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Lorentz Oscillators

The optical interactions between light and matter at the nanometer scale are important phenomena for a variety of research fields, and a rigorous understanding of these interactions requires the use of QED theory. However, nanometer-scale structures are often too complex to be solved rigorously using only QED; in these cases, a classical approach that invokes the results of QED in a phenomenological way can be applied. PyCharge uses the Lorentz oscillator (LO) model, which is an approximation from quantum theory that can be derived (e.g., from the time-dependent Schrödinger equation or a quantum master equation) to simulate the interaction of a bound charge (e.g., an electron) with light.

In the classical model, an oscillating dipole produces EM radiation which dissipates energy and modifies the self-consistent dipole moment. The recoil force, \(\mathbf{F}_\mathrm{r}\), acting on the accelerating point charges in the dipole is called the radiation reaction or radiation damping force. The equation of motion for an undriven LO (e.g., in a vacuum) that includes the radiation reaction force is given by

\[ \label{eq:underiven_EOM} m\mathbf{\ddot r_{\rm dip}}(t) + \omega_0^2 m \mathbf{r_\mathrm{dip}}(t) = \mathbf{F}_\mathrm{r}(t), \]

where \(\mathbf{r}_{\rm dip}\) is the displacement from the LO's negative charge to positive charge and \(\mathbf{\ddot r_{\rm dip}}\) is its second derivative with respect to time, \(m\) is the effective mass of the LO (further discussed below), and \(\omega_0\) is the natural angular frequency of the LO.

The radiation reaction force, \(\mathbf{F}_\mathrm{r}\), acting on the accelerating point charges in the dipole is described by the Abraham-Lorentz formula for non-relativistic velocities:

\[ \mathbf{F}_\mathrm{r}(t)=\frac{q^2}{6\pi\epsilon_oc^3}\mathbf{\dddot r_\mathrm{dip}}(t), \]

where \(\mathbf{\dddot r_\mathrm{dip}}\) is the third derivative of the displacement between the two charges. We can perform the approximation \({\mathbf{\dddot r}_\mathrm{dip}\approx -\omega_0^2\mathbf{\dot r}_\mathrm{dip}}\) in the above equation if the damping on the point charges introduced by the radiation reaction force is negligible (i.e., \(|\mathbf{F}_\mathrm{r}| \ll \omega_0^2 m |\mathbf{r}_\mathrm{dip}|\)), such that the following condition is satisfied:

\[\begin{equation}\label{eq:dipole_condition} \frac{q^2 \omega_0}{m} \ll 6\pi \epsilon_0 c^3. \end{equation}\]

In an inhomogeneous environment, an oscillating electric dipole will experience the external electric field \(\mathbf{E}_\mathrm{d}\) as a driving force, which is the component of the total electric field in the polarization direction at the dipole's origin (center of mass) position \(\mathbf{R}\) generated by the other sources in the system and its own scattered field. If the above condition is satisfied, the equation of motion for a driven LO is

\[\begin{equation}\label{eq:LO_equation} \mathbf{\ddot d}(t) +\gamma_{0} \mathbf{\dot d}(t) +\omega_{0}^{2} {\bf d}(t)= \frac{q^2}{m} \mathbf{E}_\mathrm{d}(\mathbf{R}, t), \end{equation}\]

where \({\bf d}=q{\bf r_{\rm dip}}\) is the dipole moment, \(\mathbf{\dot d}\) and \(\mathbf{\ddot d}\) are the first and second derivatives of \(\mathbf{d}\), and \(\gamma_0\) is the free-space energy decay rate given by

\[\begin{equation} \gamma_0 = \frac{q^2\omega_0^2}{6\pi\epsilon_0c^3m}. \end{equation}\]

This equation of motion for an LO corresponds to a Lorentzian atom model with transition frequency \(\omega_0\) and linewidth \(\gamma_0\) (where \(\gamma_0 \ll \omega_0\)), and is limited to non-relativistic velocities as it does not account for relativistic mass.

The effective mass \(m\) (also called the reduced mass) of the dipole is given by

\[\begin{equation} \label{eq:eff_mass} m=\frac{m_1 m_2}{m_1+m_2}, \end{equation}\]

where \(m_1\) and \(m_2\) are the masses of the two point charges in the dipole. These charges oscillate around the center of mass position \(\mathbf{R}\), defined by

\[\begin{equation} \mathbf{R} = \frac{m_1\mathbf{r}_1+m_2\mathbf{r}_2}{m_1+m_2}, \end{equation}\]

where \(\mathbf{r}_1\) and \(\mathbf{r}_2\) are the positions of the two point charges. The point charge positions can therefore be defined in terms of the displacement between the two charges \(\mathbf{r}_\mathrm{dip}\):

\[\begin{equation} \label{eq:r_1} \mathbf{r}_1 = \mathbf{R} + \frac{m_2}{m_1+m_2}\mathbf{r}_\mathrm{dip} \end{equation}\]

and

\[\begin{equation} \label{eq:r_2} \mathbf{r}_2 = \mathbf{R} - \frac{m_1}{m_1+m_2}\mathbf{r}_\mathrm{dip}. \end{equation}\]

It is also useful to discuss how the decay dynamics of LOs are related to those of a quantum TLS, in certain limits. Specifically, in the limit of weak excitation (linear response), we can connect the quantum mechanical equations of motion for a TLS to the classical equations of motion by replacing \(q^2/m\) with \(q^2f/m\), where \(f\) is the oscillator strength, defined by

\[\begin{equation} \label{eq:f} f = \frac{2m \omega_0 d_0^2}{\hbar q^2}, \end{equation}\]

where \(d_0=|\mathbf{d}(t=0)|\). We thus recover the usual expression for the SE rate \(\gamma_{0,\mathrm{TLS}}\) from an excited TLS,

\[\begin{equation} \label{eq:SE0} \gamma_{0,\mathrm{TLS}} = \frac{\omega_0^3 d_0^2}{3\pi\epsilon_0 \hbar c^3}. \end{equation}\]

An alternative argument to relate the dipole moment with the radiative decay rate is to connect the total mean energy of the LO to the ground state energy of a quantized harmonic oscillator, so that

\[\begin{equation} \frac{m\omega_0^2 d_0^2}{q^2} = \frac{\hbar\omega_0}{2}, \end{equation}\]

yielding \(q^2/m = 2\omega_0d_0^2/\hbar\), as expected. As well, the decay rate can be derived using a Fermi's golden rule approach from the interaction Hamiltonian \(H_\mathrm{int} = -{\bf d} \cdot \hat{\bf E}\), which leads to the following rate equations for the populations of an isolated TLS in a vacuum:

\[\begin{equation} \label{eq:n_e} \dot n_\mathrm{e}(t) = -\gamma_0 n_\mathrm{e}(t) \end{equation}\]

and

\[\begin{equation} \label{eq:n_g} \dot n_\mathrm{g}(t) = \gamma_0 n_\mathrm{e}(t), \end{equation}\]

where \(n_g\) and \(n_e\) are the populations of the ground and excited states (\(n_g+n_e=1\)), respectively, and we neglect all other processes. In this picture, \(\gamma_0\) is also identical to the well known Einstein A coefficient. Therefore, the energy decay rate is equivalent to the population decay rate. We stress again that we can only make the connection between LO dynamics and populations of TLS states in a regime of weak excitation.

The total energy \(\mathcal{E}\) of a dipole, which is the sum of its kinetic and potential energies, is calculated by PyCharge using

\[\begin{equation} \label{eq:dipoleE} \mathcal{E}(t) = \frac{m \omega_0^2}{2q^2} d^2(t) + \frac{m}{2q^2} \dot d^2(t), \end{equation}\]

where \(\dot d=|\mathbf{\dot d}|\). Since the total energy of a dipole \({\cal E}\) is proportional to \(n_e\), the population of the excited state using the normalized total energy can be determined by PyCharge from

\[\begin{equation} \label{eq:ne_energy} n_e(t)=\frac{\mathcal{E}(t)}{\max (\mathcal{E})}. \end{equation}\]

  1. L. Novotny and B. Hecht, Principles of Nano-Optics, Chapter 8 ↩

  2. P. Milonni and J. Eberly, Lasers Physics, Chapter 3 ↩

  3. D. Griffiths, Introduction ot Electrodynamics, Chapter 11 ↩