AtomicDensityMatrix Paclet Symbol

# RotatingWaveApproximation

 RotatingWaveApproximation[sys, h, {, tr}] performs the rotating-wave approximation on a Hamiltonian h for atomic system sys, assuming an optical field with angular frequency acting on transitions specified by tr. RotatingWaveApproximation[sys, h, {{1, tr1}, {2, tr2}, ...}] performs the RWA assuming optical fields with angular frequencies i acting on transitions specified by tri. RotatingWaveApproximation[sys, h, {{1, tr1, 1}, {2, tr2, 2}, ...}]writes optical frequencies in terms of their detunings from resonance i after performing the RWA. RotatingWaveApproximation[sys, h, ]performs the RWA using an automatically generated transformation appropriate for a single optical field. RotatingWaveApproximation[sys, h, {1, 2, ...}, TransformMatrix→u]performs the RWA using an explicitly specified unitary transformation matrix.
• The atomic system sys is specified as a list of AtomicState objects.
• Transitions are specified by critlowercritupper, where critlower and critupper are state-selection criteria for the lower and upper states involved in the transition, respectively. (critlower and critupper can each designate multiple states.)
• State-selection criteria are specified as for StatePosition: Boolean expressions involving the state parameters, such as StateLabel=2&&F=1. As a shorthand, if all states with a particular label are to be selected, just the label itself can be specified.
• Putting the lower (in energy) states first in the transition specification corresponds to the assumption that the optical frequencies i are positive. If the upper states are put first, the result will be as if the optical frequency is negative.
• A list of transitions {tri, 1, tri, 2, ...} can be supplied instead of a single transition in place of any of the input parameters tri in the function definition above.
• The transform matrix is constructed so as to shift the levels involved in the transitions so as to eliminate the corresponding optical frequencies, working in the order that the transitions are specified.
• If not all of the optical frequencies can be eliminated (because the states in the transition have already been shifted to eliminate earlier frequencies in the list), the remaining frequencies are written in terms of differences from the eliminated frequencies prior to dropping the off-resonant terms. This allows the elimination of all but the optical difference frequencies from the Hamiltonian.
• Specifying a detuning parameter, as in {1, tr1, 1}, amounts to making the substitution 1Energy[upper]-Energy[lower]+1 in the final result, where Energy[upper] and Energy[lower] are the energies of the upper and lower states in the transition, respectively.
• The following options can be given:
 Method Automatic the method to use for performing the RWA TimeVariable Automatic symbol used to represent the time variable TransformMatrix Automatic unitary transformation to the rotating frame
• The option TransformMatrix can be used to explicitly a matrix to be used for transformation to the rotating frame.
• If RotatingWaveApproximation[sys, h, ] is called with , the transform matrix is generated by assuming that states with Energy→0 are lower states, and all other states are upper states, coupled to the lower states by a field of frequency .
Define a two-level system.
 Out[2]=
Find the Hamiltonian with states coupled by an optical field with angular frequency .
 Out[3]//MatrixForm=
Apply the rotating-wave approximation, specifying that the field of frequency couples the lower state with label 1 to the upper state with label 2.
 Out[4]//MatrixForm=
Specifying the detuning is equivalent to making the replacement Energy[2]+.
 Out[5]//MatrixForm=
For this system, RotatingWaveApproximation can guess that state 1 is the lower state and state 2 is the upper state, so it is not necessary to specify the transition.
 Out[6]//MatrixForm=

Here is an example with three levels and two fields.
Define a three level system.
 Out[2]=
Find the Hamiltonian with two optical fields. We make the assumption that the field 1 only interacts with the ab transition, while the field 2 only interacts with the bc transition, by setting the additional terms to zero.
 Out[3]//MatrixForm=
Here we apply the RWA assuming that Energy[a]<Energy[b]<Energy[c].
 Out[4]//MatrixForm=
The two-photon transition between states a and c is resonant when the energies of the two states in the rotating-frame Hamiltonian are equal; in this case, when 0=Energy[c]-1-2, or 1+2=Energy[c]. This corresponds to the level diagram for the system:
 Out[5]=
If, instead, we assume that Energy[a]<Energy[c]<Energy[b], we obtain the rotating-frame Hamiltonian:
 Out[6]//MatrixForm=
Here, the two-photon resonance condition is 1-2=Energy[c], corresponding to the level diagram:
 Out[7]=
 Scope   (4)
 Options   (1)
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