383 lines
24 KiB
Matlab
383 lines
24 KiB
Matlab
%% 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; % Or simply 4*pi*VacuumPermittivity*BohrRadius^3
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% Dy specific constants
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Dy164Mass = 163.929174751*AtomicMassUnit;
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Dy164IsotopicAbundance = 0.2826;
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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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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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as = 102.515 * BohrRadius; % Scattering length
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Trapsize = 7.5815 * lz; % Trap is assumed to be a box of finite extent , given here in units of the 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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MeanWidth = 5.7304888515 * lz; % Mean width of Gaussian ansatz
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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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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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[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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figure(1)
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set(gcf,'Position',[50 50 950 750])
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xvals = (k .* add);
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yvals = EpsilonK ./ PlanckConstant;
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plot(xvals, yvals,LineWidth=2.0)
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title(horzcat(['$a_s = ',num2str(round(1/eps_dd,3)),'a_{dd}, '], ['na_{dd}^2 = ',num2str(round(AtomNumberDensity * add^2,4)),'$']),'fontsize',16,'interpreter','latex')
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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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%% For different interaction strengths
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AtomNumber = 1E5; % Total atom number in the system
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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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Trapsize = 7.5815 * lz; % Trap is assumed to be a box of finite extent , given here in units of the 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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MeanWidth = 5.7304888515 * lz; % Mean width of Gaussian ansatz
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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 = [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 = 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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[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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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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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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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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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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%% For 3 points on the roton instability boundary
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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, 5.7296048721, 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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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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%% Mean widths of the variational Gaussian ansatz - extremize the total mean field energy per particle wrt to the variational parameter
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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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TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
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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 = [7];
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nonlcon = [];
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fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
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sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
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fprintf(['Variational width of Gaussian ansatz = ' num2str(sigma) ' * lz \n'])
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%% Mean widths of the variational Gaussian ansatz - extremize the total mean field energy per particle wrt to the variational parameter
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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 = 0.74:0.001:0.79;
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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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figure(4)
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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, var_widths); % 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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% ====================================================================================================================================================== %
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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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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(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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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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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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%%
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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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gamma4 = 1/(sqrt(2*pi) * MeanWidth);
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Fka = (3 * cos(deg2rad(alpha))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(alpha))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(alpha))^2));
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Ukk = (gs + (gdd * Fka)) * gamma4;
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end
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function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced)
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eps_dd = add/as; % Relative interaction strength
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MeanWidth = x * lz;
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gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2)); % Quantum Fluctuations term
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gamma4 = 1/(sqrt(2*pi) * MeanWidth);
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gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
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gQF = gamma5 * gammaQF;
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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 |