Rotating-wave approximation: Difference between revisions

Revision as of 19:53, 18 February 2014

Revision as of 11:03, 15 March 2014

Line 3:

The '''rotating wave approximation''' is an approximation used in [[atom optics]] and [[Nuclear magnetic resonance|magnetic resonance]]. In this approximation, terms in a [[Hamiltonian (quantum mechanics)|Hamiltonian]] which oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic transition, and the intensity is low.<ref name="WuYang2007">{{cite journal|last1=Wu|first1=Ying|last2=Yang|first2=Xiaoxue|title=Strong-Coupling Theory of Periodically Driven Two-Level Systems|journal=Physical Review Letters|volume=98|issue=1|year=2007|issn=0031-9007|doi=10.1103/PhysRevLett.98.013601}}</ref> Explicitly, terms in the Hamiltonians which oscillate with frequencies <math>\omega_L+\omega_0 </math> are neglected, while terms which oscillate with frequencies <math>\omega_L-\omega_0 </math> are kept, where <math> \omega_L </math> is the light frequency and <math> \omega_0</math> 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 [[bra-ketbra–ket notation|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-state quantum system|two-level atomic system]] with [[ground state|ground]] and [[excited state|excited]] states <math>|\text{g}\rangle</math> and <math>|\text{e}\rangle</math>, respectively (using the [[bra-ketbra–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>.