一种光学通道开放且适合构建晶格的静电阱

李胜强; 张梦芝; 杨亮亮

doi:doi:10.3788/AOS201737.1202001

光学学报, 2017, 37 (12): 1202001, 网络出版: 2018-09-06

一种光学通道开放且适合构建晶格的静电阱下载： 565次

Electrostatic Trap Suitable for Construction of Lattices with Opened Optical Access

论文大纲

李胜强 ^*张梦芝杨亮亮

作者单位

盐城师范学院新能源与电子工程学院, 江苏盐城 224007

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摘要

提出了一种用三个带电球电极形成一个光学通道开放、针对弱场搜寻态冷极性分子的静电囚禁方案。给出了方案图,用镜像法推导了空间电场分布的解析解,并利用有限元软件得到了电场分布的数值解。采用经典蒙特卡罗法模拟了冷极性分子被装载和囚禁于静电阱的动力学过程。研究了入射分子束速度和装载时刻对装载效率的影响,给出了被囚禁的冷分子的温度。讨论了所提方案在芯片表面囚禁,尤其是在静电晶格方面的潜在应用。结果表明,装载效率可以达到47.4%, 阱中冷分子的温度为25.4 mK。

Abstract

An electrostatic trapping scheme using three charged spherical electrodes with opened optical access for cold polar molecules in the weak-seeking states is proposed, and the schematic drawing is given. The analytical expressions for the space electrical field distribution are derived with the image charge method. The numerical solution of electrical field distribution is calculated by the finite element software. Dynamic process of the loading and trapping cold polar molecules in an electrostatic trap is simulated with the classical Monte Carlo method. The influences of velocity of the incident molecular beam and loading time on the loading efficiency are studied, and the temperature of the trapped cold molecules is given. Some potential applications of the scheme in the electrostatic surface trapping especially the electrostatic lattices for cold polar molecules are discussed. Results show that the loading efficiency can reach 47.4%, and the temperature of trapped cold molecules is 25.4 mK.

1 引言

近年来, 越来越多的研究小组对冷极性分子的操控产生浓厚兴趣。由于具有固有电偶极矩的极性分子在非均匀电场中受到偶极梯度力的作用,所以可以用多种方式, 如利用静电反射镜^[1]、静电导引^[2-6]、静电分束器^[7-8]、静电Stark减速^[9-12]、静电存储环^[13-16]和三维静电阱^[10,17-23]等操控极性分子束。光学通道开放的静电阱可以方便激光探测和操控被囚禁的分子,甚至可以用于冷分子的光学势蒸发冷却^[22]。2007年,Schnell等^[23]提出了一种针对冷极性分子的半开放交流阱方案。2013年,Wang 等^[22]提出了一种光学通道开放的静电阱方案,但考虑到冷分子的装载需要增加两个平板,这样会破坏其光学通道开放的性能。尽管该研究小组又提出了两种改进方案^[24-25],但阱中心高度不可灵活操控,而且很难将这些方案移植到芯片表面,尤其不适合用于形成静电晶格。光学晶格可广泛用于研究Sisyphus冷却、偏振梯度冷却与囚禁动力学^[26-27]、Raman和绝热冷却^[28-29]、拉比振动和布洛赫振荡^[30-31]、量子传输和量子隧道效应^[32-33]以及通过原子光学晶格的Bragg衍射^[34]等。类似地,原子磁晶格也广泛应用于各种磁囚禁行为,波包动力学、量子传输与隧道效应研究以及采用冷原子的量子计算^[35],甚至可用于制备一维、二维和三维光子晶体等^[36-39]。而关于冷极性分子的静电晶格却几近无人问津。光晶格存在光子散射效应,磁晶格存在导线的冷却问题,相比而言,静电晶格有着自己的优势和巨大的潜在应用价值。

本文提出在双球电极静电阱方案^[22]的基础上,通过增加第三个球电极成功解决分子束的装载问题。相比双球电极静电阱方案,新方案继续保持光学通道开放的特性,且阱中心的高度也可灵活操控。此方案具有微型化、集成化、方便移植到芯片表面的特点,可构建一维和二维静电晶格。本文给出了方案模型,推导了电场的解析解,用有限元软件给出了空间电场的数值解;用经典的蒙特卡罗方法模拟了冷极性分子被装载和囚禁的动力学过程,以证实方案的可行性;讨论了此方案在静电表面囚禁方面的应用,如单阱、一维和二维静电晶格。

2 静电阱方案

2.1 公式推导

如图1(a)所示,囚禁方案是由三个半径为r的球形电极构成。上面两球的球心距为 2d ,下面球的球心坐标为(0,0,-d)。上面两球所加电压均为 U₁ ,下面球所加电压为U₂。

如图1(b)所示, O₁、 O₂ 和 O₉ 分别是三个球电极的球心。这里 O₃~O₈、O₁₀、O₁₁ 是镜像电荷位置。上面两个球电极和下面一个球电极的电荷量分别是q 和q'。在O₁~O₁₁等处的电荷量分别表示为 Q₁~Q₁₁。根据平面几何知识与镜像法原理,得到 O₁~O₁₁的坐标和Q₁~Q₁₁表达式如表1所示。

图 1. (a) 三个带电球电极囚禁分子的方案图; (b) 镜像法的原理图

Fig. 1. (a) Schematic for trapping molecules using three charged spherical electrodes; (b) schematic of the image method

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表 1. 电荷量和坐标

Table 1. Charges and coordinates

Charge	Expression	Coordinate
Q₁	q+ $\begin{matrix} \frac{r}{2 d} \end{matrix}$ q+ $\begin{matrix} \frac{r}{\sqrt[]{2} d} \end{matrix}$ q'	$\begin{matrix} (\begin{matrix} 0, & - d, & 0 \end{matrix}) \end{matrix}$
Q₂	q+ $\begin{matrix} \frac{r}{2 d} \end{matrix}$ q+ $\begin{matrix} \frac{r}{\sqrt[]{2} d} \end{matrix}$ q'	$\begin{matrix} (\begin{matrix} 0, & d, & 0 \end{matrix}) \end{matrix}$
Q₃	- $\begin{matrix} \frac{r}{2 d} \end{matrix}$ q	$\begin{matrix} (\begin{matrix} 0, & d - r^{2} / (2 d), & 0 \end{matrix}) \end{matrix}$
Q₄	- $\begin{matrix} \frac{r}{2 d} \end{matrix}$ q	$\begin{matrix} (\begin{matrix} 0, & - d + r^{2} / (2 d), & 0 \end{matrix}) \end{matrix}$
Q₅	$\begin{matrix} \frac{r^{2} q}{2 d \sqrt[]{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}} \end{matrix}$	$\begin{matrix} [0, - \frac{r^{2} (d - \frac{r^{2}}{2 d})}{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}, - d + \frac{r^{2} d}{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}] \end{matrix}$
Q₆	- $\begin{matrix} \frac{r}{\sqrt[]{2} d} \end{matrix} \begin{matrix} (q + \frac{r}{2 d} q + \frac{r}{\sqrt[]{2} d} q') \end{matrix}$	$\begin{matrix} (0, - \frac{r^{2}}{2 d}, - d + \frac{r^{2}}{2 d}) \end{matrix}$
Q₇	$\begin{matrix} \frac{r^{2} q}{2 d \sqrt[]{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}} \end{matrix}$	$\begin{matrix} [0, \frac{r^{2} (d - \frac{r^{2}}{2 d})}{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}, - d + \frac{r^{2} d}{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}] \end{matrix}$
Q₈	- $\begin{matrix} \frac{r}{\sqrt[]{2} d} \end{matrix} \begin{matrix} (q + \frac{r}{2 d} q + \frac{r}{\sqrt[]{2} d} q') \end{matrix}$	$\begin{matrix} (0, \frac{r^{2}}{2 d}, - d + \frac{r^{2}}{2 d}) \end{matrix}$
Q₉	q'+ $\begin{matrix} \frac{2 r}{\sqrt[]{2} d} \end{matrix} \begin{matrix} (q + \frac{r}{2 d} q + \frac{r}{\sqrt[]{2} d} q') \end{matrix}$ - $\begin{matrix} \frac{r^{2} q}{d \sqrt[]{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}} \end{matrix}$
Q₁₀	- $\begin{matrix} \frac{r}{\sqrt[]{2} d} \end{matrix}$ q'	$\begin{matrix} (0, - d + \frac{r^{2}}{2 d}, - \frac{r^{2}}{2 d}) \end{matrix}$
Q₁₁	- $\begin{matrix} \frac{r}{\sqrt[]{2} d} \end{matrix}$ q'	$\begin{matrix} (0, d - \frac{r^{2}}{2 d}, - \frac{r^{2}}{2 d}) \end{matrix}$

查看所有表

由表1可以得到电势的表达式为

\begin{matrix} \begin{matrix} φ = \\ \frac{1}{4 π ε_{0}} \{[\frac{q + \frac{rq}{2 d}}{\sqrt[]{x^{2} + {(y + d)}^{2} + z^{2}}} + \frac{q + \frac{rq}{2 d}}{\sqrt[]{x^{2} + {(y - d)}^{2} + z^{2}}} + \\ \begin{matrix} \frac{- \frac{r}{2 d} q}{\sqrt[]{x^{2} + {(y - d + \frac{r^{2}}{2 d})}^{2} + z^{2}}} + \frac{- \frac{r}{2 d} q}{\sqrt[]{x^{2} + {(y + d - \frac{r^{2}}{2 d})}^{2} + z^{2}}}] + \end{matrix} \\ \frac{\frac{r^{2} q}{2 d \sqrt[]{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}}}{\sqrt[]{x^{2} + {[y + \frac{r^{2} (d - \frac{r^{2}}{2 d})}{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}]}^{2} + {[z + d - \frac{r^{2} d}{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}]}^{2}}} + \\ \frac{\frac{r^{2} q}{2 d \sqrt[]{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}}}{\sqrt[]{x^{2} + {[y - \frac{r^{2} (d - \frac{r^{2}}{2 d})}{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}]}^{2} + {[z + d - \frac{r^{2} d}{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}]}^{2}}} + \\ \frac{- \frac{r}{\sqrt[]{2} d} (q + \frac{r}{2 d} q)}{\sqrt[]{x^{2} + {(y + \frac{r^{2}}{2 d})}^{2} + {(z + d - \frac{r^{2}}{2 d})}^{2}}} + \frac{- \frac{r}{\sqrt[]{2} d} (q + \frac{r}{2 d} q)}{\sqrt[]{x^{2} + {(y - \frac{r^{2}}{2 d})}^{2} + {(z + d - \frac{r^{2}}{2 d})}^{2}}} \times \\ \frac{- \frac{2 r^{2} q}{2 d \sqrt[]{{(d - \frac{r^{2}}{2 d})}^{2} + d^{2}}} + \frac{2 r}{\sqrt[]{2} d} (q + \frac{rq}{2 d})]}{\sqrt[]{x^{2} + y^{2} + {(z + d)}^{2}}} + \\ \{\frac{\frac{rq'}{\sqrt[]{2} d}}{\sqrt[]{x^{2} + {(y + d)}^{2} + z^{2}}} + \frac{\frac{rq'}{\sqrt[]{2} d}}{\sqrt[]{x^{2} + {(y - d)}^{2} + z^{2}}} + \frac{- \frac{r}{\sqrt[]{2} d} (\frac{r}{\sqrt[]{2} d} q')}{\sqrt[]{x^{2} + {(y + \frac{r^{2}}{2 d})}^{2} + {(z + d - \frac{r^{2}}{2 d})}^{2}}} + \\ \frac{- \frac{r}{\sqrt[]{2} d} (\frac{r}{\sqrt[]{2} d} q')}{\sqrt[]{x^{2} + {(y - \frac{r^{2}}{2 d})}^{2} + {(z + d - \frac{r^{2}}{2 d})}^{2}}} + \frac{[q' + \frac{2 r}{\sqrt[]{2} d} (\frac{rq'}{\sqrt[]{2} d})]}{\sqrt[]{x^{2} + y^{2} + {(z + d)}^{2}}} + \\ \frac{- \frac{rq'}{\sqrt[]{2} d}}{\sqrt[]{x^{2} + {(y - d + \frac{r^{2}}{2 d})}^{2} + {(z + \frac{r^{2}}{2 d})}^{2}}} + \frac{- \frac{rq'}{\sqrt[]{2} d}}{\sqrt[]{x^{2} + {(y + d + \frac{r^{2}}{2 d})}^{2} + {(z + \frac{r^{2}}{2 d})}^{2}}}\}\}, (1) \end{matrix} \end{matrix}

这里令

\begin{matrix} \begin{matrix} \frac{U_{1}}{A_{1}} = \frac{q}{4 π ε_{0}}, (2) \\ \frac{U_{2}}{A_{2}} = \frac{q'}{4 π ε_{0}}, (3) \\ E_{x} = - \frac{\partial φ}{\partial x}, (4) \\ E_{y} = - \frac{\partial φ}{\partial y}, (5) \\ E_{z} = - \frac{\partial φ}{\partial z}, (6) \\ E = \sqrt[]{E_{x}^{2} + E_{y}^{2} + E_{z}^{2}}, (7) \end{matrix} \end{matrix}

式中A₁、A₂为中间变量,ε₀为真空介电常数。

2.2 解析解和数值解的比对

通过(1)~(7) 式,可以得到电场的等高线分布。为进行对比,用有限元软件(Ansoft Maxwell)建模得到电场分布的数值解。在数值模拟中,文献[ 22]的方法是用一个半径为200 mm的空心球壳罩在三个球形电极外面,并且让该球壳接地,以提供边界条件。参数选取如下:U₁=40 kV, U₂=20 kV, r=0.004 m, d=0.01 m。通过对E_y 和 E_z进行积分,得到A₁≈301.8359,A₂≈3497.4190。

图 2. 囚禁时YOZ、XOZ、XOY平面的电场等高线分布的解析解和数值解。(a)(d) YOZ平面; (b)(e) XOZ平面; (c)(f) XOY平面

Fig. 2. Analytical solutions and numerical solutions of contour distributions of trapping field at YOZ, XOZ and XOY planes. (a)(d) YOZ plane; (b)(e) XOZ plane; (c)(f) XOY plane

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图2(a)~(c)分别表示 YOZ、XOZ、XOY 等平面的电场等高线分布解析解。这里将无穷远处的电势视为0。图2(d)~(f)分别表示用有限元软件得到的 YOZ、 XOZ、 XOY 等平面的电场等高线分布的数值解。由于有限元软件很消耗内存,鉴于工作站内存只有64 GB, 所以在模拟中用一个接地且有限大(半径为200 mm)的空心球壳罩在三个球形电极外面作为边界条件。采用这种近似得到的结果表明解析解和数值解吻合得较好。从图2拓展出一个三维封闭的静电阱,该静电阱可以用于囚禁弱场搜寻态的冷极性分子。

2.3 阱的灵活操控性

通过改变电压U₂,可以灵活操控阱中心的位置。保持U₁=40 kV 不变,研究阱中心位置和U₂的关系。如图3所示,当U₂从 20 kV 变化到 -10 kV,阱中心的位置沿着Z方向从 0.2 mm 增加到2.8 mm,可见阱中心位置是可以灵活操控的。

图 3. 阱中心位置与电压U2的关系

Fig. 3. Relationship between position of trap center and voltage U2

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2.4 ND3 分子的装载和囚禁

选择量子态为|J,K,M>=|1,1,-1>的重氨(ND₃)作为测试分子。当ND₃ 分子在非均匀电场中运动时,将受到电场偶极梯度力作用(F=-μ·ÑE,其中μ为分子的电偶极矩,ÑE为电场强度梯度), 因此它们可以被囚禁在三维静电阱中。相应的Stark囚禁势为

\begin{matrix} \begin{matrix} W_{Stark}^{(1)} = - μE \frac{KM}{J (J + 1)}, (8) \\ W_{Stark}^{(2)} = \frac{μ^{2} E^{2}}{2 h B} \{\frac{(J^{2} - K^{2}) (J^{2} - M^{2})}{J^{3} (2 J - 1) (2 J + 1)} - \frac{[{(J + 1)}^{2} - K^{2}] [{(J + 1)}^{2} - M^{2}]}{{(J + 1)}^{3} (2 J + 1) (2 J + 3)}\}, (9) \end{matrix} \end{matrix}

式中J为转动量子数, K和M分别为J在分子轴方向和外电场方向的投影,h为普朗克常数,B为转动常数。总的Stark囚禁势W_Stark= $\begin{matrix} W_{Stark}^{(1)} \end{matrix}$ + $\begin{matrix} W_{Stark}^{(2)} \end{matrix}$ 。这里用有限元软件计算了沿着Z 方向的电场强度以及对于ND₃ 的囚禁势。冷分子束沿着-Z 方向装载。在装载过程中选择U₁=3 kV, U₂=-15 kV。

如图4(a)所示,电场沿着Z方向呈不对称分布。尽可能将阱中心上方的势垒降低,以方便入射分子束进入静电阱。当分子束沿着-Z 方向运动时,将受到一个与其运动方向相反的电场偶极梯度力作用而减速。当它们到达阱中心时,立刻将电压切换为U₁=40 kV,U₂=20 kV,如图4(b)所示,这样就可以将冷分子囚禁在静电阱中。

图 4. 沿Z方向的电场分布及对处于量子态|J,K,M>=|1,1,-1>的ND3分子的Stark势能。 (a)装载; (b)囚禁

Fig. 4. Electric field distributions in Z direction and Stark potential for ND3 molecules in state (|J,K,M>=|1,1,-1>). (a) Loading; (b) trapping

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2.5 蒙特卡罗模拟

采用经典的蒙特卡罗法进行模拟。先通过随机数选取得到所需数目且初始位置和初始速度分布均满足高斯分布的初始分子束。在模拟中,所用的分子数目为10⁵。入射ND₃ 分子束的位置分布和速度分布均为高斯分布, X、 Y、Z 三个方向位置分布的半峰全宽分别为4, 4, 8 mm。X、 Y、Z三个方向分子束速度分布的半峰全宽均为 3 m/s,其中X、 Y方向分子束的中心速度均为 0 m/s。在分子束的装载和囚禁过程中,将分子看作经典粒子,分子在非均匀电场中受到偶极梯度力的作用,其运动满足牛顿运动方程:

\begin{matrix} mr = F (r) = - \nabla W_{Stark} (r), (10) \end{matrix}

式中m为质量,r为位置矢量,F(r)为此位置处ND₃分子所受到的偶极梯度力。如图5所示,装载效率受到从装载过程到囚禁过程的切换时刻t_load的影响。可以发现:分子束初始速度一定时,有一个最佳装载时刻和相应的最大装载效率。当入射分子束的速度分别为10, 11, 12 m/s时,最佳装载时刻分别为1.8087, 1.7147, 1.6284 ms, 相应的最大装载效率分别为 46.709%、 47.361%和45.311%;装载效率不仅与 t_load有关,还取决于入射分子束的初始速度。因此接下来探究初始速度如何影响装载效率。

如图6所示,随着分子束入射速度的增加,最大装载效率先增大,达到峰值后开始减小。值得一提的是,当初始速度为 11 m/s时,装载效率最大,可以达到约 47.4%。

图 5. 在三个不同分子束入射速度下,装载效率与装载时间的关系

Fig. 5. Relationship between loading efficiency and loading time under different velocities of incident molecular beam

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图 6. 装载效率与入射分子束中心速度的关系

Fig. 6. Relationship between loading efficiency and center velocity of incident molecular beam

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取入射分子束Z方向的中心速度为11 m/s, X或Y方向的中心速度均为 0 m/s,三个方向的速度分布半峰全宽均为3 m/s。切换时刻t_load为1.7147 ms,模拟的分子数目为10⁵。图7给出ND₃分子束初始和囚禁在静电阱中100 ms后的速度分布情况。最终囚禁在阱中的ND₃分子在X、Y、Z三个方向的中心速度均为 0 m/s ,速度分布半峰全宽分别为7.11,8.10,7.69 m/s。被囚禁在阱中的冷分子的温度约为 25.4 mK。

图 7. 入射分子束的初始速度分布及囚禁在阱中的分子速度(VX、VY、VZ)分布。 (a) VX; (b) VY; (c) VZ

Fig. 7. Velocity distributions of incident molecular beam before loading process and the trapped cold molecules. (a) VX; (b) VY; (c) VZ

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3 潜在应用与讨论

此方案可延伸性强,下面将三球静电阱方案微型化,并将其移植到芯片表面,以形成静电表面单阱、一维和二维静电晶格等。

3.1 静电表面单阱

如图8所示,接地的不锈钢金属板上方是两层聚四氟乙烯绝缘介质层(介质的相对介电常数ε_r=2.1) 。两层绝缘介质的厚度均为10 μm。每个球电极的半径均为 4 um。上面两球电极的球心间距为40 um。将上面两球半埋入上面的绝缘介质层, 将下面的球埋入两个介质层之间。加在上面两球的电压为U₁,加在下面球的电压为U₂。图9(a)~(c)分别给出了 YOZ、 XOZ、 XOY三个平面的电场等高线分布图。这里取 U₁=200 V,U₂=0 V。可以发现,此时形成了一个三维封闭的静电阱。

图 8. 用三个带电球电极构建静电表面阱方案图

Fig. 8. Schematic for trapping molecules on a chip with three charged spherical electrodes

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图 9. 囚禁电场的等高线分布。 (a) YOZ平面; (b) XOZ 平面; (c) XOY平面

Fig. 9. Contour distributions of trapping field. (a) YOZ plane; (b) XOZ plane; (c) XOY plane

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3.2 一维静电晶格

如果沿着Y轴重复这种三球结构,可形成一维阵列,得到一维静电晶格,如图10所示。这里以 1×9 晶格为例。

图11为两个不同位置(图10中虚线方框位置)处的电场等高线分布。图11(a)~(c)和图11(d)~(f)

图 10. 一系列三球电极实现在芯片表面对冷分子静电囚禁(一维静电晶格)

Fig. 10. Schematic for trapping molecules on a chip with a series of three charged spherical electrodes (one-dimensional electrostatic lattice)

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图 11. 在两个不同位置处[图9(a)虚框位置]电场的等高线分布。 (a)(d) XOZ平面; (b)(e) XOY平面; (c)(f) YOZ平面

Fig. 11. Contour distributions of trapping field at two different locations [marked with dashed box in Fig. 9(a)]. (a)(d) XOZ plane; (b)(e) XOY plane; (c)(f) YOZ plane

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分别为阵列边缘处和中心处三个平面(XOZ、XOY、YOZ)的电场分布情况,得到了9个三维封闭的静电阱,即一个1×9 的静电晶格。

3.3 二维静电晶格

同样地,如果将一维静电晶格沿着X方向重复,球形电极所形成的阵列如图12所示,这样就可以得到一个二维静电晶格,但需保证沿着X方向的相邻两行电压相反,具体电压设置如图13所示。

图 12. 利用一系列三球电极在芯片表面实现对冷分子的静电囚禁(二维静电晶格)

Fig. 12. Schematic for electrostatic trapping of cold molecules on a chip with a series of three charged spherical electrodes (two-dimensional electrostatic lattice)

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图 13. 二维静电晶格俯视图

Fig. 13. Top view of two-dimensional electrostatic lattice

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图 14. 三个不同位置[图13中虚框位置]电场的等高线分布。 (a) XOZ平面; (b) XOY平面; (c) YOZ平面

Fig. 14. Contour distributions of electric field in three different locations (marked with dashed box in Fig. 13) . (a) XOZ plane; (b) XOY plane; (c) YOZ plane

下载图片查看所有图片

图14给出了所有球形电极的俯视图(在XOY平面的投影),下面用颜色来区分电极电压以及位置。所有的黑色圆圈表示位于两个绝缘层之间的球电极, 红色和绿色圆圈表示半埋入上面一层绝缘介质层的电极。红色、绿色、黑色圆圈代表的电极所加电压分别为U₁、 U₂和 U₃。当 U₁=220 V、 U₂=-220 V和 U₃=0 V时,可以得到一个5×5 的静电晶格。图14给出了三个不同位置(图13中虚线方框位置)的电场分布情况。图14(a)为阵列中左上角处的电场等高线分布,图14(b)为阵列中间处的电场等高线分布,图14(c)为阵列中右下角处的电场等高线分布。由图14可以得到25个三维封闭的静电阱,即一个5×5 的静电晶格。

4 结论

提出了用三个带电球电极构建一个光学通道开放、针对弱场搜寻态的冷极性分子静电阱,给出了空间电场分布的解析解和数值解。阱中心的高度是可以灵活操控的。通过经典的蒙特卡罗法进行模拟,探究了影响装载效率的因素。计算结果表明:当分子束的初始速度为11 m/s、相应的装载时刻为1.7147 ms时,装载效率最大可达47.4%。将所提静电阱移植到芯片表面, 不仅可以形成静电单阱,而且可以形成一维和二维静电晶格。借助于囚禁在静电阱中的冷分子,可以研究处于电子激发态、振动激发态的分子寿命。被囚禁的冷极性分子还可用于研究量子信息处理和冷碰撞等。

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李胜强, 张梦芝, 杨亮亮. 一种光学通道开放且适合构建晶格的静电阱[J]. 光学学报, 2017, 37(12): 1202001. Shengqiang Li, Mengzhi Zhang, Liangliang Yang. Electrostatic Trap Suitable for Construction of Lattices with Opened Optical Access[J]. Acta Optica Sinica, 2017, 37(12): 1202001.

一种光学通道开放且适合构建晶格的静电阱 下载： 565次

1 引言

2 静电阱方案

2.1 公式推导

图 1. (a) 三个带电球电极囚禁分子的方案图; (b) 镜像法的原理图

Fig. 1. (a) Schematic for trapping molecules using three charged spherical electrodes; (b) schematic of the image method

表 1. 电荷量和坐标

Table 1. Charges and coordinates

图 2. 囚禁时YOZ、XOZ、XOY平面的电场等高线分布的解析解和数值解。(a)(d) YOZ平面; (b)(e) XOZ平面; (c)(f) XOY平面

Fig. 2. Analytical solutions and numerical solutions of contour distributions of trapping field at YOZ, XOZ and XOY planes. (a)(d) YOZ plane; (b)(e) XOZ plane; (c)(f) XOY plane

2.3 阱的灵活操控性

图 3. 阱中心位置与电压U2的关系

Fig. 3. Relationship between position of trap center and voltage U2

2.4 ND3 分子的装载和囚禁

图 4. 沿Z方向的电场分布及对处于量子态|J,K,M>=|1,1,-1>的ND3分子的Stark势能。 (a)装载; (b)囚禁

Fig. 4. Electric field distributions in Z direction and Stark potential for ND3 molecules in state (|J,K,M>=|1,1,-1>). (a) Loading; (b) trapping

图 5. 在三个不同分子束入射速度下,装载效率与装载时间的关系

Fig. 5. Relationship between loading efficiency and loading time under different velocities of incident molecular beam

图 6. 装载效率与入射分子束中心速度的关系

Fig. 6. Relationship between loading efficiency and center velocity of incident molecular beam

图 7. 入射分子束的初始速度分布及囚禁在阱中的分子速度(VX、VY、VZ)分布。 (a) VX; (b) VY; (c) VZ

Fig. 7. Velocity distributions of incident molecular beam before loading process and the trapped cold molecules. (a) VX; (b) VY; (c) VZ

3 潜在应用与讨论

图 8. 用三个带电球电极构建静电表面阱方案图

Fig. 8. Schematic for trapping molecules on a chip with three charged spherical electrodes

图 9. 囚禁电场的等高线分布。 (a) YOZ平面; (b) XOZ 平面; (c) XOY平面

Fig. 9. Contour distributions of trapping field. (a) YOZ plane; (b) XOZ plane; (c) XOY plane

3.2 一维静电晶格

图 10. 一系列三球电极实现在芯片表面对冷分子静电囚禁(一维静电晶格)

Fig. 10. Schematic for trapping molecules on a chip with a series of three charged spherical electrodes (one-dimensional electrostatic lattice)

图 11. 在两个不同位置处[图9(a)虚框位置]电场的等高线分布。 (a)(d) XOZ平面; (b)(e) XOY平面; (c)(f) YOZ平面

Fig. 11. Contour distributions of trapping field at two different locations [marked with dashed box in Fig. 9(a)]. (a)(d) XOZ plane; (b)(e) XOY plane; (c)(f) YOZ plane

3.3 二维静电晶格

图 12. 利用一系列三球电极在芯片表面实现对冷分子的静电囚禁(二维静电晶格)

Fig. 12. Schematic for electrostatic trapping of cold molecules on a chip with a series of three charged spherical electrodes (two-dimensional electrostatic lattice)

图 13. 二维静电晶格俯视图

Fig. 13. Top view of two-dimensional electrostatic lattice

图 14. 三个不同位置[图13中虚框位置]电场的等高线分布。 (a) XOZ平面; (b) XOY平面; (c) YOZ平面

Fig. 14. Contour distributions of electric field in three different locations (marked with dashed box in Fig. 13) . (a) XOZ plane; (b) XOY plane; (c) YOZ plane

4 结论

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一种光学通道开放且适合构建晶格的静电阱下载： 565次