Rotating Wave Approximation - Mathematical Formulation

Mathematical Formulation

For simplicity consider a two-level atomic system with ground and excited states and, respectively (using the Dirac bracket notation). Let the energy difference between the states be so that is the transition frequency of the system. Then the unperturbed Hamiltonian of the atom can be written as

.

Suppose the atom experiences an external classical electric field of frequency, given by, e.g. a plane wave propagating in space. Then under the dipole approximation the interaction Hamiltonian between the atom and the electric field 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 complex conjugate). The interaction Hamiltonian can then be shown to be (see the Derivation section below)

H_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}|

where is the Rabi frequency and is the counter-rotating frequency. To see why the terms are called `counter-rotating' consider a unitary transformation to the interaction or Dirac picture where the transformed Hamiltonian is given by

H_{1,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}|,

where is the detuning between the light field and the atom.

Read more about this topic:  Rotating Wave Approximation

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