Added plotting of dispersion on the roton boundary.
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@ -24,9 +24,9 @@ HartreeEnergy = ElectronCharge^2 / (4 * pi * VacuumPermittivity *
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AtomicUnitOfPolarizability = (ElectronCharge^2 * BohrRadius^2) / HartreeEnergy; % Or simply 4*pi*VacuumPermittivity*BohrRadius^3
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% Dy specific constants
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Dy164Mass = 163.929174751*1.660539066E-27;
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Dy164Mass = 163.929174751*AtomicMassUnit;
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Dy164IsotopicAbundance = 0.2826;
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DyMagneticMoment = 9.93*9.274009994E-24;
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DyMagneticMoment = 9.93*BohrMagneton;
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%% Bogoliubov excitation spectrum for quasi-2D dipolar gas with QF correction
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AtomNumber = 1E5; % Total atom number in the system
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@ -37,7 +37,9 @@ Trapsize = 7.5815 * lz;
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alpha = 0; % Polar angle of dipole moment
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phi = 0; % Azimuthal angle of momentum vector
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MeanWidth = 5.7304888515 * lz; % Mean width of Gaussian ansatz
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k = linspace(0, 3e6, 1000); % Vector of magnitudes of k vector
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k = linspace(0, 2e6, 1000); % Vector of magnitudes of k vector
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% no = 2.0429e+15, eps_dd = 1.2755, as = 5.4249e-09
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AtomNumberDensity = AtomNumber / Trapsize^2; % Areal density of atoms
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add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
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@ -74,17 +76,17 @@ Trapsize = 7.5815 * lz;
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alpha = 0; % Polar angle of dipole moment
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phi = 0; % Azimuthal angle of momentum vector
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MeanWidth = 5.7304888515 * lz; % Mean width of Gaussian ansatz
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k = linspace(0, 3e6, 1000); % Vector of magnitudes of k vector
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k = linspace(0, 2e6, 1000); % Vector of magnitudes of k vector
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AtomNumberDensity = AtomNumber / Trapsize^2; % Areal density of atoms
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add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
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ScatteringLengths = [];
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eps_dds = [];
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EpsilonKs = [];
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for a = linspace(131,102.515,5)
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ScatteringLengths = [108.5, 105.9, 103.3, 102.5150];
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eps_dds = zeros(1, length(ScatteringLengths));
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EpsilonKs = zeros(length(k), length(ScatteringLengths));
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for idx = 1:length(ScatteringLengths)
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as = a * BohrRadius; % Scattering length
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as = ScatteringLengths(idx) * BohrRadius; % 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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@ -100,19 +102,19 @@ for a = linspace(131,102.515,5)
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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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ScatteringLengths(end+1) = as;
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eps_dds(end+1) = eps_dd;
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EpsilonKs(end+1,:) = EpsilonK;
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eps_dds(idx) = eps_dd;
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EpsilonKs(:,idx) = EpsilonK;
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end
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figure(2)
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clf
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set(gcf,'Position',[50 50 950 750])
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xvals = (k .* add);
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yvals = EpsilonKs(1, :) ./ PlanckConstant;
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yvals = EpsilonKs(:, 1) ./ PlanckConstant;
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plot(xvals, yvals,LineWidth=2.0, DisplayName=['$a_s = ',num2str(round(1/eps_dds(1),3)),'a_{dd}$'])
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hold on
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for idx = 2:length(ScatteringLengths)
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yvals = EpsilonKs(idx, :) ./ PlanckConstant;
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yvals = EpsilonKs(:, idx) ./ PlanckConstant;
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plot(xvals, yvals,LineWidth=2.0, DisplayName=['$a_s = ',num2str(round(1/eps_dds(idx),3)),'a_{dd}$'])
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end
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title(['$na_{dd}^2 = ',num2str(round(AtomNumberDensity * add^2,4)),'$'],'fontsize',16,'interpreter','latex')
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@ -121,6 +123,58 @@ ylabel('$\epsilon(k_{\rho})/h$ (Hz)','fontsize',16,'interpreter','latex')
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grid on
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legend('location', 'northwest','fontsize',16, 'Interpreter','latex')
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%% Bogoliubov excitation spectrum for quasi-2D dipolar gas with QF correction
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wz = 2 * pi * 72.4; % 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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alpha = 0; % Polar angle of dipole moment
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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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nadd2s = [0.0844, 0.0978, 0.123];
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as_to_add = [0.7730, 0.7840, 0.7819];
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var_widths = [4.97165, 5.72960, 5.93178];
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add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
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EpsilonKs = zeros(length(k), length(nadd2s));
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for idx = 1:length(nadd2s)
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AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
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as = (as_to_add(idx) * 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(idx) * lz; % Mean width of Gaussian ansatz
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[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, 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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EpsilonKs(:,idx) = EpsilonK;
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end
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figure(3)
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clf
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set(gcf,'Position',[50 50 950 750])
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xvals = (k .* add);
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yvals = EpsilonKs(:, 1) ./ PlanckConstant;
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plot(xvals, yvals,LineWidth=2.0, DisplayName=['$a_s = ',num2str(round(as_to_add(1),4)),'a_{dd}, na_{dd}^2 = ',num2str(round(nadd2s(1),4)),'$'])
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hold on
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for idx = 2:length(nadd2s)
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yvals = EpsilonKs(:, idx) ./ PlanckConstant;
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plot(xvals, yvals,LineWidth=2.0, DisplayName=['$a_s = ',num2str(round(as_to_add(idx),4)),'a_{dd}, na_{dd}^2 = ',num2str(round(nadd2s(idx),4)),'$'])
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end
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xlabel('$k_{\rho}a_{dd}$','fontsize',16,'interpreter','latex')
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ylabel('$\epsilon(k_{\rho})/h$ (Hz)','fontsize',16,'interpreter','latex')
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grid on
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legend('location', 'northwest','fontsize',16, 'Interpreter','latex')
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%%
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function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, 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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