Lasers and Magnetic Fields

The MuscleMuseum framework provides comprehensive tools for modeling lasers and magnetic fields in atomic, molecular, and optical (AMO) physics experiments. These fundamental field classes serve as building blocks for more complex potential and trapping systems.

Overview

The laser and magnetic field framework consists of several core components:

• Laser systems: Monochromatic laser beams with full polarization and spatial control

• Gaussian beams: Focused laser beams with spatial intensity profiles

• Magnetic fields: Static and spatially varying magnetic field configurations

• Modulation: Frequency control via acousto-optic and electro-optic modulators

These field classes can be combined with atomic properties to create various potential systems (covered in a separate documentation page).

The typical workflow for using these components is shown below:

Laser Systems

The laser framework provides precise control over all aspects of monochromatic laser beams, from basic parameters to advanced spatial and temporal phase functions.

Laser Class Hierarchy

Core Laser Class

The Laser class serves as the foundation for all optical field modeling. It encapsulates the essential properties of monochromatic light:

Physical Parameters:

• Wavelength/Frequency: Optical wavelength \(\lambda\) [m] or frequency \(f\) [Hz]

• Polarization: Jones polarization vector \((E_x,E_y,E_z)\)

• Direction/Angle: Propagation direction as Cartesian unit vector or spherical angles \((\theta, \phi)\)

• Phase: Optical phase \(\phi\) [rad]

• Intensity/Power: Light intensity \(I\) [W/m²] or total power \(P\) [W]

Derived Quantities:

• Wavevector: \(\mathbf{k} = k \hat{\mathbf{k}}\) [rad/m]

• AngularFrequency: \(\omega = 2\pi f\) [rad/s]

• ElectricFieldAmplitude: \(|E| = \sqrt{2 Z_0 I}\) [V/m]

Basic Usage:

% Create a laser with specific parameters
laser = Laser(...
    frequency = 3.84e14, ...        % 780 nm (Rb D2 line)
    polarization = [1; 0; 0], ...   % Linear polarization along x
    direction = [0; 0; 1], ...      % Propagating along z
    intensity = 1e3, ...            % 1 kW/m²
    phase = 0 ...                   % Zero phase
);

% Access derived properties
wavelength = laser.Wavelength;      % 780.24e-9 m
k_vector = laser.AngularWavevector; % [0; 0; 8.04e6] rad/m
omega = laser.AngularFrequency;     % 2.41e15 rad/s

Advanced Features:

The Laser class provides sophisticated phase function generators for space-time dependent calculations:

% Generate phase functions
spacePhase = laser.spacePhaseFunc();    % e^(i(φ - k·r))
timePhase = laser.timePhaseFunc();      % e^(iωt)
fullPhase = laser.spaceTimePhaseFunc(); % e^(i(ωt + φ - k·r))

% Use in calculations
position = [0; 0; 1e-6];  % 1 μm along z
time = 1e-9;              % 1 ns

spatial_factor = spacePhase(position);
temporal_factor = timePhase(time);
full_factor = fullPhase(position, time);

Beam Manipulation:

% Rotate laser direction and polarization
euler_angles = [pi/4, pi/6, 0];  % ZYZ Euler angles
laser.rotate(euler_angles);

% Rotate to specific spherical angles
target_angles = [pi/3, pi/4];    % [θ, φ]
laser.rotateToAngle(target_angles);

Gaussian Beams

The GaussianBeam class extends Laser to model focused \(\mathrm{TEM}_{00}\) Gaussian laser beams, essential for optical trapping applications.

Additional Properties:

• Waist: \(1/e^2\) beam radii \((w_x, w_y)\) [m]

• Center: Beam center position \((x, y, z)\) [m]

• RayleighRange: Rayleigh range \(z_R = \pi w_x w_y / \lambda\) [m]

• IntensityAveraged: Average intensity \(I = P/(\pi w_x w_y)\) [W/m²]

Usage Example:

% Create a focused Gaussian beam for optical trapping
gaussianBeam = GaussianBeam(...
    waist = [50e-6; 60e-6], ...     % 50×60 μm waist
    center = [0; 0; 0], ...         % Centered at origin
    wavelength = 1064e-9, ...       % 1064 nm (Nd:YAG)
    power = 100e-3 ...              % 100 mW
);

% Calculate beam properties
rayleigh_range = gaussianBeam.RayleighRange;  % ~8.8 mm
peak_intensity = gaussianBeam.Intensity;      % Peak intensity
avg_intensity = gaussianBeam.IntensityAveraged; % Average intensity

Power-Intensity Relationship:

For Gaussian beams, power and intensity are automatically linked:

% Setting power automatically calculates intensity
beam.Power = 50e-3;  % 50 mW
I_peak = beam.Intensity;  % Automatically calculated

% Setting intensity automatically calculates power
beam.Intensity = 1e6;  % 1 MW/m²
P_total = beam.Power;  % Automatically calculated

Magnetic Fields

The magnetic field framework provides flexible modeling of static and spatially varying magnetic field configurations commonly used in atomic physics experiments.

Magnetic Field Class Structure

MagneticField Class

The MagneticField class models magnetic field distributions using polynomial expansions or arbitrary functions.

Field Components:

• Bias: Uniform bias field \(\mathbf{B}_0 = (B_x, B_y, B_z)\) [T]

• Gradient: Linear gradient matrix \(\partial B_i/\partial x_j\) [T/m]

• Quadratic: Quadratic terms (not yet implemented)

• ArbitraryDistribution: Custom spatial distribution \(\mathbf{B}(\mathbf{r})\)

Mathematical Representation:

The magnetic field at position \(\mathbf{r}\) is calculated as:

\[\mathbf{B}(\mathbf{r}) = \mathbf{B}_0 + \nabla \mathbf{B} \cdot \mathbf{r} + \text{quadratic terms}\]

Basic Usage:

% Uniform magnetic field
uniform_field = MagneticField(bias = [0; 0; 1e-4]); % 1 G along z

% Quadrupole magnetic trap
grad_matrix = diag([10, 10, -20]) * 1e-2; % [T/m]
quadrupole_trap = MagneticField(...
    bias = [0; 0; 5e-4], ...     % 5 G bias
    gradient = grad_matrix ...
);

% Custom field distribution
custom_field = MagneticField(...
    distribution = @(r) [0; 0; 1e-4] .* exp(-sum(r.^2)/1e-6) ...
);

Unit Conversions:

The framework provides convenient unit conversions between Tesla and Gauss:

field = MagneticField(bias = [0; 0; 1e-4]); % 1 G in Tesla

% Access in different units
bias_tesla = field.Bias;        % [0; 0; 1e-4] T
bias_gauss = field.BiasLu;      % [0; 0; 1] G

% Gradients
grad_tesla_per_m = field.Gradient;    % [T/m]
grad_gauss_per_cm = field.GradientLu; % [G/cm]

Spatial Field Functions:

% Generate spatial field function
field_func = field.spaceFunc();

% Evaluate at specific positions
positions = [0, 1e-3, 2e-3; 0, 0, 0; 0, 0, 0]; % x positions
B_values = field_func(positions);

% Find field zero (for diagonal gradients)
field_zero_pos = field.FieldZero; % Position where B = 0

Common Magnetic Trap Configurations:

% Ioffe-Pritchard trap
ioffe_trap = MagneticField(...
    bias = [0; 0; 1e-4], ...           % Axial bias
    gradient = [20e-2, 0, 0; ...       % Radial gradients
                0, 20e-2, 0; ...
                0, 0, -2e-2] ...       % Weak axial gradient
);

% Anti-Helmholtz coil configuration
anti_helmholtz = MagneticField(...
    gradient = diag([10, 10, -20]) * 1e-2 ... % [T/m]
);

Integration with Potential Systems

The MagneticField class serves as the foundation for creating magnetic potentials that describe Zeeman energy shifts. When combined with atomic properties, magnetic fields can create trapping or anti-trapping potentials depending on the magnetic quantum numbers of the atomic state.

The conversion from magnetic fields to atomic potentials involves:

• Low Field Regime (Linear Zeeman): Energy shifts proportional to \(|\mathbf{B}|\)

• High Field Regime (Paschen-Back): Decoupled nuclear and electronic contributions

• Automatic regime selection: Based on field strength vs. hyperfine splitting

These potential calculations are handled by specialized classes covered in the potential systems documentation.

Integration with Optical Potential Systems

The Laser and GaussianBeam classes provide the electromagnetic field foundations for creating optical potentials. When combined with atomic polarizability data, these laser fields can create:

• Optical lattices: Periodic potentials from standing wave interference

• Optical dipole traps: Localized trapping from focused Gaussian beams

• Optical tweezers: Highly focused beams for single atom manipulation

The interaction strength depends on:

• Laser intensity: Higher intensity creates deeper potentials

• Detuning: Red-detuned light creates attractive potentials, blue-detuned creates repulsive

• Atomic polarizability: Species and state-dependent coupling strength

• Beam geometry: Waist size and shape determine spatial extent

These optical potential calculations and applications are covered in detail in the potential systems documentation.

Laser Frequency Control

Precise frequency control is achieved using acousto-optic modulators (AOMs) and electro-optic modulators (EOMs).

Acousto-Optic Modulators (AOM)

AOMs provide frequency shifts through acoustic wave interactions:

% Create AOM with 80 MHz drive frequency
aom = Aom(80); % MHz

% Single-pass configuration
shift_sp_plus1 = aom.shiftSP(+1);  % +80 MHz
shift_sp_minus1 = aom.shiftSP(-1); % -80 MHz

% Double-pass configuration (retro-reflected)
shift_dp_plus1 = aom.shiftDP(+1);  % +160 MHz
shift_dp_minus1 = aom.shiftDP(-1); % -160 MHz

Typical Applications:

% Laser cooling setup
cooling_aom = Aom(200);  % 200 MHz AOM
red_detuned = cooling_aom.shiftSP(-1); % -200 MHz (red detuning)

% Optical pumping
pumping_aom = Aom(78.5);  % Specific frequency for state selection
pump_shift = pumping_aom.shiftDP(+1); % +157 MHz

Electro-Optic Modulators (EOM)

EOMs generate sidebands for phase modulation and frequency stabilization:

% Create EOM with 40 MHz modulation
eom = Eom(40); % MHz

% Generate sidebands
first_order_red = eom.shift(-1);   % -40 MHz
carrier = eom.shift(0);            % 0 MHz (carrier)
first_order_blue = eom.shift(+1);  % +40 MHz

Frequency Stabilization:

% PDH (Pound-Drever-Hall) stabilization
pdh_eom = Eom(25); % 25 MHz modulation

% Generate error signal from sideband interference
red_sideband = pdh_eom.shift(-1);
blue_sideband = pdh_eom.shift(+1);

Advanced Field Applications

Multi-Beam Laser Systems

Creating complex laser configurations with multiple beams:

% Create array of laser beams for lattice formation
beam1 = Laser(wavelength=1064e-9, intensity=1e6, direction=[1;0;0]);
beam2 = Laser(wavelength=1064e-9, intensity=1e6, direction=[-1;0;0]);
beam3 = Laser(wavelength=1064e-9, intensity=1e6, direction=[0;1;0]);

% Counter-propagating beams for 1D lattice
lattice_beams = [beam1, beam2];

% Calculate interference pattern (simplified)
for i = 1:length(lattice_beams)
    phase_func = lattice_beams(i).spacePhaseFunc();
    % Use phase functions to calculate standing wave patterns
end

Complex Magnetic Field Configurations

Designing sophisticated magnetic trap geometries:

% Ioffe-Pritchard trap with custom curvature
custom_field = MagneticField(...
    distribution = @(r) ioffe_pritchard_field(r) ...
);

function B = ioffe_pritchard_field(r)
    % Custom implementation of Ioffe-Pritchard field
    x = r(1,:); y = r(2,:); z = r(3,:);

    % Bias field along z
    B0 = 1e-4; % 1 G

    % Radial gradients
    grad_radial = 20e-2; % 20 G/cm

    % Axial curvature
    curvature = 1e-2; % G/cm²

    Bx = grad_radial * x;
    By = grad_radial * y;
    Bz = B0 + curvature * (x.^2 + y.^2 - 2*z.^2);

    B = [Bx; By; Bz];
end

Time-Varying Fields

Implementing dynamic field control:

% Time-dependent magnetic field for evaporative cooling
function create_evap_sequence()
    initial_field = MagneticField(bias=[0;0;2e-4]); % 2 G

    % Create time-dependent field function
    evap_field = @(t) time_dependent_field(t, initial_field);

    % Use in experimental sequence
    times = linspace(0, 10, 100); % 10 second evaporation
    for i = 1:length(times)
        current_field = evap_field(times(i));
        % Apply field at this time step
    end
end

function field = time_dependent_field(t, initial_field)
    % Exponential decay for evaporative cooling
    decay_rate = 0.1; % 1/s
    scale_factor = exp(-decay_rate * t);

    field = MagneticField(bias = initial_field.Bias * scale_factor);
end

Best Practices

Parameter Consistency - Always specify either wavelength OR frequency, not both - Use consistent unit systems throughout calculations - Validate field gradients for physical realizability - Check that laser powers and magnetic field strengths are experimentally reasonable

Numerical Considerations - Cache expensive phase function calculations when used repeatedly - Use vectorized operations for spatial field evaluations - Consider computational cost vs. accuracy trade-offs for complex field distributions

Physical Validation - Verify magnetic field configurations are experimentally achievable - Check that optical powers are within safe operating limits - Validate that field gradients don’t exceed hardware capabilities

Error Handling - Implement bounds checking for physical parameters - Handle edge cases (e.g., zero magnetic fields, undefined directions) - Validate input parameters before expensive calculations

Performance Optimization - Pre-calculate commonly used phase functions and field distributions - Use appropriate spatial sampling for field evaluations - Consider parallel computation for parameter sweeps and multi-beam systems

The laser and magnetic field framework provides the fundamental electromagnetic field modeling capabilities needed for AMO physics experiments. These classes serve as building blocks that can be combined with atomic properties to create sophisticated trapping and manipulation systems.