Relocation, restructuring of scripts
This commit is contained in:
parent
de6940f2e1
commit
fbb0b493e3
@ -0,0 +1,290 @@
|
||||
%% Physical constants
|
||||
PlanckConstant = 6.62607015E-34;
|
||||
PlanckConstantReduced = 6.62607015E-34/(2*pi);
|
||||
FineStructureConstant = 7.2973525698E-3;
|
||||
ElectronMass = 9.10938291E-31;
|
||||
GravitationalConstant = 6.67384E-11;
|
||||
ProtonMass = 1.672621777E-27;
|
||||
AtomicMassUnit = 1.660539066E-27;
|
||||
BohrRadius = 5.2917721067E-11;
|
||||
BohrMagneton = 9.274009994E-24;
|
||||
BoltzmannConstant = 1.38064852E-23;
|
||||
StandardGravityAcceleration = 9.80665;
|
||||
SpeedOfLight = 299792458;
|
||||
StefanBoltzmannConstant = 5.670373E-8;
|
||||
ElectronCharge = 1.602176634E-19;
|
||||
VacuumPermeability = 1.25663706212E-6;
|
||||
DielectricConstant = 8.8541878128E-12;
|
||||
ElectronGyromagneticFactor = -2.00231930436153;
|
||||
AvogadroConstant = 6.02214076E23;
|
||||
ZeroKelvin = 273.15;
|
||||
GravitationalAcceleration = 9.80553;
|
||||
VacuumPermittivity = 1 / (SpeedOfLight^2 * VacuumPermeability);
|
||||
HartreeEnergy = ElectronCharge^2 / (4 * pi * VacuumPermittivity * BohrRadius);
|
||||
AtomicUnitOfPolarizability = (ElectronCharge^2 * BohrRadius^2) / HartreeEnergy; % Or simply 4*pi*VacuumPermittivity*BohrRadius^3
|
||||
|
||||
% Dy specific constants
|
||||
Dy164Mass = 163.929174751*AtomicMassUnit;
|
||||
Dy164IsotopicAbundance = 0.2826;
|
||||
DyMagneticMoment = 9.93*BohrMagneton;
|
||||
|
||||
%% 2-D DDI Potential in k-space, with Gaussian ansatz width determined by constrained minimization
|
||||
|
||||
wz = 2 * pi * 300; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
|
||||
% Number of grid points in each direction
|
||||
Params.Nx = 128;
|
||||
Params.Ny = 128;
|
||||
|
||||
% Dimensions (in units of l0)
|
||||
w0 = 2*pi*61.6316; % Angular frequency unit [s^-1]
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass*w0)); % Defining a harmonic oscillator length
|
||||
Params.Lx = 150*l0;
|
||||
Params.Ly = 150*l0;
|
||||
[Transf] = setupSpace(Params);
|
||||
|
||||
nadd2s = 0.05:0.001:0.25;
|
||||
as_to_add = 0.74:0.001:0.79;
|
||||
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
%%
|
||||
var_widths = zeros(length(as_to_add), length(nadd2s));
|
||||
|
||||
x0 = 5;
|
||||
Aineq = [];
|
||||
Bineq = [];
|
||||
Aeq = [];
|
||||
Beq = [];
|
||||
lb = [1];
|
||||
ub = [10];
|
||||
nonlcon = [];
|
||||
fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
|
||||
sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
|
||||
var_widths(jdx, idx) = sigma;
|
||||
end
|
||||
end
|
||||
|
||||
figure(10)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
imagesc(nadd2s, as_to_add, var_widths); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
colorbar; % Add a colorbar
|
||||
xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
|
||||
|
||||
%% Chosen values of interaction, density and tilt
|
||||
|
||||
nadd = 0.110;
|
||||
asadd = 0.782;
|
||||
Params.alpha = 0;
|
||||
Params.phi = 0;
|
||||
|
||||
[~, nadd2sidx] = min(abs(nadd2s - nadd));
|
||||
[~, asaddidx] = min(abs(as_to_add - asadd));
|
||||
|
||||
AtomNumberDensity = nadd2s(nadd2sidx) / add^2; % Areal density of atoms
|
||||
as = (as_to_add(asaddidx) * add); % Scattering length
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(asaddidx, nadd2sidx)*lz;
|
||||
|
||||
% == 2-D DDI Potential in k-space == %
|
||||
VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth);
|
||||
VDk_fftshifted = fftshift(VDk);
|
||||
|
||||
figure(11)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
imagesc(fftshift(Transf.kx)*l0, fftshift(Transf.ky)*l0, VDk_fftshifted); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
cbar1 = colorbar;
|
||||
cbar1.Label.Interpreter = 'latex';
|
||||
xlabel('$k_x l_o$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$k_y l_o$','fontsize',16,'interpreter','latex');
|
||||
title(['$\theta = ',num2str(Params.alpha), '; \phi = ', num2str(Params.phi),'$'],'fontsize',16,'interpreter','latex')
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
EpsilonK = zeros(length(Transf.ky), length(Transf.kx));
|
||||
gs_tilde = gs / (sqrt(2*pi) * MeanWidth);
|
||||
|
||||
% == Dispersion relation == %
|
||||
for idx = 1:length(Transf.kx)
|
||||
for jdx = 1:length(Transf.ky)
|
||||
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));
|
||||
EpsilonK(jdx, idx) = sqrt(((PlanckConstantReduced^2 .* (Transf.kx(idx).^2 + Transf.ky(jdx).^2)) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
end
|
||||
end
|
||||
|
||||
EpsilonK = double(imag(EpsilonK) ~= 0); % 'isreal' returns 0 for complex numbers and 1 for real numbers, so we negate it
|
||||
|
||||
figure(12)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
imagesc(fftshift(Transf.kx)*l0, fftshift(Transf.ky)*l0, EpsilonK); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
cbar1 = colorbar;
|
||||
cbar1.Label.Interpreter = 'latex';
|
||||
xlabel('$k_x l_o$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$k_y l_o$','fontsize',16,'interpreter','latex');
|
||||
title(['$\theta = ',num2str(Params.alpha), '; \phi = ', num2str(Params.phi),'$'],'fontsize',16,'interpreter','latex')
|
||||
|
||||
%%
|
||||
|
||||
nadd = 0.110;
|
||||
asadd = 0.782;
|
||||
% Define values for alpha and phi
|
||||
alpha_values = 0:5:90; % Range of alpha values (you can modify this)
|
||||
phi_values = 0:2:90; % Range of phi values (you can modify this)
|
||||
|
||||
[~, nadd2sidx] = min(abs(nadd2s - nadd));
|
||||
[~, asaddidx] = min(abs(as_to_add - asadd));
|
||||
|
||||
AtomNumberDensity = nadd2s(nadd2sidx) / add^2; % Areal density of atoms
|
||||
as = (as_to_add(asaddidx) * add); % Scattering length
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(asaddidx, nadd2sidx)*lz;
|
||||
|
||||
% Set up VideoWriter object
|
||||
v = VideoWriter('potential_movie', 'MPEG-4'); % Create a video object
|
||||
v.FrameRate = 5; % Frame rate of the video
|
||||
open(v); % Open the video file
|
||||
|
||||
% Loop over alpha and phi values
|
||||
for alpha = alpha_values
|
||||
for phi = phi_values
|
||||
% Update Params with current alpha and phi
|
||||
Params.alpha = alpha;
|
||||
Params.phi = phi;
|
||||
|
||||
% Compute the potential for the current alpha and phi
|
||||
% == 2-D DDI Potential in k-space == %
|
||||
VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth);
|
||||
VDk_fftshifted = fftshift(VDk);
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
EpsilonK = zeros(length(Transf.ky), length(Transf.kx));
|
||||
gs_tilde = gs / (sqrt(2*pi) * MeanWidth);
|
||||
|
||||
% == Dispersion relation == %
|
||||
for idx = 1:length(Transf.kx)
|
||||
for jdx = 1:length(Transf.ky)
|
||||
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));
|
||||
EpsilonK(jdx, idx) = sqrt(((PlanckConstantReduced^2 .* (Transf.kx(idx).^2 + Transf.ky(jdx).^2)) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
end
|
||||
end
|
||||
|
||||
EpsilonK = double(imag(EpsilonK) ~= 0); % 'isreal' returns 0 for complex numbers and 1 for real numbers, so we negate it
|
||||
|
||||
% Plot the result
|
||||
figure(12)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
imagesc(fftshift(Transf.kx)*l0, fftshift(Transf.ky)*l0, EpsilonK); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
cbar1 = colorbar;
|
||||
cbar1.Label.Interpreter = 'latex';
|
||||
xlabel('$k_x$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$k_y$','fontsize',16,'interpreter','latex');
|
||||
title(['$\theta = ',num2str(Params.alpha), '; \phi = ', num2str(Params.phi),'$'],'fontsize',16,'interpreter','latex')
|
||||
|
||||
% Capture the frame and write to video
|
||||
frame = getframe(gcf); % Capture the current figure
|
||||
writeVideo(v, frame); % Write the frame to the video
|
||||
end
|
||||
end
|
||||
|
||||
% Close the video file
|
||||
close(v);
|
||||
|
||||
%%
|
||||
function [Transf] = setupSpace(Params)
|
||||
|
||||
Transf.Xmax = 0.5*Params.Lx;
|
||||
Transf.Ymax = 0.5*Params.Ly;
|
||||
|
||||
Nx = Params.Nx;
|
||||
Ny = Params.Ny;
|
||||
|
||||
% Fourier grids
|
||||
x = linspace(-0.5*Params.Lx,0.5*Params.Lx-Params.Lx/Params.Nx,Params.Nx);
|
||||
Kmax = pi*Params.Nx/Params.Lx;
|
||||
kx = linspace(-Kmax,Kmax,Nx+1);
|
||||
kx = kx(1:end-1);
|
||||
dkx = kx(2)-kx(1);
|
||||
kx = fftshift(kx);
|
||||
|
||||
y = linspace(-0.5*Params.Ly,0.5*Params.Ly-Params.Ly/Params.Ny,Params.Ny);
|
||||
Kmax = pi*Params.Ny/Params.Ly;
|
||||
ky = linspace(-Kmax,Kmax,Ny+1);
|
||||
ky = ky(1:end-1);
|
||||
dky = ky(2)-ky(1);
|
||||
ky = fftshift(ky);
|
||||
|
||||
[Transf.X,Transf.Y] = ndgrid(x,y);
|
||||
[Transf.KX,Transf.KY] = ndgrid(kx,ky);
|
||||
Transf.x = x;
|
||||
Transf.y = y;
|
||||
Transf.kx = kx;
|
||||
Transf.ky = ky;
|
||||
Transf.dx = x(2)-x(1);
|
||||
Transf.dy = y(2)-y(1);
|
||||
Transf.dkx = dkx;
|
||||
Transf.dky = dky;
|
||||
end
|
||||
|
||||
function VDk = compute2DPotentialInMomentumSpace(Transf, Params, MeanWidth)
|
||||
% == Calculating the DDI potential in Fourier space with appropriate cutoff == %
|
||||
|
||||
% Interaction in K space
|
||||
QX = Transf.KX*MeanWidth/sqrt(2);
|
||||
QY = Transf.KY*MeanWidth/sqrt(2);
|
||||
|
||||
Qsq = QX.^2 + QY.^2;
|
||||
absQ = sqrt(Qsq);
|
||||
QDsq = QX.^2*cos(Params.phi)^2 + QY.^2*sin(Params.phi)^2;
|
||||
|
||||
% Bare interaction
|
||||
Fpar = -1 + 3*sqrt(pi)*QDsq.*erfcx(absQ)./absQ; % Scaled complementary error function erfcx(x) = e^(x^2) * erfc(x)
|
||||
Fperp = 2 - 3*sqrt(pi).*absQ.*erfcx(absQ);
|
||||
Fpar(absQ == 0) = -1;
|
||||
|
||||
% Full DDI
|
||||
VDk = (Fpar*sin(Params.alpha)^2 + Fperp*cos(Params.alpha)^2);
|
||||
end
|
||||
|
||||
function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced)
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
MeanWidth = x * lz;
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2)); % Quantum Fluctuations term
|
||||
gamma4 = 1/(sqrt(2*pi) * MeanWidth);
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
Energy_AxialComponent = (PlanckConstantReduced * wz) * ((lz^2/(4 * MeanWidth^2)) + (MeanWidth^2/(4 * lz^2)));
|
||||
Energy_TransverseComponent = (0.5 * (gs + (2*gdd)) * gamma4 * AtomNumberDensity) + ((2/5) * gQF * AtomNumberDensity^(3/2));
|
||||
ret = (Energy_AxialComponent + Energy_TransverseComponent) / (PlanckConstantReduced * wz);
|
||||
end
|
||||
|
||||
|
||||
|
||||
@ -0,0 +1,157 @@
|
||||
%% Physical constants
|
||||
PlanckConstant = 6.62607015E-34;
|
||||
PlanckConstantReduced = 6.62607015E-34/(2*pi);
|
||||
FineStructureConstant = 7.2973525698E-3;
|
||||
ElectronMass = 9.10938291E-31;
|
||||
GravitationalConstant = 6.67384E-11;
|
||||
ProtonMass = 1.672621777E-27;
|
||||
AtomicMassUnit = 1.660539066E-27;
|
||||
BohrRadius = 5.2917721067E-11;
|
||||
BohrMagneton = 9.274009994E-24;
|
||||
BoltzmannConstant = 1.38064852E-23;
|
||||
StandardGravityAcceleration = 9.80665;
|
||||
SpeedOfLight = 299792458;
|
||||
StefanBoltzmannConstant = 5.670373E-8;
|
||||
ElectronCharge = 1.602176634E-19;
|
||||
VacuumPermeability = 1.25663706212E-6;
|
||||
DielectricConstant = 8.8541878128E-12;
|
||||
ElectronGyromagneticFactor = -2.00231930436153;
|
||||
AvogadroConstant = 6.02214076E23;
|
||||
ZeroKelvin = 273.15;
|
||||
GravitationalAcceleration = 9.80553;
|
||||
VacuumPermittivity = 1 / (SpeedOfLight^2 * VacuumPermeability);
|
||||
HartreeEnergy = ElectronCharge^2 / (4 * pi * VacuumPermittivity * BohrRadius);
|
||||
AtomicUnitOfPolarizability = (ElectronCharge^2 * BohrRadius^2) / HartreeEnergy; % Or simply 4*pi*VacuumPermittivity*BohrRadius^3
|
||||
|
||||
% Dy specific constants
|
||||
Dy164Mass = 163.929174751*AtomicMassUnit;
|
||||
Dy164IsotopicAbundance = 0.2826;
|
||||
DyMagneticMoment = 9.93*BohrMagneton;
|
||||
|
||||
%% k_roton at the instability boundary for tilted dipoles
|
||||
|
||||
wz = 2 * pi * 500; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
% nadd2s = 0.2:0.005:0.75;
|
||||
% as_to_add = 0.4:0.002:0.5;
|
||||
|
||||
nadd2s = 0.05:0.005:0.25;
|
||||
as_to_add = 0.50:0.001:0.80;
|
||||
|
||||
var_widths = zeros(length(as_to_add), length(nadd2s));
|
||||
|
||||
x0 = 5;
|
||||
Aineq = [];
|
||||
Bineq = [];
|
||||
Aeq = [];
|
||||
Beq = [];
|
||||
lb = [1];
|
||||
ub = [10];
|
||||
nonlcon = [];
|
||||
fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
|
||||
sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
|
||||
var_widths(jdx, idx) = sigma;
|
||||
end
|
||||
end
|
||||
|
||||
% ====================================================================================================================================================== %
|
||||
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
instability_boundary = zeros(length(as_to_add), length(nadd2s));
|
||||
k_roton = zeros(length(as_to_add), length(nadd2s));
|
||||
ScatteringLengths = zeros(length(as_to_add), 1);
|
||||
AtomNumber = zeros(length(nadd2s), 1);
|
||||
w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
|
||||
tsize = 10 * l0;
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
AtomNumber(idx) = AtomNumberDensity*tsize^2;
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
ScatteringLengths(jdx) = as/BohrRadius;
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
instability_boundary(jdx, idx) = ~isreal(EpsilonK);
|
||||
k_roton_indices = find(imag(EpsilonK) ~= 0);
|
||||
if ~isempty(k_roton_indices)
|
||||
k_roton(jdx, idx) = k(k_roton_indices(1));
|
||||
else
|
||||
k_roton(jdx, idx) = NaN;
|
||||
end
|
||||
end
|
||||
end
|
||||
%
|
||||
k_roton_vals = (k_roton .* add);
|
||||
%
|
||||
figure(8)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
imagesc(AtomNumber*1E-5, ScatteringLengths, k_roton_vals); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
cbar1 = colorbar;
|
||||
cbar1.Label.Interpreter = 'latex';
|
||||
% ylabel(cbar1,'$$','FontSize',16,'Rotation',270)
|
||||
xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex');
|
||||
ylabel('Scattering length ($\times a_0$)','fontsize',16,'interpreter','latex');
|
||||
title('Roton instability boundary','fontsize',16,'interpreter','latex')
|
||||
%
|
||||
% Get the size of the matrix
|
||||
k_roton_vals = flipud(k_roton_vals);
|
||||
[rows, cols] = size(k_roton_vals);
|
||||
|
||||
first_nonnan_row = zeros(1, cols);
|
||||
|
||||
% Loop through each column
|
||||
for col = 1:cols
|
||||
nonnan_rows = find(~isnan(k_roton_vals(:, col)));
|
||||
|
||||
if ~isempty(nonnan_rows)
|
||||
first_nonnan_row(col) = nonnan_rows(1);
|
||||
else
|
||||
first_nonnan_row(col) = NaN; % Use NaN to represent no non-zero elements in this column
|
||||
end
|
||||
end
|
||||
|
||||
% Create column indices (1 to number of columns)
|
||||
column_indices = 1:cols;
|
||||
%
|
||||
% Use row and column indices to extract the first non-zero elements
|
||||
k_roton_instability_boundary = arrayfun(@(r, c) k_roton_vals(r, c), first_nonnan_row(~isnan(first_nonnan_row)), column_indices(~isnan(first_nonnan_row)));
|
||||
|
||||
figure(9)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
xvals = AtomNumber*1E-5;
|
||||
yvals = k_roton_instability_boundary;
|
||||
plot(xvals', yvals,LineWidth=2.0)
|
||||
xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$k_{\rho}a_{dd}$','fontsize',16,'interpreter','latex')
|
||||
title('$k_{roton}$ at the instability boundary','fontsize',16,'interpreter','latex')
|
||||
grid on
|
||||
@ -0,0 +1,149 @@
|
||||
%% Physical constants
|
||||
PlanckConstant = 6.62607015E-34;
|
||||
PlanckConstantReduced = 6.62607015E-34/(2*pi);
|
||||
FineStructureConstant = 7.2973525698E-3;
|
||||
ElectronMass = 9.10938291E-31;
|
||||
GravitationalConstant = 6.67384E-11;
|
||||
ProtonMass = 1.672621777E-27;
|
||||
AtomicMassUnit = 1.660539066E-27;
|
||||
BohrRadius = 5.2917721067E-11;
|
||||
BohrMagneton = 9.274009994E-24;
|
||||
BoltzmannConstant = 1.38064852E-23;
|
||||
StandardGravityAcceleration = 9.80665;
|
||||
SpeedOfLight = 299792458;
|
||||
StefanBoltzmannConstant = 5.670373E-8;
|
||||
ElectronCharge = 1.602176634E-19;
|
||||
VacuumPermeability = 1.25663706212E-6;
|
||||
DielectricConstant = 8.8541878128E-12;
|
||||
ElectronGyromagneticFactor = -2.00231930436153;
|
||||
AvogadroConstant = 6.02214076E23;
|
||||
ZeroKelvin = 273.15;
|
||||
GravitationalAcceleration = 9.80553;
|
||||
VacuumPermittivity = 1 / (SpeedOfLight^2 * VacuumPermeability);
|
||||
HartreeEnergy = ElectronCharge^2 / (4 * pi * VacuumPermittivity * BohrRadius);
|
||||
AtomicUnitOfPolarizability = (ElectronCharge^2 * BohrRadius^2) / HartreeEnergy; % Or simply 4*pi*VacuumPermittivity*BohrRadius^3
|
||||
|
||||
% Dy specific constants
|
||||
Dy164Mass = 163.929174751*AtomicMassUnit;
|
||||
Dy164IsotopicAbundance = 0.2826;
|
||||
DyMagneticMoment = 9.93*BohrMagneton;
|
||||
|
||||
%% Roton instability boundary for tilted dipoles
|
||||
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
nadd2s = 0.05:0.001:0.25;
|
||||
as_to_add = 0.76:0.001:0.81;
|
||||
var_widths = zeros(length(as_to_add), length(nadd2s));
|
||||
|
||||
x0 = 5;
|
||||
Aineq = [];
|
||||
Bineq = [];
|
||||
Aeq = [];
|
||||
Beq = [];
|
||||
lb = [1];
|
||||
ub = [10];
|
||||
nonlcon = [];
|
||||
fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
|
||||
sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
|
||||
var_widths(jdx, idx) = sigma;
|
||||
end
|
||||
end
|
||||
|
||||
% ====================================================================================================================================================== %
|
||||
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
instability_boundary = zeros(length(as_to_add), length(nadd2s));
|
||||
ScatteringLengths = zeros(length(as_to_add), 1);
|
||||
AtomNumber = zeros(length(nadd2s), 1);
|
||||
w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
|
||||
tsize = 10 * l0;
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
AtomNumber(idx) = AtomNumberDensity*tsize^2;
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
ScatteringLengths(jdx) = as/BohrRadius;
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
instability_boundary(jdx, idx) = ~isreal(EpsilonK);
|
||||
end
|
||||
end
|
||||
|
||||
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];
|
||||
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];
|
||||
|
||||
figure(6)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
%
|
||||
|
||||
imagesc(nadd2s, as_to_add, instability_boundary); % Specify x and y data for axes
|
||||
hold on
|
||||
plot(nadd2s_from_figure, as_to_add_from_figure, 'r*-', 'LineWidth', 2); % Plot the curve (red line)
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
colorbar; % Add a colorbar
|
||||
xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
|
||||
|
||||
%{
|
||||
imagesc(AtomNumber*1E-5, ScatteringLengths, instability_boundary); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
cbar1 = colorbar;
|
||||
cbar1.Label.Interpreter = 'latex';
|
||||
ylabel(cbar1,'$(\times 10^{-31})$','FontSize',16,'Rotation',270)
|
||||
xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex');
|
||||
ylabel('Scattering length ($\times a_0$)','fontsize',16,'interpreter','latex');
|
||||
%}
|
||||
|
||||
title('Roton instability boundary','fontsize',16,'interpreter','latex')
|
||||
|
||||
%%
|
||||
function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi)
|
||||
Go = sqrt(pi) * (k * MeanWidth/sqrt(2)) .* exp((k * MeanWidth/sqrt(2)).^2) .* erfc((k * MeanWidth/sqrt(2)));
|
||||
gamma4 = 1/(sqrt(2*pi) * MeanWidth);
|
||||
Fka = (3 * cos(deg2rad(alpha))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(alpha))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(alpha))^2));
|
||||
Ukk = (gs + (gdd * Fka)) * gamma4;
|
||||
end
|
||||
|
||||
function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced)
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
MeanWidth = x * lz;
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2)); % Quantum Fluctuations term
|
||||
gamma4 = 1/(sqrt(2*pi) * MeanWidth);
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
Energy_AxialComponent = (PlanckConstantReduced * wz) * ((lz^2/(4 * MeanWidth^2)) + (MeanWidth^2/(4 * lz^2)));
|
||||
Energy_TransverseComponent = (0.5 * (gs + (2*gdd)) * gamma4 * AtomNumberDensity) + ((2/5) * gQF * AtomNumberDensity^(3/2));
|
||||
ret = (Energy_AxialComponent + Energy_TransverseComponent) / (PlanckConstantReduced * wz);
|
||||
end
|
||||
|
||||
|
||||
|
||||
@ -0,0 +1,335 @@
|
||||
%% Physical constants
|
||||
PlanckConstant = 6.62607015E-34;
|
||||
PlanckConstantReduced = 6.62607015E-34/(2*pi);
|
||||
FineStructureConstant = 7.2973525698E-3;
|
||||
ElectronMass = 9.10938291E-31;
|
||||
GravitationalConstant = 6.67384E-11;
|
||||
ProtonMass = 1.672621777E-27;
|
||||
AtomicMassUnit = 1.660539066E-27;
|
||||
BohrRadius = 5.2917721067E-11;
|
||||
BohrMagneton = 9.274009994E-24;
|
||||
BoltzmannConstant = 1.38064852E-23;
|
||||
StandardGravityAcceleration = 9.80665;
|
||||
SpeedOfLight = 299792458;
|
||||
StefanBoltzmannConstant = 5.670373E-8;
|
||||
ElectronCharge = 1.602176634E-19;
|
||||
VacuumPermeability = 1.25663706212E-6;
|
||||
DielectricConstant = 8.8541878128E-12;
|
||||
ElectronGyromagneticFactor = -2.00231930436153;
|
||||
AvogadroConstant = 6.02214076E23;
|
||||
ZeroKelvin = 273.15;
|
||||
GravitationalAcceleration = 9.80553;
|
||||
VacuumPermittivity = 1 / (SpeedOfLight^2 * VacuumPermeability);
|
||||
HartreeEnergy = ElectronCharge^2 / (4 * pi * VacuumPermittivity * BohrRadius);
|
||||
AtomicUnitOfPolarizability = (ElectronCharge^2 * BohrRadius^2) / HartreeEnergy; % Or simply 4*pi*VacuumPermittivity*BohrRadius^3
|
||||
|
||||
% Dy specific constants
|
||||
Dy164Mass = 163.929174751*AtomicMassUnit;
|
||||
Dy164IsotopicAbundance = 0.2826;
|
||||
DyMagneticMoment = 9.93*BohrMagneton;
|
||||
|
||||
%% Bogoliubov excitation spectrum for quasi-2D dipolar gas with QF correction
|
||||
AtomNumber = 1E5; % Total atom number in the system
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
as = 102.515 * BohrRadius; % Scattering length
|
||||
Trapsize = 7.5815 * lz; % Trap is assumed to be a box of finite extent , given here in units of the harmonic oscillator length
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
MeanWidth = 5.7304888515 * lz; % Mean width of Gaussian ansatz
|
||||
k = linspace(0, 2e6, 1000); % Vector of magnitudes of k vector
|
||||
|
||||
% no = 2.0429e+15, eps_dd = 1.2755, as = 5.4249e-09
|
||||
|
||||
AtomNumberDensity = AtomNumber / Trapsize^2; % Areal density of atoms
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
|
||||
figure(1)
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
xvals = (k .* add);
|
||||
yvals = EpsilonK ./ PlanckConstant;
|
||||
plot(xvals, yvals,LineWidth=2.0)
|
||||
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')
|
||||
xlabel('$k_{\rho}a_{dd}$','fontsize',16,'interpreter','latex')
|
||||
ylabel('$\epsilon(k_{\rho})/h$ (Hz)','fontsize',16,'interpreter','latex')
|
||||
grid on
|
||||
|
||||
%% For different interaction strengths
|
||||
|
||||
AtomNumber = 1E5; % Total atom number in the system
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
Trapsize = 7.5815 * lz; % Trap is assumed to be a box of finite extent , given here in units of the harmonic oscillator length
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
MeanWidth = 5.7304888515 * lz; % Mean width of Gaussian ansatz
|
||||
k = linspace(0, 2e6, 1000); % Vector of magnitudes of k vector
|
||||
|
||||
AtomNumberDensity = AtomNumber / Trapsize^2; % Areal density of atoms
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
|
||||
ScatteringLengths = [108.5, 105.9, 103.3, 102.5150];
|
||||
eps_dds = zeros(1, length(ScatteringLengths));
|
||||
EpsilonKs = zeros(length(k), length(ScatteringLengths));
|
||||
for idx = 1:length(ScatteringLengths)
|
||||
|
||||
as = ScatteringLengths(idx) * BohrRadius; % Scattering length
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
|
||||
eps_dds(idx) = eps_dd;
|
||||
EpsilonKs(:,idx) = EpsilonK;
|
||||
end
|
||||
|
||||
figure(2)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
xvals = (k .* add);
|
||||
yvals = EpsilonKs(:, 1) ./ PlanckConstant;
|
||||
plot(xvals, yvals,LineWidth=2.0, DisplayName=['$a_s = ',num2str(round(1/eps_dds(1),3)),'a_{dd}$'])
|
||||
hold on
|
||||
for idx = 2:length(ScatteringLengths)
|
||||
yvals = EpsilonKs(:, idx) ./ PlanckConstant;
|
||||
plot(xvals, yvals,LineWidth=2.0, DisplayName=['$a_s = ',num2str(round(1/eps_dds(idx),3)),'a_{dd}$'])
|
||||
end
|
||||
title(['$na_{dd}^2 = ',num2str(round(AtomNumberDensity * add^2,4)),'$'],'fontsize',16,'interpreter','latex')
|
||||
xlabel('$k_{\rho}a_{dd}$','fontsize',16,'interpreter','latex')
|
||||
ylabel('$\epsilon(k_{\rho})/h$ (Hz)','fontsize',16,'interpreter','latex')
|
||||
grid on
|
||||
legend('location', 'northwest','fontsize',16, 'Interpreter','latex')
|
||||
|
||||
%% For 3 points on the roton instability boundary
|
||||
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
|
||||
nadd2s = [0.0844, 0.0978, 0.123];
|
||||
as_to_add = [0.7730, 0.7840, 0.7819];
|
||||
var_widths = [4.97165, 5.7296048721, 5.93178];
|
||||
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
EpsilonKs = zeros(length(k), length(nadd2s));
|
||||
ScatteringLengths = zeros(length(as_to_add), 1);
|
||||
AtomNumber = zeros(length(nadd2s), 1);
|
||||
w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
|
||||
tsize = 10 * l0;
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
AtomNumber(idx) = AtomNumberDensity*tsize^2;
|
||||
as = (as_to_add(idx) * add); % Scattering length
|
||||
ScatteringLengths(idx) = as/BohrRadius;
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
EpsilonKs(:,idx) = EpsilonK;
|
||||
end
|
||||
|
||||
figure(3)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
xvals = (k .* add);
|
||||
yvals = EpsilonKs(:, 1) ./ PlanckConstant;
|
||||
plot(xvals, yvals,LineWidth=2.0, DisplayName=['$a_s = ',num2str(round(as_to_add(1),4)),'a_{dd}, na_{dd}^2 = ',num2str(round(nadd2s(1),4)),'$'])
|
||||
hold on
|
||||
for idx = 2:length(nadd2s)
|
||||
yvals = EpsilonKs(:, idx) ./ PlanckConstant;
|
||||
plot(xvals, yvals,LineWidth=2.0, DisplayName=['$a_s = ',num2str(round(as_to_add(idx),4)),'a_{dd}, na_{dd}^2 = ',num2str(round(nadd2s(idx),4)),'$'])
|
||||
end
|
||||
xlabel('$k_{\rho}a_{dd}$','fontsize',16,'interpreter','latex')
|
||||
ylabel('$\epsilon(k_{\rho})/h$ (Hz)','fontsize',16,'interpreter','latex')
|
||||
grid on
|
||||
legend('location', 'northwest','fontsize',16, 'Interpreter','latex')
|
||||
|
||||
%% Mean widths of the variational Gaussian ansatz - extremize the total mean field energy per particle wrt to the variational parameter
|
||||
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
AtomNumberDensity = 0.0978 / add^2;
|
||||
as = 0.784 * add; % Scattering length
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
|
||||
|
||||
x0 = 5;
|
||||
Aineq = [];
|
||||
Bineq = [];
|
||||
Aeq = [];
|
||||
Beq = [];
|
||||
lb = [1];
|
||||
ub = [7];
|
||||
nonlcon = [];
|
||||
fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
|
||||
sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
|
||||
fprintf(['Variational width of Gaussian ansatz = ' num2str(sigma) ' * lz \n'])
|
||||
|
||||
%% Mean widths of the variational Gaussian ansatz - extremize the total mean field energy per particle wrt to the variational parameter
|
||||
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
nadd2s = 0.05:0.001:0.25;
|
||||
as_to_add = 0.74:0.001:0.79;
|
||||
var_widths = zeros(length(as_to_add), length(nadd2s));
|
||||
|
||||
x0 = 5;
|
||||
Aineq = [];
|
||||
Bineq = [];
|
||||
Aeq = [];
|
||||
Beq = [];
|
||||
lb = [1];
|
||||
ub = [10];
|
||||
nonlcon = [];
|
||||
fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
|
||||
sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
|
||||
var_widths(jdx, idx) = sigma;
|
||||
end
|
||||
end
|
||||
|
||||
figure(4)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
imagesc(nadd2s, as_to_add, var_widths); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
colorbar; % Add a colorbar
|
||||
xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
|
||||
|
||||
% ====================================================================================================================================================== %
|
||||
|
||||
alpha = 0; % Polar angle of dipole moment
|
||||
phi = 0; % Azimuthal angle of momentum vector
|
||||
k = linspace(0, 2.25e6, 1000); % Vector of magnitudes of k vector
|
||||
instability_boundary = zeros(length(as_to_add), length(nadd2s));
|
||||
ScatteringLengths = zeros(length(as_to_add), 1);
|
||||
AtomNumber = zeros(length(nadd2s), 1);
|
||||
w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
|
||||
tsize = 10 * l0;
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
AtomNumber(idx) = AtomNumberDensity*tsize^2;
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
ScatteringLengths(jdx) = as/BohrRadius;
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
MeanWidth = var_widths(jdx, idx) * lz; % Mean width of Gaussian ansatz
|
||||
|
||||
[Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi); % DDI potential in k-space
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
|
||||
% == Dispersion relation == %
|
||||
DeltaK = ((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) + ((2 * AtomNumberDensity) .* Ukk) + (3 * gQF * AtomNumberDensity^(3/2));
|
||||
EpsilonK = sqrt(((PlanckConstantReduced^2 .* k.^2) ./ (2 * Dy164Mass)) .* DeltaK);
|
||||
instability_boundary(jdx, idx) = ~isreal(EpsilonK);
|
||||
end
|
||||
end
|
||||
|
||||
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];
|
||||
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];
|
||||
|
||||
figure(5)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
|
||||
|
||||
imagesc(nadd2s, as_to_add, instability_boundary); % Specify x and y data for axes
|
||||
hold on
|
||||
plot(nadd2s_from_figure, as_to_add_from_figure, 'r*-', 'LineWidth', 2); % Plot the curve (red line)
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
colorbar; % Add a colorbar
|
||||
xlabel('$na_{dd}^2$','fontsize',16,'interpreter','latex');
|
||||
ylabel('$a_s/a_{dd}$','fontsize',16,'interpreter','latex');
|
||||
|
||||
%{
|
||||
imagesc(AtomNumber*1E-5, ScatteringLengths, instability_boundary); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
cbar1 = colorbar;
|
||||
cbar1.Label.Interpreter = 'latex';
|
||||
ylabel(cbar1,'$(\times 10^{-31})$','FontSize',16,'Rotation',270)
|
||||
xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex');
|
||||
ylabel('Scattering length ($\times a_0$)','fontsize',16,'interpreter','latex');
|
||||
title('Roton instability boundary','fontsize',16,'interpreter','latex')
|
||||
%}
|
||||
|
||||
%%
|
||||
function [Go,gamma4,Fka,Ukk] = computePotentialInMomentumSpace(k, gs, gdd, MeanWidth, alpha, phi)
|
||||
Go = sqrt(pi) * (k * MeanWidth/sqrt(2)) .* exp((k * MeanWidth/sqrt(2)).^2) .* erfc((k * MeanWidth/sqrt(2)));
|
||||
gamma4 = 1/(sqrt(2*pi) * MeanWidth);
|
||||
Fka = (3 * cos(deg2rad(alpha))^2 - 1) + ((3 * Go) .* ((sin(deg2rad(alpha))^2 .* sin(deg2rad(phi))^2) - cos(deg2rad(alpha))^2));
|
||||
Ukk = (gs + (gdd * Fka)) * gamma4;
|
||||
end
|
||||
|
||||
function ret = computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced)
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
MeanWidth = x * lz;
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2)); % Quantum Fluctuations term
|
||||
gamma4 = 1/(sqrt(2*pi) * MeanWidth);
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
Energy_AxialComponent = (PlanckConstantReduced * wz) * ((lz^2/(4 * MeanWidth^2)) + (MeanWidth^2/(4 * lz^2)));
|
||||
Energy_TransverseComponent = (0.5 * (gs + (2*gdd)) * gamma4 * AtomNumberDensity) + ((2/5) * gQF * AtomNumberDensity^(3/2));
|
||||
ret = (Energy_AxialComponent + Energy_TransverseComponent) / (PlanckConstantReduced * wz);
|
||||
end
|
||||
|
||||
|
||||
|
||||
@ -0,0 +1,89 @@
|
||||
%% Physical constants
|
||||
PlanckConstant = 6.62607015E-34;
|
||||
PlanckConstantReduced = 6.62607015E-34/(2*pi);
|
||||
FineStructureConstant = 7.2973525698E-3;
|
||||
ElectronMass = 9.10938291E-31;
|
||||
GravitationalConstant = 6.67384E-11;
|
||||
ProtonMass = 1.672621777E-27;
|
||||
AtomicMassUnit = 1.660539066E-27;
|
||||
BohrRadius = 5.2917721067E-11;
|
||||
BohrMagneton = 9.274009994E-24;
|
||||
BoltzmannConstant = 1.38064852E-23;
|
||||
StandardGravityAcceleration = 9.80665;
|
||||
SpeedOfLight = 299792458;
|
||||
StefanBoltzmannConstant = 5.670373E-8;
|
||||
ElectronCharge = 1.602176634E-19;
|
||||
VacuumPermeability = 1.25663706212E-6;
|
||||
DielectricConstant = 8.8541878128E-12;
|
||||
ElectronGyromagneticFactor = -2.00231930436153;
|
||||
AvogadroConstant = 6.02214076E23;
|
||||
ZeroKelvin = 273.15;
|
||||
GravitationalAcceleration = 9.80553;
|
||||
VacuumPermittivity = 1 / (SpeedOfLight^2 * VacuumPermeability);
|
||||
HartreeEnergy = ElectronCharge^2 / (4 * pi * VacuumPermittivity * BohrRadius);
|
||||
AtomicUnitOfPolarizability = (ElectronCharge^2 * BohrRadius^2) / HartreeEnergy; % Or simply 4*pi*VacuumPermittivity*BohrRadius^3
|
||||
|
||||
% Dy specific constants
|
||||
Dy164Mass = 163.929174751*AtomicMassUnit;
|
||||
Dy164IsotopicAbundance = 0.2826;
|
||||
DyMagneticMoment = 9.93*BohrMagneton;
|
||||
|
||||
%% Scaling of the QF term
|
||||
|
||||
wz = 2 * pi * 72.4; % Trap frequency in the tight confinement direction
|
||||
lz = sqrt(PlanckConstantReduced/(Dy164Mass * wz)); % Defining a harmonic oscillator length
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
add = VacuumPermeability*DyMagneticMoment^2*Dy164Mass/(12*pi*PlanckConstantReduced^2); % Dipole length
|
||||
gdd = VacuumPermeability*DyMagneticMoment^2/3;
|
||||
|
||||
nadd2s = 0.05:0.01:0.25;
|
||||
as_to_add = 0.76:0.01:0.81;
|
||||
|
||||
QF = zeros(length(as_to_add), length(nadd2s));
|
||||
ScatteringLengths = zeros(length(as_to_add), 1);
|
||||
AtomNumber = zeros(length(nadd2s), 1);
|
||||
w0 = 2 * pi * 61.6316; % Trap frequency in the tight confinement direction
|
||||
l0 = sqrt(PlanckConstantReduced/(Dy164Mass * w0)); % Defining a harmonic oscillator length
|
||||
tsize = 10 * l0;
|
||||
|
||||
x0 = 5;
|
||||
Aineq = [];
|
||||
Bineq = [];
|
||||
Aeq = [];
|
||||
Beq = [];
|
||||
lb = [1];
|
||||
ub = [10];
|
||||
nonlcon = [];
|
||||
fminconopts = optimoptions(@fmincon,'Display','off', 'StepTolerance', 1.0000e-11, 'MaxIterations',1500);
|
||||
|
||||
for idx = 1:length(nadd2s)
|
||||
for jdx = 1:length(as_to_add)
|
||||
AtomNumberDensity = nadd2s(idx) / add^2; % Areal density of atoms
|
||||
AtomNumber(idx) = AtomNumberDensity*tsize^2;
|
||||
as = (as_to_add(jdx) * add); % Scattering length
|
||||
gs = 4 * pi * PlanckConstantReduced^2/Dy164Mass * as; % Contact interaction strength
|
||||
ScatteringLengths(jdx) = as/BohrRadius;
|
||||
TotalEnergyPerParticle = @(x) computeTotalEnergyPerParticle(x, as, AtomNumberDensity, wz, lz, gs, add, gdd, PlanckConstantReduced);
|
||||
sigma = fmincon(TotalEnergyPerParticle, x0, Aineq, Bineq, Aeq, Beq, lb, ub, nonlcon, fminconopts);
|
||||
eps_dd = add/as; % Relative interaction strength
|
||||
|
||||
% == Quantum Fluctuations term == %
|
||||
MeanWidth = sigma * lz;
|
||||
gammaQF = (32/3) * gs * (as^3/pi)^(1/2) * (1 + ((3/2) * eps_dd^2));
|
||||
gamma5 = sqrt(2/5) / (sqrt(pi) * MeanWidth)^(3/2);
|
||||
gQF = gamma5 * gammaQF;
|
||||
QF(jdx, idx) = 3 * gQF * AtomNumberDensity^(3/2);
|
||||
end
|
||||
end
|
||||
|
||||
figure(7)
|
||||
clf
|
||||
set(gcf,'Position',[50 50 950 750])
|
||||
imagesc(AtomNumber*1E-5, ScatteringLengths, QF * 1E31); % Specify x and y data for axes
|
||||
set(gca, 'YDir', 'normal'); % Correct the y-axis direction
|
||||
cbar1 = colorbar;
|
||||
cbar1.Label.Interpreter = 'latex';
|
||||
ylabel(cbar1,'$(\times 10^{-31})$','FontSize',16,'Rotation',270)
|
||||
xlabel(' Atom number for a trap area of 100$\mu m^2 ~ (\times 10^5)$','fontsize',16,'interpreter','latex');
|
||||
ylabel('Scattering length ($\times a_0$)','fontsize',16,'interpreter','latex');
|
||||
title('Scaling of the quantum fluctuations term','fontsize',16,'interpreter','latex')
|
||||
Loading…
Reference in New Issue
Block a user