Localizing spin dynamics in a spin-1 Bose-Einstein condensate via magnetic pulses

Spin-exchange interaction between atoms in a spin-1 Bose-Einstein condensate causes atomic spin evolving periodically under the single-spatial-mode approximation in the mean-field theory. By applying fast magnetic pulses according to a two-step or a four-step control protocol, we find analytically that the spin dynamics is significantly suppressed for an arbitrary initial state. Numerical calculations under single-mode approximation are carried out to confirm the validity and robustness of these protocols. This localization method can be readily utilized to improve the sensitivity of a magnetometer based on spin-1 Bose-Einstein condensates.

Figure 1

(a) Typical trajectories in the ${\rho }_0\text{−}\theta$ plane for $\delta /\left|c\right|=-0.5$ (dashed line), $-0.39,-0.1,0.2,0.7,1.5,1.59$ (solid lines), and $1.8$ (dotted line), from bottom to top. All the trajectories with solid lines evolve in a clockwise direction. The initial state (asterisk) is ${\rho }_0\left(0\right)={\rho }_{0i}=0.8$ and $\theta \left(0\right)={\theta }_i=0$. (b) Dependence of the oscillation amplitude $A$ (blue solid line) and the period $T$ (green dashed line) of ${\rho }_0$ on $\delta$. The running phase modes correspond to regions I and IV, and the oscillatory modes correspond to regions II and III. The red dotted lines marked by ${\delta }_0$ and ${\delta }_{\pm }$ denote, respectively, the ground-state quadratic Zeeman energy (the initial state coincides with the ground state and $A$ is zero) and the resonant quadratic Zeeman energy ($T$ is infinite and ${\rho }_0\to 0$ or 1).

Figure 2

(a) Typical trajectories under the modulation of $\delta \left(t\right)$. Each modulation cycle includes a free evolution with $\delta =0$ (dashed line) and a controlled evolution with $d/\left|c\right|=1.5,2.0,3.0$ (solid lines, from bottom to top). The initial state (red asterisk) is ${\rho }_{0i}=0.8$ and ${\theta }_i=-0.1\pi$. (b) Dependence of the amplitude (blue solid line) and period (green dashed line) of ${\rho }_0$ on $d$, the nonzero quadratic Zeeman splitting. Circles and diamonds are the period and amplitude calculated analytically with Eqs. (7) and (8) for large $d\text{"}\mathrm{s}$, respectively.

Figure 3

(a) Schematic of a magnetic pulse sequence. (b) Typical controlled trajectory of ${\rho }_0\left(t\right)$ and $\theta \left(t\right)$ under a four-step pulse sequence of $\delta \left(t\right)$ for a four-step protocol. The red asterisk marks the initial state. The parameters are ${\rho }_{0i}=0.8,{\theta }_i=-0.2\pi ,d/\left|c\right|=10$, and ${\tau }_1={\tau }_3=\tau =0.05\pi$. The values of ${\tau }_2$ and ${\tau }_4$ are calculated, ${\tau }_2\approx 0.0072\pi$ and ${\tau }_4\approx 0.0226\pi$. (c) Amplitude of ${\rho }_0$ under four-step pulse sequences. Better localization of ${\rho }_0$ (smaller $A$) is achieved for larger $d$ and smaller $\tau$.

Figure 4

(a) Fidelity after 1 cycle (blue solid line), 10 cycles (red dashed line), 100 cycles (black dash-dotted line), and 200 cycles (purple dotted line) for the two-step protocol with 5% relative uncertainty in $\delta$. For clarity, each curve is shifted up by $0.01$ from bottom to top. (b) Fidelity after 1 cycle [blue (dark gray) solid line], 10 cycles [red (light ray) solid line], and 100 cycles (black lines) while $\tau /\pi =0.01$ (dash-dotted line), $0.03$ (dashed line), and $0.05$ (solid lines) for the four-step protocol with 5% relative uncertainty in $\delta$. (c) Fidelity after 1 cycle (blue solid line), 10 cycles (red dashed line), 100 cycles (black dash-dotted line), and 200 cycles (purple dotted line) for the two-step protocol with 5% relative uncertainty in ${\theta }_i$. Each curve is also shifted up by $0.01$ from bottom to top. The inset shows a zoom-in view near the dip in the main panel. The results show that both the two-step protocol and the four-step protocol are robust against errors.