AtomicDensityMatrix

mathematica package

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about : installation : documentation : release notes : feedback

AtomicDensityMatrix is a package for Mathematica version 6 and later that facilitates analytic and numerical density-matrix calculations in atomic and related systems. It is intended to be both general and user-friendly, and to be useful to the working physicist as well as to students.

AtomicDensityMatrix is open source software, licensed under the GPLv3.

The package has been updated for compatibility with Mathematica 10. Please send bug reports to simon@rochesterscientific.com.

Features:

• Supports systems with an arbitrary number of states, automatically generating Zeeman and hyperfine substructure. Also supports calculations for model systems that neglect angular momentum
• Automatic generation of Hamiltonians including the effects of arbitrary static or dynamic electric, magnetic, and optical fields
• Routines for automatic application of the rotating-wave approximation
• Automatic generation of terms describing relaxation effects, including spontaneous decay
• Formulates and solves density matrix evolution equations, either analytically or numerically, as a function of time or in the steady state
• Routines implementing the matrix-continued fraction method for efficient numerical computation of periodic solutions for the case of modulated fields
• Supports generation and simultaneous solution of coupled density matrices describing different velocity classes and physical regions in the system
• Finds the effect of atomic polarization on light polarization, using various light polarization parameterization schemes
• Performs irreducible tensor decomposition of the density matrix
• General implementation of irreducible tensor algebra
• Performs arbitrary rotations of operators and irreducible tensors
• Automatic generation of level diagrams showing level structure, electromagnetic-field induced coupling, and atomic sublevel populations from the density matrix
• Plots angular-momentum probability surfaces: a 3D geometrical representation of atomic polarization
• An associated graphics package aids the generation of plots and animations for visualization of the physical processes
• A growing collection of basic and advanced examples, showing the application of the package to real atomic physics problems
• User extensible

First download ADM_yy.mm.dd.zip.

Installing a Mathematica package consists entirely of placing the package files in a location where Mathematica can find them. You can do this using one of the following two methods:

1. Using the File > Install menu item (Mathematica version 8 and up):
• Start Mathematica.
• Select the menu item File > Install.
• In the dialog box, for "Type of Item to Install" choose "Application" (not "Package").
• For "Source" choose "From File..."
• When the file selector appears, choose the zip file that you have downloaded, and select "Open".
• Leave the "Install Name" as it's given, and click "OK".
• Mathematica will extract the contents of the zip file, and copy them to the correct install location.
2. Manually (all versions):
• Locate your Mathematica base directory (for installation for all users) or user base directory (for installation for the current user only). You can find these directories by evaluating $BaseDirectory or $UserBaseDirectory in Mathematica. In Mathematica 7 or later, you can open the directory in a file browser by evaluating

  SystemOpen[$BaseDirectory]

  or

  SystemOpen[$UserBaseDirectory]

  (Note that in Windows or Mac OS X, these directories may be hidden in the file system by default.)
• Unzip the contents of the downloaded zip archive into the "Applications" subdirectory of the base directory.
• The package files should now be in the directory

  [BaseDirectory]/Applications/AtomicDensityMatrix

  where [BaseDirectory] is the directory you chose in the previous step.

To load and test the package in Mathematica, evaluate

<<AtomicDensityMatrix`

and then

$ADMVersion

to see the package version number,

AMPSPlot[DiagonalMatrix[{1, 0, 1}]]

to see a simple angular-momentum-probability surface, and

LevelDiagram[ZeemanSystem[1]]

to plot the level diagram for a J=1 atomic system. There are many other examples in the documentation.

The package documentation is growing, but far from complete. It is available in the Mathematica Documentation Center when the package is installed.

You may now have to restart Mathematica to see the AtomicDensityMatrix documentation appear in the Documentation Center.

To view the documentation, select Help > Documentation Center, and click "Add-Ons and Packages" at the bottom of the page (for version 6, "Installed Add-Ons" on the bottom right). You will reach a page of links to the documentation for installed packages. Alternatively, you can just enter "adm" in the Documentation Center search bar—one of the first few results should be for the ADM package guide page. (This shortcut appears to be broken in Mathematica 10.1, but it's working again in Mathematica 10.2.)

The documentation is also available online in html format.

• v17.02.12
  • Updated for compatibility with Mathematica 11.
  • Renamed Region to ExperimentalRegion. User notebooks that use Region will need to be updated.
  • Extended StateLabelQ and StateMatchQ functions, removed deprecated syntax for StatePosition and StateMatchQ, added StateEvaluate function.
  • Rewrote and improved LevelDiagram function.
  • Added Fluorescence function.
  • Fixed FluorescenceOperator for toy systems.
  • Set BranchingRatio to zero for same-parity states.
  • Added symbol documentation page for LevelDiagram.
  • Renamed GetLine to GetLines.
  • Other minor fixes and improvements.
• v15.08.27
  • Moved Arrow3D and PlotAndFit packages into the AtomicDensityMatrix application. These packages no longer have to be loaded separately. The Graphics and DataAnalysis folders have been removed from the ADM application.
  • Speed improvements in WignerEckart, Hamiltonian, and OpticalRepopulation.
  • Updates to Arrow3D for Mathematica 10. Tweaked default graphics options for Axes3D and AMPSPlot.
  • Fixed options handling in PlotAndFit for Mathematica 10.
  • Updated documentation for compatibility with Mathematica 10 graphics.
  • Added "Magneto-Optical Rotation in an Alkali Atom" tutorial.
• v15.04.24
  • Added GPL license info to source files.
  • Removed dummy AtomicPhysics` package.
  • For compatibility with Mathematica 10, renamed Void as VoidRegion. User notebooks that use the Void command will need to be changed accordingly.
  • Adopted WWBCommon package for cross-version-compatible documentation.
  • Higher-res banner and logo images.
  • Speed improvements in FixTicks and AlignedGraphicsGrid.
  • Reorganized and improved unit tests.
• v14.02.28
  • Fixed behavior of BranchingRatio for setting branching ratios for inverse decays to zero.
  • Modified OpticalRepopulation to give more reasonable result when BranchingRatios are not specified.
  • Added rule to MatrixMultiply to allow multiplying a scalar operator by a (Cartesian) vector operator.
  • Changed possible values for Interaction option of Hamiltonian to the strings "Internal", "ElectricDipole", "MagneticDipole", and "Polarizability"
  • Updated (and renamed) the "Electromagnetically Induced Transparency and Atomic Polarization" example.
  • Speed optimiziations.
  • Dummy position variables Xx, Yy, and Zz are defined only if the VectorAnalysis` package has not been loaded. If using VectorAnalysis` package, load it first to avoid name conflict warnings.
  • Modified OpticalField to produce simpler expressions for arbitrary polarization and propagation vectors.
  • Added and updated symbol documentation pages.
• v13.07.19
  • Updated for compatibility with Mathematica 9:
    • In Mathematica 9, the functionality of PhysicalConstants` and Units` has been completely redesigned, and merged with the kernel. However, I don't think loading these packages causes any conflict with built-in functionality, so for backwards compatibility I'm going to continue loading these packages (and suppress the obsolete package warning). Please let me know if you encounter any conflicts.
    • The functionality of the VectorAnalysis` package has been redesigned and incorporated into the kernel. There was very little use of it in the ADM package, so I was able to remove the dependency on the VectorAnalysis` package altogether.
    • There is a new TensorProduct function in Mathematica 9, so the ADM TensorProduct function has been renamed SphericalTensorProduct
    • Renamed Label as StateLabel; the option Label for DensityMatrix has been renamed DMLabel. User notebooks that use the Label command will need to be changed correspondingly.
    • Fixed problems stemming from new (fixed) ComplexExpand behavior.
    • Native documentation was restyled in version 9, producing some ugly layout problems in the ADM documentation. Cross-version documentation has now been implemented using the method described here.
  • Updated documentation.
  • Fixed a bug blocking the package from showing up in the "Add-ons" section of the Documentation Center.
  • Fixed web links in the ADM documentation, making them point to the online version of the documentation.
  • Made functions for obtaining atomic parameters automatically thread over atomic systems.
  • Speed optimizations.
  • Changed optical field parameterization specifications to strings.
• v13.07.07
  • Added "Nonlinear Magneto-optical Rotation with Frequency-modulated Light" example.
  • Updated "EIT and Atomic Polarization", "Breit-Rabi", "Rotating-wave Approximation", and other assorted tutorials.
  • Added symbol pages for RotatingWaveApproximation and F.
  • Modified DropFastTerms (and RotatingWaveApproximation) so that sums of two different optical frequencies are not dropped. Also extended to allow some more complicated expressions as frequency arguments.
  • Extended StatePosition, SelectStates, and related functions to allow specification of the selection criteria Label == lab as just lab.
  • Extended RotatingWaveApproximation, RotatingWaveTransform and RotatingWaveTransformMatrix to automatically generate the RWA transform matrix given a list of optical frequencies and the associated levels that they couple. Also allow the posibility to specify the optical detuning parameters along with the optical frequencies.
  • Defined new function ReplaceMatrixElement that applies replacement rules to particular matrix elements in an operator selected according to their cooresponding AtomicState parameters.
  • Defined new function StateMatchQ
• v13.06.09
  • Corrected the units for quantities in AtomicData by changing Hertz to 1/Second (all quantities are in omega units, rad/sec, but were incorrectly labelled as being Hertz). If your application depends on the AtomicData package, it may require modification.
  • Extended RotatingWaveTransformMatrix to allow manual specification of level shifts
  • Changed ToContravariant and ToCovariant to use Dual rather than GeneralizedConjugateTranspose when converting between co- and contravariant quantities in order to correctly handle complex vectors
  • Included SphericalPhase parameterization in OpticalField
  • Modified BranchingRatio so that it returns zero if it can tell that the reverse branching ratio is nonzero.
  • Fixed links in documentation.
  • Updated documentation.
  • Updated installation instructions
• v12.08.15
  • Added parities for alkali states to AtomicData
  • Added ParityOffset option to LevelDiagram
  • Updated and added usage messages.
• v12.06.21
  • Documented the Voigt profile in the "Linear Absorption Fitting" example
  • Modified SublevelMultiplicity so that it returns (2F+1) when F is defined (suggested by Guobin Liu)
  • Modified DopplerWidth to work if lower and upper states are given in either order (suggested by Guobin Liu)
  • WignerD is defined as a system function in Mathematica 8. Therefore the ADM version of WignerD has been renamed ADMWignerD, and the built-in WignerD has been overloaded so that it calls ADMWignerD when given arguments that match the ADMWignerD format.
  • Fixed linear absorption for Toy systems
  • SelectStates now allows boolean operators in the selection criteria
  • Speed enhancements
  • Split more functions into subpackages
  • Removed outer matrix in return value of PolarizationMatrices with numerical argument
  • LevelDiagram improvements
  • Assorted documentation improvements
• v10.07.17
  • Package now loads with <<AtomicDensityMatrix` rather than <<AtomicPhysics`. A dummy AtomicPhysics package that calls the AtomicDensityMatrix package is provided for backwards compatibility.
  • Updated documentation
• v10.07.09
  • Standardized WignerRotate function
  • Standardized AMPSPlot for polarization-moment expansions
  • Added "Quantum-mechanical rotations" tutorial
  • Updated documentation
• v10.06.28
  • Added options to Ket function
  • Updated some tutorials
  • Split LevelDiagram and perturbative solve functions into subpackages
  • Fixed bug in LevelDiagram for Mathematica 6.0 and made speed improvements
• v10.04.28
  • Speed improvements in SteadyStatePerturbativeSolve, OpticalRepopulation
• v10.04.20
  • Modified package context structure
  • Speed optimizations
  • Migrated development to Wolfram Workbench
  • Added DopplerWidth and MostProbableSpeed functions
  • Added "Linear Absorption Fitting" example
  • Fixed Component function for tensor decomposition matrix input
  • Combined DMElementPattern and DMElementPartPattern functions
• v09.12.08
  • Corrected sign error in optical-field azimuthal angle
  • Flipped rotating wave transform from active to passive
  • Changed UnitaryTransformation to EffectiveHamiltonian
  • Changed DMRotating to DMFromRotatingDM
  • Included PlotAndFit package in distribution
• v09.09.07
  • Added potassium to AtomicData
  • Added AtomicTransition function to AtomicData package
• v09.08.07
  • Added ExcitedStates function
  • Added LinearAbsorption function
  • Improved LevelDiagram function
  • Improved SphericalTensorQ function
  • Extended SublevelMultiplicity function
  • Change in internals of WignerEckart function
  • Fixed bug in RotatingWaveTransform
• v09.06.09
  • Internal change in ExprToReIm
  • Fixed bug in DMElementPattern
  • Fixed bug in FluorescenceOperator for angle-integrated case
• v08.12.09
  • Improved LevelDiagram function
  • Renamed FluorescenceOperator as OpticalRepopulationOperator, added new FluorescenceOperator function that calculates fluorescence into a solid angle
  • Wrote usage messages for some symbols.
  • Updated values in AtomicData and added Na
• v08.10.30
  • Improved TensorForm, Component, PolarizationMoments, LevelDiagram functions
  • Added PolarizationMoments representation to OpticalRepopulation function
  • Set up LiouvilleEquation function to produce Bloch equations in terms of polarization moments
  • Extended ToCartesian function to apply to operators and arbitrary rank spherical tensors
  • Added "EIT and Atomic Polarization" example
• v08.10.23
  • Worked on LevelDiagram function
  • Separated out AtomicDensityMatrix`Common` package
  • Changed behavior of F parameter: it now returns J if F is not defined
  • Changed labeling of density matrix elements in Toy systems: labels are no longer enclosed in lists
  • Generalized tensor operations to include imaginary/nonhermitian tensors
  • Improved TensorForm function
  • Updated "MatrixNotation", "Matrix Multiplication" tutorials
• v08.10.07
  • Fixed bug in Recomposition function
  • Improved LevelDiagram function
  • Updated "Angular-momentum Probability surfaces" example
  • Improved "Collapse and Revival Quantum Beats" example
• v08.09.22
  • Expanded "Two-level system", and "NMOR" examples
  • Updated "Breit-Rabi Diagram", "Collapse and Revival", "Three-level System", "Angular-momentum Probability Surfaces" examples
  • Improved LevelDiagram, TransitRepopulation, DensityMatrix, DMVariables, Recomposition functions
  • Changed polarization moment expansion coefficients to contravariant to match Varshalovich
  • Fixed bug in Component function
  • Added CovariantDecompositionQ, ContravariantDecompositionQ, CovariantDecompositionMatrixQ, ContravariantDecompositionMatrixQ, TensorForm functions
  • Added link to source code in documentation
• v08.07.07
  • Added "The AC Stark Effect" example
  • Added "The Wigner-Eckart Theorem", "Constructing the Hamiltonian", "The Rotating-wave Approximation" tutorials
  • Improved LevelDiagram function
• v08.06.29
  • Got documentation working in 6.0.3
  • Revamped Guide page
  • Added "Displaying the Density Matrix" tutorial
  • Added "Collapse and Revival Quantum Beats" example
  • Added symbol reference pages for DensityMatrix, DMVariables, Ket, AtomicState, L, S
  • Added $ADMVersion symbol.
• v08.05.26
  • Streamlined internals of OpticalField
  • Improved Region input to Hamiltonian
  • Eliminated FractionalDensity Region parameter
  • Included case of multiple ground states in TransitRepopulation
  • Added "Isolating the Hexadecapole Moment" example
  • Added GFactor parameter to AtomicState
  • Added LevelDiagram function (prototype)
  • Added "The Stark Effect" example
  • Added "Hyperfine Structure: Breit-Rabi Diagram" example

Please send questions, suggestions, and bug reports to Simon Rochester at simon@rochesterscientific.com.