Interaction of 3-level atom with radiation

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Interactionof3–levelatomwithradiation
L.Accardi,K.Imafuku,S.V.Kozyrev
February1,2008
Abstract
Theinteractionof3–levelsystemwithaquantumfieldinanon–equilibriumstateisconsidered.Wedescribeaclassofstatesofthequantumfieldforwichastationarystatedrivesthesystemtoinversepopulatedstate.WefindthatthequotientofthepopulationoftheenergylevelsinthesimplestcaseisdescribedbythedoubleEinsteinformulawhichinvolvesproductsoftwoEinsteinemission/absorptionrelations.Emissionandabsorptionofradiationby3–levelatominnon–equlibriumstationarystateisdescribed.
arXiv:quant-ph/0104118v124Apr2001
1Introduction
Inthepresentpaperweconsidera3–levelsystem(forexampleanatominteractingwithradiation.Inthework[1]weapplythetechniquedevelopedheretodescribestimulatedemissionfor3–levelatominteractingwithradiationinanon–equilibriumstate.Inthework[2]a2–levelquantumsysteminteractingwithquantumfieldwasconsidered.In[3]astationarynonequilibriumstateforann–levelsysteminteractingwithanonequilibriumquantumfieldwasobtained.
Weshowthatforsuchasystemapplicationofthestochasticlimitallowstoobtainaninterestingeffectofinversionofpopulation:foraspecialchoiceofthestateofreservoirthesystemrelaxestostationarystatewherethepopulationofthelevelofthesystemwithhigherenergywillbelargerthanthepopulationofthelevelwithlowerenergy.
Weobtainanequationforthenumberofphotonsemittedandabsorbedbythesystem.
Weinvestigatetheexamplesof2–leveland3–levelsystems.Weshowthatfora2–levelatomtheemissioninthestationaryregimeequalstoabsorption.Fora3–levelsystememissionandabsorptionofradiationarecontrolledbythestateofthefield.Wefindthatfora3–levelsysteminastationarynonequilibriumstatetworegimesarepossible:theemissionandtheabsorptionregime.Intheemissionregimethetotalnumberofquantainthesystemincreases,andintheabsorptionregimeitdecreases.Theseregimesarecontrolledbythefunctionβ(ωin(3.Forexamplea3–levelsystemwithenergylevelsε1<ε2<ε3isinemissionregimewhen
β(ε2ε1+β(ε3ε2>β(ε3ε1
andisinabsorptionregimewhentheoppositeinequalityholds.
Intheemissionregimethe3–levelsystemconvertsradiationwiththefrequencyω2intoradi-ationwiththefrequenciesω1andω3andviceversaintheabsorptionregime.
Thismeansthatthisstationarystateofthe3–levelsystemgivesanexampleofdissipativestructureinthePrigoginesense[5].
1

Foranequilibriumstateofthefield,i.e.whenthefunctionβ(ωislinear,thesystemisinequilibriumwithradiation.
TheinteractionofaquantumsystemwithaquantumfieldisdescribedbyanHamiltonian
H=HS+HR+λHI
ThesystemdegreesoffreedomaredescribedbythesystemHamiltonianHS.TheradiationdegreesoffreedomaredescribedbytheHamiltonian
HR=

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