AtomicDensityMatrix`
AtomicDensityMatrix`

Hamiltonian

Hamiltonian[sys]

returns the matrix Hamiltonian for atomic system sys assuming the absence of external fields.

Hamiltonian[sys,MagneticField{bx,by,bz},ElectricField{ex,ey,ez}]

returns the matrix Hamiltonian including interaction terms with external classical magnetic and/or electric fields.

Details and Options

  • The atomic system sys is specified as a list of AtomicState objects.
  • Hamiltonian calls WignerEckart to evaluate the matrix elements for the necessary operators.
  • Hamiltonian[sys] returns a diagonal Hamiltonian with diagonal terms determined by the Energy parameters (and the HyperfineA and HyperfineB parameters for hyperfine-Zeeman systems) of the corresponding atomic states.
  • Different options applying to different fields can be specified using the form Hamiltonian[sys,{{MagneticField,opts},{MagneticField,}},].
  • The following options can be given:
  • ElectricField 0external electric field
    MagneticField 0external magnetic field
    Interaction Automaticwhich types of interactions to include
    AllowedCouplings Allwhich transitions to allow nonzero coupling on
  • The setting for the Interaction option may be given as a list of one or more of the following interaction types:
  • "Internal"eigenenergies due to internal interactions
    "MagneticDipole"magnetic dipole interaction with external field
    "ElectricDipole"electric dipole interaction with external field
    "Polarizability"effective Hamiltonian due to atomic polarizability
  • Additional possible settings are Automatic, which is equivalent to {"Internal","MagneticDipole","ElectricDipole"}, and All, meaning all interactions.
  • The magnetic-dipole Hamiltonian is given by , where is the electronic magnetic moment operator and is the magnetic field.
  • The electric-dipole Hamiltonian is given by , where is the electric dipole operator and is the electric field.
  • Under the -coupling approximation, employed in the AtomicDensityMatrix package, magnetic coupling is allowed between two different -states only if they are components of the fine structure of one term. This can be specified by labeling the two states {term, lab1} and {term, lab2}, where term, lab1, and lab2 are arbitrary labels.
  • For a toy-type atomic system (no angular momentum), the magnetic field is disregarded, and the electric field is converted to a scalar by taking the norm, if there is more than one nonzero Cartesian component, or by taking the nonzero Cartesian component, if there is only one.

Examples

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Basic Examples  (6)

Internal and electric-field-interaction Hamiltonian for a toy system:

Magnetic-field-interaction Hamiltonian for a Zeeman system subject to an arbitrarily directed magnetic field:

Stark-polarizability Hamiltonian for a Zeeman system subject to an x-directed electric field:

Hamiltonian for a toy system subject to an optical field:

Hamiltonian for a Zeeman system subject to a linearly (x)-polarized optical field and static magnetic field:

Internal Hamiltonian for a hyperfine-Zeeman system:

Scope  (1)

Hamiltonian due to magnetic-dipole coupling between two fine-structure states:

Generalizations & Extensions  (1)

Hamiltonian for two optical fields, each restricted to act on a particular transition:

Options  (7)

ElectricField  (2)

Hamiltonian for a Zeeman system subject to an electric field:

For a toy-type system, the first nonzero Cartesian component of the electric field is used:

Interaction  (1)

Define a system:

Electric and magnetic fields to be applied:

Internal interactions only:

Electric dipole interaction only:

Magnetic dipole interaction only:

Effective electric-polarizability interaction only:

Internal, electric-dipole, and magnetic-dipole interactions:

All interactions:

MagneticField  (3)

Hamiltonian for a Zeeman system subject to a magnetic field:

Hamiltonian for M1 interaction between two fine-structure states:

For a toy-type system, the magnetic field is disregarded:

AllowedCouplings  (1)

Restrict the magnetic dipole coupling to act only between the two fine-structure states, and not within them:

Options  (7)

ElectricField  (2)

Hamiltonian for a Zeeman system subject to an electric field:

For a toy-type system, the first nonzero Cartesian component of the electric field is used:

Interaction  (1)

Define a system:

Electric and magnetic fields to be applied:

Internal interactions only:

Electric dipole interaction only:

Magnetic dipole interaction only:

Effective electric-polarizability interaction only:

Internal, electric-dipole, and magnetic-dipole interactions:

All interactions:

MagneticField  (3)

Hamiltonian for a Zeeman system subject to a magnetic field:

Hamiltonian for M1 interaction between two fine-structure states:

For a toy-type system, the magnetic field is disregarded:

AllowedCouplings  (1)

Restrict the magnetic dipole coupling to act only between the two fine-structure states, and not within them: