Experimento

Coherent Interactions of Fields and Atoms

In this chapter, the tools and perspectives developed earlier are applied to predict the dynamics of many different systems, starting from the simplest and progressing to more complicated ones. Systems in each successive subsection are representative of a class of problem, and only one change is introduced at each step. In this way a veritable catalog of possibilities is assembled, all described with the density matrix and the same interaction Hamiltonian using the semiclassical approach. The problems in which it becomes necessary to consider the quantum structure of the electromagnetic field itself are deferred to Chapter 6 on “Quantized Fields and Coherent States.”

5.1 Stationary atoms

5.1.1 Stationary two-level atoms in a traveling wave

Consider a system of stationary two-level atoms as shown in Fig. 5.1, characterized by a single resonance frequency ω0≡ ω2− ω1.

V (t) =−1

2µ¯· ¯E0e^iωt+ c.c. =−1

2
Ωe^iωt+ c.c. (5.1.1) According to Eqs. (3.6.19) and (3.8.1), the density matrix equations of motion are

i
˙ρ11= V₁₂ρ₂₁− ρ12V₂₁+ i
γ21ρ₂₂, (5.1.2) i
˙ρ22=−V12ρ21+ ρ12V21− i
γ21ρ22, (5.1.3) i
˙ρ12=−
ω0ρ12+ (V12ρ22− ρ11V12)− i
Γ12ρ12, (5.1.4)

ρ₂₁= ρ^∗₁₂. (5.1.5)

Note that either a plus or a minus sign may apply to the phenomenological population decay terms on the far right of Eqs. (5.1.2) and (5.1.3). The choice of sign must reflect whether the population is “arriving” (+) in a given level from more energetic states above it, or “leaving” (−). On the other hand, the sign for the decay of the off-diagonal

w₀

2

1

Figure 5.1 Energy levels and resonant transition frequency in a two-level atom. The system is subjected to traveling plane wave excitation of the form.

matrix element (the “coherence”) in Eq. (5.1.4) is always negative, reflecting the fact that coherence of a system never spontaneously increases. When there is no driving field, it always decreases. In addition the dephasing rate constant is real, so that Γ12= Γ21. Although the subscripts on Γ12, Γ21 seem unnecessary at this stage, they do indicate the levels connected by a particular coherence. Hence we shall retain them in anticipation of later situations in which there may be multiple coherences between more than two levels.

For steady-state behavior we assume that the response of the electron, described by off-diagonal element ρ₁₂, follows only one frequency component of the driving field, namely the positive one. This is called the rotating wave approximation (RWA).

Rather than perform the time-integration of Eq. (5.1.4), which is necessary to find transient solutions, we begin by simply assuming the charge oscillation is at the optical frequency, so that the solution has the form

ρ₁₂= ˜ρ₁₂e^iωt. (5.1.6)

(This is justified by direct integration of the equation of motion for the off-diagonal density matrix element ρ12 in Appendix G.) Upon substitution of Eq. (5.1.6) into Eq. (5.1.4), one immediately finds

[Γ₁₂+ i (ω− ω0)] ˜ρ₁₂+ ˙˜ρ₁₂= iΩ₁₂(ρ₁₁− ρ22) /2. (5.1.7) We now make the further assumption that ˜ρ12 varies much more slowly than the optical period. This corresponds to the assumption that d ˜V12(t^)/dt^∼= 0 in Appendix G and is called the slowly varying envelope approximation (SVEA). It is implemented by setting ˙˜ρ₁₂= 0, which yields

˜ ρ12=

Ω12/2

∆ + iΓ



(ρ11− ρ²²) . (5.1.8)

Note that we have dropped the subscripts on the dephasing rate for simplicity, since Γ12= Γ21≡ Γ is a real decay rate and there are only two energy levels in the system, so there is no possible ambiguity as to the polarization decay to which it refers. The steady-state solution for ρ₁₂(t) is therefore

ρ12(t) =

Ω12/2

∆ + iΓ



(ρ11− ρ22) e^iωt. (5.1.9)

Stationary atoms 103 For systems in which emission and coherence properties are not of interest, often it is only the temporal development of populations that is needed to describe basic system dynamics. For this purpose, population equations that do not require knowledge of the off-diagonal elements of the density matrix suffice. For example, absorption of the system, which is proportional to the population difference ρ11− ρ22, can be predicted without knowing ρ₁₂. This can be demonstrated by substituting Eq. (5.1.9) and its conjugate into the density matrix equations for ˙ρ₁₁ and ˙ρ₂₂.

Exercise: Show that substitution of Eqs. (5.1.9) and (5.1.5) into Eqs. (5.1.2) and (5.1.3) results in the population rate equations:

˙

ρ₁₁=− Γ/2

∆²+ Γ²|Ω12|²(ρ₁₁− ρ22) + γ₂₁ρ₂₂, (5.1.10)

˙

ρ22= Γ/2

∆²+ Γ²|Ω12|²(ρ11− ρ22)− γ21ρ22. (5.1.11) These are coupled equations for the populations in levels 1 and 2 that can be solved exactly and include intensity-dependent dynamics. Without having to evaluate ρ₁₂ explicitly, they can be used to describe the bleaching of absorbing systems like colored filters by intense light beams.

The next step is to solve for the steady-state populations. Let us start with ρ₂₂, using Eq. (5.1.9) in Eq. (5.1.3). Setting ˙ρ₂₂= 0 for steady-state response, this procedure yields

γ21ρ22= i

(V12ρ21− ρ12V21)

=−(iΩ¹²ρ˜21− iΩ²¹ρ˜12)/2

=|Ω12/2|²[L + L^∗] (ρ11− ρ22) , (5.1.12) where L≡ (i∆ + Γ)⁻¹. Solving for the excited state occupation in terms of that of the ground state, one finds

ρ22=

(L + L^∗)|Ω12/2|²/γ21

1 + (L + L^∗)|Ω12/2|²/γ₂₁

ρ11. (5.1.13)

Since the total occupation probability must be unity (ρ11+ ρ22= 1), one also obtains

ρ₁₁=

1 + (L + L^∗)|Ω12/2|²/γ₂₁ 1 + 2(L + L^∗)|Ω¹²/2|²/γ21

. (5.1.14)

Exercise: Verify that ρ₁₁, ρ₂₂ are purely real quantities that tend to appropriate limiting values as Ω→ ∞ (i.e., at high intensity)

The absorption of light depends on the number of absorbers and their distribution among the available states. Hence it is proportional to N (ρ11− ρ22), where the population difference is given by Eqs. (5.1.13) and (5.1.14) as

ρ₁₁− ρ22= [1 + 2(L + L^∗)|Ω/2|²/γ₂₁]⁻¹

=

1 + Γ|Ω|² (∆²+ Γ²)γ₂₁

₋₁

. (5.1.15)

Exercise: Find the limiting value of absorption as Ω→ ∞ (high intensity limit)? Does the result in Eq. (5.1.15) make physical sense in this limit?

Equation (5.1.15) has the form [1 + I/Isat]⁻¹, where the intensity Isat at which absorption drops to half its maximum, a quantity known as the saturation intensity, is defined by

Isat ≡
²γ21µ0c/(|µ12|²[L + L^∗])

=


²γ₂₁µ₀c(∆²+ Γ²)

|µ12|²2Γ

(5.1.16)

Exercise: At a positive detuning from resonance equal to the linewidth (i.e., ∆ = Γ), does the intensity required to saturate the system increase or decrease, and by what factor? Does the saturation intensity depend on whether the detuning ∆ is negative or positive?