AtomicDensityMatrix Package

AtomicDensityMatrix is a package for Mathematica version 6+ that facilitates density-matrix calculations in atomic and related systems. Note: The documentation is incomplete; there may be missing symbol reference pages. Look at the tutorials to get started. Documentation is currently an active part of package development.

Atomic Systems

AtomicState — expression representing an atomic state

Sublevels — generate the hyperfine and Zeeman sublevels for a list of atomic states

J  ▪  L  ▪  S  ▪  M  ▪  NuclearSpin  ▪  F  ▪  Energy  ▪  HyperfineA  ▪  HyperfineB  ▪  NaturalWidth  ▪  GFactor  ▪  StateLabel  ▪  BranchingRatio  ▪  Parity  ▪  AtomicSystemQ  ▪  AtomicSystemType  ▪  StatePosition  ▪  SelectStates  ▪  DeleteStates  ▪  ExcitedStates  ▪  GroundStates  ▪  RemoveZeemanSublevels

Atomic Properties

AtomicData — spectroscopic data for a selection of atomic states

Density Matrices and State Vectors

DensityMatrix — create density matrix for an atomic system

DMVariables  ▪  Ket

Operators and Hamiltonians

WignerEckart — matrix elements of spherical tensor operators

Hamiltonian — the Hamiltonian for an atomic system subject to external classical fields

RotatingWaveApproximation — perform the rotating-wave approximation on a Hamiltonian

RotatingWaveTransformMatrix  ▪  EffectiveHamiltonian

Relaxation and Repopulation

IntrinsicRelaxation — relaxation matrix for spontaneous decay

TransitRelaxation  ▪  TransitRepopulation

Atomic Evolution

LiouvilleEquation — evolution equations for an atomic system

InitialConditions — specify the density matrix at an initial time

Light Propagation

OpticalField — generate the expression for an optical field

Observables — the change in optical parameters upon propagation through an optically thin medium

LinearAbsorption — formula for absorption of weak light

Visualization

LevelDiagram — draw a level diagram for an atomic system, Hamiltonian, and density matrix

AMPSPlot — draw the angular-momentum probability surface for a density matrix

Tensor Algebra

ToCartesian — convert a spherical tensor into the Cartesian basis

Decomposition — decompose a reducible operator into a sum of polarization moments

Rank  ▪  Component  ▪  Recomposition  ▪  TensorForm

Tutorials

Introduction

  • The State Vector and the Schrödinger Equation
  • The Density Matrix and the Liouville Equation

    • Components

  • Matrix Notation
  • Inner Products and Matrix Multiplication
  • Tensor Component Operations and Extraction
  • Conversion Between the Cartesian and Spherical Bases
  • Specifying the Atomic System
  • Displaying the Density Matrix
  • The Wigner-Eckart Theorem
  • Quantum-Mechanical Rotations
  • Constructing the Hamiltonian
  • The Rotating-Wave Approximation
  • Angular-Momentum Probability Surfaces

    • Basic Examples

  • Two-Level System: the Rabi Frequency and the Saturation Parameter
  • Three-Level System: Electromagnetically Induced Transparency
  • Zeeman Structure: Nonlinear Magneto-Optical Rotation
  • Hyperfine Structure: Breit-Rabi Diagram
  • Linear Absorption Fitting
  • Stark Mixing
  • Crossed Electric and Magnetic Fields

    • Advanced Examples

  • Visualizing Polarization Moments and Quantum Beats
  • Electromagnetically Induced Transparency and Atomic Polarization
  • Pump—Probe Magnetometry
  • Nonlinear Magneto-Optical Rotation in an Alkali Atom
  • Nonlinear Magneto-Optical Rotation with Frequency-Modulated Light
  • Nonlinear Magneto-Optical Rotation in a Radio-Frequency Field
  • Collapse and Revival Quantum Beats
  • The AC Stark Effect
  • Isolating the Hexadecapole Moment
  • Light Polarization Demonstration

    • Related Links

    ▪ AtomicDensityMatrix web page

    Copyright and License:

     Copyright (C) 2004—2023 Simon M. Rochester <simon@rochesterscientific.com> This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.