Rotating wave approximation - AbsoluteAstronomy.com

The rotating wave approximation is an approximation used in atom optics

Atom optics

Atom optics is the area of physics which deals with beams of cold, slowly moving neutral atoms, as a special case of a particle beam....

and magnetic resonance

Magnetic resonance

Magnetic resonance can mean:*Nuclear magnetic resonance*Electron spin resonance*Magnetic resonance imaging *Functional magnetic resonance imaging *Muon spin spectroscopy...

. In this approximation, terms in a Hamiltonian

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

which oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic resonance, and the intensity is low. Explicitly, terms in the Hamiltonians which oscillate with frequencies

are neglected, while terms which oscillate with frequencies

are kept, where

is the light frequency and

is a transition frequency.

The name of the approximation stems from the form of the Hamiltonian in the interaction picture

Interaction picture

In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...

, as shown below. By switching to this picture the evolution of an atom due to the corresponding atomic Hamiltonian is absorbed into the system ket

Bra-ket notation

Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

, leaving only the evolution due to the interaction of the atom with the light field to consider. It is in this picture that the rapidly-oscillating terms mentioned previously can be neglected. Since in some sense the interaction picture can be thought of as rotating with the system ket only that part of the electromagnetic wave that approximately co-rotates is kept; the counter-rotating component is discarded.

Mathematical formulation

For simplicity consider a two-level atomic system with excited

Excited state

Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state....

and ground states

and

respectively (using the Dirac bracket notation

Bra-ket notation

). Let the energy difference between the states be

so that

is the transition frequency of the system. Then the unperturbed Hamiltonian

Hamiltonian (quantum mechanics)

of the atom can be written as

Suppose the atom is placed at

in an external (classical) electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

of frequency

, given by

(so that the field contains both positive- and negative-frequency modes in general). Then under the dipole approximation the interaction Hamiltonian can be expressed as

where

is the dipole moment operator of the atom. The total Hamiltonian for the atom-light system is therefore

The atom does not have a dipole moment when it is in an energy eigenstate, so

This means that defining

allows the dipole operator to be written as

(with `

' denoting the Hermitian conjugate). The interaction Hamiltonian can then be shown to be (see the Derivations section below)

where

is the Rabi frequency

Rabi frequency

The Rabi frequency is the frequency of population oscillation for a given atomic transition in a given light field. It is associated with the strength of the coupling between the light and the transition. Rabi flopping between the levels of a 2-level system illuminated with resonant light, will...

and

is the counter-rotating frequency. To see why the

terms are called `counter-rotating' consider a unitary transformation

Unitary transformation

In mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation....

to the interaction or Dirac picture

Interaction picture

where the transformed Hamiltonian

is given by

where

is the detuning of the light field.

Making the approximation

This is the point at which the rotating wave approximation is made. The dipole approximation has been assumed, and for this to remain valid the electric field must be near resonance

Resonance

In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

with the atomic transition. This means that

and the complex exponentials multiplying

and

can be considered to be rapidly oscillating. Hence on any appreciable time scale the oscillations will quickly average to 0. The rotating wave approximation is thus the claim that these terms are negligible and the Hamiltonian can be written in the interaction picture as

Finally, in the Schrödinger picture

Schrödinger picture

In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...

the Hamiltonian is given by

At this point the rotating wave approximation is complete. A common first step beyond this is to remove the remaining time dependence in the Hamiltonian via another unitary transformation.

Derivations

Given the above definitions the interaction Hamiltonian is

as stated. The next stage is to find the Hamiltonian in the interaction picture,

The unitary operator required for the transformation is

and an arbitrary state

transforms to

The Schrödinger equation

Schrödinger equation

The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....

must still hold in this new picture, so

where a dot denotes the time derivative

Time derivative

A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t\,.-Notation:...

. This shows that the new Hamiltonian is given by

Using a Taylor series expansion of the exponential,

and operating on

Operating from the left on the second term of

above yields zero by orthogonality of

and

, and the same results apply to the operation of the second exponential from the right. Thus, the new Hamiltonian becomes

The penultimate equality can be easily seen from the series expansion

Series expansion

In mathematics, a series expansion is a method for calculating a function that cannot be expressed by just elementary operators . The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function...

of the exponential map and the fact that

for i and j each equal to e or g (and

the Kronecker delta).

The final step is to transform the approximate Hamiltonian back to the Schrödinger picture. The first line of the previous calculation shows that

, so in the same manner as the last calculation,

The atomic Hamiltonian was unaffected by the approximation, so the total Hamiltonian in the Schrödinger picture under the rotating wave approximation is

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