Implementation of a double-path multimode interferometer using a spinor Bose-Einstein condensate

We realize a double-path multimode matter wave interferometer with spinor Bose-Einstein condensate and observe clear spatial interference fringes as well as a periodic change of the visibility in the time domain, which we refer to as the time domain interference and which is different from the traditional double-path interferometer. By changing the relative phase of the two paths, we find that the spatial fringes first lose coherence and recover. As the number of modes increases, the time domain interference signal with the narrower peaks is observed, which is beneficial to the improvement of the resolution of the phase measurement. We also investigated the influence of initial phase configuration and phase evolution rate between different modes in the two paths. With enhanced resolution, the sensitivity of interferometric measurements of physical observables can also be improved by properly assigning measurable quantities to the relative phase between two paths.

Figure 1

The schematic of a double-path multimode interferometer. An initial BEC in panel (a) is converted to two coherent wave packets serving as paths $|\text{I}\rangle$ and $|\text{II}\rangle$ in panel (b) (the momenta of which are $p^{\left(\text{I}\right)}$ and $p^{\left(\text{II}\right)}$, respectively) through a Majorana transition and an evolution of time $T_d$ in the gradient magnetic field, which we refer to as step 1 (S1), or the double-path operation. In our step 2, each wave packet is converted to a superposition of several Zeeman sublevels by a rf pulse. After the evolution in the gradient magnetic field with different phase evolution rate ${\omega }_{m_F}^{(\text{I},\text{II})}$ during time $T_N$, we got the superposition states in panel (c). Finally, after the time of flight (TOF) we observe the interference (d) of two superposition states by absorption imaging, where $\mathrm{\Delta }\varphi (T_d,T_N)$ is the relative phase between adjacent mode fringes. The fringe in each color represents the interference between the two wave packets of a single mode.

Figure 2

(a1–a3) Single-shot spatial interference pattern with $N=5$ interference modes after $T_{\text{TOF}}=26\mathrm{ms}$, in which (a1) $T_d=8.0\mu \mathrm{s},T_N=133.0\mu \mathrm{s}$; (a2) $T_d=18.0\mu \mathrm{s},T_N=146.0\mu \mathrm{s}$; and (a3) $T_d=28.0\mu \mathrm{s},T_N=146.0\mu \mathrm{s}$, with different relative phases between adjacent modes. (b1–b3) Data (black points) are got by integrating the image in panels (a1)–(a3) along the $z$ direction and they are fitted by Eq. (5) (red solid lines). The visibilities are 0.55, 0.24, and 0.05, respectively.

Figure 3

The oscillation of visibility $V_N$ with $\mathrm{\Delta }\varphi$ with (a) $N=3$ modes, (b) $N=4$ modes, and (c) $N=5$ modes, with $T_d=42.5\mu \mathrm{s}$. The points and the dashed lines show the experimental data and fit, respectively. Each data point is a statistical average of four repeated experiments and the error bar shows the standard deviation. In the fitting, free parameters are $b_{m_F}^{\left(\text{II}\right)}$, $b_{m_F}^{\left(\text{I}\right)}$, and $\mathrm{\Delta }{\varphi }_{m_F}$. The theoretical result of a two-mode situation is shown by the black solid line in panel (a) for comparison. We set the phase to be zero at $T_N=110.0\mu \mathrm{s}$ as a reference and the data are measured from $T_N=110.0$ to $140.0\mu \mathrm{s}$. $\mathrm{\Delta }\omega =0.31\pm 0.03\mathrm{rad}/\mu \mathrm{s}$ is used in all three measurements. (d) The TOF image near the main peak with different number of modes $N$. The images in A, B, and C correspond to points A, B, and C in (a), (b), and (c), respectively.

Figure 4

The dependence of visibility on the number of modes $N$ and initial relative phase of the same mode in two paths $\mathrm{\Delta }{\theta }_l$. (a) Dependence on $N$ in a situation that the relative initial phases $\mathrm{\Delta }{\theta }_l$ are all zero. The full width at half maximum $\mathrm{\Delta }{\tau }_N$ of the $N$-mode fringe is $2/N$ times that of the two-mode fringe. (b) Dependence on $\mathrm{\Delta }{\theta }_l$ using $N=4$ as an example. The green dashed line, red solid line, and purple dotted line show the fringes with $\left\{\mathrm{\Delta }{\theta }_l\right\}=\left\{\mathrm{\Delta }{\theta }_1,\mathrm{\Delta }{\theta }_2,\mathrm{\Delta }{\theta }_3,\mathrm{\Delta }{\theta }_4\right\}=\left\{0,0,0,0\right\}$, $\left\{\mathrm{\Delta }{\theta }_l\right\}=\left\{0,0,\pi ,\pi \right\}$, and $\left\{\mathrm{\Delta }{\theta }_l\right\}=\left\{0.7\pi ,0.2\pi ,0.5\pi ,\pi \right\}$, respectively.

Figure 5

The oscillation of visibility with $\mathrm{\Delta }\varphi$ under the conditions of different $\mathrm{\Delta }\omega$ at $N=4$. Dashed lines show the fit results. The green (purple) points represent $T_d=42.5\mu \mathrm{s}$ ($T_d=17.0\mu \mathrm{s}$) and corresponding $\mathrm{\Delta }\omega =0.31\pm 0.02\mathrm{rad}/\mu \mathrm{s}$ ($\mathrm{\Delta }\omega =0.12\pm 0.02\mathrm{rad}/\mu \mathrm{s}$). Each data point is a statistical average of four repeated experiments and the error bar shows the standard deviation. We set the phase of the green line at $T_N=110.0\mu \mathrm{s}$ to be zero as a reference, and the data are measured from 110.0 to $140.0\mu \mathrm{s}$. For purple points we set the valley phase to be the same as the valley value of the green line around $2\pi$ and the data are measured from 100.0 to $166.0\mu \mathrm{s}$.

Figure 6

(a) Averaged interference fringes of 42 consecutive measurements. The black points are the experimental data integrated along the $z$ direction, and the red line is the fitted curve by Eq. (5), and we get the visibility 0.32. (b) The percentage ratio of the first ten principle components (PCs). The gray dashed line shows the level of background noise ($\simeq 5\%$) and indicates the threshold for distinguishing important PCs. PCs of interest are highlighted with different colors, which correspond to panels (c1)–(c4) from left to right. Note that the first four PCs are significantly reduced as the order of PC increases. (c1) Average of interference fringes. (c2, c3) Spatial fluctuations along $y$ and $z$ axes, respectively. They together represent atom cloud position fluctuations in the $y–z$ plane. (c4) Noise sources from imaging light.

Figure 7

The visibility $V_N$ vs $T_d$ when $T_N=146.0\mu \mathrm{s}$ and $N=5$. The yellow points are the experimental data with average of three shots, and the black solid line is obtained by fitting the curve using Eq. (A1).