由於優異的傳播速度及較不受環境干擾的特性,光子適合用於量子資訊的傳播媒介。但由於量子態相當脆弱,因此操控光子的量子態在量子資訊科技來說是一個重要而具挑戰的課題。至今,許多不同的光量子操控的方案都相繼問世,而其中一個相當有潛力的操控方式則是電磁波引發透明機制。由於運作在共振的條件下,電磁波引發透明機制能夠大幅增強光與物質的交互作用。在本論文中,我們將探討數個與量子資訊科學相關的議題,並提出基於電磁波引發透明的解決方案。第一個工作中,我們提出一全光學相位調製的新方案,並在實驗上觀察到了僅由平均光子數八顆的光脈衝經由另一相同光子數的脈衝所引發的pi的相位變化。而根據我們的理論預測,只要有相位同調的光子對做為光子源,此方案理論上能也夠在單光子的範疇下運作。而在第二個工作,我們在理論及實驗上證實了一個共振四波混頻在光反向對打的結構下,具有高的轉換效率。實驗上,我們由光學密度48及相位不匹配量0.273pi的系統下觀察到63%的轉換效率。據我們所知,這不只是第一次在共振四波混頻系統發現能夠突破25%效率的限制,還是目前已知在共振條件下的最高效率。另外,我們也證實了相位不匹配能夠由電磁波引發透明下的雙光子失諧所引發的相位變化補償,並在光學密度52的系統下藉由補償進一步觀測到76%的效率。接著我們進一步以全量子化的理論模型證明只要在100%的轉換效率下,此方案確實為一量子轉頻系統。而在最後一個部分,我們同樣也用全量子的方式討論基於電磁波引發透明下的窄頻寬光子對產生的模型。根據此模型,在系統同調破壞率夠小的情況下此方案能夠產生具有kHz頻寬等級的光子對。

Photons are suitable carriers in quantum information technologies because of their fast propagation speed and weak interaction with the environment. The manipulation of the quantum states of individual photons is hence an essential, however, a challenging task in quantum information technologies because of the fragility of the quantum states. To date, various issues for optical-quantum manipulation have been proposed in a variety of schemes. Among them, electromagnetically induced transparency (EIT) paves a potential way in manipulation of photons. Owing to the resonant process, EIT largely increases the interaction strength and allows the narrowband interaction. In this dissertation, we focus on several tasks in quantum information science and propose the schemes based on EIT in the medium of the cold atomic ensemble. In the first work, we propose an all-optical cross-phase modulation scheme based on EIT. In the experiment, we have observed the pi phase shift of one light pulse induced by the other, with both containing eight photons in each pulse. According to the theoretical model, it is possible to reach the goal of $pi$ phase modulation per photon if one can prepare two phase-coherent single-photon pulses. In the second work, we demonstrate an efficient frequency conversion scheme using resonant four-wave mixing (FWM) based on EIT by arranging the applied laser beams in a backward configuration, both theoretically and experimentally. In the experiment, we observe 63% conversion efficiency (CE) with optical depth (OD) of 48 under the phase-mismatch value of 0.273pi. Moreover, we show that the phase mismatch can be compensated by the two-photon-detuning induced phase shift in EIT, and we observe the CE of 76\% using an optical depth (OD) of 52 after compensation. To our knowledge, this is not only the first time ever that observes the CE exceeding 25% in resonant the FWM, but also the highest record of CE under the resonant condition. Moreover, we also demonstrate that the scheme is indeed a quantum frequency conversion process if the CE reaches 100%, and discuss the generation of the narrowband bi-photon source based on EIT using the fully-quantum approach in the last work. According to the model, it is possible to generate the paired photons with kHz-order bandwidth if the dephasing rate of a system is small enough.

[1] J. Hecht, “A short history of laser development,” Applied optics, vol. 49, no. 25, pp. F99–F122, 2010.
[2] W. F. Brinkman, D. E. Haggan, and W. W. Troutman, “A history of the invention of the transistor and where it will lead us,” IEEE Journal of Solid-State Circuits, vol. 32, no. 12, pp. 1858–1865, 1997.
[3] L. Łukasiak and A. Jakubowski, “History of semiconductors,” Journal of Telecommunications and information technology, pp. 3–9, 2010.
[4] J. P. Dowling and G. J. Milburn, “Quantum technology: the second quantum revolution,” Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, vol. 361, no. 1809, pp. 1655–1674, 2003.
[5] W. Heisenberg, “Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik,” in Original Scientific Papers Wissenschaftliche Originalarbeiten, pp. 478–504, Springer, 1985.
[6] W. Greiner, Quantum mechanics: an introduction. Springer Science & Business Media, 2011.
[7] D. H. Menzel, Fundamental formulas of physics, vol. 1. Courier Corporation, 1960.
[8] R. P. Feynman, “Simulating physics with computers,” International journal of theoretical physics, vol. 21, no. 6, pp. 467–488, 1982.
[9] D. Deutsch, “Quantum theory, the church–turing principle and the universal quantum computer,” Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, vol. 400, no. 1818, pp. 97–117, 1985.
[10] L. K. Grover, “A fast quantum mechanical algorithm for database search,” arXiv preprint quant-ph/9605043, 1996.
[11] P. W. Shor, “Algorithms for quantum computation: Discrete logarithms and factoring,” in Proceedings 35th annual symposium on foundations of computer science, pp. 124–134, Ieee, 1994.
[12] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, “Quantum computers,” nature, vol. 464, no. 7285, p. 45, 2010.
[13] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, “Experimental issues in coherent quantum-state manipulation of trapped atomic ions,” Journal of Research of the National Institute of Standards and Technology, vol. 103, no. 3, p. 259, 1998.
[14] R. Blatt and D. Wineland, “Entangled states of trapped atomic ions,” Nature, vol. 453, no. 7198, p. 1008, 2008.
[15] M. Anderlini, P. J. Lee, B. L. Brown, J. Sebby-Strabley, W. D. Phillips, and J. V. Porto, “Controlled exchange interaction between pairs of neutral atoms in an optical lattice,” Nature, vol. 448, no. 7152, p. 452, 2007.
[16] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. Yavuz, T. Walker, and M. Saffman, “Observation of rydberg blockade between two atoms,” Nature Physics, vol. 5, no. 2, p. 110, 2009.
[17] A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the rydberg blockade regime,” Nature Physics, vol. 5, no. 2, p. 115, 2009.
[18] N. A. Gershenfeld and I. L. Chuang, “Bulk spin-resonance quantum computation,” science, vol. 275, no. 5298, pp. 350–356, 1997.
[19] D. G. Cory, A. F. Fahmy, and T. F. Havel, “Ensemble quantum computing by nmr spectroscopy,” Proceedings of the National Academy of Sciences, vol. 94, no. 5, pp. 1634–1639, 1997.
[20] M. Mehring, J. Mende, and W. Scherer, “Entanglement between an electron and a nuclear spin 1 2,” Physical review letters, vol. 90, no. 15, p. 153001, 2003.
[21] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha, and L. M. Vandersypen, “Spins in few-electron quantum dots,” Reviews of modern physics, vol. 79, no. 4, p. 1217, 2007.
[22] R. Hanson, V. Dobrovitski, A. Feiguin, O. Gywat, and D. Awschalom, “Coherent dynamics of a single spin interacting with an adjustable spin bath,” Science, vol. 320, no. 5874, pp. 352–355, 2008.
[23] S. Takahashi, R. Hanson, J. Van Tol, M. S. Sherwin, and D. D. Awschalom, “Quenching spin decoherence in diamond through spin bath polarization,” Physical review letters, vol. 101, no. 4, p. 047601, 2008.
[24] Y. Nakamura, Y. A. Pashkin, and J. Tsai, “Coherent control of macroscopic quantum states in a single-cooper-pair box,” nature, vol. 398, no. 6730, p. 786, 1999.
[25] I. Chiorescu, Y. Nakamura, C. M. Harmans, and J. Mooij, “Coherent quantum dynamics of a superconducting flux qubit,” Science, vol. 299, no. 5614, pp. 1869–1871, 2003.
[26] J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina, “Rabi oscillations in a large josephson-junction qubit,” Physical review letters, vol. 89, no. 11, p. 117901, 2002.
[27] D. Castelvecchi, “Quantum computers ready to leap out of the lab in 2017,” Nature News, vol. 541, no. 7635, p. 9, 2017.
[28] J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, et al., “State preservation by repetitive error detection in a superconducting quantum circuit,” Nature, vol. 519, no. 7541, p. 66, 2015.
[29] Y. Wang, Y. Li, Z.-q. Yin, and B. Zeng, “16-qubit ibm universal quantum computer can be fully entangled,” npj Quantum Information, vol. 4, no. 1, p. 46, 2018.
[30] G. G. Guerreschi and J. Park, “Two-step approach to scheduling quantum circuits,” Quantum Science and Technology, vol. 3, no. 4, p. 045003, 2018.
[31] M. W. Johnson, M. H. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, et al., “Quantum annealing with manufactured spins,” Nature, vol. 473, no. 7346, p. 194, 2011.
[32] A. D. King, J. Carrasquilla, J. Raymond, I. Ozfidan, E. Andriyash, A. Berkley, M. Reis, T. Lanting, R. Harris, F. Altomare, et al., “Observation of topological phenomena in a programmable lattice of 1,800 qubits,” Nature, vol. 560, no. 7719, p. 456, 2018.
[33] D. P. DiVincenzo, “The physical implementation of quantum computation,” Fortschritte der Physik: Progress of Physics, vol. 48, no. 9-11, pp. 771–783, 2000.
[34] A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nature photonics, vol. 3, no. 12, p. 706, 2009.
[35] M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Physics Today, vol. 65, no. 11, p. 32, 2012.
[36] E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” nature, vol. 409, no. 6816, p. 46, 2001.
[37] L.-M. Duan and H. Kimble, “Scalable photonic quantum computation through cavityassisted interactions,” Physical review letters, vol. 92, no. 12, p. 127902, 2004.
[38] I. I. Ryabtsev, D. B. Tretyakov, and I. I. Beterov, “Applicability of rydberg atoms to quantum computers,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 38, no. 2, p. S421, 2005.
[39] G. Alzetta, “Induced transparency,” Phys. Today, vol. 50, no. 7, pp. 36–42, 1997.
[40] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Reviews of modern physics, vol. 77, no. 2, p. 633, 2005.
[41] K.-J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Physical Review Letters, vol. 66, no. 20, p. 2593, 1991.
[42] Y.-F. Hsiao, P.-J. Tsai, H.-S. Chen, S.-X. Lin, C.-C. Hung, C.-H. Lee, Y.-H. Chen, Y.-F. Chen, A. Y. Ite, and Y.-C. Chen, “Highly efficient coherent optical memory based on electromagnetically induced transparency,” Physical review letters, vol. 120, no. 18, p. 183602, 2018.
[43] Y. Dudin, L. Li, and A. Kuzmich, “Light storage on the time scale of a minute,” Physical Review A, vol. 87, no. 3, p. 031801, 2013.
[44] P. Vernaz-Gris, K. Huang, M. Cao, A. S. Sheremet, and J. Laurat, “Highly-efficient quantum memory for polarization qubits in a spatially-multiplexed cold atomic ensemble,” Nature communications, vol. 9, no. 1, p. 363, 2018.
[45] D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a förster resonance,” Physical review letters, vol. 113, no. 5, p. 053602, 2014.
[46] H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Singlephoton transistor mediated by interstate rydberg interactions,” Physical review letters, vol. 113, no. 5, p. 053601, 2014.
[47] W. Chen, K. M. Beck, R. Bücker, M. Gullans, M. D. Lukin, H. Tanji-Suzuki, and V. Vuletić, “All-optical switch and transistor gated by one stored photon,” Science, vol. 341, no. 6147, pp. 768–770, 2013.
[48] G. J. Milburn, “Quantum optical fredkin gate,” Physical Review Letters, vol. 62, no. 18, p. 2124, 1989.
[49] I. L. Chuang and Y. Yamamoto, “Simple quantum computer,” Physical Review A, vol. 52, no. 5, p. 3489, 1995.
[50] G. J. Milburn, “Photons as qubits,” Physica Scripta, vol. T137, p. 014003, 2009.
[51] S. E. Harris, J. Field, and A. Imamoğlu, “Nonlinear optical processes using electromagnetically induced transparency,” Physical Review Letters, vol. 64, no. 10, p. 1107, 1990.
[52] M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Physical review letters, vol. 84, no. 22, p. 5094, 2000.
[53] M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: Dark-state polaritons,” Physical Review A, vol. 65, no. 2, p. 022314, 2002.
[54] J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson, “Stopped light with storage times greater than one second using electromagnetically induced transparency in a solid,” Physical review letters, vol. 95, no. 6, p. 063601, 2005.
[55] S. Marcinkevičius, A. Gushterov, and J. Reithmaier, “Transient electromagnetically induced transparency in self-assembled quantum dots,” Applied Physics Letters, vol. 92, no. 4, p. 041113, 2008.
[56] P. Light, F. Benabid, F. Couny, M. Maric, and A. Luiten, “Electromagnetically induced transparency in rb-filled coated hollow-core photonic crystal fiber,” Optics letters, vol. 32, no. 10, pp. 1323–1325, 2007.
[57] J. Zhang, S. Xiao, C. Jeppesen, A. Kristensen, and N. A. Mortensen, “Electromagnetically induced transparency in metamaterials at near-infrared frequency,” Optics express, vol. 18, no. 16, pp. 17187–17192, 2010.
[58] G. Grynberg, A. Aspect, and C. Fabre, Introduction to quantum optics: from the semiclassical approach to quantized light. Cambridge university press, 2010.
[59] F. Bloch and A. Siegert, “Magnetic resonance for nonrotating fields,” Physical Review, vol. 57, no. 6, p. 522, 1940.
[60] P. R. Berman and V. S. Malinovsky, Principles of laser spectroscopy and quantum optics. Princeton University Press, 2010.
[61] V. Weisskopf and E. Wigner, “Berechnung der natürlichen linienbreite auf grund der diracschen lichttheorie,” in Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics, pp. 30–49, Springer, 1997.
[62] T. Y. Abi-Salloum, “Electromagnetically induced transparency and autler-townes splitting: Two similar but distinct phenomena in two categories of three-level atomic systems,” Physical Review A, vol. 81, no. 5, p. 053836, 2010.
[63] H.-C. Chen, “Theoretical study of nonlinear optics by quantum interference,” Master’s thesis, National Chen Kung Unuversity, 2011.
[64] J. S. Toll, “Causality and the dispersion relation: logical foundations,” Physical review, vol. 104, no. 6, p. 1760, 1956.
[65] R. d. L. Kronig, “On the theory of dispersion of x-rays,” Josa, vol. 12, no. 6, pp. 547–557, 1926.
[66] S. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Physical Review Letters, vol. 82, no. 23, p. 4611, 1999.
[67] D. A. Steck, “Rubidium 87 d line data,” 2001.
[68] Y.-W. Cheng, “Setup and optimization of rubidium magneto-optical trap,” Master’s thesis, National Chen Kung Unuversity, 2009.
[69] W. D. Phillips, “Nobel lecture: Laser cooling and trapping of neutral atoms,” Reviews of Modern Physics, vol. 70, no. 3, p. 721, 1998.
[70] M. Fox, Quantum optics: an introduction, vol. 15. OUP Oxford, 2006.
[71] C.-K. Chiu, “Studies on eit-based four-wave mixing at low-light levels,” Master’s thesis, National Chen Kung Unuversity, 2013.
[72] C.-S. Fang, “Studies on oscillation behavior of eit-based light storage and retrival,” Master’s thesis, National Chen Kung Unuversity, 2013.
[73] R. Lang, “Injection locking properties of a semiconductor laser,” IEEE Journal of Quantum Electronics, vol. 18, no. 6, pp. 976–983, 1982.
[74] W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous-force optical trap,” Physical review letters, vol. 70, no. 15, p. 2253, 1993.
[75] M. Fox, Quantum optics: an introduction, vol. 15. OUP Oxford, 2006.
[76] Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Physical Review Letters, vol. 75, no. 25, p. 4710, 1995.
[77] I. Fushman, D. Englund, A. Faraon, N. Stoltz, P. Petroff, and J. Vučković, “Controlled phase shifts with a single quantum dot,” science, vol. 320, no. 5877, pp. 769 772, 2008.
[78] H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, A. Y. Ite, and Y.-F. Chen,“Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Physical Review A, vol. 83, no. 4, p. 041804, 2011.
[79] A. Feizpour, M. Hallaji, G. Dmochowski, and A. M. Steinberg, “Observation of the nonlinear phase shift due to single post-selected photons,” Nature Physics, vol. 11, no. 11, p. 905, 2015.
[80] V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nature Photonics, vol. 7, no. 2, p. 138, 2013.
[81] M. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Physical Review Letters, vol. 84, no. 7, p. 1419, 2000.
[82] B.-W. Shiau, M.-C. Wu, C.-C. Lin, and Y.-C. Chen, “Low-light-level cross-phase modulation with double slow light pulses,” Physical review letters, vol. 106, no. 19, p. 193006, 2011.
[83] M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature, vol. 426, no. 6967, p. 638, 2003.
[84] Y.-H. Chen, M.-J. Lee, W. Hung, Y.-C. Chen, Y.-F. Chen, and A. Y. Ite, “Demonstration of the interaction between two stopped light pulses,” Physical review letters, vol. 108, no. 17, p. 173603, 2012.
[85] M. Mücke, E. Figueroa, J. Bochmann, C. Hahn, K. Murr, S. Ritter, C. J. Villas-Boas, and G. Rempe,“Electromagnetically induced transparency with single atoms in a cavity,” Nature, vol. 465, no. 7299, p. 755, 2010.
[86] Y. Zhu, “Large kerr nonlinearities on cavity-atom polaritons,” Optics letters, vol. 35, no. 3, pp. 303–305, 2010.
[87] I. Friedler, D. Petrosyan, M. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Physical Review A, vol. 72, no. 4, p. 043803, 2005.
[88] A. Mohapatra, T. Jackson, and C. Adams, “Coherent optical detection of highly excited rydberg states using electromagnetically induced transparency,” Physical review letters, vol. 98, no. 11, p. 113003, 2007.
[89] J. D. Pritchard, D. Maxwell, A. Gauguet, K. J. Weatherill, M. Jones, and C. S. Adams, “Cooperative atom-light interaction in a blockaded rydberg ensemble,” Physical review letters, vol. 105, no. 19, p. 193603, 2010.
[90] B. He, A. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent rydberg atomic ensemble,” Physical review letters, vol. 112, no. 13, p. 133606, 2014.
[91] S. Baur, D. Tiarks, G. Rempe, and S. Dürr, “Single-photon switch based on rydberg blockade,” Physical review letters, vol. 112, no. 7, p. 073901, 2014.
[92] K. M. Beck, M. Hosseini, Y. Duan, and V. Vuletić, “Large conditional single-photon cross-phase modulation,” Proceedings of the National Academy of Sciences, vol. 113, no. 35, pp. 9740–9744, 2016.
[93] D. Tiarks, S. Schmidt, G. Rempe, and S. Dürr, “Optical phase shift created with a single-photon pulse,” Science Advances, vol. 2, no. 4, p. e1600036, 2016.
[94] Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, A. Y. Ite, Y.-C. Chen, and Y.-F. Chen, “Large cross-phase modulations at the few-photon level,” Physical review letters, vol. 117, no. 20, p. 203601, 2016.
[95] C.-K. Chiu, Y.-H. Chen, Y.-C. Chen, A. Y. Ite, Y.-C. Chen, and Y.-F. Chen, “Lowlight-level four-wave mixing by quantum interference,” Physical Review A, vol. 89, no. 2, p. 023839, 2014.
[96] H.-Y. Lo, P.-C. Su, Y.-W. Cheng, P.-I. Wu, and Y.-F. Chen, “Femtowatt-light-level phase measurement of slow light pulses via beat-note interferometer,” Optics express, vol. 18, no. 17, pp. 18498–18505, 2010.
[97] H. J. Kimble, “The quantum internet,” Nature, vol. 453, no. 7198, p. 1023, 2008.
[98] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for the road ahead,” Science, vol. 362, no. 6412, p. eaam9288, 2018.
[99] J. James, Fiber Optic Cable: A Light Guide. ABC Teletraining Inc., 1991.
[100] R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nature photonics, vol. 3, no. 12, p. 696, 2009.
[101] W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature, vol. 299, no. 5886, p. 802, 1982.
[102] P. Kumar, “Quantum frequency conversion,” Optics letters, vol. 15, no. 24, pp. 1476–1478, 1990.
[103] J. Huang and P. Kumar, “Observation of quantum frequency conversion,” Physical review letters, vol. 68, no. 14, p. 2153, 1992.
[104] S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature, vol. 437, no. 7055, p. 116, 2005.
[105] S. Ates, I. Agha, A. Gulinatti, I. Rech, M. T. Rakher, A. Badolato, and K. Srinivasan, “Two-photon interference using background-free quantum frequency conversion of single photons emitted by an inas quantum dot,” Physical review letters, vol. 109, no. 14, p. 147405, 2012.
[106] S. Zaske, A. Lenhard, C. A. Keßler, J. Kettler, C. Hepp, C. Arend, R. Albrecht, W.-M. Schulz, M. Jetter, P. Michler, et al., “Visible-to-telecom quantum frequency conversion of light from a single quantum emitter,” Physical review letters, vol. 109, no. 14, p. 147404, 2012.
[107] M. T. Rakher, L. Ma, O. Slattery, X. Tang, and K. Srinivasan, “Quantum transduction of telecommunications-band single photons from a quantum dot by frequency upconversion,” Nature Photonics, vol. 4, no. 11, p. 786, 2010.
[108] M. Allgaier, V. Ansari, L. Sansoni, C. Eigner, V. Quiring, R. Ricken, G. Harder, B. Brecht, and C. Silberhorn, “Highly efficient frequency conversion with bandwidth compression of quantum light,” Nature communications, vol. 8, p. 14288, 2017.
[109] K. A. Fisher, D. G. England, J.-P. W. MacLean, P. J. Bustard, K. J. Resch, and B. J. Sussman, “Frequency and bandwidth conversion of single photons in a room-temperature diamond quantum memory,” Nature communications, vol. 7, p. 11200, 2016.
[110] A. S. Clark, S. Shahnia, M. J. Collins, C. Xiong, and B. J. Eggleton, “High-efficiency frequency conversion in the single-photon regime,” Optics letters, vol. 38, no. 6, pp. 947–949, 2013.
[111] Q. Li, M. Davanço, and K. Srinivasan, “Efficient and low-noise single-photonlevel frequency conversion interfaces using silicon nanophotonics,” Nature Photonics, vol. 10, no. 6, p. 406, 2016.
[112] J. L. O’brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nature Photonics, vol. 3, no. 12, p. 687, 2009.
[113] M. Jain, H. Xia, G. Yin, A. Merriam, and S. Harris, “Efficient nonlinear frequency conversion with maximal atomic coherence,” Physical review letters, vol. 77, no. 21, p. 4326, 1996.
[114] M. Lukin, P. Hemmer, M. Löffler, and M. Scully, “Resonant enhancement of parametric processes via radiative interference and induced coherence,” Physical review letters, vol. 81, no. 13, p. 2675, 1998.
[115] A. J. Merriam, S. Sharpe, M. Shverdin, D. Manuszak, G. Yin, and S. Harris, “Efficient nonlinear frequency conversion in an all-resonant double- system,” Physical review letters, vol. 84, no. 23, p. 5308, 2000.
[116] C. McCormick, V. Boyer, E. Arimondo, and P. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Optics letters, vol. 32, no. 2, pp. 178-180, 2007.
[117] Y. Du, Y. Zhang, C. Zuo, C. Li, Z. Nie, H. Zheng, M. Shi, R. Wang, J. Song, K. Lu, et al., “Controlling four-wave mixing and six-wave mixing in a multi-zeeman-sublevel atomic system with electromagnetically induced transparency,” Physical Review A, vol. 79, no. 6, p. 063839, 2009.
[118] Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Temporal and spatial interference between four-wave mixing and six-wave mixing channels,” Physical review letters, vol. 102, no. 1, p. 013601, 2009.
[119] N. B. Phillips, A. V. Gorshkov, and I. Novikova, “Slow light propagation and amplification via electromagnetically induced transparency and four-wave mixing in an optically dense atomic vapor,” Journal of Modern Optics, vol. 56, no. 18-19, pp. 1916–1925, 2009.
[120] N. B. Phillips, A. V. Gorshkov, and I. Novikova, “Light storage in an optically thick atomic ensemble under conditions of electromagnetically induced transparency and four-wave mixing,” Physical Review A, vol. 83, no. 6, p. 063823, 2011.
[121] N. B. Phillips, I. Novikova, E. E. Mikhailov, D. Budker, and S. Rochester, “Controllable steep dispersion with gain in a four-level n-scheme with four-wave mixing,” Journal of Modern Optics, vol. 60, no. 1, pp. 64–72, 2013.
[122] Z. Zhang, F. Wen, J. Che, D. Zhang, C. Li, Y. Zhang, and M. Xiao, “Dressed gain from the parametrically amplified four-wave mixing process in an atomic vapor,” Scientific reports, vol. 5, p. 15058, 2015.
[123] Z. Zhang, H. Zheng, X. Yao, Y. Tian, J. Che, X. Wang, D. Zhu, Y. Zhang, and M. Xiao, “Phase modulation in rydberg dressed multi-wave mixing processes,” Scientific reports, vol. 5, p. 10462, 2015.
[124] D. T. Kutzke, O. Wolfe, S. M. Rochester, D. Budker, I. Novikova, and E. E. Mikhailov, “Tailorable dispersion in a four-wave mixing laser,” Optics letters, vol. 42, no. 14, pp. 2846–2849, 2017.
[125] A. Radnaev, Y. Dudin, R. Zhao, H. Jen, S. Jenkins, A. Kuzmich, and T. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nature Physics, vol. 6, no. 11, p. 894, 2010.
[126] K.-F. Chang, T.-P. Wang, C.-Y. Chen, Y.-H. Chen, Y.-S. Wang, Y.-F. Chen, Y.-C. Chen, and I. A. Yu, “Low-loss high-fidelity frequency-mode hadamard gates based on electromagnetically induced transparency,” arXiv preprint arXiv:1907.03393, 2019.
[127] M. Payne and L. Deng, “Consequences of induced transparency in a double-lambda scheme: Destructive interference in four-wave mixing,” Physical Review A, vol. 65, no. 6, p. 063806, 2002.
[128] J.-Y. Juo, J.-K. Lin, C.-Y. Cheng, Z.-Y. Liu, A. Y. Ite, and Y.-F. Chen, “Demonstration of spatial-light-modulation-based four-wave mixing in cold atoms,” Physical Review A, vol. 97, no. 5, p. 053815, 2018.
[129] H. Kang, G. Hernandez, J. Zhang, and Y. Zhu, “Backward four-wave mixing in a fourlevel medium with electromagnetically induced transparency,” JOSA B, vol. 23, no. 4, pp. 718–722, 2006.
[130] H. Kang, G. Hernandez, and Y. Zhu, “Resonant four-wave mixing with slow light,” Physical Review A, vol. 70, no. 6, p. 061804, 2004.
[131] Z.-Y. Liu, J.-T. Xiao, J.-K. Lin, J.-J. Wu, J.-Y. Juo, C.-Y. Cheng, and Y.-F. Chen, “Highefficiency backward four-wave mixing by quantum interference,” Scientific reports, vol. 7, no. 1, p. 15796, 2017.
[132] S. P. Tewari and G. Agarwal, “Control of phase matching and nonlinear generation in dense media by resonant fields,” Physical review letters, vol. 56, no. 17, p. 1811, 1986.
[133] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature, vol. 397, no. 6720, p. 594, 1999.
[134] X. Qiang, X. Zhou, J. Wang, C. M. Wilkes, T. Loke, S. O’Gara, L. Kling, G. D. Marshall, R. Santagati, T. C. Ralph, et al., “Large-scale silicon quantum photonics implementing arbitrary two-qubit processing,” Nature photonics, vol. 12, no. 9, p. 534, 2018.
[135] T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Physical Review Letters, vol. 84, no. 20, p. 4729, 2000.
[136] A. Beveratos, R. Brouri, T. Gacoin, A. Villing, J.-P. Poizat, and P. Grangier, “Single photon quantum cryptography,” Physical review letters, vol. 89, no. 18, p. 187901, 2002.
[137] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature, vol. 390, no. 6660, p. 575, 1997.
[138] X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature, vol. 518, no. 7540, p. 516, 2015.
[139] C. A. Kocher and E. D. Commins, “Polarization correlation of photons emitted in an atomic cascade,” Physical Review Letters, vol. 18, no. 15, p. 575, 1967.
[140] S. Harris, M. Oshman, and R. Byer, “Observation of tunable optical parametric fluorescence,” Physical Review Letters, vol. 18, no. 18, p. 732, 1967.
[141] D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Physical Review Letters, vol. 25, no. 2, p. 84, 1970.
[142] X.-H. Bao, Y. Qian, J. Yang, H. Zhang, Z.-B. Chen, T. Yang, and J.-W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Physical review letters, vol. 101, no. 19, p. 190501, 2008.
[143] M. Scholz, L. Koch, and O. Benson, “Statistics of narrow-band single photons for quantum memories generated by ultrabright cavity-enhanced parametric downconversion,” Physical review letters, vol. 102, no. 6, p. 063603, 2009.
[144] M. Scholz, L. Koch, R. Ullmann, and O. Benson, “Single-mode operation of a high-brightness narrow-band single-photon source,” Applied Physics Letters, vol. 94, no. 20, p. 201105, 2009.
[145] J. Fekete, D. Rieländer, M. Cristiani, and H. de Riedmatten, “Ultranarrow-band photon-pair source compatible with solid state quantum memories and telecommunication networks,” Physical review letters, vol. 110, no. 22, p. 220502, 2013.
[146] M. Rambach, A. Nikolova, T. J. Weinhold, and A. G. White, “Sub-megahertz linewidth single photon source,” APL Photonics, vol. 1, no. 9, p. 096101, 2016.
[147] M. Rambach, A.Nikolova, T. J.Weinhold, andA.G.White, “Erratum“: sub-megahertz linewidth single photon source”[apl photonics 1, 096101 (2016)],” Apl Photonics, vol. 2, no. 11, p. 119901, 2017.
[148] P.-J. Tsai and Y.-C. Chen, “Ultrabright, narrow-band photon-pair source for atomic quantum memories,” Quantum Science and Technology, vol. 3, no. 3, p. 034005, 2018.
[149] C.-S. Chuu, G. Yin, and S. Harris, “A miniature ultrabright source of temporally long, narrowband biphotons,” Applied Physics Letters, vol. 101, no. 5, p. 051108, 2012.
[150] V. Balić, D. A. Braje, P. Kolchin, G. Yin, and S. E. Harris, “Generation of paired photons with controllable waveforms,” Physical review letters, vol. 94, no. 18, p. 183601, 2005.
[151] P. Kolchin, S. Du, C. Belthangady, G. Yin, and S. Harris, “Generation of narrowbandwidth paired photons: use of a single driving laser,” Physical review letters, vol. 97, no. 11, p. 113602, 2006.
[152] P. Kolchin, “Electromagnetically-induced transparency-based paired photon generation,” Physical Review A, vol. 75, no. 3, p. 033814, 2007.
[153] S. Du, P. Kolchin, C. Belthangady, G. Yin, and S. Harris, “Subnatural linewidth biphotons with controllable temporal length,” Physical review letters, vol. 100, no. 18, p. 183603, 2008.
[154] S. Du, J. Wen, and M. H. Rubin, “Narrowband biphoton generation near atomic resonance,” JOSA B, vol. 25, no. 12, pp. C98–C108, 2008.
[155] L. Zhao, X. Guo, C. Liu, Y. Sun, M. Loy, and S. Du, “Photon pairs with coherence time exceeding 1 s,” Optica, vol. 1, no. 2, pp. 84–88, 2014.
[156] Z. Han, P. Qian, L. Zhou, J. Chen, and W. Zhang, “Coherence time limit of the biphotons generated in a dense cold atom cloud,” Scientific reports, vol. 5, p. 9126, 2015.
[157] L. Zhao, Y. Su, and S. Du, “Narrowband biphoton generation in the group delay regime,” Physical Review A, vol. 93, no. 3, p. 033815, 2016.
[158] H. Yan, S. Zhang, J. Chen, M. M. Loy, G. K. Wong, and S. Du, “Generation of narrowband hyperentangled nondegenerate paired photons,” Physical review letters, vol. 106, no. 3, p. 033601, 2011.
[159] K. Liao, H. Yan, J. He, S. Du, Z.-M. Zhang, and S.-L. Zhu, “Subnatural-linewidth polarization-entangled photon pairs with controllable temporal length,” Physical review letters, vol. 112, no. 24, p. 243602, 2014.
[160] T.-M. Zhao, Y. S. Ihn, and Y.-H. Kim, “Direct generation of narrow-band hyperentangled photons,” Physical review letters, vol. 122, no. 12, p. 123607, 2019.
[161] S.-Y. Lee and E. J. Heller, “Time-dependent theory of raman scattering,” The Journal of chemical physics, vol. 71, no. 12, pp. 4777–4788, 1979.
[162] C.-H. Wu, T.-Y. Wu, Y.-C. Yeh, P.-H. Liu, C.-H. Chang, C.-K. Liu, T. Cheng, and C.-S. Chuu, “Bright single photons for light-matter interaction,” Physical Review A, vol. 96, no. 2, p. 023811, 2017.
[163] L. Mandel and E. Wolf, Optical coherence and quantum optics. Cambridge university press, 1995.
[164] W. H. Louisell and W. H. Louisell, Quantum statistical properties of radiation, vol. 7. Wiley New York, 1973.
[165] J. F. Clauser and M. A. Horne, “Experimental consequences of objective local theories,” Physical review D, vol. 10, no. 2, p. 526, 1974.
[166] M. H. Rubin, D. N. Klyshko, Y. Shih, and A. Sergienko, “Theory of two-photon entanglement in type-ii optical parametric down-conversion,” Physical Review A, vol. 50, no. 6, p. 5122, 1994.
[167] S. Du, C. Belthangady, P. Kolchin, G. Yin, and S. Harris, “Observation of optical precursors at the biphoton level,” Optics letters, vol. 33, no. 18, pp. 2149–2151, 2008.
[168] A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Reviews of modern physics, vol. 81, no. 1, p. 299, 2009.
[169] S. Chu and S. Wong, “Linear pulse propagation in an absorbing medium,” Physical review letters, vol. 48, no. 11, p. 738, 1982.
[170] J. Marangos, “Faster than a speeding photon,” Nature, vol. 406, no. 6793, p. 243, 2000.
[171] M. D. Stenner, D. J. Gauthier, and M. A. Neifeld, “The speed of information in a ‘fast-light’optical medium,” Nature, vol. 425, no. 6959, p. 695, 2003.
[172] R. Kubo, “The fluctuation-dissipation theorem,” Reports on progress in physics, vol. 29, no. 1, p. 255, 1966.
[173] J. Garrison and R. Chiao, Quantum optics. Oxford University Press, 2008.
[174] C. W. Gardiner et al., Handbook of stochastic methods, vol. 3. springer Berlin, 1985.
[175] D. A. Braje, V. Balić, S. Goda, G. Yin, and S. Harris, “Frequency mixing using electromagnetically induced transparency in cold atoms,” Physical review letters, vol. 93, no. 18, p. 183601, 2004.