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We study the non-adiabatic dynamics of a 2D p+ip superfluid following aquantum quench of the BCS coupling constant. The model describes a topologicalsuperconductor with a non-trivial BCS (trivial BEC) phase appearing at weak(strong) coupling strengths. We extract the exact long-time asymptotics of theorder parameter \Delta(t) by exploiting the integrability of the classicalp-wave Hamiltonian, which we establish via a Lax construction. Three differenttypes of behavior can occur depending upon the strength and direction of thequench. In phase I, the order parameter asymptotes to zero. In phase II,\Delta(t) goes to a non-zero constant. Phase III is characterized by persistentoscillations of \Delta(t). For quenches within I and II, we determine thetopological character of the asymptotic states. We show that two differentformulations of the bulk topological winding number, although equivalent in theground state, must be regarded as independent out of equilibrium. The firstnumber Q characterizes the Anderson pseudospin texture of the initial state; weshow that it is conserved. For non-zero Q, this leads to the prediction of a"gapless topological" state when \Delta(t) goes to zero. The presence orabsence of Majorana edge modes in a sample with a boundary is encoded in thesecond winding number W, formulated in terms of the retarded Green's function.We show that W can change following a quench across the quantum critical point.We discuss the implications for the (dis)appearance of Majorana edge modes.Finally, we show that the parity of zeros in the bulk out-of-equilibrium Cooperpair distribution function constitutes a Z2-valued quantum number, which isnon-zero whenever W differs from Q. The pair distribution can in principle bemeasured using RF spectroscopy in an ultracold atom realization, allowingdirect experimental detection of the bulk Z2 number.

Quantum quench in a p + i p superfluid: Winding numbers and topological states far from equilibrium