Tools

This page provides conversion between different units and physical quantities for quantum emitters and cavity modes. It uses math.js to support unitful input, so quantities can be entered with a wide range of units (see reference, plus debye = D), and also using supported physical constants, e.g., you can enter “0.3934303 elementaryCharge bohrRadius” to get $1~D$ for $\mu$. See below for a detailed description of the different quantities and of the conventions we use.

Energy $\hbar\omega$, angular frequency $\omega$, frequency $f$, oscillation period $T_{\mathrm{osc}}$, wavelength $\lambda$, (angular) wavenumber $k=2\pi/\lambda$, (spectroscopic) wave number $\nu = 1/\lambda$, and equivalent temperature $T$ are related by $\hbar\omega = h f = h/T_{\mathrm{osc}} = \hbar c k = h c \nu = h c/\lambda = k_B T$.

The decay rate $\gamma$ is the population (or probability) decay rate, with associated lifetime $\tau=1/\gamma$, leading to $P(t) \propto e^{-\gamma t} = e^{-t/\tau}$. The decay rate is then equivalent to the full width half maximum (FWHM) of the corresponding Lorentzian in spectral measurements. With these conventions, the corresponding complex energy in effective non-Hermitian descriptions is equal to $E_c = \hbar\omega - \frac{i}{2} \hbar\gamma$. The quality factor is given by $Q = \omega/\gamma$.

For a dipolar emitter with dipole moment $\mu$, the free-space decay rate $\gamma$ is given by $\gamma = \frac{\omega^3 \mu^2}{3\pi \epsilon_0 \hbar c^3}$. The oscillator strength is given by $f_{\mathrm{osc}} = \frac{2 m_e \omega \mu^2}{3 e^2 \hbar}$ and is unitless.

For cavity modes, the effective mode volume $V_{\mathrm{eff}}$ and maximum single-photon field strength $E_{\mathrm{1ph}}$ are related by $E_{\mathrm{1ph}} = \sqrt{\frac{\hbar\omega}{2\epsilon_0 V_\mathrm{eff}}}$, with corresponding single-photon intensity $I_{\mathrm{1ph}} = \frac{1}{2} \epsilon_0 c E_{\mathrm{1ph}}^2$. We also give the coupling parameter $\lambda_c = \sqrt{\frac{1}{\epsilon_0 V_{\mathrm{eff}}}}$ that is sometimes used in the literature. Since it does not have “nice” units, we calculate its value in Hartree atomic units where $4\pi\epsilon_0 = 1$ and thus $\lambda_c [\mathrm{a.u.}] = \sqrt{\frac{4\pi}{V_{\mathrm{eff}}[\mathrm{a.u.}]}} = \sqrt{\frac{4\pi a_0^3}{V_{\mathrm{eff}}}}$.

The emitter-mode coupling strength is given by $\hbar g = \vec{E}_{\mathrm{1ph}} \cdot \vec{\mu}$, with Rabi frequency $\Omega_R = 2g$ (calculated for perfect alignment, $\hbar\Omega_R = 2E_{\mathrm{1ph}} \mu$). The cooperativity parameter is $\eta = \frac{4g^2}{\gamma_c \gamma_e} = \frac{\Omega_R^2}{\gamma_c \gamma_e}$, where $\gamma_c$ and $\gamma_e$ are the decay rates (=FWHM linewidths) of the cavity and emitter, respectively. We also calculate the parameters $C_1 = \frac{\Omega_R}{|\gamma_c-\gamma_e|/2}$ and $C_2 = \frac{\Omega_R}{\sqrt{\gamma_c^2/2+\gamma_e^2/2}}$, with values $>1$ indicating that the strong coupling regime is reached, depending on which convention is used (also see below). These quantities can be understood by using that the system is described by the non-Hermitian Hamiltonian

\begin{equation} H = \hbar\begin{pmatrix}\omega_c - \frac{i}{2} \gamma_c & \Omega_R/2\\ \Omega_R/2 & \omega_e - \frac{i}{2} \gamma_e\end{pmatrix}. \end{equation} with eigenvalues \begin{equation} E_{\pm} = \hbar\left(\frac{\omega_{c}+\omega_{e}}{2} - \frac{i}{2} \frac{\gamma_{c}+\gamma_{e}}{2} \pm \frac{1}{2} \sqrt{\Omega_{R}^2 + (\omega_{c}-\omega_{e} - \frac{i}{2} (\gamma_{c}-\gamma_{e}))^2}\right). \end{equation}

Both $C_1$ and $C_2$ assume that cavity and emitter are on resonance, $\omega_c=\omega_e$. The condition $C_1>1$ indicates that the term in the square root becomes positive, such that the eigenvalues of $H$ show real energy splitting, i.e., $\mathrm{Re}(E_+ - E_-) \not= 0$, while the imaginary parts become equal $\mathrm{Im} E_+ = \mathrm{Im} E_- = \hbar \frac{\gamma_c+\gamma_e}{4}$. In contrast, $C_2>1$ corresponds to the more stringent criterion that this splitting is larger than the FWHM linewidth $\hbar\frac{\gamma_c+\gamma_e}{2}$ of the two new modes, i.e., that $(E_+-E_-)^2 > \left(\hbar\frac{\gamma_c+\gamma_e}{2}\right)^2$. We note further that $C_1$ and $C_2$ correspond exactly to the strong coupling conditions 2 and 3 discussed in a recent review on strong coupling in molecular systems. Comparison to that review also highlights the infinity of notations used in the field: The decay rates there are amplitude decay rates, such that “our” rates $\gamma_{c,e}$ correspond to twice “their” rates $\kappa,\gamma$. Similarly, they use $\Omega$ for the “actual” Rabi splitting $\mathrm{Re}(E_+ - E_-)/\hbar$, while we define it simply as $\Omega_R = 2g$ (the two definitions are equal when $\gamma_c=\gamma_e$).

Note finally that for the emitter-cavity coupling calculations, we are only taking into account the radiative linewidth of the emitter (determined by the dipole moment). In many experimental realizations, the emitter linewidth will be dominated by other effects (dephasing, inhomogeneous broadening, …).