Collective Rabi Oscillations and Solitons in a Time-Dependent BCS Pairing Problem

Motivated by recent efforts to achieve cold fermions pairing, we study the nonadiabatic regime of the Bardeen-Cooper-Schrieffer state formation. After the interaction is turned on, at times shorter than the quasiparticle energy relaxation time, the system oscillates between the superfluid and normal state. The collective nonlinear evolution of the BCS-Bogoliubov amplitudes $u_p$, $v_p$, along with the pairing function $\Delta$, is shown to be an integrable dynamical problem which admits single soliton and soliton train solitons. We interpret the collective oscillations as Bloch precession of Anderson pseudospins, where each soliton causes a pseudospin $2\pi$ Rabi rotation.

Figure 1

Coherent BCS dynamics. Above: the soliton solutions oscillating in the limits ${\Delta }_{-}\le \Delta (t)\le {\Delta }_{+}$, (23). Below: trajectories of individual Cooper pair states on the Bloch sphere (14). Note that each state completes a full $2\pi$ Rabi cycle per soliton. The dashed-line and solid-line curves correspond to the energies ${ϵ}_{\mathbf{p}}$ above and below the Fermi level.

Figure 2

Bloch dynamics ${\dot{\mathbf{r}}}_{\mathbf{p}}=2{\mathbf{b}}_{\mathbf{p}}^{\mathrm{e}\mathrm{f}\mathrm{f}}\times {\mathbf{r}}_{\mathbf{p}}$ for ${10}^3$ spins with a constant density of states. Damping is included via ${\mathbf{b}}_{\mathbf{p}}^{\mathrm{e}\mathrm{f}\mathrm{f}}={\mathbf{b}}_{\mathbf{p}}-\gamma {\mathbf{b}}_{\mathbf{p}}\times {\mathbf{r}}_{\mathbf{p}}$. Initial conditions $r_{3,\mathbf{p}}=\tanh (\frac{1}{2}\beta {ϵ}_{\mathbf{p}})$, $(r_1+i{r}_2{)}_{\mathbf{p}}=e^{i{\varphi }_{\mathbf{p}}}(1-r_3^2{)}^{1/2}$, with ${\beta }^{-1}\equiv T=0.1{\Delta }_0$ and random uncorrelated ${\varphi }_{\mathbf{p}}$ were used to model thermal noise.