Abstract

The time-frequency degree of freedom is a powerful resource for implementing high-dimensional quantum information processing. In particular, field-orthogonal pulsed temporal modes offer a flexible framework compatible with both long-distance fiber networks and integrated waveguide devices. In order for this architecture to be fully utilized, techniques to reliably generate diverse quantum states of light and accurately measure complex temporal waveforms must be developed. To this end, nonlinear processes mediated by spectrally shaped pump pulses in group-velocity engineered waveguides and crystals provide a capable toolbox. In this review, we examine how tailoring the phase-matching conditions of parametric downconversion and sum-frequency generation allows for highly pure single-photon generation, flexible temporal-mode entanglement, and accurate measurement of time-frequency photon states. We provide an overview of experimental progress towards these goals and summarize challenges that remain in the field.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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V. Ansari, E. Roccia, M. Santandrea, M. Doostdar, C. Eigner, L. Padberg, I. Gianani, M. Sbroscia, J. M. Donohue, L. Mancino, M. Barbieri, and C. Silberhorn, “Heralded generation of high-purity ultrashort single photons in programmable temporal shapes,” Opt. Express 26, 2764–2774 (2018).
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D. V. Reddy and M. G. Raymer, “High-selectivity quantum pulse gating of photonic temporal modes using all-optical Ramsey interferometry,” Optica 5, 423–428(2018).

M. Allgaier, V. Ansari, C. Eigner, V. Quiring, R. Ricken, J. M. Donohue, T. Czerniuk, M. Aßmann, M. Bayer, B. Brecht, and C. Silberhorn, “Streak camera imaging of single photons at telecom wavelength,” Appl. Phys. Lett. 112, 031110 (2018).
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M. Grimau Puigibert, G. Aguilar, Q. Zhou, F. Marsili, M. Shaw, V. Verma, S. Nam, D. Oblak, and W. Tittel, “Heralded single photons based on spectral multiplexing and feed-forward control,” Phys. Rev. Lett. 119, 083601 (2017).
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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,” Nat. Commun. 8, 14288 (2017).
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V. Ansari, G. Harder, M. Allgaier, B. Brecht, and C. Silberhorn, “Temporal-mode measurement tomography of a quantum pulse gate,” Phys. Rev. A 96, 063817 (2017).
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P. Erker, M. Krenn, and M. Huber, “Quantifying high dimensional entanglement with two mutually unbiased bases,” Quantum 1, 22 (2017).
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J. A. Jaramillo-Villegas, P. Imany, O. D. Odele, D. E. Leaird, Z.-Y. Ou, M. Qi, and A. M. Weiner, “Persistent energy–time entanglement covering multiple resonances of an on-chip biphoton frequency comb,” Optica 4, 655–658 (2017).
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J. M. Lukens and P. Lougovski, “Frequency-encoded photonic qubits for scalable quantum information processing,” Optica 4, 8–16 (2017).
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Y.-S. Ra, C. Jacquard, A. Dufour, C. Fabre, and N. Treps, “Tomography of a mode-tunable coherent single-photon subtractor,” Phys. Rev. X 7, 031012 (2017).
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M. Silver, P. Manurkar, Y.-P. Huang, C. Langrock, M. M. Fejer, P. Kumar, and G. S. Kanter, “Spectrally multiplexed upconversion detection with C-band pump and signal wavelengths,” IEEE Photon. Technol. Lett. 29, 1097–1100 (2017).
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P. J. Bustard, D. G. England, K. Heshami, C. Kupchak, and B. J. Sussman, “Quantum frequency conversion with ultra-broadband tuning in a Raman memory,” Phys. Rev. A 95, 053816 (2017).
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J. Shi, G. Patera, M. I. Kolobov, and S. Han, “Quantum temporal imaging by four-wave mixing,” Opt. Lett. 42, 3121–3124 (2017).
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L. J. Wright, M. Karpiński, C. Söller, and B. J. Smith, “Spectral shearing of quantum light pulses by electro-optic phase modulation,” Phys. Rev. Lett. 118, 023601 (2017).
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S. Mittal, V. V. Orre, A. Restelli, R. Salem, E. A. Goldschmidt, and M. Hafezi, “Temporal and spectral manipulations of correlated photons using a time lens,” Phys. Rev. A 96, 043807 (2017).
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M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
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P. Manurkar, N. Jain, P. Kumar, and G. S. Kanter, “Programmable optical waveform reshaping on a picosecond timescale,” Opt. Lett. 42, 951–954 (2017).
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A. Shahverdi, Y. M. Sua, L. Tumeh, and Y.-P. Huang, “Quantum parametric mode sorting: beating the time-frequency filtering,” Sci. Rep. 7, 6495 (2017).
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D. V. Reddy and M. G. Raymer, “Engineering temporal-mode-selective frequency conversion in nonlinear optical waveguides: from theory to experiment,” Opt. Express 25, 12952–12966 (2017).
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H. Rütz, K.-H. Luo, H. Suche, and C. Silberhorn, “Quantum frequency conversion between infrared and ultraviolet,” Phys. Rev. Appl. 7, 024021 (2017).
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N. Maring, P. Farrera, K. Kutluer, M. Mazzera, G. Heinze, and H. de Riedmatten, “Photonic quantum state transfer between a cold atomic gas and a crystal,” Nature 551, 485–488 (2017).
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C. Chen, C. Bo, M. Y. Niu, F. Xu, Z. Zhang, J. H. Shapiro, and F. N. C. Wong, “Efficient generation and characterization of spectrally factorable biphotons,” Opt. Express 25, 7300–7312 (2017).
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F. Graffitti, D. Kundys, D. T. Reid, A. M. Brańczyk, and A. Fedrizzi, “Pure down-conversion photons through sub-coherence-length domain engineering,” Quantum Sci. Technol. 2, 035001 (2017).
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G. Harder, V. Ansari, T. J. Bartley, B. Brecht, and C. Silberhorn, “Harnessing temporal modes for multi-photon quantum information processing based on integrated optics,” Philos. Trans. A Math. Phys. Eng. Sci. 375, 20160244 (2017).
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V. Averchenko, D. Sych, G. Schunk, U. Vogl, C. Marquardt, and G. Leuchs, “Temporal shaping of single photons enabled by entanglement,” Phys. Rev. A 96, 043822 (2017).
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S. Lemieux, M. Manceau, P. R. Sharapova, O. V. Tikhonova, R. W. Boyd, G. Leuchs, and M. V. Chekhova, “Engineering the frequency spectrum of bright squeezed vacuum via group velocity dispersion in an SU(1, 1) interferometer,” Phys. Rev. Lett. 117, 183601 (2016).
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G. Harder, T. J. Bartley, A. E. Lita, S. W. Nam, T. Gerrits, and C. Silberhorn, “Single-mode parametric-down-conversion states with 50 photons as a source for mesoscopic quantum optics,” Phys. Rev. Lett. 116, 143601 (2016).
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M. M. Weston, H. M. Chrzanowski, S. Wollmann, A. Boston, J. Ho, L. K. Shalm, V. B. Verma, M. S. Allman, S. W. Nam, R. B. Patel, S. Slussarenko, and G. J. Pryde, “Efficient and pure femtosecond-pulse-length source of polarization-entangled photons,” Opt. Express 24, 10869–10879 (2016).
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A. Dosseva, Ł. Cincio, and A. M. Brańczyk, “Shaping the joint spectrum of down-converted photons through optimized custom poling,” Phys. Rev. A 93, 013801 (2016).
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J.-L. Tambasco, A. Boes, L. G. Helt, M. J. Steel, and A. Mitchell, “Domain engineering algorithm for practical and effective photon sources,” Opt. Express 24, 19616–19626 (2016).
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P. Manurkar, N. Jain, M. Silver, Y.-P. Huang, C. Langrock, M. M. Fejer, P. Kumar, and G. S. Kanter, “Multidimensional mode-separable frequency conversion for high-speed quantum communication,” Optica 3, 1300–1307 (2016).
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F. Steinlechner, N. Hermosa, V. Pruneri, and J. P. Torres, “Frequency conversion of structured light,” Sci. Rep. 6, 21390 (2016).
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S. J. Nowierski, N. N. Oza, P. Kumar, and G. S. Kanter, “Tomographic reconstruction of time-bin-entangled qudits,” Phys. Rev. A 94, 042328 (2016).
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Z. Huang, L. Maccone, A. Karim, C. Macchiavello, R. J. Chapman, and A. Peruzzo, “High-dimensional entanglement certification,” Sci. Rep. 6, 27637 (2016).
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N. Matsuda, “Deterministic reshaping of single-photon spectra using cross-phase modulation,” Sci. Adv. 2, e1501223 (2016).
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M. Karpiński, M. Jachura, L. J. Wright, and B. J. Smith, “Bandwidth manipulation of quantum light by an electro-optic time lens,” Nat. Photonics 11, 53–57 (2016).
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K. A. G. 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,” Nat. Commun. 7, 11200 (2016).
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N. Quesada and J. E. Sipe, “High efficiency in mode-selective frequency conversion,” Opt. Lett. 41, 364–367 (2016).
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J. M. Donohue, M. Mastrovich, and K. J. Resch, “Spectrally engineering photonic entanglement with a time lens,” Phys. Rev. Lett. 117, 1–15 (2016).
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2015 (12)

T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, “Photon-efficient quantum key distribution using time-energy entanglement with high-dimensional encoding,” New J. Phys. 17, 022002 (2015).
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Z. Xie, T. Zhong, S. Shrestha, X. Xu, J. Liang, Y.-X. Gong, J. C. Bienfang, A. Restelli, J. H. Shapiro, F. N. Wong, and C. W. Wong, “Harnessing high-dimensional hyperentanglement through a biphoton frequency comb,” Nat. Photonics 9, 536–542 (2015).
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J. M. Donohue, M. D. Mazurek, and K. J. Resch, “Theory of high-efficiency sum-frequency generation for single-photon waveform conversion,” Phys. Rev. A 91, 1–11 (2015).
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G. Patera and M. I. Kolobov, “Temporal imaging with squeezed light,” Opt. Lett. 40, 1125–1128 (2015).
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Z. Zheng, O. Mishina, N. Treps, and C. Fabre, “Atomic quantum memory for multimode frequency combs,” Phys. Rev. A 91, 031802 (2015).
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Y.-H. Cheng, T. Thomay, G. S. Solomon, A. L. Migdall, and S. V. Polyakov, “Statistically background-free, phase-preserving parametric up-conversion with faint light,” Opt. Express 23, 18671–18678 (2015).
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D. V. Reddy, M. G. Raymer, and C. J. McKinstrie, “Sorting photon wave packets using temporal-mode interferometry based on multiple-stage quantum frequency conversion,” Phys. Rev. A 91, 012323 (2015).
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B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: a complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
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N. Quesada and J. E. Sipe, “Time-ordering effects in the generation of entangled photons using nonlinear optical processes,” Phys. Rev. Lett. 114, 093903 (2015).
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C. Baune, J. Gniesmer, A. Schönbeck, C. E. Vollmer, J. Fiurášek, and R. Schnabel, “Strongly squeezed states at 532  nm based on frequency up-conversion,” Opt. Express 23, 16035–16041 (2015).
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W. Liu, N. Wang, Z. Li, and Y. Li, “Quantum frequency up-conversion of continuous variable entangled states,” Appl. Phys. Lett. 107, 231109 (2015).
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S. Gerke, J. Sperling, W. Vogel, Y. Cai, J. Roslund, N. Treps, and C. Fabre, “Full multipartite entanglement of frequency-comb Gaussian states,” Phys. Rev. Lett. 114, 050501 (2015).
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2014 (17)

J. Roslund, R. M. de Araújo, S. Jiang, C. Fabre, and N. Treps, “Wavelength-multiplexed quantum networks with ultrafast frequency combs,” Nat. Photonics 8, 109–112 (2014).
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M. Chen, N. C. Menicucci, and O. Pfister, “Experimental realization of multipartite entanglement of 60 modes of a quantum optical frequency comb,” Phys. Rev. Lett. 112, 120505 (2014).
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K. Wakui, Y. Eto, H. Benichi, S. Izumi, T. Yanagida, K. Ema, T. Numata, D. Fukuda, M. Takeoka, and M. Sasaki, “Ultrabroadband direct detection of nonclassical photon statistics at telecom wavelength,” Sci. Rep. 4, 4535 (2014).
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T. Lutz, P. Kolenderski, and T. Jennewein, “Demonstration of spectral correlation control in a source of polarization-entangled photon pairs at telecom wavelength,” Opt. Lett. 39, 1481–1484 (2014).
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C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Händchen, J. Fiurášek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532  nm,” Phys. Rev. Lett. 112, 073602 (2014).
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R.-B. Jin, R. Shimizu, K. Wakui, M. Fujiwara, T. Yamashita, S. Miki, H. Terai, Z. Wang, and M. Sasaki, “Pulsed Sagnac polarization-entangled photon source with a PPKTP crystal at telecom wavelength,” Opt. Express 22, 11498–11507 (2014).
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J. M. Lukens, A. Dezfooliyan, C. Langrock, M. M. Fejer, D. E. Leaird, and A. M. Weiner, “Orthogonal spectral coding of entangled photons,” Phys. Rev. Lett. 112, 133602 (2014).
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B. Brecht, A. Eckstein, R. Ricken, V. Quiring, H. Suche, L. Sansoni, and C. Silberhorn, “Demonstration of coherent time-frequency Schmidt mode selection using dispersion-engineered frequency conversion,” Phys. Rev. A 90, 030302 (2014).
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A. S. Kowligy, P. Manurkar, N. V. Corzo, V. G. Velev, M. Silver, R. P. Scott, S. J. B. Yoo, P. Kumar, G. S. Kanter, and Y.-P. Huang, “Quantum optical arbitrary waveform manipulation and measurement in real time,” Opt. Express 22, 27942–27957 (2014).
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N. Quesada and J. E. Sipe, “Effects of time ordering in quantum nonlinear optics,” Phys. Rev. A 90, 1–13 (2014).
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R. Ikuta, T. Kobayashi, S. Yasui, S. Miki, T. Yamashita, H. Terai, M. Fujiwara, T. Yamamoto, M. Koashi, M. Sasaki, Z. Wang, and N. Imoto, “Frequency down-conversion of 637  nm light to the telecommunication band for non-classical light emitted from NV centers in diamond,” Opt. Express 22, 11205–11214 (2014).
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J. M. Donohue, J. Lavoie, and K. J. Resch, “Ultrafast time-division demultiplexing of polarization-entangled photons,” Phys. Rev. Lett. 113, 163602 (2014).
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V. A. Averchenko, V. Thiel, and N. Treps, “Non-linear photon subtraction from a multimode quantum field,” Phys. Rev. A 89, 063808 (2014).
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2013 (10)

J. M. Donohue, M. Agnew, J. Lavoie, and K. J. Resch, “Coherent ultrafast measurement of time-bin encoded photons,” Phys. Rev. Lett. 111, 153602 (2013).
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2012 (7)

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M. Pysher, Y. Miwa, R. Shahrokhshahi, R. Bloomer, and O. Pfister, “Parallel generation of quadripartite cluster entanglement in the optical frequency comb,” Phys. Rev. Lett. 107, 030505 (2011).
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2010 (9)

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V. Ansari, B. Brecht, G. Harder, and C. Silberhorn, “Probing spectral-temporal correlations with a versatile integrated source of parametric down-conversion states,” arXiv:1404.7725 (2014).

F. Graffitti, P. Barrow, M. Proietti, D. Kundys, and A. Fedrizzi, “Independent high-purity photons created in domain-engineered crystals,” arXiv:1712.07140 (2017).

V. Ansari, J. M. Donohue, M. Allgaier, L. Sansoni, B. Brecht, J. Roslund, N. Treps, G. Harder, and C. Silberhorn, “Tomography and purification of the temporal-mode structure of quantum light,” arXiv 1607.03001 (2018).

J. H. Munns, S. Park, B. Brecht, and I. A. Walmsley, “Temporal amplitude & phase: algorithmic reconstruction via time-domain interferometry (teAPARTI),” in Frontiers in Optics (Optical Society of America, 2017), paper JW3A-34.

D. V. Reddy and M. G. Raymer, “Photonic temporal-mode multiplexing by quantum frequency conversion in a dichroic-finesse cavity,” arXiv:1708.01705 (2017).

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Figures (11)

Fig. 1.
Fig. 1. Temporal-mode encodings visualized in time-frequency space. Orthogonal temporal mode bases can be constructed through slicing bins in time or frequency, as in (a) and (b), or through intensity-overlapping but field-orthogonal pulsed temporal modes, such as the Hermite–Gauss modes in (c).
Fig. 2.
Fig. 2. Joint spectral amplitude, temporal modes, and Schmidt coefficients of a non-engineered PDC process. (a) Outline of a PDC process with the three involved fields. (b) The JSA and its marginal distributions which is the product of pump (dashed lines) and phase matching (solid lines) functions and, in this case, exhibits frequency anti-correlations between signal and idler frequencies. The Schmidt decomposition of this Gaussian JSA is given by Hermite–Gaussian functions, with the first three TM pairs shown in (c)–(e). (f) The first seven Schmidt coefficients λk. The decomposition of this example yields an effective mode number of K3.14.
Fig. 3.
Fig. 3. Three different group-velocity matching condition. The JSA of each case is plotted on the left side, with the respective group velocities uj of the pump, signal, and idler fields plotted on the right side. The group velocities (normalized over the speed of light in vacuum) are exemplary for TE- and TM-polarized light in a z-cut KTP crystal. (a) Typically without dispersion engineering, the long-wavelength signal and idler photons both have a larger group velocity than the pump (ξ>0). This leads to a negative phase-matching angle and consequently to a correlated JSA as shown on the left. In this example, ξ0.4. (b) In the case of aGVM (ξ0), one photon (here the signal) propagates at the same velocity as the pump. This yields a phase-matching function that is aligned with the signal or idler frequency axis. If the pump spectral bandwidth is larger than the phase-matching bandwidth, a separable JSA is generated. (c) For sGVM (ξ1), the group velocity of the pump lies between the group velocities of signal and idler. This leads to a +45° phase-matching angle and, given that the pump spectral bandwidth matches the phase-matching bandwidth, a separable JSA with potentially indistinguishable signal and idler.
Fig. 4.
Fig. 4. Joint spectral amplitudes (absolute value) with standard periodic poling and filters on the individual photons. (a) In an aGVM source, the idler can be filtered to remove the side lobes and herald pure signal photons. However, filtering on the signal arm cannot be used to remove the side lobes. (b) In sGVM sources, the JSA is symmetric. Filtering either signal or idler leaves the other with a purity of about 94%.
Fig. 5.
Fig. 5. Orchestrating Schmidt modes via group-velocity matching and pump pulse shaping. (a)–(c) JSAs for a PDC source with an aGVM setting. The weights of the first five Schmidt modes λk are shown under each JSA. The state remains single-mode regardless of the pump shape. The only significant Schmidt modes of signal A0 and idler B0 photons are shown at the bottom, where we plot TM amplitudes versus frequency. The idler photon shape is invariant to the pump, while the TM of the signal photon reflects the TM of the pump field. (e)–(g) A sGVM PDC can be used to control the exact number of excited TMs. For example, driving the source with a first-order Hermite–Gaussian pump pulse as in (e) results in exactly two TMs. This can be extended with higher orders of Hermite–Gaussian pulses as in (f), but the different Schmidt modes are not occupied with the same probability. A balanced Schmidt-mode distribution can be achieved when the source is pumped with time-bin superpositions, as in (g).
Fig. 6.
Fig. 6. Frequency conversion process and its transfer function. (a) Outline of a general frequency conversion process with pump, input and output fields. (b)–(d) Sum-frequency conversion transfer functions F(ωin,ωout) with its marginal distributions (left) and its first few Schmidt coefficients λk. (b) A non-engineered SFG with significant frequency correlations and a K3.7. (c) and (d) present a tailored SFG process with aGVM condition with pump functions α(ωoutωin) of Gaussian and first-order Hermite–Gauss, respectively, and a K1.01.
Fig. 7.
Fig. 7. Absolute value of the temporal (left) and spectral (right) transfer functions for broadband frequency conversion. The left column shows the mapping from input times tin to output times tout for increasing pump powers (top to bottom), corresponding to increasing conversion efficiencies. The relative pump energy P, selectivity S, and separability σ0 are printed on top-right corner of each row. This leads to simultaneous forward and backward conversion, which is reflected by the oscillations in the mapping function. The functions were calculated by numerically solving the Heisenberg equations for the input and output field operators. The right column shows the respective spectral mapping functions. It can be seen that the general shape of the function broadens and that additional correlations are introduced for stronger pump powers. These correlations do not show up in a perturbative approach.
Fig. 8.
Fig. 8. Group-velocity mismatch contrast ξ (such that 0 is perfectly matched) for processes in lithium niobate (LN) waveguides, potassium titanyl phosphate (KTP) waveguides, and bulk bismuth borate (BiBO), as the input signal is detuned from the optimal group-velocity matching. The grey dashed line corresponds to the type-II process in LN, where GVM is found for a 1550 nm signal, 875 nm pump, and 560 nm upconverted [99]. All other processes have degenerate signal and QPG pump for group-velocity matching, and IR (NIR) corresponds to 1550 nm (800 nm) signal and QPG pump. Signal detuning or noncollinear geometry is necessary in all cases except for type-II LN to overcome the second harmonic of the QPG pump.
Fig. 9.
Fig. 9. Generic experimental situation for a quantum pulse gate. A TM-encoded single photon or weak coherent state is prepared through PDC or through shaping a spectrally broad input pulse and attenuating with a neutral density (ND) filter. A strong QPG pump is prepared using similar pulse shaping methods, or through electro-optic modulation (EOM) of a strong cw laser to produce a frequency comb, which is modulated in a tooth-by-tooth fashion by a pulse shaper [100]. The two are mixed in a group-velocity-matched χ(2) waveguide, and the upconverted signal in the register mode is measured. For temporal-mode interferometry (TMI) [97], the QPG is split into two 50% efficient steps with phase shifts in between.
Fig. 10.
Fig. 10. Experimental spectral-intensity transfer functions for the first four Hermite–Gaussian temporal modes (top–bottom), as measured in the experimental apparatus of Ref. [165]. The QPG in question was built from a 17-mm-long PPLN waveguide phasematched for a type-II interaction (875 nm + 1540 nm to 555.7 nm), with the group-velocity matching necessary to produce highly separable SFG transfer functions.
Fig. 11.
Fig. 11. Spectral field amplitudes spanning a complete set of mutually-unbiased bases for Hermite-Gauss modes in two (top) and five (bottom) dimensions [165,137]. In order to completely access the Hilbert space, effective projections on all of these states must be realizable. The normalised spectral intensity is shown in grey and the red line corresponds to the spectral phase (on the interval 02π).

Equations (34)

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H^PDC=Bdωsdωif(ωs,ωi)a^(ωs)b^(ωi)+h.c.,
|ψPDC=Bdωsdωif(ωs,ωi)a^(ωs)b^(ωi)|vac,
f(ωs,ωi)=α(ωs+ωi)ϕ(ωs,ωi).
ϕ(ωs,ωi)=0Ldzχ(z)exp[ιΔk(ωs,ωi)z],
ϕ(ωs,ωi)=1Lsinc(Δk(ωs,ωi)L2)eιΔk(ωs,ωi)L2.
f(ωs,ωi)=kλkgk(s)(ωs)hk(i)(ωi),
A^k=dωsgk(s)(ωs)a^(ωs),
B^k=dωihk(i)(ωi)b^(ωi),
|ψPDC=kλkA^kB^k|0,
ρ^s=Tri(ρ^PDC)=kλk|AkAk|
Ps=tr(ρ^s2)=1K.
Δk(ωs,ωi)(us1up1)ωs+(ui1up1)ωi,
ξ=us1up1ui1up1.
f(ωs,ωi)α(ωs+ωi)ϕ(ωi).
f(ωs,ωi)α(ωs+ωi)ϕ(ωsωi).
|ψBell=12(|0s|1i+eıϕ|1s|0i),
a^(ωin)dωinGaa(ωin,ωin)a^(ωin)+dωoutGac(ωin,ωout)c^(ωout),
c^(ωout)dωinGca(ωout,ωin)a^(ωin)+dωoutGcc(ωin,ωout)c^(ωout).
H^int=θdωindωoutF(ωin,ωout)a^(ωin)c^(ωout)+h.c.,
F(ωin,ωout)=α(ωoutωin)ϕ(ωin,ωout).
H^int=θk=0λkA^kC^k+h.c.,
A^kcos(λkθ)A^k+sin(λkθ)C^k,
C^kcos(λkθ)C^ksin(λkθ)A^k.
Δk(ωin,ωout)(uin1up1)ωin(uout1up1)ωout.
F(ωin,ωout)α˜(ωin)ϕ˜(ωout).
H^QPG=θA^0C^0+h.c.,
A^0cos(θ)A^0+sin(θ)C^0,
C^0cos(θ)C^0sin(θ)A^0,
A^kA^kfor  k0,
C^kC^kfor  k0.
ξ=uin1up1uout1up1.
S=η0·η0k=0ηk1,
σj=ηjk=0dηk1.
E.R.j(dB)=10log10ηjmaxkjηk,

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