  The recent experimental observation of dissipation-induced structural
instability provides new opportunities for exploring the competition mechanism
between stationary and nonstationary dynamics [Science 366, 1496 (2019)]. In
that study, two orthogonal quadratures of cavity field are coupled to two
different Zeeman states of a spinor Bose-Einstein condensate (BEC). Here we
propose a novel scheme to couple two density-wave degrees of freedom of a BEC
to two quadratures of the cavity field. Being drastically different from
previous studies, the light-matter quadratures coupling in our model is endowed
with a tunable coupling angle. Apart from the uniform and self-organized
phases, we unravel a dynamically unstable state induced by the cavity
dissipation. Interestingly, the dissipation defines a particular coupling
angle, across which the instabilities disappear. Moreover, at this critical
coupling angle, one of the two atomic density waves can be independently
excited without affecting one another. It is also found that our system can be
mapped into a reduced three-level model under the commonly used
low-excitation-mode approximation. However, the effectiveness of this
approximation is shown to be broken by the dissipation nature for some special
system parameters, hinting that the low-excitation-mode approximation is
insufficient in capturing some dissipation-sensitive physics. Our work enriches
the quantum simulation toolbox in the cavity-quantum-electrodynamics system and
broadens the frontiers of light-matter interaction.
