Quantum theory of the inverse Faraday effect

We provide a quantum theoretical description of the magnetic polarization induced by intense circularly polarized light in a material. Such effect---commonly referred to as the inverse Faraday effect---is treated using beyond-linear response theory, considering the applied electromagnetic field as external perturbation. An analytical time-dependent solution of the Liouville--von Neumann equation to second order is obtained for the density matrix and used to derive expressions for the optomagnetic polarization. Two distinct cases are treated, the long-time adiabatic limit of polarization imparted by continuous wave irradiation, and the full temporal shape of the transient magnetic polarization induced by a short laser pulse. We further derive expressions for the Verdet constants for the inverse, optomagnetic Faraday effect and for the conventional, magneto-optical Faraday effect and show that they are in general different. Additionally, we derive expressions for the Faraday and inverse Faraday effects within the Drude-Lorentz theory and demonstrate that their equality does not hold in general, but only for dissipationless media. As an example, we perform initial quantum mechanical calculations of the two Verdet constants for a hydrogenlike atom and we extract the trends. We observe that one reason for a large inverse Faraday effect in heavy atoms is the spatial extension of the wave functions rather than the spin-orbit interaction, which nonetheless contributes positively.