Collective excitation of a trapped Bose-Einstein condensate with spin-orbit coupling

We investigate the collective excitations of a Raman-induced spin-orbit coupled Bose-Einstein condensate confined in a quasi-one-dimensional harmonic trap using the Bogoliubov method. By tuning the Raman coupling strength, three phases of the system can be identified. By calculating the transition strength, we are able to classify various excitation modes that are experimentally relevant. We show that the three quantum phases possess distinct features in their collective excitation properties. In particular, the spin dipole and the spin breathing modes can be used to clearly map out the phase boundaries. We confirm these predictions by direct numerical simulations of the quench dynamics that excites the relevant collective modes.

Figure 1

(a) Bogoliubov spectrum as functions of the Raman coupling strength $\mathrm{\Omega }$. Hollow circles and triangles denote the frequency of the dipole (D) and the breathing (B) mode, respectively; while solid diamonds and disks correspond to the spin dipole (SD) and the spin breathing (SB) frequencies. Furthermore, black hollow squares label the near zero-energy mode in the ST phase. Panels (b1), (b2), (c), and (d) correspond to the normalized transition strength ${\mathrm{\Gamma }}_n$ of the dipole, the breathing, the spin dipole, and the spin breathing modes, respectively. Panels (b)–(d) share the same color map at the right side of (d). In our calculations, we take $g_{\uparrow \downarrow }=0.7g$, where $g$ is calculated with the ${}^{87}\mathrm{Rb}$ BEC in a condition that ${\omega }_x=2\pi \times 45$ Hz, $a=101.8a_B$, with $a_B$ being the Bohr radius, and total atom number $N=2000$.

Figure 2

Spin dipole (SD) and the spin breathing (SB) mode frequency as functions of the Raman coupling strength $\mathrm{\Omega }$. These two modes are not well defined in the PW phase. Other parameters are the same as in Fig. 1.

Figure 3

(a) Upper row: ground-state density profiles in three phases, where $\ell ={\omega }_x^{-1/2}$ is the harmonic-oscillator length. Lower row: density profiles in the presence of a magnetic gradient with $d={\omega }_x^2\ell /4$. In the calculation, we take $\mathrm{\Omega }=E_L,3.5E_L$, and $4.5E_L$ to represent the ST, the PW, and the ZM phase, respectively. The evolution of the spin separation $\mathcal{D}\left(t\right)$ after the sudden quench of the magnetic gradient for the three phases are illustrated in (b), (c), and (d), respectively. (b) Evolution of $\mathcal{D}\left(t\right)$ in the ST phase. (c) Evolution of $\mathcal{D}\left(t\right)$ (black solid line) and polarization $P$ (red dotted line) in the PW phase. The two insets show the typical density profiles before and after the jump which occurs around $t=60$ trap periods. (d) Evolution of $\mathcal{D}\left(t\right)$ in the ZM phase, with (black solid line) and without (red dotted line) the SO coupling, respectively. Other parameters are the same as in Fig. 1.