Aharonov-Bohm phase for an electromagnetic wave background

The canonical Aharonov-Bohm effect is usually studied with time-independent potentials. In this work, we investigate the Aharonov-Bohm phase acquired by a charged particle moving in {\it time-dependent} potentials . In particular, we focus on the case of a charged particle moving in the time varying field of a plane electromagnetic wave. We work out the Aharonov-Bohm phase using both the potential ({\it i.e.} $\oint A_{\mu }d{x}^{\mu }$) and field ({\it i.e.} $\frac{1}{2}\int F_{\mu \nu }d{\sigma }^{\mu \nu }$) forms of the Aharanov-Bohm phase. We give conditions in terms of the parameters of the system (frequency of the electromagnetic wave, the size of the space-time loop, amplitude of the electromagnetic wave) under which the time varying Aharonov-Bohm effect could be observed.