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.
  • 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:
ElectricField0external electric field
MagneticField0external magnetic field
InteractionAutomaticwhich types of interactions to include
RestrictCouplingAllwhich 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 L S-coupling approximation, employed in the AtomicDensityMatrix package, magnetic coupling is allowed between two different J-states only if they are components of the fine structure of one term. This is 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 first nonzero Cartesian component.
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Internal and electric-field-interaction Hamiltonian for a toy system:
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Magnetic-field-interaction Hamiltonian for a Zeeman system subject to an arbitrarily directed magnetic field:
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Stark-polarizability Hamiltonian for a Zeeman system subject to an x-directed electric field:
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Hamiltonian for a toy system subject to an optical field:
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Hamiltonian for a Zeeman system subject to a linearly (x)-polarized optical field and static magnetic field:
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Internal Hamiltonian for a hyperfine-Zeeman system:
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