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利用Kapitza-Dirac脉冲操控简谐势阱中冷原子测量重力加速度

何天琛 李吉

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利用Kapitza-Dirac脉冲操控简谐势阱中冷原子测量重力加速度

何天琛, 李吉

Measurement of gravity acceleration by cold atoms in a harmonic trap using Kapitza-Dirac pulses

He Tian-Chen, Li Ji
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  • 利用Feynman路径积分研究了简谐势阱中冷原子受到两次Kapitza-Dirac脉冲的干涉机制. 理论研究表明: 当初始态为简谐势阱的基态时, 外场使不同模式的演化路径相对于没有外场情况下的路径发生偏离; 同时外场强度和测量时刻相邻模式的相位差成线性关系; 当外场为重力场时, 测量重力加速度的精度可达10–9; 当初始态为简谐势阱和外场作用下的共同基态时, 测量精度会下降, 同时原子间排斥和吸引相互作用的增强也会导致测量精度提高.
    The interferometry of two Kapitza-Dirac (KD) pulses acting on cold atoms in a harmonic oscillator potential well is investigated by Feynman path integral method. The wave function and density distribution function of the system at a given time are calculated analytically by using the propagator under the action of an external field. The first KD pulse acts on cold atoms to produce a large number of modes in the harmonic oscillator potential well. The maximum value of wave packet of mode 0 is larger than those of other modes. These modes evolve along different paths. The external field changes the phase of each mode and makes the evolution path of the mode deviate from that without the external field. When the second KD pulse is added, it splits the mode of the first KD pulse, and thus generates more modes. These modes will evolve along different paths under the action of external field and harmonic potential well, and interfere with each other. At the moment of measurement, all the wave packets are separated without overlapping. The effect of the external field does not change the magnitude of the density distribution at the time of measurement, but makes the wave packet of each mode shift. The phase difference between adjacent modes decreases linearly with the increase of field intensity. When the external field is a gravity field, we calculate the Fisher information and the Cramer-Rao lowér bound. The Fisher information is proportional to the mass of atoms and inversely to the third power of harmonic potential well frequency. We can improve the measurement accuracy of interferometer by reducing the frequency of harmonic potential well and increasing atomic mass. When the initial state is the ground state of the harmonic potential well, the accuracy of the gravity acceleration measured by the interference device can be obtained to be 10–9 by using the experimental parameters. The initial state is the ground state of the harmonic potential well and the external field, and the calculation result indicates that the measurement accuracy will decrease. At the same time, the enhancement of inter-atomic repulsion and attraction interaction will also lead the measurement accuracy to increase.
      通信作者: 李吉, liji163love@163.com
    • 基金项目: 山西省高等学校科技创新项目 (批准号: 2019L0785, 2019L0813)和北京市自然科学基金 (批准号: 1182009)资助的课题.
      Corresponding author: Li Ji, liji163love@163.com
    • Funds: Project supported by the Scientific and Technologial Innovation Program of Higher Education Institutions in Shanxi, China (Grant Nos. 2019L0785, 2019L0813) and the Beijing Natural Science Foundation, China (Grant No. 1182009).
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    Jaffe M, Xu V, Haslinger P, Müller H, Hamilton P 2018 Phys. Rev. Lett. 121 040402Google Scholar

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    Carey M, Saywell J, Elcock D, Belal M, Freegarde T 2019 Phys. Rev. A 99 023631Google Scholar

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    Wigley P B, Hardman K S, Freier C, Everitt P J, Legge S, Manju P, Close J D, Robins N P 2019 Phys. Rev. A 99 023615Google Scholar

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    Arias A, Lochead G, Wintermantel T M, Helmrich S, Whitlock S 2019 Phys. Rev. Lett. 122 053601Google Scholar

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    Zhou L, Xiong Z Y, Yang W, Tang B, Peng W C, Wang Y B, Xu P, Wang J, Zhan M S 2011 Chin. Phys. Lett. 28 013701Google Scholar

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    Wu B, Cheng B, Fu Z J, Zhu D, Zhou Y, Wong K X, Wang X L, Lin Q 2018 Acta Phys. Sin. 67 190302Google Scholar

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    McGuirk J M, Foster G T, Fixler J B, Snadden M J, Kasevich M A 2002 Phys. Rev. A 65 033608Google Scholar

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    Fixler J B, Foster G T, McGuirk J M, Kasevich M A 2007 Science 315 74Google Scholar

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    Li W D, He T, Smerzi A 2014 Phys. Rev. Lett. 113 023003Google Scholar

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    杨志安 2013 物理学报 62 110302Google Scholar

    Yang Z A 2013 Acta Phys. Sin. 62 110302Google Scholar

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    He T C, Niu P B 2017 Phys. Lett. A 381 108

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    Cheng R, He T, Li W, Smerzi A 2016 J. Mod. Phys. 7 2043Google Scholar

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    Sapiro R E, Zhang R, Raithel G 2009 Phys. Rev. A 79 043630Google Scholar

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    Keller C, Schmiedmayer J, Zeilinger A, Nonn T, Dürr S, Rempe G 1999 Appl. Phys. B: Lasers Opt. 69 303Google Scholar

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    Dong G J, Zhu J, Zhang W P, Malomed B A 2013 Phys. Rev. Lett. 110 250401Google Scholar

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    Zhu J, Dong G J, Shneider M N, Zhang W P 2011 Phys. Rev. Lett. 106 210403Google Scholar

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    Chwedenczuk J, Piazza F, Smerzi A 2013 Phys. Rev. A 87 033607Google Scholar

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    Shin Y, Saba M, Schirotzek A, Pasquini T A, Leanhardt A E, Pritchard D E, Ketterle W 2004 Phys. Rev. Lett. 92 150401Google Scholar

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    Lehtovaara L, Toivanen J, Eloranta J 2007 J. Comput. Phys. 221 148Google Scholar

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    Fattori M, D’Errico C, Roati G, Zaccanti M, Jona-Lasinio M, Modugno M, Inguscio M, Modugno G 2008 Phys. Rev. Lett. 100 080405Google Scholar

  • 图 1  简谐势阱中冷原子受到两次KD脉冲的示意图 在测量时刻${t_{\rm{f}}} ={\text{π}}/\omega $, 不同模式相干叠加, $\beta $为外场

    Fig. 1.  Diagram of cold atoms with two KD pulses in harmonic oscillator potential. At the measurement time ${t_{\rm{f}}} = {\text{π}}/\omega $, the coherent superposition of different modes occurs. $\beta $ is the external field.

    图 2  没有外场的情况下($\beta = 0$), 系统态密度分布函数的演化规律 无量纲参数为$V = 1,\; k = 10$, 测量时刻在$t = {t_{\rm{f}}} = {\text{π}}$, 两次KD脉冲都用红色箭头表示, 态密度演化图右边的彩色条表示态密度分布函数值由低(蓝色)变到高(红色)

    Fig. 2.  In the absence of an external field ($\beta = 0$), the density distribution function of system varies with time. The dimensionless parameter are $V = 1,\; k = 10$. The measurement time is at $t = {t_{\rm{f}}} ={\text{π}}$. Both KD pulses are represented by red arrows. The color bar on the right side of the density of states evolution diagram indicates that the value of the density distribution function changes from low (blue) to high (red).

    图 3  在外场$\beta = 1$的情况下, 系统态密度分布函数的演化规律 图中除$\beta $之外的其他无量纲参数与图2相同, 红色箭头为KD脉冲, 测量时刻在$t = {t_{\rm{f}}} = {\text{π}}$

    Fig. 3.  In the case of external field $\beta = 1$, the evolution of density distribution function of the system is obtained. The dimensionless parameters in this figure are the same as those in Figure 2. The red arrow are KD pulses. The measurement time is at $t = {t_{\rm{f}}} = {\text{π}}$.

    图 4  在外场$\beta = 2$的情况下, 系统态密度分布函数的演化规律 图中除$\beta $之外的其他无量纲参数与图2相同, 红色的箭头为KD脉冲, 测量时刻在$t = {t_{\rm{f}}} = {\text{π}}$

    Fig. 4.  In the case of external field $\beta = 2$, the evolution of density distribution function of the system is obtained. The dimensionless parameters in this figure are the same as those in Figure 2. The red arrow are KD pulses. The measurement time is at $t = {t_{\rm{f}}} = {\text{π}}$.

    图 5  图2图3图4中测量时刻${t_{\rm{f}}} = {\text{π}}$态密度的分布规律

    Fig. 5.  The density distribution functions at measuring time${t_{\rm{f}}} = {\text{π}}$ for figure 2, 3, and 4.

    图 6  外场$\beta = 1$的情况下系统基态的态密度分布函数, 不同的颜色对应不同非线性相互作用下的态密度分布函数

    Fig. 6.  In the case of external field $\beta = 1$, the density distribution function of the ground state of the system. Different colors correspond to the density distribution function under different non-linear interactions.

    图 7  在外场$\beta = 1$的情况下系统基态的态密度分布函数, 不同的颜色对应不同非线性相互作用下的态密度分布函数

    Fig. 7.  In the case of external field $\beta = 1$, the density distribution function of the ground state of the system. Different colors correspond to the density distribution function under different non-linear interactions.

    图 8  $V = 1,\;k = 10,\; \beta = {\rm{1}}$的情况下测量时刻的态密度分布函数, 不同的颜色对应不同非线性相互作用下的态密度分布函数

    Fig. 8.  In the case of $V = 1,\;k = 10,\; \beta = {\rm{1}}$, the density distribution functions at measuring time. Different colors correspond to the density distribution function under different non-linear interactions.

    图 9  图8中0模式的放大图

    Fig. 9.  Detailed diagram of 0 mode in Fig. 8.

    图 10  测量时刻系统态密度的分布函数, 除了非线性参数以外的其他无量纲参数和图8相同

    Fig. 10.  The density distribution functions at measuring time, dimensionless parameters other than non-linear parameters are the same as those in Fig. 8.

    图 11  图10中0模式的放大图

    Fig. 11.  Detailed diagram of 0 mode in Fig. 10.

    图 12  系统的测量精度随非线性参数的变化规律, 除了非线性参数以外的其他无量纲参数和图8相同

    Fig. 12.  The variation of measuring accuracy of the system with nonlinear parameters, dimensionless parameters other than non-linear parameters are the same as those in Fig. 8.

  • [1]

    Fang J, Hu J, Chen X, Zhu H, Zhou L, Zhong J, Wang J, Zhan M 2018 Opt. Express 26 1586Google Scholar

    [2]

    Lu S B, Yao Z W, Li R B, Luo J, Barthwal S, Chen H H, Lu Z X, Wang J, Zhan M S 2018 Opt. Commun. 429 158Google Scholar

    [3]

    Fang B, Mielec N, Savoie D, Altorio M, Landragin A, Geiger R 2018 New J. Phys. 20 023020Google Scholar

    [4]

    Jaffe M, Xu V, Haslinger P, Müller H, Hamilton P 2018 Phys. Rev. Lett. 121 040402Google Scholar

    [5]

    Carey M, Saywell J, Elcock D, Belal M, Freegarde T 2019 Phys. Rev. A 99 023631Google Scholar

    [6]

    Wigley P B, Hardman K S, Freier C, Everitt P J, Legge S, Manju P, Close J D, Robins N P 2019 Phys. Rev. A 99 023615Google Scholar

    [7]

    Arias A, Lochead G, Wintermantel T M, Helmrich S, Whitlock S 2019 Phys. Rev. Lett. 122 053601Google Scholar

    [8]

    Zhou L, Xiong Z Y, Yang W, Tang B, Peng W C, Wang Y B, Xu P, Wang J, Zhan M S 2011 Chin. Phys. Lett. 28 013701Google Scholar

    [9]

    Imanishi Y, Sato T, Higashi T, Sun W, Okubo S 2004 Science 306 476Google Scholar

    [10]

    Kibble B P 1991 Proc. IEE: Science, Measurement and Technology 138 187Google Scholar

    [11]

    Blakely R J, Jachens R C 1991 Geological Society of America Bulletin 103 795Google Scholar

    [12]

    Peters A, Chung K Y, Chu S 2001 Metrologia 38 25Google Scholar

    [13]

    Rasel E M, Oberthaler M K, Batelaan H, Schmiedmayer J, Zeilinger A 1995 Phys. Rev. Lett. 75 2633Google Scholar

    [14]

    Giltner D M, McGowan R W, Lee S A 1995 Phys. Rev. Lett. 75 2638Google Scholar

    [15]

    吴彬, 程冰, 付志杰, 朱栋, 周寅, 翁堪兴, 王肖隆, 林强 2018 物理学报 67 190302Google Scholar

    Wu B, Cheng B, Fu Z J, Zhu D, Zhou Y, Wong K X, Wang X L, Lin Q 2018 Acta Phys. Sin. 67 190302Google Scholar

    [16]

    McGuirk J M, Foster G T, Fixler J B, Snadden M J, Kasevich M A 2002 Phys. Rev. A 65 033608Google Scholar

    [17]

    Fixler J B, Foster G T, McGuirk J M, Kasevich M A 2007 Science 315 74Google Scholar

    [18]

    Li W D, He T, Smerzi A 2014 Phys. Rev. Lett. 113 023003Google Scholar

    [19]

    杨志安 2013 物理学报 62 110302Google Scholar

    Yang Z A 2013 Acta Phys. Sin. 62 110302Google Scholar

    [20]

    He T C, Niu P B 2017 Phys. Lett. A 381 108

    [21]

    Cheng R, He T, Li W, Smerzi A 2016 J. Mod. Phys. 7 2043Google Scholar

    [22]

    Sapiro R E, Zhang R, Raithel G 2009 Phys. Rev. A 79 043630Google Scholar

    [23]

    Keller C, Schmiedmayer J, Zeilinger A, Nonn T, Dürr S, Rempe G 1999 Appl. Phys. B: Lasers Opt. 69 303Google Scholar

    [24]

    Dong G J, Zhu J, Zhang W P, Malomed B A 2013 Phys. Rev. Lett. 110 250401Google Scholar

    [25]

    Zhu J, Dong G J, Shneider M N, Zhang W P 2011 Phys. Rev. Lett. 106 210403Google Scholar

    [26]

    Chwedenczuk J, Piazza F, Smerzi A 2013 Phys. Rev. A 87 033607Google Scholar

    [27]

    Shin Y, Saba M, Schirotzek A, Pasquini T A, Leanhardt A E, Pritchard D E, Ketterle W 2004 Phys. Rev. Lett. 92 150401Google Scholar

    [28]

    Lehtovaara L, Toivanen J, Eloranta J 2007 J. Comput. Phys. 221 148Google Scholar

    [29]

    Bandrauk A D, Shen H 1994 J. Phys. A 27 7147Google Scholar

    [30]

    Fattori M, D’Errico C, Roati G, Zaccanti M, Jona-Lasinio M, Modugno M, Inguscio M, Modugno G 2008 Phys. Rev. Lett. 100 080405Google Scholar

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出版历程
  • 收稿日期:  2019-05-17
  • 修回日期:  2019-08-06
  • 上网日期:  2019-10-01
  • 刊出日期:  2019-10-20

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