%% Physical constants PlanckConstant = 6.62607015E-34; PlanckConstantReduced = 6.62607015E-34/(2*pi); FineStructureConstant = 7.2973525698E-3; ElectronMass = 9.10938291E-31; GravitationalConstant = 6.67384E-11; ProtonMass = 1.672621777E-27; AtomicMassUnit = 1.660539066E-27; BohrRadius = 5.2917721067E-11; BohrMagneton = 9.274009994E-24; BoltzmannConstant = 1.38064852E-23; StandardGravityAcceleration = 9.80665; SpeedOfLight = 299792458; StefanBoltzmannConstant = 5.670373E-8; ElectronCharge = 1.602176634E-19; VacuumPermeability = 1.25663706212E-6; DielectricConstant = 8.8541878128E-12; ElectronGyromagneticFactor = -2.00231930436153; AvogadroConstant = 6.02214076E23; ZeroKelvin = 273.15; GravitationalAcceleration = 9.80553; VacuumPermittivity = 1 / (SpeedOfLight^2 * VacuumPermeability); HartreeEnergy = ElectronCharge^2 / (4 * pi * VacuumPermittivity * BohrRadius); AtomicUnitOfPolarizability = (ElectronCharge^2 * BohrRadius^2) / HartreeEnergy; % Or simply 4*pi*VacuumPermittivity*BohrRadius^3 % Dy specific constants Dy164Mass = 163.929174751*AtomicMassUnit; Dy164IsotopicAbundance = 0.2826; DyMagneticMoment = 9.93*BohrMagneton; %% 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 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 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; end end % ====================================================================================================================================================== % alpha = 0; % 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)); 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 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 % == 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) 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'); %{ imagesc(AtomNumber*1E-5, ScatteringLengths, instability_boundary); % Specify x and y data for axes set(gca, 'YDir', 'normal'); % Correct the y-axis direction 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) Go = sqrt(pi) * (k * MeanWidth/sqrt(2)) .* exp((k * MeanWidth/sqrt(2)).^2) .* erfc((k * MeanWidth/sqrt(2))); gamma4 = 1/(sqrt(2*pi) * MeanWidth); Fka = (3 * cos(deg2rad(alpha))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(alpha))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(alpha))^2)); Ukk = (gs + (gdd * Fka)) * gamma4; end function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced) eps_dd = add/as; % Relative interaction strength MeanWidth = x * lz; gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2)); % Quantum Fluctuations term gamma4 = 1/(sqrt(2*pi) * MeanWidth); gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2); gQF = gamma5 * gammaQF; 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