  We theoretically investigate quantum tunnelling escape of a spin-orbit (SO)
coupled Bose-Einstein condensate (BEC) from a trapping well. The condensate is
initially prepared in a quasi-one-dimensional harmonic trap. Depending on the
system parameters, the ground state can fall in different phases --- single
minimum, separated or stripe. Then, suddenly the trapping well is opened at one
side. The subsequent dynamics of the condensate is studied by solving nonlinear
Schr\"{o}dinger equations. We found that the diverse phases will greatly change
the tunneling escape behavior of SO coupled BECs. In single minimum and
separated phases, the condensate escapes the trapping well continuously, while
in stripe phase it escapes the well as an array of pulses. We also found that
SO coupling has a suppressing effect on the tunnelling escape of atoms.
Especially, for BECs without inter-atom interaction, the tunnelling escape can
be almost completely eliminated when the system is tuned near the transition
point between single minimum and stripe phase. Our work thus suggests that SO
coupling may be a useful tool to control the tunnelling dynamic of BECs, and
potentially be applied in realization of atom lasers and matter wave switches.
