Spin dynamics and domain formation of a spinor Bose-Einstein condensate in an optical cavity

We consider a ferromagnetic spin-1 Bose-Einstein condensate (BEC) dispersively coupled to a unidirectional ring cavity. We show that the ability of the cavity to modify, in a highly nonlinear fashion, matter-wave phase shifts adds an additional dimension to the study of spinor condensates. In addition to demonstrating strong matter-wave bistability as in our earlier publication [L. Zhou et al., Phys. Rev. Lett. 103, 160403 (2009)], we show that the interplay between atomic and cavity fields can greatly enrich both the physics of critical slowing down in spin-mixing dynamics and the physics of spin-domain formation in spinor condensates.

Figure 1

Schematic diagram showing the system under consideration. An $F=1$ spinor condensate is trapped inside the cavity using an optical dipole trap. The population of different spin components can exchange via spin mixing. The cavity is coherently driven by an external laser with amplitude ${\varepsilon }_p$ and decays with a rate $\kappa$. The cavity field is $\pi$ polarized and is dispersively coupled to the atomic system.

Figure 2

(a) Mean intracavity photon number $\left|\alpha \right|{}^2$ and (b) the normalized spin-0 population ${\rho }_0$ versus cavity-pump detuning ${\overline{\delta }}_c$ for a steady-state solution with $\theta =0$. The ones represented by the red dashed lines correspond to dynamically unstable solutions. The vertical dotted lines indicate the position of the first-order transition which occurs at ${\overline{\delta }}_c=-4.75$.

Figure 3

(Top panel) Period of spin oscillations as a function of cavity-pump detuning ${\overline{\delta }}_c$. (Middle panel) The anharmonic time evolution of ${\rho }_0$ for the three peaks marked in the top panel. (Bottom panel) From left to right, the phase-space contour plot of $H$ corresponding, respectively, to the peak 1, 2, and 3 marked in the top panel. The black dots refer to the initial state of the system, while the white dots refer to dynamically unstable fixed points.

Figure 4

(a) The gray area shows the range of wave vector $k$ corresponding to the unstable excitations of the self-consistent ground state of a homogeneous rubidium condensate-cavity interacting system. The red lines (top) refer to those with the maximum instability growth rate. (b) Spin-domain width versus cavity-pump detuning ${\overline{\delta }}_c$

Figure 5

The ground density profile of a ${}^{87}\mathrm{Rb}$ condensate trapped in a unidirectional ring cavity. Here the distance $z$ is scaled with $a_s=\sqrt{\hslash /m_a{\omega }_z}$ and the parameters used are specified in the main text.