diff --git a/Estimations/UniformDipolarBECRotonInstabilityBoundary.m b/Estimations/ReproduceBlairBlakieResults.m similarity index 75% rename from Estimations/UniformDipolarBECRotonInstabilityBoundary.m rename to Estimations/ReproduceBlairBlakieResults.m index 2157a4e..e40489f 100644 --- a/Estimations/UniformDipolarBECRotonInstabilityBoundary.m +++ b/Estimations/ReproduceBlairBlakieResults.m @@ -138,15 +138,21 @@ var_widths = [4.97165, 5.7296048721, 5.93178]; add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length EpsilonKs = zeros(length(k), length(nadd2s)); +ScatteringLengths = zeros(length(as_to_add), 1); +AtomNumber = zeros(length(nadd2s), 1); +w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction +l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length +tsize = 10 * l0; for idx = 1:length(nadd2s) - - AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms - as = (as_to_add(idx) * add); % Scattering length - eps_dd = add/as; % Relative interaction strength - gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength - gdd = VacuumPermeability*DyMagneticMoment^2/3; - MeanWidth = var_widths(idx) * lz; % Mean width of Gaussian ansatz + AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms + AtomNumber(idx) = AtomNumberDensity*tsize^2; + as = (as_to_add(idx) * add); % Scattering length + ScatteringLengths(idx) = as/BohrRadius; + eps_dd = add/as; % Relative interaction strength + gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength + gdd = VacuumPermeability*DyMagneticMoment^2/3; + MeanWidth = var_widths(idx) * lz; % Mean width of Gaussian ansatz [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space @@ -181,11 +187,11 @@ legend('location', 'northwest','fontsize',16, 'Interpreter','latex') wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length -gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length gdd = VacuumPermeability*DyMagneticMoment^2/3; AtomNumberDensity = 0.0978 / add^2; as = 0.784 * add; % Scattering length +gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced); x0 = 5; @@ -204,7 +210,6 @@ fprintf(['Variational width of Gaussian ansatz = ' num2str(sigma) ' * lz \n']) wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length -gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length gdd = VacuumPermeability*DyMagneticMoment^2/3; @@ -226,6 +231,7 @@ for idx = 1:length(nadd2s) for jdx = 1:length(as_to_add) AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms as = (as_to_add(jdx) * add); % Scattering length + gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced); sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts); var_widths(jdx, idx) = sigma; @@ -247,15 +253,22 @@ alpha = 0; phi = 0; % Azimuthal angle of momentum vector k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector instability_boundary = zeros(length(as_to_add), length(nadd2s)); +ScatteringLengths = zeros(length(as_to_add), 1); +AtomNumber = zeros(length(nadd2s), 1); +w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction +l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length +tsize = 10 * l0; for idx = 1:length(nadd2s) for jdx = 1:length(as_to_add) - AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms - as = (as_to_add(jdx) * add); % Scattering length - eps_dd = add/as; % Relative interaction strength - gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength - gdd = VacuumPermeability*DyMagneticMoment^2/3; - MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz + AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms + AtomNumber(idx) = AtomNumberDensity*tsize^2; + as = (as_to_add(jdx) * add); % Scattering length + ScatteringLengths(jdx) = as/BohrRadius; + eps_dd = add/as; % Relative interaction strength + gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength + gdd = VacuumPermeability*DyMagneticMoment^2/3; + MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space @@ -277,6 +290,8 @@ as_to_add_from_figure = [0.76383, 0.76766, 0.76974, 0.77543, 0.77675, 0.77828, figure(5) clf set(gcf,'Position',[50 50 950 750]) + + imagesc(nadd2s, as_to_add, instability_boundary); % Specify x and y data for axes hold on plot(nadd2s_from_figure, as_to_add_from_figure, 'r*-', 'LineWidth', 2); % Plot the curve (red line) @@ -284,84 +299,17 @@ set(gca, 'YDir', 'normal'); % Correct the y-axis direction colorbar; % Add a colorbar xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex'); ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex'); -title('Roton instability boundary','fontsize',16,'interpreter','latex') -%% Roton instability boundary for tilted dipoles - -wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction -lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length -gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength -add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length -gdd = VacuumPermeability*DyMagneticMoment^2/3; - -nadd2s = 0.05:0.001:0.25; -as_to_add = 0.76:0.001:0.81; -var_widths = zeros(length(as_to_add), length(nadd2s)); - -x0 = 5; -Aineq = []; -Bineq = []; -Aeq = []; -Beq = []; -lb = [1]; -ub = [10]; -nonlcon = []; -fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500); - -for idx = 1:length(nadd2s) - for jdx = 1:length(as_to_add) - AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms - as = (as_to_add(jdx) * add); % Scattering length - TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced); - sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts); - var_widths(jdx, idx) = sigma; - end -end - -% ====================================================================================================================================================== % - -alpha = 45; % Polar angle of dipole moment -phi = 0; % Azimuthal angle of momentum vector -k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector -instability_boundary = zeros(length(as_to_add), length(nadd2s)); - -for idx = 1:length(nadd2s) - for jdx = 1:length(as_to_add) - AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms - as = (as_to_add(jdx) * add); % Scattering length - eps_dd = add/as; % Relative interaction strength - gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength - gdd = VacuumPermeability*DyMagneticMoment^2/3; - MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz - - [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, 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; - - % == Dispersion relation == % - DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2)); - EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK); - instability_boundary(jdx, idx) = ~isreal(EpsilonK); - end -end - -nadd2s_from_figure = [0.04974, 0.05383, 0.05655, 0.06609, 0.06916, 0.07291, 0.07836, 0.08517, 0.09063, 0.0978, 0.10459, 0.11345, 0.11822, 0.12231, 0.12674, 0.13117, 0.13560, 0.14003, 0.14548, 0.15127, 0.15775, 0.16660, 0.17546, 0.18364, 0.19557, 0.20579, 0.21839, 0.23850, 0.25144]; -as_to_add_from_figure = [0.76383, 0.76766, 0.76974, 0.77543, 0.77675, 0.77828, 0.78003, 0.78178, 0.78288, 0.7840, 0.78474, 0.78540, 0.78562, 0.78572, 0.78583, 0.78583, 0.78583, 0.78583, 0.78567, 0.78551, 0.78529, 0.78485, 0.78441, 0.78386, 0.78310, 0.78233, 0.78135, 0.77970, 0.77861]; - -figure(6) -clf -set(gcf,'Position',[50 50 950 750]) -imagesc(nadd2s, as_to_add, instability_boundary); % Specify x and y data for axes -hold on -plot(nadd2s_from_figure, as_to_add_from_figure, 'r*-', 'LineWidth', 2); % Plot the curve (red line) +%{ +imagesc(AtomNumber*1E-5, ScatteringLengths, instability_boundary); % Specify x and y data for axes set(gca, 'YDir', 'normal'); % Correct the y-axis direction -colorbar; % Add a colorbar -xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex'); -ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex'); +cbar1 = colorbar; +cbar1.Label.Interpreter = 'latex'; +ylabel(cbar1,'$(\times 10^{-31})$','FontSize',16,'Rotation',270) +xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex'); +ylabel('Scattering length ($\times a_0$)','fontsize',16,'interpreter','latex'); title('Roton instability boundary','fontsize',16,'interpreter','latex') +%} %% function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi) @@ -381,4 +329,7 @@ function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, g Energy_AxialComponent = (PlanckConstantReduced * wz) * ((lz^2/(4 * MeanWidth^2)) + (MeanWidth^2/(4 * lz^2))); Energy_TransverseComponent = (0.5 * (gs + (2*gdd)) * gamma4 * AtomNumberDensity) + ((2/5) * gQF * AtomNumberDensity^(3/2)); ret = (Energy_AxialComponent + Energy_TransverseComponent) / (PlanckConstantReduced * wz); -end \ No newline at end of file +end + + +