Long range dipole-dipole interaction in low-density atomic vapors probed by double-quantum two-dimensional coherent spectroscopy

Shaogang Yu; Shaogang Yu; Shaogang Yu; Michael Titze; Yifu Zhu; Xiaojun Liu; Hebin Li

doi:10.1364/OE.27.028891

Long range dipole-dipole interaction in low-density atomic vapors probed by double-quantum two-dimensional coherent spectroscopy

Shaogang Yu, Michael Titze, Yifu Zhu, Xiaojun Liu, and Hebin Li

Optics Express
Vol. 27,
Issue 20,
pp. 28891-28901
(2019)
•https://doi.org/10.1364/OE.27.028891

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Shaogang Yu, Michael Titze, Yifu Zhu, Xiaojun Liu, and Hebin Li, "Long range dipole-dipole interaction in low-density atomic vapors probed by double-quantum two-dimensional coherent spectroscopy," Opt. Express 27, 28891-28901 (2019)

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Table of Contents Category
- Atomic and Molecular Physics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: June 26, 2019
- Revised Manuscript: August 26, 2019
- Manuscript Accepted: September 12, 2019
- Published: September 24, 2019

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Open Access

Abstract

Optical double-quantum two-dimensional coherent spectroscopy (2DCS) was implemented to probe interatomic dipole-dipole interactions in both potassium and rubidium atomic vapors. The dipole-dipole interaction was detected at densities of $4.81 \times 10^8$ cm$^{-3}$ and $8.40 \times 10^9$ cm$^{-3}$ for potassium and rubidium, respectively, corresponding to a mean interatomic separation of 15.8 $\mu$m or $3.0\times 10^5a_0$ for potassium and 6.1 $\mu$m or $1.2\times 10^5a_0$ for rubidium, where $a_0$ is the Bohr radius. The experimental results confirm the long range nature of the dipole-dipole interaction, which is critical for understanding many-body physics in atoms/molecules. The long range interaction also has implications in atom-based applications involving many-body interactions. Additionally, we demonstrated that double-quantum 2DCS is sufficiently sensitive to probe dipole-dipole interaction at densities that can be achieved with cold atom in a magneto-optical trap, paving the way for double-quantum 2DCS studies of cold atoms and molecules. The method can also open a new avenue to study long-range interactions in solid state systems such as quantum dots and color centers in diamonds.

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References

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Publication

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[Crossref]
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[Crossref]
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[Crossref]
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[Crossref]
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[Crossref]
H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletic, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551(7682), 579–584 (2017).
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[Crossref]
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[Crossref]
J. Szudy and W. Baylis, “Unified Franck-Condon treatment of pressure broadening of spectral lines,” J. Quant. Spectrosc. Radiat. Transfer 15(7-8), 641–668 (1975).
[Crossref]
A. Ben-Reuven, “Spectral line shapes in gases in the binary-collision approximation,” Adv. Chem. Phys. 33, 235–293 (1975).
[Crossref]
E. W. Smith, J. Cooper, and L. J. Roszman, “An analysis of the unified and scalar additivity theories of spectral line broadening,” J. Quant. Spectrosc. Radiat. Transfer 13(12), 1523–1538 (1973).
[Crossref]
A. W. Ali and H. R. Griem, “Theory of Resonance Broadening of Spectral Lines by Atom-Atom Impacts,” Phys. Rev. 140(4A), A1044–A1049 (1965).
[Crossref]
P. W. Anderson, “Pressure Broadening in the Mircrowave and Infra-Red Regions,” Phys. Rev. 76(5), 647–661 (1949).
[Crossref]
E. Lewis, “Collisional relaxation of atomic excited states, line broadening and interatomic interactions,” Phys. Rep. 58(1), 1–71 (1980).
[Crossref]
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[Crossref]
G. Nardin, T. M. Autry, K. L. Silverman, and S. T. Cundiff, “Multidimensional coherent photocurrent spectroscopy of a semiconductor nanostructure,” Opt. Express 21(23), 28617 (2013).
[Crossref]
P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. 127(21), 214307 (2007).
[Crossref]
G. Nardin, T. M. Autry, G. Moody, R. Singh, H. Li, and S. T. Cundiff, “Multi-dimensional coherent optical spectroscopy of semiconductor nanostructures: Collinear and non-collinear approaches,” J. Appl. Phys. 117(11), 112804 (2015).
[Crossref]
L. Bruder, M. Binz, and F. Stienkemeier, “Efficient isolation of multiphoton processes and detection of collective resonances in dilute samples,” Phys. Rev. A 92(5), 053412 (2015).
[Crossref]
S. Yu, M. Titze, Y. Zhu, X. Liu, and H. Li, “Observation of scalable and deterministic multi-atom dicke states in an atomic vapor,” Opt. Lett. 44(11), 2795–2798 (2019).
[Crossref]
A. Perdomo-Ortiz, J. R. Widom, G. A. Lott, A. Aspuru-Guzik, and A. H. Marcus, “Conformation and electronic population transfer in membrane-supported self-assembled porphyrin dimers by 2d fluorescence spectroscopy,” J. Phys. Chem. B 116(35), 10757–10770 (2012).
[Crossref]
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[Crossref]
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[Crossref]
L. Bruder, A. Eisfeld, U. Bangert, M. Binz, M. Jakob, D. Uhl, M. Schulz-Weiling, E. R. Grant, and F. Stienkemeier, “Delocalized excitons and interaction effects in extremely dilute thermal ensembles,” Phys. Chem. Chem. Phys. 21(5), 2276–2282 (2019).
[Crossref]
S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15(1), 1–89 (1943).
[Crossref]

2019 (2)

L. Bruder, A. Eisfeld, U. Bangert, M. Binz, M. Jakob, D. Uhl, M. Schulz-Weiling, E. R. Grant, and F. Stienkemeier, “Delocalized excitons and interaction effects in extremely dilute thermal ensembles,” Phys. Chem. Chem. Phys. 21(5), 2276–2282 (2019).
[Crossref]

S. Yu, M. Titze, Y. Zhu, X. Liu, and H. Li, “Observation of scalable and deterministic multi-atom dicke states in an atomic vapor,” Opt. Lett. 44(11), 2795–2798 (2019).
[Crossref]

2018 (3)

P. Malý and T. Mančal, “Signatures of exciton delocalization and exciton-exciton annihilation in fluorescence-detected two-dimensional coherent spectroscopy,” J. Phys. Chem. Lett. 9(19), 5654–5659 (2018).
[Crossref]

M. Schröter, T. Pullerits, and O. Kühn, “Using fluorescence detected two-dimensional spectroscopy to investigate initial exciton delocalization between coupled chromophores,” J. Chem. Phys. 149(11), 114107 (2018).
[Crossref]

B. Lomsadze and S. T. Cundiff, “Frequency-comb based double-quantum two-dimensional spectrum identifies collective hyperfine resonances in atomic vapor induced by dipole-dipole interactions,” Phys. Rev. Lett. 120(23), 233401 (2018).
[Crossref]

2017 (4)

B. Lomsadze and S. T. Cundiff, “Frequency combs enable rapid and high-resolution multidimensional coherent spectroscopy,” Science 357(6358), 1389–1391 (2017).
[Crossref]

H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletic, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551(7682), 579–584 (2017).
[Crossref]

A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, “A cold-atom Fermi-Hubbard antiferromagnet,” Nature 545(7655), 462–466 (2017).
[Crossref]

M. Titze and H. Li, “Interpretation of optical three-dimensional coherent spectroscopy,” Phys. Rev. A 96(3), 032508 (2017).
[Crossref]

2016 (1)

F. Gao, S. T. Cundiff, and H. Li, “Probing dipole–dipole interaction in a rubidium gas via double-quantum 2D spectroscopy,” Opt. Lett. 41(13), 2954–2957 (2016).
[Crossref]

2015 (2)

G. Nardin, T. M. Autry, G. Moody, R. Singh, H. Li, and S. T. Cundiff, “Multi-dimensional coherent optical spectroscopy of semiconductor nanostructures: Collinear and non-collinear approaches,” J. Appl. Phys. 117(11), 112804 (2015).
[Crossref]

L. Bruder, M. Binz, and F. Stienkemeier, “Efficient isolation of multiphoton processes and detection of collective resonances in dilute samples,” Phys. Rev. A 92(5), 053412 (2015).
[Crossref]

2013 (1)

G. Nardin, T. M. Autry, K. L. Silverman, and S. T. Cundiff, “Multidimensional coherent photocurrent spectroscopy of a semiconductor nanostructure,” Opt. Express 21(23), 28617 (2013).
[Crossref]

2012 (3)

A. Perdomo-Ortiz, J. R. Widom, G. A. Lott, A. Aspuru-Guzik, and A. H. Marcus, “Conformation and electronic population transfer in membrane-supported self-assembled porphyrin dimers by 2d fluorescence spectroscopy,” J. Phys. Chem. B 116(35), 10757–10770 (2012).
[Crossref]

S. Trotzky, Y. A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, “Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas,” Nat. Phys. 8(4), 325–330 (2012).
[Crossref]

X. Dai, M. Richter, H. Li, A. D. Bristow, C. Falvo, S. Mukamel, and S. T. Cundiff, “Two-Dimensional Double-Quantum Spectra Reveal Collective Resonances in an Atomic Vapor,” Phys. Rev. Lett. 108(19), 193201 (2012).
[Crossref]

2011 (1)

M. D. Swallows, M. Bishof, Y. Lin, S. Blatt, M. J. Martin, A. M. Rey, and J. Ye, “Suppression of collisional shifts in a strongly interacting lattice clock,” Science 331(6020), 1043–1046 (2011).
[Crossref]

2009 (1)

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, and M. O. Scully, “Excitation dependence of resonance line self-broadening at different atomic densities,” J. Phys. B: At., Mol. Opt. Phys. 42(6), 065203 (2009).
[Crossref]

2008 (4)

J. Eden, B. Ricconi, Y. Xiao, F. Shen, and A. Senin, “Interactions Between Thermal Ground or Excited Atoms in the Vapor Phase: Many-Body Dipole-Dipole Effects, Molecular Dissociation, and Photoassociation Probed By Laser Spectroscopy,” Adv. At., Mol., Opt. Phys. 56, 49–118 (2008).
[Crossref]

V. O. Lorenz, S. Mukamel, W. Zhuang, and S. T. Cundiff, “Ultrafast optical spectroscopy of spectral fluctuations in a dense atomic vapor,” Phys. Rev. Lett. 100(1), 013603 (2008).
[Crossref]

H. Li, T. Varzhapetyan, V. Sautenkov, Y. Rostovtsev, H. Chen, D. Sarkisyan, and M. Scully, “Improvement of spectral resolution by using the excitation dependence of dipole-dipole interaction in a dense atomic gas,” Appl. Phys. B: Lasers Opt. 91(2), 229–231 (2008).
[Crossref]

A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt, M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W. Thomsen, S. M. Foreman, J. Ye, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, Y. Le Coq, Z. W. Barber, N. Poli, N. D. Lemke, K. M. Beck, and C. W. Oates, “Sr lattice clock at $1 \times 10^{-16}$1×10−16 fractional uncertainty by remote optical evaluation with a Ca clock,” Science 319(5871), 1805–1808 (2008).
[Crossref]

2007 (3)

F. Shen, J. Gao, A. A. Senin, C. J. Zhu, J. R. Allen, Z. H. Lu, Y. Xiao, and J. G. Eden, “Many-Body Dipole-Dipole Interactions between Excited Rb Atoms Probed by Wave Packets and Parametric Four-Wave Mixing,” Phys. Rev. Lett. 99(14), 143201 (2007).
[Crossref]

P. F. Tekavec, G. A. Lott, and A. H. Marcus, “Fluorescence-detected two-dimensional electronic coherence spectroscopy by acousto-optic phase modulation,” J. Chem. Phys. 127(21), 214307 (2007).
[Crossref]

G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mančal, Y.-C. Cheng, R. E. Blankenship, and G. R. Fleming, “Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems,” Nature 446(7137), 782–786 (2007).
[Crossref]

2006 (1)

K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne, “Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering,” Rev. Mod. Phys. 78(2), 483–535 (2006).
[Crossref]

2005 (1)

V. Lorenz and S. Cundiff, “Non-Markovian Dynamics in a Dense Potassium Vapor,” Phys. Rev. Lett. 95(16), 163601 (2005).
[Crossref]

2002 (1)

S. T. Cundiff, “Time domain observation of the Lorentz-local field,” Laser Phys. 12, 1073–1078 (2002).

1999 (1)

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71(1), 1–85 (1999).
[Crossref]

1997 (1)

H. van Kampen, V. A. Sautenkov, A. M. Shalagin, E. R. Eliel, and J. P. Woerdman, “Dipole-dipole collision-induced transport of resonance excitation in a high-density atomic vapor,” Phys. Rev. A 56(5), 3569–3575 (1997).
[Crossref]

1996 (1)

V. A. Sautenkov, H. van Kampen, E. R. Eliel, and J. P. Woerdman, “Dipole-Dipole Broadened Line Shape in a Partially Excited Dense Atomic Gas,” Phys. Rev. Lett. 77(16), 3327–3330 (1996).
[Crossref]

1994 (2)

J. A. Leegwater and S. Mukamel, “Self-broadening and exciton line shifts in gases: Beyond the local-field approximation,” Phys. Rev. A 49(1), 146–155 (1994).
[Crossref]

R. A. Cline, J. D. Miller, and D. J. Heinzen, “Study of ${\mathrm {rb}}_{2}$rb2 long-range states by high-resolution photoassociation spectroscopy,” Phys. Rev. Lett. 73(5), 632–635 (1994).
[Crossref]

1991 (1)

J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and nonlinear optical measurements of the Lorentz local field,” Phys. Rev. Lett. 67(8), 972–975 (1991).
[Crossref]

1980 (1)

E. Lewis, “Collisional relaxation of atomic excited states, line broadening and interatomic interactions,” Phys. Rep. 58(1), 1–71 (1980).
[Crossref]

1975 (2)

J. Szudy and W. Baylis, “Unified Franck-Condon treatment of pressure broadening of spectral lines,” J. Quant. Spectrosc. Radiat. Transfer 15(7-8), 641–668 (1975).
[Crossref]

A. Ben-Reuven, “Spectral line shapes in gases in the binary-collision approximation,” Adv. Chem. Phys. 33, 235–293 (1975).
[Crossref]

1973 (1)

E. W. Smith, J. Cooper, and L. J. Roszman, “An analysis of the unified and scalar additivity theories of spectral line broadening,” J. Quant. Spectrosc. Radiat. Transfer 13(12), 1523–1538 (1973).
[Crossref]

1965 (1)

A. W. Ali and H. R. Griem, “Theory of Resonance Broadening of Spectral Lines by Atom-Atom Impacts,” Phys. Rev. 140(4A), A1044–A1049 (1965).
[Crossref]

1949 (1)

P. W. Anderson, “Pressure Broadening in the Mircrowave and Infra-Red Regions,” Phys. Rev. 76(5), 647–661 (1949).
[Crossref]

1943 (1)

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15(1), 1–89 (1943).
[Crossref]

1939 (1)

G. W. King and J. H. van Vleck, “Dipole-Dipole Resonance Forces,” Phys. Rev. 55(12), 1165–1172 (1939).
[Crossref]

1932 (1)

V. Weisskopf, “Zur theorie der kopplungsbreite und der stoßdämpfung,” Eur. Phys. J. A 75(5-6), 287–301 (1932).
[Crossref]

Ahn, T.-K.

Ali, A. W.

A. W. Ali and H. R. Griem, “Theory of Resonance Broadening of Spectral Lines by Atom-Atom Impacts,” Phys. Rev. 140(4A), A1044–A1049 (1965).
[Crossref]

Allen, J. R.

Anderson, P. W.

P. W. Anderson, “Pressure Broadening in the Mircrowave and Infra-Red Regions,” Phys. Rev. 76(5), 647–661 (1949).
[Crossref]

Aspuru-Guzik, A.

Autry, T. M.

G. Nardin, T. M. Autry, K. L. Silverman, and S. T. Cundiff, “Multidimensional coherent photocurrent spectroscopy of a semiconductor nanostructure,” Opt. Express 21(23), 28617 (2013).
[Crossref]

Bagnato, V. S.

J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, “Experiments and theory in cold and ultracold collisions,” Rev. Mod. Phys. 71(1), 1–85 (1999).
[Crossref]

Bangert, U.

Barber, Z. W.

Baylis, W.

J. Szudy and W. Baylis, “Unified Franck-Condon treatment of pressure broadening of spectral lines,” J. Quant. Spectrosc. Radiat. Transfer 15(7-8), 641–668 (1975).
[Crossref]

Beck, K. M.

Ben-Reuven, A.

A. Ben-Reuven, “Spectral line shapes in gases in the binary-collision approximation,” Adv. Chem. Phys. 33, 235–293 (1975).
[Crossref]

Bernien, H.

Binz, M.

L. Bruder, M. Binz, and F. Stienkemeier, “Efficient isolation of multiphoton processes and detection of collective resonances in dilute samples,” Phys. Rev. A 92(5), 053412 (2015).
[Crossref]

Bishof, M.

Blankenship, R. E.

Blatt, S.

Bloch, I.

Boyd, M. M.

Boyd, R. W.

J. J. Maki, M. S. Malcuit, J. E. Sipe, and R. W. Boyd, “Linear and nonlinear optical measurements of the Lorentz local field,” Phys. Rev. Lett. 67(8), 972–975 (1991).
[Crossref]

Bristow, A. D.

Bruder, L.

L. Bruder, M. Binz, and F. Stienkemeier, “Efficient isolation of multiphoton processes and detection of collective resonances in dilute samples,” Phys. Rev. A 92(5), 053412 (2015).
[Crossref]

Calhoun, T. R.

Campbell, G. K.

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15(1), 1–89 (1943).
[Crossref]

Chen, H.

Chen, Y. A.

Cheng, Y.-C.

Chiu, C. S.

Choi, S.

Cline, R. A.

Cooper, J.

Cundiff, S.

V. Lorenz and S. Cundiff, “Non-Markovian Dynamics in a Dense Potassium Vapor,” Phys. Rev. Lett. 95(16), 163601 (2005).
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A. Thorne, U. Litzén, S. Johansson, and U. Litzen, Spectrophysics: Principles and Applications (Springer, 1999).

A. N. Nesmeyanov, Vapor pressure curve of chemical elements (Elsevier, New York, 1963).

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Fig. 1. (a) Experimental schematic. (b) Time ordering of the excitation pulse sequence. (c) Energy diagram of two two-level atoms. (d) Six of the twelve excitation pathways contributing to the double-quantum signal. Exchanging

$s$

and

$s'$

gives the other six.

Download Full Size | PPT Slide | PDF

Fig. 2. (a) Relevant energy levels of single (solid lines) and two (dashed lines) K atoms. Double-quantum 2D spectra of a K vapor at densities (b)

$N=4.81\times 10^8$

$^{-3}$

and (c)

$N=1.12\times 10^{10}$

$^{-3}$

. (d) Relevant energy levels of single (solid lines) and two (dashed lines) Rb atoms. Double-quantum 2D spectra of a Rb vapor at densities (e)

$N=8.40\times 10^9$

$^{-3}$

and (f)

$N=2.29\times 10^{10}$

$^{-3}$

. In all 2D spectra, the amplitude is plotted.

Download Full Size | PPT Slide | PDF

Fig. 3. (a) Simulated double-quantum 2D spectrum of Rb. The amplitude of the spectrum is plotted here. Comparison of slices taken from experimental and simulated 2D spectra along the directions denoted as (b) Slice 1, (c) Slice 2 and (d) Slice 3. The black dots are experimental data and the red lines are simulation.

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Fig. 4. The extracted dephasing rates when different number of interacting atoms are accounted for. The black dots are the extracted values and the red line is an exponential fit.

Download Full Size | PPT Slide | PDF

Fig. 5. Sensitivity of detecting dipole-dipole interaction in atomic vapors by various techniques. Atomic density (solid lines) and mean interatomic separation (dashed lines) at different temperatures are shown for K (blue) and Rb (orange) vapors. The reported lowest density is marked with diamond for TFWM, triangle for selective reflection, pentagon for quantum beating, hexagon for MQC, star for boxcar 2DCS, and circle for collinear 2DCS.

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Equations (5)

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V_{A B} = \frac{1}{4 π ε_{0} R^{3}} [{\vec{μ}}_{A} \cdot {\vec{μ}}_{B} - \frac{3 ({\vec{μ}}_{A} \cdot \vec{R}) ({\vec{μ}}_{B} \cdot \vec{R})}{R^{2}}],

\begin{aligned} ρ_{s s, I}^{(4)} & = \frac{- i μ_{s g}}{2 ℏ} e^{- i k_{D} \cdot r} \int_{- \infty}^{t_{e}} d t^{⁗} e^{- Γ_{s s} (t_{e} - t^{⁗})} E_{D}^{*} δ (t^{⁗} - t_{4}) e^{i ω t^{⁗}} \\ \cdot \frac{i μ_{d s}}{2 ℏ} e^{- i k_{A} \cdot r} \int_{- \infty}^{t^{⁗}} d t^{‴} e^{- i (ω_{s g} - i Γ_{s g}) (t^{⁗} - t^{‴})} E_{A}^{*} δ (t^{‴} - t_{3}) e^{i ω t^{‴}} \\ \cdot \frac{i μ_{d g}}{2 ℏ} e^{i k_{C} \cdot r} \int_{- \infty}^{t^{‴}} d t^{″} e^{- i (ω_{d g} - i Γ_{d g}) (t^{‴} - t^{″})} E_{C} δ (t^{″} - t_{2}) e^{- i ω t^{″}} \\ \cdot \frac{i μ_{s g}}{2 ℏ} e^{i k_{B} \cdot r} \int_{- \infty}^{t^{″}} d t^{'} e^{- i (ω_{s g} - i Γ_{s g}) (t^{″} - t^{'})} E_{B} δ (t^{'} - t_{1}) e^{- i ω t^{'}} \cdot ρ_{00}^{(0)} \\ = \frac{- μ_{s g}^{2} μ_{d s} μ_{d g}}{16 ℏ^{4}} e^{i (- k_{A} + k_{B} + k_{C} - k_{D}) \cdot r} E_{A} E_{B} E_{C} E_{D} ρ_{00}^{(0)} Θ (τ) Θ (T) Θ (t) Θ (t_{e}) \\ \cdot e^{- Γ_{s s} t_{e}} e^{- i (ω_{s g} - ω) t - Γ_{s g} t} e^{- i (ω_{d g} - 2 ω) T - Γ_{d g} T} e^{- i (ω_{s g} - ω) τ - Γ_{s g} τ}, \end{aligned}

S_{I}^{(4)} (ω_{t}, ω_{T}) = S_{0} μ_{s g}^{2} μ_{d s} μ_{d g} \frac{1}{ω_{T} - ω_{d g} + i Γ_{d g}} \times \frac{1}{ω_{t} - ω_{s g} + i Γ_{s g}},

\begin{aligned} S^{(4)} (ω_{t}, ω_{T}) = \frac{S_{0} μ_{10}^{2} μ_{21}^{2}}{ω_{T} - ω_{20} + i Γ_{20}} (\frac{1}{ω_{t} - ω_{10} + i Γ_{10}} - \frac{1}{ω_{t} - ω_{21} + i Γ_{21}}) + \\ \frac{4 S_{0} μ_{10}^{4}}{ω_{T} - ω_{d g} + i Γ_{d g}} (\frac{1}{ω_{t} - ω_{s g} + i Γ_{s g}} - \frac{1}{ω_{t} - ω_{d s} + i Γ_{d s}} - \frac{1}{ω_{t} - ω_{d s^{'}} + i Γ_{d s^{'}}} + \\ \frac{1}{ω_{t} - ω_{s^{'} g} + i Γ_{s^{'} g}}), \end{aligned}

P (r) = \frac{3}{a} (\frac{r}{a})^{2} e^{- (\frac{r}{a})^{3}},