Name change
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@ -138,15 +138,21 @@ var_widths = [4.97165, 5.7296048721, 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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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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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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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(idx) * add); % Scattering length
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ScatteringLengths(idx) = 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(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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@ -181,11 +187,11 @@ legend('location', 'northwest','fontsize',16, 'Interpreter','latex')
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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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gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
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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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AtomNumberDensity = 0.0978 / add^2;
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as = 0.784 * 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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x0 = 5;
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@ -204,7 +210,6 @@ fprintf(['Variational width of Gaussian ansatz = ' num2str(sigma) ' * lz \n'])
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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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gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
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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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@ -226,6 +231,7 @@ 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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@ -247,15 +253,22 @@ alpha = 0;
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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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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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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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[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
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@ -277,6 +290,8 @@ as_to_add_from_figure = [0.76383, 0.76766, 0.76974, 0.77543, 0.77675, 0.77828,
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figure(5)
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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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hold on
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plot(nadd2s_from_figure, as_to_add_from_figure, 'r*-', 'LineWidth', 2); % Plot the curve (red line)
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@ -284,84 +299,17 @@ 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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%% 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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lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
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gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
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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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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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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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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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% ====================================================================================================================================================== %
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alpha = 45; % 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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instability_boundary = zeros(length(as_to_add), length(nadd2s));
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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(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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instability_boundary(jdx, idx) = ~isreal(EpsilonK);
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end
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end
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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];
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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];
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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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hold on
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plot(nadd2s_from_figure, as_to_add_from_figure, 'r*-', 'LineWidth', 2); % Plot the curve (red line)
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%{
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imagesc(AtomNumber*1E-5, ScatteringLengths, 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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cbar1 = colorbar;
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cbar1.Label.Interpreter = 'latex';
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ylabel(cbar1,'$(\times 10^{-31})$','FontSize',16,'Rotation',270)
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xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex');
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ylabel('Scattering length ($\times a_0$)','fontsize',16,'interpreter','latex');
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title('Roton instability boundary','fontsize',16,'interpreter','latex')
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%}
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%%
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function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi)
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@ -381,4 +329,7 @@ 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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