17.3.3_Behavior_of_the_density_matrix_in_time

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1、17.3ThedensitymatrixandopticalabsorptionSlides:Video17.3.3BehaviorofthedensitymatrixintimeTextreference:QuantumMechanicsforScientistsandEngineersSection14.5fromEq.14.35throughtheparagraphafterEq.14.40ThedensitymatrixandopticalabsorptionBehaviorofthedensitymatrixintimeQuant

2、ummechanicsforscientistsandengineersDavidMillerBehaviorofthedensitymatrixintimeWehave,from//,tiwiththedefinitionsHˆ1112ˆˆˆE1-EdandHHHop-EE2122d2diHˆˆHdtiEE-E-E111211dd1112-EEE-E2122dd222

3、122iEEdd12211122EE2112EEdd2211EE12212112BehaviorofthedensitymatrixintimeTakingthe“2–1”elementofbothsidesindiEEdd12211122EE2112dtEEdd2211EE12212112withEEgives2121di21d

4、1122EEdEE2121i2121i1122dtBehaviorofthedensitymatrixintimeFromthediagonalelementsindiEEdd12211122EE2112dtEEdd2211EE12212112wecanexaminethepopulationdifference–1122betweenthelowerandupperstatesUsingtheHerm

5、iticityofwhichtellsusthat1221ddwehave2iE11222121dtBehaviorofthedensitymatrixintimedSolving21iid2121E1122dtddand11222iE2121dtcoversanypossiblebehaviorofthisidealizedsystemNote:thisisnotaperturbationtheoryanalysisDensitymatrixand

6、relaxationtimesConsiderafractionalpopulationdifference1122betweenthe“lower”and“upper”statesSupposethat,inequilibrium,withnoappliedfieldsthisdifferencewouldhaveavalue()1122oThenexperiencemighttellusthatbecauseofmechanismssuchascollisionswiththewallsofaboxorwithotherat

7、omsorbyspontaneousemissionsuchsystemsoftensettlebackdownagainto()1122owithanexponentialdecaywithsometimeconstantT1DensitymatrixandrelaxationtimesThenwecouldhypothesizethatwecouldaddatermtodd11222iE2121dttogived11221122do11222iE2121

8、dtT1ForE=0,thisexpressionwouldgiveexponentialdecaybackto11221122owithtimecons