  At thermal equilibrium, the concept of effective central charge for massive
deformations of two-dimensional conformal field theories (CFT) is well
understood, and can be defined by comparing the partition function of the
massive model to that of a CFT. This temperature-dependent effective charge
interpolates monotonically between the central charge values corresponding to
the IR and UV fixed points at low and high temperatures, respectively. We
propose a non-equilibrium, time-dependent generalization of the effective
central charge for integrable models after a quantum quench, $c_{\rm eff}(t)$,
obtained by comparing the return amplitude to that of a CFT quench. We study
this proposal for a large mass quench of a free boson, where the charge is seen
to interpolate between $c_{\rm eff}=0$ at $t=0$, and $c_{\rm eff}\sim 1$ at
$t\to\infty$, as is expected. We use our effective charge to define an "Ising
to Tricritical Ising" quench protocol, where the charge evolves from $c_{\rm
eff}=1/2$ at $t=0$, to $c_{\rm eff}=7/10$ at $t\to\infty$, the corresponding
values of the first two unitary minimal CFT models. We then argue that the
inverse "Tricritical Ising to Ising" quench is impossible with our methods.
These conclusions can be generalized for quenches between any two adjacent
unitary minimal CFT models. We finally study a large mass quench into the
"staircase model" (sinh-Gordon with a particular complex coupling). At short
times after the quench, the effective central charge increases in a discrete
"staircase" structure, where the values of the charge at the steps can be
computed in terms of the central charges of unitary minimal CFT models. When
the initial state is a pure state, one always finds that $c_{\rm
eff}(t\to\infty)\geq c_{\rm eff}(t=0)$, though $c_{\rm eff}(t)$, generally
oscillates at finite times. We explore how this constraint may be related to RG
flow irreversibility.
