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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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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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  • 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.
      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

    Figure 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

    Figure 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

    Figure 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

    Figure 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

    Figure 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

    Figure 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)$R = 0.5$

    Figure 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)$R = 0.5$

    Figure 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$
    DownLoad: 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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Publishing process
  • Received Date:  20 June 2019
  • Accepted Date:  21 October 2019
  • Published Online:  20 January 2020

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