## 3 Controlled Energy Transfer
### Optically Controlled EET  
Another strategy to produce a modification to the efficiency of energy transfer involves the input of an off-resonant (i.e., a non-absorbing) laser beam of sufficient intensity. Depending on a number of factors, the rate of transfer can be enhanced or reduced [53; 54]. The equation representing an observable for this case is again derived from the Fermi rate expression; it now includes an extra term that corresponds to an overall fourth-order interaction with respect to D and A. This is written as \(\Gamma^{\prime}\sim\left|M_{fi}+M_{fi}^{\text{dyn}}\right|^{2}\), in which \(M_{fi}^{\text{dyn}}\) is a correction term that arises due to the input of the dynamic electromagnetic field. In detail, again presuming that D and A are in close proximity, \(M_{fi}^{\text{dyn}}\) is representative of the energy transfer interactions taking place as part of a concerted quantum process. This involves four possible fourth-order interactions: second-order molecular interactions at D and A, in which the input beam is absorbed at D and emitted back into the beam at A or vice versa (the former is shown in Figure 4a), or third-order molecular interactions at either D or A, where the input beam is absorbed and emitted at the same molecule (Figure 4b displays the case when this occurs at D). Therefore, in this near-zone region that is again represented by the static limit, the explicit form of \(M_{fi}^{\text{dyn}}\) is given by;  
\[ \begin{split} M_{fi}^{\text{dyn}}&=\bigg{(}\frac{I }{2\varepsilon_{0}c}\bigg{)}e_{i}\bar{e_{l}}V_{jk}(0,\mathbf{R})\Big{(}S_{ij}^{ \prime\downarrow}(\text{D})S_{lk}^{\prime\uparrow}(\text{A})+S_{ij}^{\prime \uparrow}(\text{A})S_{lk}^{\prime\downarrow}(\text{D})\\ &\quad+T_{ijl}^{\downarrow}(\text{D})\mu_{k}^{\uparrow}(\text{A}) +T_{ijl}^{\uparrow}(\text{A})\mu_{k}^{\downarrow}(\text{D})\Big{)}\end{split}  \tag{10}\]  
Here, \(S^{\prime}\) is a transition polarisability, explicitly written as \(\mathbf{S^{\prime}}^{\prime\downarrow}(\text{D})\equiv\mathbf{S^{\prime}}^{\downarrow }(-c(k+k^{\prime});ck^{\prime})\) and \(\mathbf{S^{\prime}}^{\uparrow}(\text{A})\equiv\mathbf{S^{\prime}}^{\uparrow}(-ck^{ \prime};c(k+k^{\prime}))\), where \(k^{\prime}\) is the wave-vector of the input photons. Moreover, \(T_{ijl}\) is a transition hyperpolarisability, explicitly given by \(T_{ijl}^{\downarrow}(\text{D})\equiv T^{\prime}_{ijl}(-ck,-ck^{\prime};ck^{ \prime})\) and \(T_{ijl}^{\uparrow}(\text{A})\equiv T^{\prime}_{ijl}{}^{\uparrow}(-ck^{\prime} ;ck,ck^{\prime})\), while \(I\) and \(\mathbf{e}\) are the intensity and polarisation of the throughput beam, respectively. For entirely nonpolar donor and acceptor molecules, the rate expression for the optically controlled EET (otherwise known as laser-assisted RET or LARET) is then written as;  
\[ \begin{split}\Gamma_{\text{LARET}}&=V_{jk}(0,\mathbf{R} )V_{mn}(0,\mathbf{R})\Big{\{}\mu_{j}^{\downarrow}(\text{D})\mu_{k}^{\uparrow}(\text {A})\mu_{m}^{\downarrow}(\text{D})\mu_{n}^{\uparrow}(\text{A})+\Big{(}\frac{I }{2\varepsilon_{0}}\Big{)}e_{i}\bar{e_{l}}\\ &\times\mu_{m}^{\downarrow}(\text{D})\mu_{n}^{\uparrow}(\text{A}) \Big{(}S_{ij}^{\prime\downarrow}(\text{D})S_{lk}^{\prime\uparrow}(\text{ A})+S_{ij}^{\prime\uparrow}(\text{A})S_{lk}^{\prime\downarrow}(\text{D}) \Big{)}+\cdots\Big{\}}\end{split}  \tag{11}\]  
Here, the first term is the EET rate that denotes an input-field-independent contribution, and the second term (which has the next largest magnitude since its linear, not quadratic, in \(I\)) is based on a quantum interference between the EET term and the fourth-order input beam terms. Note that the transition hyperpolarisability terms are omitted because of the assumption that the molecules are nonpolar. With the very rare exception of molecules with icosahedral symmetry, all transitions that are single-quantum electric dipole (E1) allowed are also allowed in a three-quantum (E13) process [55].