- Rotating wave approximation
The rotating wave approximation is an approximation used in
atom optics andmagnetic resonance . In this approximation, terms in a Hamiltonian 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 omega_L+omega_0 are neglected, while terms which oscillate with frequencies omega_L-omega_0 are kept, where omega_L is the light frequency and omega_0 is a transition frequency.The name of the approximation stems from the form of the Hamiltonian in the
interaction picture , 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, 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 and ground states ext{e} angle and ext{g} angle respectively (using the Dirac bracket notation). Let the energy difference between the states be hbaromega_0 so that omega_0 is the transition frequency of the system. Then the unperturbed
Hamiltonian of the atom can be written asH_0=hbaromega_0| ext{e} anglelangle ext{e}|
Suppose the atom is placed at z=0 in an external (classical)
electric field of frequency omega_L, given by vec{E}(z,t)=vec{E}_0(z)e^{-iomega_Lt}+vec{E}_0^*(z)e^{iomega_Lt} (so that the field contains both positive- and negative-frequency modes in general). Then under thedipole approximation theinteraction Hamiltonian can be expressed asH_I=-vec{d}cdotvec{E}
where vec{d} is the dipole moment operator of the atom. The total Hamiltonian for the atom-light system is therefore H=H_0+H_I. The atom does not have a dipole moment when it is in an
energy eigenstate , so langle ext{e}|vec{d}| ext{e} angle=langle ext{g}|vec{d}| ext{g} angle=0. This means that defining vec{d}_{ ext{eg:=langle ext{e}|vec{d}| ext{g} angle allows the dipole operator to be written asvec{d}=vec{d}_{ ext{eg| ext{e} anglelangle ext{g}|+vec{d}_{ ext{eg^*| ext{g} anglelangle ext{e}|
(with `' denoting the
Hermitian conjugate ). The interaction Hamiltonian can then be shown to be (see the Derivations section below)H_I=-hbarleft(Omega e^{-iomega_Lt}+ ilde{Omega}e^{iomega_Lt} ight)| ext{e} anglelangle ext{g}
-hbarleft( ilde{Omega}^*e^{-iomega_Lt}+Omega^*e^{iomega_Lt} ight)| ext{g} anglelangle ext{e}|where Omega is the
Rabi frequency and ilde{Omega}:=hbar^{-1}vec{d}_ ext{eg}cdotvec{E}_0^* is the counter-rotating frequency. To see why the ilde{Omega} terms are called `counter-rotating' consider aunitary transformation to the interaction or Dirac picture where the transformed Hamiltonian ar{H} is given byar{H}=-hbarleft(Omega e^{-iDelta t}+ ilde{Omega}e^{i(omega_L+omega_0)t} ight)| ext{e} anglelangle ext{g}
-hbarleft( ilde{Omega}^*e^{-i(omega_L+omega_0)t}+Omega^*e^{iDelta t} ight)| ext{g} anglelangle ext{e}|,where Delta:=omega_L-omega_0 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 with the atomic transition. This means that Deltallomega_L+omega_0 and the complex exponentials multiplying ilde{Omega} and ilde{Omega}^* 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 asar{H}_ ext{RWA}=-hbarOmega e^{-iDelta t}| ext{e} anglelangle ext{g}
-hbarOmega^*e^{iDelta t}| ext{g} anglelangle ext{e}|.Finally, in the
Schrödinger picture the Hamiltonian is given byH_ ext{RWA}=hbaromega_0| ext{e} anglelangle ext{e}
-hbarOmega e^{-iomega_Lt}| ext{e} anglelangle ext{g}
-hbarOmega^*e^{iomega_Lt}| ext{g} anglelangle ext{e}|.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
egin{align}H_I &= -vec{d}cdotvec{E} \&=-left(vec{d}_ ext{eg}| ext{e} anglelangle ext{g}|+vec{d}_ ext{eg}^*| ext{g} anglelangle ext{e}| ight) cdotleft(vec{E}_0e^{-iomega_Lt}+vec{E}_0^*e^{iomega_Lt} ight) \&=-left(vec{d}_ ext{eg}cdotvec{E}_0e^{-iomega_Lt} +vec{d}_ ext{eg}cdotvec{E}_0^*e^{iomega_Lt} ight)| ext{e} anglelangle ext{g}
-left(vec{d}_ ext{eg}^*cdotvec{E}_0e^{-iomega_Lt} +vec{d}_ ext{eg}^*cdotvec{E}_0^*e^{iomega_Lt} ight)| ext{g} anglelangle ext{e}| \&=-hbarleft(Omega e^{-iomega_Lt}+ ilde{Omega}e^{iomega_Lt} ight)| ext{e} anglelangle ext{g}
-hbarleft( ilde{Omega}^*e^{-iomega_Lt}+Omega^*e^{iomega_Lt} ight)| ext{g} anglelangle ext{e}|,end{align}as stated. The next stage is to find the Hamiltonian in the interaction picture, ar{H}. The unitary operator required for the transformation isU=e^{iH_0t/hbar},and an arbitrary state psi angle transforms to ar{psi} angle=U|psi angle. The
Schrödinger equation must still hold in this new picture, soar{H}|ar{psi} angle=ihbarpartial_t|ar{psi} angle=ihbardot{U}|psi angle+Uihbarpartial_t|psi angle=left(ihbardot{U}+UH ight)|psi angle=left(ihbardot{U}U^dagger+UHU^dagger ight)|ar{psi} angle,
where a dot denotes the
time derivative . This shows that the new Hamiltonian is given byegin{align}ar{H}&=ihbardot{U}U^dagger+UHU^dagger=ihbarleft(frac{i}{hbar}UH_0 ight)U^dagger+U(H_0+H_I)U^dagger=UH_IU^dagger \&=-e^{iomega_gt| ext{g} anglelangle ext{g}| + iomega_et| ext{e} anglelangle ext{e}left( hbarleft(Omega e^{-iomega_Lt}+ ilde{Omega}e^{iomega_Lt} ight)| ext{e} anglelangle ext{g}
+hbarleft( ilde{Omega}^*e^{-iomega_Lt}+Omega^*e^{iomega_Lt} ight)| ext{g} anglelangle ext{e}| ight) e^{-iomega_gt| ext{g} anglelangle ext{g}| - iomega_et| ext{e} anglelangle ext{e}\&=-e^{i,0,t| ext{g} anglelangle ext{g}| + iomega_0t| ext{e} anglelangle ext{e}left( hbarleft(Omega e^{-iomega_Lt}+ ilde{Omega}e^{iomega_Lt} ight)| ext{e} anglelangle ext{g}
+hbarleft( ilde{Omega}^*e^{-iomega_Lt}+Omega^*e^{iomega_Lt} ight)| ext{g} anglelangle ext{e}| ight) e^{-i,0,t| ext{g} anglelangle ext{g}| - iomega_0t| ext{e} anglelangle ext{e}\&=-e^{iomega_0t| ext{e} anglelangle ext{e}left( hbarleft(Omega e^{-iomega_Lt}+ ilde{Omega}e^{iomega_Lt} ight)| ext{e} anglelangle ext{g}
+hbarleft( ilde{Omega}^*e^{-iomega_Lt}+Omega^*e^{iomega_Lt} ight)| ext{g} anglelangle ext{e}| ight) e^{-iomega_0t| ext{e} anglelangle ext{e}\end{align}Using a Taylor series expansion of the exponential, egin{align}e^{iomega_0t| ext{e} anglelangle ext{e} = 1 + iomega_0t| ext{e} anglelangle ext{e}| + ldotsend{align}
and operating on egin{align}| ext{e} anglelangle ext{g}| ,end{align}
egin{align}e^{iomega_0t| ext{e} anglelangle ext{e}| ext{e} anglelangle ext{g}| &= (1 + iomega_0t| ext{e} anglelangle ext{e}| + ldots)| ext{e} anglelangle ext{g}| \&= | ext{e} anglelangle ext{g}| + iomega_0t| ext{e} anglelangle ext{g}| + ldots \&= (1 + iomega_0t + ldots)| ext{e} anglelangle ext{g}| \&= e^{iomega_0t}| ext{e} anglelangle ext{g}| . \end{align}
Operating from the left on the second term of H above yields zero by orthogonality of ext{g} angle and ext{e} angle, and the same results apply to the operation of the second exponential from the right. Thus, the new Hamiltonian becomes
egin{align}ar{H}&=-hbarleft(Omega e^{-iomega_Lt}+ ilde{Omega}e^{iomega_Lt} ight)e^{iomega_0t}| ext{e} anglelangle ext{g}
-hbarleft( ilde{Omega}^*e^{-iomega_Lt}+Omega^*e^{iomega_Lt} ight)| ext{g} anglelangle ext{e}|e^{-iomega_0t} \&=-hbarleft(Omega e^{-iDelta t}+ ilde{Omega}e^{i(omega_L+omega_0)t} ight)| ext{e} anglelangle ext{g}
-hbarleft( ilde{Omega}^*e^{-i(omega_L+omega_0)t}+Omega^*e^{iDelta t} ight)| ext{g} anglelangle ext{e}| .end{align}The penultimate equality can be easily seen from the
series expansion of the exponential map and the fact thatlangle ext{i}| ext{j} angle=delta_ ext{ij} for i and j each equal to e or g (and delta_ ext{ij} theKronecker delta ).The final step is to transform the approximate Hamiltonian back to the Schrödinger picture. The first line of the previous calculation shows thatar{H}=UH_IU^dagger, so in the same manner as the last calculation,
egin{align}H_{I, ext{RWA&=U^daggerar{H}_ ext{RWA}U=e^{-iomega_0t| ext{e} anglelangle ext{e} left(-hbarOmega e^{-iDelta t}| ext{e} anglelangle ext{g}
-hbarOmega^*e^{iDelta t}| ext{g} anglelangle ext{e}| ight) e^{iomega_0t| ext{e} anglelangle ext{e} \&=-hbarOmega e^{-iDelta t}e^{-iomega_0t}| ext{e} anglelangle ext{g}
-hbarOmega^*e^{iDelta t}| ext{g} anglelangle ext{e}|e^{iomega_0t} \&=-hbarOmega e^{-iomega_Lt}| ext{e} anglelangle ext{g}
-hbarOmega^*e^{iomega_Lt}| ext{g} anglelangle ext{e}|.end{align}The atomic Hamiltonian was unaffected by the approximation, so the total Hamiltonian in the Schrödinger picture under the rotating wave approximation is
H_ ext{RWA}=H_0+H_{I, ext{RWA = hbaromega_0| ext{e} anglelangle ext{e}
-hbarOmega e^{-iomega_Lt}| ext{e} anglelangle ext{g}
-hbarOmega^*e^{iomega_Lt}| ext{g} anglelangle ext{e}|.
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