We present a dynamic displacement sensor based on a quadrature phase interferometer, providing a real-time analog output of the differential displacement between two mobile surfaces. The sensor offers a sub-picometer resolution with a sensitivity essentially uniform over a distance range extending to several micrometers, and can be used to measure the amplitude and phase of very small oscillations in nano-mechanical testing. We demonstrate its use in nano-rheology, by studying the flow in nanometric liquid films at very small strain rates.