nadd2s = 0.005:0.005:0.5;
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as_to_add = 0.35:0.0001:1.15;
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nadd2s = 0.05:0.001:0.25;
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as_to_add = 0.76:0.001:0.81;
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var_widths = zeros(length(as_to_add), length(nadd2s));
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data_struct = struct;
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wz_values = [150, 300, 500, 750];
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theta_values = 0:5:45; % Range of theta values (you can modify this)
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phi = 0; % Azimuthal angle of momentum vector
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kvec = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
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x0 = 5;
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Aineq = [];
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Bineq = [];
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Aeq = [];
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Beq = [];
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lb = [1];
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ub = [10];
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nonlcon = [];
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fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
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for mainloop_idx = 1:length(wz_values)
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format long
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for idx = 1:length(nadd2s)
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for jdx = 1:length(as_to_add)
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AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
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as = (as_to_add(jdx) * add); % Scattering length
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gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
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TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
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sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
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var_widths(jdx, idx) = sigma;
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end
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end
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PlanckConstantReduced = 6.62607015E-34/(2*pi);
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AtomicMassUnit = 1.660539066E-27;
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Dy164Mass = 163.929174751*AtomicMassUnit;
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VacuumPermeability = 1.25663706212E-6;
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BohrMagneton = 9.274009994E-24;
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DyMagneticMoment = 9.93*BohrMagneton;
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%% ====================================================================================================================================================== %
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phi = 0; % Azimuthal angle of momentum vector
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k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
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instability_boundary = zeros(length(as_to_add), length(nadd2s));
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ScatteringLengths = zeros(length(as_to_add), 1);
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AtomNumber = zeros(length(nadd2s), 1);
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w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
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l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
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tsize = 10 * l0;
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for idx = 1:length(nadd2s)
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for jdx = 1:length(as_to_add)
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AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
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AtomNumber(idx) = AtomNumberDensity*tsize^2;
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as = (as_to_add(jdx) * add); % Scattering length
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ScatteringLengths(jdx) = as/BohrRadius;
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eps_dd = add/as; % Relative interaction strength
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gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
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wz = 2 * pi * wz_values(mainloop_idx); % Trap frequency in the tight confinement direction
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lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
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add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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var_widths = zeros(length(as_to_add), length(nadd2s));
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[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
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% == Quantum Fluctuations term == %
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gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
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gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
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gQF = gamma5 * gammaQF;
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% == Dispersion relation == %
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DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
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EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
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instability_boundary(jdx, idx) = ~isreal(EpsilonK);
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end
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end
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figure(6)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(nadd2s, as_to_add, instability_boundary); % Specify x and y data for axes
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set(gca, 'YDir', 'normal'); % Correct the y-axis direction
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colorbar; % Add a colorbar
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xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
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ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
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title('Roton instability boundary','fontsize',16,'interpreter','latex')
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%%
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matrix = instability_boundary;
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% Initialize arrays to store row and column indices of transitions
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row_indices = [];
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col_indices = [];
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% Loop through the matrix to find transitions from 0 to 1
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[rows, cols] = size(matrix);
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for j = 1:cols
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for i = 2:rows
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if matrix(i-1, j) == 1 && matrix(i, j) == 0
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row_indices = [row_indices; i];
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col_indices = [col_indices; j];
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break; % Stop after the first transition in the column
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x0 = 5;
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Aineq = [];
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Bineq = [];
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Aeq = [];
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Beq = [];
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lb = [1];
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ub = [10];
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nonlcon = [];
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fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
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for idx = 1:length(nadd2s)
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for jdx = 1:length(as_to_add)
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AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
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as = (as_to_add(jdx) * add); % Scattering length
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gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
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TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
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sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
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var_widths(jdx, idx) = sigma;
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end
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end
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eps_dd_values = zeros(length(theta_values), 1);
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n_values = zeros(length(theta_values), 1);
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k_roton_values = zeros(length(theta_values), 1);
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for idx = 1:length(theta_values)
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theta = theta_values(idx);
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[eps_dd_values(idx), n_values(idx), k_roton_values(idx)] = extractFromBoundaryCurve(theta, phi, nadd2s, as_to_add, var_widths, wz, lz, kvec);
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end
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data_struct(mainloop_idx).wz_value = wz / (2 * pi);
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data_struct(mainloop_idx).theta_values = theta_values;
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data_struct(mainloop_idx).eps_dd_values = eps_dd_values;
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data_struct(mainloop_idx).n_values = n_values;
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data_struct(mainloop_idx).k_roton_values = k_roton_values;
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%{
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figure(13)
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clf
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set(gcf,'Position',[50 50 950 750])
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plot(theta_values, eps_dd_values, '-o', LineWidth=2.0)
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xlabel('$\theta$','fontsize',16,'interpreter','latex');
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ylabel('$\epsilon_{dd}$','fontsize',16,'interpreter','latex');
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% title([''],'fontsize',16,'interpreter','latex')
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grid on
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figure(14)
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clf
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set(gcf,'Position',[50 50 950 750])
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plot(theta_values, (1./eps_dd_values) * (add/BohrRadius), '-o', LineWidth=2.0)
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xlabel('$\theta$','fontsize',16,'interpreter','latex');
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ylabel('$a_s (\times a_o)$','fontsize',16,'interpreter','latex');
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% title([''],'fontsize',16,'interpreter','latex')
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grid on
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figure(15)
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clf
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set(gcf,'Position',[50 50 950 750])
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plot(theta_values, n_values * 1E-15, '-o', LineWidth=2.0)
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xlabel('$\theta$','fontsize',16,'interpreter','latex');
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ylabel('$n (\times 10^{3} \mu m^{-2})$','fontsize',16,'interpreter','latex');
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% title([''],'fontsize',16,'interpreter','latex')
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grid on
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figure(16)
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clf
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set(gcf,'Position',[50 50 950 750])
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plot(theta_values, k_roton_values * 1E-6, '-o', LineWidth=2.0)
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xlabel('$\theta$','fontsize',16,'interpreter','latex');
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ylabel('$k_{roton} (\mu m^{-1})$','fontsize',16,'interpreter','latex');
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% title([''],'fontsize',16,'interpreter','latex')
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grid on
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%}
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end
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% Now extract the values from the corresponding vectors
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xvals = zeros(length(col_indices), 1);
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yvals = zeros(length(row_indices), 1);
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for k = 1:length(row_indices)
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row = row_indices(k);
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col = col_indices(k);
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xvals(k) = nadd2s(col);
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yvals(k) = as_to_add(row);
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end
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% Plot the extracted values
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figure(7);
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plot(xvals, yvals, '-o');
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title('Instability Boundary');
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xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
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ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex')
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save('ExtractingKRoton_Result.mat', 'data_struct');
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%%
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function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi)
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@ -164,4 +155,161 @@ function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, g
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Energy_AxialComponent = (PlanckConstantReduced * wz) * ((lz^2/(4 * MeanWidth^2)) + (MeanWidth^2/(4 * lz^2)));
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Energy_TransverseComponent = (0.5 * (gs + (2*gdd)) * gamma4 * AtomNumberDensity) + ((2/5) * gQF * AtomNumberDensity^(3/2));
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ret = (Energy_AxialComponent + Energy_TransverseComponent) / (PlanckConstantReduced * wz);
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end
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end
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function [eps_dd, AtomNumberDensity, k_roton] = extractFromBoundaryCurve(theta, phi, nadd2s, as_to_add, var_widths, wz, lz, kvec)
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format long
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PlanckConstantReduced = 6.62607015E-34/(2*pi);
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AtomicMassUnit = 1.660539066E-27;
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Dy164Mass = 163.929174751*AtomicMassUnit;
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VacuumPermeability = 1.25663706212E-6;
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BohrMagneton = 9.274009994E-24;
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DyMagneticMoment = 9.93*BohrMagneton;
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add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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phase_diagram = zeros(length(as_to_add), length(nadd2s));
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w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
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l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
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for idx = 1:length(nadd2s)
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for jdx = 1:length(as_to_add)
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AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
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as = (as_to_add(jdx) * add); % Scattering length
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eps_dd = add/as; % Relative interaction strength
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gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
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[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(kvec, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
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% == Quantum Fluctuations term == %
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gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
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gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
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gQF = gamma5 * gammaQF;
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% == Dispersion relation == %
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DeltaK = ((PlanckConstantReduced^2 .* kvec.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
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EpsilonK = sqrt(((PlanckConstantReduced^2 .* kvec.^2) ./ (2 * Dy164Mass)) .* DeltaK);
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phase_diagram(jdx, idx) = ~isreal(EpsilonK);
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end
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end
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%{
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figure(11)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(nadd2s, as_to_add, phase_diagram); % Specify x and y data for axes
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set(gca, 'YDir', 'normal'); % Correct the y-axis direction
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colorbar; % Add a colorbar
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xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
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ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
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title(['$\theta = ',num2str(theta), '; \phi = 0','$', '(Along Y)'],'fontsize',16,'interpreter','latex')
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%}
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%-------------%
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matrix = phase_diagram;
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% Initialize arrays to store row and column indices of transitions
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row_indices = [];
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col_indices = [];
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% Loop through the matrix to find transitions from 0 to 1
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[rows, cols] = size(matrix);
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for j = 1:cols
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for i = 2:rows
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if matrix(i-1, j) == 1 && matrix(i, j) == 0
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row_indices = [row_indices; i-1];
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col_indices = [col_indices; j];
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break; % Stop after the first transition in the column
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end
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end
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end
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% Now extract the values from the corresponding vectors
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xvals = zeros(length(col_indices), 1);
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yvals = zeros(length(row_indices), 1);
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for k = 1:length(row_indices)
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row = row_indices(k);
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col = col_indices(k);
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xvals(k) = nadd2s(col);
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yvals(k) = as_to_add(row);
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end
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instability_boundary = [xvals, yvals];
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%-------------%
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% Degree of the polynomial to fit
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n = 5; % For a quadratic fit
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% Fit the polynomial
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p = polyfit(xvals, yvals, n);
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% Display the polynomial coefficients
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% disp('Polynomial coefficients:');
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% disp(p);
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% Evaluate the polynomial at points in x
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y_fit = polyval(p, xvals);
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%{
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% Plot the original data and the fitted polynomial curve
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figure(12);
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clf
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set(gcf,'Position',[50 50 950 750])
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plot(xvals, yvals, 'o', 'LineWidth', 2.0, 'DisplayName', 'Extracted boundary points'); % Original data
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hold on;
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plot(xvals, y_fit, '-r','LineWidth', 2.0, 'DisplayName', ['Polynomial Fit (degree ' num2str(n) ')']); % Fitted curve
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ylim([min(as_to_add) max(as_to_add)])
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xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
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ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex')
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title(['$\theta = ',num2str(theta), '; \phi = 0','$', '(Along Y)'],'fontsize',16,'interpreter','latex')
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legend('show');
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grid on;
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%}
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[val, idx] = max(y_fit);
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% Round down to 4 decimal places
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rounded_val = floor(val * 10^4) / 10^4;
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||||
% Find nearest from original vector of boundary points
|
||||
[~, nearest_idx] = min(abs(instability_boundary(:, 2) - rounded_val));
|
||||
nearest_val = instability_boundary(nearest_idx, 2);
|
||||
% Choose the scalar value between the two
|
||||
if ~isscalar(nearest_val)
|
||||
val = rounded_val;
|
||||
else
|
||||
val = nearest_val;
|
||||
end
|
||||
|
||||
AtomNumberDensity = xvals(idx) / add^2; % Areal density of atoms
|
||||
as = val * add; % Scattering length
|
||||
eps_dd = 1/val; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
x0 = 5;
|
||||
Aineq = [];
|
||||
Bineq = [];
|
||||
Aeq = [];
|
||||
Beq = [];
|
||||
lb = [1];
|
||||
ub = [10];
|
||||
nonlcon = [];
|
||||
fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
|
||||
TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
|
||||
sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
|
||||
MeanWidth = sigma * lz; % Mean width of Gaussian ansatz
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(kvec, gs, gdd, MeanWidth, theta, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
DeltaK = ((PlanckConstantReduced^2 .* kvec.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* kvec.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
k_roton_indices = find(imag(EpsilonK) ~= 0);
|
||||
if ~isempty(k_roton_indices)
|
||||
k_roton = median(kvec(k_roton_indices));
|
||||
else
|
||||
k_roton = NaN;
|
||||
end
|
||||
end
|
||||
|
@ -62,7 +62,7 @@ end
|
||||
|
||||
%% ====================================================================================================================================================== %
|
||||
|
||||
figure(6)
|
||||
figure(7)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 1850 750])
|
||||
|
||||
@ -181,7 +181,7 @@ v.FrameRate = 5; % Frame rate of the video
|
||||
open(v); % Open the video file
|
||||
|
||||
for theta = theta_values
|
||||
figure(6)
|
||||
figure(7)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 1850 750])
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
|
@ -76,7 +76,7 @@ for idx = 1:length(nadd2s)
|
||||
end
|
||||
end
|
||||
|
||||
figure(7)
|
||||
figure
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
imagesc(AtomNumber*1E-5, ScatteringLengths, QF * 1E31); % Specify x and y data for axes
|
||||
|
Loading…
Reference in New Issue
Block a user