291 lines
13 KiB
Mathematica
291 lines
13 KiB
Mathematica
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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; % 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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%% 2-D DDI Potential in k-space, with Gaussian ansatz width determined by constrained minimization
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wz = 2 * pi * 300; % 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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% Number of grid points in each direction
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Params.Nx = 128;
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Params.Ny = 128;
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% Dimensions (in units of l0)
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w0 = 2*pi*61.6316; % Angular frequency unit [s^-1]
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l0 = sqrt(PlanckConstantReduced/(Dy164Mass*w0)); % Defining a harmonic oscillator length
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Params.Lx = 150*l0;
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Params.Ly = 150*l0;
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[Transf] = setupSpace(Params);
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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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add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
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gdd = VacuumPermeability*DyMagneticMoment^2/3;
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%%
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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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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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end
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end
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figure(10)
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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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%% Chosen values of interaction, density and tilt
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nadd = 0.110;
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asadd = 0.782;
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Params.alpha = 0;
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Params.phi = 0;
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[~, nadd2sidx] = min(abs(nadd2s - nadd));
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[~, asaddidx] = min(abs(as_to_add - asadd));
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AtomNumberDensity = nadd2s(nadd2sidx) / add^2; % Areal density of atoms
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as = (as_to_add(asaddidx) * 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(asaddidx, nadd2sidx)*lz;
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% == 2-D DDI Potential in k-space == %
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VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth);
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VDk_fftshifted = fftshift(VDk);
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figure(11)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*l0, fftshift(Transf.ky)*l0, VDk_fftshifted); % Specify x and y data for axes
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set(gca, 'YDir', 'normal'); % Correct the y-axis direction
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cbar1 = colorbar;
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cbar1.Label.Interpreter = 'latex';
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xlabel('$k_x l_o$','fontsize',16,'interpreter','latex');
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ylabel('$k_y l_o$','fontsize',16,'interpreter','latex');
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title(['$\theta = ',num2str(Params.alpha), '; \phi = ', num2str(Params.phi),'$'],'fontsize',16,'interpreter','latex')
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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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EpsilonK = zeros(length(Transf.ky), length(Transf.kx));
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gs_tilde = gs / (sqrt(2*pi) * MeanWidth);
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% == Dispersion relation == %
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for idx = 1:length(Transf.kx)
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for jdx = 1:length(Transf.ky)
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DeltaK = ((PlanckConstantReduced^2 .* (Transf.kx(idx).^2 + Transf.ky(jdx).^2)) ./ (2 * Dy164Mass)) + (2 * AtomNumberDensity * gs_tilde) + ((2 * AtomNumberDensity) .* VDk_fftshifted(jdx, idx)) + (3 * gQF * AtomNumberDensity^(3/2));
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EpsilonK(jdx, idx) = sqrt(((PlanckConstantReduced^2 .* (Transf.kx(idx).^2 + Transf.ky(jdx).^2)) ./ (2 * Dy164Mass)) .* DeltaK);
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end
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end
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EpsilonK = double(imag(EpsilonK) ~= 0); % 'isreal' returns 0 for complex numbers and 1 for real numbers, so we negate it
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figure(12)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*l0, fftshift(Transf.ky)*l0, EpsilonK); % Specify x and y data for axes
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set(gca, 'YDir', 'normal'); % Correct the y-axis direction
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cbar1 = colorbar;
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cbar1.Label.Interpreter = 'latex';
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xlabel('$k_x l_o$','fontsize',16,'interpreter','latex');
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ylabel('$k_y l_o$','fontsize',16,'interpreter','latex');
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title(['$\theta = ',num2str(Params.alpha), '; \phi = ', num2str(Params.phi),'$'],'fontsize',16,'interpreter','latex')
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%%
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nadd = 0.110;
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asadd = 0.782;
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% Define values for alpha and phi
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alpha_values = 0:5:90; % Range of alpha values (you can modify this)
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phi_values = 0:2:90; % Range of phi values (you can modify this)
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[~, nadd2sidx] = min(abs(nadd2s - nadd));
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[~, asaddidx] = min(abs(as_to_add - asadd));
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AtomNumberDensity = nadd2s(nadd2sidx) / add^2; % Areal density of atoms
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as = (as_to_add(asaddidx) * 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(asaddidx, nadd2sidx)*lz;
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% Set up VideoWriter object
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v = VideoWriter('potential_movie', 'MPEG-4'); % Create a video object
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v.FrameRate = 5; % Frame rate of the video
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open(v); % Open the video file
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% Loop over alpha and phi values
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for alpha = alpha_values
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for phi = phi_values
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% Update Params with current alpha and phi
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Params.alpha = alpha;
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Params.phi = phi;
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% Compute the potential for the current alpha and phi
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% == 2-D DDI Potential in k-space == %
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VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth);
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VDk_fftshifted = fftshift(VDk);
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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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EpsilonK = zeros(length(Transf.ky), length(Transf.kx));
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gs_tilde = gs / (sqrt(2*pi) * MeanWidth);
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% == Dispersion relation == %
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for idx = 1:length(Transf.kx)
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for jdx = 1:length(Transf.ky)
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DeltaK = ((PlanckConstantReduced^2 .* (Transf.kx(idx).^2 + Transf.ky(jdx).^2)) ./ (2 * Dy164Mass)) + (2 * AtomNumberDensity * gs_tilde) + ((2 * AtomNumberDensity) .* VDk_fftshifted(jdx, idx)) + (3 * gQF * AtomNumberDensity^(3/2));
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EpsilonK(jdx, idx) = sqrt(((PlanckConstantReduced^2 .* (Transf.kx(idx).^2 + Transf.ky(jdx).^2)) ./ (2 * Dy164Mass)) .* DeltaK);
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end
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end
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EpsilonK = double(imag(EpsilonK) ~= 0); % 'isreal' returns 0 for complex numbers and 1 for real numbers, so we negate it
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% Plot the result
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figure(12)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*l0, fftshift(Transf.ky)*l0, EpsilonK); % Specify x and y data for axes
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set(gca, 'YDir', 'normal'); % Correct the y-axis direction
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cbar1 = colorbar;
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cbar1.Label.Interpreter = 'latex';
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xlabel('$k_x$','fontsize',16,'interpreter','latex');
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ylabel('$k_y$','fontsize',16,'interpreter','latex');
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title(['$\theta = ',num2str(Params.alpha), '; \phi = ', num2str(Params.phi),'$'],'fontsize',16,'interpreter','latex')
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% Capture the frame and write to video
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frame = getframe(gcf); % Capture the current figure
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writeVideo(v, frame); % Write the frame to the video
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end
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end
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% Close the video file
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close(v);
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%%
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function [Transf] = setupSpace(Params)
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Transf.Xmax = 0.5*Params.Lx;
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Transf.Ymax = 0.5*Params.Ly;
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Nx = Params.Nx;
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Ny = Params.Ny;
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% Fourier grids
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x = linspace(-0.5*Params.Lx,0.5*Params.Lx-Params.Lx/Params.Nx,Params.Nx);
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Kmax = pi*Params.Nx/Params.Lx;
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kx = linspace(-Kmax,Kmax,Nx+1);
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kx = kx(1:end-1);
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dkx = kx(2)-kx(1);
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kx = fftshift(kx);
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y = linspace(-0.5*Params.Ly,0.5*Params.Ly-Params.Ly/Params.Ny,Params.Ny);
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Kmax = pi*Params.Ny/Params.Ly;
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ky = linspace(-Kmax,Kmax,Ny+1);
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ky = ky(1:end-1);
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dky = ky(2)-ky(1);
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ky = fftshift(ky);
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[Transf.X,Transf.Y] = ndgrid(x,y);
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[Transf.KX,Transf.KY] = ndgrid(kx,ky);
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Transf.x = x;
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Transf.y = y;
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Transf.kx = kx;
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Transf.ky = ky;
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Transf.dx = x(2)-x(1);
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Transf.dy = y(2)-y(1);
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Transf.dkx = dkx;
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Transf.dky = dky;
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end
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function VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth)
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% == Calculating the DDI potential in Fourier space with appropriate cutoff == %
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% Interaction in K space
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QX = Transf.KX*MeanWidth/sqrt(2);
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QY = Transf.KY*MeanWidth/sqrt(2);
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Qsq = QX.^2 + QY.^2;
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absQ = sqrt(Qsq);
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QDsq = QX.^2*cos(Params.phi)^2 + QY.^2*sin(Params.phi)^2;
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% Bare interaction
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Fpar = -1 + 3*sqrt(pi)*QDsq.*erfcx(absQ)./absQ; % Scaled complementary error function erfcx(x) = e^(x^2) * erfc(x)
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Fperp = 2 - 3*sqrt(pi).*absQ.*erfcx(absQ);
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Fpar(absQ == 0) = -1;
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% Full DDI
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VDk = (Fpar*sin(Params.alpha)^2 + Fperp*cos(Params.alpha)^2);
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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
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