AtomicDensityMatrix`
AtomicDensityMatrix`

RotatingWaveApproximation

RotatingWaveApproximation[sys,h,{ω,tr}]

performs the rotatingwave approximation on a Hamiltonian h for atomic system sys, assuming an optical field with angular frequency ω acting on transitions specified by tr.

RotatingWaveApproximation[sys,h,{{ω1,tr1},{ω2,tr2},...}]

performs the RWA assuming optical fields with angular frequencies ωi acting on transitions specified by tri.

RotatingWaveApproximation[sys,h,{{ω1,tr1,Δ1},{ω2,tr2,Δ2},...}]

writes optical frequencies in terms of their detunings from resonance Δi after performing the RWA.

RotatingWaveApproximation[sys,h,ω]

performs the RWA using an automatically generated transformation appropriate for a single optical field.

RotatingWaveApproximation[sys,h,{ω1,ω2,...},TransformMatrixu]

performs the RWA using an explicitly specified unitary transformation matrix.

Details and Options

• 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 StateLabel2&&F1. 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 ωi 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.
• RotatingWaveApproximation performs the rotating-wave approximation by applying a unitary transformation, such as obtained from RotatingWaveTransformMatrix, to the Hamiltonian, as done by EffectiveHamiltonian. Far-off-resonant terms are then dropped, as done by DropFastTerms.
• 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 {ω1,tr1,Δ1}, amounts to making the substitution ω1Energy[upper]-Energy[lower]+Δ1 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,ω] is called with , the transform matrix is generated by assuming that states with Energy0 are lower states, and all other states are upper states, coupled to the lower states by a field of frequency ω.

Examples

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Basic Examples(2)

Define a two-level system:

Find the Hamiltonian with states coupled by an optical field with angular frequency ω:

Apply the rotating-wave approximation, specifying that the field of frequency ω couples the lower state with label 1 to the upper state with label 2:

Specifying the detuning Δ is equivalent to making the replacement ωEnergy[2]+Δ:

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:

Here is an example with three levels and two fields.

Define a three level system:

Write the Hamiltonian with two optical fields, one acting only on the ab transition, and acting only on the bc transition:

Here we apply the RWA assuming that Energy[a]<Energy[b]<Energy[c]:

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 0Energy[c]-ω1-ω2, or ω1+ω2Energy[c]. This corresponds to the level diagram for the system:

If, instead, we assume that Energy[a]<Energy[c]<Energy[b], we obtain the rotating-frame Hamiltonian:

Here, the two-photon resonance condition is ω1-ω2Energy[c], corresponding to the level diagram:

Scope(5)

A system where not all of the optical frequencies can be eliminated from the Hamiltonian.

Define a system with four levels coupled by four optical fields. We assume that each field interacts with only one transition additional coupling terms are set to zero:

Plot the level diagram:

Looking at the level diagram, we see that the "closed loop" of optical transitions means that, in general, not all of the optical frequencies can be eliminated from the Hamiltonian through a unitary transform. The position of one of the states is arbitrary, and after the remaining three are shifted to eliminate optical frequencies, there will be one uneliminated frequency left over, unless the optical frequencies are such that the four-photon transition frequency happens to be zero.

Apply the RWA, eliminating optical frequencies in the order specified. The last frequency, ω4, acting on the 34 transition, is not be eliminated, but it is shifted to become -ω1+ω2-ω3+ω4:

If the four-photon resonance condition -ω1+ω2-ω3+ω40 is satisfied, all time dependence is eliminated from the rotating-frame Hamiltonian:

Specifying the frequencies and transitions in a different order results in a different rotating-frame Hamiltonian, in which the left-over time dependence is found in different coupling terms:

The result can also be written in terms of the corresponding detunings from the one-photon optical resonances:

A system in which a single field acts on multiple transitions.

Define a three level system:

Define the Hamiltonian, assuming an optical field of amplitude E0 and frequency ω. The field interacts with both the 12 and 23 transitions:

Apply the RWA assuming that Energy[1]<Energy[2]<Energy[3], and specifying that the field acts on both transitions:

A system with two optical fields acting on the same transition.

Define a two-level system:

The Hamiltonian for the system subject to two optical fields with frequencies ω1 and ω2, respectively:

The rotating-wave approximation shifts all of the optical frequencies by the same amount. To facilitate this, we write ω1 and ω2 in terms of a reference frequency ω0:

Now we do the RWA, removing the oscillation at ω0 from the system:

One possibility is to set ω0 equal to the transition frequency Energy[2]. This results in the Hamiltonian in what is called the interaction frame:

Or it might be more convenient to set ω0=ω1, removing one of the light frequencies (ω1) from the system, and leaving the oscillation at the difference frequency Δ=ω2-ω1:

A system with Zeeman and hyperfine structure.

Define the atomic system, with and upper and lower states having :

Find the Hamiltonian with states coupled by a circularly polarized optical field with angular frequency ω:

Some of the matrix elements in the Hamiltonian represent resonant couplings, and others off resonant couplings. The LevelDiagram suppresses the off-resonant couplings by default:

Applying the rotating-wave approximation explicitly removes the off-resonant couplings from the Hamiltonian:

Using ν-units for frequency.

Define a two-level system:

Find the Hamiltonian and apply the RWA with states coupled by an optical field with frequency ν in cycles per second. OpticalField and RotatingWaveApproximation take angular frequencies as inputs these can be specified as 2π ν:

Options(3)

Method(1)

With , RotatingWaveApproximation shifts the energy of the upper state to eliminate the optical frequency:

Method->"ShiftLowerState" can be used to instead shift the energy of the lower state:

Method->"KeepFastTerms" can be used to make the transformation to the rotating frame without dropping the fast-oscillating terms:

TimeVariable(1)

Write the Hamiltonian and perform the RWA using a different time variable:

TransformMatrix(1)

Manually specify the transformation matrix.

Define a three level system:

Write the Hamiltonian with two optical fields, one acting only on the ab transition, and acting only on the bc transition:

We have defined the energy of state b to be zero. Here we will assume that the energies of states a and c are negative, and we wish to perform the RWA by shifting the lower states up. We manually define a unitary transformation u to accomplish this:

Perform the RWA using our predefined matrix u:

If we instead perform the RWA by specifying the transition for each optical frequency, we get a slightly different result, because by default the transformation is defined to shift the upper states down:

This effective Hamiltonian differs only by an overall energy shift from the one that we obtained with the custom transformation matrix:

We can obtain the same result by specifying Method"ShiftLowerState":

Properties & Relations(1)

Define a system with Zeeman structure:

Find the Hamiltonian assuming a single optical field of frequency ω:

Plot the level diagram.

Perform the RWA directly using RotatingWaveApproximation:

The unitary transformation used is returned by RotatingWaveTransformMatrix:

This matrix can be supplied to the function EffectiveHamiltonian to transform the Hamiltonian to the rotating frame without dropping the fast-oscillating terms:

This can also be accomplished in one step using RotatingWaveApproximation with the option Method->"KeepFastTerms":

Applying DropFastTerms to the transformed matrix results in the same Hamiltonian returned by the initial use of RotatingWaveApproximation above: