252 lines
12 KiB
Matlab
252 lines
12 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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%% 2-D DDI Potential in k-space, with Gaussian ansatz width determined by constrained minimization
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% With user-defined values of interaction, density and tilt
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% w0 = 2*pi*61.6316; % Angular frequency unit [s^-1]
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% l0 = sqrt(PlanckConstantReduced/(Dy164Mass*w0));
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% % Defining a harmonic oscillator length - heredue to the choice of w0, l0
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% is 1 micrometer
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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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Params.Lx = 150*1e-6;
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Params.Ly = 150*1e-6;
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[Transf] = setupSpace(Params);
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nadd2s = 0.110; % Number density * add^2
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as_to_add = 0.782; % 1/edd
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Params.theta = 57; % Polar angle of dipole moment
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Params.eta = 0; % Azimuthal angle of dipole moment
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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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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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AtomNumberDensity = nadd2s / add^2; % Number density of atoms
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as = as_to_add * 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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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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MeanWidth = sigma * 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)*1e-6, fftshift(Transf.ky)*1e-6, 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(['2-D DDI Potential: $\theta = ',num2str(Params.theta), '; \eta = ', num2str(Params.eta),'$'],'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)*1e-6, fftshift(Transf.ky)*1e-6, 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(['2-D Dispersion: $\theta = ',num2str(Params.theta), '; \eta = ', num2str(Params.eta),'$'],'fontsize',16,'interpreter','latex')
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%% Cycle through angles
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% Define values for theta and eta
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theta_values = 0:10:90; % Range of theta values (you can modify this)
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eta_values = 0:10:90; % Range of eta values (you can modify this)
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% Set up VideoWriter object to produce a movie
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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 theta and eta values
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for theta = theta_values
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for eta = eta_values
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% Update Params with current theta and eta
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Params.theta = theta;
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Params.eta = eta;
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% Compute the potential for the current theta and eta
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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(13)
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clf
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set(gcf,'Position',[50 50 950 750])
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imagesc(fftshift(Transf.kx)*1e-6, fftshift(Transf.ky)*1e-6, 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(['2-D Dispersion: $\theta = ',num2str(Params.theta), '; \eta = ', num2str(Params.eta),'$'],'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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% Angles of the dipole moment are defined in and away from the X-Z plane
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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.eta)^2 + QY.^2*sin(Params.eta)^2; % eta here is the azimuthal angle of the dipole moment (angle from the x-axis)
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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.theta)^2 + Fperp*cos(Params.theta)^2); % theta here is the polar angle of the dipole moment (angle from the z-axis)
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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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