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三能级钾原子气体三维傅里叶变换频谱的解析解

赵超樱 谭维翰

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三能级钾原子气体三维傅里叶变换频谱的解析解

赵超樱, 谭维翰

Analytical solution of three-dimensional Fourier transform frequency spectrum for three-level potassium atomic gas

Zhao Chao-Ying, Tan Wei-Han
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  • 利用投影切片定理、傅里叶位移定理和误差函数给出三能级钾原子气体三维傅里叶变换频谱在T = 0界面的解析解. 固定均匀线宽, 非均匀展宽和对角线相关系数可以定量地识别, 通过在适当方向上拟合三维傅里叶变换频谱谱峰的切片来确定. 结果表明, 非均匀展宽增大, 频谱图沿着对角线方向延伸, 对角线相关系数增大, 频谱图逐渐变圆, 振幅也逐渐变小.
    With the development of laser technology in the field of optics, ultra-fast optics has become an important research field. Compared with the traditional technology, ultrafast optics can be realized not only under shorter pulse function, but also on a smaller scale, which can more quickly reflect the dynamic process. We present an analytical calculation of the full three-dimensional (3D) coherent spectrum with a finite duration two-dimensional (2D) Gaussian pulse envelope. Our starting point is the solution of the optical Bloch equations for three-level potassium atomic gas in the 3D time domain by using the projection-slice theorem, error function and Fourier-shift theorem of 3D Fourier transform. These principles are used to calculate and simplify the third-order polarization equation generated by the device, and the analytical calculation of three-dimensional Fourier transform frequency spectrum at T = 0 is obtained. We simulate the analytic solution by using mathematics software. By comparing the simulations with the experimental results, with the homogeneous line-width fixed, we can obtain the relationship among the in-homogeneous broadening, the correlation diagonal coefficients and the three-dimensional spectrum characteristics, which can be identified quantitatively by fitting the slices of three-dimensional Fourier transform spectrum peaks in an appropriate direction. The results show that the three-dimensional Fourier transform spectrum will extend along the diagonal direction with the increasing of the in-homogeneous broadening, and the spectrogram progressively becomes a circle with the increasing of the diagonal correlation coefficient, and the amplitude also gradually turns smaller. According to the analytical solution, we give a complete two-dimensional spectrum of the T = 0 interface. The results can be fit to the experimental 3D coherent spectrum for arbitrary inhomogeneity.
      通信作者: 赵超樱, zchy49@hdu.edu.cn
    • 基金项目: 国家自然科学基金青年科学基金(批准号: 11504074)和教育部量子光学重点实验室(批准号: KF201801)资助的课题
      Corresponding author: Zhao Chao-Ying, zchy49@hdu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China(Grant No. 11504074) and the Key Laboratory of Quantum Optics, Ministry of Education, China (Grant No. KF201801)
    [1]

    Ernst R R, Bodenhausen G, Wokaun A 1987 Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford: Clarendon Press)

    [2]

    Jonas D M 2003 Annu. Rev. Phys. Chem. 54 425Google Scholar

    [3]

    Siemens M E, Moody G, Li H B, Bristow A D, Cundiff S T 2010 Opt. Express 18 17699Google Scholar

    [4]

    Fecko C J, Eaves J D, Loparo J J, Tokmakoff A, Geissler P L 2003 Science 301 1698Google Scholar

    [5]

    Turner D B, Wen P, Arias D H, Nelson K A, Li H B, Moody G, Siemens M E, Cundiff S T 2012 Phys. Rev. B 85 201303Google Scholar

    [6]

    Cundiff S T, Bristow A D, Siemen M, Li H B, Moody G, Karaiskaj D, Dai X C, Zhang T H 2012 IEEE J. Sel. Top Quant. 18 318Google Scholar

    [7]

    Nardin G, Moody G, Singh R, Autry T M, Li H B, Morier-Genoud F, Cundiff S T 2014 Phys. Rev. Lett. 112 046402Google Scholar

    [8]

    Moody G, Akimov I A, Li H B, Singh R, Yakovlev D R, Karczewski G, Wiater M, Wojtowicz T, Bayer M, Cundiff S T 2014 Phys. Rev. Lett. 112 097401Google Scholar

    [9]

    Li H B, Bristow A D, Siemens M E, Moody G, Cundiff S T 2013 Nat. Commun. 4 1390Google Scholar

    [10]

    Bell J D, Conrad R, Siemens M E 2015 Opt. Lett. 4 1157

    [11]

    Titze M, Li H B 2017 Phys. Rev. A 96 032508Google Scholar

    [12]

    Dai X C, Bristow A D, Cundiff S T 2010 Phys. Rev. A 82 052503Google Scholar

    [13]

    Dai X C, Richter M, Li H B, Bristow A D, Falvo C, Mukamel S, Cundiff S T 2012 Phys. Rev. Lett. 108 193201Google Scholar

    [14]

    赵威, 周肇宇, 杨金新, 戴星灿 2015 物理学进展 35 177

    Zhao W, Zhou Z Y, Yang J X, Dai X C 2015 Prog. Phys. 35 177

    [15]

    Zhu W D, Wang R, Zhang C F, Wang G D, Liu Y L, Zhao W, Dai X C, Wang X Y, Cerullo G, Cundiff S T, Xiao M 2017 Opt. Express 25 21115Google Scholar

    [16]

    Zhao W, Qin Z Y, Zhang C F, Wang G D, Li B, Dai X C, Xiao M 2019 J. Phys. Chem. Lett. 10 1251Google Scholar

    [17]

    Huang T Y, Li X H, Shum P P, Wang Q J, Shao X G, Wang L L, Li H Z, Wu Z F, Dong X Y 2015 Opt. Express 23 340Google Scholar

    [18]

    Wang L, Li X H, Wang C, Luo W F, Feng T C, Zhang Y, Zhang H 2019 Chem. Nanomater. Bio. 5 1233

    [19]

    Liu J S, Li X H, Guo Y X, Qyyum A, Shi Z J, Feng T C, Zhang Y, Jiang C X, Liu X F 2019 Small 15 1902811Google Scholar

    [20]

    Zhao Y, Guo P L, Li X H, Jin Z W 2019 Carbon 149 336Google Scholar

    [21]

    Garrett-Roe S, Hamm P 2009 J. Chem. Phys. 130 164510Google Scholar

    [22]

    Mukherjee S S, Skoff D R, Middleton C T, Zanni M T 2013 J. Chem. Phys. 139 144205Google Scholar

    [23]

    李淳飞 2009 非线性光学 (北京: 电子工业出版社) 第57页

    Li C F 2009 Nonlinear Optics (Beijing: Electronics industry Press) p57 (in Chinese)

  • 图 1  四波混频原理图

    Fig. 1.  Four wave mixing schematic.

    图 2  (a) 二维时域; (b) 光子回波信号的频率坐标; (c) 二维时域投影在对应于沿${\hat \omega _{{t'}}}$的切片的对角线上; (d)沿${\hat \omega _{{\tau '}}}$的切片对应的交叉对角线上的二维时域投影

    Fig. 2.  (a) 2D time; (b) frequency coordinates for photon echo signals; (c) 2D time projection onto the diagonal corresponding to a slice along ${\hat \omega _{{t'}}}$; (d) 2D time projection onto the cross diagonal corresponding to a slice along ${\hat \omega _{{\tau '}}}$.

    图 3  ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{ THz}}$, ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$, $R = 1$${S_{C{\rm{1}}}}\left( {{\omega _t}, {\omega _\tau }} \right)$频谱图 (a)实部; (b)虚部; (c)模

    Fig. 3.  The three-dimensional Fourier transform spectrum ${S_{C{\rm{1}}}}\left( {{\omega _t}, {\omega _\tau }} \right)$ with ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{ THz}}$, ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$, $R = 1$: (a) Real part; (b) imaginary part; (c) module.

    图 4  ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$, ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$, $R = 0.5$${S_{C2}}\left( {{\omega _t}, {\omega _\tau }} \right)$频谱图 (a)实部; (b)虚部; (c)模

    Fig. 4.  The three-dimensional Fourier transform spectrum ${S_{C2}}\left( {{\omega _t}, {\omega _\tau }} \right)$ with ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{ THz}}$, ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$, $R = 0.5$: (a) Real part; (b) imaginary part; (c) module.

    图 5  ${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$, ${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$, ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05 \;{\rm{THz}}$, $R = 1$${S_{C3, E{\rm{3}}}}\left( {{\omega _t}, {\omega _\tau }} \right)$频谱图 (a)实部; (b)虚部; (c)模

    Fig. 5.  The three-dimensional Fourier transform spectrum ${S_{C3, E{\rm{3}}}}\left( {{\omega _t}, {\omega _\tau }} \right)$ with ${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$, ${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$, ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = $ 0.05 THz, $R = 1$: (a) Real part; (b) imaginary part; (c) module.

    图 6  ${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$, ${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$, ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$, $R = 0.5$${S_{C4, E4}}\left( {{\omega _t}, {\omega _\tau }} \right)$频谱图 (a)实部; (b)虚部; (c)模

    Fig. 6.  The three-dimensional Fourier transform spectrum ${S_{C4, E4}}\left( {{\omega _t}, {\omega _\tau }} \right)$ with ${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$, ${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$, ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = $ 0.05 THz, $R = 0.5$ (a) Real part; (b) imaginary part; (c) module.

    图 7  三维傅里叶转换频谱图 (a) 参考文献[11]中的图5(a), ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05 \;{\rm{THz}}$, ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$, (b)$R = 1$; (c)$R = 0.5$

    Fig. 7.  Three-dimensional Fourier transform spectrum: (a) Fig. 5(a) in Ref. [11], ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05 \;{\rm{THz}}$, ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}} = $ 0.2 THz; (b)$R = 1$; (c)$R = 0.5$.

    图 8  R不同时, 三维傅里叶转换频谱, ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$, ${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$, ${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ (a)$R = 1$; (b)$R = 0.5$

    Fig. 8.  The three-dimensional Fourier transform spectrum with ${\varGamma _{10}} = {\varGamma _{{\rm{2}}0}} = 0.05\;{\rm{THz}}$, ${\text{δ}} {\omega _{10}} = 0.3\;{\rm{THz}}$, ${\text{δ}} {\omega _{20}} = 0.2\;{\rm{THz}}$ for different R: (a) $R = 1$; (b) $R = 0.5$.

    表 1  非均匀展宽和对角线相关系数之间的关系

    Table 1.  The relation between in-homogeneous line-width and the diagonal correlation coefficient.

    xyz
    ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}}$, $R = {\rm{1}}$0${\rm{4}}{\text{δ}} \omega _{10}^2$0
    ${\text{δ}} {\omega _{10}} = {\text{δ}} {\omega _{20}}$, $R \ne {\rm{1}}$$2\left( {1 - R} \right){\text{δ}} \omega _{10}^2$$2\left( {{\rm{1}} + R} \right){\text{δ}} \omega _{10}^2$0
    ${\text{δ} } {\omega _{10} } = m{\text{δ} } {\omega _{20} },$$R = {\rm{1}}$${\left(1 - \dfrac{1}{m}\right)^2}{\text{δ}} \omega _{10}^2$${\left(1 + \dfrac{1}{m}\right)^2}{\text{δ}} \omega _{10}^2$$\left(1- \dfrac{1}{{{m^2}}}\right){\text{δ}} \omega _{10}^2$
    ${\text{δ} } {\omega _{10} } = m{\text{δ} } {\omega _{20} },$$R \ne {\rm{1}}$$\dfrac{{({m^2} - 2 Rm + 1)}}{{{m^2}}}{\text{δ}} \omega _{10}^2$$\dfrac{{({m^2} + 2 Rm + 1)}}{{{m^2}}}{\text{δ}} \omega _{10}^2$$\left(1 - \dfrac{1}{{{m^2}}}\right){\text{δ}} \omega _{10}^2$
    下载: 导出CSV
  • [1]

    Ernst R R, Bodenhausen G, Wokaun A 1987 Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford: Clarendon Press)

    [2]

    Jonas D M 2003 Annu. Rev. Phys. Chem. 54 425Google Scholar

    [3]

    Siemens M E, Moody G, Li H B, Bristow A D, Cundiff S T 2010 Opt. Express 18 17699Google Scholar

    [4]

    Fecko C J, Eaves J D, Loparo J J, Tokmakoff A, Geissler P L 2003 Science 301 1698Google Scholar

    [5]

    Turner D B, Wen P, Arias D H, Nelson K A, Li H B, Moody G, Siemens M E, Cundiff S T 2012 Phys. Rev. B 85 201303Google Scholar

    [6]

    Cundiff S T, Bristow A D, Siemen M, Li H B, Moody G, Karaiskaj D, Dai X C, Zhang T H 2012 IEEE J. Sel. Top Quant. 18 318Google Scholar

    [7]

    Nardin G, Moody G, Singh R, Autry T M, Li H B, Morier-Genoud F, Cundiff S T 2014 Phys. Rev. Lett. 112 046402Google Scholar

    [8]

    Moody G, Akimov I A, Li H B, Singh R, Yakovlev D R, Karczewski G, Wiater M, Wojtowicz T, Bayer M, Cundiff S T 2014 Phys. Rev. Lett. 112 097401Google Scholar

    [9]

    Li H B, Bristow A D, Siemens M E, Moody G, Cundiff S T 2013 Nat. Commun. 4 1390Google Scholar

    [10]

    Bell J D, Conrad R, Siemens M E 2015 Opt. Lett. 4 1157

    [11]

    Titze M, Li H B 2017 Phys. Rev. A 96 032508Google Scholar

    [12]

    Dai X C, Bristow A D, Cundiff S T 2010 Phys. Rev. A 82 052503Google Scholar

    [13]

    Dai X C, Richter M, Li H B, Bristow A D, Falvo C, Mukamel S, Cundiff S T 2012 Phys. Rev. Lett. 108 193201Google Scholar

    [14]

    赵威, 周肇宇, 杨金新, 戴星灿 2015 物理学进展 35 177

    Zhao W, Zhou Z Y, Yang J X, Dai X C 2015 Prog. Phys. 35 177

    [15]

    Zhu W D, Wang R, Zhang C F, Wang G D, Liu Y L, Zhao W, Dai X C, Wang X Y, Cerullo G, Cundiff S T, Xiao M 2017 Opt. Express 25 21115Google Scholar

    [16]

    Zhao W, Qin Z Y, Zhang C F, Wang G D, Li B, Dai X C, Xiao M 2019 J. Phys. Chem. Lett. 10 1251Google Scholar

    [17]

    Huang T Y, Li X H, Shum P P, Wang Q J, Shao X G, Wang L L, Li H Z, Wu Z F, Dong X Y 2015 Opt. Express 23 340Google Scholar

    [18]

    Wang L, Li X H, Wang C, Luo W F, Feng T C, Zhang Y, Zhang H 2019 Chem. Nanomater. Bio. 5 1233

    [19]

    Liu J S, Li X H, Guo Y X, Qyyum A, Shi Z J, Feng T C, Zhang Y, Jiang C X, Liu X F 2019 Small 15 1902811Google Scholar

    [20]

    Zhao Y, Guo P L, Li X H, Jin Z W 2019 Carbon 149 336Google Scholar

    [21]

    Garrett-Roe S, Hamm P 2009 J. Chem. Phys. 130 164510Google Scholar

    [22]

    Mukherjee S S, Skoff D R, Middleton C T, Zanni M T 2013 J. Chem. Phys. 139 144205Google Scholar

    [23]

    李淳飞 2009 非线性光学 (北京: 电子工业出版社) 第57页

    Li C F 2009 Nonlinear Optics (Beijing: Electronics industry Press) p57 (in Chinese)

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出版历程
  • 收稿日期:  2019-06-20
  • 修回日期:  2019-10-21
  • 刊出日期:  2020-01-20

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