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In this paper, the quantum phase transition of cold atoms in a two-mode photomechanical cavity with nonlinear coupling between the optical field (mode 1) and the mechanical oscillator is studied on the basis of the two-mode Dicke model. The functional of the ground state energy of the system is obtained by spin coherent states and variational method. By solving and judging the stability, the phase transformation point and ground state phase diagram are obtained. It is found that there are bistable state of normal phase and reverse normal phase, coexistent state of superradiation phase and reversed normal phase, and reversed normal phase that exists alone. The different interaction strengths between atoms and two-mode light fields greatly affect the value of the phase transition point. There is a quantum phase transition from a normal phase through a phase transition point to a superradiant phase. The light-phonon nonlinear coupling has no effect on the phase transition point, but induces the collapse of the superradiant phase. There is a turning point through which the quantum phase transition from the superradiant phase to the reversed normal phase can be realized. The region of the superradiation phase decreases with the increase of the photon-phonon coupling, and it shrinks to zero at the critical value of the coupling, that is, the turning point and the phase transition point coincide, and there may be a reversal of the atomic population between the two normal phases. The nonlinear coupling of the light-phonon also produces an unstable non-zero photon state, which corresponds to the superradiation state. In the absence of mechanical oscillators, the results of the two-mode Dicke model are returned.
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Keywords:
- two model optomechanical cavity /
- light-phonon nonlinear coupling /
- quantum phase transition /
- superradiation phase collapse
[1] Dicke R H 1954 Phys. Rev. 93 99
Google Scholar
[2] Hepp K, Lieb E H 1973 Ann. Phys. 76 360
Google Scholar
[3] Wang Y K, Hioe F T 1973 Phys. Rev. A 7 831
Google Scholar
[4] Hioe F T 1973 Phys. Rev. A 8 1440
Google Scholar
[5] Baumann K, Guerlin C, Brennecke F, Esslinger T 2010 Nature 464 1301
Google Scholar
[6] Baumann K, Mottl R, Brennecke F, Esslinger T 2011 Phys. Rev. Lett. 107 140402
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[7] Emary C, Brandes T 2003 Phys. Rev. Lett. 90 044101
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[8] Chen G, Li J Q, Liang J Q 2006 Phys. Rev. A 74 054101
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[9] Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391
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[10] Brennecke F, Ritter S, Donner T, Esslinger T 2008 Science 322 235
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[11] Gröblacher S, Hammerer K, Vanner M R, Aspelmeyer M 2009 Nature 460 724
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[12] Anetsberger G, Arcizet O, Unterreithmeier Q P, Rivière R, Schliesser A, Weig E M, Kotthaus J P, Kippenberg T J 2009 Nat. Phys. 5 909
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[13] Chan J, Alegre T P M, Safavi-Naeini A H, Hill J T, Krause A, Gröblacher S, Aspelmeyer M, Painter O 2011 Nature 478 89
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[14] 陈华俊, 米贤武 2011 物理学报 60 124206
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Chen H J, Mi X W 2011 Acta Phys. Sin. 60 124206
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[15] 严晓波, 杨柳, 田雪冬, 刘一谋, 张岩 2014 物理学报 63 204201
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Yan X B, Yang L, Tian X D, Liu Y M, Zhang Y 2014 Acta Phys. Sin. 63 204201
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Google Scholar
Han M, Gu K H, Liu Y M, Zhang Y, Wang X C, Tian X D, Fu C B, Cui C L 2014 Acta Phys. Sin. 63 094206
Google Scholar
[17] Brooks D W C, Botter T, Schreppler S, Purdy T P, Brahms N, Stamper-Kurn D M 2012 Nature 488 476
Google Scholar
[18] Ian H, Gong Z R, Liu Y X, Sun C P, Nori F 2008 Phys. Rev. A 78 013824
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[19] Jiang C, Bian X T, Cui Y S, Chen G B 2016 J. Opt. Soc. Am. B: Opt. Phys. 33 2099
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[20] Wang Z M, Lian J L, Liang J Q, Yu Y M, Liu W M 2016 Phys. Rev. A 93 033630
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Google Scholar
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Google Scholar
[34] Arecchi F T, Courtens E, Gilmore R, Thomas H 1972 Phys. Rev. A 6 2211
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[35] Fox R F 1999 Phys. Rev. A 59 3241
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 双模光机械腔的示意图. 在水平方向上, 机械谐振子的频率是$ {\omega _{\text{b}}} $, 腔模的频率是$ {\omega _1} $; 在垂直方向上, 腔模的频率是$ {\omega _2} $
Figure 1. Schematic diagram of a two-mode opto-mechanical cavity. $ {\omega _{\text{b}}} $ is the frequency of the mechanical harmonic oscillator in the horizontal direction, $ {\omega _1} $ is the frequency of the cavity mode, and $ {\omega _2} $ is the frequency of the cavity mode in the vertical direction.
图 2 $ {g_1}/{\omega _{\mathrm{a}}} $ -$ {g_2}/{\omega _{\text{a}}} $的平面被分成五个区域, 其中$ {g_1}/ {\omega _{\text{a}}} $$ \lt 1, $ $ g_1^2/\omega _{\text{a}}^2 + g_2^2/\omega _{\text{a}}^2 \lt 1 $在区域I; $ {g_1}/{\omega _{\text{a}}} \leqslant 1, {\text{ }}{g_2}/{\omega _{\text{a}}} \leqslant 1, $$ g_1^2/\omega _{\text{a}}^2 + g_2^2/\omega _{\text{a}}^2 \geqslant 1 $在区域II; $ {g_1}/{\omega _{\text{a}}} \lt 1, {\text{ }}{g_2}/{\omega _{\text{a}}} \gt 1 $在区域III; $ {g_1}/{\omega _{\text{a}}} \gt 1, {\text{ }}{g_2}/{\omega _{\text{a}}} \gt 1 $在区域IV; $ {g_1}/{\omega _{\text{a}}} \gt 1, ~~{g_2}/{\omega _{\text{a}}} $$ \lt 1 $在区域V
Figure 2. Plane $ {g_1}/{\omega _{\text{a}}} {\text{-}} {g_2}/{\omega _{\text{a}}} $ is divided into five regions, $ {g_1}/{\omega _{\text{a}}} \lt 1 $, $ g_1^2/\omega _{\text{a}}^2 + g_2^2/\omega _{\text{a}}^2 \lt 1 $ in region I; $ {g_1}/{\omega _{\text{a}}} \leqslant 1, $$ {\text{ }}{g_2}/{\omega _{\text{a}}} \leqslant 1,~~ g_1^2/\omega _{\text{a}}^2 + g_2^2/\omega _{\text{a}}^2 \geqslant 1 $ in region II; $ {g_1}/{\omega _{\text{a}}} \lt 1 $, $ {g_2}/{\omega _{\text{a}}} \gt 1 $ in region III; $ {g_1}/{\omega _{\text{a}}} \gt 1 $, $ {g_2}/{\omega _{\text{a}}} \gt 1 $ in region IV; $ {g_1}/{\omega _{\text{a}}} \gt 1 $, $ {g_2}/{\omega _{\text{a}}} \lt 1 $ in region V.
图 3 平均光子数${n_{{\text{p}}1}}$ (a), ${n_{{\text{p}}2}}$ (b), 原子布居数差分布$\Delta {n_{\text{a}}}$ (c)和平均基态能量$\varepsilon $ (d)
Figure 3. Average photon number ${n_{{\text{p}}1}}$ (a), ${n_{{\text{p}}2}}$ (b), atomic population difference $\Delta {n_{\text{a}}}$ (c) and average energy $\varepsilon $ (d).
图 4 $ g/{\omega _{\text{a}}}{\text{ - }}\zeta /{\omega _{\text{a}}} $平面中的相图 $ \left( {\text{a}} \right){\text{ }}\delta = - {1}. {\text{0;}} $ $ \left(\text{b}\right)\text{ }\delta =-0.5; $ $ \left( {\text{c}} \right){\text{ }}\delta = 0.0; $ $ \left( {\text{d}} \right){\text{ }}\delta = 0.5 $
Figure 4. Phase diagram in a plane $ g/{\omega _{\text{a}}}{\text{ - }}\zeta /{\omega _{\text{a}}} $: $ \left( {\text{a}} \right){\text{ }}\delta = - {1}. {\text{0;}} $ $ \left( {\text{b}} \right){\text{ }}\delta = - 0.5; $ $ \left( {\text{c}} \right){\text{ }}\delta = 0.0; $ $ \left( {\text{d}} \right){\text{ }}\delta = 0.5 $
表 1 光-声子的非线性参量$ \zeta = 0 $、平均基态能量为$ {\varepsilon _ - } $时, 相应点的计算值
Table 1. Calculated values of the corresponding point when nonlinear parameters of the light-phonon $ \zeta = 0 $ and the average ground state energy is $ {\varepsilon _ - } $
区域 $ \left( {\dfrac{{{g_1}}}{{{\omega _{\text{a}}}}}, \dfrac{{{g_2}}}{{{\omega _{\text{a}}}}}} \right) $ $ (\overline{{\gamma }_{1}^{2}}, \text{ }\overline{{\gamma }_{2}^{2}}) $ $ {\varepsilon _ - } $ $ {{\boldsymbol{H}}_ - } = \left[ {\begin{array}{*{20}{c}} {\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial \gamma _1^2}}}&{\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial {\gamma _1}\partial {\gamma _2}}}} \\ {\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial {\gamma _2}\partial {\gamma _1}}}}&{\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial \gamma _2^2}}} \end{array}} \right] $ 特征值$ \left[ {{H_ - }} \right] $ I (0.3, 0.4) (0, 0) –0.5 $ \left[ {\begin{array}{*{20}{c}} {1.82}&{ - 0.24} \\ { - 0.24}&{1.68} \end{array}} \right] $ (2, 1.5) II (0.8, 0.7) (0.0347, 0.0266) –0.5037 $ \left[ {\begin{array}{*{20}{c}} {1.1129}&{ - 0.7762} \\ { - 0.7762}&{1.3208} \end{array}} \right] $ (2, 0.4337) III (0.6, 1.6) (0.0794, 0.5649) –0.8156 $ \left[ {\begin{array}{*{20}{c}} {1.9711}&{ - 0.0711} \\ { - 0.0711}&{1.7943} \end{array}} \right] $ (2, 1.7654) IV (1.5, 1.5) (0.5347, 0.5347) –1.1806 $ \left[ {\begin{array}{*{20}{c}} {1.9506}&{ - 0.0494} \\ { - 0.0494}&{1.9506} \end{array}} \right] $ (2, 1.9012) V (1.5, 0.5) (0.4725, 0.0525) –0.725 $ \left[ {\begin{array}{*{20}{c}} {1.7119}&{ - 0.0960} \\ { - 0.0960}&{1.9680} \end{array}} \right] $ (2, 1.6800) 
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表 2 光-声子的非线性参量$ \zeta /{\omega _{\text{a}}} = 1.0 $、平均基态能量为$ {\varepsilon _ - } $时, 相应点的计算值
Table 2. Calculated values of the corresponding point when nonlinear parameters of the light-phonon $ \zeta /{\omega _{\text{a}}} = 1.0 $ and the average ground state energy is $ {\varepsilon _ - } $.
区域 $ \left( {\dfrac{{{g_1}}}{{{\omega _{\text{a}}}}}, \dfrac{{{g_2}}}{{{\omega _{\text{a}}}}}} \right) $ $ (\overline {\gamma _1^2} , {\text{ }}\overline {\gamma _2^2} ) $ $ {\varepsilon _ - } $ $ {{\boldsymbol{H}}_ - } = \left[ \begin{array}{*{20}{c}} {\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial \gamma _1^2}}}&{\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial {\gamma _1}\partial {\gamma _2}}}} \\ {\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial {\gamma _2}\partial {\gamma _1}}}}&{\dfrac{{{\partial ^2}{\varepsilon _ - }}}{{\partial \gamma _2^2}}} \end{array} \right] $ 特征值$ \left[ {{H_ - }} \right] $ I (0.4, 0.3) (4.5925, 0.01716) 1.2585 $ \left[ \begin{array}{*{20}{c}} { - 3.5436}&{ - 0.0326} \\ { - 0.0326}&{1.9674} \end{array} \right] $ (–3.5438, 1.9676) II (0.7, 0.8) $ \left\{ {\begin{aligned} &{\left( {4.1771, 0.1478} \right)} \\ &{\left( {0.0273, 0.0353} \right)} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{0.7714} \\ &{ - 0.5038} \end{aligned}} \right. $ $ \left\{ \begin{aligned} {\left[ \begin{array}{*{20}{c}} { - 3.033}&{ - 0.0237} \\ { - 0.0237}&{1.9730} \end{array} \right]} \\ {\left[ \begin{array}{*{20}{c}} {1.2929}&{ - 0.7707} \\ { - 0.7707}&{1.1192} \end{array} \right]} \end{aligned} \right. $ $ \left\{ {\begin{aligned} &{\left( { - 0.0334, 1.9731} \right)} \\ &{\left( {1.9816, 0.4305} \right)} \end{aligned}} \right. $ III (0.8, 1.2) $ \left\{ {\begin{aligned} &{\left( {4.0258, 0.3439} \right)} \\ &{\left( {0.1303, 0.2781} \right)} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{0.3866} \\ & { - 0.6418} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} {\left[ \begin{array}{*{20}{c}} { - 2.8431}&{ - 0.0182} \\ { - 0.0182}&{1.9727} \end{array} \right]} \\ {\left[ \begin{array}{*{20}{c}} {1.7048}&{ - 0.2082} \\ { - 0.2082}&{1.6878} \end{array} \right]} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{\left( { - 2.8432, 1.9728} \right)} \\ &{\left( {1.9046, 1.4880} \right)} \end{aligned}} \right. $ IV (1.5, 1.5) $ \left\{ {\begin{aligned} &{\left( {2.7669, 0.5519} \right)} \\ &{\left( {0.7439, 0.5390} \right)} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{ - 1.0907} \\ & { - 1.2190} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} {\left[ \begin{array}{*{20}{c}} { - 1.3319}&{ - 0.0116} \\ { - 0.0116}&{1.9884} \end{array} \right]} \\ {\left[ \begin{array}{*{20}{c}} {1.7048}&{ - 0.2082} \\ { - 0.2082}&{1.6878} \end{array} \right]} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{\left( { - 1.3220, 1.9884} \right)} \\ & {\left( {1.9632, 1.0673} \right)} \end{aligned}} \right. $ V (1.2, 0.8) $ \left\{ {\begin{aligned} &{\left( {3.4015, 0.1540} \right)} \\ &{\left( {0.3252, 0.1263} \right)} \end{aligned}} \right. $ $ \left\{ {\begin{aligned}& { - 0.1776} \\ &{ - 0.6491} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{\left[ \begin{array}{*{20}{c}} { - 2.1072}&{ - 0.0141} \\ { - 0.0141}&{1.9907} \end{array} \right]} \\ &{\left[ \begin{array}{*{20}{c}} {1.3319}&{ - 0.1853} \\ { - 0.1853}&{1.8765} \end{array} \right]} \end{aligned}} \right. $ $ \left\{ {\begin{aligned} &{\left( { - 2.1073, 1.9907} \right)} \\ &{\left( {1.9336, 1.2748} \right)} \end{aligned}} \right. $
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表 3 模式2光场$ {\gamma _2} = 0 $、光-声子的非线性参量$ \zeta /{\omega _{\text{a}}} = 1.0 $、基态能量取$ {\varepsilon _ + } $时, 相应点的计算值
Table 3. Calculated values of the corresponding point when the light field of mode 2 $ {\gamma _2} = 0 $, nonlinear parameters of the light-phonon $ \zeta /{\omega _{\text{a}}} = 1.0 $ and the average ground state energy is $ {\varepsilon _ + } $.
$ {{{g_1}}}/{{{\omega _{\text{a}}}}} $ $ \overline {\gamma _1^2} $ $ {\varepsilon _ + } $ $ {{{\partial ^2}{\varepsilon _ + }}}/{{\partial \gamma _1^2}} $ 0.4 5.0910 3.531 –4.0728 0.8 5.0614 4.3676 –4.0491 1.2 5.0437 5.2408 –4.0350 1.6 5.0336 6.1243 –4.0269
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[1] Dicke R H 1954 Phys. Rev. 93 99
Google Scholar
[2] Hepp K, Lieb E H 1973 Ann. Phys. 76 360
Google Scholar
[3] Wang Y K, Hioe F T 1973 Phys. Rev. A 7 831
Google Scholar
[4] Hioe F T 1973 Phys. Rev. A 8 1440
Google Scholar
[5] Baumann K, Guerlin C, Brennecke F, Esslinger T 2010 Nature 464 1301
Google Scholar
[6] Baumann K, Mottl R, Brennecke F, Esslinger T 2011 Phys. Rev. Lett. 107 140402
Google Scholar
[7] Emary C, Brandes T 2003 Phys. Rev. Lett. 90 044101
Google Scholar
[8] Chen G, Li J Q, Liang J Q 2006 Phys. Rev. A 74 054101
Google Scholar
[9] Aspelmeyer M, Kippenberg T J, Marquardt F 2014 Rev. Mod. Phys. 86 1391
Google Scholar
[10] Brennecke F, Ritter S, Donner T, Esslinger T 2008 Science 322 235
Google Scholar
[11] Gröblacher S, Hammerer K, Vanner M R, Aspelmeyer M 2009 Nature 460 724
Google Scholar
[12] Anetsberger G, Arcizet O, Unterreithmeier Q P, Rivière R, Schliesser A, Weig E M, Kotthaus J P, Kippenberg T J 2009 Nat. Phys. 5 909
Google Scholar
[13] Chan J, Alegre T P M, Safavi-Naeini A H, Hill J T, Krause A, Gröblacher S, Aspelmeyer M, Painter O 2011 Nature 478 89
Google Scholar
[14] 陈华俊, 米贤武 2011 物理学报 60 124206
Google Scholar
Chen H J, Mi X W 2011 Acta Phys. Sin. 60 124206
Google Scholar
[15] 严晓波, 杨柳, 田雪冬, 刘一谋, 张岩 2014 物理学报 63 204201
Google Scholar
Yan X B, Yang L, Tian X D, Liu Y M, Zhang Y 2014 Acta Phys. Sin. 63 204201
Google Scholar
[16] 韩明, 谷开慧, 刘一谋, 张岩, 王晓畅, 田雪冬, 付长宝, 崔淬砺 2014 物理学报 63 094206
Google Scholar
Han M, Gu K H, Liu Y M, Zhang Y, Wang X C, Tian X D, Fu C B, Cui C L 2014 Acta Phys. Sin. 63 094206
Google Scholar
[17] Brooks D W C, Botter T, Schreppler S, Purdy T P, Brahms N, Stamper-Kurn D M 2012 Nature 488 476
Google Scholar
[18] Ian H, Gong Z R, Liu Y X, Sun C P, Nori F 2008 Phys. Rev. A 78 013824
Google Scholar
[19] Jiang C, Bian X T, Cui Y S, Chen G B 2016 J. Opt. Soc. Am. B: Opt. Phys. 33 2099
Google Scholar
[20] Wang Z M, Lian J L, Liang J Q, Yu Y M, Liu W M 2016 Phys. Rev. A 93 033630
Google Scholar
[21] Lian J L, Liu N, Liang J Q, Chen G, Jia S T 2013 Phys. Rev. A 88 043820
Google Scholar
[22] Zhao X Q, Liu N, Bai X M, Liang J Q 2017 Ann. Phys. 378 448
Google Scholar
[23] Santos J P, Semião F L, Furuya K 2010 Phys. Rev. A 82 063801
Google Scholar
[24] Sankey J C, Yang C, Zwickl B M, Jayich A M, Harris J G E 2010 Nat. Phys. 6 707
Google Scholar
[25] Clerk A A, Marquardt F, Harris J G E 2010 Phys. Rev. Lett. 104 213603
Google Scholar
[26] Purdy T P, Brooks D W C, Botter T, Brahms N, Ma Z Y, Stamper-Kurn D M 2010 Phys. Rev. Lett. 105 133602
Google Scholar
[27] Wang B, Nori F, Xiang Z X 2024 Phys. Rev. Lett. 132 053601
Google Scholar
[28] Léonard J, Morales A, Zupancic P, Esslinger T, Donner T 2017 Nature 543 87
Google Scholar
[29] Léonard J, Morales A, Zupancic P, Donner T, Esslinger T 2017 Science 358 1415
Google Scholar
[30] Zhang G Q, Chen Z, You J Q 2020 Phys. Rev. A 102 032202
Google Scholar
[31] Quezada L F, Nahmad-Achar E 2017 Phys. Rev. A 95 013849
Google Scholar
[32] Liu N, Zhao X Q, Liang J Q 2019 Int. J. Theor. Phys. 58 558
Google Scholar
[33] 赵秀琴, 张文慧, 王红梅 2024 物理学报 73 160302
Google Scholar
Zhao X Q, Zhang W H, Wang H M 2024 Acta Phys. Sin. 73 160302
Google Scholar
[34] Arecchi F T, Courtens E, Gilmore R, Thomas H 1972 Phys. Rev. A 6 2211
Google Scholar
[35] Fox R F 1999 Phys. Rev. A 59 3241
Google Scholar
[36] 黄洪斌 1991 物理学报 40 1396
Google Scholar
Huang H B 1991 Acta Phys. Sin. 40 1396
Google Scholar
[37] Zhu W S, Rabitz H 1998 Phys. Rev. A 58 4741
Google Scholar
[38] Stannigel K, Komar P, Habraken S J M, Bennett S D, Lukin M D, Zoller P, Rabl P 2012 Phys. Rev. Lett. 109 013603
Google Scholar
[39] Bell S, Crighton J S, Fletcher R 1981 Chem. Phys. Lett. 82 122
Google Scholar
[40] Vallone G, Cariolaro G, Pierobon G 2019 Phys. Rev. A 99 023817
Google Scholar
[41] Frueholz R P, Camparo J C 1996 Phys. Rev. A 54 3499
Google Scholar
[42] Aftalion A, Mason P 2016 Phys. Rev. A 94 023616
Google Scholar
[43] Schlittler T M, Mosseri R, Barthel T 2017 Phys. Rev. B 96 195142
Google Scholar
[44] Deshpande A, Gorshkov A V, Fefferman B 2022 PRX Quantum 3 040327
Google Scholar
[45] Tolkunov D, Solenov D 2007 Phys. Rev. B 75 024402
Google Scholar
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