Optical quantum states based on hot atomic ensembles and their applications

Kai Zhang; Shengshuai Liu; Yingxuan Chen; Xutong Wang; Jietai Jing

doi:10.3788/PI.2022.R06

[1] J.-W. Pan et al. Multiphoton entanglement and interferometry. Rev. Mod. Phys., 84, 777(2012).

[2] S. L. Braunstein, P. van Loock. Quantum information with continuous variables. Rev. Mod. Phys., 77, 513(2005).

[3] P. G. Kwiat et al. New high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett., 75, 4337(1995).

[4] A. MacRae et al. Tomography of a high-purity narrowband photon from a transient atomic collective excitation. Phys. Rev. Lett., 109, 033601(2012).

[5] Z. Qin et al. Complete temporal characterization of a single photon. Light Sci. Appl., 4, e298(2015).

[6] M. H. Devoret, R. J. Schoelkopf. Superconducting circuits for quantum information: an outlook. Science, 339, 1169(2013).

[7] M. Kjaergaard et al. Superconducting qubits: current state of play. Annu. Rev. Condens. Matter Phys., 11, 369(2020).

[8] C. Silberhorn et al. Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the Kerr nonlinearity in an optical fiber. Phys. Rev. Lett., 86, 4267(2001).

[9] J. Jing et al. Experimental demonstration of tripartite entanglement and controlled dense coding for continuous variables. Phys. Rev. Lett., 90, 167903(2003).

[10] C. Fabre, N. Treps. Modes and states in quantum optics. Rev. Mod. Phys., 92, 035005(2020).

[11] C. Eichler et al. Observation of two-mode squeezing in the microwave frequency domain. Phys. Rev. Lett., 107, 113601(2011).

[12] H. Zhong et al. 12-Photon entanglement and scalable scattershot boson sampling with optimal entangled-photon pairs from parametric down-conversion. Phys. Rev. Lett., 121, 250505(2018).

[13] Y. Huang et al. Experimental generation of an eight-photon Greenberger-Horne-Zeilinger state. Nat. Commun., 2, 546(2011).

[14] C. Reimer et al. Generation of multiphoton entangled quantum states by means of integrated frequency combs. Science, 351, 1176(2016).

[15] A. Mair et al. Entanglement of the orbital angular momentum states of photons. Nature, 412, 313(2001).

[16] A. C. Data et al. Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities. Nat. Phys., 7, 677(2011).

[17] M. Malik et al. Multi-photon entanglement in high dimensions. Nat. Photon., 10, 248(2016).

[18] D. Ding et al. High-dimensional entanglement between distant atomic-ensemble memories. Light Sci. Appl., 5, e16157(2016).

[19] M. Kues et al. On-chip generation of high-dimensional entangled quantum states and their coherent control. Nature, 546, 622(2017).

[20] Y. Zhang et al. Engineering two-photon high-dimensional states through quantum interference. Sci. Adv., 2, e1501165(2016).

[21] X. Wang et al. 18-Qubit entanglement with six photons’ three degrees of freedom. Phys. Rev. Lett., 120, 260502(2018).

[22] W. Zhang et al. Experimental realization of entanglement in multiple degrees of freedom between two quantum memories. Nat. Commun., 7, 13514(2016).

[23] C. Reimer et al. High-dimensional one-way quantum processing implemented on d-level cluster states. Nat. Phys., 15, 148(2019).

[24] S. Liao et al. Satellite-to-ground quantum key distribution. Nature, 549, 43(2017).

[25] J. Yin et al. Satellite-based entanglement distribution over 1200 kilometers. Science, 356, 1140(2017).

[26] J. Yin et al. Entanglement-based secure quantum cryptography over 1,120 kilometers. Nature, 582, 501(2020).

[27] W. Asavanant et al. Generation of time-domain-multiplexed two-dimensional cluster state. Science, 366, 373(2019).

[28] M. V. Larsen et al. Deterministic generation of a two-dimensional cluster state. Science, 366, 369(2019).

[29] X. Pan et al. Orbital-angular-momentum multiplexed continuous-variable entanglement from four-wave mixing in hot atomic vapor. Phys. Rev. Lett., 123, 070506(2019).

[30] U. L. Andersen et al. Hybrid discrete- and continuous-variable quantum information. Nat. Phys., 11, 713(2015).

[31] R. E. Slusher et al. Observation of squeezed states generated by four-wave mixing in an optical cavity. Phys. Rev. Lett., 55, 2409(1985).

[32] L. A. Wu et al. Generation of squeezed states by parametric down conversion. Phys. Rev. Lett., 57, 2520(1986).

[33] R. M. Shelby et al. Broad-band parametric deamplification of quantum noise in an optical fiber. Phys. Rev. Lett., 57, 691(1986).

[34] H. Vahlbruch et al. Detection of 15 dB squeezed states of light and their application for the absolute calibration of photoelectric quantum efficiency. Phys. Rev. Lett., 117, 110801(2016).

[35] S. Shi et al. Detection and perfect fitting of 13.2 dB squeezed vacuum states by considering green-light-induced infrared absorption. Opt. Lett., 43, 5411(2018).

[36] The LIGO. A gravitational wave observatory operating beyond the quantum shot-noise limit. Nat. Phys., 7, 962(2011).

[37] The LIGO. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light. Nat. Photon., 7, 613(2013).

[38] H. Grote et al. First long-term application of squeezed states of light in a gravitational-wave observatory. Phys. Rev. Lett., 110, 181101(2013).

[39] N. Treps et al. A quantum laser pointer. Science, 301, 940(2003).

[40] R. C. Pooser, B. Lawie. Ultrasensitive measurement of microcantilever displacement below the shot-noise limit. Optica, 2, 393(2015).

[41] K. Liu et al. Squeezing-enhanced rotating-angle measurement beyond the quantum limit. Appl. Phys. Lett., 113, 261103(2018).

[42] B. Lamine, C. Fabre, N. Treps. Quantum improvement of time transfer between remote clocks. Phys. Rev. Lett., 101, 123601(2008).

[43] A. Einstein, B. Podolsky, N. Rosen. Can quantum mechanical description of physical reality be considered complete. Phys. Rev., 47, 777(1935).

[44] Z. Y. Ou et al. Realization of the Einstein-Podolsky-Rosen paradox for continuous-variables. Phys. Rev. Lett., 68, 3663(1992).

[45] M. D. Reid. Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification. Phys. Rev. A, 40, 913(1989).

[46] L. M. Duan et al. Inseparability criterion for continuous variable systems. Phys. Rev. Lett., 84, 2722(2000).

[47] R. Simon. Peres-horodecki separability criterion for continuous variable systems. Phys. Rev. Lett., 84, 2726(2000).

[48] Y. Zhang et al. Experimental generation of bright two-mode quadrature squeezed light from a narrow-band nondegenerate optical parametric amplifier. Phys. Rev. A, 62, 023813(2000).

[49] P. van Loock, S. L. Braunstein. Multipartite entanglement for continuous variables: a quantum teleportation network. Phys. Rev. Lett., 84, 3482(2000).

[50] P. van Loock, S. L. Braunstein. Telecloning of continuous quantum variables. Phys. Rev. Lett., 87, 247901(2001).

[51] T. Aoki et al. Experimental creation of a fully inseparable tripartite continuous-variable state. Phys. Rev. Lett., 91, 080404(2003).

[52] H. Yonezawa, T. Aoki, A. Furusawa. Demonstration of a quantum teleportation network for continuous variables. Nature, 431, 430(2004).

[53] S. Koike et al. Demonstration of quantum telecloning of optical coherent states. Phys. Rev. Lett., 96, 060504(2006).

[54] X. Su et al. Experimental preparation of quadripartite cluster and Greenberger-Horne-Zeilinger entangled states for continuous variables. Phys. Rev. Lett., 98, 070502(2007).

[55] X. Su et al. Experimental preparation of eight-partite cluster state for photonic qumodes. Opt. Lett., 37, 5178(2012).

[56] F. A. S. Barbosa et al. Robustness of bipartite Gaussian entangled beams propagating in lossy channels. Nat. Photon., 4, 858(2010).

[57] A. S. Villar et al. Generation of bright two-color continuous variable entanglement. Phys. Rev. Lett., 95, 243603(2005).

[58] F. A. S. Barbosa et al. Hexapartite entanglement in an above-threshold optical parametric oscillator. Phys. Rev. Lett., 121, 073601(2018).

[59] L. F. Muñoz-Martínez et al. Exploring six modes of an optical parametric oscillator. Phys. Rev. A, 98, 023823(2018).

[60] S. Yokoyama et al. Ultra-large-scale continuous-variable cluster states multiplexed in the time domain. Nat. Photon., 7, 982(2013).

[61] S. Armstrong et al. Programmable multimode quantum networks. Nat. Commun., 3, 1026(2012).

[62] M. Chen, N. C. Menicucci, O. Pfister. Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb. Phys. Rev. Lett., 112, 120505(2014).

[63] Z. Yang et al. A squeezed quantum microcomb on a chip. Nat. Commun., 12, 4781(2021).

[64] J. Roslund et al. Wavelength-multiplexed quantum networks with ultrafast frequency combs. Nat. Photon., 8, 109(2014).

[65] Y. Cai et al. Multimode entanglement in reconfigurable graph states using optical frequency combs. Nat. Commun., 8, 15645(2017).

[66] S. L. Braunstein, H. J. Kimble. Teleportation of continuous quantum variables. Phys. Rev. Lett., 80, 869(1998).

[67] A. Furusawa et al. Unconditional quantum teleportation. Science, 282, 706(1998).

[68] N. Lee et al. Teleportation of nonclassical wave packets of light. Science, 332, 330(2011).

[69] M. Huo et al. Deterministic quantum teleportation through fiber channels. Sci. Adv., 4, eaas9401(2018).

[70] X. Li et al. Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam. Phys. Rev. Lett., 88, 047904(2002).

[71] X. Jia et al. Experimental demonstration of unconditional entanglement swapping for continuous variables. Phys. Rev. Lett., 93, 250503(2004).

[72] C. F. McCormick et al. Strong relative intensity squeezing by four-wave mixing in rubidium vapor. Opt. Lett., 32, 178(2007).

[73] R. C. Pooser et al. Quantum correlated light beams from nondegenerate four-wave mixing in an atomic vapor: the D1 and D2 lines of ⁸⁵Rb and ⁸⁷Rb. Opt. Express, 17, 16722(2009).

[74] J. D. Swaim, R. T. Glasser. Squeezed-twin-beam generation in strongly absorbing media. Phys. Rev. A, 96, 033818(2017).

[75] M. Guo et al. Experimental investigation of high-frequency-difference twin beams in hot cesium atoms. Phys. Rev. A, 89, 033813(2014).

[76] R. Ma et al. Generating quantum correlated twin beams by four-wave mixing in hot cesium vapor. Phys. Rev. A, 96, 043843(2017).

[77] A. M. Marino, V. Boyer, P. D. Lett. Violation of the Cauchy-Schwarz inequality in the macroscopic regime. Phys. Rev. Lett., 100, 233601(2008).

[78] H. Vahlbruch et al. Observation of squeezed light with 10-dB quantum-noise reduction. Phys. Rev. Lett., 100, 033602(2008).

[79] D. Zhang et al. Enhanced intensity-difference squeezing via energy-level modulations in hot atomic media. Phys. Rev. A, 96, 043847(2017).

[80] Q. Glorieux et al. Generation of pulsed bipartite entanglement using four-wave mixing. New J. Phys., 14, 123024(2012).

[81] D. Budker, M. Romalis. Optical magnetometry. Nat. Phys., 3, 227(2007).

[82] A. M. Marino, P. D. Lett. Absolute calibration of photodiodes with bright twin beams. J. Mod. Opt., 58, 328(2011).

[83] C. F. McCormick et al. Strong low-frequency quantum correlations from a four-wave-mixing amplifier. Phys. Rev. A, 78, 043816(2008).

[84] M. C. Wu et al. Twin-beam intensity-difference squeezing below 10 Hz. Opt. Express, 27, 4769(2019).

[85] M. I. Kolobov. Quantum Imaging(2006).

[86] V. Boyer, A. M. Marino, P. D. Lett. Generation of spatially broadband twin beams for quantum imaging. Phys. Rev. Lett., 100, 143601(2008).

[87] M. W. Holtfrerich, A. M. Marino. Control of the size of the coherence area in entangled twin besms. Phys. Rev. A, 93, 063821(2016).

[88] C. S. Embrey et al. Observation of localized multi-spatial-mode quadrature squeezing. Phys. Rev. X, 5, 031004(2015).

[89] Q. Glorieux et al. Temporally multiplexed storage of images in a gradient echo memory. Opt. Express, 20, 12350(2012).

[90] D.-S. Ding et al. Image transfer through two sequential four-wave mixing processes in hot atomic vapor. Phys. Rev. A, 85, 053815(2012).

[91] J. B. Clark et al. Imaging using quantum noise properties of light. Opt. Express, 20, 17050(2012).

[92] E. Brambilla et al. High-sensitivity imaging with multi-mode twin beams. Phys. Rev. A, 77, 053807(2008).

[93] V. Boyer et al. Entangled images from four-wave mixing. Science, 321, 544(2008).

[94] B. J. Lawrie, R. C. Pooser. Toward real-time quantum imaging with a single pixel camera. Opt. Express, 21, 7549(2013).

[95] A. M. Marino et al. Tunable delay of Einstein–Podolsky–Rosen entanglement. Nature, 457, 859(2009).

[96] V. Boyer et al. Ultraslow propagation of matched pulses by four-wave mixing in an atomic vapor. Phys. Rev. Lett., 99, 143601(2007).

[97] R. T. Glasser, U. Vogl, P. D. Lett. Stimulated generation of superluminal light pulses via four-wave mixing. Phys. Rev. Lett., 108, 173902(2012).

[98] U. Vogl et al. Advanced quantum noise correlations. New J. Phys., 16, 013011(2014).

[99] J. B. Clark et al. Quantum mutual information of an entangled state propagating through a fast-light medium. Nat. Photon., 8, 515(2014).

[100] N. V. Corzo et al. Noiseless optical amplifier operating on hundreds of spatial modes. Phys. Rev. Lett., 109, 043602(2012).

[101] N. Corzo et al. Multi-spatial-mode single-beam quadrature squeezed states of light from four-wave mixing in hot rubidium vapor. Opt. Express, 19, 21358(2011).

[102] S. Liu, Y. Lou, J. Jing. Interference-induced quantum squeezing enhancement in a two-beam phase-sensitive amplifier. Phys. Rev. Lett., 123, 113602(2019).

[103] S. Liu, Y. Lou, J. Jing. Phase manipulated two-mode entangled state from a phase-sensitive amplifier. Opt. Express, 29, 38971(2021).

[104] O. Danaci, C. Rios, R. T. Glasser. All-optical mode conversion via spatially multimode four-wave mixing. New J. Phys., 18, 073032(2016).

[105] S. Kim, A. M. Marino. Atomic resonant single-mode squeezed light from four-wave mixing through feedforward. Opt. Lett., 44, 4630(2019).

[106] S. Kim, A. M. Marino. Generation of ⁸⁷Rb resonant bright two-mode squeezed light with four-wave mixing. Opt. Express, 26, 33366(2018).

[107] P. Gupta et al. Effect of imperfect homodyne visibility on multi-spatial-mode two-mode squeezing measurements. Opt. Express, 28, 652(2020).

[108] U. Vogl et al. Experimental characterization of Gaussian quantum discord generated by four-wave mixing. Phys. Rev. A, 87, 010101(R)(2013).

[109] A. Kumar, A. M. Marino. Spatial squeezing in bright twin beams generated with four-wave mixing: constraints on characterization with an electron-multiplying charge-coupled-device camera. Phys. Rev. A, 100, 063828(2019).

[110] A. Kumar, G. Nirala, A. M. Marino. Einstein–Podolsky–Rosen paradox with position–momentum entangled macroscopic twin beams. Quantum Sci. Technol., 6, 045016(2021).

[111] A. Kumar, H. Nunley, A. M. Marino. Comparison of coherence-area measurement techniques for bright entangled twin beams. Phys. Rev. A, 98, 043853(2018).

[112] B. J. Lawrie et al. Robust and compact entanglement generation from diode-laser-pumped four-wave mixing. Appl. Phys. Lett., 108, 151107(2016).

[113] A. M. Guerrero et al. Quantum noise correlations of an optical parametric oscillator based on a nondegenerate four wave mixing process in hot alkali atoms. Phys. Rev. Lett., 125, 083601(2020).

[114] C. M. Caves. Quantum-mechanical noise in an interferometer. Phys. Rev. D, 23, 1693(1981).

[115] F. Wolfgramm et al. Squeezed-light optical magnetometry. Phys. Rev. Lett., 105, 053601(2010).

[116] B. Li et al. Quantum enhanced optomechanical magnetometry. Optica, 5, 850(2018).

[117] B. J. Lawrie et al. Quantum sensing with squeezed light. ACS Photonics, 6, 1307(2019).

[118] W. Fan, B. J. Lawrie, R. C. Pooser. Quantum plasmonic sensing. Phys. Rev. A, 92, 053812(2015).

[119] R. C. Pooser, B. J. Lawrie. Plasmonic trace sensing below the photon shot noise limit. ACS Photonics, 3, 8(2016).

[120] M. Dowran et al. Quantum-enhanced plasmonic sensing. Optica, 5, 628(2018).

[121] B. J. Lawrie, P. G. Evans, R. C. Pooser. Extraordinary optical transmission of multimode quantum correlations via localized surface plasmons. Phys. Rev. Lett., 110, 156802(2013).

[122] M. W. Holtfreric et al. Toward quantum plasmonic networks. Optica, 3, 985(2016).

[123] M. C. Wu et al. Two-beam coupling in the production of quantum correlated images by four-wave mixing. Opt. Express, 29, 16665(2021).

[124] J. Jing et al. Realization of a nonlinear interferometer with parametric amplifiers. Appl. Phys. Lett., 99, 011110(2011).

[125] A. M. Marino, N. V. Corzo Trejo, P. D. Lett. Effect of losses on the performance of an SU(1,1) interferometer. Phys. Rev. A, 86, 023844(2012).

[126] F. Hudelist et al. Quantum metrology with parametric amplifier-based photon correlation interferometers. Nat. Commun., 5, 3049(2014).

[127] B. E. Anderson et al. Phase sensing beyond the standard quantum limit with a variation on the SU(1,1) interferometer. Optica, 4, 752(2017).

[128] R. C. Pooser et al. Truncated nonlinear interferometry for quantum-enhanced atomic force microscopy. Phys. Rev. Lett., 124, 230504(2020).

[129] J. M. Lukens, N. A. Peters, R. C. Pooser. Naturally stable Sagnac–Michelson nonlinear interferometer. Opt. Lett., 41, 5438(2016).

[130] X. Guo et al. Distributed quantum sensing in a continuous-variable entangled network. Nat. Phys., 16, 281(2020).

[131] Y. Xia et al. Demonstration of a reconfigurable entangled radio-frequency photonic sensor network. Phys. Rev. Lett., 124, 150502(2020).

[132] Y. Zhou et al. Quantum secret sharing among four players using multipartite bound entanglement of an optical field. Phys. Rev. Lett., 121, 150502(2018).

[133] A. S. Coelho et al. Three-color entanglement. Science, 326, 823(2009).

[134] X. Jia et al. Experimental realization of three-color entanglement at optical fiber communication and atomic storage wavelengths. Phys. Rev. Lett., 109, 253604(2012).

[135] Z. Qin et al. Experimental generation of multiple quantum correlated beams from hot rubidium vapor. Phys. Rev. Lett., 113, 023602(2014).

[136] L. Cao et al. Experimental generation of quadruple quantum-correlated beams from hot rubidium vapor by cascaded four-wave mixing using spatial multiplexing. Phys. Rev. A, 95, 023803(2017).

[137] H. Wang, C. Fabre, J. Jing. Single-step fabrication of scalable multimode quantum resources using four-wave mixing with a spatially structured pump. Phys. Rev. A, 95, 051802(2017).

[138] K. Zhang et al. Reconfigurable hexapartite entanglement by spatially multiplexed four-wave mixing processes. Phys. Rev. Lett., 124, 090501(2020).

[139] S. Liu, H. Wang, J. Jing. Two-beam pumped cascaded four-wave-mixing process for producing multiple-beam quantum correlation. Phys. Rev. A, 97, 043846(2018).

[140] S. Liu, Y. Lou, J. Jing. Experimental characterization of multiple quantum correlated beams in two-beam pumped cascaded four-wave mixing process. Opt. Express, 27, 37999(2019).

[141] J. D. Swaim et al. Multimode four-wave mixing with a spatially structured pump. Opt. Lett., 43, 2716(2018).

[142] E. M. Knutson et al. Phase-sensitive amplification via multi-phase-matched four-wave mixing. Opt. Express, 28, 22748(2020).

[143] E. M. Knutson et al. Optimal mode configuration for multiple phase-matched four-wave-mixing processes. Phys. Rev. A, 98, 013828(2018).

[144] M. E. J. Friese et al. Optical angular-momentum transfer to trapped absorbing particles. Phys. Rev. A, 54, 1593(1996).

[145] G. Gibson et al. Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express, 12, 5448(2004).

[146] M. F. Andersen et al. Quantized rotation of atoms from photons with orbital angular momentum. Phys. Rev. Lett., 97, 170406(2006).

[147] L. Allen et al. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A, 45, 8185(1992).

[148] R. Fickler et al. Quantum entanglement of high angular momenta. Science, 338, 640(2012).

[149] A. M. Marino et al. Delocalized correlations in twin light beams with orbital angular momentum. Phys. Rev. Lett., 101, 093602(2008).

[150] J. Wang et al. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat. Photon., 6, 488(2012).

[151] N. Bozinovic et al. Terabit-scale orbital angular momentum mode division multiplexing in fibers. Science, 340, 1545(2013).

[152] A. Trichili et al. Encoding information using Laguerre Gaussian modes over free space turbulence media. Opt. Lett., 41, 3086(2016).

[153] R. C. Devlin et al. Arbitrary spin-to–orbital angular momentum conversion of light. Science, 358, 896(2017).

[154] X. Wang et al. Quantum teleportation of multiple degrees of freedom of a single photon. Nature, 518, 516(2015).

[155] T.-M. Zhao, Y. S. Ihn, Y.-H. Kim. Direct generation of narrow-band hyperentangled photons. Phys. Rev. Lett., 122, 123607(2019).

[156] Y. Zhang et al. Simultaneous entanglement swapping of multiple orbital angular momentum states of light. Nat. Commun., 8, 632(2017).

[157] A. Vaziri, G. Weihs, A. Zeilinger. Experimental two-photon, three-dimensional entanglement for quantum communication. Phys. Rev. Lett., 89, 240401(2002).

[158] S. Franke-Arnold et al. Uncertainty principle for angular position and angular momentum. New J. Phys., 6, 103(2004).

[159] S. S. R. Oemrawsingh et al. Experimental demonstration of fractional orbital angular momentum entanglement of two photons. Phys. Rev. Lett., 95, 240501(2005).

[160] G. Molina-Terriza, J. P. Torres, L. Torner. Twisted photons. Nat. Phys., 3, 305(2007).

[161] J. Leach et al. Quantum correlations in optical angle–orbital angular momentum variables. Science, 329, 662(2010).

[162] A. M. Yao. Angular momentum decomposition of entangled photons with an arbitrary pump. New J. Phys., 13, 053048(2011).

[163] D. S. Ding et al. Quantum storage of orbital angular momentum entanglement in an atomic ensemble. Phys. Rev. Lett., 114, 050502(2015).

[164] Z. Q. Zhou et al. Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal. Phys. Rev. Lett., 115, 070502(2015).

[165] Z. Y. Zhou et al. Orbital angular momentum-entanglement frequency transducer. Phys. Rev. Lett., 117, 103601(2016).

[166] B. C. Hiesmayr, M. J. A. de Dood, W. Löffler. Observation of four-photon orbital angular momentum entanglement. Phys. Rev. Lett., 116, 073601(2016).

[167] M. Mafu et al. Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases. Phys. Rev. A, 88, 032305(2013).

[168] A. HamadouIbrahim et al. Orbital-angular-momentum entanglement in turbulence. Phys. Rev. A, 88, 012312(2013).

[169] A. Sit et al. High-dimensional intracity quantum cryptography with structured photons. Optica, 4, 1006(2017).

[170] F. Bouchard, R. Fickler, E. Karimi. High-dimensional quantum cloning and applications to quantum hacking. Sci. Adv., 3, e1601915(2017).

[171] M. Krenn et al. Generation and confirmation of a (100 × 100)-dimensional entangled quantum system. Proc. Natl. Acad. Sci. U.S.A., 111, 6243(2014).

[172] S. P. Walborn et al. Entanglement and conservation of orbital angular momentum in spontaneous parametric down-conversion. Phys. Rev. A, 69, 023811(2004).

[173] C. K. Law, J. H. Eberly. Analysis and interpretation of high transverse entanglement in optical parametric down conversion. Phys. Rev. Lett., 92, 127903(2004).

[174] L. K. Shalm et al. Three-photon energy–time entanglement. Nat. Phys., 9, 19(2013).

[175] C. Weedbrook et al. Gaussian quantum information. Rev. Mod. Phys., 84, 621(2012).

[176] S. Li et al. Deterministic generation of orbital-angular-momentum multiplexed tripartite entanglement. Phys. Rev. Lett., 124, 083605(2020).

[177] W. Wang, K. Zhang, J. Jing. Large-scale quantum network over 66 orbital angular momentum optical modes. Phys. Rev. Lett., 125, 140501(2020).

[178] S. Tanzilli et al. On the genesis and evolution of integrated quantum optics. Laser Photonics Rev., 6, 115(2012).

[179] L. Caspani et al. Integrated sources of photon quantum states based on nonlinear optics. Light Sci. Appl., 6, e17100(2017).

[180] M. Kues et al. Quantum optical microcombs. Nat. Photon., 13, 170(2019).

[181] F. Lenzini et al. Integrated photonic platform for quantum information with continuous variables. Sci. Adv., 4, eaat9331(2018).

[182] C. H. Bennett et al. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett., 70, 1895(1993).

[183] D. Bouwmeester et al. Experimental quantum teleportation. Nature, 390, 575(1997).

[184] M. A. Nielsen, E. Knill, R. Laflamme. Complete quantum teleportation using nuclear magnetic resonance. Nature, 396, 52(1998).

[185] I. Marcikic et al. Long-distance teleportation of qubits at telecommunication wavelengths. Nature, 421, 509(2003).

[186] M. Riebe et al. Deterministic quantum teleportation with atoms. Nature, 429, 734(2004).

[187] M. D. Barrett et al. Deterministic quantum teleportation of atomic qubits. Nature, 429, 737(2004).

[188] J. F. Sherson et al. Quantum teleportation between light and matter. Nature, 443, 557(2006).

[189] S. Olmschenk et al. Quantum teleportation between distant matter qubits. Science, 323, 486(2009).

[190] L. Steffen et al. Deterministic quantum teleportation with feed-forward in a solid state system. Nature, 500, 319(2013).

[191] J. Yin et al. Quantum teleportation and entanglement distribution over 100-kilometre free-space channels. Nature, 488, 185(2012).

[192] X.-S. Ma et al. Quantum teleportation over 143 kilometres using active feed-forward. Nature, 489, 269(2012).

[193] J.-G. Ren et al. Ground-to-satellite quantum teleportation. Nature, 549, 70(2017).

[194] W. Pfaff et al. Unconditional quantum teleportation between distant solid-state quantum bits. Science, 345, 532(2014).

[195] Y.-H. Luo et al. Quantum teleportation in high dimensions. Phys. Rev. Lett., 123, 070505(2019).

[196] M. Yukawa, H. Benichi, A. Furusawa. High-fidelity continuous-variable quantum teleportation toward multistep quantum operations. Phys. Rev. A, 77, 022314(2008).

[197] L. Vaidman. Teleportation of quantum states. Phys. Rev. A, 49, 1473(1994).

[198] T. C. Ralph. All-optical quantum teleportation. Opt. Lett., 24, 348(1999).

[199] S. Liu, Y. Lou, J. Jing. Orbital angular momentum multiplexed deterministic all-optical quantum teleportation. Nat. Commun., 11, 3875(2020).

[200] S. Pirandola et al. Fundamental limits of repeaterless quantum communications. Nat. Commun., 8, 15043(2017).

[201] Z.-S. Yuan et al. Experimental demonstration of a BDCZ quantum repeater node. Nature, 454, 1098(2008).

[202] M. Żukowski et al. “Event-ready-detectors” Bell experiment via entanglement swapping. Phys. Rev. Lett., 71, 4287(1993).

[203] J.-W. Pan et al. Experimental entanglement swapping: entangling photons that never interacted. Phys. Rev. Lett., 80, 3891(1998).

[204] T. Jennewein et al. Experimental nonlocality proof of quantum teleportation and entanglement swapping. Phys. Rev. Lett., 88, 017903(2001).

[205] X.-S. Ma et al. Experimental delayed-choice entanglement swapping. Nat. Phys., 8, 479(2012).

[206] M. Halder et al. Entangling independent photons by time measurement. Nat. Phys., 3, 692(2007).

[207] W. Ning et al. Deterministic entanglement swapping in a superconducting circuit. Phys. Rev. Lett., 123, 060502(2019).

[208] R. E. S. Polkinghorne, T. C. Ralph. Continuous variable entanglement swapping. Phys. Rev. Lett., 83, 2095(1999).

[209] S. M. Tan. Confirming entanglement in continuous variable quantum teleportation. Phys. Rev. A, 60, 2752(1999).

[210] P. van Loock, S. L. Braunstein. Unconditional teleportation of continuous-variable entanglement. Phys. Rev. A, 61, 010302(R)(1999).

[211] N. Takei et al. High-fidelity teleportation beyond the no-cloning limit and entanglement swapping for continuous variables. Phys. Rev. Lett., 94, 220502(2005).

[212] S. Liu et al. All-optical entanglement swapping. Phys. Rev. Lett., 128, 060503(2022).

[213] R. C. Pooser et al. Low-noise amplification of a continuous-variable quantum state. Phys. Rev. Lett., 103, 010501(2009).

[214] W. K. Wooters, W. H. Zurek. A single quantum cannot be cloned. Nature, 299, 802(1982).

[215] D. Dieks. Communication by EPR devices. Phys. Lett. A, 92, 271(1982).

[216] V. Bužek, M. Hillery. Quantum copying: beyond the no-cloning theorem. Phys. Rev. A, 54, 1844(1996).

[217] A. Lamas-Linares et al. Experimental quantum cloning of single photons. Science, 296, 712(2002).

[218] W. T. M. Irvine et al. Optimal quantum cloning on a beam splitter. Phys. Rev. Lett., 92, 047902(2004).

[219] S. Fasel et al. Quantum cloning with an optical fiber amplifier. Phys. Rev. Lett., 89, 107901(2002).

[220] E. Nagali et al. Experimental optimal cloning of four-dimensional quantum states of photons. Phys. Rev. Lett., 105, 073602(2010).

[221] E. Nagali et al. Optimal quantum cloning of orbital angular momentum photon qubits through Hong–Ou–Mandel coalescence. Nat. Photon., 3, 720(2009).

[222] U. L. Andersen, V. Josse, G. Leuchs. Unconditional quantum cloning of coherent states with linear optics. Phys. Rev. Lett., 94, 240503(2005).

[223] J. Y. Haw et al. Surpassing the no-cloning limit with a heralded hybrid linear amplifier for coherent states. Nat. Commun., 7, 13222(2016).

[224] M. Sabuncu, U. L. Andersen, G. Leuchs. Experimental demonstration of continuous variable cloning with phase-conjugate inputs. Phys. Rev. Lett., 98, 170503(2007).

[225] J. Fiurášek. Optical implementation of continuous-variable quantum cloning machines. Phys. Rev. Lett., 86, 4942(2001).

[226] S. L. Braunstein et al. Optimal cloning of coherent states with a linear amplifier and beam splitters. Phys. Rev. Lett., 86, 4938(2001).

[227] S. Liu et al. All-optical optimal N-to-M quantum cloning of coherent states. Phys. Rev. Lett., 126, 060503(2021).

[228] H. J. Kimble. The quantum internet. Nature, 453, 1023(2008).

[229] F. Bussières et al. Quantum teleportation from a telecom-wavelength photon to a solid-state quantum memory. Nat. Photon., 8, 775(2014).

[230] S. Takeda, A. Furusawa. Toward large-scale fault-tolerant universal photonic quantum computing. APL Photonics, 4, 060902(2019).

[231] K. Mattle et al. Dense coding in experimental quantum communication. Phys. Rev. Lett., 76, 4656(1996).

[232] M. Hillery, V. Bužek, A. Berthiaume. Quantum secret sharing. Phys. Rev. A, 59, 1829(1999).

[233] Y. Lou, S. Liu, J. Jing. Experimental demonstration of a multifunctional all-optical quantum state transfer machine. Phys. Rev. Lett., 126, 210507(2021).

[234] S. L. Braunstein, H. J. Kimble. Dense coding for continuous variables. Phys. Rev. A, 61, 042302(2000).

[235] J. Zhang, K. Peng. Quantum teleportation and dense coding by means of bright amplitude-squeezed light and direct measurement of a Bell state. Phys. Rev. A, 62, 064302(2000).

[236] M. Ban. Quantum dense coding via a two-mode squeezed-vacuum state. J. Opt. B, 1, L9(1999).

[237] T. C. Ralph, E. H. Huntington. Unconditional continuous-variable dense coding. Phys. Rev. A, 66, 042321(2002).

[238] J. Mizuno et al. Experimental demonstration of entanglement-assisted coding using a two-mode squeezed vacuum state. Phys. Rev. A, 71, 012304(2005).

[239] X. Hu et al. Beating the channel capacity limit for superdense coding with entangled ququarts. Sci. Adv., 4, eaat9304(2018).

[240] S. Shi et al. Demonstration of channel multiplexing quantum communication exploiting entangled sideband modes. Phys. Rev. Lett., 125, 070502(2020).

[241] Y. Chen et al. Orbital angular momentum multiplexed quantum dense coding. Phys. Rev. Lett., 127, 093601(2021).

[242] V. Giovannetti, S. Lloyd, L. Maccone. Quantum metrology. Phys. Rev. Lett., 96, 010401(2006).

[243] B. Yurke, S. L. McCall, J. R. Klauder. SU(2) and SU(1,1) interferometers. Phys. Rev. A, 33, 4033(1986).

[244] G. Frascella et al. Wide-field SU(1,1) interferometer. Optica, 6, 1233(2019).

[245] S. Lemieux et al. Engineering the frequency spectrum of bright squeezed vacuum via group velocity dispersion in an SU(1,1) interferometer. Phys. Rev. Lett., 117, 183601(2016).

[246] P. R. Sharapova et al. Bright squeezed vacuum in a nonlinear interferometer: Frequency and temporal Schmidt-mode description. Phys. Rev. A, 97, 053827(2018).

[247] A. A. Michelson, E. W. Morley. On the relative motion of the earth and the luminiferous ether. Am. J. Sci., s3-34, 333(1887).

[248] S. L. Braunstein. Quantum limits on precision measurements of phase. Phys. Rev. Lett., 69, 3598(1992).

[249] K. Goda et al. A quantum-enhanced prototype gravitational-wave detector. Nat. Phys., 4, 472(2008).

[250] (for the, G. M. Harry. Advanced LIGO: the next generation of gravitational wave detectors. Class. Quantum Grav., 27, 084006(2010).

[251] M. Xiao, L. A. Wu, H. J. Kimble. Precision measurement beyond the shot-noise limit. Phys. Rev. Lett., 59, 278(1987).

[252] K. McKenzie et al. Experimental demonstration of a squeezing-enhanced power-recycled michelson interferometer for gravitational wave detection. Phys. Rev. Lett., 88, 231102(2002).

[253] H. Vahlbruch et al. Demonstration of a squeezed-light-enhanced power- and signal-recycled michelson interferometer. Phys. Rev. Lett., 95, 211102(2005).

[254] A. Kolkiran, G. S. Agarwal. Quantum interferometry using coherent beam stimulated parametric down-conversion. Opt. Express, 16, 6479(2008).

[255] M. W. Mitchell, J. S. Lundeen, A. M. Steinberg. Super-resolving phase measurements with a multiphoton entangled state. Nature, 429, 161(2004).

[256] J. P. Dowling. Quantum optical metrology—the lowdown on high-N00N states. Contemp. Phys., 49, 125(2008).

[257] G. Y. Xiang et al. Entanglement-enhanced measurement of a completely unknown optical phase. Nat. Photon., 5, 43(2010).

[258] A. Kuzmich, L. Mandel. Sub-shot-noise interferometric measurements with two-photon states. Quantum Semiclass. Opt., 10, 493(1998).

[259] A. N. Boto et al. Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit. Phys. Rev. Lett., 85, 2733(2000).

[260] P. M. Anisimov et al. Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit. Phys. Rev. Lett., 104, 103602(2010).

[261] A. Rivas, A. Luis. Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes. Phys. Rev. Lett., 105, 010403(2010).

[262] D. Li et al. Phase sensitivity at the Heisenberg limit in an SU(1,1) interferometer via parity detection. Phys. Rev. A, 94, 063840(2016).

[263] B. E. Anderson et al. Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers. Phys. Rev. A, 95, 063843(2017).

[264] S. S. Szigeti, R. J. Lewis-Swan, S. A. Haine. Pumped-up SU(1,1) interferometry. Phys. Rev. Lett., 118, 150401(2017).

[265] M. Manceau et al. Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer. Phys. Rev. Lett., 119, 223604(2017).

[266] S. Liu et al. Quantum enhancement of phase sensitivity for the bright-seeded SU(1,1) interference with direct intensity detection. Phys. Rev. Appl., 10, 064046(2018).

[267] Y. Shang et al. Continuous variable entanglement enhancement and manipulation by a subthreshold type II optical parametric amplifier. Opt. Lett., 35, 853(2010).

[268] Z. Yan et al. Cascaded entanglement enhancement. Phys. Rev. A, 85, 040305(2012).

[269] J. Xin, J. Qi, J. Jing. Enhancement of entanglement using cascaded four-wave mixing processes. Opt. Lett., 42, 366(2017).

[270] S. S. Szigeti et al. Squeezed-light-enhanced atom interferometry below the standard quantum limit. Phys. Rev. A, 90, 063630(2014).

[271] Z. Y. Ou. Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer. Phys. Rev. A, 85, 023815(2012).

[272] R. Schnabel et al. Quantum metrology for gravitational wave astronomy. Nat. Commun., 1, 121(2010).

[273] J. Kong et al. Experimental investigation of the visibility dependence in a nonlinear interferometer using parametric amplifiers. Appl. Phys. Lett., 102, 011130(2013).

[274] J. Kong et al. Cancellation of internal quantum noise of an amplifier by quantum correlation. Phys. Rev. Lett., 111, 033608(2013).

[275] J. Xin, H. Wang, J. Jing. The effect of losses on the quantum-noise cancellation in the SU(1,1) interferometer. Appl. Phys. Lett., 109, 051107(2016).

[276] T. Kim et al. Precision measurement scheme using a quantum interferometer. Phys. Rev. A, 72, 055801(2005).

[277] J. Dunningham, T. Kim. Using quantum interferometers to make measurements at the Heisenberg limit. J. Mod. Opt., 53, 557(2006).

[278] P. Gupta et al. Optimized phase sensing in a truncated SU(1,1) interferometer. Opt. Express, 26, 391(2018).

[279] D. Rugar et al. Single spin detection by magnetic resonance force microscopy. Nature, 430, 329(2004).

[280] J. Arlett, E. Myers, M. Roukes. Comparative advantages of mechanical biosensors. Nat. Nanotechnol., 6, 203(2011).

[281] C. LeMieux et al. Polymeric nanolayers as actuators for ultrasensitive thermal bimorphs. Nano Lett., 6, 730(2006).

[282] U. B. Hoff et al. Quantum-enhanced micromechanical displacement sensitivity. Opt. Lett., 38, 1413(2013).

[283] C. A. Putman et al. A detailed analysis of the optical beam deflection technique for use in atomic force microscopy. Appl. Phys., 72, 6(1992).

[284] D. Smith. Limits of force microscopy. Rev. Sci. Instrum., 66, 3191(1995).

[285] S. Kawata, Y. Inouye, P. Verma. Plasmonics for near-field nano-imaging and superlensing. Nat. Photon., 3, 388(2009).

[286] E. Ozbay. Plasmonics: merging photonics and electronics at nanoscale dimensions. Science, 311, 189(2006).

[287] T. Ebbesen et al. Extraordinary optical transmission through sub-wavelength hole arrays. Nature, 391, 667(1998).

[288] A. V. Zayats, I. I. Smolyaninov. Near-field photonics: surface plasmon polaritons and localized surface plasmons. J. Opt. A, 5, S16(2003).

[289] C. Lee et al. Quantum plasmonics with a metal nanoparticle array. Phys. Rev. A, 85, 063823(2012).

[290] D. Ballester et al. Long-range surface-plasmon-polariton excitation at the quantum level. Phys. Rev. A, 79, 053845(2009).

[291] E. Altewischer, M. van Exter, J. Woerdman. Plasmon-assisted transmission of entangled photons. Nature, 418, 304(2002).

[292] G.-Y. Chen et al. Surface plasmons in a metal nanowire coupled to colloidal quantum dots: scattering properties and quantum entanglement. Phys. Rev. B, 84, 045310(2011).

[293] A. Huck et al. Demonstration of quadrature-squeezed surface plasmons in a gold waveguide. Phys. Rev. Lett., 102, 246802(2009).

[294] Z. Jacob, V. Shalaev. Plasmonics goes quantum. Science, 334, 463(2011).

[295] D. Wang et al. Feedback-optimized extraordinary optical transmission of continuous-variable entangled states. Phys. Rev. B, 91, 121406(2015).

[296] R. Nichols et al. Multiparameter Gaussian quantum metrology. Phys. Rev. A, 98, 012114(2018).

[297] L. Bao et al. Multi-channel quantum parameter estimation. Sci. China Inf. Sci., 65, 200505(2022).