3-1.1 Isotropic model
As can be seen from Eq. (2.46), the dynamics of the spontaneous emission strongly depends on the detuning Δc = ω21-ωc. Using the explicit form for A(t) in Eq. (2.46), we can calculate the probability P(t) = |A(t)|2 of the atom on the excited state in isotropic model and plot it on a time scale of the order of β.
Figure 3-1 shows the atomic population on the excited state as a function of the scaled time for various values of the atomic detuning outside the photonic band gap (Δc＞0).
Outside the photonic band gap which means in the allowed band, the atomic population vanishes in the long-time limit, regardless how close the atomic frequency is to the band-edge frequency. In other words, the excited-state population eventually decays to zero (there is no population trapped on the upper level) due to the propagating state in the allowed band. The population decay becomes exponential for sufficiently large detuning into the allowed band, where the atom emits to the continuum modes with a decay rate proportional to the density of states. The closer photonic band edge, the density of states is bigger. Therefore, we can see that as increasing Δc (the atomic level is resonant father away from the band edge), the atom would decay very rapidly.
Figure 3-2 depicts the variation of Δc with respect to the excited-state population inside the band gap (Δc＜0) and at the band edge(Δc = 0). We observe the excited-state population exhibits decay and oscillation behavior before reaching a nonzero steady-state
value due to photon localization. In other words, the spontaneously emitted photon will tunnel through the photonic crystal for a short length before being Bragg reflected back to the emitting atom to re-excite it. The result is a strongly coupled eigenstates of the electronic degrees of freedom of the atom and the electromagnetic modes of the dielectric.
This is the photon-atom bound state and is the optical analogue of an electronic impurity level bound state in the gap of a semiconductor [22]. We note that the degree of localization of the upper state population for ω21 within the gap is influenced by the density of states in the continuum of modes. This accounts for the absence of a completely localized state for ω21 deep in the gap within our model.
Moreover, spontaneous emission in free space produces monotonic and irreversible decay of upper-level amplitude, whereas here we find the so-called Rabi oscillation and the generalized Rabi frequency is defined as Ω = ∆ +n 2 4ng2 , where Δ is atomic detuning, n is the number of photons and g is coupling strength which is proportional to the overlap integral of the atom and the confined photon field. Hence, as |Δc| is increased, the population oscillates faster due to the photon field being more confined, and reaches its steady-state value more quickly. We also find that there is no unphysical photon-atom bound state, as described in Refs. [11], in the allowed band.
For comparison, the probability PS(t)=|AS(t)|2 derived from the “cut-off smoothing”
density of states in Eq. (2.55) for ∆c/β = 0.3 (near-band edge condition) were plotted in Fig.
3-3 with ε = 0 (i.e., the case of singular density of states as the solid curve), 10-5 (dash curve), and 10-3 (dot-dash curve), respectively. It also shows no unphysical photon-atom bound state with small oscillatory behavior in the short time regime and approaches zero in the long time limit. The probability PS(t) of ε = 0 is basically identical with that of ε
=10-5 and slightly differs forε=10-3. Note that Fig. 3 of Ref. [12] was plotted for ∆c =
0.3γc = 0.3Cε−1/2 = 10β 3/2, which may be still far from the band edge in the allowed band, therefore, exhibits much faster decay. It is hard to tell whether the excited-state probability derived in [12] for the near bandedge (0 < ∆c < β) would decay to zero in the long time limit or not.
By calculating the probabilities contributed from three Xn’s from Eq. (2.39) to Eq.
(2.41) separately for ∆c/β =0.01, we found these probabilities show decaying characteristics and only small oscillation in the short time regime even closer to the band edge. Indeed, it can be analytically shown that A(t) of Eq. (2.46) will always approach zero as t approaches infinity for positive detuning (∆c > 0) due to the first term,
1 2
( , )
t 2 n
E − X , and the second term, X en X tn2 , in the square bracket of Eq. (2.45) will asymptotically cancel out each other as t→ ∞ . Thus, there is neither interference effect being involved in the decaying probability of the excited state nor photon-atom bound states existing near the allowed band. It might be that the photon will not strongly interact with atom in the allowed band. Therefore, in the allowed band, the atomic frequency shift may not provide enough strength to form the photon-atom bound state.
0 5 10 15 20 25 30
Fig. 3-1. Atomic population on the excited atomic state, P(t) = |A(t)|2, as a function of βI t, for various values of the atomic detuning inside the band gap (Δc/βI＞0).
Fig. 3-2. Atomic population on the excited atomic state, P(t) = |A(t)|2, as a function of βIt, for various values of the atomic detuning inside the band gap (Δc/βI＜0) and at the band edge (Δc/βI = 0).
In anisotropic density of states, no singularity occurring like the isotropic density of states, the photon-atom bound states might also exist near the allowed band due to the shifted atomic frequency excitations. The temporal evolution of the excited atomic population for anisotropic model is given in Eqs. (2.71) and Eq. (2.74).
As shown in Fig. 3-4, when the atomic level is within the allowed band (Δc＞0), the propagating electromagnetic modes are present here. Hence, the atomic population vanishes in the long-time limit like in isotropic model. However, the population for Δc
= 0.01 (blue line) and Δc = 0.1 (green line) decay faster in anisotropic model than in
0 2 4 6 8 10
isotropic model. This is because that the anisotropic model is more realistic causing imperfect localization.
Clearly, if the atomic level is inside the photonic band gap (Δc＜0) which is plotted in Fig. 3-5, the population also exhibits decay and oscillation behavior before reaching a nonzero steady-state value for Δc/βA = −1 and Δc/βA = −5 due to the number of stable localized states which is intimately connected to the behavior of the system in the long-time limit. One such state gives rise to a steady-state population in the excited level.
These phenomena are the same as in the isotropic model except that the atomic population decay to zero for Δc/βA = −0.1 and at band edge (Δc/βA = 0). It is possible to realistically investigate the immediate neighborhood of the band-edge frequency in anisotropic model. When the atomic frequency is detuned into the band gap, a superposition of the continuum states and a bound state leads the emitted photon to the atom. Therefore, the atomic population again displays fractionalized inversion for relatively small values of the atomic detuning. However, this phenomenon should not be due to Lamb shift, which is only in order of 10-7 [23], the Lamb shift may not provide enough strength to push the atomic frequency to the gap to form the photon-atom bound state.
0 1 2 3 4 5
Fig. 3-4. Atomic population on the excited atomic state in allowed band, P(t) = |A(t)|2, as a function of βA t, for various values of the atomic detuning from Δc/βA= 0.01 (blue
Fig. 3-5. Atomic population on the excited atomic state in the photonic band gap, P(t)
= |A(t)|2, as a function of βA t, for various values of the atomic detuning from Δc/βA= −5
(black line) to Δc/βA= −0.01 (blue line).
3-1.3 Summary
In this section, we have reported the properties of spontaneous emission of an atom under dipole approximation in an isotropic and anisotropic model. We find a propagating state corresponding to transition frequency outside the photonic gap, therefore the population vanishes in the long-time limit in the isotropic and anisotropic model. On the other hand, the dressed state corresponding to transition frequency in side the gap is a no decaying photon-atom bound state except that the atomic population decays to zero when the detuning frequency is close to photonic band-edge.