User:Inaginni/sandbox

=EIT vs. ATS=

Electromagnetically induced transparency (EIT) and the Autler-Townes effect (ATS) are often considered separate effects. This is generally presented as EIT representing a coherent superposition of two states resulting in what is called a dark state, while ATS represents an incoherent sum of two separate states with a Lorentzian distribution (see AC Stark effect). This view is too narrow and came about due to the two effects being discovered separately. By viewing the Hamiltonian for two situations, "Lambda" and "Vee", the connection between the two is established. This connections creates two different regimes to distinguish the two effects, high intensity and low intensity.

History


Electromagnetically induced transparency is a coherent optical effect used to improve the transparency or a medium and is also used in the creation of slow light. It's name was coined in 1989 and is an extension on the theory of coherent population trapping.The method involves the application of two lasers, a weak probe and a strong pump turned on adiabatically, to a material with at least three energy levels. The wavelength of these two lasers is such that they correspond to separate transitions in the material, with the additional constraint that these two end/ beginning states do not form an allowed transition. The effect appears through the creation of a dark state, with the level which the two lasers are exciting to/from playing this role. This level is then slaved to the remaining states, such that any atoms excited to this state do not spend long in this level. This causes once opaque materials to become transparent, through the act of absorbing and then rapidly re-emitting.

In contrast to EIT, Autler-Townes splitting, discovered in 1955 by Stanley H. Autler and Charles H. Townes, has been described as an incoherent sum of two Lorentzians and is considered a result of the AC Stark effect. The experimental setup is very similar to the one for EIT, although generally involving a stronger pump.

Lambda


The initial Hamiltonian is (in the rotating wave approximation, RWA):
 * $$\hat{\mathbf{H}}=\mathbf{\hat{H}_A}+\mathbf{\hat{H}_{AL}}$$

Where $$\mathbf{\hat{H}_A}$$ is the Hamiltonian due to the atom described as, in Dirac notation:
 * $$\mathbf{\hat{H}_A}=E_1 \mid 1 \rangle\langle 1\mid + (E_2-i\hbar\frac{\Gamma_{21}}{2}) \mid 2 \rangle\langle 2\mid + (E_3-i\hbar\frac{\Gamma_{32}+\Gamma_{31}}{2}) \mid 3 \rangle\langle 3\mid $$

And $$\mathbf{\hat{H}_{AL}}$$ is the atom-laser interaction Hamiltonian described as:


 * $$\mathbf{\hat{H}_{AL}}=\frac{\hbar\Omega_c}{2}(\mid 3 \rangle\langle 2\mid e^{-i\omega_c t}+\mid 2 \rangle\langle 3\mid e^{i\omega_c t})+\frac{\hbar\Omega_p}{2}(\mid 3 \rangle\langle 1\mid e^{-i\omega_p t}+\mid 1 \rangle\langle 3\mid e^{i\omega_p t})$$

With $$ E_i$$ the energy value for each level, $$\Gamma_{ij}$$ the decay rate from state i to j, $$ \hbar$$ the reduced Planck's constant, $$\Omega_p$$ and $$\Omega_c$$ the Rabi frequencies associated with the weak probe and strong coupling lasers, respectively.

A unitary transformation, equivalent to an effective rotating frame, is then applied to remove the time dependence:


 * $$\hat{\mathbf{U}}=\Sigma_{j=1} ^3 e^{-i\lambda_j t} \mid j \rangle\langle j\mid$$

With the Hamiltonian in the rotating frame then being described by:


 * $$\mathbf{\hat{H}^{RWA}}=\mathbf{\hat{H}_A}+\mathbf{\hat{U}^\dagger\hat{H}_{AL}\hat{U}}+i\hbar\frac{\partial \mathbf{\hat {U}^\dagger}}{\partial t}\mathbf{\hat {U}}$$

Where $$\mathbf{\hat{U}^\dagger}$$ is the Hermitian transpose of $$\mathbf{\hat{U}}$$.

Focusing on the $$\mathbf{\hat{H_{AL}}}$$ term first gives:


 * $$\mathbf{\hat{U}^\dagger\hat{H}_{AL}\hat{U}}=\frac{\hbar\Omega_c}{2}(\mid 3 \rangle\langle 2\mid e^{-i(\omega_c+\lambda_2-\lambda_3) t}+\mid 2 \rangle\langle 3\mid e^{i(\omega_c+\lambda_2-\lambda_3) t})+\frac{\hbar\Omega_p}{2}(\mid 3 \rangle\langle 1\mid e^{-i(\omega_p+\lambda_1-\lambda_3) t}+\mid 1 \rangle\langle 3\mid e^{i(\omega_p+\lambda_1-\lambda_3) t})$$

The first and third terms of $$\mathbf{\hat{H}^{RWA}}$$ give the Hamiltonian of the atom in the rotating frame:


 * $$\mathbf{\hat{H}_{A}^{RWA}}=(E_1-\hbar\lambda_1) \mid 1 \rangle\langle 1\mid + (E_2-i\hbar\frac{\Gamma_{21}}{2}-\hbar\lambda_2) \mid 2 \rangle\langle 2\mid + (E_3-i\hbar\frac{\Gamma_{32}+\Gamma_{31}}{2}-\hbar\lambda_3) \mid 3 \rangle\langle 3\mid $$

With the additional requirement that the zero energy level is moved to $$E_1$$, along with removing the time dependence, the following $$\lambda$$'s provide the solution:


 * $$\lambda_1 = \frac{E_1}{\hbar}$$
 * $$\lambda_2 = \omega_p-\omega_c+\frac{E_1}{\hbar}$$
 * $$\lambda_3 = \omega_p+\frac{E_1}{\hbar}$$

$$\mathbf{\hat{H}^{RWA}}$$ is then (using the fact that $$E_3-E_2=\hbar\omega_c$$, see figure):


 * $$\mathbf{\hat{H}^{RWA}}=(-i\hbar\frac{\Gamma_{21}}{2}-\hbar\delta_p) \mid 2 \rangle\langle 2\mid + (-i\hbar\frac{\Gamma_{32}+\Gamma_{31}}{2}-\hbar\delta_p) \mid 3 \rangle\langle 3\mid +\frac{\hbar\Omega_c}{2}(\mid 3 \rangle\langle 2\mid+\mid 2 \rangle\langle 3\mid)+\frac{\hbar\Omega_p}{2}(\mid 3 \rangle\langle 1\mid+\mid 1 \rangle\langle 3\mid)$$


 * $$\delta_p=\omega_p-\frac{E_3-E_1}{\hbar}$$

With the Hamiltonian now completed, the density matrix can be determined.

Recall that the Schrödinger equation is,


 * $$i\hbar\frac{d}{dt}\mid\psi^{RWA}\rangle=\mathbf{\hat{H}^{RWA}}\mid\psi^{RWA}\rangle$$

And that the density matrix, $$\mathbf{\hat{\rho}^{RWA}}$$, is defined as:


 * $$\mathbf{\hat{\rho}^{RWA}}=\mid\psi^{RWA}\rangle\langle\psi^{RWA}\mid$$

Define:


 * $$\mid\psi^{RWA}\rangle=\begin{bmatrix}c_1 \\c_2\\c_3 \end{bmatrix}$$

Using the time evolution described in the Schrödinger equation, this gives:


 * $$\dot c_1=-i\frac{\omega_p}{2}c_3$$
 * $$\dot c_2=(i\delta_p-\frac{\Gamma_{21}}{2})c_2-i\frac{\Omega_c}{2}c_3$$
 * $$\dot c_3=(i\delta_p-\frac{\Gamma_{31}+\Gamma_{32}}{2})c_3-i\frac{\Omega_c}{2}c_2-i\frac{\Omega_p}{2}c_1$$

The time derivative description is not as simple as the above equation for $$\mathbf{\hat{\rho}^{RWA}}$$ would suggest. Since there are decay terms involved in this Hamiltonian it is necessary to add feeding terms to the density matrix. These feeding terms are then:


 * $$\mathcal{L}_{feed}=(\Gamma_{31}\rho_{33}+\Gamma_{21}\rho_{22})\mid 1\rangle\langle 1 \mid+\Gamma_{32}\rho_{33}\mid 2\rangle\langle 2 \mid$$

The time derivative of the density matrix is then described as:




 * $$\dot\rho_{ij}=\dot {c}_ic_j^*+c_i\dot {c}_j^*+\langle i \mid\mathcal{L}_{feed}\mid j\rangle$$

This gives the following elements of $$\mathbf{\hat{\rho}^{RWA}}$$:


 * $$\dot \rho_{11}=-i\frac{\Omega_p}{2}(\rho_{31}-\rho_{13})+\Gamma_{21}\rho_{22}+\Gamma_{31}\rho_{33}$$
 * $$\dot \rho_{22}=-\Gamma_{21}\rho_{22}-i\frac{\Omega_c}{2}(\rho_{32}-\rho_{23})+\Gamma_{32}\rho_{33}$$
 * $$\dot \rho_{33}=-(\Gamma_{31}+\Gamma_{32})\rho_{33}-i\frac{\Omega_c}{2}(\rho_{23}-\rho_{32})-i\frac{\Omega_p}{2}(\rho_{13}-\rho_{31})$$
 * $$\dot \rho_{12}=-i\frac{\Omega_p}{2}\rho_{32}-(i\delta_p+\frac{\Gamma_{21}}{2})\rho_{12}+i\frac{\Omega_c}{2}\rho_{13}$$
 * $$\dot \rho_{13}=-i\frac{\Omega_p}{2}(\rho_{33}-\rho_{11})-(i\delta_p+\frac{\Gamma_{32}+\Gamma_{31}}{2})\rho_{13}+i\frac{\Omega_c}{2}\rho_{12}$$
 * $$\dot \rho_{23}=-\frac{\Gamma_{21}+\Gamma_{31}+\Gamma_{32}}{2}\rho_{23}-i\frac{\Omega_c}{2}(\rho_{33}-\rho_{22})+i\frac{\Omega_p}{2}\rho_{21}$$

Note that, as expected, the trace of $$\mathbf{\dot\hat{\rho}^{RWA}}$$ is 0. These 6 equations represent all terms since $$\mathbf{\hat{\rho}^{RWA}}$$ is a Hermitian matrix.

For the conditions of $$ \Omega_c >> \Omega_p$$ and state 3 the dark state ($$\rho_{22}\approx\rho_{33}\approx 0,\rho_{11}\approx 1$$), these equations can be solved in steady-state. The term of interest is $$\rho_{13}$$, as this describes the probe transition. This becomes


 * $$\rho_{13}\propto \frac{\delta_p-i\gamma_{12}}{\mid\Omega_c \mid ^2/4-(\delta_p-i\gamma_{13})(\delta_p-i\gamma_{12})}$$

Where:


 * $$\gamma_{nm}=\Sigma_{t=1}^3 \Gamma_{nt}+\Gamma_{mt}$$

A plot of the imaginary part of this term, corresponding to the absorption is shown. This plot identifies an intensity threshold, above which is seen the typical separated Lorentzians of ATS and below which is seen the interference effect of EIT. This intensity threshold is defined as (for the Lambda configuration) :


 * $$\Omega_t=\gamma_{13}-\gamma_{12}$$

With this, the connection between ATS and EIT is now established.

Vee
This connection is not straightforward, however. To see this it is necessary to analyze the Vee level configuration.

The Hamiltonian, in RWA, for this configuration is:


 * $$\mathbf{\hat{H}}=E_1\mid 1\rangle\langle 1\mid+(E_2-i\hbar\frac{\Gamma_{21}}{2})\mid 2\rangle\langle 2\mid+(E_3-i\hbar\frac{\Gamma_{31}+\Gamma_{32}}{2})\mid 3\rangle\langle 3\mid+\hbar\frac{\Omega_p}{2}(\mid 1\rangle\langle 3\mid e^{i\omega_p t}+\mid 3\rangle\langle 1\mid e^{-i\omega_p t})+\hbar\frac{\Omega_c}{2}(\mid 1\rangle\langle 2\mid e^{i\omega_c t}+\mid 2\rangle\langle 1\mid e^{-i\omega_c t})$$

When performing a unitary transformation to the effective rotating frame, the $$\lambda$$'s are (with the same conditions as in the Lambda configuration):


 * $$\lambda_1 = \frac{E_1}{\hbar}$$
 * $$\lambda_2 = \omega_c+\frac{E_1}{\hbar}$$
 * $$\lambda_3 = \omega_p+\frac{E_1}{\hbar}$$

This gives the following Hamiltonian in the rotating frame:


 * $$\mathbf{\hat{H}^{RWA}}=-i\frac{\Gamma_{21}}{2}\mid 2\rangle\langle 2\mid-\hbar(\delta_p+i\frac{\Gamma_{31}+\Gamma_{32}}{2})\mid 3\rangle\langle 3\mid+\hbar\frac{\Omega_p}{2}(\mid 1\rangle\langle 3\mid+\mid 3\rangle\langle 1\mid)+\hbar\frac{\Omega_c}{2}(\mid 1\rangle\langle 2\mid+\mid 2\rangle\langle 1\mid)$$

Using Schrödinger's equation gives:


 * $$\dot c_1=-i\frac{\omega_p}{2}c_3-i\frac{\omega_c}{2}c_2$$
 * $$\dot c_2=-\frac{\Gamma_{21}}{2}c_2-i\frac{\Omega_c}{2}c_1$$
 * $$\dot c_3=(i\delta_p-\frac{\Gamma_{31}+\Gamma_{32}}{2})c_3-i\frac{\Omega_p}{2}c_1$$

With feed terms:


 * $$\mathcal{L}_{feed}=(\Gamma_{31}\rho_{33}+\Gamma_{21}\rho_{22})\mid 1\rangle\langle 1 \mid+\Gamma_{32}\rho_{33}\mid 2\rangle\langle 2 \mid$$

Solving for the time derivative of the density matrix gives:




 * $$\dot \rho_{11}=-i\frac{\Omega_p}{2}(\rho_{31}-\rho_{13})-i\frac{\Omega_c}{2}(\rho_{21}-\rho_{12})+\Gamma_{21}\rho_{22}+\Gamma_{31}\rho_{33}$$
 * $$\dot \rho_{22}=-\Gamma_{21}\rho_{22}-i\frac{\Omega_c}{2}(\rho_{12}-\rho_{21})+\Gamma_{32}\rho_{33}$$
 * $$\dot \rho_{33}=-(\Gamma_{31}+\Gamma_{32})\rho_{33}-i\frac{\Omega_p}{2}(\rho_{13}-\rho_{31})$$
 * $$\dot \rho_{12}=-i\frac{\Omega_p}{2}\rho_{32}-i\frac{\Omega_c}{2}(\rho_{22}-\rho_{11})-\frac{\Gamma_{21}}{2}\rho_{12}$$
 * $$\dot \rho_{13}=-i\frac{\Omega_p}{2}(\rho_{33}-\rho_{11})-(i\delta_p+\frac{\Gamma_{32}+\Gamma_{31}}{2})\rho_{13}-i\frac{\Omega_c}{2}\rho_{23}$$
 * $$\dot \rho_{23}=-(i\delta_p+\frac{\Gamma_{21}+\Gamma_{31}+\Gamma_{32}}{2})\rho_{23}-i\frac{\Omega_c}{2}\rho_{13}+i\frac{\Omega_p}{2}\rho_{21}$$

Solving these equations simultaneously in steady-state, $$\Omega_c>>\Omega_p$$ and state 1 being the dark state ($$\rho_{22}\approx\rho_{11}\approx 0,\rho_{33}\approx 1$$), gives :


 * $$\rho_{13}\propto \frac{(\delta_p-i\gamma_{13})\frac{\mid\Omega_c\mid ^2}{4\gamma_{12}^2}+\delta_p-i\gamma_{23}}{\mid\Omega_c\mid ^2/4-(\delta_p-i\gamma_{23})(\delta_p-i\gamma_{13})}$$

A plot of the imaginary part of this term can be seen. For this case, the threshold is defined as :


 * $$\Omega_t = \gamma_{12}$$

Instead of seeing EIT effects, at low intensities for the Vee configuration the whole absorption profile is lowered. This demonstrates that although there exists a direct connection between EIT and ATS there is more to discover on the EIT side of this intensity transition.