User:Wgopu/resonance fluorescence

Resonance Fluorescence , as the name suggests is the fluorescence or spontaneous emission from an atom or molecule when it interacts with a continuous electromagnetic field. It is an important phenomena coming under a much broader field of interactions of photon with atoms and molecules. This article greatly deals with resonance fluorescence of a two level atom interacting with light which is mostly monochromatic. The classical regime of the same has been greatly studied as a dipole oscillating in a driven oscillating field, and damped by radiation, which is the classical rabi problem. More exciting results are uncovered when the fully quantum regime is explored, i.e taking quantized electromagnetic field instead of classical.

== Scattering of light by a two level atom == The Hamiltonian of such a system can be written as below. The Electric field operator is given by the quantized field which can be found in quantization of electromagnetic field. The free hamiltonians of atom and field is as shown below.



$$\begin{align} H= H_A + H_f + H_{AF}  \quad \text{where } \end{align} $$

$$H_A = \sum E_i | u_i\rangle\langle u_i| \text{ is the atom hamiltonian}$$

$$H_f = \sum_{k,\lambda} \hbar \omega_k (a_{k,\lambda}^\dagger a_{k,\lambda} + \frac{1}{2} ) \text{is the field hamiltonian}$$

$$H_{AF} = - \hat d. \hat E (\overrightarrow R) =(\sum_{k,\lambda} \hbar g_{k,\lambda} a_{k,\lambda} \hat\sigma_+ + \hbar g^*_{k,\lambda} a_{k,\lambda}^\dagger \hat\sigma_-) + (\sum_{k,\lambda} \hbar g_{k,\lambda} a_{k,\lambda} \hat\sigma_- + \hbar g^*_{k,\lambda} a_{k,\lambda}^\dagger \hat\sigma_+) \text{ is the interaction hamiltonian}$$

Now after applying the rotating wave approximation the second term of $$H_{AF}$$ which is the counter rotating gets neglected. The total hamiltonian becomes

$$H = \frac{\hbar \omega_0}{ 2} \sigma_z+ \sum_{k,\lambda} \hbar \omega_k (a_{k,\lambda}^\dagger a_{k,\lambda} + \frac{1}{2} ) + (\sum_{k,\lambda} \hbar g_{k,\lambda} a_{k,\lambda} \hat\sigma_+ + \hbar g^*_{k,\lambda} a_{k,\lambda}^\dagger \hat\sigma_-)$$

The laser beam is in one mode and has a single wavelength $$\omega_l$$. The initial state of the combined quantum system is the coherent state in that single mode and vacuum in all other modes. We expect the solution to be some sort of spontaneous emission by the atom in any mode. Solving the full quantum mechanical problem will give an insight to the mode and wavelength of the emitted light. The initial state is $$|\Psi\rangle = |\psi\rangle_A \otimes |\alpha_l e^{-i \omega_l t} \rangle_{k=k_l} \otimes |0\rangle_{k\neq k_l}$$. It is common in solving Schrodinger equation to do a unitary transformation and simplify the problem. Heisenberg picture, Interaction picture are examples of this. In solving the above hamiltonian B R Mollow came up with a transformation now known as mollow transformation. In this picture the coherent state is displaced back to vacuum by acting the dagger of the displacement operator.

$$|\Psi\rangle \Rightarrow D^\dagger ( \alpha_l) |\Psi\rangle \text{ where } \hat D_{k_l} ( \alpha_l) = e^{(\alpha_l \hat a^\dagger_{k_l} - \alpha_l^* \hat a) }   $$ so that the operators change according to $$\hat \Theta \Rightarrow \hat D_{k_l}^\dagger ( \alpha_l) \hat \Theta \hat D_{k_l} ( \alpha_l)  $$.$$\text{Intial state } |\psi\rangle \text{ becomes } |\psi\rangle_A \otimes |0\rangle_{k} \text{ and } \hat a \Rightarrow \hat a + \alpha_l e^{-i \omega_l t}  \qquad \hbar g_{k_l} \alpha_l = \frac{\hbar \Omega}{2} ,\quad \Omega \text{ is the rabi frequency}  $$.

The hamiltonian after this transformation becomes $$H= \frac{\hbar \omega_0}{2} \hat \sigma_z + \frac{\hbar \Omega}{2} ( \hat \sigma_+ e^{-i \omega_l t} + \hat\sigma_- e^{i \omega_l t} ) \quad +  \sum_{k,\lambda} \hbar \omega_k \hat a^\dagger_{k,\lambda} \hat a_{k,\lambda} +   \sum_{k,\lambda} \hbar ( g_{k,\lambda} \hat \sigma_+ \hat a_{k,\lambda} + g^*_{k,\lambda} \hat a^\dagger \hat\sigma_-)$$. The first two term correspond to the hamiltonian of semi classical approach to this problem when we treat the atom to be quantum but with classical electromagnetic field. Where we have the oscillating dipole and the dipole radiation. Last two term represent the interaction of the atom with the vacuum. The formal solution for this hamiltonian is as follows

$$\hat a_{k,\lambda}(t) = \hat a_{k,\lambda}(0) e^{-i \omega_k t} + \int^t_0 dt^' e^{-i \omega_k (t-t^')} \sqrt{\frac{2\pi\hbar \omega_k}{V}} \overrightarrow d_{eg}. \overrightarrow \epsilon_{k,\lambda} \hat\sigma_-(t^')$$

$$ \begin{align} \Rightarrow \overrightarrow \hat E^+(\overrightarrow r,t) = \overrightarrow \hat E^+_{vac}(\overrightarrow r,t) +\overrightarrow \hat E^+_{source}(\overrightarrow r,t) = \sum_{k,\lambda} \sqrt{\frac{2\pi\hbar \omega_k}{V}} \hat a_{k,\lambda} (0) \overrightarrow \epsilon_{k,\lambda} e^{i(\overrightarrow k.\overrightarrow r-\omega_k t)} +  \eta \hat\sigma_- ( t-\frac{r}{c})  \\ \text{where } \eta= - \frac{\omega^2_0}{c^2} \frac{(\overrightarrow d_{eg})_{\perp}}{r} \end{align} $$

As we discussed the whole problem is the scattering of a light by a two level atom. The scattered light can be in any mode or in some wavelength according to the solutions given by the above equations. The scattering can be broadly divided into two categories viz. elastic and inelastic scattering.

Elastic and inelastic scattering
The elastic or coherent scattering conserves the energy and therefore has all of its frequency equal to the driving frequency $$\omega_l$$. All other frequency component come from the inelastic scattering, where the process involved is not just the absorption and re-emission of a photon from the laser field. As we will see in the following discussion the mode in which the light is emitted is not completely random as we have in the classical picture. There is a definite state which gives the probability to be in each mode.

The detector kept around the atom detects the intensity of light scattered at different directions at different times $$(\overrightarrow r,t)$$. We can calculate the intensity of scattered light.

$$\begin{align} I(\overrightarrow r,t) = \langle \overrightarrow \hat E^{(-)} (\overrightarrow r,t) \overrightarrow \hat E^{(+)} (\overrightarrow r,t) \rangle = G^{(1)} (\overrightarrow r,t;\overrightarrow r,t) = \eta^2 \langle \hat \sigma_+(t-\frac{r}{c}) \hat \sigma_-(t- \frac{r}{c}) \rangle \\ \eta^2 \langle (|e\rangle\langle g | ) (|g\rangle\langle e | ) (t-\frac{r}{c})\rangle =\eta^2 P_e(t-\frac{r}{c}). \end{align}$$

$$p_e$$ is the probability of finding the atom in excited state at a retarded time which accounts for the time taken by the light to reach the detector. The atom has to be first in the excited state so as to emit a photon and then comes the probability of that being in the direction of the detector. That explains the form of Intensity we obtained. Now the heisenberg evolution of $$\hat \sigma_-(t)$$ has its mean value component and the fluctuations around it. so that $$\hat \sigma_-(t) = \langle \hat \sigma_-(t) \rangle + \delta \hat \sigma_-(t) $$. This splitting helps us to explain the two parts of scattered light elastic/coherent (induced by the first term) and inelastic/incoherent (second term or the fluctuation). Going to a frame which rotates with frequency of incident laser light and substituting it in the intensity equation, we obtain

$$I(\overrightarrow r,t) = \eta^2 \langle \hat\sigma_+(t-\frac{r}{c}) \hat\sigma_-(t-\frac{r}{c}) \rangle_{RF} + \eta^2 \langle \delta\hat\sigma_+(t-\frac{r}{c}) \delta\hat\sigma_-(t-\frac{r}{c}) \rangle_{RF} = I_{coh} + I_{incoh}$$

From the steady state solution of optical bloch equations with the saturation parameter s defined as $$s=\frac{\frac{\Omega^2}{2}}{\Delta^2 + \frac{\Gamma^2}{4}}$$, the intensities can be found out

$$I = P_e = \frac{s}{2 (1+s)} \qquad \frac{1}{\eta^2}I_{coh} = |\langle\hat\sigma_+\rangle|^2 = \frac{s}{2 (1+s)^2} \qquad \Rightarrow   \frac{1}{\eta^2}I_{incoh} = I-I_{coh} = \frac{1}{2} \frac{s^2}{(1+s)^2}$$



In low saturation regime (weak incident field or high detuning) it can be seen that the coherent or elastic scattering (with frequency same as incident laser beam) dominates over incoherent part. It grows linearly where as incoherent increases quadratic. It crosses when s=1 and in high saturation regime (strong field or near resonance) the incoherent part which is due to the fluctuations dominates. We will see this in detail in the next section.

Spectrum of resonance fluorescence
The spectrum is the total intensity of light scattered as a function of its frequency. It is obtained by integrating the two time correlation function for all time. Intensity is the two time correlation function at same time. $$G^{(1)} (\tau) = \langle \overrightarrow \hat E^{(-)} (\overrightarrow r,t+\tau) \overrightarrow \hat E^{(+)} (\overrightarrow r,t) \rangle$$ and $$S(\omega) = \eta^2 \int^\infty_{-\infty} d\tau  G^{(1)}(\tau)  e^{-i\omega \tau}$$. In the same way as the intensity in the previous section spectrum can be split into two parts coherent and incoherent. We get $$S_{coh}= I_{coh}  \delta(\omega_l - \omega) $$. So the coherent spectrum fully consist of scattered light at the frequency of the laser beam. This is the elastic scattering. The elastic scattered light is exactly similar to the linear response of a classical dipole. So it is the "classical" part of the scattered light. It will show the common phenomena of classical light such as interference. Needless to say it is the incoherent part of the spectrum that give rise to the "quantumness" of the light scattered. $$S_{incoh}(\omega) = \frac{\eta^2}{2\pi} \int^\infty_{-\infty} d\tau e^{i (\omega_l-\omega) \tau} \langle \delta\hat\sigma_+(\tau) \delta\hat\sigma_-(0) \rangle_{RF}$$ ,  is the incoherent part and is not monochromatic as the elastic scattering. In order to know about the frequencies involved we will have to solve for the expectation value inside the spectrum. Under the markoff approximation the quantum regression theorem on the optical bloch equations which has the form $$\frac{d}{dt} \langle \hat \sigma_i \rangle = \sum_j B_{i,j} \langle \hat \sigma_j \rangle $$, gives following results

$$\frac{d}{dt} \langle \delta\hat\sigma_i(\tau) \delta\hat\sigma_-(0) \rangle = \sum_j B_{i,j} \langle\delta\hat\sigma_i(\tau) \delta\hat\sigma_j(0) \rangle $$

It has an exponential solution and therefore the frequency of the scattered light depends on the eigenvalues of B matrix. We will now explore in detail the two cases of low saturation (weak field limit) and high saturation (strong field limit).

Far off resonance, Low saturation

In this regime we have $$\Delta \gg \Omega, \Gamma, S\ll 1$$. The saturation parameter is very small and as we discussed earlier the coherent part of the scattered light dominates of the incoherent part. The eigenvalues of B in the case are $$\lambda_1=-2\Gamma, \lambda_2 = -i(\omega_l-\omega_0) -\Gamma, \lambda_3=-i(\omega_0-\omega_l) -\Gamma $$, and the frequencies are $$\omega_l- Im(\lambda_i)$$. So we have three peaks, one of width $$2\Gamma$$ at $$\omega_l$$ and two of width $$\Gamma$$ at $$\omega_0$$ and $$2\omega_l - \omega_0$$. It is surprising to note that there are two side bands besides the expected central peak. The perturbation picture can be used to sort of understand where these peaks come from. The first order correction come from the two photon process where the atom after absorbing photon from the laser field does not decay back to the ground state but some other state and thus emitting a photon of lower frequency (lower than that of the laser). The atom then absorbs one more photon and excite to excited state and decays back to the ground state which emits a photon of higher frequency. The central peak comes from the second order correction. Instead of going back to the initial state after the absorption of second photon the atom can again come to the previous state emitting a photon of laser frequency and then again repeating the process to emit a third photon of higher frequency. It accounts for the fact that the central peak is much lower in height than the side bands since 2nd order corrections are smaller.



On resonance, High saturation: Mollow triplet
This is the regime opposite to the one discussed in previous section. We have $$\Delta=0, \Omega\gg \Gamma, S\gg1$$. This was first studied and calculated by B.R Mollow, in his original paper. The eigenvalues in this case are $$\lambda_0=-\Gamma, \lambda_{\pm}=\pm i\Omega -\frac{3\Gamma}{2}$$. So we have three peaks at $$\omega_l, \omega_l \pm \Omega$$, each of width $$\Gamma, \frac{3\Gamma}{2}$$ respectively. Notice that here we have the central peak narrower and higher than the side bands. The figure shows the Mollow triplet for different rabi frequency.



When off resonance the rabi frequency gets modified by the detuning. SInce there is strong interaction between atom and laser the higher order corrections are not negligible and therefore perturbation picture is of no use here to explain it. The dressed state picture by Cohen-Tannoudji of the jaynes cummings model give the dressed states as shown in figure. The figure itself explains the existance of the three peaks. Apart from the inelastic part of the scattering light there are Other phenomena like photon antibunching, sub poisson distribution etc are also indications of "quantum" light.