26CHAPTER2.MAXWELL-BLOCHEOUATIONSwhere E(t) = Eejut. By using the ansatz r(t) = ejwt, we obtain for thecomplex amplitude of the dipole moment p = eor(t) = peiwtE(2.35)P:(w-w2)+2iwForthe susceptibility,wegetNEImEo(2.36)x(w) :(wg - w2) + 2jw号and thuswg(2.37)x(w) =(%-w)+2iwwith the plasma frequency wp, determined by w, = Nea/meo.Figure 2.1shows the real part and imaginary part of the classical susceptiblity (2.37).1.00.6Q=100.4 020Z0.500.0(o),x-0.20.40.02.00.00.51.01.5el.Figure2.l:Realpartand imaginarypart of thesusceptibilityofthe classicaloscillator model for the electric polarizability.Note, there is a small resonance shift due to the loss.Off resonance,the imaginary part approaches very quickly zero. Not so the real part, itapproaches a constant value wp/w below resonance, and approaches zero forabove resonance,but slower than the real part,i.e. offresonancethere is stilla contribution to the index but practically no loss
26 CHAPTER 2. MAXWELL-BLOCH EQUATIONS where E(t) = Eeˆ jωt. By using the ansatz x (t)=ˆxejωt, we obtain for the complex amplitude of the dipole moment p = e0x(t)=ˆpejωt pˆ = e2 0 m (ω2 0 − ω2) + 2jω0 Q ω E. ˆ (2.35) For the susceptibility, we get χ(ω) = N e2 0 m 1 0 (ω2 0 − ω2) + 2jω ω0 Q (2.36) and thus χ(ω) = ω2 p (ω2 0 − ω2) + 2jω ω0 Q , (2.37) with the plasma frequency ωp, determined by ω2 p = Ne2 0/m0. Figure 2.1 shows the real part and imaginary part of the classical susceptiblity (2.37). 1.0 0.5 0.0 χ ''(ω ) *2 Q/ 0.0 0.5 1.0 1.5 2.0 ω / ω 0 0.6 0.4 0.2 0.0 -0.2 -0.4 χ '(ω ) *2 Q/ 2 Q Q=10 Figure 2.1: Real part and imaginary part of the susceptibility of the classical oscillator model for the electric polarizability. Note, there is a small resonance shift due to the loss. Off resonance, the imaginary part approaches very quickly zero. Not so the real part, it approaches a constant value ω2 p/ω2 0 below resonance, and approaches zero for above resonance, but slower than the real part, i.e. off resonance there is still a contribution to the index but practically no loss
272.3.BLOCHEQUATIONS2.3Bloch EquationsAtoms in lowconcentration show line spectra as found in gas-,dye-and somesolid-state laser media. Usually, there are infinitely many energy eigenstatesin an atomic, molecular or solid-state medium and the spectral lines areassociated with allowed transitions between two of these energy eigenstatesFor nany physical considerations it is already sufficient to take only two ofthe possible energy eigenstates into account, for example those which arerelated to the laser transition. The pumping of the laser can be describedby phenomenological relaxation processes into the upper laser level and outof the lower laser level.The resulting simple model is often called a two-level atom, which is mathematically also equivalent to a spin1/2particlein an external magnetic field, because the spin can only be parallel or anti-parallel to the field, i.e. it has two energy levels and energy eigenstates. Theinteraction of the two-level atom or the spin with the electric or magneticfield is described by the Bloch equations.2.3.1The Two-Level ModelAn atom having only two energy eigenvalues is described by a two-dimensionalstate space spanned by the two energy eigenstates le > and lg >. The twostates constitute a complete orthonormal system.The corresponding energyeigenvalues are Ee and Eg (Fig. 2.2).E+EeEgFigure 2.2: Two-level atomIn the position-, i.e. x-representation, these states correspond to the wave
2.3. BLOCH EQUATIONS 27 2.3 Bloch Equations Atoms in low concentration show line spectra as found in gas-, dye- and some solid-state laser media. Usually, there are infinitely many energy eigenstates in an atomic, molecular or solid-state medium and the spectral lines are associated with allowed transitions between two of these energy eigenstates. For many physical considerations it is already sufficient to take only two of the possible energy eigenstates into account, for example those which are related to the laser transition. The pumping of the laser can be described by phenomenological relaxation processes into the upper laser level and out of the lower laser level. The resulting simple model is often called a twolevel atom, which is mathematically also equivalent to a spin 1/2 particle in an external magnetic field, because the spin can only be parallel or antiparallel to the field, i.e. it has two energy levels and energy eigenstates. The interaction of the two-level atom or the spin with the electric or magnetic field is described by the Bloch equations. 2.3.1 The Two-Level Model An atom having only two energy eigenvalues is described by a two-dimensional state space spanned by the two energy eigenstates |e > and |g >. The two states constitute a complete orthonormal system. The corresponding energy eigenvalues are Ee and Eg (Fig. 2.2). Figure 2.2: Two-level atom In the position-, i.e. x-representation, these states correspond to the wave
28CHAPTER2.MAXWELL-BLOCHEOUATIONSfunctions(2.38)e() =<le>,and Φg(r) =<rlg > :The Hamiltonian of the atom is given by(2.39)HA=Eele><el+Eglg><glIn this two-dimensional state space only2× 2=4 linearly independent linearoperators are possible.Apossible choiceforan operator base in this space is1 = le><el+ lg><gl,(2.40)(2.41)oz = Je><el-lg><glo+ = le><g,(2.42)(2.43)α- = lg ><el.The non-Hermitian operators α+ could be replaced by the Hermitian oper-ators r.y(2.44)ar = ++a-,(2.45)Qy = -jo++jo-.The physical meaning of these operators becomes obvious, if we look at theaction when applied to an arbitrary state(2.46)[b>= cglg>+cele>Weobtain(2.47)a+[ > =cgle>,(2.48)o-b > =celg>,(2.49)ob > =cele>-cglg>:The operator o+generates a transition from the ground to the excited state.and o- does the opposite. In contrast to α+ and o-, is a Hermitianoperator, and its expectation value is an observable physical quantity withexpectation value(2.50)<g[b>=|ce[2-Jca/2=w,the inversion w of the atom, since |ce/? and Ic/? are the probabilities forfinding the atom in state Je > or |g > upon a corresponding measurement
28 CHAPTER 2. MAXWELL-BLOCH EQUATIONS functions ψe(x) =< x|e >, and ψg(x) =< x|g>. (2.38) The Hamiltonian of the atom is given by HA = Ee|e >< e| + Eg|g >< g|. (2.39) In this two-dimensional state space only 2×2=4 linearly independent linear operators are possible. A possible choice for an operator base in this space is 1 = |e >< e| + |g >< g|, (2.40) σz = |e >< e| − |g >< g|, (2.41) σ+ = |e >< g|, (2.42) σ− = |g >< e|. (2.43) The non-Hermitian operators σ± could be replaced by the Hermitian operators σx,y σx = σ+ + σ−, (2.44) σy = −jσ+ + jσ−. (2.45) The physical meaning of these operators becomes obvious, if we look at the action when applied to an arbitrary state |ψ >= cg|g > + ce|e>. (2.46) We obtain σ+|ψ > = cg|e >, (2.47) σ−|ψ > = ce|g >, (2.48) σz|ψ > = ce|e > −cg|g>. (2.49) The operator σ+ generates a transition from the ground to the excited state, and σ− does the opposite. In contrast to σ+ and σ−, σz is a Hermitian operator, and its expectation value is an observable physical quantity with expectation value < ψ|σz|ψ >= |ce| 2 − |cg| 2 = w, (2.50) the inversion w of the atom, since |ce| 2 and |cg| 2 are the probabilities for finding the atom in state |e > or |g > upon a corresponding measurement
292.3.BLOCHEQUATIONSIf we consider an ensemble of N atoms the total inversion would be =N < / >. If we separate from the Hamiltonian (2.38) the term (Ee +Eg)/2 -1, where 1 denotes the unity matrix, we rescale the energy valuescorrespondingly and obtainfortheHamiltonian of the two-level system1(2.51)HA=hwegozSwith the transition frequency(2.52)- Eg),(EWeg方This form of the Hamiltonian is favorable. There are the following commu-tatorrelationsbetweenoperators(2.41)to(2.43)(2.53)[ot,o] = az][g+,a] = -2g+,(2.54)(2.55)[g-,0] = 2g-,and anti-commutator relations, respectively[ot,a-]+ = 1,(2.56)[o+,a]+ = 0,(2.57)[-,α]+ = 0,(2.58)[α-,a-]+ = [α+,a+]+ =0.(2.59)The operators r,u,,fulfill the angular momentum commutator relations(2.60)[0a,Qy] = 2jo2,(2.61)[oy,o:] = 2jor,(2.62)[o,Q] = 2joy.The two-dimensional state space can be represented as vectors in C2 accord-ing to the rule:Ce[b >=cele>+cglg > -→(2.63)
2.3. BLOCH EQUATIONS 29 If we consider an ensemble of N atoms the total inversion would be σ = N<ψ|σz|ψ >. If we separate from the Hamiltonian (2.38) the term (Ee + Eg)/2 ·1, where 1 denotes the unity matrix, we rescale the energy values correspondingly and obtain for the Hamiltonian of the two-level system HA = 1 2 ~ωegσz, (2.51) with the transition frequency ωeg = 1 ~ (Ee − Eg). (2.52) This form of the Hamiltonian is favorable. There are the following commutator relations between operators (2.41) to (2.43) [σ+,σ−] = σz, (2.53) [σ+,σz] = −2σ+, (2.54) [σ−,σz]=2σ−, (2.55) and anti-commutator relations, respectively [σ+,σ−]+ = 1, (2.56) [σ+,σz]+ = 0, (2.57) [σ−,σz]+ = 0, (2.58) [σ−,σ−]+ = [σ+, σ+]+ = 0. (2.59) The operators σx, σy, σz fulfill the angular momentum commutator relations [σx,σy] = 2jσz, (2.60) [σy,σz] = 2jσx, (2.61) [σz,σx] = 2jσy. (2.62) The two-dimensional state space can be represented as vectors in C2 according to the rule: |ψ >= ce|e > + cg|g > → µ ce cg ¶ . (2.63)
30CHAPTER2.MAXWELL-BLOCHEQUATIONSThe operators are then represented by matrices(2.64)(2.65)(2.66)dz1(2.67)2.3.2The Atom-Field Interaction In Dipole Approxi-mationThe dipole moment of an atom p is essentially determined by the positionoperatorxvia(2.68)p= -eo x.Then the expectation value for the dipole moment of an atom in state (2.46)is(2.69)<>=-eo(cej2<exe>+cec,<gxe>+ Cgc<exg >+|cgl<gx[g>).For simplicity, we may assume that that the medium is an atomic gas. Theatoms posses inversion symmetry, therefore, energy eigenstates must be sym-metric or anti-symmetric, i.e. < exle >=< gxg >= 0. We obtain(2.70)<pb>=-eo (cec<ge>+cgc<ge>*). (Note, this means, there is no permanent dipole moment in an atom, whichis in an energy eigenstate. Note, this might not be the case in a solid. Theatoms consituting the solid are oriented in a lattice, which may break thesymmetry.If so,there arepermanent dipole moments and consequently thematrix elements < exe > and < gxg > would not vanish. If so, thereare also crystal fields, which then imply level shifts, via the linear Starkeffect.) Thus an atom does only exhibit a dipole moment in the average, ifthe product cecg + O, i.e. the state of the atom is in a superposition of statesle > and Ig >
30 CHAPTER 2. MAXWELL-BLOCH EQUATIONS The operators are then represented by matrices σ+ → µ 0 1 0 0 ¶ , (2.64) σ− → µ 0 0 1 0 ¶ , (2.65) σz → µ 1 0 0 −1 ¶ , (2.66) 1 → µ 1 0 0 1 ¶ . (2.67) 2.3.2 The Atom-Field Interaction In Dipole Approximation The dipole moment of an atom p˜ is essentially determined by the position operator x via p = −e0 x. (2.68) Then the expectation value for the dipole moment of an atom in state (2.46) is < ψ|p|ψ > = −e0(|ce| 2 < e|x|e > +cec∗ g < g|x|e > (2.69) + cgc∗ e < e|x|g > +|cg| 2 < g|x|g >). For simplicity, we may assume that that the medium is an atomic gas. The atoms posses inversion symmetry, therefore, energy eigenstates must be symmetric or anti-symmetric, i.e. < e|x|e >=< g|x|g >= 0. We obtain < ψ|p|ψ >= −e0 (cec∗ g < g|x|e > +cgc∗ e < g|x|e >∗ ). (2.70) (Note, this means, there is no permanent dipole moment in an atom, which is in an energy eigenstate. Note, this might not be the case in a solid. The atoms consituting the solid are oriented in a lattice, which may break the symmetry. If so, there are permanent dipole moments and consequently the matrix elements < e|x|e > and < g|x|g > would not vanish. If so, there are also crystal fields, which then imply level shifts, via the linear Stark effect.) Thus an atom does only exhibit a dipole moment in the average, if the product cec∗ g 6= 0, i.e. the state of the atom is in a superposition of states |e > and |g >