近簡(jiǎn)并V_型三能級(jí)系統(tǒng)的自由感應(yīng)衰減現(xiàn)象_英文_

1、Optical free induction decay in an almostdegenerate V -Type Three -Level system HUANG Ting,ZHANG Chun -fa,LIAO Yan -lin,CAO Zhuo -liang(Departmen t of Physics,Anhui Universi ty,Hefei 230039,ChinaAbstract:A op tical free induction decay is studied semiclassically in the V-type three-level sys -tem wi

2、 th two upper levels being almost degenerate in Bloch form.It is discussed that the systematic relax -ation has effect on the signal field of the free-induction decay.The decay behavior has three contribu -tions:a homogeneous part wi th transverse relax ati on ti me and two inhomogeneous parts with

3、different timeconstants that reflect the velocity bandwid th excited.Key words:op tical free-induction decay;bloch equation;V-type three-level systemCLC number:O434.14 Document code:A Article ID:1000-2162(200201-0046-050 IntroductionIn the past years,much interest has been focused on the quantum-bea

4、t three-level atomic sys -tem 1-5.This system has novel properties 1,2:it can lase without inversion,invert without lasing.Pulse propagation in a resonant medium of a two -level syste m can yield many interesting phenome -na 6,7.Most of them origina te from the transient response of the medium to th

5、e coherent pulsed e xcita -tions.Transient coherent effects are the most fascinating subjects of resonant interaction of light wave with matter.It has been shown in recent papers that the self-induced transparency phenomenon 3,optical nutation and the optical free-induction decay 4occur in the +-typ

6、e degenerate quantum-beat system driven by a coherent micro wave field.In the present paper,semiclassically using the cou -pled Schrodinger-Max well equations,we shall treat optical free induction deca y,a coherent transient effect in V-type quantum-beat three-level syste m with two upper levels bei

7、ng almost degenera te.Our procedure is to first solve the Schrodinger equation,in Bloch form for an atomic (or molecularsystem of a gas subject to a pulse or continuous wave coherent optical field.This preparative stage gen -erates a sample polarization that can radiate coherent light even after the

8、 preparation has passed.The final polarization must be averaged over Doppler velocity distribution.Received data:2001-01-11Foundation item:Project(99022019supported by Key Foundation of Anhui Province,project by the Natural Sc-i ence Foundation of Anhui Province and project (98J L010by the Education

9、 Commission Foundation of Anhui Province,China.Biography:HUANG Ting(1978-,female,now is studying for M aster program in Anhui Universi ty.Her current re -search interest includes quan tum optics,nonlinear optics and quan tum in formation.2002年3月1 The model and optical Bloch equationThe syste m under

10、 consideration is a molecular (atomicsystem with one lower level and two upperlevels being almost degenerate with initial populations Q 033=0,Q 022=0,Q 011=1as shown in Fig.1.The molecular (atomicgas sample encounters a laser fieldthree-level system E x (z ,t= e x E 0cos (8t -kz (1of frequency 8and

11、a mplitude E 0,polarized along thex-a xis and propa gating in the z-direction.The Hamilton-ian of the system is given by (set Ü=0H =-X |3431|+D (|3433|-2432|+H I (2ahere,E 3=D ,E 2=-D ,and E 1=-X are the energies ofthe system,respectively.The molecular-optical field inter -ac tion term8H I =- L

12、 # E x (z ,t(2bhas the electric -dipole matrix elements,31|L x |34=L 13,31|L x |24=L 12,32|L x |34=0.Since we haveassumed that levels|24and |34are almost degenerate,it is reasonable 5to neglect second term with the other terms in (2.The time -dependent behavior of the density matrix is given by the

13、Schrodinger equation of motioni ÛQ =H ,Q +relaxation terms(3For this three-level problem,equation (3becomesÛQ 33=i V 13(Q 13-Q 31cos (8t -kz -Q 33-Q 033S,ÛQ 22=i V 12(Q 12-Q 21cos (8t -kz -Q 22-Q 022S ,ÛQ 13=i V 13(Q 33-Q 11cos (8t -kz +i X -1T 1Q 13+i V 12Q 23cos (8t -kz ,Û

14、Q 12=i V 12(Q 22-Q 11cos (8t -kz +i X -1T 1Q 12+i V 13Q32cos (8t -kz ,ÛQ 23=i(V 12Q 13-V 13Q 21cos (8t -kz -Q 23T 2.(4where we introduced the phenomenological transverse rela xation times T 1,T 2,for the off-diagonal e-l ements Q 13(Q 12,Q 23and longitudinal relaxation time S for Q 33(Q 22.V 13

15、=L 13E 0and V 12=L 12E 0are two Rabi flopping frequencies.Using the variablesQ 13= Q 13e i(8t-kz=12(u 13+iv 13e i (8t-k z ,Q 12= Q 12e i (8t -kz=12(u 12+iv 12e i(8t-kz,Q 23=u 23+iv 23,w 31=Q 33-Q 11,w 21=Q 22-Q 11,w 031=Q 033-Q 011,w 021=Q 022-Q 011(547第1期HUANG Ti ng,et al.:Optical free i nducti on

16、decay i n an al most degenerate V-Type Three-Level sys te min the rotating-wave approximation,we may write equations(4as the Bloch equationsÛu13=-$v13-V12v23-u13T1,Ûv13=$u13+V13w31+V12u23-v13T1,Ûu12=-$v12-V13v23-u12T1,Ûv12=$u12+V12w21+V13u23-v12T1,Ûu23=-14(V12v13+V13v12-u23T

17、2,Ûv23=14(V12u13-V13u12-v12T2,Ûw31=-V13v13-12V12v12-w31-w31S,Ûw21=-12V13v13-V12v12-w21-w21S.(6where$=X-8-kv z,kv z is the Doppler shift.Note that at resonance$=0.Basing on the equa-tions(6,we will study the optical free induction decay in the V-type quantum-beat three-level system wit

18、h two upper levels being almost degenerate in follo wing.2Free induction decayThe light waveE x(z,t,induces in a sample of molec ular density N the polarization.This po-larization,in turn,is the origin of a signal fieldE s(z,t= e xE s(z,te i(8t-kz+ c.c(7 which can be calculated from Maxwell.s equati

19、ons9E s9z=-2P ik N3L13 Q13+L12 Q124=-P ikN(3L13u13+L23u124+i3L13v13+L12v124(8 The bracket34denotes an average over the Maxwellian molecular velocity distribution3L13u13+L12u124=1ku PQ+-e-$ku2(L13u13+L1u12d$,3L13v13+L12v124=1ku PQ+-e-$ku2(L13v13+L1v12d$.(9 u is the rms velocity.The molecular sample i

20、s resonantly e xcited under steady state conditions by a laser beam and sud-denly the e xcitation is terminated.Under these conditions,the sa mple will radiate an intense coherent bea m of light kno wn as optical free induction decay.The preparative stage ends by switching the molecular transition f

21、requency out resonance with the laser frequency.The preparative steady solutions for the period t0can be derived from the Bloch equation(6 by setting the time L13u13(0+ L13v13(0+1048安徽大學(xué)學(xué)報(bào)(自然科學(xué)版第26卷wherea=1+T1SV22-T21T2T V432,b=14S+T2T1+T2T V282+S-T2T1#3S+T2T1-T2(2S-T2V24J,J=(L213-L2122L4,T=S-J(t-T2

22、,V=L E0,L=L213+L212.At time t=0,we assume that the molecular transition frequency has been shifted by D X so that$y $c=$+D X and the sample is sufficiently far out of resonance with the laser light that we may set V13=V12=0.For t>0,the solutions of the Bloch equations(6are given byL13u13(t+L12u12

23、(t=(L13u13(0+L12u12(0cos$c t-(L13v13(0+L12v12(0sin$c t#e-t T 1 ,L13v13(t+L12v12(t=(L13u13(0+L12u12(0sin$c t+(L13v13(0+L12v12(0sin$c t#e-tT1.(11According to(8,the amplitude of the signal field that exits the sa mple of length L isE s(L,t=-P3NL LV2ue-$1ku21+ (4-T1T2V2(4+T1T V2-16a16bT21V21a-b V2T21-1e

24、-a-b V2T21+1tT1+1-(4-T1T2V2(4+T1T V2-16a16bT21V21a+b V2T21-1e-a+b V2T21+1t T1cos D X t(12For simplicity,here have we assumed that the excitation is centered near the Doppler peak at$=$1 and that the bandwidth excited is narro w compared to the Doppler width ku so that the Gaussian may be taken outsi

25、de the integral.The total field,including that of the laser,isE T E s+E02e i(8t-kz+ c.c.(13The intensity,therefore,c ontains a cross term or beat(E2beat=2E s E0=-P3NL LV2u e-$1ku21+ (4-T1T2V2(4+T1T V2-16a16b T21V21a-b V2T21-1e-a-b V2T21+1t T1+1-(4-T1T2V2(4+T1T V2-16a16bT21V21a+b V2T21-1e-a+b V2T21+1

26、t T1cos D X t(14The decay behavior of(14has three contributions:a homogeneous part with time c onstant T2and two inhomogeneous parts with time constants T1/49第1期HUANG Ti ng,et al.:Optical free i nducti on decay i n an al most degenerate V-Type Three-Level sys te m50安徽大學(xué)學(xué)報(bào)(自然科學(xué)版第26卷which reflect the

27、velocity bandwidth excited.At moderately high laser intensity,the inhomogeneous dephasing can be dominant,the free induction signal will then decay rapidly and appear modulated at a frequency D X.The free induction signal field is superposition of two parts with different decay times,because the int

28、eraction of the laser field with the three-level molecular system can be regarded as the interactions of the laser with two two-level syste ms.If two two-level syste ms are not coher-ent,the intensity of the free induction signal field is the superposition of two intensities without inter-ference.Ho

29、 wever the transitions between|34and|14and|24and|14are induced by the laser field. The transitions establish c oherence(known as quantum-beatbetween the two upper levels.The am-plitude of the total free induction signal field(12shall be superposition of the two amplitudes.This leads to interference.Therefore,the interference results from the coherence between the two upper