Quench-Induced Floquet Topological $p$-Wave Superfluids

Ultracold atomic gases in two dimensions tuned close to a $p$-wave Feshbach resonance were expected to exhibit topological superfluidity, but these were found to be experimentally unstable. We show that one can induce a topological Floquet superfluid if weakly interacting atoms are brought suddenly close (“quenched”) to such a resonance, in the time before the instability kicks in. The resulting superfluid possesses Majorana edge modes, yet differs from a conventional Floquet system as it is not driven externally. Instead, the periodic modulation is self-generated by the dynamics.

Figure 1

Phase diagram showing the three regimes (I–III) of nonequilibrium superfluidity reached after a quantum quench in a $p$-wave gas [17]. Each point in this phase diagram represents a particular quench, wherein one takes an initial state with order parameter amplitude ${\mathrm{\Delta }}_i$ and subsequently ramps the strength of attractive atom-atom interactions to weaker or stronger pairing. The initial state is specified via the vertical axis. The horizontal axis measures ${\mathrm{\Delta }}_f$, which is the amplitude one would find in the ground state of the postquench Hamiltonian. The diagonal line ${\mathrm{\Delta }}_i={\mathrm{\Delta }}_f$ is the case of no quench; ${\mathrm{\Delta }}_{\mathsf{QCP}}$ locates the BCS-BEC ground-state transition (see the Supplemental Material [33]). Each off-diagonal point to the left (right) of this line denotes a particular quench from stronger-to-weaker (weaker-to-stronger) pairing. The regions labeled I, II, III denote three different regimes of nonequilibrium superfluid dynamics. For a strong-to-weak quench in I, the order parameter $\mathrm{\Delta }(t)$ decays to zero. A quench in II leads to a nonzero steady-state order parameter amplitude. A weak-to-strong quench in III induces persistent oscillations in $|\mathrm{\Delta }(t)|$. $W$ denotes the winding number described in the text. Quenches in II with $W=1$ and in III produce topological states. Point A specifies a quench from very weak initial pairing that produces a Floquet topological state, which could be accessible experimentally.

Figure 2

Majorana edge modes for a quench-induced time-dependent state of $p$-wave superfluidity. The Floquet spectrum (top) of $\ln U(T)$ for a system on a finite cylinder is plotted in the large time asymptotic regime for a quench in region III, point A in Fig. 1. The horizontal axis represents the momentum along the boundary, and the vertical axis represents the quasienergies multiplied by the period of oscillations, both ranging from $-\pi$ to $\pi$. The edge states can clearly be seen crossing in the center of the figure. The bottom left shows the orbit swept by ${\mathrm{\Delta }}_{\infty }(t)$ in the complex $\mathrm{\Delta }$ plane at this point in region III. The bottom right plots the orbital maximum ${\mathrm{\Delta }}_{\max }\equiv \max |\mathrm{\Delta }(t)|$ as a function of ${\mathrm{\Delta }}_f$ for quenches from very weak initial pairing (${\mathrm{\Delta }}_i\to 0$) in III. The dashed line is ${\mathrm{\Delta }}_{\max }={\mathrm{\Delta }}_f$.