  The interaction of ultra-intense laser pulses with an underdense plasma is
used in laser-plasma acceleration to create compact sources of ultrashort
pulses of relativistic electrons and X-rays. The accelerating structure is a
plasma wave, or wakefield, that is excited by the laser ponderomotive force, a
force that is usually assumed to depend solely on the laser envelope and not on
its exact waveform. Here, we use near-single-cycle laser pulses with a
controlled carrier-envelope-phase (CEP) to show that the actual waveform of the
laser field has a clear impact on the plasma response. We measure relativistic
electron beams that are found to be strongly CEP dependent, implying that we
achieve waveform control of electron dynamics in underdense laser-plasma
interaction. Our results pave the way to high precision, sub-cycle control of
electron injection in plasma accelerators, enabling the production of
attosecond relativistic electron bunches and X-rays.
