# Off-Resonance Control and All-Optical Switching: Expanded Dimensions in Nonlinear Optics
## 2 Conventional Nonlinear Optics  
From the viewpoint of quantum electrodynamics (QED) [43; 44], all optical interactions occur through the annihilation and creation of photons. A central motivation to utilizing the quantized radiation field over classical wave theory is the ability to account for the spontaneous generation of new modes of light [45]. This has obvious significance in the field of nonlinear optics, where new modes of light are invariably produced in processes such as harmonic generation and hyper-Raman scattering. One principle flaw of using classical wave theory to describe specific nonlinear processes is that it suggests the ability of generating a second harmonic even at the intensity level of a single photon. Moreover, for spectroscopic interactions involving quantum transitions, the interplay of energetics and selection rules based on angular momentum can be understood in no other satisfactory way. Within the realm of nonlinear optics, there are numerous further reasons why a photon-based description of nonlinear optical phenomena is advisable over the classical wave picture of light (described elsewhere [45; 46; 47]). For all these reasons, casting light-matter interactions in QED offers the best route to take in this review to fully develop and describe the mechanisms of nonlinear optical interaction. The most suitable Hamiltonian to describe non-relativistic light-matter interactions is the fully quantized Power-Zienau-Woolley (PZW) multipolar Hamiltonian [48]. Its merits include gauge-independence, its optical elements being directly cast in terms of electromagnetic fields only, casting the interactions between light and matter through a Hamiltonian operator \(H_{\text{int}}\) as couplings  
one photon back into the same optical mode (classically, one would recognize this as arising from cos \(\omega t\) term in the identity \((\cos\omega t)^{3}=\frac{1}{4}(\cos\omega t+\cos 3\omega t)\).) However, such contributions would normally be associated with a cubic dependence on the electric field strength, rendering the effect comparatively insignificant at any intensity below the ultra-high intensity level at which both perturbation theory and molecules themselves break down. In the next section, we demonstrate how the involvement of two throughput beams dramatically changes the significance of the third-order term. First, to briefly provide an indicative example of the use of QED for conventional _non_linear optics, we derive the quantum amplitude and corresponding rate and radiant intensity of SHG. Second harmonic generation is a frequency-doubling, three-photon process, fundamentally involving the pairwise annihilation of input photons of frequency \(\omega\) (wave-vector \(\mathbf{k}\) and polarization \(\eta\)) correlated with the creation of single photons of frequency \(2\omega\) (\(\mathbf{k}^{\prime},\eta^{\prime}\)). The coherent, linear momentum-conserving process dictates the relationship \(2\mathbf{k}=\mathbf{k}^{\prime}\) between the input photons and output photon wave-vector. The matrix element is obtained using third-order perturbation theory, corresponding to the third term of Equation (4), where the initial state expressed in Dirac ket notation is \(|i_{\text{mol}};i_{\text{rad}}\rangle=\left|E_{0};n(k,\eta)\right\rangle\), and the final state is \(\left|f_{\text{mol}};f_{\text{rad}}\right\rangle=\left|E_{0};(n-2)(\mathbf{k},\eta ),1(\mathbf{k}^{\prime},\eta^{\prime})\right\rangle\). Inserting Equation (1) into the third term of Equation (4), and working through the detailed analysis, delivers the matrix element  
\[M_{fi}=-\frac{i}{2}\left(\frac{hck}{\varepsilon_{0}V}\right)^{\frac{3}{2}}[n( n-1)]^{\frac{1}{2}}\overline{e}_{i}e_{j}e_{k}\sum_{\xi_{i}}\beta_{ijk}^{(\xi_{i})} (-2\omega;\omega,\omega) \tag{5}\]  
where, in this specific case, \(\mathbf{e}\) and \(\mathbf{e}^{\prime}\) are unit polarization vectors of the incident and emitted photons, respectively. The molecule-level response is denoted by \(\beta_{ijk}\), the frequency-dependent hyperpolarizability tensor  
\[