Holographic Approach to Nonequilibrium Dynamics of Moving Mirrors Coupled to Quantum Critical Theories

We employ the holographic method to study fluctuations and dissipation of an $n$-dimensional moving mirror coupled to quantum critical theories in $d$ spacetime dimensions. The bulk counterpart of the mirror with perfect reflection is a D$(n+1)$ brane in the Lifshitz geometry of $d+1$ dimensions. The motion of the mirror can be realized from the dynamics of the brane at the boundary of the bulk. The excited modes of the brane in the bulk render the mirror undergoing Brownian motion. For small displacement of the mirror, we derive the analytical results of the correlation functions and response functions. The dynamics of the mirror due to small fluctuations around the brane vacuum state in the bulk is found supraohmic so that after initial growth, the velocity fluctuations approach a saturated value at late time with a power-law behavior. On the contrary, in the Lifshitz black hole background, the mirror in thermal fluctuations shows that its relaxation dynamics becomes ohmic, and the saturation of velocity fluctuations is reached exponentially in time. Finally a comparison is made with the result of a moving mirror driven by free fields.