集體原子系綜在量子資訊與通訊中扮演了重要角色，杜昂-盧金-西拉克-佐勒 (DLCZ)協議通過生成遠距節點間的糾纏，實現了長距離量子通訊。作為中繼 網絡中的量子節點，Λ-型原子系綜因其減少光子損失並提升網絡效能的能力而受到 重視。儘管該協議已提出超過二十年，但一個能全面描述其動態特性、特別是在非 穩態條件下的完整開放量子系統模型，尚待系統性地建立。這一限制對某些實驗結 果的定量分析與解釋帶來挑戰，並使基於 DLCZ 的系統最佳化更加複雜。本研究利 用海森堡-朗之萬運算子方法，結合麥克斯韋-薛丁格方程，建立了一個描述 DLCZ 協 議中由自發拉曼散射驅動的斯托克斯光子生成過程的開放量子系統模型。此框架能 同時描述原子相干性、居量動態及輻射光場的時空演化。我們推導出光子生成率的 解析表達式，並提供了比以往方法更精確的實驗條件表徵。這一進展不僅實現了更 廣參數範圍的模擬，還為寫入過程提供了更深入的洞見，並奠定了優化基於 DLCZ的量子通訊系統的理論基礎。

Collective atomic ensembles have become essential in quantum information and communi- cation, with the Duan-Lukin-Cirac-Zoller (DLCZ) protocol enabling long-distance quantum communication via entanglement generation between distant nodes. The Λ-level atomic en- sembles used as quantum nodes in repeater networks are valued for mitigating photon loss and enhancing network performance. Although the protocol was introduced over two decades ago, a fully realized open quantum system model that comprehensively describes its dynam- ics, particularly under non-steady-state conditions, remains to be systematically established. This limitation has posed challenges for the quantitative analysis and interpretation of certain experimental results, making the optimization of DLCZ-based systems more complex. In this study, we develop an open quantum system model for Stokes photon generation in the DLCZ protocol, driven by spontaneous Raman scattering, using the Heisenberg-Langevin operator approach in conjunction with Maxwell-Schrödinger equations. This framework allows us to simultaneously describe the atomic coherence, population dynamics, and the spatiotem- poral evolution of the emitted light field. By deriving an analytical expression for the pho- ton generation rate, our model offers a more precise representation of realistic experimental conditions compared to previous approaches. This advancement enables simulations across diverse parameter ranges, providing deeper insights into the writing process and establishing a theoretical foundation for optimizing DLCZ-based quantum communication systems.

[1] F.Arute,K.Arya,R.Babbush,D.Bacon,J.C.Bardin,R.Barends,R.Biswas,S.Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lind- mark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Marti- nis. Quantum supremacy using a programmable superconducting processor. Nature, 574(7779):505–510, 2019.
[2] R. G. Bartle and D. R. Sherbert. Introduction to real analysis, volume 2. Wiley New York, 2000.
[3] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters. Tele- porting an unknown quantum state via dual classical and einstein-podolsky-rosen chan- nels. Phys. Rev. Lett., 70:1895–1899, Mar 1993.
[4] E. T. Campbell, B. M. Terhal, and C. Vuillot. Roads towards fault-tolerant universal quantum computation. Nature, 549(7671):172–179, 2017.
[5] J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi. Quantum state transfer and en- tanglement distribution among distant nodes in a quantum network. Physical Review Letters, 78(16):3221, 1997.
[6] J.-P.Dou,A.-L.Yang,M.-Y.Du,D.Lao,J.Gao,L.-F.Qiao,H.Li,X.-L.Pang,Z.Feng, H. Tang, and X.-M. Jin. A broadband dlcz quantum memory in room-temperature atoms. Communications Physics, 1(1):55, 2018.
[7] L.-M.Duan,J.I.Cirac,andP.Zoller.Three-dimensionaltheoryforinteractionbetween atomic ensembles and free-space light. Physical Review A, 66(2):023818, 2002.
[8] L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller. Long-distance quantum communi- cation with atomic ensembles and linear optics. Nature, 414(6862):413–418, 2001.
[9] W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller. Quantum repeaters based on entangle- ment purification. Phys. Rev. A, 59:169–181, Jan 1999.
[10] A. K. Ekert. Quantum cryptography based on bell’s theorem. Phys. Rev. Lett., 67:661– 663, Aug 1991.
[11] M. Fleischhauer, A. Imamoglu, and J. P. Marangos. Electromagnetically induced trans- parency: Optics in coherent media. Rev. Mod. Phys., 77:633–673, Jul 2005.
[12] M.FleischhauerandM.D.Lukin.Dark-statepolaritonsinelectromagneticallyinduced transparency. Phys. Rev. Lett., 84:5094–5097, May 2000.
[13] M. Fleischhauer and M. D. Lukin. Quantum memory for photons: Dark-state polaritons. Phys. Rev. A, 65:022314, Jan 2002.
[14] M. Fleischhauer, A. B. Matsko, and M. O. Scully. Quantum limit of optical magnetom- etry in the presence of ac stark shifts. Phys. Rev. A, 62:013808, Jun 2000.
[15] C. Gardiner and P. Zoller. The quantum world of ultra-cold atoms and light book II: the physics of quantum-optical devices, volume 4. World Scientific Publishing Company, 2015.
[16] J. Garrison and R. Chiao. Quantum optics. OUP Oxford, 2008.
[17] G. Grynberg, A. Aspect, and C. Fabre. Introduction to quantum optics: from the semi-classical approach to quantized light. Cambridge university press, 2010.
[18] P. S. Gupta and B. K. Mohanty. On the quantum theory of stimulated raman scattering. Czechoslovak Journal of Physics B, 30(10):1127–1139, 1980.
[19] A. W. Harrow, A. Hassidim, and S. Lloyd. Quantum algorithm for linear systems of equations. Phys. Rev. Lett., 103:150502, Oct 2009.
[20] P.Kolchin.Electromagnetically-induced-transparency-basedpairedphotongeneration. Physical Review A—Atomic, Molecular, and Optical Physics, 75(3):033814, 2007.
[21] D. A. Lidar and T. A. Brun. Quantum error correction. Cambridge university press, 2013.
[22] C. R. Ooi, Y. Huang, and J. W. Lee. Spatial-temporal dynamics of stimulated raman scattering: Effects of populations and two-photon detuning. Physics Open, 19:100211, 2024.
[23] Y.-F.Pu,S.Zhang,Y.-K.Wu,N.Jiang,W.Chang,C.Li,andL.-M.Duan.Experimental demonstration of memory-enhanced scaling for entanglement connection of quantum repeater segments. Nature Photonics, 15(5):374–378, 2021.
[24] M. G. Raymer and J. Mostowski. Stimulated raman scattering: Unified treatment of spontaneous initiation and spatial propagation. Phys. Rev. A, 24:1980–1993, Oct 1981.
[25] J. J. Sakurai and J. Napolitano. Modern quantum mechanics. Cambridge University Press, 2020.
[26] N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin. Quantum repeaters based on atomic ensembles and linear optics. Rev. Mod. Phys., 83:33–80, Mar 2011.
[27] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal on Computing, 26(5):1484–1509, 1997.
[28] S. Van Enk, J. Cirac, and P. Zoller. Photonic channels for quantum communication. Science, 279(5348):205–208, 1998.
[29] J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan. Satellite-based entanglement distribution over 1200 kilometers. Science, 356(6343):1140–1144, 2017.
[30] L. You, J. Mostowski, and J. Cooper. Cone emission from laser-pumped two-level atoms. i. quantum theory of resonant light propagation. Phys. Rev. A, 46:2903–2924, Sep 1992.
[31] 楊智閔 et al. 基於 DLCZ 機制產生窄頻史托克光子之研究. PhD thesis, 2021.