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Collision clock shift of two Fermi atoms in harmonic potentials

## Collision clock shift of two Fermi atoms in harmonic potentials

Chen Ze-Rui, Liu Guang-Cun, Yu Zhen-Hua
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• #### Abstract

Atomic clocks provide the most accurate definition for time. The precision of atomic clock has been improved by many orders of magnitude since the first atomic clock was built. However, the interatomic interaction usually suppress the precision of atomic clock. As a result, it is especially meaningful to study the interaction effect in atomic clock, which is considered to be helpful in improving the precision and accuracy of atomic clock. In order to characterize the collision effect induced clock shift, we theoretically study the collision clock shift in the Rabi spectrum, caused by the short-range interaction between two Fermi atoms in harmonic potential. Given that the short-range interatomic interaction is generally weak, and that the parameters of external lattice laser field are in the Lamb-Dicke regime, we make an approximation that the spatial wave-function of the Fermi atoms does not change, and then derive the motion equation for the internal wave-function under the external Rabi driving field. We solve the equation of motion by the perturbative method, and obtain the solution to first order, and thus derive the expression of the collision clock shift of the Rabi spectrum in terms of the interatomic interaction and the external Rabi driving laser field parameters for specific spatial wave-functions of atoms. Finally, we use the exact expression of the Green’s function in harmonic potential to obtain the averaged clock shift of collision at finite temperatures. Our results relate the atomic interaction with atomic clock shift, and provide a unified description of all partial waves of atomic interaction induced clock shift. Therefore, it becomes much more convenient to study the contributions of different partial waves to atomic clock shift. On the other hand, our results indicate that through precisely measuring the clock shift, the information about the interatomic interactions can also be obtained. In addition, our results for two interacting atoms can inspire the future study of real many-body interacting system which will be the next research topic.

#### References

 [1] 阮军, 王叶兵, 常宏, 刘涛, 董瑞芳, 张首刚 2015 物理学报 64 160308 Ruan G, Wang Y B, Chang H, Jiang H F, Liu T, Dong R F, Zhang S G 2015 Acta Phys. Sin. 64 160308 [2] Ludlow A D, Boyd M M, Ye J, Peik E, Schmidt P O 2015 Rev. Mod. Phys. 87 637 [3] Blatt S, Thomsen J W, Campbell G K, et al. 2009 Phys. Rev. A 80 052703 [4] Zhu B, Gadway B, Foss-Feig M, Schachenmayer J, Wall M L, Hazzard K R A, Yan B, Moses S A, Covey J P, Jin D S, Ye J, Holland M, Rey A M 2014 Phys. Rev. Lett. 112 070404 [5] Moses S A, Covey J P, Miecnikowski M T, Yan B, Gadway B, Ye J, Jin D S 2015 Science 350 659 [6] Labuhn H, Barredo D, Ravets S, de Léséleuc S, Macrì T, Lahaye T, Browaeys A 2016 Nature 534 667 [7] Aikawa K, Baier S, Frisch A, Mark M, Ravensbergen C, Ferlaino F 2014 Science 345 1484 [8] Burdick N Q, Tang Y, Lev B L 2016 Phys. Rev. X 6 031022 [9] Kadau H, Schmitt M, Wenzel M, Wink C, Maier T, Ferrier-Barbut I, Pfau T 2016 Nature 530 194 [10] Heller E J, Falconer I S, Dewar R L 2002 Proceedings of the XVIII International Conference on Atomic Physics: the Expanding Frontier of Atomic Physics Cambridge, Massachusett, USA, July 28–August 2, 2002 p363 [11] Zwierlein M W, Hadzibabic Z, Gupta S, Ketterle W 2003 Phys. Rev. Lett. 91 250404 [12] Campbell G K, Boyd M M, Thomsen J W, et al. 2009 Science 324 360 [13] Harber D M, Lewandowski H J, Mcguirk J M, Cornell E A 2002 Phys. Rev. A 66 053616 [14] Fuchs J N, Gangardt D M, Laloe F 2002 Phys. Rev. Lett. 88 230404 [15] Kadio D, Band Y B 2006 Phys. Rev. A 74 053609 [16] Kurt G 2009 Phys. Rev. Lett. 103 113202 [17] Hazlett E L, Zhang Y, Stites R W, Gibble K, O'Hara K M 2013 Phys. Rev. Lett. 110 160801 [18] Lemke N D, Stecher J V, Sherman J A, Rey A M, Oates C W, Ludlow A D 2011 Phys. Rev. Lett. 107 103902 [19] Rey A M, Martin M J, Swallows M D, Bishof M, Benko C, Blatt S, Stecher J Von, Gorshkov A, Ye J 2012 IEEE International Frequency Control Symposium (FCS): Probing Many-body Spin Interactions with an Optical Lattice Clock Baltimore, Maryland, USA, May 21–24, 2012 p1 [20] Campbell S L, Hutson R B, Marti G E, Goban A, Oppong N D, Mcnally R L, Sonderhouse L, Robinson J M, Zhang W, Bloom B J 2017 Science 358 90 [21] Liu G C, Huang Y, Cheng Z, Chen Z R, Yu Z H 2020 Phys. Rev. A 101 012504

#### Cited By

• 图 1  光钟频移系数(a) $A\;(\delta_{\rm R}/\varOmega)$与(b) $B\;(\delta_{\rm R}/\varOmega)$

Figure 1.  Coefficient (a) $A\;(\delta_{\rm R}/\varOmega)$ and (b) $B\;(\delta_{\rm R}/\varOmega)$ for clock shift.

图 2  系数$C_\ell$随温度的变化　(a) C1; (b) C2

Figure 2.  Coefficient $C_\ell$ versus temperature: (a) C1; (b) C2.

图 3  当温度较高时, 系数$C_\ell$随温度的变化, 这里横轴和纵轴都取对数　(a)线性拟合表达式为$y = 5.54203 x- 3.83918$; (b)线性拟合表达式为$y = 6.5296 x-0.119146$

Figure 3.  Coefficient $C_\ell$ versus temperature for higher temperature. The horizontal coordinate and the vertical coordinate are logarithmic here: (a) Linear fitting function is $y = 5.54203 x-3.83918$; (b) linear fitting function is $y = 6.5296 x-0.119146$.

图 C1  同时考虑s波至d波时的钟频移${\bar\delta}_{\rm s}\;(\ell = 2)$与只考虑s波和p波时的钟频移${\bar\delta}_{\rm s}\;(\ell = 1)$的相对误差

Figure C1.  Relative error of clock shifts between considering s, p, d partial waves (${\bar\delta}_{\rm s}\;(\ell = 2)$) and only considering s, p partial waves (${\bar\delta}_{\rm s}\; (\ell = 1)$).

•  [1] 阮军, 王叶兵, 常宏, 刘涛, 董瑞芳, 张首刚 2015 物理学报 64 160308 Ruan G, Wang Y B, Chang H, Jiang H F, Liu T, Dong R F, Zhang S G 2015 Acta Phys. Sin. 64 160308 [2] Ludlow A D, Boyd M M, Ye J, Peik E, Schmidt P O 2015 Rev. Mod. Phys. 87 637 [3] Blatt S, Thomsen J W, Campbell G K, et al. 2009 Phys. Rev. A 80 052703 [4] Zhu B, Gadway B, Foss-Feig M, Schachenmayer J, Wall M L, Hazzard K R A, Yan B, Moses S A, Covey J P, Jin D S, Ye J, Holland M, Rey A M 2014 Phys. Rev. Lett. 112 070404 [5] Moses S A, Covey J P, Miecnikowski M T, Yan B, Gadway B, Ye J, Jin D S 2015 Science 350 659 [6] Labuhn H, Barredo D, Ravets S, de Léséleuc S, Macrì T, Lahaye T, Browaeys A 2016 Nature 534 667 [7] Aikawa K, Baier S, Frisch A, Mark M, Ravensbergen C, Ferlaino F 2014 Science 345 1484 [8] Burdick N Q, Tang Y, Lev B L 2016 Phys. Rev. X 6 031022 [9] Kadau H, Schmitt M, Wenzel M, Wink C, Maier T, Ferrier-Barbut I, Pfau T 2016 Nature 530 194 [10] Heller E J, Falconer I S, Dewar R L 2002 Proceedings of the XVIII International Conference on Atomic Physics: the Expanding Frontier of Atomic Physics Cambridge, Massachusett, USA, July 28–August 2, 2002 p363 [11] Zwierlein M W, Hadzibabic Z, Gupta S, Ketterle W 2003 Phys. Rev. Lett. 91 250404 [12] Campbell G K, Boyd M M, Thomsen J W, et al. 2009 Science 324 360 [13] Harber D M, Lewandowski H J, Mcguirk J M, Cornell E A 2002 Phys. Rev. A 66 053616 [14] Fuchs J N, Gangardt D M, Laloe F 2002 Phys. Rev. Lett. 88 230404 [15] Kadio D, Band Y B 2006 Phys. Rev. A 74 053609 [16] Kurt G 2009 Phys. Rev. Lett. 103 113202 [17] Hazlett E L, Zhang Y, Stites R W, Gibble K, O'Hara K M 2013 Phys. Rev. Lett. 110 160801 [18] Lemke N D, Stecher J V, Sherman J A, Rey A M, Oates C W, Ludlow A D 2011 Phys. Rev. Lett. 107 103902 [19] Rey A M, Martin M J, Swallows M D, Bishof M, Benko C, Blatt S, Stecher J Von, Gorshkov A, Ye J 2012 IEEE International Frequency Control Symposium (FCS): Probing Many-body Spin Interactions with an Optical Lattice Clock Baltimore, Maryland, USA, May 21–24, 2012 p1 [20] Campbell S L, Hutson R B, Marti G E, Goban A, Oppong N D, Mcnally R L, Sonderhouse L, Robinson J M, Zhang W, Bloom B J 2017 Science 358 90 [21] Liu G C, Huang Y, Cheng Z, Chen Z R, Yu Z H 2020 Phys. Rev. A 101 012504
•  Citation:
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• Abstract views:  321
• Cited By: 0
##### Publishing process
• Received Date:  01 February 2021
• Accepted Date:  17 May 2021
• Available Online:  07 June 2021
• Published Online:  20 September 2021

## Collision clock shift of two Fermi atoms in harmonic potentials

###### Corresponding author: Yu Zhen-Hua, yuzhh5@mail.sysu.edu.cn
• Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing, School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai 519082, China

Abstract: Atomic clocks provide the most accurate definition for time. The precision of atomic clock has been improved by many orders of magnitude since the first atomic clock was built. However, the interatomic interaction usually suppress the precision of atomic clock. As a result, it is especially meaningful to study the interaction effect in atomic clock, which is considered to be helpful in improving the precision and accuracy of atomic clock. In order to characterize the collision effect induced clock shift, we theoretically study the collision clock shift in the Rabi spectrum, caused by the short-range interaction between two Fermi atoms in harmonic potential. Given that the short-range interatomic interaction is generally weak, and that the parameters of external lattice laser field are in the Lamb-Dicke regime, we make an approximation that the spatial wave-function of the Fermi atoms does not change, and then derive the motion equation for the internal wave-function under the external Rabi driving field. We solve the equation of motion by the perturbative method, and obtain the solution to first order, and thus derive the expression of the collision clock shift of the Rabi spectrum in terms of the interatomic interaction and the external Rabi driving laser field parameters for specific spatial wave-functions of atoms. Finally, we use the exact expression of the Green’s function in harmonic potential to obtain the averaged clock shift of collision at finite temperatures. Our results relate the atomic interaction with atomic clock shift, and provide a unified description of all partial waves of atomic interaction induced clock shift. Therefore, it becomes much more convenient to study the contributions of different partial waves to atomic clock shift. On the other hand, our results indicate that through precisely measuring the clock shift, the information about the interatomic interactions can also be obtained. In addition, our results for two interacting atoms can inspire the future study of real many-body interacting system which will be the next research topic.

Reference (21)

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