  Linear response theory lies at the heart of quantum many-body physics because
it builds up connections between the dynamical response to an external probe
and correlation functions at equilibrium. Here we consider the dynamical
response of a Hermitian system to a non-Hermitian probe, and we develop a
non-Hermitian linear response theory that can also relate this dynamical
response to equilibrium properties. As an application of our theory, we
consider the real-time dynamics of momentum distribution induced by one-body
and two-body dissipations. We find that, for many cases, the dynamics of
momentum occupation and the width of momentum distribution obey the same
universal function, governed by the single-particle spectral function. We also
find that, for critical state with no well-defined quasi-particles, the
dynamics are slower than normal state and our theory provides a model
independent way to extract the critical exponent. We apply our results to
analyze recent experiment on the Bose-Hubbard model and find surprising good
agreement between theory and experiment. We also propose to further verify our
theory by carrying out a similar experiment on a one-dimensional Luttinger
liquid.
