Most readers familiar with the behaviour of EM waves in materials will recognise the following equation:

\(\chi (\omega) = \frac{\omega_p^2}{(\omega_0^2-\omega^2) \pm i\omega \gamma} \)

Which is the susceptibility of a material approximated by the Drude-Lorentz model. This model treats the electrons as classical charged particles bound to their respective atoms/ chemical bonds by a spring force. Hence, the electrons are just classical harmonic oscillators where an external electric field provides the driving force and they all experience some form of ‘resistance’ in the form of \(\gamma\). The \(\pm\) sign depends on \(e^{\pm j\omega t}\) convention and \(\omega_p^2 = \frac{Ne^2}{\epsilon_0 m_e}\), the plasma frequency of the material. The plasma frequency increases with the number density(\(N\))of the harmonic oscillators involved and typically this is in the THz to visible range for most materials. However, the behaviour of electrons is often quantum mechanical in nature and the question is given just the energy levels involved, could we derive from first principles the complex susceptibility of the system without assuming that the potential is harmonic(which often fails if the amplitudes are too large). In this post I seek to show the reader how the weak field excitation of the quantum system will give similar properties to the drude-lorentz mode.

The 2 level system and incoherence.

Consider a 2-level system with eigenstates \(|1\rangle , |2\rangle\)  (ground and excited states respectively). Assuming that there is only one excited state for the polarisation involved and the excitation is purely sinusoidal, then the Hamiltonian in the basis is:

\(H = E_1 |1 \rangle \langle 1| + E_2|2 \rangle \langle 2|+V(t)|1 \rangle \langle 2| + V(t)|2 \rangle \langle 1|\)

Where \(V(t) = \hbar \Omega cos(\omega t)\). For the case of most atomic excitations, \(\hbar\Omega = eE_0\langle 2|z |1 \rangle = eE_0\langle 1|z |2 \rangle = pE_0\) and \(p\) is the transition dipole moment between the states. Like the classical model, decoherence causes damping and for the case of weak field excitations, \(T_2\) is the dominant cause. The evolution of the quantum system is described by the evolution of the density operator since the system is not fully coherent:

\(\frac{d\hat{\rho}}{dt} = \frac{i}{\hbar}[\hat{\rho},H] + [\frac{d\hat{\rho}}{dt}]_{relax}, \gamma = 1/T_2\)

For the measurement of the dipole moment of the system, the operator \(\hat{\mu}\) is defined as:

\(-p|1 \rangle \langle 2| – p|2 \rangle \langle 1| \)

And the expected value of the dipole moment is given by: \(Tr[\hat{\rho}]\). Diving into the calculations and finding the DE for the off-diagonal terms gives:

\(\dot{\rho}_{12} = i\Omega cos(\omega t)(\rho_{11}-\rho_{22}) + i\omega_0\rho_{12}-\gamma\rho_{12}\)

\(\dot{\rho}_{21} = i\Omega cos(\omega t)(\rho_{22}-\rho_{11}) – i\omega_0\rho_{21}-\gamma\rho_{21}\)

Let \(\rho_{12} = \rho_0e^{i\omega t}, \rho_{21} = \rho_0^*e^{-i\omega t}\). In the weak field approximation, inversion is typically almost absent, and the population of excited states is low. Hence, \(\rho_{11}\approx 1\) and \(\rho_{22}\approx 0\). The cosine excitation can be expanded in terms of complex sinusoids: \(cos(\omega t) = \frac{1}{2}(e^{i\omega t}+e^{-i\omega t})\).  And now, for each equation, factoring out the relevant complex term and ignoring the remaining \(e^{\pm 2i\omega t}\) term (as this averages to 0 very quickly) gives:

\([i(\omega – \omega_0)+\gamma]\rho_0\approx\frac{i\Omega}{2}, [-i(\omega – \omega_0)+\gamma]\rho_0^*\approx-\frac{i\Omega}{2}\)

Both of which are consistent and give:

\(\rho_0 = \frac{\Omega}{2(\Delta – i\gamma)}, \Delta = \omega – \omega_0\)

Hence, the dipole moment is:

\(p_z = -\frac{p\Omega}{2}[\frac{e^{i\omega t}}{\Delta – i\gamma}+\frac{e^{-i\omega t}}{\Delta +i\gamma}] = -\frac{p\Omega}{\sqrt{\Delta^2+\gamma^2}}cos(\omega t + tan^{-1}\frac{\gamma}{\Delta})\)

With the expression for \(\Omega\) given earlier, the complex polarizability of the system is:

\(\alpha = \frac{-p^2}{\hbar(\Delta-i\gamma)}\)

If the number density is \(n\) and if it is low, the susceptibility is given by:

\(\chi(\Delta) = \frac{-np^2}{\hbar\epsilon_0(\Delta-i\gamma)} = \frac{-np^2(\Delta+i\gamma)}{\hbar\epsilon_0(\Delta^2+\gamma^2)}\)

The complex relative permittivity is:

\(\epsilon_r= 1+\chi(\Delta)= 1-\frac{np^2(\Delta+i\gamma)}{\hbar\epsilon_0(\Delta^2+\gamma^2)}\)

For dilute cases, the refractive indices (\(\tilde{n}=n_{Re}-jn_{Im}\)) are:

\(n_{Re} = 1-\frac{np^2\Delta}{2\hbar\epsilon_0(\Delta^2+\gamma^2)}\)

\(n_{Im} = \frac{np^2\gamma}{2\hbar\epsilon_0(\Delta^2+\gamma^2)}\)

Now, there is a clearer picture for the relationship between the Drude-Lorentz model and its quantum origins. Notice that if the system was perfectly coherent without the decoherence issue, then \(\gamma\) would not exist and you would be getting rabi oscillations instead where the population of the atoms cycle in btw their excited and ground states. I am not sure if this is fully correct and I have yet to ask my tutor on the validity of my approximations. One thing I find interesting though is the difference in dependence on the detuning(\(\Delta\)). While both the classical and quantum have similar trends, their denominator is different if you notice. I do hope to verify my plot one day by measuring the refractive index of a dilute alkali metal vapour which should be an easier system to isolate 2 levels.

The faraday effect, circular birefringence and dichroism.

Now, depending on the polarisation of the photon, assuming that it travels along the z axis, it will either excite the system from the ground state(assume S orbital) to and excited states with l+1 and \(m\pm 1\) depending on the polarisation. A magnetic field will cause the different excited states to shift in energy level due to the zeeman effect. This causes the resonant frequency of the LCP and RCP excitations to be different. Thus, at the same frequency, LCP and RCP will have different detunings from their resonances which results in different refractive indices(circular birefringence). This effect is known as the faraday effect where the magnetic field induces the birefringence. I find it really interesting that the two level system is able to explain this phenomenon precisely without having to go through the classical picture of electrons moving in circular orbits around the their central atoms.