Nonadiabatic dynamics of superfluid spin-orbit-coupled degenerate Fermi gas

We study a problem of nonadiabatic superfluid dynamics of spin-orbit-coupled neutral fermions in two spatial dimensions. We focus on the two cases when the out-of-equilibrium conditions are initiated by either a sudden change of the pairing strength or the population imbalance. For the case of a zero-population imbalance and within the mean-field approximation, the nonadiabatic evolution of the pairing amplitude in a collisionless regime can be found exactly by employing the method of Lax vector construction. Our main finding is that the presence of the spin-orbit coupling significantly reduces the region in the parameter space where a steady state with periodically oscillating pairing amplitude is realized. For the collisionless dynamics initiated by a sudden disappearance of the population imbalance, we obtain an exact expression for the steady-state pairing amplitude. In the general case of quenches to a state with finite population imbalance, we show that there is a region in the steady-state phase diagram where at long times the pairing amplitude dynamics is governed by the reduced number of the equations of motion in full analogy with the exactly integrable case.

Figure 1

Dependence of the pairing amplitude $\mathrm{\Delta }$, chemical potential $\mu$, and spectral gap $E_{\mathrm{gap}}=E_{\mathbf{k}=0,\lambda =+}$ (all in arbitrary units) as a function of the Zeeman field $h_Z$ determined by the numerical solution of the self-consistency equations (2.13) and (2.14). Note that the superfluid becomes gapless while the order parameter remains finite at some critical value of the field $h_Z^{\left(c\right)}=\sqrt{{\mu }^2+{\mathrm{\Delta }}^2}$. These results correspond to the following choice of parameters: $n_c=0.125,{\varepsilon }_F=2\pi n_c=0.785$, and ${\alpha }_{\text{SO}}=0.752$.

Figure 2

Steady-state diagram for the case $h_Z=0$. In region I $\mathrm{\Delta }(t\to \infty )=0$. In region II $\mathrm{\Delta }(t\to \infty )={\mathrm{\Delta }}_{\infty }$. In region III $\mathrm{\Delta }\left(t\right)$ varies periodically with time. The parameters are $n_c=0.125$ and ${\alpha }_{\text{SO}}=0.75$. In the limit of zero spin-orbit coupling the red line inside region III is absent.

Figure 3

Imaginary parts of the roots of ${\mathcal{L}}^2\left(u\right)=0$ and superfluid order parameter ${\mathrm{\Delta }}_{0i}$ in the initial state plotted as a function of the initial value of the Zeeman field $h_{Zi}=h_Z$ for the quenches $h_{Zi}\to h_{Zf}=0$. Note that the imaginary parts of both roots essentially coincide with each other for the initial conditions with ${\mathrm{\Delta }}_{0i}\to 0$. At $h_Z=h_{c2}$ the second complex root appears. Thus, $h_{c2}$ separates the steady states with constant and periodically oscillating superfluid order parameters. These results correspond to the following choice of parameters: $n_c=0.125,{\varepsilon }_F=0.785$, and ${\alpha }_{\text{SO}}=0.752$.

Figure 4

Results of the numerical solution of the equations of motion for $\mathrm{\Delta }\left(t\right)$ in the exactly integrable case when $h_{Zf}=0$. These results correspond to the following choice of parameters: $n_c=0.125,{\varepsilon }_F=0.785$, and $h_{c2}=1.02{\varepsilon }_F$.

Figure 5

Dependence of the critical Zeeman field $h_{c2}$, which separates the steady states with constant and periodically oscillating pairing amplitudes, on the strength of the spin-orbit coupling. Note that for the case of weak spin-orbit coupling, the pairing amplitude will always go to a constant at long times $\mathrm{\Delta }(t\to \infty )={\mathrm{\Delta }}_{\infty }$. The inset shows the dependence of the real part of the second root on ${\alpha }_{\text{SO}}$. These results correspond to the following choice of parameters: $n_c=0.125$ and ${\varepsilon }_F=0.785$.

Figure 6

Dependence of the roots $e_1,e_2$, and $e_3$ of the qubic polynomial $P_3\left(w\right)$ [Eq. (4.25)] on the value of the imbalance $h_Z$. The parameters are $n_c=0.125$ and ${\varepsilon }_F=0.785$.

Figure 7

Results of the numerical solution of the equations of motion for $\mathrm{\Delta }\left(t\right)$ in the general, i.e., nonintegrable, case $h_{Zf}\ne 0$ with (a) $h_{Zf}=0.9{\varepsilon }_F$, (b) $h_{Zf}=0.5{\varepsilon }_F$, (c) $h_{Zf}=0.25{\varepsilon }_F$, and (d) $h_{Zf}=0.1{\varepsilon }_F$. The values of the remaining parameters are $h_{Zi}=1.85{\varepsilon }_F,n_c=0.125$, and ${\varepsilon }_F=0.785$.

Figure 8

Same as Fig. 7 with (a) $h_{Zf}=1.25{\varepsilon }_F$, (b) $h_{Zf}=1.15{\varepsilon }_F$, (c) $h_{Zf}=1.1{\varepsilon }_F$, and (d) $h_{Zf}=0.95{\varepsilon }_F$.

Figure 9

Energy dependence of $\vec{L}\left(\varepsilon \right)$ and $T\left(\varepsilon \right)$ at time $t\delta =2.05$ $(\delta$ is a level spacing). The values of the remaining parameters are $h_{Zi}=1.85{\varepsilon }_F,h_{Zf}=0.5{\varepsilon }_F,n_c=0.125,\mathrm{\Lambda }=10{\varepsilon }_F$, and ${\varepsilon }_F=0.785$.

Figure 10

Same as Fig. 9 with $h_{Zf}=1.25{\varepsilon }_F$. In contrast to Fig. 9, we see that all components of $\vec{L}(\varepsilon ,t\to \infty )$ and $T(\varepsilon ,t\to \infty )$ are vanishingly small for all single-particle energies.