Herramientas y tecnologías a utilizar en el desarrollo de la aplicación

Third-order interactions are by no means restricted to the specialized geometry of NDFWM analyzed in the previous section. In fact, third-order interactions may take place when only one or two beams are incident on a sample. Hence they are more com-mon than might be expected, and researchers using optical characterization techniques not only need to be able to anticipate when saturation effects need to be taken into account, but also be aware of simple two-beam measurements in which the third-order susceptibility χ⁽³⁾ dominates the analysis, such as pump-probe experiments.

To illustrate this, let us now discuss the pump-probe scenario of Fig. 5.18. It is common practice to saturate systems with a strong pump wave and then monitor bleaching or recovery dynamics with a probe wave. The objective is usually to obtain information about time-dependent processes in completely unknown systems by forcing it slightly away from equilibrium. There are two basic approaches. The probe may be delayed in time or detuned in frequency. The first technique measures the dynamics directly in the temporal domain. The second measures dynamics in the corresponding Fourier space by frequency-domain spectroscopy.

What is not so obvious at first is that both types of experiment involve a third-order interaction. In this section, we address the equations governing this general class of measurement and then analyze the frequency-domain spectrum in detail, to show that indeed three input fields determine the response just as in four-wave mixing [5.9]. To broaden the applicability of the analysis, we extend it to three-level systems however using an approach that combines exact and perturbative analysis. At the end, we shall point out that pump-probe experiments measure the same basic quantities as NDFWM spectroscopy of the last section.

In a typical pump-probe experiment, a strong (saturating) pump wave and a weak probe wave intersect at a small angle within the sample. To investigate the probe transmission spectrum versus pump-probe detuning, we ignore the directions of the two input waves. This means that we ignore phase-matching, and simply find the changes induced by the pump wave in the real and imaginary parts of the refractive

Sample

Probe Pump

Detector

Figure 5.18 Typical geometry of a pump-probe experiment in which a strong wave (dark arrow) partially saturates an absorptive transition or induces a parametric change in the birefringence or dispersion of a sample. The transmission of a probe wave is monitored with a detector.

Coherent multiple field interactions 131

3

1

∆ 2

w₀ w₁ w₂

Figure 5.19 Energy levels and detunings of a three-level system driven by two optical waves of similar frequency.

index of the sample. These changes may be probed from any angle and depend on the relaxation rates among various levels in the system, so it may not seem too surprising that the resulting probe spectrum contains the same basic information about the atomic system that was provided by NDFWM spectroscopy. However, the analysis of the last section was strictly perturbative, whereas in this section the pump wave is considered intense enough that an exact treatment that includes saturation effects is desired [5.10]. One consequence of this is that new high field features appear (Rabi sidebands).

The component equations of the density matrix corresponding to the three-level system in Fig. 5.19 are

˙

ρ11=−(i/
)(V12ρ21− ρ12V21) + γ31ρ33+ γ21ρ22, (5.4.15)

˙

ρ22= +(i/
)(V¹²ρ21− ρ¹²V21)− (γ²¹+ γ23)ρ22, (5.4.16)

˙

ρ₃₃=−γ31ρ₃₃+ γ₂₃ρ₂₂, (5.4.17)

˙

ρ12=−(i/
)V12(ρ22− ρ11)− (i/
)(H11ρ12− ρ12H22)− Γρ12, (5.4.18)

ρ21= ρ^∗₁₂. (5.4.19)

In this problem, the optical interaction is separated into two parts. The first is due to a strong pump wave E₁ and the second is due to a weak probe wave E₂. The pump wave is a traveling wave of the usual form

E₁(t) = 1

2E₁₀exp(iω₁t) + c.c, (5.4.20) and the probe is similar,

E₂(t) = 1

2E₂₀exp(iω₂t) + c.c., (5.4.21) except that E₂₀<< E₁₀.

The interaction Hamiltonian is written as the sum of a zeroth-order pump interac-tion and a first-order probe interacinterac-tion. That is,

V = V⁽⁰⁾+ V⁽¹⁾, (5.4.22)

where

V⁽⁰⁾=−1

2
Ω1exp(iω₁t) + c.c., (5.4.23) V⁽¹⁾=−1

2
Ω2exp(iω₂t) + c.c., (5.4.24) and Ω₁≡ µE10/
and Ω2≡ µE20/
.

The effect of V⁽⁰⁾ may be taken into account exactly, by the same procedure as in Section 5.1.2. This yields the results

˜

ρ⁽⁰⁾₁₂ = (Ω₁/2)

∆ + iΓ(ρ⁽⁰⁾₁₁ − ρ⁽⁰⁾22), (5.4.25)

ρ⁽⁰⁾₁₁ = 1 + Γ[2γ₂(∆²+ Γ²)]⁻¹|Ω1|²

1 + Γ[2γ2(∆²+ Γ²)]⁻¹(2 + γ23/γ31)|Ω1|², (5.4.26)

ρ⁽⁰⁾₂₂ = Γ[2γ2(∆²+ Γ²)]⁻¹|Ω1|²

1 + Γ[2γ₂(∆²+ Γ²)]⁻¹(2 + γ₂₃/γ₃₁)|Ω1|². (5.4.27) Although the population expressions in Eqs. (5.4.26) and (5.4.27) are exact, in the next step of the calculation they are taken as the zeroth-order solutions in a perturbation expansion (see Appendix D). The first-order equation of motion that includes the effect of the weak probe wave has the form

i
˙ρ⁽¹⁾= [H₀, ρ⁽¹⁾] + [V⁽⁰⁾, ρ⁽¹⁾] + [V⁽¹⁾, ρ⁽⁰⁾] + i
˙ρ⁽¹⁾relax. (5.4.28) The various components are

˙

ρ⁽¹⁾₁₁ = i(Ω₁/2) ˜ρ^∗(1)₁₂ + c.c. + i(Ω₂/2) ˜ρ^∗(0)₁₂ + c.c. + γ₂₁ρ⁽¹⁾₂₂ + γ₃₁ρ⁽¹⁾₃₃, (5.4.29)

˙

ρ⁽¹⁾₂₂ =−i(Ω1/2) ˜ρ^∗(1)₁₂ + c.c.− i(Ω2/2) ˜ρ^∗(0)₁₂ + c.c.− (γ21+ γ23)ρ⁽¹⁾₂₂, (5.4.30)

˙

ρ⁽¹⁾₃₃ = γ₃₁ρ⁽¹⁾₃₃ + γ₃₂ρ⁽¹⁾₂₂, (5.4.31)

˙

ρ⁽¹⁾₁₂ = i(Ω1/2) exp(iω1t)(ρ⁽¹⁾₂₂ − ρ⁽¹⁾11)

+ i(Ω₂/2) exp(iω₂t)(ρ⁽⁰⁾₂₂ − ρ⁽⁰⁾11) + iω₀ρ⁽¹⁾₁₂ − Γρ⁽¹⁾12. (5.4.32) Because the pump and probe waves are detuned, their combined driving effect can produce pulsations in population and coherence at frequency ±δ. Consequently, the solutions acquire additional time dependences and are assumed to have the

Coherent multiple field interactions 133

forms

ρ₁₁= ρ⁽⁰⁾₁₁ + ρ⁽¹⁾₁₁ = ρ⁽⁰⁾₁₁ + ρ⁽⁺⁾₁₁ exp(iδt) + ρ⁽₁₁⁻⁾exp(−iδt), (5.4.33) ρ22= ρ⁽⁰⁾₂₂ + ρ⁽¹⁾₂₂ = ρ⁽⁰⁾₂₂ + ρ⁽⁺⁾₂₂ exp(iδt) + ρ⁽₂₂⁻⁾exp(−iδt), (5.4.34) ρ₃₃= ρ⁽⁰⁾₃₃ + ρ⁽¹⁾₃₃ = ρ⁽⁰⁾₃₃ + ρ⁽⁺⁾₃₃ exp(iδt) + ρ⁽₃₃⁻⁾exp(−iδt), (5.4.35)

˜

ρ12= ˜ρ⁽⁰⁾₁₂ + ˜ρ⁽¹⁾₁₂ = ˜ρ⁽⁰⁾₁₂ + ˜ρ⁽⁺⁾₁₂ exp(iδt) + ˜ρ⁽₁₂⁻⁾exp(−iδt). (5.4.36) The time derivatives on the left side of Eqs. (5.4.29)–(5.4.32) acquire new “beat”

terms, even if we restrict our solutions to steady-state conditions by assuming that

˙

ρ⁽⁺⁾₁₁ = ˙ρ⁽₁₁⁻⁾= ˙ρ⁽⁺⁾₂₂ = ˙ρ⁽₂₂⁻⁾= ˙ρ⁽⁺⁾₃₃ = ˙ρ⁽₃₃⁻⁾= 0, (5.4.37) and

˙˜

ρ⁽⁺⁾₁₂ = ˙˜ρ⁽₁₂⁻⁾= 0. (5.4.38) For example, by substituting Eqs. (5.4.34) and (5.4.35) into Eq. (5.4.31) and setting the coefficients of individual frequency components equal to zero, we find

ρ⁽⁰⁾₃₃ = (γ23/γ31)ρ⁽⁰⁾₂₂, (5.4.39) ρ⁽⁺⁾₃₃ = [γ₂₃/(γ₃₁+ iδ)]ρ⁽⁺⁾₂₂ , (5.4.40) ρ⁽₃₃⁻⁾= [γ23/(γ31− iδ)]ρ⁽22⁻⁾. (5.4.41) Now, using the results for ρ⁽₃₃^±) given in Eqs. (5.4.40) and (5.4.41), together with the first-order closure relation

ρ⁽¹⁾₁₁ + ρ⁽¹⁾₂₂ + ρ⁽¹⁾₃₃ = 0, (5.4.42) one finds

ρ⁽⁺⁾₁₁ =−

 1 +

 γ₂₃ γ₃₁+ iδ

'

ρ⁽⁺⁾₂₂ , (5.4.43)

ρ⁽₁₁⁻⁾=−

 1 +

 γ23

γ₃₁− iδ

'

ρ⁽₂₂⁻⁾. (5.4.44) After substitution of these results in Eq. (5.4.29), expressions for the first-order excited state amplitudes are obtained.

ρ⁽⁺⁾₂₂ = i(γ2+ iδ)⁻¹ (

(Ω^∗₁/2) ˜ρ⁽⁺⁾₁₂ − (Ω1/2) ˜ρ^∗(−)₁₂ − (Ω2/2) ˜ρ^∗(0)₁₂ )

, (5.4.45)

ρ⁽₂₂⁻⁾ = ρ⁽⁺⁾₂₂ ^∗. (5.4.46)

Next, by substituting Eq. (5.4.36) into Eq. (5.4.32), one can also find the coherence amplitudes

˜

ρ⁽⁺⁾₁₂ = i[(∆− δ) + iΓ][(Ω1/2)(ρ⁽⁺⁾₁₁ − ρ⁽⁺⁾22 ) + (Ω2/2)(ρ⁽⁰⁾₁₁ − ρ⁽⁰⁾22)], (5.4.47)

˜

ρ⁽₁₂⁻⁾ = [(∆ + δ) + iΓ]⁻¹[(Ω1/2)(ρ⁽₁₁⁻⁾− ρ⁽22⁻⁾)]. (5.4.48) Finally, substitution of these last two relations in Eq. (5.4.45) yields a general expres-sion for ρ⁽⁺⁾₂₂ , from which all other quantities may be determined using Eqs. (5.4.43)–

(5.4.48).

ρ⁽⁺⁾₂₂ = (δ− 2iΓ)(Ω^∗1Ω₂/4)(ρ⁽⁰⁾₁₁ − ρ⁽⁰⁾22)

(δ− iγ²)[(∆ + δ)− iΓ][(∆ − δ) + iΓ] + 2[2 + γ²³/(γ31+ iδ)][δ− iΓ] |Ω¹/2|². (5.4.49) With this result in hand, the coherence in Eq. (5.4.36) is fully determined to first order. The probe polarization and susceptibility can therefore be determined for any value of the pump intensity, resonance detuning ∆, and pump-probe detuning δ from the polarization P (ω + δ) = µ12ρ21(ω + δ) + µ21ρ12(ω + δ).

Results are shown in Fig. 5.20a–c for the absorption spectrum of the probe at low, moderate, and high pump intensities, respectively. The pump detuning and decay parameters were identical for these comparisons. At low intensities, weak dispersive beam coupling is seen at zero detuning and an absorption peak is present at a pump-probe detuning of δ/Γ = 10. Just above saturation, the dispersive feature at zero detuning reveals two resonances, one for each decay process from the excited states, and the absorption peak at δ/Γ = 10 decreases in strength. At twenty-times saturation (Ω/Γ = 20), power-broadening smears out the narrowest central resonance and two Rabi sidebands appear at the generalized Rabi frequency splitting of δ/Γ =±22.3 in Fig. 5.20c. The central dispersive feature in each spectrum is a weak field response present at intensities below the saturation intensity. When magnified, the central feature shows two overlapping resonances whose widths are determined by the decay rates of levels 2 and 3. Consequently, experimental measurements of their widths provide direct determinations of the excited state decay rates, just as in NDFWM spectroscopy described in the previous section. Power broadening can obscure the central feature, as shown in Fig. 5.20c where the second resonance disappears at high intensity.

Pump-probe spectroscopy is frequently performed with short pulses to monitor changes in sample transmission versus time. System dynamics are then revealed in the time domain as “differential transmission” instead of in the frequency domain as calculated here and portrayed in Fig. 5.20. Experimentally, measurements may be made by delaying the probe pulse with respect to the pump pulse using a mechanical delay line [5.11, 5.12, 5.13]. Theoretically, analysis requires direct integration of the time-dependent density matrix Eqs. (5.4.29)–(5.4.32). The outcome of analyzing the temporal profile of the transient transmission spectrum is however analogous to frequency-domain measurements. The results reflect the decay rates of dynamic processes in excited states of the system.

Coherent multiple field interactions 135

1 (a)

(b)

(c)

0.01

0.009

0.008

–0.02

0.05

0.02 0 0.5

0.001

0

0

0.00004

–0.00004 0 –0.001

0.002

0

–1 –0.5 0 0.5 1.0

0 –40

–40 0 40

–30 –20 –10

Pump–probe detuning (d/Γ)

Pump–probe detuning (d/Γ)

Pump–probe detuning (d/Γ)

Probe absorption (a.u.)Probe absorption (a.u.)Probe absorption (a.u.)

0 10 20 30 40

–40 –30 –20 –10 0 10 20 30 40

Figure 5.20 Nearly degenerate pump-probe transmission spectra of three-level atoms with γ31= 0.0001, γ2= 0.1, and γ23= 0.05 at (a) low intensity (Ω/Γ = 0.1), (b) moderate inten-sity (Ω/Γ = 3.0), and (c) high inteninten-sity (Ω/Γ = 20). The insets show enlargements of the central detuning region in each case. (After Ref. [5.10].)