New script to generate an animation of the roton softening.
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Karthik
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Estimations/RotonInstability/AnimateRotonSoftening.m
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215
Estimations/RotonInstability/AnimateRotonSoftening.m
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% === Preliminaries ===
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clear; clc;
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%% Physical constants
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PlanckConstant = 6.62607015E-34;
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PlanckConstantReduced = 6.62607015E-34/(2*pi);
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FineStructureConstant = 7.2973525698E-3;
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ElectronMass = 9.10938291E-31;
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GravitationalConstant = 6.67384E-11;
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ProtonMass = 1.672621777E-27;
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AtomicMassUnit = 1.660539066E-27;
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BohrRadius = 5.2917721067E-11;
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BohrMagneton = 9.274009994E-24;
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BoltzmannConstant = 1.38064852E-23;
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StandardGravityAcceleration = 9.80665;
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SpeedOfLight = 299792458;
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StefanBoltzmannConstant = 5.670373E-8;
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ElectronCharge = 1.602176634E-19;
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VacuumPermeability = 1.25663706212E-6;
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DielectricConstant = 8.8541878128E-12;
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ElectronGyromagneticFactor = -2.00231930436153;
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AvogadroConstant = 6.02214076E23;
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ZeroKelvin = 273.15;
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GravitationalAcceleration = 9.80553;
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VacuumPermittivity = 1 / (SpeedOfLight^2 * VacuumPermeability);
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HartreeEnergy = ElectronCharge^2 / (4 * pi * VacuumPermittivity * BohrRadius);
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AtomicUnitOfPolarizability = (ElectronCharge^2 * BohrRadius^2) / HartreeEnergy;
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% Dy specific constants
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Dy164Mass = 163.929174751*AtomicMassUnit;
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DyMagneticMoment = 9.93 * BohrMagneton;
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% === System Parameters ===
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AtomNumber = 1E5;
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wz = 2 * pi * 72.4;
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lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz));
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Trapsize = 7.5815 * lz;
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theta = 0;
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phi = 0;
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MeanWidth = 5.7304888515 * lz;
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k = linspace(0, 3e6, 1000);
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AtomNumberDensity = AtomNumber / Trapsize^2;
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add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2);
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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% === eps_dd sweep parameters ===
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eps_dd_values = linspace(1.15, 1.2755, 40);
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max_kval = max(k .* add);
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max_yval = 200; % Hz - manually set this based on known safe upper bound
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frames(length(eps_dd_values)) = struct('cdata', [], 'colormap', []);
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pauseFrameIdx = 0;
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% History for trail
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xHist = {};
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yHist = {};
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% === Plotting ===
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figure(1)
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set(gcf,'Position',[100 100 950 750])
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set(gcf, 'Color', 'w'); % Set figure background to white
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set(gca, 'Color', 'w'); % Set axes background to white
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for idx = 1:length(eps_dd_values)
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eps_dd = eps_dd_values(idx);
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as = add / eps_dd;
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gs = 4 * pi * PlanckConstantReduced^2 / Dy164Mass * as;
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% Compute DDI potential
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[~, ~, ~, Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi);
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% Quantum fluctuations
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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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% Check for instability
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if any(~isreal(DeltaK))
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disp(['Instability reached at eps_dd = ', num2str(eps_dd)]);
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break
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end
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EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
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% Store current curve
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xvals = k .* add;
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yvals = EpsilonK ./ PlanckConstant;
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xHist{end+1} = xvals;
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yHist{end+1} = yvals;
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% Plot all past curves
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clf;
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hold on;
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Nfade = length(xHist);
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cmap = coolwarm(Nfade); % Use custom coolwarm colormap
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for n = 1:5:Nfade-1
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plot(xHist{n}, yHist{n}, 'Color', [cmap(n,:), 0.2 + 0.6*(n/Nfade)], 'LineWidth', 1.5);
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end
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% Highlight current curve
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minDelta = min(DeltaK);
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[~, minIdx] = min(DeltaK);
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if (minDelta < 0.0001 * max(DeltaK)) && (minIdx > 10)
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plot(xvals, yvals, 'Color', [180, 4, 38]/255, 'LineWidth', 5); % Orange
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% Arrow
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x_arrow = xvals(minIdx);
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y_arrow = yvals(minIdx);
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dx = 0;
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dy = -0.1 * max_yval;
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quiver(x_arrow, y_arrow - dy/2, dx, dy, ...
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'AutoScale', 'off', 'MaxHeadSize', 1.5, ...
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'Color', 'k', 'LineWidth', 2);
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text(x_arrow, y_arrow - dy/2 + 0.05 * max_yval, ...
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'$2\pi/\lambda_{\mathrm{rot}}$', ...
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'Interpreter', 'latex', 'FontSize', 16, 'Color', 'k', ...
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'HorizontalAlignment', 'center');
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pauseFrameIdx = idx;
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else
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plot(xvals, yvals, 'Color', [59, 76, 192]/255, 'LineWidth', 5); % Blue
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end
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% Axes
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% title(['$a_s = ', num2str(round(1/eps_dd,3)), 'a_{dd}, \quad na_{dd}^2 = ', num2str(round(AtomNumberDensity * add^2,4)), '$'], ...
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% 'fontsize', 16, 'interpreter', 'latex')
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% xlabel('$k_{\rho}a_{dd}$','fontsize',26,'interpreter','latex')
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% ylabel('$\epsilon(k_{\rho})/h$ (Hz)','fontsize',26,'interpreter','latex')
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xlim([0, max_kval])
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ylim([0, max_yval])
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set(gca, 'XTick', [], 'YTick', [], 'Color', 'w', 'XColor', 'none', 'YColor', 'none');
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% Arrow positions (in normalized figure coordinates)
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ax = gca;
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ax.Units = 'normalized';
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pos = ax.Position;
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% X-axis arrow (horizontal)
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annotation('arrow', ...
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[pos(1), pos(1) + pos(3)*0.95], ...
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[pos(2), pos(2)], ...
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'Color', 'k', 'LineWidth', 2);
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% Y-axis arrow (vertical)
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annotation('arrow', ...
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[pos(1), pos(1)], ...
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[pos(2), pos(2) + pos(4)*0.95], ...
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'Color', 'k', 'LineWidth', 2);
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% x-label
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text(pos(1) + pos(3)*0.5, pos(2) - 0.15, 'Momentum', ...
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'Units', 'normalized', ...
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'HorizontalAlignment', 'center', ...
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'VerticalAlignment', 'top', ...
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'FontName', 'Bahnschrift' , ...
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'FontSize', 26);
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% y-label
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text(pos(1) - 0.2, pos(2) + pos(4)*0.5, 'Energy', ...
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'Units', 'normalized', ...
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'HorizontalAlignment', 'center', ...
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'VerticalAlignment', 'middle', ...
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'FontName', 'Bahnschrift' , ...
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'FontSize', 26, ...
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'Rotation', 90);
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drawnow;
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frames(idx) = getframe(gcf);
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end
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%% === Save as GIF ===
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filename = 'roton_dispersion.gif';
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delayTime = 0.15;
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pauseRepeats = 10;
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for i = 1:length(frames)
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[A, map] = rgb2ind(frame2im(frames(i)), 256);
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if i == 1
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imwrite(A, map, filename, 'gif', 'LoopCount', Inf, 'DelayTime', delayTime);
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elseif i == pauseFrameIdx
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for j = 1:pauseRepeats
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imwrite(A, map, filename, 'gif', 'WriteMode', 'append', 'DelayTime', delayTime);
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end
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else
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imwrite(A, map, filename, 'gif', 'WriteMode', 'append', 'DelayTime', delayTime);
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end
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end
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disp(['GIF saved as ', filename]);
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%% === Helper Function ===
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function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, theta, phi)
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Go = sqrt(pi) * (k * MeanWidth/sqrt(2)) .* exp((k * MeanWidth/sqrt(2)).^2) .* erfc((k * MeanWidth/sqrt(2)));
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gamma4 = 1/(sqrt(2*pi) * MeanWidth);
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Fka = (3 * cos(deg2rad(theta))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(theta))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(theta))^2));
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Ukk = (gs + (gdd * Fka)) * gamma4;
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end
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function cmap = coolwarm(n)
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% Generates a coolwarm colormap with n colors
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c1 = [59, 76, 192]/255;
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c2 = [180, 4, 38]/255;
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r = linspace(c1(1), c2(1), n)';
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g = linspace(c1(2), c2(2), n)';
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b = linspace(c1(3), c2(3), n)';
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cmap = [r g b];
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end
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