Abstract

Collective coherent scattering of laser light induces strong light forces between polarizable point particles. These dipole forces are strongly enhanced in magnitude and distance within the field of an optical waveguide so that at low temperature the particles self-order in strongly bound regular patterns. The stationary configurations typically exhibit super-radiant scattering with strong particle and light confinement. Here we study collective excitations of such self-consistent crystalline particle-light structures as function of particle number and pump strength. Multiple scattering and absorption modify the collective particle-field eigenfrequencies and create eigenmodes of surprisingly complex nature. For larger arrays this often leads to dynamical instabilities and disintegration of the structures even if additional damping is present.

© 2015 Optical Society of America

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  1. G. Birkl, M. Gatzke, I. Deutsch, S. Rolston, and W. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
    [Crossref] [PubMed]
  2. A. T. Black, H. W. Chan, and V. Vuletić, “Observation of collective friction forces due to spatial self-organization of atoms: from rayleigh to bragg scattering,” Phys. Rev. Lett. 91, 203001 (2003).
    [Crossref] [PubMed]
  3. N. Piovella, R. Bonifacio, B. McNeil, and G. Robb, “Superradiant light scattering and grating formation in cold atomic vapours,” Opt. Commun. 187, 165–170 (2001).
    [Crossref]
  4. V. Demergis and E.-L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
    [Crossref] [PubMed]
  5. H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
    [Crossref]
  6. G. Labeyrie, E. Tesio, P. M. Gomes, G.-L. Oppo, W. J. Firth, G. R. Robb, A. S. Arnold, R. Kaiser, and T. Ackemann, “Optomechanical self-structuring in a cold atomic gas,” Nat. Photonics 8, 321–325 (2014).
    [Crossref]
  7. 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] [PubMed]
  8. T. Grießer and H. Ritsch, “Light-induced crystallization of cold atoms in a 1d optical trap,” Phys. Rev. Lett. 111, 055702 (2013).
    [Crossref]
  9. D. Holzmann, M. Sonnleitner, and H. Ritsch, “Self-ordering and collective dynamics of transversely illuminated point-scatterers in a 1d trap,” Eur. Phys. J. D 68, 352 (2014).
    [Crossref]
  10. E. Shahmoon, I. Mazets, and G. Kurizki, “Giant vacuum forces via transmission lines,” Proc. Natl. Acad. Sci. USA 111, 10485–10490 (2014).
    [Crossref] [PubMed]
  11. K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys. 82, 1767 (2010).
    [Crossref]
  12. W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B 20, 1568–1574 (2003).
    [Crossref]
  13. J.-M. R. Fournier, G. Boer, G. Delacretaz, P. M. Jacquot, J. Rohner, and R. P. Salathe, “Building optical matter with binding and trapping forces,” Proc. SPIE 5514, 309–317 (2004).
    [Crossref]
  14. D. Chang, L. Jiang, A. Gorshkov, and H. Kimble, “Cavity qed with atomic mirrors,” New J. Phys. 14, 063003 (2012).
    [Crossref]
  15. J. Asboth and P. Domokos, “Comment on “Coupled dynamics of atoms and radiation-pressure-driven interferometers” and “Superstrong coupling regime of cavity quantum electrodynamics”,” Phys. Rev. A 76, 057801 (2007).
    [Crossref]
  16. J. Asbóth, H. Ritsch, and P. Domokos, “Collective excitations and instability of an optical lattice due to unbalanced pumping,” Phys. Rev. Lett. 98, 203008 (2007).
    [Crossref] [PubMed]
  17. S. Ostermann, M. Sonnleitner, and H. Ritsch, “Scattering approach to two-colour light forces and self-ordering of polarizable particles,” New J. Phys. 16, 043017 (2014).
    [Crossref]
  18. J. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
    [Crossref]
  19. M. Sonnleitner, M. Ritsch-Marte, and H. Ritsch, “Optomechanical deformation and strain in elastic dielectrics,” New J. Phys. 14, 103011 (2012).
    [Crossref]
  20. A. Xuereb, P. Domokos, J. Asbóth, P. Horak, and T. Freegarde, “Scattering theory of cooling and heating in optomechanical systems,” Phys. Rev. A 79, 053810 (2009).
    [Crossref]
  21. M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
    [Crossref]
  22. T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
    [Crossref] [PubMed]
  23. G. Hétet, L. Slodička, M. Hennrich, and R. Blatt, “Single atom as a mirror of an optical cavity,” Phys. Rev. Lett. 107, 133002 (2011).
    [Crossref] [PubMed]

2014 (5)

D. Holzmann, M. Sonnleitner, and H. Ritsch, “Self-ordering and collective dynamics of transversely illuminated point-scatterers in a 1d trap,” Eur. Phys. J. D 68, 352 (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] [PubMed]

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

S. Ostermann, M. Sonnleitner, and H. Ritsch, “Scattering approach to two-colour light forces and self-ordering of polarizable particles,” New J. Phys. 16, 043017 (2014).
[Crossref]

T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
[Crossref] [PubMed]

2013 (3)

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] [PubMed]

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

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
[Crossref]

2012 (3)

D. Chang, L. Jiang, A. Gorshkov, and H. Kimble, “Cavity qed with atomic mirrors,” New J. Phys. 14, 063003 (2012).
[Crossref]

V. Demergis and E.-L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref] [PubMed]

M. Sonnleitner, M. Ritsch-Marte, and H. Ritsch, “Optomechanical deformation and strain in elastic dielectrics,” New J. Phys. 14, 103011 (2012).
[Crossref]

2011 (1)

G. Hétet, L. Slodička, M. Hennrich, and R. Blatt, “Single atom as a mirror of an optical cavity,” Phys. Rev. Lett. 107, 133002 (2011).
[Crossref] [PubMed]

2010 (1)

K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys. 82, 1767 (2010).
[Crossref]

2009 (1)

A. Xuereb, P. Domokos, J. Asbóth, P. Horak, and T. Freegarde, “Scattering theory of cooling and heating in optomechanical systems,” Phys. Rev. A 79, 053810 (2009).
[Crossref]

2008 (1)

J. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
[Crossref]

2007 (2)

J. Asboth and P. Domokos, “Comment on “Coupled dynamics of atoms and radiation-pressure-driven interferometers” and “Superstrong coupling regime of cavity quantum electrodynamics”,” Phys. Rev. A 76, 057801 (2007).
[Crossref]

J. Asbóth, H. Ritsch, and P. Domokos, “Collective excitations and instability of an optical lattice due to unbalanced pumping,” Phys. Rev. Lett. 98, 203008 (2007).
[Crossref] [PubMed]

2004 (1)

J.-M. R. Fournier, G. Boer, G. Delacretaz, P. M. Jacquot, J. Rohner, and R. P. Salathe, “Building optical matter with binding and trapping forces,” Proc. SPIE 5514, 309–317 (2004).
[Crossref]

2003 (2)

W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B 20, 1568–1574 (2003).
[Crossref]

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

2001 (1)

N. Piovella, R. Bonifacio, B. McNeil, and G. Robb, “Superradiant light scattering and grating formation in cold atomic vapours,” Opt. Commun. 187, 165–170 (2001).
[Crossref]

1999 (1)

M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
[Crossref]

1995 (1)

G. Birkl, M. Gatzke, I. Deutsch, S. Rolston, and W. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

Ackemann, T.

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

Antonoyiannakis, M. I.

M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
[Crossref]

Arnold, A. S.

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

Asboth, J.

J. Asboth and P. Domokos, “Comment on “Coupled dynamics of atoms and radiation-pressure-driven interferometers” and “Superstrong coupling regime of cavity quantum electrodynamics”,” Phys. Rev. A 76, 057801 (2007).
[Crossref]

Asbóth, J.

A. Xuereb, P. Domokos, J. Asbóth, P. Horak, and T. Freegarde, “Scattering theory of cooling and heating in optomechanical systems,” Phys. Rev. A 79, 053810 (2009).
[Crossref]

J. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
[Crossref]

J. Asbóth, H. Ritsch, and P. Domokos, “Collective excitations and instability of an optical lattice due to unbalanced pumping,” Phys. Rev. Lett. 98, 203008 (2007).
[Crossref] [PubMed]

Bernet, S.

Birkl, G.

G. Birkl, M. Gatzke, I. Deutsch, S. Rolston, and W. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

Black, A. T.

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

Blatt, R.

G. Hétet, L. Slodička, M. Hennrich, and R. Blatt, “Single atom as a mirror of an optical cavity,” Phys. Rev. Lett. 107, 133002 (2011).
[Crossref] [PubMed]

Boer, G.

J.-M. R. Fournier, G. Boer, G. Delacretaz, P. M. Jacquot, J. Rohner, and R. P. Salathe, “Building optical matter with binding and trapping forces,” Proc. SPIE 5514, 309–317 (2004).
[Crossref]

Bonifacio, R.

N. Piovella, R. Bonifacio, B. McNeil, and G. Robb, “Superradiant light scattering and grating formation in cold atomic vapours,” Opt. Commun. 187, 165–170 (2001).
[Crossref]

Brennecke, F.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
[Crossref]

Chan, H. W.

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

Chang, D.

D. Chang, L. Jiang, A. Gorshkov, and H. Kimble, “Cavity qed with atomic mirrors,” New J. Phys. 14, 063003 (2012).
[Crossref]

Chang, D. E.

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] [PubMed]

Cirac, J. I.

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] [PubMed]

Daley, A. J.

T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
[Crossref] [PubMed]

Delacretaz, G.

J.-M. R. Fournier, G. Boer, G. Delacretaz, P. M. Jacquot, J. Rohner, and R. P. Salathe, “Building optical matter with binding and trapping forces,” Proc. SPIE 5514, 309–317 (2004).
[Crossref]

Demergis, V.

V. Demergis and E.-L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref] [PubMed]

Deutsch, I.

G. Birkl, M. Gatzke, I. Deutsch, S. Rolston, and W. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

Dholakia, K.

K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys. 82, 1767 (2010).
[Crossref]

Domokos, P.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
[Crossref]

A. Xuereb, P. Domokos, J. Asbóth, P. Horak, and T. Freegarde, “Scattering theory of cooling and heating in optomechanical systems,” Phys. Rev. A 79, 053810 (2009).
[Crossref]

J. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
[Crossref]

J. Asbóth, H. Ritsch, and P. Domokos, “Collective excitations and instability of an optical lattice due to unbalanced pumping,” Phys. Rev. Lett. 98, 203008 (2007).
[Crossref] [PubMed]

J. Asboth and P. Domokos, “Comment on “Coupled dynamics of atoms and radiation-pressure-driven interferometers” and “Superstrong coupling regime of cavity quantum electrodynamics”,” Phys. Rev. A 76, 057801 (2007).
[Crossref]

Esslinger, T.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
[Crossref]

Firth, W. J.

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

Florin, E.-L.

V. Demergis and E.-L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref] [PubMed]

Fournier, J.-M. R.

J.-M. R. Fournier, G. Boer, G. Delacretaz, P. M. Jacquot, J. Rohner, and R. P. Salathe, “Building optical matter with binding and trapping forces,” Proc. SPIE 5514, 309–317 (2004).
[Crossref]

Freegarde, T.

A. Xuereb, P. Domokos, J. Asbóth, P. Horak, and T. Freegarde, “Scattering theory of cooling and heating in optomechanical systems,” Phys. Rev. A 79, 053810 (2009).
[Crossref]

Frick, M.

Gatzke, M.

G. Birkl, M. Gatzke, I. Deutsch, S. Rolston, and W. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

Gomes, P. M.

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

Gorshkov, A.

D. Chang, L. Jiang, A. Gorshkov, and H. Kimble, “Cavity qed with atomic mirrors,” New J. Phys. 14, 063003 (2012).
[Crossref]

Grießer, T.

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

Hennrich, M.

G. Hétet, L. Slodička, M. Hennrich, and R. Blatt, “Single atom as a mirror of an optical cavity,” Phys. Rev. Lett. 107, 133002 (2011).
[Crossref] [PubMed]

Hétet, G.

G. Hétet, L. Slodička, M. Hennrich, and R. Blatt, “Single atom as a mirror of an optical cavity,” Phys. Rev. Lett. 107, 133002 (2011).
[Crossref] [PubMed]

Holzmann, D.

D. Holzmann, M. Sonnleitner, and H. Ritsch, “Self-ordering and collective dynamics of transversely illuminated point-scatterers in a 1d trap,” Eur. Phys. J. D 68, 352 (2014).
[Crossref]

Horak, P.

A. Xuereb, P. Domokos, J. Asbóth, P. Horak, and T. Freegarde, “Scattering theory of cooling and heating in optomechanical systems,” Phys. Rev. A 79, 053810 (2009).
[Crossref]

Jacquot, P. M.

J.-M. R. Fournier, G. Boer, G. Delacretaz, P. M. Jacquot, J. Rohner, and R. P. Salathe, “Building optical matter with binding and trapping forces,” Proc. SPIE 5514, 309–317 (2004).
[Crossref]

Jiang, L.

D. Chang, L. Jiang, A. Gorshkov, and H. Kimble, “Cavity qed with atomic mirrors,” New J. Phys. 14, 063003 (2012).
[Crossref]

Kaiser, R.

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

Kimble, H.

D. Chang, L. Jiang, A. Gorshkov, and H. Kimble, “Cavity qed with atomic mirrors,” New J. Phys. 14, 063003 (2012).
[Crossref]

Kimble, H. J.

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] [PubMed]

Kurizki, G.

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

Labeyrie, G.

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

Mazets, I.

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

McNeil, B.

N. Piovella, R. Bonifacio, B. McNeil, and G. Robb, “Superradiant light scattering and grating formation in cold atomic vapours,” Opt. Commun. 187, 165–170 (2001).
[Crossref]

Oppo, G.-L.

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

Ostermann, S.

S. Ostermann, M. Sonnleitner, and H. Ritsch, “Scattering approach to two-colour light forces and self-ordering of polarizable particles,” New J. Phys. 16, 043017 (2014).
[Crossref]

Pendry, J. B.

M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
[Crossref]

Phillips, W.

G. Birkl, M. Gatzke, I. Deutsch, S. Rolston, and W. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

Pichler, H.

T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
[Crossref] [PubMed]

Piovella, N.

N. Piovella, R. Bonifacio, B. McNeil, and G. Robb, “Superradiant light scattering and grating formation in cold atomic vapours,” Opt. Commun. 187, 165–170 (2001).
[Crossref]

Ramos, T.

T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
[Crossref] [PubMed]

Ritsch, H.

D. Holzmann, M. Sonnleitner, and H. Ritsch, “Self-ordering and collective dynamics of transversely illuminated point-scatterers in a 1d trap,” Eur. Phys. J. D 68, 352 (2014).
[Crossref]

S. Ostermann, M. Sonnleitner, and H. Ritsch, “Scattering approach to two-colour light forces and self-ordering of polarizable particles,” New J. Phys. 16, 043017 (2014).
[Crossref]

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

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
[Crossref]

M. Sonnleitner, M. Ritsch-Marte, and H. Ritsch, “Optomechanical deformation and strain in elastic dielectrics,” New J. Phys. 14, 103011 (2012).
[Crossref]

J. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
[Crossref]

J. Asbóth, H. Ritsch, and P. Domokos, “Collective excitations and instability of an optical lattice due to unbalanced pumping,” Phys. Rev. Lett. 98, 203008 (2007).
[Crossref] [PubMed]

Ritsch-Marte, M.

M. Sonnleitner, M. Ritsch-Marte, and H. Ritsch, “Optomechanical deformation and strain in elastic dielectrics,” New J. Phys. 14, 103011 (2012).
[Crossref]

W. Singer, M. Frick, S. Bernet, and M. Ritsch-Marte, “Self-organized array of regularly spaced microbeads in a fiber-optical trap,” J. Opt. Soc. Am. B 20, 1568–1574 (2003).
[Crossref]

Robb, G.

N. Piovella, R. Bonifacio, B. McNeil, and G. Robb, “Superradiant light scattering and grating formation in cold atomic vapours,” Opt. Commun. 187, 165–170 (2001).
[Crossref]

Robb, G. R.

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

Rohner, J.

J.-M. R. Fournier, G. Boer, G. Delacretaz, P. M. Jacquot, J. Rohner, and R. P. Salathe, “Building optical matter with binding and trapping forces,” Proc. SPIE 5514, 309–317 (2004).
[Crossref]

Rolston, S.

G. Birkl, M. Gatzke, I. Deutsch, S. Rolston, and W. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

Salathe, R. P.

J.-M. R. Fournier, G. Boer, G. Delacretaz, P. M. Jacquot, J. Rohner, and R. P. Salathe, “Building optical matter with binding and trapping forces,” Proc. SPIE 5514, 309–317 (2004).
[Crossref]

Shahmoon, E.

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

Singer, W.

Slodicka, L.

G. Hétet, L. Slodička, M. Hennrich, and R. Blatt, “Single atom as a mirror of an optical cavity,” Phys. Rev. Lett. 107, 133002 (2011).
[Crossref] [PubMed]

Sonnleitner, M.

D. Holzmann, M. Sonnleitner, and H. Ritsch, “Self-ordering and collective dynamics of transversely illuminated point-scatterers in a 1d trap,” Eur. Phys. J. D 68, 352 (2014).
[Crossref]

S. Ostermann, M. Sonnleitner, and H. Ritsch, “Scattering approach to two-colour light forces and self-ordering of polarizable particles,” New J. Phys. 16, 043017 (2014).
[Crossref]

M. Sonnleitner, M. Ritsch-Marte, and H. Ritsch, “Optomechanical deformation and strain in elastic dielectrics,” New J. Phys. 14, 103011 (2012).
[Crossref]

Tesio, E.

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

Vuletic, V.

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

Xuereb, A.

A. Xuereb, P. Domokos, J. Asbóth, P. Horak, and T. Freegarde, “Scattering theory of cooling and heating in optomechanical systems,” Phys. Rev. A 79, 053810 (2009).
[Crossref]

Zemánek, P.

K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys. 82, 1767 (2010).
[Crossref]

Zoller, P.

T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
[Crossref] [PubMed]

Eur. Phys. J. D (1)

D. Holzmann, M. Sonnleitner, and H. Ritsch, “Self-ordering and collective dynamics of transversely illuminated point-scatterers in a 1d trap,” Eur. Phys. J. D 68, 352 (2014).
[Crossref]

J. Opt. Soc. Am. B (1)

Nano Lett. (1)

V. Demergis and E.-L. Florin, “Ultrastrong optical binding of metallic nanoparticles,” Nano Lett. 12, 5756–5760 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

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

New J. Phys. (3)

D. Chang, L. Jiang, A. Gorshkov, and H. Kimble, “Cavity qed with atomic mirrors,” New J. Phys. 14, 063003 (2012).
[Crossref]

M. Sonnleitner, M. Ritsch-Marte, and H. Ritsch, “Optomechanical deformation and strain in elastic dielectrics,” New J. Phys. 14, 103011 (2012).
[Crossref]

S. Ostermann, M. Sonnleitner, and H. Ritsch, “Scattering approach to two-colour light forces and self-ordering of polarizable particles,” New J. Phys. 16, 043017 (2014).
[Crossref]

Opt. Commun. (1)

N. Piovella, R. Bonifacio, B. McNeil, and G. Robb, “Superradiant light scattering and grating formation in cold atomic vapours,” Opt. Commun. 187, 165–170 (2001).
[Crossref]

Phys. Rev. A (3)

J. Asbóth, H. Ritsch, and P. Domokos, “Optomechanical coupling in a one-dimensional optical lattice,” Phys. Rev. A 77, 063424 (2008).
[Crossref]

A. Xuereb, P. Domokos, J. Asbóth, P. Horak, and T. Freegarde, “Scattering theory of cooling and heating in optomechanical systems,” Phys. Rev. A 79, 053810 (2009).
[Crossref]

J. Asboth and P. Domokos, “Comment on “Coupled dynamics of atoms and radiation-pressure-driven interferometers” and “Superstrong coupling regime of cavity quantum electrodynamics”,” Phys. Rev. A 76, 057801 (2007).
[Crossref]

Phys. Rev. B (1)

M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60, 2363–2374 (1999).
[Crossref]

Phys. Rev. Lett. (7)

T. Ramos, H. Pichler, A. J. Daley, and P. Zoller, “Quantum spin dimers from chiral dissipation in cold-atom chains,” Phys. Rev. Lett. 113, 237203 (2014).
[Crossref] [PubMed]

G. Hétet, L. Slodička, M. Hennrich, and R. Blatt, “Single atom as a mirror of an optical cavity,” Phys. Rev. Lett. 107, 133002 (2011).
[Crossref] [PubMed]

J. Asbóth, H. Ritsch, and P. Domokos, “Collective excitations and instability of an optical lattice due to unbalanced pumping,” Phys. Rev. Lett. 98, 203008 (2007).
[Crossref] [PubMed]

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] [PubMed]

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

G. Birkl, M. Gatzke, I. Deutsch, S. Rolston, and W. Phillips, “Bragg scattering from atoms in optical lattices,” Phys. Rev. Lett. 75, 2823 (1995).
[Crossref] [PubMed]

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

Proc. Natl. Acad. Sci. USA (1)

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

Proc. SPIE (1)

J.-M. R. Fournier, G. Boer, G. Delacretaz, P. M. Jacquot, J. Rohner, and R. P. Salathe, “Building optical matter with binding and trapping forces,” Proc. SPIE 5514, 309–317 (2004).
[Crossref]

Rev. Mod. Phys. (2)

K. Dholakia and P. Zemánek, “Colloquium: Gripped by light: Optical binding,” Rev. Mod. Phys. 82, 1767 (2010).
[Crossref]

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553–601 (2013).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 A 1D array of point particles scattering light in and out of an optical nano-structure can be modelled as a collection of beam splitters interacting with a plane wave.

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Fig. 2
Fig. 2 Phonon frequency as a function of the phonon mode ν for ζ = 0 beginning with ν = 2. The blue line corresponds to N = 3, the red line to N = 5, the green line to N = 10, the violet line to N = 20, the orange line to N = 40 and the grey line to N = 50.

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Fig. 3
Fig. 3 Ten particles evolving over time with only transverse pump and ζ = 0. In Fig. 3(a) we have chosen d = 2 N − 1 2 N λ (n = N) as initial condition slightly perturbed by ξinitial = 0.1λ, while in Fig. 3(b) the particles are initially positioned at an unstable zero force distance d = N − 1 N λ (n = N − 1). τ2,0 is defined as τ 2 , 0 = 2 π ω 2 , 0.

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Fig. 4
Fig. 4 Figure 4(a) shows the phonon frequency ων as a function of the real part of the coupling constant ζr. The blue line represents non absorbing configurations with ζi = 0, while the red line with ζ i = 1 9 and the green line to ζ i = 1 2 include increasing absorption and scattering to non propagating modes. Figure 4(b): Same as Fig. 4(a) as a function of the imaginary part of the coupling constant ζi. The blue line corresponds to ζr = 0, the red line to ζ r = 1 9 and the green line to ζ r = 1 2. We can see that the function goes to ω 2 , 0 2 for large values of ζ. For the real part of ζ we can find a maximum for ζr > 0.

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Fig. 5
Fig. 5 Two particles evolving over time with only transverse pump oscillating after a small perturbation with maximal frequency at ζ = 0.618 (Fig. 5(a)), and minimal frequency at ζ = −1.618 (Fig. 5(b)).

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Fig. 6
Fig. 6 Three particles evolving over time with only transverse pump, ζ = 1 9 (Fig. 6(a)), ζ = 1 + i 9 (Fig. 6(b)) and a small perturbation ξinitial = 0.1λ.

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Fig. 7
Fig. 7 Ten particles evolving over time with only transverse pump, ζ = 1 9 (Fig. 7(a)), ζ = 1 + i 9 (Fig. 7(b)) and a small perturbation of ζinitial = 0.001λ. Figure (Fig. 7(c)) shows the oscillation of the particles. Here we magnify the trajectories of Fig. 7(a) and plotted a range of 0.01λ. In the second case the particles, first, start to oscillate around their equilibrium positions, but the perturbation leads to antidamping, which, in the end, destroys the configuration. The eigenvalues of the first case are all negative and real, while the eigenvalues of the second case acquire positive and negative imaginary parts.

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Fig. 8
Fig. 8 Maximum real part of the nonzero eigenvalues as function of real and imaginary part of ζ. Figure 8(a) corresponds to N = 5 particles, Fig. 8(b) to N = 10, Fig. 8(c) to N = 15 and Fig. 8(d) to N = 20. Black lines show the zero value contour lines.

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Equations (28)

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( A j B j η ) = 1 t ( t 2 − r 2 r 1 2 ( t − r ) − r 1 − 1 2 0 0 t ) ( C j D j η ) = ( 1 + i ζ i ζ 1 2 ( 1 − i ζ ) − i ζ 1 − i ζ 1 2 ( i ζ − 1 ) 0 0 1 ) ( C j D j η ) ,
F j = ε 0 2 ( | A j | 2 + | B j | 2 − | C j | 2 − | D j | 2 ) .
m x ¨ j = − μ x ˙ j + F j ( x 1 , … , x N ) .
m ξ ¨ j = − μ ξ ˙ j + ∑ l = 1 N D j l ξ l ,
D j l = ∂ ∂ x l F j ( x 1 = x 1 ( 0 ) , … , x N = x N ( 0 ) ) = lim ξ → 0 1 ξ F j ( x ν = x v ( 0 ) + δ l v ξ , v = 1 , … , N ) .
ω ν = i μ ± − μ 2 − 4 m λ ν 2 m ,
ξ j = ( A j e i − μ 2 − 4 m λ ν 2 m t + B j e − i − μ 2 − 4 m λ ν 2 m t ) e − μ t 2 m .
m ( ℑ ( λ ν ) ) 2 ≤ − μ 2 ℜ ( λ ν ) .
ξ j = A j e i − λ ν m + B j e − i − λ ν m
M TM = ( M BS ⋅ P ( d ) ) N − 1 ⋅ M BS = M N − 1 ⋅ M BS .
I ol = I or = I η 2 ( sin ( N k d 2 ) sin ( k d 2 ) ) 2
F j = − P η cos ( N k d / 2 ) sin ( ( 2 j − N − 1 ) k d / 2 ) sin ( k d / 2 ) ,
( A j B j η ) = M N − j + 1 P ( − d ) ( ( 0 D N η ) + ε ( 0 b 0 ) ) ,
( A j B j η ) = M l − j P ( ε k ) M BS P ( − ε k ) P ( d ) M N − l P ( − d ) ( ( 0 D N η ) + ε ( 0 b 0 ) ) .
F j ( d = 2 N − 1 2 N λ ) = { P η ε sin ( | j − l | π N ) , for   j ≠ l − P η ε cos ( π 2 N ) sin ( π 2 N ) , for   j = l ,
D j l = P η k ( sin ( | j − l | π N ) − δ j l cot ( π 2 N ) ) .
λ ν = − 2 P η k cot ( π 2 N ) sin 2 ( π ( ν − 1 ) N ) cos ( π N ) − cos ( 2 π ( v − 1 ) N ) , z → ν = ( e 2 π i ( ν − 1 ) ( j − 1 ) N ) j = 1 N .
ω ν = P η k m ( cot ( π 2 N ) + sin ( π N ) cos ( π N ) − cos ( 2 π ( ν − 1 ) N ) ) = ω 2 , 0 cot ( π 2 N ) sin 2 ( π ( ν − 1 ) N ) cos ( π N ) − cos ( 2 π ( ν − 1 ) N ) ,
ω ν ≈ 2 ( ν − 1 ) ω 2 , 0 N π ( 3 + 4 ν ( ν − 2 ) ) .
F j = P η k ( ∑ l ≠ j sin ( | j − l | π N ) ( ξ l − ξ j ) ) ,
λ ν = P η k 2 ( 2 n − 1 ) 2 4 ( ν − 1 ) 2 − ( 2 n − 1 ) 2 .
F 1 = − F 2 = P η | 1 − i ζ | 2 cos ( k d ) 4 ( | ζ | 2 + ζ i ) cos 2 ( k d 2 ) + 2 ζ r sin ( k d ) + 1 ,
D j l = ( − 1 ) l − j P η k | 1 − i ζ | 2 ( sin ( k d ) ( 2 | ζ | 2 + 2 ζ i + 1 ) + 2 ζ r ) ( 2 ( cos ( k d ) + 1 ) ( | ζ | 2 + ζ i ) + 2 sin ( k d ) ζ r + 1 ) 2 ,
λ 2 = 2 k P η | 1 − i ζ | 2 1 + 2 ( | ζ | 2 + ζ i − ζ r ) ,
ω = ω 2 , 0 | 1 − i ζ | 2 1 + 2 ( | ζ | 2 + ζ i − ζ r ) .
F 1 = − F 3 = ≃ P η ( cos ( k d ) + cos ( 2 k d ) − ζ ( 2 sin ( 2 k d ) + sin ( 3 k d ) + sin ( 4 k d ) ) ) + O [ ζ ] 2 , F 2 = 0
z 1 = 1 3 ( 1 1 1 ) , z 2 = ( 0.287 − 0.914 0.287 ) , z 3 = 1 2 ( − 1 0 1 ) .
z 1 = 1 3 ( 1 1 1 ) ,       z 2 = ( 0.349 − 0.869 0.349 ) ,     z 3 = 1 2 ( − 1 0 1 ) .

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