Atoms in a Classical Light Field

Transcription

1 Chapter 2 Atoms i a Classical Light Field Semicoductors like all crystals are periodic arrays of oe or more types of atoms. A prototype of a semicoductor is a lattice of group IV atoms, e.g. Si or Ge, which have four electros i the outer electroic shell. These electros participate i the covalet bidig of a give atom to its four earest eighbors which sit i the corers of a tetrahedro aroud the give atom. The bodig states form the valece bads which are separated by a eergy gap from the eergetically ext higher states formig the coductio bad. I order to uderstad the similarities ad the differeces betwee optical trasitios i a semicoductor ad i a atom, we will first give a elemetary treatmet of the optical trasitios i a atom. This chapter also serves to illustrate the differece betwee a quatum mechaical derivatio of the polarizatio ad the classical theory of Chap Atomic Optical Susceptibility The statioary Schrödiger equatio of a sigle electro i a atom is H 0 ψ (r) = ɛ ψ (r), (2.1) where ɛ ad ψ are the eergy eigevalues ad the correspodig eigefuctios, respectively. For simplicity, we discuss the example of the hydroge atom which has oly a sigle electro. The Hamiltoia H 0 is the give by the sum of the kietic eergy operator ad the Coulomb potetial i the form H 0 = 2 2 2m 0 e2 r. (2.2) 17

2 18 Quatum Theory of the Optical ad Electroic Properties of Semicoductors A optical field couples to the dipole momet of the atom ad itroduces time-depedet chages of the wave fuctio i ψ(r,t) t with =[H 0 + H I (t)]ψ(r,t) (2.3) H I (t) = exe(t) = de(t). (2.4) Here, d is the operator for the electric dipole momet ad we assumed that the homogeeous electromagetic field is polarized i x-directio. Expadig the time-depedet wave fuctios ito the statioary eigefuctios of Eq. (2.1) ψ(r,t)= a m (t)e iɛmt ψ m (r), (2.5) m isertig ito Eq. (2.3), multiplyig from the left by ψ(r) ad itegratig over space, we fid for the coefficiets a the equatio where i da dt = E(t) m ɛ m = ɛ m ɛ e iɛmt d m a m, (2.6) (2.7) is the frequecy differece ad d m = d 3 rψ(r)dψ m (r) d m (2.8) is the electric dipole matrix elemet. We assume that the electro was iitially at t i the state l, i.e., a (t )=δ,l. (2.9) Now we solve Eq. (2.6) iteratively takig the field as perturbatio. For this purpose, we itroduce the smalless parameter Δ ad expad a = a (0) +Δa(1) +... (2.10) ad E(t) ΔE(t). (2.11)

3 Atoms i a Classical Light Field 19 Isertig (2.10) ad (2.11) ito Eq. (2.6), we obtai i order Δ 0 da (0) =0, dt which is satisfied by a (0) = δ,l. I first order of Δ, wehave (2.12) (2.13) i da(1) = E(t)d l e iɛlt. (2.14) dt For = l there is o field-depedet cotributio, i.e., a (i) l 0 for i 1, sice d ll =0. Itegratig Eq. (2.14) for l from to t yields a (1) 1 (t) = i t dt E(t )d l e iɛ lt, (2.15) where a (1) (t = ) =0has bee used. This coditio is valid sice we assumed that the electro is i state l without field, Eq. (2.9). To solve the itegral i Eq. (2.15), we express the field through its Fourier trasform dω E(t) = lim γ 0 2π E(ω)e iωt e γt. (2.16) Here, we itroduced the adiabatic switch-o factor exp(γt), to assure that E(t) 0 whe t. We will see below that the switch-o parameter γ plays the same role as the ifiitesimal dampig parameter of Chap. 1. The existece of γ makes sure that the resultig optical susceptibility has poles oly i the lower half of the complex plae, i.e., causality is obeyed. For otatioal simplicity, we will drop the lim γ 0 i frot of the expressios, but it is uderstood that this limit is always implied. Isertig Eq. (2.16) ito Eq. (2.15) we obtai a (1) (t) = d l dω e i(ω+ɛl)t E(ω) 2π ω + ɛ l + iγ, (2.17) whereweletγ 0 i the expoet after the itegratio. If we wat to geerate results i higher-order perturbatio theory, we have to cotiue the iteratio by isertig the first-order result ito the RHS of (2.6) ad calculate this way a (2) etc. These higher-order terms

4 20 Quatum Theory of the Optical ad Electroic Properties of Semicoductors cotai quadratic ad higher powers of the electric field. However, we are limitig ourselves to the terms liear i the field, i.e., we employ liear respose theory. The total wave fuctio (2.5) is ow [ ψ(r,t)=e iɛ lt ψ l (r) m l d ml dω ψ m(r) 2π E(ω) e iωt ω + ɛ lm + iγ ] + O(E 2 ). (2.18) The field-iduced polarizatio is give as the expectatio value of the dipole operator P(t) = 0 d 3 rψ (r,t)dψ(r,t), (2.19) where 0 is the desity of the mutually idepedet (ot iteractig) atoms i the system. Isertig the wave fuctio (2.18) ito Eq. (2.19), ad keepig oly terms which are first order i the field, we obtai the polarizatio as d lm 2 [ dω e iωt P(t) = 0 E(ω) 2π ω + ɛ m lm + iγ + e iωt ] E (ω). ω + ɛ lm iγ (2.20) I the itegral over the last term, we substitute ω ω ad use E ( ω) = E(ω), which is valid sice E(t) is real. This way we get d lm 2 P(t) = 0 = m [ dω 2π E(ω)e iωt 1 ω + ɛ lm + iγ 1 ω ɛ lm + iγ dω 2π P(ω)e iωt. (2.21) ]

5 Atoms i a Classical Light Field 21 This equatio yields P(ω) =χ(ω)e(ω) with the optical susceptibility χ(ω) = 0 ( d lm 2 m atomic optical susceptibility 2.2 Oscillator Stregth 1 ω + ɛ lm + iγ 1 ω ɛ lm + iγ ). (2.22) If we compare the atomic optical susceptibility, Eq. (2.22), with the result of the oscillator model, Eq. (1.7), we see that both expressios have similar structures. However, i compariso with the oscillator model the atom is represeted ot by oe but by may oscillators with differet trasitio frequecies ɛ l. To see this, we rewrite the expressio (2.22), pullig out the same factors which appear i the oscillator result, Eq. (1.7), χ(ω) = 0e 2 2m 0 ( f l ɛ l 1 ω ɛ l + iγ 1 ω + ɛ l + iγ Hece, each partial oscillator has the stregth of ). (2.23) f l = 2m 0 x l 2 ɛ l. (2.24) oscillator stregth Here, we used d l 2 = e 2 x l 2. Addig the stregths of all oscillators by summig over all the fial states, we fid f l = 2m 0 x l l x (ɛ ɛ l ). (2.25) Usig the Schrödiger equatio H 0 = ɛ, wecawrite l x (ɛ ɛ l )= 1 l [x, H 0] = 1 l [H 0,x], (2.26)

6 22 Quatum Theory of the Optical ad Electroic Properties of Semicoductors where [H 0,x]=H 0 x xh 0 is the commutator of H 0 ad x. Isertig (2.26) ito (2.25) ad usig the completeess relatio =1we get f l = 2m 0 2 l [H 0,x]x l. (2.27) Alteratively to (2.26), we ca also maipulate the first term i Eq. (2.25) by writig x l (ɛ ɛ l )= 1 [H 0,x] l, (2.28) so that f l = 2m 0 2 l x[h 0,x] l. (2.29) Addig Eqs. (2.27) ad (2.29) ad dividig by two shows that the sum over the oscillator stregth is give by a double commutator f l = m 0 2 l [x, [H 0,x]] l = m 0 2 l [[x, H 0],x] l. (2.30) The double commutator ca be evaluated easily usig ) [x, H 0 ]= (x 2 d2 2m 0 dx 2 d2 dx 2 x = 2 d m 0 dx = i p x (2.31) m 0 ad to get [p x,x]= i f l =1. oscillator stregth sum rule (2.32) (2.33) Eq. (2.33) is the oscillator stregth sum rule showig that the total trasitio stregth i a atom ca be viewed as that of oe oscillator which is distributed over may partial oscillators, each havig the stregth f l.

7 Atoms i a Classical Light Field 23 Writig the imagiary part of the dielectric fuctio of the atom as ɛ (ω) = 4πχ (ω), usig χ(ω) from Eq. (2.23) ad employig the Dirac idetity, Eq. (1.69), we obtai ɛ (ω) =ωpl 2 π f l [δ(ω ɛ l ) δ(ω ɛ l )], (2.34) 2 ɛ l with ω 2 pl =4π 0e 2 /m 0. Sice l is the occupied iitial state ad are the fial states, we see that the first term i Eq. (2.34) describes light absorptio. Eergy coservatio requires ɛ = ω + ɛ l, (2.35) i.e., a optical trasitio from the lower state l to the eergetically higher state takes place if the eergy differece ɛ l is equal to the eergy ω of a light quatum, called a photo. I other words, a photo is absorbed ad the atom is excited from the iitial state l to the fial state. This iterpretatio of our result is the correct oe, but to be fully appreciated it actually requires also the quatum mechaical treatmet of the light field. The secod term o the RHS of Eq. (2.34) describes egative absorptio causig amplificatio of the light field, i.e., optical gai. This is the basis of laser actio. I order to produce optical gai, the system has to be prepared i a state l which has a higher eergy tha the fial state, because the eergy coservatio expressed by the delta fuctio i the secod term o the RHS of (2.34) requires ɛ l = ω + ɛ. (2.36) If the eergy of a light quatum equals the eergy differece ɛ l, stimulated emissio occurs. I order to obtai stimulated emissio i a real system, oe has to ivert the system so that it is iitially i a excited state rather tha i the groud state. 2.3 Optical Stark Shift Util ow we have oly calculated ad discussed the liear respose of a atom to a weak light field. For the case of two atomic levels iteractig with the light field, we will ow determie the respose at arbitrary field itesities. Callig these two levels =1, 2 with ɛ 2 >ɛ 1, (2.37)

8 24 Quatum Theory of the Optical ad Electroic Properties of Semicoductors we get from Eq. (2.6) the followig two coupled differetial equatios: i da 1 dt = E(t)e iɛ21t d 12 a 2, (2.38) i da 2 dt = E(t)eiɛ21t d 21 a 1, (2.39) where we used d ii =0. Assumig a simple moochromatic field of the form E(t) = 1 2 E(ω)(e iωt + c.c.) (2.40) yields i da 1 dt = d E(ω) [ 12 e i(ω+ɛ21)t + e i(ω ɛ21)t] a 2 2, (2.41) i da 2 dt = d E(ω) [ 21 e i(ω ɛ21)t + e i(ω+ɛ21)t] a 1, (2.42) 2 where ɛ 12 = ɛ 21 has bee employed. These two coupled differetial equatios are ofte called the optical Bloch equatios. If we are iterested oly i the light-iduced chages aroud the resoace, ω ɛ 2 ɛ 1, (2.43) we see that the expoetial factor exp[i(ω ɛ 21 )t] is almost timeidepedet, whereas the secod expoetial exp[i(ω +ɛ 21 )t] oscillates very rapidly. If we keep both terms, we fid that exp[i(ω ɛ 21 )t] leads to the resoat term proportioal to 1 (ω ɛ 21 )+iγ P 1 ω ɛ 21 iπδ(ω ɛ 21 ) (2.44) i the susceptibility, whereas exp[i(ω + ɛ 21 )t] leads to the oresoat term proportioal to 1 (ω + ɛ 21 )+iγ P 1 iπδ(ω + ɛ 21 ). (2.45) ω + ɛ 21 For optical frequecies satisfyig (2.43), the δ-fuctio i (2.45) caot be satisfied sice ɛ 2 >ɛ 1, ad the pricipal value gives oly a weak cotributio

9 Atoms i a Classical Light Field 25 to the real part. Hece, oe ofte completely igores the oresoat parts so that Eqs. (2.41) ad (2.42) simplify to i da 1 dt = d 12E(ω) e i(ω ɛ21)t a 2, (2.46) 2 i da 2 dt = d 21E(ω) e i(ω ɛ21)t a 1. (2.47) 2 This approximatio is also called the rotatig wave approximatio (RWA). This ame origiates from the fact that the periodic time developmet i Eqs. (2.46) ad (2.47) ca be represeted as a rotatio of the Bloch vector (see Chap. 5). If oe trasforms these simplified Bloch equatios ito a time frame which rotates with the frequecy differece ω ɛ 21,the eglected term is ω out of phase ad more or less average to zero for loger times. To solve Eqs. (2.46) ad (2.47), we first treat the case of exact resoace, ω = ɛ 21. Differetiatig Eq. (2.47) ad isertig (2.46) we get d 2 a 2 dt 2 = id 21E(ω) da 1 2 dt = d 12 E(ω) 2 2 a 2 = ω2 R 4 a 2, (2.48) whereweusedd 21 = d 12 ad itroduced the Rabi frequecy as ω R = d 21E. Rabi frequecy The solutio of (2.48) is of the form (2.49) a 2 (t) =a 2 (0)e ±iωrt/2. (2.50) For a 1 (t) we get the equivalet result. Isertig the solutios for a 1 ad a 2 back ito Eq. (2.5) yields ψ(r,t)=a 1 (0)e i(ɛ1±ωr/2)t ψ 1 (r)+a 2 (0)e i(ɛ2±ωr/2)t ψ 2 (r), (2.51) showig that the origial frequecies ɛ 1 ad ɛ 2 have bee chaged to ɛ 1 ± ω R /2 ad ɛ 2 ± ω R /2, respectively. Hece, as idicated i Fig. 2.1, oe has ot just oe but three optical frequecies ɛ 21,adɛ 21 ± ω R, respectively. I other words, uder the ifluece of the light field the sigle trasitio

10 26 Quatum Theory of the Optical ad Electroic Properties of Semicoductors e + w / 2 2 R e 2 e 1 e -w / 2 2 R e + w / 2 1 R e -w / 2 1 R Fig. 2.1 Schematic drawig of the frequecy scheme of a two-level system without the light field (left part of Figure) ad light-field iduced level splittig (right part of Figure) for the case of a resoat field, i.e., zero detuig. The vertical arrows idicate the possible optical trasitios betwee the levels. possible i a two-level atom splits ito a triplet, the mai trasitio at ɛ 21 ad the Rabi sidebads at ɛ 21 ± ω R. Eq. (2.49) shows that the splittig is proportioal to the product of field stregth ad electric dipole momet. Therefore, Rabi sidebads ca oly be observed for reasoably strog fields, where the Rabi frequecy is larger tha the lie broadeig, which is always preset i real systems. The two-level model ca be solved also for the case of a fiite detuig ν = ɛ 21 ω. I this situatio, Eqs. (2.46) ad (2.47) ca be writte as da 1 dt = d 12E(ω) ie iνt a 2 2 (2.52) da 2 dt = d 21E(ω) ieiνt a 1 2. (2.53) Takig the time derivative of Eq. (2.52) d 2 a 1 dt 2 = d ( 12E(ω) e iνt a 2 ν + i da ) 2 2 dt, (2.54) ad expressig a 2 ad da 2 /dt i terms of a 1 we get d 2 a 1 dt 2 = iν da 1 dt ω2 R a 1 4 with the solutio (2.55) a 1 (t) =a 1 (0)e iωt (or a 1 (t) =a 1 (0)e iωt ), (2.56)

11 Atoms i a Classical Light Field 27 where Ω= ν 2 ± 1 2 ν 2 + ωr 2. (2.57) Similarly, we obtai a 2 (t) =a 2 (0)e iωt (or a 2 (t) =a 2 (0)e iωt ). (2.58) Hece, we agai get split ad shifted levels ɛ 2 Ω 2 ɛ 2 +Ω=ɛ 2 ν 2 ± 1 2 ɛ 1 Ω 1 ɛ 1 Ω=ɛ 1 + ν 2 ± 1 2 ν 2 + ω 2 R (2.59) ν 2 + ωr 2. (2.60) The coheret modificatio of the atomic spectrum i the electric field of a light field resembles the Stark splittig ad shiftig i a static electric field. It is therefore called optical Stark effect. The modified or, as oe also says, the reormalized states of the atom i the itese light field are those of a dressed atom. While the optical Stark effect has bee well-kow for a log time i atoms, it has bee see relatively recetly i semicoductors, where the dephasig times are ormally much shorter tha i atoms, as will be discussed i more detail i later chapters of this book. REFERENCES For the basic quatum mechaical theory used i this chapter we recommed: A.S. Davydov, Quatum Mechaics, Pergamo, New York (1965) L.I. Schiff, Quatum Mechaics, 3rded., McGraw Hill, New York (1968) The optical properties of two-level atoms are treated extesively i: L. Alle ad J.H. Eberly, Optical Resoace ad Two-Level Atoms, Wiley ad Sos, New York (1975) P. Meystre ad M. Sarget III, Elemets of Quatum Optics, Spriger, Berli (1990)

12 28 Quatum Theory of the Optical ad Electroic Properties of Semicoductors M. SargetIII, M.O. Scully, adw.e. Lamb, Jr., Laser Physics, Addiso Wesley, Readig, MA (1974) PROBLEMS Problem 2.1: To describe the dielectric relaxatio i a dielectric medium, oe ofte uses the Debye model where the polarizatio obeys the equatio dp dt = 1 τ [P(t) χ 0E(t)]. (2.61) Here, τ is the relaxatio time ad χ 0 is the static dielectric susceptibility. The iitial coditio is P(t = ) =0. Compute the optical susceptibility. Problem 2.2: Compute the oscillator stregth for the trasitios betwee the states of a quatum mechaical harmoic oscillator. Verify the sum rule, Eq. (2.33).