312.3.BLOCHEQUATIONSWith the dipole matrix elementsM = eo<gxe >(2.71)the expectation value for the dipole moment can be written as<Pl >= -(cec,M + cgcM*) =-<l(α+M* +α-M) >. (2.72)Since this is true for an arbitrary state, the dipole operator (2.68) is repre-sented byp= p++p-=-M*o+- Mo-(2.73)Therefore,theoperators+and -areproportional to the complexdipolemoment operators p+ and p-, respectively.The energy of an electric dipole in an electric field isHA-F =-P·E(A,t).(2.74)The electric field at the position of the atom, A, can be written asE(正A,t) =(E(t)(+) +E(t)(一) =((t)(+)ejut +E(t)(-)e-it), (2.75)where E(t)(+) denotes the slowly varying complex field envelope with w ~weg. In the Rotating-Wave Approximation (RWA), we only keep the slowlyvarying components in the interaction Hamiltonian. As we will see later, ifthere is no field the operator α+ evolves like o+(t) = +(0)ejwegt, thus weobtainin RWAHA-F = -p.E(A,t)~(2.76)HRWAIM*E(t)(-)o+ + h.c.(2.77)~2The Schrodinger Equation for the two-level atom in a classical field is thengiven byd(2.78)一> =(HA+HA-F)|Φ>dt(HA +HRWA)Ib >:(2.79)~
2.3. BLOCH EQUATIONS 31 With the dipole matrix elements M = e0 < g|x|e > (2.71) the expectation value for the dipole moment can be written as < ψ|p|ψ >= −(cec∗ gM + cgc∗ eM ∗ ) = − < ψ|(σ+M ∗ + σ−M )|ψ>. (2.72) Since this is true for an arbitrary state, the dipole operator (2.68) is represented by p = p+ + p− = −M ∗ σ+ − M σ−. (2.73) Therefore, the operators σ+ and σ− are proportional to the complex dipole moment operators p+ and p−, respectively. The energy of an electric dipole in an electric field is HA−F = −p · E (xA, t). (2.74) The electric field at the position of the atom, xA, can be written as E (xA, t) = 1 2 ³ E (t) (+) + E (t) (−) ´ = 1 2 ³ ˆ E (t) (+)ejωt + ˆ E (t) (−) e−jωt´ , (2.75) where ˆ E (t)(+) denotes the slowly varying complex field envelope with ω ≈ ωeg. In the Rotating-Wave Approximation (RWA), we only keep the slowly varying components in the interaction Hamiltonian. As we will see later, if there is no field the operator σ+ evolves like σ+(t) = σ+(0)ejωegt , thus we obtain in RWA HA−F = −p · E (xA, t) ≈ (2.76) ≈ HRWA A−F = 1 2 M ∗ E (t) (−) σ+ + h.c. (2.77) The Schrödinger Equation for the two-level atom in a classical field is then given by j~ d dt|ψ > = (HA + HA−F )|ψ > (2.78) ≈ (HA + HRWA A−F )|ψ>. (2.79)
32CHAPTER2.MAXWELL-BLOCHEQUATIONSWritten in the energy representation, weobtaindWegc(2.80)e-jre-jdtCeCo2dWegCg - jre+jutce,(2.81)dt= +i*2with the Rabi-frequency defined asM·E2r:(2.82)2hFor the time being, we assume that the the Rabi-frequency is real. If this isnot the case, a transformation including a phase shift in the amplitudes Ca,bwould be necessary to eliminate this phase. As expected the field couples theenergy eigenstates.2.3.3Rabi-OscillationsIf the incident light has a constant field amplitude E Eqs: (2.80) and (2.81)can be solved and we observe an oscillation in the population difference, theRabi-oscillation [1]. To show this we introduce the detuning between fieldandatomicresonancewab-w△=(2.83)2and the new probability amplitudesCe = Ceeigt(2.84)Cg = Cge-igt.(2.85)This leads to the new system of equations with constant coefficientsdce(2.86)=-jACe-j2,Cgdtd(2.87)αC。 = +jAC,-j2,Ce.Note, these are coupling of mode equations in time. Now, the modes areelectronic ones instead of photonic modes. But otherwise everything is the
32 CHAPTER 2. MAXWELL-BLOCH EQUATIONS Written in the energy representation, we obtain d dtce = −j ωeg 2 ce − jΩre−jωtcg, (2.80) d dtcg = +jωeg 2 cg − jΩre+jωtce, (2.81) with the Rabi-frequency defined as Ωr = M ∗ ˆ E 2~ . (2.82) For the time being, we assume that the the Rabi-frequency is real. If this is not the case, a transformation including a phase shift in the amplitudes ca,b would be necessary to eliminate this phase. As expected the field couples the energy eigenstates. 2.3.3 Rabi-Oscillations If the incident light has a constant field amplitude ˆ E Eqs. (2.80) and (2.81) can be solved and we observe an oscillation in the population difference, the Rabi-oscillation [1]. To show this we introduce the detuning between field and atomic resonance ∆ = ωab − ω 2 (2.83) and the new probability amplitudes Ce = ceej ω 2 t , (2.84) Cg = cge−j ω 2 t . (2.85) This leads to the new system of equations with constant coefficients d dtCe = −j∆Ce − jΩrCg, (2.86) d dtCg = +j∆Cg − jΩrCe. (2.87) Note, these are coupling of mode equations in time. Now, the modes are electronic ones instead of photonic modes. But otherwise everything is the
2.3.BLOCHEQUATIONS33same. For the case of vanishing detuning it is especially easy to eliminateOneof thevariablesandwearriveatdCe-n2Ce(2.88)一dt2d(2.89)=-22Cgdt2CgThe solution to this set of equations are the oscillations we are looking for. Ifthe atom is at time t = 0 in the ground-state, i.e. Cg(O) = 1 and Ce(0) = 0,respectively.wearriveatCg(t) = cos (2rt)(2.90)(2.91)Ce(t) = -jsin(Qrt)Then, the probabilities for finding the atom in the ground or excited stateare[co(t)2 = cos2 (2rt)(2.92)[ca(t)2 = sin2 (2rt),(2.93)as shown inFig.2.3.For theexpectationvalue of thedipole operator underthe assumption of a real dipole matrix element M = M* we obtain<p> = -Mcec,+c.c.(2.94)= -M sin (22rt) sin (wegt) .(2.95)
2.3. BLOCH EQUATIONS 33 same. For the case of vanishing detuning it is especially easy to eliminate one of the variables and we arrive at d2 dt2Ce = −Ω2 rCe (2.88) d2 dt2Cg = −Ω2 rCg. (2.89) The solution to this set of equations are the oscillations we are looking for. If the atom is at time t = 0 in the ground-state, i.e. Cg(0) = 1 and Ce(0) = 0, respectively, we arrive at Cg(t) = cos (Ωrt) (2.90) Ce(t) = −j sin (Ωrt). (2.91) Then, the probabilities for finding the atom in the ground or excited state are |cb(t)| 2 = cos2 (Ωrt) (2.92) |ca(t)| 2 = sin2 (Ωrt), (2.93) as shown in Fig. 2.3. For the expectation value of the dipole operator under the assumption of a real dipole matrix element M = M ∗ we obtain < p > = −Mc ec∗ g + c.c. (2.94) = −M sin (2Ωrt) sin (ωegt). (2.95)
34CHAPTER2.MAXWELL-BLOCHEOUATIONSPOI2Pe2rt元元2rtZI2QrtFigure 2.3:Evolution of occupation probabilities of ground and excited stateand the average dipole moment of a two-level atom in resonant interactionwith a coherent classical field.The coherent external field drives the population of the atomic systembetween the two available states with a period T, =/r.Applying the fieldonly over half of this period leads to a complete inversion of the population.These Rabi-oscillations have been observed in various systems ranging fromgases to semiconductors. Interestingly, the light emitted from the coherentlydriven two-level atom is not identical in frequency to the driving field. Ifwe look at the Fourier spectrum of the polarization according to Eq.(2.95)we obtain lines at frequencies w+ = weg ± 22r. This is clearly a nonlinearoutput and the sidebands are called Mollow-sidebands [2].Most importantfor theexistence of these oscillations is thecoherence of the atomic systemover at least one Rabi-oscillation.If this coherence is destroyed fast enough,the Rabi-oscillations cannot happen and it is then impossible to generateinversion in a two-level system by interaction with light. This is the case fora large class of situations inlight-matterinteraction.So we areinterestedwhat happens in the case of loss of coherence due to additional interaction
34 CHAPTER 2. MAXWELL-BLOCH EQUATIONS Figure 2.3: Evolution of occupation probabilities of ground and excited state and the average dipole moment of a two-level atom in resonant interaction with a coherent classical field. The coherent external field drives the population of the atomic system between the two available states with a period Tr = π/Ωr. Applying the field only over half of this period leads to a complete inversion of the population. These Rabi-oscillations have been observed in various systems ranging from gases to semiconductors. Interestingly, the light emitted from the coherently driven two-level atom is not identical in frequency to the driving field. If we look at the Fourier spectrum of the polarization according to Eq.(2.95), we obtain lines at frequencies ω± = ωeg ± 2Ωr. This is clearly a nonlinear output and the sidebands are called Mollow-sidebands [2] . Most important for the existence of these oscillations is the coherence of the atomic system over at least one Rabi-oscillation. If this coherence is destroyed fast enough, the Rabi-oscillations cannot happen and it is then impossible to generate inversion in a two-level system by interaction with light. This is the case for a large class of situations in light-matter interaction. So we are interested what happens in the case of loss of coherence due to additional interaction