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Calculations/Estimations/LinearizeScatteringLengthScan.m

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FR_choice = 1;
ABKG_choice = 1;
a_start = 86; % initial scattering length (a_0)
a_end = 85; % final scattering length (a_0)
T = 0.030; % ramp duration (s)
Nt = 1000; % number of time points
ResonanceRange = [1.2, 2.6];
% Options for ramp generation
options = struct( ...
'smoothingMethod', 'sgolay', ... % 'sgolay', 'lowpass', or 'none'
'sgolayOrder', 3, ...
'sgolayFrameLength', 51, ...
'maxRampRate', 5, ...
'Bmin', 0.5, ...
'Bmax', 3.5, ...
'rampShape', 'linear' ... % Choose: 'linear', 'exponential', 'sigmoid'
);
[t, B_ramp, a_check] = generateSmoothBRamp(FR_choice, ABKG_choice, a_start, a_end, ResonanceRange, T, Nt, options);
% ----- Plot results -----
figure(1); clf
subplot(2,1,1);
plot(t, B_ramp, 'b-', 'LineWidth', 2);
xlabel('Time (s)');
ylabel('Magnetic field B(t) (G)');
title('Generated magnetic field ramp B(t)');
grid on;
set(gca, 'FontSize', 14);
subplot(2,1,2);
plot(t, a_check, 'r-', 'LineWidth', 2);
xlabel('Time (s)');
ylabel('Scattering length a_s(t) (a_0)');
legend('Resulting a_s(t)');
title('Scattering length ramp');
grid on;
set(gca, 'FontSize', 14);
% Visualize full resonance curve and selected window
[B_full, a_full] = extractBetweenResonances(FR_choice, ABKG_choice, ResonanceRange);
[B_curve, a_curve] = fullResonanceCurve(FR_choice, ABKG_choice, ResonanceRange);
[B_all, a_all] = getFullFeschbachSpectrum(FR_choice, ABKG_choice);
figure(2);
clf
plot(B_all, a_all , 'g-', 'LineWidth', 1); hold on;
plot(B_curve, a_curve, 'k-', 'LineWidth', 1);
% Plot all unfiltered resonances in black
plot(B_full, a_full, 'b-', 'LineWidth', 2); % zoomed-in region
plot(B_ramp, a_check, 'r-', 'LineWidth', 2); % actual ramp
plot(B_ramp([1 end]), a_check([1 end]), 'ro', 'MarkerSize', 8); % endpoints
xline(min(B_ramp), '--b', 'B_{min}');
xline(max(B_ramp), '--b', 'B_{max}');
yline(min(a_check), '--r', 'a_{min}');
yline(max(a_check), '--r', 'a_{max}');
xlabel('B field (G)');
ylabel('Scattering length a_s (a_0)');
title('Feshbach spectrum with selected B and a_s range', 'Interpreter','tex');
legend('Full a(B)', 'Modified a(B)', 'Zoomed Region', 'Ramp a_s(t)', 'Ramp Endpoints', 'Location', 'NorthWest');
xlim([1.15 2.7])
ylim([0 150])
set(gca, 'FontSize', 14)
grid on;
%% Isolate from spectrum manually
FR_choice = 1;
ABKG_choice = 1;
a_start = 124; % initial scattering length (a_0)
a_end = 117; % final scattering length (a_0)
T = 0.010; % ramp duration (s)
Nt = 15000; % number of time points
ResonanceRange = [1.2, 2.6];
% Get the full spectrum
[B, a_s] = getFullFeschbachSpectrum(1, 2); % Using new FR params and middle a_bkg
% Define the plotting range
x_limits = [1.15, 2.7];
y_limits = [0, 150];
% Find indices within our x-range
mask = (B >= x_limits(1)) & (B <= x_limits(2));
B_plot = B(mask);
a_s_plot = a_s(mask);
% Identify resonances to mask (looking at the spectrum, we'll mask around 2.174G and 2.336G)
resonances_to_mask = [2.174, 2.336];
mask_width = 0.1; % Width to mask around each resonance (in G)
% Create a mask for the regions to keep (1 = keep, 0 = mask)
keep_mask = true(size(B_plot));
for res = resonances_to_mask
keep_mask = keep_mask & (B_plot < (res - mask_width) | B_plot > (res + mask_width));
end
% Create masked version
B_masked = B_plot(keep_mask);
a_s_masked = a_s_plot(keep_mask);
% Interpolate over the masked regions
a_s_interp = interp1(B_masked, a_s_masked, B_plot, 'pchip');
% --- Find the remaining resonances (peaks) in the interpolated data ---
% Use findpeaks to detect peaks (adjust MinPeakProminence as needed)
[peaks, locs] = findpeaks(abs(a_s_interp), 'MinPeakProminence', 20); % Tune this!
remaining_resonances = B_plot(locs); % Positions of remaining resonances
% --- Select the two resonances of interest (1.3G and 2.59G) ---
% If automatic detection fails, manually define them:
target_resonances = [1.3, 2.59]; % Manually specified (adjust as needed)
% OR use the closest detected resonances:
% target_resonances = remaining_resonances(abs(remaining_resonances - 1.3) < 0.2 & abs(remaining_resonances - 2.59) < 0.2);
% --- Extract the curve BETWEEN these two resonances ---
res1 = min(target_resonances); % Lower resonance (1.3G)
res2 = max(target_resonances); % Upper resonance (2.59G)
between_mask = (B_plot > res1) & (B_plot < res2); % No masking here, since resonances are already removed
B_between = B_plot(between_mask);
a_s_between = a_s_interp(between_mask);
% Plotting
figure(3);
hold on;
% Plot the original spectrum
plot(B_plot, a_s_plot, 'b-', 'DisplayName', 'Original spectrum');
% Plot the masked regions in red
for res = resonances_to_mask
mask_region = (B_plot >= (res - mask_width)) & (B_plot <= (res + mask_width));
plot(B_plot(mask_region), a_s_plot(mask_region), 'r-', 'LineWidth', 2, 'DisplayName', 'Unwanted Resonance');
end
% Plot the interpolated curve
plot(B_plot, a_s_interp, 'g--', 'LineWidth', 1.5, 'DisplayName', 'Interpolated');
% Formatting
xlim(x_limits);
ylim(y_limits);
xlabel('Magnetic Field (G)');
ylabel('Scattering Length (a_0)');
title('Dy-164 Feshbach Spectrum with Masked Resonances');
legend('Location', 'best');
grid on;
hold off;
% --- Plotting ---
figure(4);
hold on;
% Plot the interpolated spectrum (without 2.174G and 2.336G)
plot(B_plot, a_s_interp, 'k-', 'LineWidth', 1, 'DisplayName', 'Interpolated spectrum');
% Highlight the region between 1.3G and 2.59G
plot(B_between, a_s_between, 'm-', 'LineWidth', 2, 'DisplayName', 'Between 1.3G and 2.59G');
% Formatting
xlim(x_limits);
ylim([0, 150]);
xlabel('Magnetic Field (G)');
ylabel('Scattering Length (a_0)');
title('Interpolated Spectrum: Region Between 1.3G and 2.59G');
legend('Location', 'best');
grid on;
hold off;
% Generate the ramp
[t, B_ramp, a_check] = generateLinearBRamp(B_between, a_s_between, a_start, a_end, T, Nt);
% Plot results
figure;
subplot(2,1,1);
plot(t, B_ramp, 'b');
xlabel('Time (s)');
ylabel('B Field (G)');
title('Generated B-Field Ramp');
grid on;
subplot(2,1,2);
plot(t, a_check, 'r');
xlabel('Time (s)');
ylabel('a_s (a_0)');
title('Achieved Scattering Length');
grid on;
%% Helper functions
function [t, B_ramp, a_check] = generateSmoothBRamp(FR_choice, ABKG_choice, a_start, a_end, selectedResRange, T, Nt, opts)
% Time array
t = linspace(0, T, Nt);
if strcmp(opts.rampShape, 'linear')
% Directly call helper for linear LUT ramp generation
[t, B_ramp, a_check] = generateLinearBRampUsingLUT(FR_choice, ABKG_choice, a_start, a_end, selectedResRange, T, Nt);
return
end
% Get target a_s(t) based on rampShape choice
a_target = getTargetScatteringLength(t, T, a_start, a_end, opts.rampShape);
% --- a(B) interpolation ---
[B_between, a_between] = extractBetweenResonances(FR_choice, ABKG_choice, selectedResRange);
valid_idx = a_between > 0 & a_between < 150;
[a_sorted, sort_idx] = sort(a_between(valid_idx));
B_sorted = B_between(valid_idx);
B_sorted = B_sorted(sort_idx);
B_of_a = @(a) interp1(a_sorted, B_sorted, a, 'linear', 'extrap');
B_raw = B_of_a(a_target);
% --- Smoothing ---
switch opts.smoothingMethod
case 'sgolay'
B_smooth = sgolayfilt(B_raw, opts.sgolayOrder, opts.sgolayFrameLength);
case 'lowpass'
dt = T / (Nt - 1); Fs = 1 / dt;
B_smooth = lowpass(B_raw, Fs / 20, Fs);
case 'none'
B_smooth = B_raw;
otherwise
error('Unknown smoothing method');
end
% --- Bound the ramp ---
B_smooth = min(max(B_smooth, opts.Bmin), opts.Bmax);
% --- Enforce max dB/dt ---
dt = T / (Nt - 1);
for i = 2:Nt
delta = B_smooth(i) - B_smooth(i-1);
if abs(delta/dt) > opts.maxRampRate
delta = sign(delta) * opts.maxRampRate * dt;
B_smooth(i) = B_smooth(i-1) + delta;
end
end
B_ramp = B_smooth;
% --- Verify a_s(t) from B_ramp ---
[a_bkg, resonanceB, resonancewB] = getResonanceParams(FR_choice, ABKG_choice, selectedResRange);
a_of_B = @(B) arrayfun(@(b) ...
a_bkg * prod(1 - resonancewB ./ (b - resonanceB)), B);
a_check = a_of_B(B_ramp);
end
function a_target = getTargetScatteringLength(t, T, a_start, a_end, rampShape)
% If rampShape is a function handle, use it directly (for flexibility)
if isa(rampShape, 'function_handle')
a_target = rampShape(t);
return
end
switch lower(rampShape)
case 'exponential'
tau = T / 3;
base = (1 - exp(-t / tau)) / (1 - exp(-T / tau));
a_target = a_start + (a_end - a_start) * base;
case 'sigmoid'
s = 10 / T; center = T / 2;
sigmoid = @(x) 1 ./ (1 + exp(-s * (x - center)));
base = (sigmoid(t) - sigmoid(t(1))) / (sigmoid(t(end)) - sigmoid(t(1)));
a_target = a_start + (a_end - a_start) * base;
otherwise
error('Unknown ramp shape: %s', rampShape);
end
end
function [B_between, a_between] = extractBetweenResonances(FR_choice, ABKG_choice, selectedRange)
[a_bkg, resonanceB, resonancewB] = getResonanceParams(FR_choice, ABKG_choice, selectedRange);
[~, idx] = sort(resonancewB, 'descend');
B1 = resonanceB(idx(1)); B2 = resonanceB(idx(2));
w1 = resonancewB(idx(1)); w2 = resonancewB(idx(2));
Bvis = linspace(min(B1, B2) - 20*min(w1,w2), max(B1, B2) + 20*min(w1,w2), 2000);
a_of_B = @(B) arrayfun(@(b) ...
a_bkg * prod(1 - resonancewB ./ (b - resonanceB)), B);
avis = a_of_B(Bvis);
between_idx = Bvis >= min(B1,B2) & Bvis <= max(B1,B2);
B_between = Bvis(between_idx);
a_between = avis(between_idx);
end
function [B_range, a_values] = fullResonanceCurve(FR_choice, ABKG_choice, selectedRange)
[a_bkg, resonanceB, resonancewB] = getResonanceParams(FR_choice, ABKG_choice, selectedRange);
B_range = linspace(min(resonanceB)-0.2, max(resonanceB)+0.2, 3000);
a_of_B = @(B) arrayfun(@(b) ...
a_bkg * prod(1 - resonancewB ./ (b - resonanceB)), B);
a_values = a_of_B(B_range);
end
function [a_bkg, resonanceB, resonancewB] = getResonanceParams(FR_choice, ABKG_choice, selectedRange)
if FR_choice == 1
a_bkg_list = [85.5, 93.5, 77.5];
resonanceB = [1.295, 1.306, 2.174, 2.336, 2.591, 2.740, 2.803, ...
2.780, 3.357, 4.949, 5.083, 7.172, 7.204, 7.134, 76.9];
resonancewB = [0.009, 0.010, 0.0005, 0.0005, 0.001, 0.0005, 0.021, ...
0.015, 0.043, 0.0005, 0.130, 0.024, 0.0005, 0.036, 3.1];
else
a_bkg_list = [87.2, 95.2, 79.2];
resonanceB = [1.298, 2.802, 3.370, 5.092, 7.154, 2.592, 2.338, 2.177];
resonancewB = [0.018, 0.047, 0.048, 0.145, 0.020, 0.008, 0.001, 0.001];
end
a_bkg = a_bkg_list(ABKG_choice);
% --- Filter resonanceB and resonancewB if selectedRange is provided ---
if nargin >= 3 && ~isempty(selectedRange)
minB = min(selectedRange); maxB = max(selectedRange);
keep_idx = (resonanceB >= minB) & (resonanceB <= maxB);
% Keep only the lowest and highest resonance in the selected range
if sum(keep_idx) >= 2
B_sub = resonanceB(keep_idx);
w_sub = resonancewB(keep_idx);
[~, idx_lo] = min(B_sub);
[~, idx_hi] = max(B_sub);
resonanceB = [B_sub(idx_lo), B_sub(idx_hi)];
resonancewB = [w_sub(idx_lo), w_sub(idx_hi)];
else
error('Selected resonance range does not include at least two resonances.');
end
end
end
function [t, B_ramp, a_check] = generateLinearBRampUsingLUT(FR_choice, ABKG_choice, a_start, a_end, selectedResRange, T, Nt)
% Time vector
t = linspace(0, T, Nt);
% 1) Generate LUT of B and a_s(B)
[B_between, a_between] = extractBetweenResonances(FR_choice, ABKG_choice, selectedResRange);
% Restrict to physically meaningful range (optional)
valid_idx = a_between > 0 & a_between < 150;
B_lut = B_between(valid_idx);
a_lut = a_between(valid_idx);
% 2) Generate linear a_s ramp in time
a_target = linspace(a_start, a_end, Nt);
% 3) Interpolate B(t) from LUT a_s -> B
% Make sure a_lut is sorted ascending
[a_lut_sorted, idx_sort] = sort(a_lut);
B_lut_sorted = B_lut(idx_sort);
B_ramp = interp1(a_lut_sorted, B_lut_sorted, a_target, 'linear', 'extrap');
% 4) Compute resulting a_s(t) for verification
[a_bkg, resonanceB, resonancewB] = getResonanceParams(FR_choice, ABKG_choice, selectedResRange);
a_of_B = @(B) arrayfun(@(b) ...
a_bkg * prod(1 - resonancewB ./ (b - resonanceB)), B);
a_check = a_of_B(B_ramp);
end
function [B, a_s] = getFullFeschbachSpectrum(FR_choice, ABKG_choice)
% plotScatteringLength(FR_choice, ABKG_choice)
% Plots the Dy-164 scattering length vs B field based on PhysRevX.9.021012 Suppl. Fig. S5
%
% Inputs:
% FR_choice: 1 = new resonance parameters, 2 = old
% ABKG_choice: 1 = lower a_bkg, 2 = mid, 3 = upper
%
% Example:
% plotScatteringLength(1, 1);
if nargin < 1, FR_choice = 1; end
if nargin < 2, ABKG_choice = 1; end
% Choose background scattering length
if FR_choice == 1 % New values
switch ABKG_choice
case 1, a_bkg = 85.5;
case 2, a_bkg = 93.5;
case 3, a_bkg = 77.5;
end
% Resonance positions (G) and widths (G)
resonanceB = [1.295, 1.306, 2.174, 2.336, 2.591, 2.740, 2.803, 2.780, ...
3.357, 4.949, 5.083, 7.172, 7.204, 7.134, 76.9];
resonancewB = [0.009, 0.010, 0.0005, 0.0005, 0.0010, 0.0005, 0.021, ...
0.015, 0.043, 0.0005, 0.130, 0.024, 0.0005, 0.036, 3.1];
else % Old values
switch ABKG_choice
case 1, a_bkg = 87.2;
case 2, a_bkg = 95.2;
case 3, a_bkg = 79.2;
end
% Resonance positions (G) and widths (G)
resonanceB = [1.298, 2.802, 3.370, 5.092, 7.154, 2.592, 2.338, 2.177];
resonancewB = [0.018, 0.047, 0.048, 0.145, 0.020, 0.008, 0.001, 0.001];
end
% Magnetic field range for plotting (G)
B = linspace(0.5, 8, 2000);
% Compute scattering length using product formula
a_s = a_bkg * ones(size(B));
for j = 1:length(resonanceB)
a_s = a_s .* (1 - resonancewB(j) ./ (B - resonanceB(j)));
end
end
function [t, B_ramp, a_check] = generateLinearBRamp(B_between, a_s_between, a_start, a_end, T, Nt)
% Generates a B-field ramp (B_ramp) to produce a linear a_s ramp from a_start to a_end.
% Uses precomputed LUT: B_between (magnetic field) and a_s_between (scattering length).
%
% Inputs:
% B_between - Magnetic field values (G) between resonances [vector]
% a_s_between - Corresponding scattering lengths (a_0) [vector]
% a_start, a_end - Target scattering length range (a_0)
% T - Total ramp time (s)
% Nt - Number of time steps
%
% Outputs:
% t - Time vector (s)
% B_ramp - Generated B-field ramp (G)
% a_check - Verified a_s(t) using interpolation (a_0)
% --- 1. Time vector ---
t = linspace(0, T, Nt);
% --- 2. Ensure LUT is sorted and unique ---
[a_s_sorted, sort_idx] = unique(a_s_between); % Remove duplicates and sort
B_sorted = B_between(sort_idx);
% --- 3. Generate target linear a_s ramp ---
a_target = linspace(a_start, a_end, Nt);
% --- 4. Interpolate B(t) from a_s -> B ---
B_ramp = interp1(a_s_sorted, B_sorted, a_target, 'pchip', 'extrap');
% --- 5. Verify a_s(t) by re-interpolating ---
a_check = interp1(B_sorted, a_s_sorted, B_ramp, 'pchip');
end
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