Strong coupling regime and weak coupling regime

The quantum treatment between light and atoms is usually developed in terms of the two-level atom approximation [34]. This approximation is applicable when the frequency of the light coincides with one of the optical transitions of the atom. The atom will have many quantum levels, and there will be many possible optical transitions between them. However, in the two-level atom approximation we only consider the specific transition that satisfies eq.(2.1-1) and ignore all the other levels.

It is customary to label the lower and upper levels as 1 and 2, respectively.

1 two-level atom in the presence of the light. In other words, we must solve:

t In the case of a two-level atom, the general solution to the time-dependent Schrödinger equation reduces to On substituting this wave function into eq.(2.1-2) with H given by eq.(2.1-3), ˆ₀

we obtain:

To proceed further we must consider the explicit form of the perturbationVˆ t( ). In the semi-classical approach, the light–atom interaction is given by the energy shift of the atomic dipole in the electric field of the light, and we arbitrarily choose the x-axis as the direction of the polarization so that we can write

)

where ₀is the amplitude of the light wave and the perturbation matrix elements are given by: where u is the dipole matrix element given by _ij

|

) atom in the light field. It turns out that there are two distinct types of solution that can be found, which correspond to the weak-field limit and the strong-field limit respectively.

We consider the weak-field limit first. The weak-field limit applies to low-intensity light sources. We assume that the atom is initially in the lower level and that the lamp is turned on at t = 0. This implies that c1(0) = 1 and c2(0) = 0. With a low-intensity source, the electric field amplitude will be small and the perturbation weak. The number of transitions expected is therefore small, and it will always be the case that c1(t)>>c2(t). In these conditions we can put c1(t) = 1 for all t, so that eq.

According to the rotating wave approximation, we now neglect the second term in eq.(2.1-14). This is justified by the fact that since δω<<(ω + ω0), the second term is much smaller than the first. After some manipulation we find:

2

We know in fact that all spectral lines have a finite width Δω. Furthermore, we

We therefore integrate eq.(2.1-16) the spectral line:



We now make the approximation that the spectral line is sharp compared to the broad-band spectrum of the lamp, so that u(ω) does not vary significantly within the integral. This allows us to replace u(ω) by a constant value u(ω0), and thus to evaluate the integral. The limiting value for tΔω→∞ is u(ω0)2πt. Hence we finally obtain:

)t the atom is in the upper level increases linearly with time.

We now wish to return to the more general case in which the population of the upper level is significant. It is intuitively obvious that this condition applies when the light–atom interaction is strong. In other words, we are dealing with the case of strong electric fields, such as those found in powerful laser beams. In order to find a solution to eq.(2.1-12) in the strong-field limit we make two simplifications. First, we apply the rotating wave approximation to neglect the terms that oscillate at ±(ω +ω₀), as in

the previous section. Second, we only consider the case of exact resonance with δω = 0. With these simplifications, eq.(2.1-11) reduces to:

) The time dependence of these probabilities is shown in Fig. 2.1. At t = π/ΩR the electron is in the upper level, whereas at t = 2π/ΩR it is back in the lower level. The process then repeats itself with a period equal to 2π/Ω_R. The electron thus oscillates back and forth between the lower and upper levels at a frequency equal to Ω_R/2π. This oscillatory behavior in response to the strong-field is called Rabi oscillation or Rabi flopping. When the light is not exactly resonant with the transition, it can be shown that the second line of eq.(2.1-24) is modified to

))

Where ² _R² w², w being the detuning. This shows that the frequency of the Rabi oscillations increases but their amplitude decreases as the light is tuned away from resonance.

At low powers, the oscillation period is longer than the radiative lifetime, and we would expect random spontaneous emission events to destroy the coherence of the superposition states, and hence curtail the oscillations. We thus have to work at higher powers to shorten the Rabi flopping period, which can be difficult to achieve in practice. The damping processes for coherent phenomena such as Rabi flopping are traditionally characterized by two time constants, T1 and T2. These two types of damping are sometimes called longitudinal relaxation and transverse relaxation, respectively. In physical terms, T1 damping is essentially determined by population decay, whereas T2 damping is related to dephasing processes.

Having considered the processes that cause dephasing in quantum systems, we can now study the detailed effects of damping on Rabi oscillations. It can be shown that if the damping rate is γ, the probability that the electron is in the upper level,

Fig 2.2 shows graphs of |c2(t)|² from eq.(2.1-25) for three different values of the damping constant. The dotted line shows the undamped case with γ = 0. The two other graphs correspond to light damping (γ/ΩR = 0.1) and strong damping (γ/ΩR = 1), respectively. Let us consider the case of light damping first. The electron performs a few damped oscillations and then approaches the asymptotic limit with |c₁|² = |c₂|² = 1/2. This asymptotic limit is exactly the behavior we would have expected from the Einstein analysis of a pure two-level system in the strong-field limit. At high optical power levels the spontaneous emission rate is negligible and the rates of stimulated emission and absorption eventually equal out, leading to identical upper and lower level populations.

Now consider the behavior for strong damping. This is effectively equivalent to the weak-field limit, because we can always make γ/ΩR large by turning down the electric field of the light beam. (See eq. (2.1-25)) No oscillations are observed, and the asymptotic value of |c2|² for very large damping rates (i.e. ξ>> 1) is given by:

2 understand the evolution of the behavior as the electric field strength is increased. At low fields, we are in the strongly damped regime where there are discrete transitions and the Einstein analysis is valid. As the field is increased, the ratio of the damping rate to the Rabi frequency decreases, and we can eventually reach the case where the oscillations are observable.

2.2 What are microcavity polaritons