Interaction quenches in nonzero-temperature fermionic condensates

We revisit the study of amplitude oscillations in a pair condensate of fermions after an interaction quench, and generalize it to nonzero temperature. For small variations of the order parameter, we show that the energy transfer during the quench determines both the asymptotic pseudoequilibrated value of the order parameter and the magnitude of the oscillations, after multiplication by, respectively, the static response of the order parameter and spectral weight of the pair-breaking threshold. Since the energy transferred to the condensed pairs decreases with temperature as the superfluid contact, the oscillations eventually disappear at the critical temperature. For deeper quenches, we generalize the regimes of persistent oscillations and monotonic decay to nonzero temperatures, and explain how they become more abrupt and are more easily entered at high temperatures when the ratio of the initial to final gap either diverges, when quenching toward the normal phase, or tends to zero, when quenching toward the superfluid phase. Our results are directly relevant for existing and future experiments on the nonequilibrium evolution of Fermi superfluids near the phase transition.

Figure 1

The asymptotic change of the order parameter ${\mathrm{\Delta }}_{\infty }-{\mathrm{\Delta }}_i$ measured relative to the change ${\mathrm{\Delta }}_f-{\mathrm{\Delta }}_i$ under an adiabatic evolution, as a function of temperature at unitarity, in the BCS and BEC limits. At $T=0, {\mathrm{\Delta }}_{\infty }={\mathrm{\Delta }}_f$ despite the nonadiabatic nature of the quench. Near $T_c$, we indicate the asymptotic behavior in the BCS limit, Eq. (10). (Inset) The change ${\mathrm{\Delta }}_{\infty }-{\mathrm{\Delta }}_i$ (in units of ${\epsilon }_F$) relative to the change in the scattering length $1/k_F{a}_f-1/k_F{a}_i$ throughout the BEC-BCS crossover at $T=0$ (red curve) and $T\to T_c$ (black curve). In both cases, a maximum is reached near unitarity. Note that the change remains nonzero (in units of ${\epsilon }_F$) on the BCS side ($\mu >0$), which means the linear approximation breaks down (if the quench depth $1/k_F{a}_f-1/k_F{a}_i$ is kept independent of temperature).

Figure 2

The spectral weight of the pair-breaking edge $f_{\mathrm{th}}$ [relative to the static response $f\left(0\right)]$ as a function of the interaction strength at $T=0$ (solid curves) and $T\to T_c$ (dashed curves). In red is the BCS regime where the edge exhibits a square root divergence [upper line of Eq. (11)], and in blue is the BEC regime where instead this edge is a square root cancellation [lower line of Eq. (11)]. Note that the boundary between those two regimes (shaded area) shifts from $1/k_{F}a\simeq 0.55$ at $T=0$ to $1/k_{F}a\simeq 0.68$ at $T\to T_c$.

Figure 3

(a) The onset of regime III (persistent oscillations) in a quench from $1/\left(k_F{a}_i\right)=-0.18$ to $1/\left(k_F{a}_f\right)=0$ when raising the initial temperature. (b) Effect of temperature on the persistent oscillations for a quench from $1/\left(k_F{a}_i\right)=-2$ to $1/\left(k_F{a}_f\right)=0$ belonging to regime III at all temperatures. To make the curves oscillate in phase, we introduce here the frequency ${\omega }_{po}=0.6\left(2{\mathrm{\Delta }}_f\right)$ at zero temperature and $0.13\left(2{\mathrm{\Delta }}_f\right)$ at $T=0.88T_{c,i}$. (c) Illustration of the quenches studied in (a) and (b) in the $[1/\left(k_{F}a\right), T/T_F]$ plot.

Figure 4

(a) The onset of regime I (overdamped evolution) when raising the initial temperature of quenches from $1/\left(k_F{a}_i\right)=0$ to $1/\left(k_F{a}_f\right)=-0.18$. (b) The decay of $\mathrm{\Delta }\left(t\right)$ becomes more abrupt at higher temperature, as shown here for a quench from $1/\left(k_F{a}_i\right)=0$ to $1/\left(k_F{a}_f\right)=-1.5$ belonging to regime I at all temperatures. (c) Illustration of the quenches studied in (a) and (b) in the $[1/\left(k_{F}a\right), T/T_F]$ plot.