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The quantum oscillator model plays a significant role in quantum optics and quantum information and has been one of the hot topics in related research fields. Inspired by the single-mode linear harmonic oscillator and the two-mode entangled state representation, we construct a two-mode coupled harmonic oscillator in this work. Different from the quantum transformation method used in previous literature, the entangled state representation is directly used in this work to solve its energy eigenvalues and eigenfunctions easily. The energy eigenvalues and eigenfunctions of this two-mode coupled harmonic oscillator are continuous compared with those of the one-mode harmonic oscillator. Using the matrix theory of quantum operators, we derive the transformation and inverse transformation of the time evolution operator corresponding to the two-mode coupled harmonic oscillator. In addition, using the entangled state representation, the specific form of the time evolution of the two-mode vacuum state under the action of the oscillator is obtained. Through the analysis of quantum fidelity, it is found that the fidelity of the output quantum state decreases with the oscillator frequency increasing, and the fidelity eventually tends to zero with the increase of time. When analyzing the orthogonal squeezing properties of the output quantum state, this type of two-mode oscillator does not have the orthogonal squeezing effect, but it has a strong quantum dissipation effect instead. This conclusion is further verified by the quasi-probability distribution Q function of the quantum state phase space. Therefore, the two-mode coupled harmonic oscillator has a major reference value in quantum control such as quantum decoherence and quantum information transmission. Like the two-mode squeezed vacuum state, the photon distribution of the output quantum light field corresponding to the two-mode harmonic oscillator presents a super-Poisson distribution, and the photons exhibit a strong anti-bunching effect. Using the three-dimensional discrete plot of the photon number distribution, the super-Poisson distribution and quantum dissipation effect of the output quantum state are intuitively demonstrated. Finally, the SV, which is an entanglement criterion, is used to determine that the output quantum state has a high degree of entanglement. Further numerical analysis shows that the degree of entanglement increases with the action time and the oscillator frequency. In summary, the two-mode coupled harmonic oscillator constructed in this work can be used to prepare highly entangled quantum states through a complete quantum dissipation process. This provides theoretical support for experimental preparing quantum entangled states based on dissipative mechanisms. -
Keywords:
- quantum oscillator /
- quantum dissipation /
- super-poisson distribution /
- quantum entanglement
[1] 徐秀玮, 任廷琦, 刘姝延, 董永绵, 赵继德 2006 物理学报 55 535
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图 1 双模真空态保真度随时间的变化曲线
Figure 1. Curves for the change of fidelity with two-mode vacuum state over time.
图 2 量子态$\left| \tau \right\rangle $的相空间分布Q函数 (a) $\omega = 0.5,\; t = 0$; (b) $\omega = 0.5,\; t = 0.5$; (c) $\omega = 0.5, \;t = 1.5$; (d) $\omega = 1.5,\; t = 1.5$
Figure 2. Q-distribution function of quantum state $\left| \tau \right\rangle $ in phase space: (a) $\omega = 0.5,\; t = 0$; (b) $\omega = 0.5, \;t = 0.5$; (c) $\omega = 0.5, \;t = 1.5$; (d) $\omega = 1.5,\; t = 1.5$.
图 3 量子态$\left| \tau \right\rangle $的Q参数随时间的变化曲线
Figure 3. Curves for the change of Q-parameter with quantum state $\left| \tau \right\rangle $over time.
图 4 量子态$\left| \tau \right\rangle $的${R_{ab}}$函数随时间的变化曲线
Figure 4. Curves for the change of ${R_{ab}}$ function with quantum state $\left| \tau \right\rangle $over time.
图 5 量子态$\left| \tau \right\rangle $中光子分布概率 (a) $\omega = 0.5,\; t = 0.5$; (b) $\omega = 0.5,\; t = 2.5$; (c) $t = 0.5, \;\omega = 0.1$; (d) $t = 0.5,\; \omega = 2.5$
Figure 5. Photon number distribution probability for the quantum state $\left| \tau \right\rangle $: (a) $\omega = 0.5,\; t = 0.5$; (b) $\omega = 0.5,\; t = 2.5$; (c) $t = 0.5,\; \omega = 0.1$; (d) $t = 0.5,\; \omega = 2.5$.
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[1] 徐秀玮, 任廷琦, 刘姝延, 董永绵, 赵继德 2006 物理学报 55 535
Google Scholar
Xu X W, Ren T Q, Liu S Y, Dong Y M, Zhao J D 2006 Acta Phys. Sin. 55 535
Google Scholar
[2] Qu L C, Chen J, Liu Y X 2022 Phys. Rev. D 105 126015
Google Scholar
[3] Hou B P, Wang S J, Yu W L, Sun W L, Wang G 2004 Chin. Phys. Lett. 21 2334
Google Scholar
[4] Mechler M, Man'ko M A, Man'ko V I, Adam P 2024 J. Russ. Laser Res. 45 1
Google Scholar
[5] Schrödinger E 1926 Phys. Rev. 28 1049
Google Scholar
[6] 张秀兰, 刘恒, 余海军, 张文海 2011 物理学报 60 040303
Google Scholar
Zhang X L, Liu H, Yu H J, Zhang W H 2011 Acta Phys. Sin. 60 040303
Google Scholar
[7] Zhong Z R, Sheng J Q, Lin L H, Zheng S B 2019 Opt. Lett. 44 1726
Google Scholar
[8] Glauber R J 1963 Phys. Rev. 130 2529
Google Scholar
[9] Cardoso F R, Rossatto D Z, Fernandes G, Higgins G, Villas-Boas C J 2021 Phys. Rev. A 103 062405
Google Scholar
[10] Lu H L, Fan H Y 2007 Commun. Theor. Phys. 47 1024
Google Scholar
[11] De Castro A S M, Dodonov V V 2001 J. Opt. B 3 228
Google Scholar
[12] 姜磊, 赖莉, 蔚涛, 罗懋康 2021 物理学报 70 130501
Google Scholar
Jiang L, Lai L, Yu T, Luo M K 2021 Acta Phys. Sin. 70 130501
Google Scholar
[13] Einstein A, Podolsky B, Rosen N 1935 Phys. Rev. 47 777
Google Scholar
[14] Fan H Y 2002 Chin. Phys. Lett. 19 897
Google Scholar
[15] Zhang J D, Wang S 2024 Phys. Lett. A 502 129400
Google Scholar
[16] Caldeira A O, Leggett A J 1981 Phys. Rev. Lett. 46 211
Google Scholar
[17] Wang X B, Yu S X, Zhang Y D 1994 J. Phys. A 27 6563
Google Scholar
[18] Ghiu I, Marian P, Marian T A 2013 Phys. Scripta T153 014028
Google Scholar
[19] Tian L J, Zhu C Q, Zhang H B, Qin L G 2011 Chin. Phys. B 20 040302
Google Scholar
[20] He H, Lou Y, Xu X, Liu S, Jing J 2023 Opt. Lett. 48 1375
Google Scholar
[21] Bose S 2021 Phys. Rev. A 104 042419
Google Scholar
[22] Harrington P M, Mueller E J, Murch K W 2022 Nat. Rev. Phys. 4 660
Google Scholar
[23] Chen Y H, Shi Z C, Song J, Xia Y, Zheng S B 2017 Phys. Rev. A 96 043853
Google Scholar
[24] Sauer S, Gneiting C, Buchleitner A 2013 Phys. Rev. Lett. 111 030405
Google Scholar
[25] Krauter H, Muschik C A, Jensen K, Wasilewski W, Petersen J M, Cirac J I, Polzik E S 2011 Phys. Rev. Lett. 107 080503
Google Scholar
[26] Choi T, Lee H J 2007 Phys. Rev. A 76 012308
Google Scholar
[27] Mandel L 1979 Opt. Lett. 4 205
Google Scholar
[28] Xu X F, Wang S, Tang B 2014 Chin. Phys. B 23 024206
Google Scholar
[29] 张浩亮, 贾芳, 徐学翔, 郭琴, 陶向阳, 胡利云 2013 物理学报 62 014208
Google Scholar
Zhang H L, Jia F, Xu X X, Guo Q, Tao X Y, Hu L Y 2013 Acta Phys. Sin. 62 014208
Google Scholar
[30] Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865
Google Scholar
[31] Shchukin E V, Vogel W 2005 Phys. Rev. A 72 043808
Google Scholar
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