Rotating-wave approximation: Difference between revisions

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== Mathematical formulation ==

For simplicity consider a [[two-state quantum system|two-level atomic system]] with [[excitedground state|excitedground]] and ground[[excited state|excited]] states <math>|\text{eg}\rangle</math> and <math>|\text{ge}\rangle</math>, respectively (using the [[bra-ket notation|Dirac bracket notation]]). Let the energy difference between the states be <math>\hbar\omega_0</math> so that <math>\omega_0</math> is the transition frequency of the system. Then the unperturbed [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the atom can be written as

: <math>H_0=\hbar\omega_0|\text{e}\rangle\langle\text{e}|</math>.

Suppose the atom is placed at <math>z=0</math> inexperiences an external (classical) [[electric field]] of frequency <math>\omega_L</math>, given by
<math>\vec{E}(z,t) = \vec{E}_0(z) e^{-i\omega_Lt} +\vec{E}_0^*(z) e^{i\omega_Lt}</math>,
e.g. (soa that[[plane thewave]] field contains both positive- and negative-frequency modespropagating in general)space. Then under the [[dipole approximation]] the [[interaction Hamiltonian]] between the atom and the electric field can be expressed as

: <math>H_IH_1=-\vec{d}\cdot\vec{E}</math>,

where <math>\vec{d}</math> is the [[transition dipole moment|dipole moment operator]] of the atom. The total Hamiltonian for the atom-light system is therefore <math>H=H_0+H_IH_1.</math> The atom does not have a dipole moment when it is in an [[energy eigenstate]], so <math>\langle\text{e}|\vec{d}|\text{e}\rangle=\langle\text{g}|\vec{d}|\text{g}\rangle=0.</math> This means that defining <math>\vec{d}_{\text{eg}}:=\langle\text{e}|\vec{d}|\text{g}\rangle</math> allows the dipole operator to be written as

: <math>\vec{d}=\vec{d}_{\text{eg}}|\text{e}\rangle\langle\text{g}|+\vec{d}_{\text{eg}}^*|\text{g}\rangle\langle\text{e}|</math>

(with `<math>^*</math>' denoting the [[Hermitiancomplex conjugate]]). The interaction Hamiltonian can then be shown to be (see the DerivationsDerivation section below)

: <math>H_IH_1 = -\hbar\left(\Omega e^{-i\omega_Lt}+\tilde{\Omega}e^{i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|

-\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|</math>

where <math>\Omega=\hbar^{-1}\vec{d}_\text{eg}\cdot\vec{E}_0</math> is the [[Rabi frequency]] and <math>\tilde{\Omega}:=\hbar^{-1}\vec{d}_\text{eg}\cdot\vec{E}_0^*</math> is the counter-rotating frequency. To see why the <math>\tilde{\Omega}</math> terms are called `counter-rotating' consider a [[unitary transformation]] to the [[Interaction picture|interaction or Dirac picture]] where the transformed Hamiltonian <math>\barH_{H1,I}</math> is given by

: <math>\barH_{H1,I}=-\hbar\left(\Omega e^{-i\Delta t}+\tilde{\Omega}e^{i(\omega_L+\omega_0)t}\right)|\text{e}\rangle\langle\text{g}|

-\hbar\left(\tilde{\Omega}^*e^{-i(\omega_L+\omega_0)t}+\Omega^*e^{i\Delta t}\right)|\text{g}\rangle\langle\text{e}|,</math>

where <math>\Delta:=\omega_L-\omega_0</math> is the detuning ofbetween the light field and the atom.

=== 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]] with the atomic transition. This means that <math>\Delta\ll\omega_L+\omega_0</math> and the complex exponentials multiplying <math>\tilde{\Omega}</math> and <math>\tilde{\Omega}^*</math> 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 aremay negligiblebe neglected and thus the Hamiltonian can be written in the interaction picture as

: <math>\bar{H}_\text{RWA}=-\hbar\Omega e^{-i\Delta t}|\text{e}\rangle\langle\text{g}|

: <math>\barH_{H1,I}_^{\text{RWA}}=-\hbar\Omega e^{-i\Delta t}|\text{e}\rangle\langle\text{g}|

-\hbar\Omega^*e^{i\Delta t}|\text{g}\rangle\langle\text{e}|.</math>

Finally, intransforming back into the [[Schrödinger picture]], the Hamiltonian is given by

: <math>

H_H^\text{RWA}=\hbar\omega_0|\text{e}\rangle\langle\text{e}|

H_\text{RWA}=\hbar\omega_0|\text{e}\rangle\langle\text{e}|

-\hbar\Omega e^{-i\omega_Lt}|\text{e}\rangle\langle\text{g}|

-\hbar\Omega^*e^{i\omega_Lt}|\text{g}\rangle\langle\text{e}|.