Abstract: A quadrupole ion trap apparatus includes a main electrode, a first end-cap electrode, a second end-cap electrode, and a phase-controlled waveform synthesizer. The phase-controlled waveform synthesizer generates a main RE waveform for the main electrode. The main RE waveform includes a plurality of sinuous waveform segments each of which is a part of a sine wave, and a plurality of phase conjunction segments each of which is non-sinuous. Each of the sinuous waveform segments is bridged to another sinuous waveform segment via one of the phase conjunction segments, so as to perform ordering of micro motions of sample ions trapped by the electrodes.