Abstract

Self-organization in driven dissipative systems is an active field of research with implications in physics, chemistry, and biology. When applying a red-detuned retroreflected laser beam to a large cloud of cold atoms, we observe the spontaneous formation of 2D structures in the transverse plane corresponding to high contrast spatial modulations of both light field and atomic spins. By applying a weak magnetic field, we explore the rich resulting phase space and identify specific phases associated with both dipolar and quadrupolar terms of the atomic magnetic moment. In particular, we demonstrate spontaneous structures in optically induced ground state coherences representing magnetic quadrupoles. Our results illustrate the wealth of behavior exhibited by laser-driven atomic media with complex level structure under optical feedback.

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

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  75. See Supplement 1
  76. E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin, and J. I. Cirac, “Dissipative phase transition in a central spin system,” Phys. Rev. A 86, 012116 (2012).
    [Crossref]

2018 (2)

M. Landini, N. Dogra, K. Kroeger, L. Hruby, T. Donner, and T. Esslinger, “Formation of a spin texture in a quantum gas coupled to a cavity,” Phys. Rev. Lett. 120, 223602 (2018).
[Crossref]

I. Kresic, G. Labeyrie, G. R. M. Robb, G.-L. Oppo, P. M. Gomes, P. Griffin, R. Kaiser, and T. Ackemann, “Spontaneous light-mediated magnetism in cold atoms,” Commun. Phys. 1, 33 (2018).

2017 (7)

H. J. Lee, M. Choi, and G. S. Jeon, “Emergent incommensurate correlations in frustrated ferromagnetic spin-1 chains,” Phys. Rev. B 95, 024424 (2017).
[Crossref]

A. J. Kollar, A. T. Papageorge, V. D. Vaidya, Y. Guo, J. Keeling, and B. L. Lev, “Supermode-density-wave-polariton condensation with a Bose–Einstein condensate in a multimode cavity,” Nat. Commun. 8, 14386 (2017).
[Crossref]

J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, “Supersolid formation in a quantum gas breaking a continuous translational symmetry,” Nature 543, 87–90 (2017).
[Crossref]

J. Zeiher, J.-Y. Choi, A. Rubio-Abadal, T. Pohl, R. van Bijnen, I. Bloch, and C. Gross, “Coherent many-body spin dynamics in a long-range interacting Ising chain,” Phys. Rev. X 7, 041063 (2017).
[Crossref]

H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551, 579–584 (2017).
[Crossref]

C. Gross and I. Bloch, “Quantum simulations with ultracold atoms in optical lattices,” Science 357, 995–1001 (2017).
[Crossref]

W. J. Firth, I. Kresic, G. Labeyrie, A. Camara, and T. Ackemann, “Thick-medium model of transverse pattern formation in optically excited cold two-level atoms with a feedback mirror,” Phys. Rev. A 96, 053806 (2017).
[Crossref]

2016 (6)

H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut, and T. Pfau, “Observing the Rosensweig instability of a quantum ferrofluid,” Nature 530, 194–197 (2016).
[Crossref]

H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macr, T. Lahaye, and A. Browaeys, “Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models,” Nature 534, 667–670 (2016).
[Crossref]

Z.-X. Gong, M. F. Maghrebi, A. Hu, M. Foss-Feig, P. Richerme, C. Monroe, and A. V. Gorshkov, “Kaleidoscope of quantum phases in a long-range interacting spin-1 chain,” Phys. Rev. B 93, 205115 (2016).
[Crossref]

G. Labeyrie and R. Kaiser, “Kibble–Zurek mechanism in the self-organization of a cold atomic cloud,” Phys. Rev. Lett. 117, 275701 (2016).
[Crossref]

B. L. Schmittberger and D. J. Gauthier, “Spontaneous emergence of free-space optical and atomic patterns,” New J. Phys. 18, 103021 (2016).
[Crossref]

B. L. Schmittberger and D. J. Gauthier, “Transverse optical and atomic pattern formation,” J. Opt. Soc. Am. B 33, 1543–1551 (2016).
[Crossref]

2015 (3)

A. Camara, R. Kaiser, G. Labeyrie, W. J. Firth, G.-L. Oppo, G. R. M. Robb, A. S. Arnold, and T. Ackemann, “Optical pattern formation with a two-level nonlinearity,” Phys. Rev. A 92, 013820 (2015).
[Crossref]

C. Senko, P. Richerme, J. Smith, A. Lee, I. Cohen, A. Retzker, and C. Monroe, “Realization of a quantum integer-spin chain with controllable interactions,” Phys. Rev. X 5, 021026 (2015).
[Crossref]

N. Radwell, T. W. Clark, B. Piccirillo, S. M. Barnett, and S. Franke-Arnold, “Spatially dependent electromagnetically induced transparency,” Phys. Rev. Lett. 114, 123603 (2015).
[Crossref]

2014 (2)

M. Eslami, R. Kheradmand, D. McArthur, and G.-L. Oppo, “Complex structures in media displaying electromagnetically induced transparency: pattern multistability and competition,” Phys. Rev. A 90, 023840 (2014).
[Crossref]

G. Labeyrie, E. Tesio, P. M. Gomes, G.-L. Oppo, W. J. Firth, G. R. M. Robb, A. S. Arnold, R. Kaiser, and T. Ackemann, “Optomechanical self-structuring in a cold atomic gas,” Nat. Photonics 8, 321–325 (2014).
[Crossref]

2013 (3)

G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013).
[Crossref]

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
[Crossref]

D. Greif, T. Uehlinger, G. Jotzu, L. Tarruel, and T. Esslinger, “Short-range quantum magnetism of ultracold fermions in an optical lattice,” Science 340, 1307–1310 (2013).
[Crossref]

2012 (4)

D. Jacob, L. Shao, V. Corre, T. Zibold, L. De Sarlo, E. Mimoun, J. Dalibard, and F. Gerbier, “Phase diagram of spin-1 antiferromagnetic Bose–Einstein condensates,” Phys. Rev. A 86, 061601 (2012).
[Crossref]

J. A. Greenberg and D. J. Gauthier, “Steady-state, cavityless, multimode superradiance in a cold vapor,” Phys. Rev. A 86, 013823 (2012).
[Crossref]

J. A. Greenberg and D. J. Gauthier, “High-order optical nonlinearity at low light levels,” Europhys. Lett. 98, 24001 (2012).
[Crossref]

E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin, and J. I. Cirac, “Dissipative phase transition in a central spin system,” Phys. Rev. A 86, 012116 (2012).
[Crossref]

2011 (1)

2010 (3)

J. Kronjäger, C. Becker, P. Soltan-Panahi, K. Bongs, and K. Sengstock, “Spontaneous pattern formation in an antiferromagnetic quantum gas,” Phys. Rev. Lett. 105, 090402 (2010).
[Crossref]

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
[Crossref]

G.-L. Oppo, “Complex spatial structures due to atomic coherence,” J. Mod. Opt. 57, 1408–1416 (2010).
[Crossref]

2008 (3)

P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic rubidium vapor,” Phys. Rev. Lett. 100, 123903 (2008).
[Crossref]

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008).
[Crossref]

M. Vengalattore, S. R. Leslie, J. Guzman, and D. M. Stamper-Kurn, “Spontaneously modulated spin textures in a dipolar spinor Bose–Einstein condensate,” Phys. Rev. Lett. 100, 170403 (2008).
[Crossref]

2005 (3)

J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau, S. Giovanazzi, P. Pedri, and L. Santos, “Observation of dipole-dipole interaction in a degenerate quantum gas,” Phys. Rev. Lett. 95, 150406 (2005).
[Crossref]

A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308, 672–674 (2005).
[Crossref]

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

2004 (1)

M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M. Fortier, W. Zhang, L. You, and M. S. Chapman, “Observation of spinor dynamics in optically trapped 87Rb Bose–Einstein condensates,” Phys. Rev. Lett. 92, 140403 (2004).
[Crossref]

2003 (1)

A. T. Black, H. W. Chan, and V. Vuletic, “Observation of collective friction forces due to spatial self-organization of atoms: from Rayleigh to Bragg scattering,” Phys. Rev. Lett. 91, 203001 (2003).
[Crossref]

2002 (1)

P. Domokos and H. Ritsch, “Collective cooling and self-organization of atoms in a cavity,” Phys. Rev. Lett. 89, 253003 (2002).
[Crossref]

2000 (1)

R. Kuszelewicz, I. Ganne, I. Sagnes, and G. Slekys, “Optical self-organization in bulk and multiquantum well GaAlAs microresonators,” Phys. Rev. Lett. 84, 6006–6009 (2000).
[Crossref]

1999 (1)

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T. Tschudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos Solitons Fractals 10, 701–707 (1999).
[Crossref]

1998 (2)

J. Stenger, S. Inouye, D. Stamper-Kurn, H.-J. Miesner, A. Chikkatur, and W. Ketterle, “Spin domains in ground-state Bose–Einstein condensates,” Nature 396, 345–348 (1998).
[Crossref]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in Rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
[Crossref]

1997 (1)

A. Aumann, E. Büthe, Y. Logvin, T. Ackemann, and W. Lange, “Polarized patterns in sodium vapor with single mirror feedback,” Phys. Rev. A 56, R1709–R1712 (1997).
[Crossref]

1996 (1)

A. J. Scroggie and W. J. Firth, “Pattern formation in an alkali-metal vapor with a feedback mirror,” Phys. Rev. A 53, 2752–2764 (1996).
[Crossref]

1995 (2)

T. Ackemann, Y. A. Logvin, A. Heuer, and W. Lange, “Transition between positive and negative hexagons in optical pattern formation,” Phys. Rev. Lett. 75, 3450–3453 (1995).
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M. Le Berre, D. Leduc, E. Ressayre, A. Tallet, and A. Maître, “Simulation and analysis of the flower-like instability in the single-feedback mirror experiment with rubidium vapor,” Opt. Commun. 118, 447–456 (1995).
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1994 (7)

G. Grynberg, “Drift instability and light-induced spin waves in an alkali vapor with a feedback mirror,” Opt. Commun. 109, 483–486 (1994).
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G. Grynberg, A. Maître, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with a single feedback mirror,” Phys. Rev. Lett. 72, 2379–2382 (1994).
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T. Ackemann and W. Lange, “Non- and nearly hexagonal patterns in sodium vapor generated by single-mirror feedback,” Phys. Rev. A 50, R4468–R4471 (1994).
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J. B. Geddes, J. V. Moloney, E. M. Wright, and W. J. Firth, “Polarization patterns in a nonlinear cavity,” Opt. Commun. 111, 623–631 (1994).
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A. J. Scroggie, W. J. Firth, G. S. McDonald, M. Tlidi, R. Lefever, and L. A. Lugiato, “Pattern formation in a passive Kerr cavity,” Chaos Solitons Fractals 4, 1323–1354 (1994).
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M. Tlidi and P. Mandel, “Localized structures and localized patterns in optical bistability,” Chaos Solitons Fractals 4, 1475–1486 (1994).
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G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
[Crossref]

1993 (4)

E. Ciaramella, M. Tamburrini, and E. Santamato, “Talbot assisted hexagonal beam patterning in a thin liquid crystal film with a single feedback mirror at negative distance,” Appl. Phys. Lett. 63, 1604–1606 (1993).
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T. Honda, “Hexagonal pattern formation due to counterpropagation in KNbO3,” Opt. Lett. 18, 598–600 (1993).
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E. Pampaloni, S. Residori, and F. T. Arecchi, “Roll-hexagon transition in a Kerr-like experiment,” Europhys. Lett. 24, 647–652 (1993).
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B. Thüring, R. Neubecker, and T. Tschudi, “Transverse pattern formation in liquid crystal light valve feedback system,” Opt. Commun. 102, 111–115 (1993).
[Crossref]

1992 (2)

A. Petrossian, M. Pinard, A. Maître, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating laser beams in rubidium vapour,” Europhys. Lett. 18, 689–695 (1992).
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R. Macdonald and H. J. Eichler, “Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror,” Opt. Commun. 89, 289–295 (1992).
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1991 (3)

M. Kreuzer, H. Gottschilk, T. Tschudi, and R. Neubecker, “Structure formation and self-organization phenomena in bistable optical elements,” Mol. Cryst. Liq. Cryst. 207, 219–230 (1991).
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G. D’Alessandro and W. J. Firth, “Spontaneous hexagon formation in a nonlinear optical medium with feedback mirror,” Phys. Rev. Lett. 66, 2597–2600 (1991).
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K.-J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[Crossref]

1990 (3)

1988 (4)

S. A. Akhmanov, M. A. Vorontsov, and V. Y. Ivanov, “Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of ‘optical turbulence’,” JETP Lett. 47, 707–710 (1988).

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
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G. Grynberg, “Mirrorless four-wave mixing oscillation in atomic vapors,” Opt. Commun. 66, 321–324 (1988).
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A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
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1987 (1)

L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. 58, 2209–2211 (1987).
[Crossref]

1977 (1)

A. Omont, “Irreducible components of the density matrix. Application to optical pumping,” Prog. Quantum Electron. 5, 69–138 (1977).
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1976 (1)

E. Arimondo and G. Orriols, “Nonabsorbing atomic coherences by coherent two-photon transitions in a three-level optical pumping,” Lett. Nuovo Cimento 17, 333–338 (1976).
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Ackemann, T.

I. Kresic, G. Labeyrie, G. R. M. Robb, G.-L. Oppo, P. M. Gomes, P. Griffin, R. Kaiser, and T. Ackemann, “Spontaneous light-mediated magnetism in cold atoms,” Commun. Phys. 1, 33 (2018).

W. J. Firth, I. Kresic, G. Labeyrie, A. Camara, and T. Ackemann, “Thick-medium model of transverse pattern formation in optically excited cold two-level atoms with a feedback mirror,” Phys. Rev. A 96, 053806 (2017).
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A. Camara, R. Kaiser, G. Labeyrie, W. J. Firth, G.-L. Oppo, G. R. M. Robb, A. S. Arnold, and T. Ackemann, “Optical pattern formation with a two-level nonlinearity,” Phys. Rev. A 92, 013820 (2015).
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G. Labeyrie, E. Tesio, P. M. Gomes, G.-L. Oppo, W. J. Firth, G. R. M. Robb, A. S. Arnold, R. Kaiser, and T. Ackemann, “Optomechanical self-structuring in a cold atomic gas,” Nat. Photonics 8, 321–325 (2014).
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A. Aumann, E. Büthe, Y. Logvin, T. Ackemann, and W. Lange, “Polarized patterns in sodium vapor with single mirror feedback,” Phys. Rev. A 56, R1709–R1712 (1997).
[Crossref]

T. Ackemann, Y. A. Logvin, A. Heuer, and W. Lange, “Transition between positive and negative hexagons in optical pattern formation,” Phys. Rev. Lett. 75, 3450–3453 (1995).
[Crossref]

T. Ackemann and W. Lange, “Non- and nearly hexagonal patterns in sodium vapor generated by single-mirror feedback,” Phys. Rev. A 50, R4468–R4471 (1994).
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Akhmanov, S. A.

S. A. Akhmanov, M. A. Vorontsov, and V. Y. Ivanov, “Large-scale transverse nonlinear interactions in laser beams; new types of nonlinear waves; onset of ‘optical turbulence’,” JETP Lett. 47, 707–710 (1988).

Akulshin, A. M.

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in Rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
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Arecchi, F. T.

E. Pampaloni, S. Residori, and F. T. Arecchi, “Roll-hexagon transition in a Kerr-like experiment,” Europhys. Lett. 24, 647–652 (1993).
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Arimondo, E.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
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E. Arimondo and G. Orriols, “Nonabsorbing atomic coherences by coherent two-photon transitions in a three-level optical pumping,” Lett. Nuovo Cimento 17, 333–338 (1976).
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Arnold, A. S.

A. Camara, R. Kaiser, G. Labeyrie, W. J. Firth, G.-L. Oppo, G. R. M. Robb, A. S. Arnold, and T. Ackemann, “Optical pattern formation with a two-level nonlinearity,” Phys. Rev. A 92, 013820 (2015).
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G. Labeyrie, E. Tesio, P. M. Gomes, G.-L. Oppo, W. J. Firth, G. R. M. Robb, A. S. Arnold, R. Kaiser, and T. Ackemann, “Optomechanical self-structuring in a cold atomic gas,” Nat. Photonics 8, 321–325 (2014).
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Aspect, A.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
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Aumann, A.

A. Aumann, E. Büthe, Y. Logvin, T. Ackemann, and W. Lange, “Polarized patterns in sodium vapor with single mirror feedback,” Phys. Rev. A 56, R1709–R1712 (1997).
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M. Auzinsh, D. Budker, and S. Rochester, Optically Polarized Atoms: Understanding Light-Atom Interactions (Oxford University, 2010).

Barnett, S. M.

N. Radwell, T. W. Clark, B. Piccirillo, S. M. Barnett, and S. Franke-Arnold, “Spatially dependent electromagnetically induced transparency,” Phys. Rev. Lett. 114, 123603 (2015).
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Barredo, D.

H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macr, T. Lahaye, and A. Browaeys, “Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models,” Nature 534, 667–670 (2016).
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Barreiro, S.

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in Rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
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Barrett, M. D.

M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M. Fortier, W. Zhang, L. You, and M. S. Chapman, “Observation of spinor dynamics in optically trapped 87Rb Bose–Einstein condensates,” Phys. Rev. Lett. 92, 140403 (2004).
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Baumann, K.

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
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Becker, C.

J. Kronjäger, C. Becker, P. Soltan-Panahi, K. Bongs, and K. Sengstock, “Spontaneous pattern formation in an antiferromagnetic quantum gas,” Phys. Rev. Lett. 105, 090402 (2010).
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Bernien, H.

H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551, 579–584 (2017).
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Black, A. T.

A. T. Black, H. W. Chan, and V. Vuletic, “Observation of collective friction forces due to spatial self-organization of atoms: from Rayleigh to Bragg scattering,” Phys. Rev. Lett. 91, 203001 (2003).
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Bloch, D.

G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
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Bloch, I.

J. Zeiher, J.-Y. Choi, A. Rubio-Abadal, T. Pohl, R. van Bijnen, I. Bloch, and C. Gross, “Coherent many-body spin dynamics in a long-range interacting Ising chain,” Phys. Rev. X 7, 041063 (2017).
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C. Gross and I. Bloch, “Quantum simulations with ultracold atoms in optical lattices,” Science 357, 995–1001 (2017).
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Boller, K.-J.

K.-J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
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J. Kronjäger, C. Becker, P. Soltan-Panahi, K. Bongs, and K. Sengstock, “Spontaneous pattern formation in an antiferromagnetic quantum gas,” Phys. Rev. Lett. 105, 090402 (2010).
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Boyd, R. W.

D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,” Phys. Rev. Lett. 64, 1721–1724 (1990).
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Brambilla, M.

G.-L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and evolution of roll patterns in optical parametric oscillators,” Phys. Rev. A 49, 2028–2032 (1994).
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Brennecke, F.

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
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Browaeys, A.

H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macr, T. Lahaye, and A. Browaeys, “Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models,” Nature 534, 667–670 (2016).
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Budker, D.

M. Auzinsh, D. Budker, and S. Rochester, Optically Polarized Atoms: Understanding Light-Atom Interactions (Oxford University, 2010).

Büthe, E.

A. Aumann, E. Büthe, Y. Logvin, T. Ackemann, and W. Lange, “Polarized patterns in sodium vapor with single mirror feedback,” Phys. Rev. A 56, R1709–R1712 (1997).
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Camacho, R. M.

P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, “Storage and retrieval of multimode transverse images in hot atomic rubidium vapor,” Phys. Rev. Lett. 100, 123903 (2008).
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Camara, A.

W. J. Firth, I. Kresic, G. Labeyrie, A. Camara, and T. Ackemann, “Thick-medium model of transverse pattern formation in optically excited cold two-level atoms with a feedback mirror,” Phys. Rev. A 96, 053806 (2017).
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A. Camara, R. Kaiser, G. Labeyrie, W. J. Firth, G.-L. Oppo, G. R. M. Robb, A. S. Arnold, and T. Ackemann, “Optical pattern formation with a two-level nonlinearity,” Phys. Rev. A 92, 013820 (2015).
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Chan, H. W.

A. T. Black, H. W. Chan, and V. Vuletic, “Observation of collective friction forces due to spatial self-organization of atoms: from Rayleigh to Bragg scattering,” Phys. Rev. Lett. 91, 203001 (2003).
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Chang, M.-S.

M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M. Fortier, W. Zhang, L. You, and M. S. Chapman, “Observation of spinor dynamics in optically trapped 87Rb Bose–Einstein condensates,” Phys. Rev. Lett. 92, 140403 (2004).
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Chapman, M. S.

M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M. Fortier, W. Zhang, L. You, and M. S. Chapman, “Observation of spinor dynamics in optically trapped 87Rb Bose–Einstein condensates,” Phys. Rev. Lett. 92, 140403 (2004).
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J. Stenger, S. Inouye, D. Stamper-Kurn, H.-J. Miesner, A. Chikkatur, and W. Ketterle, “Spin domains in ground-state Bose–Einstein condensates,” Nature 396, 345–348 (1998).
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Choi, J.-Y.

J. Zeiher, J.-Y. Choi, A. Rubio-Abadal, T. Pohl, R. van Bijnen, I. Bloch, and C. Gross, “Coherent many-body spin dynamics in a long-range interacting Ising chain,” Phys. Rev. X 7, 041063 (2017).
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H. J. Lee, M. Choi, and G. S. Jeon, “Emergent incommensurate correlations in frustrated ferromagnetic spin-1 chains,” Phys. Rev. B 95, 024424 (2017).
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Choi, S.

H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551, 579–584 (2017).
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Ciaramella, E.

E. Ciaramella, M. Tamburrini, and E. Santamato, “Talbot assisted hexagonal beam patterning in a thin liquid crystal film with a single feedback mirror at negative distance,” Appl. Phys. Lett. 63, 1604–1606 (1993).
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Cirac, J. I.

E. M. Kessler, G. Giedke, A. Imamoglu, S. F. Yelin, M. D. Lukin, and J. I. Cirac, “Dissipative phase transition in a central spin system,” Phys. Rev. A 86, 012116 (2012).
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Clark, S. M.

A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308, 672–674 (2005).
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Clark, T. W.

N. Radwell, T. W. Clark, B. Piccirillo, S. M. Barnett, and S. Franke-Arnold, “Spatially dependent electromagnetically induced transparency,” Phys. Rev. Lett. 114, 123603 (2015).
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Cohen, I.

C. Senko, P. Richerme, J. Smith, A. Lee, I. Cohen, A. Retzker, and C. Monroe, “Realization of a quantum integer-spin chain with controllable interactions,” Phys. Rev. X 5, 021026 (2015).
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Cohen-Tannoudji, C.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, “Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping,” Phys. Rev. Lett. 61, 826–829 (1988).
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Corre, V.

D. Jacob, L. Shao, V. Corre, T. Zibold, L. De Sarlo, E. Mimoun, J. Dalibard, and F. Gerbier, “Phase diagram of spin-1 antiferromagnetic Bose–Einstein condensates,” Phys. Rev. A 86, 061601 (2012).
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Courtois, J. Y.

A. Petrossian, M. Pinard, A. Maître, J. Y. Courtois, and G. Grynberg, “Transverse pattern formation for counterpropagating laser beams in rubidium vapour,” Europhys. Lett. 18, 689–695 (1992).
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G. D’Alessandro and W. J. Firth, “Spontaneous hexagon formation in a nonlinear optical medium with feedback mirror,” Phys. Rev. Lett. 66, 2597–2600 (1991).
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Dalibard, J.

D. Jacob, L. Shao, V. Corre, T. Zibold, L. De Sarlo, E. Mimoun, J. Dalibard, and F. Gerbier, “Phase diagram of spin-1 antiferromagnetic Bose–Einstein condensates,” Phys. Rev. A 86, 061601 (2012).
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M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008).
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A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308, 672–674 (2005).
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H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macr, T. Lahaye, and A. Browaeys, “Tunable two-dimensional arrays of single Rydberg atoms for realizing quantum Ising models,” Nature 534, 667–670 (2016).
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D. Jacob, L. Shao, V. Corre, T. Zibold, L. De Sarlo, E. Mimoun, J. Dalibard, and F. Gerbier, “Phase diagram of spin-1 antiferromagnetic Bose–Einstein condensates,” Phys. Rev. A 86, 061601 (2012).
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M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, and T. Tschudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos Solitons Fractals 10, 701–707 (1999).
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D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, and G.-C. Guo, “Single-photon-level quantum image memory based on cold atomic ensembles,” Nat. Commun. 4, 2527 (2013).
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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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M. Landini, N. Dogra, K. Kroeger, L. Hruby, T. Donner, and T. Esslinger, “Formation of a spin texture in a quantum gas coupled to a cavity,” Phys. Rev. Lett. 120, 223602 (2018).
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G. Grynberg, E. Le Bihan, P. Verkerk, P. Simoneau, J. R. R. Leite, D. Bloch, S. Le Boiteux, and M. Ducloy, “Observation of instabilities due to mirrorless four-wave mixing oscillation in sodium,” Opt. Commun. 67, 363–366 (1988).
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R. Macdonald and H. J. Eichler, “Spontaneous optical pattern formation in a nematic liquid crystal with feedback mirror,” Opt. Commun. 89, 289–295 (1992).
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H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” Nature 551, 579–584 (2017).
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M. Landini, N. Dogra, K. Kroeger, L. Hruby, T. Donner, and T. Esslinger, “Formation of a spin texture in a quantum gas coupled to a cavity,” Phys. Rev. Lett. 120, 223602 (2018).
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J. Léonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, “Supersolid formation in a quantum gas breaking a continuous translational symmetry,” Nature 543, 87–90 (2017).
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D. Greif, T. Uehlinger, G. Jotzu, L. Tarruel, and T. Esslinger, “Short-range quantum magnetism of ultracold fermions in an optical lattice,” Science 340, 1307–1310 (2013).
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K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature 464, 1301–1306 (2010).
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M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008).
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Firth, W. J.

W. J. Firth, I. Kresic, G. Labeyrie, A. Camara, and T. Ackemann, “Thick-medium model of transverse pattern formation in optically excited cold two-level atoms with a feedback mirror,” Phys. Rev. A 96, 053806 (2017).
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Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. Principle of experiment. (a) Components of magnetic moment and symmetries: orientation w (left, dipole), alignment X (middle, quadrupole), and coherence between stretched states ϕ=u+iv (right, quadrupole). (b) Experimental setup. A detuned laser beam, linearly polarized along x, is sent through the cold cloud and retroreflected by a mirror (M). The transverse intensity distribution of the light is recorded in both circular polarization channels, using a quarter-wave plate (λ/4) + polarizing beam splitter (PBS) assembly.
Fig. 2.
Fig. 2. Magnetic phase space of spin structures (experimental). Note that Bx(By) is parallel (perpendicular) to the pump polarization axis. Bz is parallel to the pump propagation axis. (a) ByBz cross section of phase space. Three phases with different symmetries are observed as illustrated by NF images. These phases vanish for a large transverse field By. (b) BxBz cross section of phase space. An additional phase is observed when the transverse field Bx is increased. The structures typically take a few 10–100 μs to emerge and can persist for a few ms, suggesting that they are stable in the long term. Parameters: OD=80, I0=8  mW/cm2, δ=8Γ, d=20  mm.
Fig. 3.
Fig. 3. Coherence phase. (a) Bx scan showing the transition from the AFM phase (white background) to the coherence phase (shaded). Parameters, By=Bz=0; I0=12  mW/cm2; δ=8Γ; d=20  mm; (b) experimental (top) and numerical (bottom) NF patterns for Bx=1G, both in the ⊥ channel. The field of view of the experimental image is 3.3 mm; the typical pattern length scale is 170 μm.
Fig. 4.
Fig. 4. Experimental evidence for a strong spatial modulation of ϕ in the coherence phase. A σ+ weak probe is sent through the cloud shortly after the pump, and FF images are detected in the σ+ and σ channels. For Bx=0, all the spatially modulated probe light is detected in σ+. On the contrary, for Bx=1.5G, approximately 20% of the spatially modulated probe light is transferred to the σ channel.
Fig. 5.
Fig. 5. Alignment phase. (a) Modified setup with a PBS inside the feedback loop. We monitor the wave vector q of the spatial structures versus feedback distance d in both ∥ and ⊥ polarization channels. (b) q versus d in the ⊥ channel. q is nearly d-independent (no mirror feedback). (c) q versus d in the ∥ channel. Active mirror feedback hints at a polarization-preserving instability mechanism. B=0, I0=53  mW/cm2, δ=8Γ.

Equations (1)

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P±=ε0Re(χ±)[(1±34w+120X)E±+320(uiv)E],

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