# Forces in the Interaction of Light with Matter
## 2 Optical Forces--Fundamentals  
### Optical Forces on an Electric Dipole  
Optical forces on matter primarily owe their origin to dipolar interactions with light. In the quantum formulation, each particle is characterised by an electric dipole operator \(\mathbf{d}=e\mathbf{q}\), where \(e\) is charge, and \(\mathbf{q}\) is a displacement vector. The quantum operator representation provides for the description of both fluctuating and transition moments. The simplest formulation that directly embodies the salient features of light-matter interaction is a quantum electrodynamical Hamiltonian theory [35; 93; 133]. The system Hamiltonian is here represented as follows:  
\[\mathcal{H}=\frac{|\mathbf{P}|^{2}}{2M}+\mathcal{H}_{a}+\mathcal{H}_{fields}+ \mathcal{H}_{int}, \tag{3}\]  
where \(\mathbf{P}\) is the linear momentum of a particle of mass M so that the first Hamiltonian term is the kinetic energy of the centre of mass. The rest of the terms are as follows: \(\mathcal{H}_{fields}\) is the electromagnetic field Hamiltonian field; \(\mathcal{H}_{a}\) is the Hamiltonian representing the internal energetics of the particle, and \(\mathcal{H}_{int}\) is the interaction Hamiltonian, which in the electric dipole approximation is expressible as:  
\[\mathcal{H}_{int}=-\mathbf{d}\cdot\mathbf{E}+\frac{1}{2M}[\mathbf{P}\cdot( \mathbf{d}\times\mathbf{B})+(\mathbf{d}\times\mathbf{B})\cdot\mathbf{P}]. \tag{4}\]  
The velocity of the centre of mass \(\dot{R}\) follows as a Heisenberg equation  
\[\dot{\mathbf{R}}=\frac{i}{\hbar}[\mathcal{H},\mathbf{R}]=\frac{1}{M}(\mathbf{ P}+\mathbf{d}\times\mathbf{B}), \tag{5}\]  
where a superscript dot denotes a time derivative. The mass-acceleration product for the centre of mass is then given by  
\[M\ddot{\mathbf{R}}=\frac{i}{\hbar}[\mathcal{H},(\mathbf{P}+\mathbf{d}\times \mathbf{B})]=\
abla(\mathbf{d}\cdot\mathbf{E})+\frac{d}{dt}(\mathbf{d}\times \mathbf{B}). \tag{6}\]  
This expression represents the optical force \(\mathbf{f}_{d}\) acting upon an electric dipole. Note that the spatial derivatives in \(\
abla\) are taken with respect to the centre of mass variable \(\mathbf{R}\). The time derivative in Equation (6) contains two terms, and with the use of the Maxwell equation, we find  
\[\mathbf{F}=(\mathbf{d}\cdot\
abla)\mathbf{E}+\dot{\mathbf{d}}\times\mathbf{B}. \tag{7}\]  
As shown below, this result can be generalised to the optical force in the many-dipole case, forming a medium. Before we move to consider optical forces in media, we briefly comment on the two terms involved in the optical force on a single dipole, as given by Equation (7). The first term is the dominant contribution to the force on the dipole, while the second term signifies an additional force that arises from the interaction term in the Hamiltonian. The latter, called the Rontgen force, whose origin has been established within the dipole approximation [133], has been the subject of interest in a number of recent investigations [56; 133]. Surprisingly, it produces a momentum-dependent interaction that leads to high-field-seeking atoms being repelled from intensity maxima in a travelling laser pulse.