Abstract

Nonlinear optical phenomena are typically local. Here, we predict the possibility of highly nonlocal optical nonlinearities for light propagating in atomic media trapped near a nano-waveguide, where long-range interactions between the atoms can be tailored. When the atoms are in an electromagnetically induced transparency configuration, the atomic interactions are translated to long-range interactions between photons and thus to highly nonlocal optical nonlinearities. We derive and analyze the governing nonlinear propagation equation, finding a roton-like excitation spectrum for light and the emergence of order in its output intensity. These predictions open the door to studies of unexplored wave dynamics and many-body physics with highly nonlocal interactions of optical fields in one dimension.

© 2016 Optical Society of America

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2015 (7)

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
[Crossref]

G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. USA 112, 3866–3873 (2015).
[Crossref]

B. Gouraud, D. Maxein, A. Nicolas, O. Morin, and J. Laurat, “Demonstration of a memory for tightly guided light in an optical nanofiber,” Phys. Rev. Lett. 114, 180503 (2015).
[Crossref]

J. Lee, J. A. Grover, J. E. Hoffman, L. A. Orozco, and S. L. Rolston, “Inhomogeneous broadening of optical transitions of Rb87 atoms in an optical nanofiber trap,” J. Phys. B 48, 165004 (2015).
[Crossref]

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
[Crossref]

C. Sayrin, C. Clausen, B. Albrecht, P. Schneeweiss, and A. Rauschenbeutel, “Storage of fiber-guided light in a nanofiber-trapped ensemble of cold atoms,” Optica 2, 353–356 (2015).
[Crossref]

2014 (12)

E. Shahmoon, I. Mazets, and G. Kurizki, “Non-additivity in laser-illuminated many-atom systems,” Opt. Lett. 39, 3674–3677 (2014).
[Crossref]

M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
[Crossref]

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113, 093603 (2014).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonlinear theory of laser-induced dipolar interactions in arbitrary geometry,” Phys. Rev. A 89, 043419 (2014).
[Crossref]

E. Shahmoon, I. Mazets, and G. Kurizki, “Giant vacuum forces via transmission lines,” Proc. Natl. Acad. Sci. USA 111, 10485–10490 (2014).
[Crossref]

J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
[Crossref]

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref]

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref]

A. Grankin, E. Brion, E. Bimbard, R. Boddeda, I. Usmani, A. Ourjoumtsev, and P. Grangier, “Quantum statistics of light transmitted through an intracavity Rydberg medium,” New J. Phys. 16, 043020 (2014).
[Crossref]

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
[Crossref]

2013 (11)

O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref]

D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche, J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A. Jones, and C. S. Adams, “Storage and control of optical photons using Rydberg polaritons,” Phys. Rev. Lett. 110, 103001 (2013).
[Crossref]

J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
[Crossref]

D. Reitz, C. Sayrin, R. Mitsch, P. Schneeweiss, and A. Rauschenbeutel, “Coherence properties of nanofiber-trapped cesium atoms,” Phys. Rev. Lett. 110, 243603 (2013).
[Crossref]

D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013).
[Crossref]

T. Grießer and H. Ritsch, “Light-Induced crystallization of cold atoms in a 1D optical trap,” Phys. Rev. Lett. 111, 055702 (2013).
[Crossref]

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
[Crossref]

J. Otterbach, M. Moos, D. Muth, and M. Fleischhauer, “Wigner crystallization of single photons in cold Rydberg ensembles,” Phys. Rev. Lett. 111, 113001 (2013).
[Crossref]

E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87, 033831 (2013).
[Crossref]

V. Venkataraman, K. Saha, and A. L. Gaeta, “Phase modulation at the few-photon level for weak-nonlinearity-based quantum computing,” Nat. Photonics 7, 138–141 (2013).
[Crossref]

K. P. Nayak and K. Hakuta, “Photonic crystal formation on optical nanofibers using femtosecond laser ablation technique,” Opt. Express 21, 2480–2490 (2013).
[Crossref]

2012 (3)

R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
[Crossref]

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
[Crossref]

T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref]

2011 (6)

S. Sevinçli, N. Henkel, C. Ates, and T. Pohl, “Nonlocal nonlinear optics in cold Rydberg gases,” Phys. Rev. Lett. 107, 153001 (2011).
[Crossref]

E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011).
[Crossref]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref]

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[Crossref]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref]

P. Grisins and I. E. Mazets, “Thermalization in a one-dimensional integrable system,” Phys. Rev. A 84, 053635 (2011).
[Crossref]

2010 (1)

E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
[Crossref]

2009 (2)

J. Rodrguez and D. L. Andrews, “Inter-particle interaction induced by broadband radiation,” Opt. Commun. 282, 2267–2269 (2009).

A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
[Crossref]

2007 (2)

S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, “Non-equilibrium coherence dynamics in one-dimensional Bose gases,” Nature 449, 324–327 (2007).
[Crossref]

S. Skupin, M. Saffman, and W. Królikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98, 263902 (2007).
[Crossref]

2006 (1)

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006).
[Crossref]

2005 (3)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
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I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
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2004 (2)

C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
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W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
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2002 (4)

M. Fleischhauer and M. D. Lukin, “Quantum memory for photons: dark-state polaritons,” Phys. Rev. A 65, 022314 (2002).
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P. Domokos and H. Ritsch, “Collective cooling and self-organization of atoms in a cavity,” Phys. Rev. Lett. 89, 253003 (2002).
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S. Giovanazzi, D. O’Dell, and G. Kurizki, “Density modulations of Bose–Einstein condensates via laser-induced interactions,” Phys. Rev. Lett. 88, 130402 (2002).
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D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose–Einstein condensates irradiated by a laser,” Phys. Rev. Lett. 90, 110402 (2002).
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M. D. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
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T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
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J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” arXiv: 1511.00816 (2015).

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C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
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M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
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Chang, D. E.

J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
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A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
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T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
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A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
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D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013).
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J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” arXiv: 1511.00816 (2015).

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

Chiao, R. Y.

R. Y. Chiao, A. E. Kozhekin, and G. Kurizki, “Tachyonlike excitations in inverted two-level media,” Phys. Rev. Lett. 77, 1254–1257 (1996).
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Choi, K. S.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
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A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
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Choi, S.

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
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Cirac, I.

T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
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Cirac, J. I.

A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
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D. E. Chang, J. I. Cirac, and H. J. Kimble, “Self-organization of atoms along a nanophotonic waveguide,” Phys. Rev. Lett. 110, 113606 (2013).
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Cohen, O.

C. Rotschild, B. Alfassi, O. Cohen, and M. Segev, “Long-range interactions between optical solitons,” Nat. Phys. 2, 769–774 (2006).
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C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, “Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons,” Phys. Rev. Lett. 95, 213904 (2005).
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C. Conti, M. Peccianti, and G. Assanto, “Observation of optical spatial solitons in a highly nonlocal medium,” Phys. Rev. Lett. 92, 113902 (2004).
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A. Campa, T. Dauxois, and S. Ruffo, “Statistical mechanics and dynamics of solvable models with long-range interactions,” Phys. Rep. 480, 57–159 (2009).
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E. Vetsch, D. Reitz, G. Sagué, R. Schmidt, S. T. Dawkins, and A. Rauschenbeutel, “Optical interface created by laser-cooled atoms trapped in the evanescent field surrounding an optical nanofiber,” Phys. Rev. Lett. 104, 203603 (2010).
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J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
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Ding, D.

A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
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P. Domokos and H. Ritsch, “Collective cooling and self-organization of atoms in a cavity,” Phys. Rev. Lett. 89, 253003 (2002).
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R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
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J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
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T. Caneva, M. T. Manzoni, T. Shi, J. S. Douglas, I. Cirac, and D. E. Chang, “Quantum dynamics of propagating photons with strong interactions: a generalized input–output formalism,” New J. Phys. 17, 113001 (2015).
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J. S. Douglas, T. Caneva, and D. E. Chang, “Photon molecules in atomic gases trapped near photonic crystal waveguides,” arXiv: 1511.00816 (2015).

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D. Tiarks, S. Baur, K. Schneider, S. Dürr, and G. Rempe, “Single-photon transistor using a Förster resonance,” Phys. Rev. Lett. 113, 053602 (2014).
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W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
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J. A. Mlynek, A. A. Abdumalikov, C. Eichler, and A. Wallraff, “Observation of Dicke superradiance for two artificial atoms in a cavity with high decay rate,” Nat. Commun. 5, 5186 (2014).
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M. R. Sprague, P. S. Michelberger, T. F. M. Champion, D. G. England, J. Nunn, X.-M. Jin, W. S. Kolthammer, A. Abdolvand, P. St. J. Russell, and I. A. Walmsley, “Broadband single-photon-level memory in a hollow-core photonic crystal fibre,” Nat. Photonics 8, 287–291 (2014).
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R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Donner, and T. Esslinger, “Roton-type mode softening in a quantum gas with cavity-mediated long-range interactions,” Science 336, 1570–1573 (2012).
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H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
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A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-mediated interactions between distant artificial atoms,” Science 342, 1494–1496 (2013).
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J. D. Thompson, T. G. Tiecke, N. P. de Leon, J. Feist, A. V. Akimov, M. Gullans, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coupling a single trapped atom to a nanoscale optical cavity,” Science 340, 1202–1205 (2013).
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P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
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O. Firstenberg, T. Peyronel, Q.-Y. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
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T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
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I. Friedler, D. Petrosyan, M. L. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slow single-photon pulses,” Phys. Rev. A 72, 043803 (2005).
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D. H. J. O’Dell, S. Giovanazzi, and G. Kurizki, “Rotons in gaseous Bose–Einstein condensates irradiated by a laser,” Phys. Rev. Lett. 90, 110402 (2002).
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A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom–light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
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A. Goban, K. S. Choi, D. J. Alton, D. Ding, C. Lacroute, M. Pototschnig, T. Thiele, N. P. Stern, and H. J. Kimble, “Demonstration of a state-insensitive, compensated nanofiber trap,” Phys. Rev. Lett. 109, 033603 (2012).
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A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for atoms trapped along a photonic crystal waveguide,” arXiv:1503.04503 (2015).

J. D. Hood, A. Goban, A. Asenjo-Garcia, M. Lu, S.-P. Yu, D. E. Chang, and H. J. Kimble, “Atom–atom interactions around the band edge of a photonic crystal waveguide,” arXiv: 1603.02771 (2016).

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A. González-Tudela, C.-L. Hung, D. E. Chang, J. I. Cirac, and H. J. Kimble, “Subwavelength vacuum lattices and atom–atom interactions in two-dimensional photonic crystals,” Nat. Photonics 9, 320–325 (2015).
[Crossref]

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H. Gorniaczyk, C. Tresp, J. Schmidt, H. Fedder, and S. Hofferberth, “Single-photon transistor mediated by interstate Rydberg interactions,” Phys. Rev. Lett. 113, 053601 (2014).
[Crossref]

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J. S. Douglas, H. Habibian, C.-L. Hung, A. V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9, 326–331 (2015).
[Crossref]

P. Bienias, S. Choi, O. Firstenberg, M. F. Maghrebi, M. Gullans, M. D. Lukin, A. V. Gorshkov, and H. P. Büchler, “Scattering resonances and bound states for strongly interacting Rydberg polaritons,” Phys. Rev. A 90, 053804 (2014).
[Crossref]

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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1.

(a) Setup: atoms (black dots), illuminated by the EIT fields [see (b)] E ^ and Ω (thin blue arrow), are trapped at a distance r a from a nano-waveguide (gray cylinder) along z , from z = 0 to z = L . A far-detuned laser Ω L (thick orange arrow), tilted by an angle θ L from the z -axis, induces long-range interactions between the atoms, mediated by the waveguide modes [see (c)]. (b) EIT atomic configuration: the probe field E ^ is resonantly coupled to the | g | e transition, whereas the coupling field Ω is coupled to the | d | e transition with detuning δ c . Interaction between the atoms in the | d -level [see (c)] induces its energy shift δ NL , which is effectively added to the detuning δ c . (c) Laser-induced dipolar interactions: the laser with Rabi frequency Ω L and detuning δ L operates on the | d | s transition of all atoms, | s being an additional level, thus inducing a dipolar potential U ( z ) between pairs of atoms ( z apart) populating the state | d [59].

Fig. 2.
Fig. 2.

Linear susceptibility of the EIT atomic medium to the probe field as a function of its detuning Δ p [8,9] ( Δ p in units of level width γ and Ω = 2.5 γ ). (a) For total detuning of the coupling field Δ c = 0 , the absorption Im χ (red dashed line) and dispersion Re χ (blue solid line) are symmetrical and antisymmetric, respectively, with respect to Δ p , so that no (real) quadratic dispersion exists for the probe envelope centered around Δ p = 0 . (b) For Δ c 0 , Re χ is not antisymmetric, so that quadratic dispersion exists, giving rise to the term Δ c C v 2 z 2 in Eq. (1), with Δ c = δ c + δ NL [Fig. 1(b)].

Fig. 3.
Fig. 3.

Dispersion relations for EIT polaritons with waveguide-grating mediated atomic interactions ( k values presented in all plots are within the EIT transparency window; values of physical parameters used here are given in Supplement 1, Section 4). (a) Roton-like excitation spectrum (dispersion relation) ω k for the potential of Eq. (12) in the anomalous dispersion case (opposite signs of ω k 0 and U k ). The analytical results from Eq. (8) (blue solid line) agree well with those of direct numerical simulations of Eq. (1) (gray dots). Compared with the spectrum of a local interaction ( U k independent of k , dashed red line), the roton-like spectrum exhibits a dip in a narrow band of k -values around k R = k L z k B . (b) Anti-roton peak of the spectrum ω k in a narrow band around k R in the normal dispersion case (identical signs of ω k 0 and U k ). (c) Possible homodyne detection scheme: the input probe field consists of a CW field + perturbation at wavenumber k and frequency ω ( 0 ) ( k ) . The field is split before entering the EIT medium ( z = 0 ) so that a local oscillator of ω ( 0 ) ( k ) is formed (lower arm) by filtering out the CW component. Then, mixing the output signal ( z = L ) with the local oscillator reveals their phase difference, from which ω k can be inferred (see text).

Fig. 4.
Fig. 4.

Instability and self-ordering (values of physical parameters are given in Supplement 1, Section 4). (a)  γ k = Im ω k is the exponential growth rate of unstable perturbations at spatial frequency k (anomalous dispersion case). It exhibits a narrow peak around k R (blue dotted line), compared to the broadband instability of the local interaction case (red dotted line). The instability is accompanied by the generation of quantum entanglement, characterized by a narrow-band squeezing spectrum G k (blue solid line; G k < 1 quantifies entanglement between ± k photon modes), in contrast to the broadband spectrum for a local interaction (horizontal red solid line). (b) Dynamics of spatial spectrum N k ( t ) of the field intensity for different propagation times t inside the medium ( t = L / v at the output). The emergence of a large peak around k R (and k R ) out of the initial Gaussian perturbation is clearly seen. Results of both the linearized theory (solid lines) and numerical simulations of Eq. (1) (dots) are shown: slight differences at later t -values are attributed to nonlinear corrections. (c) Self-ordering of the field intensity: considering the narrow peaks of N k ( t ) around ± k R , fluctuations at these k -values become dominant, resulting in ordered intensity correlations g ( 2 ) which grow with propagation time t ( t = L / v at the output). Excellent agreement between the theory, Eq. (11) (solid lines), and numerical simulations of Eq. (1) (dots) is observed. The correlations oscillate with a period 2 π / k R 6.1    mm and a range of a few l 2.7    mm (where L = 2.68    cm and v = 4340    ms 1 ), determined by the long-range interaction U ( z ) . This is in contrast to the local interaction case, where the correlations vanish ( g ( 2 ) = 1 ) after a short distance π / q tr 1.75    mm , which is determined by the bandwidth q tr of the initial fluctuations [see inset: local (red thin line) and nonlocal (blue thick line) cases for the output field at t = L / v ].

Equations (12)

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( t + v z ) Ψ ^ ( z ) = i Δ ^ c α Ψ ^ ( z ) i Δ ^ c C v 2 z 2 Ψ ^ ( z ) , Δ ^ c = δ c + δ ^ NL , δ ^ NL = α L d z U ( z z ) Ψ ^ ( z ) Ψ ^ ( z ) .
H A F = n a d z [ i g E ^ ( z ) e i k 0 z σ ^ e g ( z ) + h.c. ] , H A C = n a d z [ i Ω e i δ c t e i k c z σ ^ e d ( z ) + h.c. ] , H D D = 1 2 n a 2 d z d z U ( z z ) σ ^ d d ( z ) σ ^ d d ( z )
σ ¯ g d ( z ) = g Ω E ^ ( z ) 1 Ω [ ( t + γ ) ( 1 Ω * ) × ( t + i δ c + i S ^ ( z ) ) σ ¯ g d ( z ) F ^ ] , ( t + c z ) E ^ ( z ) = n a g * Ω * [ t + i δ c + i S ^ ( z ) ] σ ¯ g d ( z ) ,
S ^ ( z ) = n a L d z U ( z z ) σ ¯ g d ( z ) σ ¯ g d ( z ) .
( Ψ ^ ( z ) Φ ^ ( z ) ) = ( cos θ N sin θ sin θ N cos θ ) ( E ^ ( z ) σ ¯ g d ( z ) / L ) ,
( t + c cos 2 θ z ) Ψ ^ ( z ) = sin θ cos θ c z Φ ^ ( z ) i sin θ ( δ c + S ^ ( z ) ) [ sin θ Ψ ^ ( z ) cos θ Φ ^ ( z ) ] , Φ ^ ( z ) = cos θ | Ω | 2 ( t + γ ) ( t + i δ c + i S ^ ( z ) ) × [ sin θ Ψ ^ ( z ) cos θ Φ ^ ( z ) ] + n a cos θ Ω F ^ .
ϕ ( z , t ) = e i [ φ ( α δ c + n p U 0 ) t ] [ u k e i k z e i ( ω k + k v ) t v k * e i k z e i ( ω k + k v ) t ]
ω k = ω k 0 ( ω k 0 + 2 n p U k ) , ω k 0 = ( n p U 0 / α + δ c ) C v 2 k 2 .
ω ( k ) = α δ c + n p U 0 ± ( v k + ω k ) ,
a ^ k ( t ) = e i ( n p U 0 + α δ c + k v ) t [ μ k ( t ) a ^ k ( 0 ) + e i 2 φ ν k ( t ) a ^ k ( 0 ) ] , μ k ( t ) = cos ( ω k t ) i n p U k + ω k 0 ω k sin ( ω k t ) , ν k ( t ) = i n p U k ω k sin ( ω k t ) .
g ( 2 ) ( z , z ) 1 + 2 α 2 π n p 0 d k [ | μ k | 2 N k + | ν k | 2 ( N k + 1 ) + ( 2 N k + 1 ) | μ k | | ν k | cos φ k ] cos [ k ( z z ) ] ,
U ( z ) = U L 1 2 cos ( k L z z ) cos ( k B z ) e | z | / l .

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