Abstract

We have used diffraction gratings to simplify the fabrication, and dramatically increase the atomic collection efficiency, of magneto-optical traps using micro-fabricated optics. The atom number enhancement was mainly due to the increased beam capture volume, afforded by the large area (4cm2) shallow etch (~ 200nm) binary grating chips. Here we provide a detailed theoretical and experimental investigation of the on-chip magneto-optical trap temperature and density in four different chip geometries using 87Rb, whilst studying effects due to MOT radiation pressure imbalance. With optimal initial MOTs on two of the chips we obtain both large atom number (2×107) and sub-Doppler temperatures (50 μK) after optical molasses.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  33. M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  37. J. A. Rushton, M. Aldous, and M. D. Himsworth, “The feasibility of a fully miniaturized magneto-optical trap for portable ultracold quantum technology,” Rev. Sci. Instrum. 85, 121501 (2014).
    [Crossref]
  38. J. P. McGilligan, P. F. Griffin, E. Riis, and A. S. Arnold, in preparation (2015).
  39. J. Lee, J. A. Grover, L. A. Orozco, and S. L. Rolston, “Sub-Doppler cooling of neutral atoms in a grating magneto-optical trap,” J. Opt. Soc. Am. B 30, 2869–2874 (2013).
    [Crossref]

2014 (6)

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,” Nature Phot. 8, 321–325 (2014).
[Crossref]

A. Camara, R. Kaiser, and G. Labeyrie, “Behavior of a very large magneto-optical trap,” Phys. Rev. A 90, 063404 (2014).
[Crossref]

J. Miao, J. Hostetter, G. Stratis, and M. Saffman, “Magneto-optical trapping of holmium atoms,” Phys. Rev. A 89, 041401 (2014).
[Crossref]

J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. DeMille, “Magneto-optical trapping of a diatomic molecule,” Nature 512, 286–289 (2014).
[Crossref] [PubMed]

N. Poli, M. Schioppo, S. Vogt, St. Falke, U. Sterr, Ch. Lisdat, and G. M. Tino, “A transportable strontium optical lattice clock,” Appl. Phys. B 117, 1107–1116 (2014).
[Crossref]

J. A. Rushton, M. Aldous, and M. D. Himsworth, “The feasibility of a fully miniaturized magneto-optical trap for portable ultracold quantum technology,” Rev. Sci. Instrum. 85, 121501 (2014).
[Crossref]

2013 (5)

F.-X. Esnault, E. Blanshan, E. N. Ivanov, R. E. Scholten, J. Kitching, and E. A. Donley, “Cold-atom double-Λ coherent population trapping clock,” Phys. Rev. A 88, 042120 (2013).
[Crossref]

C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

N. Radwell, G. Walker, and S. Franke-Arnold, “Cold-atom densities of more than 1012 cm−3 in a holographically shaped dark spontaneous-force optical trap,” Phys. Rev. A 88, 043409 (2013).
[Crossref]

K. Jooya, N. Musterer, K. W. Madison, and J. L. Booth, “Photon-scattering-rate measurement of atoms in a magneto-optical trap,” Phys. Rev. A 88, 063401 (2013).
[Crossref]

J. Lee, J. A. Grover, L. A. Orozco, and S. L. Rolston, “Sub-Doppler cooling of neutral atoms in a grating magneto-optical trap,” J. Opt. Soc. Am. B 30, 2869–2874 (2013).
[Crossref]

2012 (1)

L. Huet, M. Ammar, E. Morvan, N. Sarazin, J. P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, “Experimental investigation of transparent silicon carbide for atom chips,” Appl. Phys. Lett. 100, 121114 (2012).
[Crossref]

2011 (1)

S. Pollock, J. P. Cotter, A. Laliotis, F. Ramirez-Martinez, and E. A. Hinds, “Characteristics of integrated magneto-optical traps for atom chips,” New J. Phys. 13, 043029 (2011).
[Crossref]

2010 (1)

2009 (2)

2001 (1)

W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[Crossref] [PubMed]

2000 (1)

1999 (1)

J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
[Crossref]

1998 (1)

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1998).
[Crossref]

1996 (1)

1995 (1)

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[Crossref] [PubMed]

1994 (3)

M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[Crossref]

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[Crossref]

Z. Hu and H. J. Kimble, “Observation of a single atom in a magneto-optical trap,” Opt. Lett. 19, 1888–1890 (1994).
[Crossref] [PubMed]

1993 (1)

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[Crossref] [PubMed]

1992 (2)

C. D. Wallace, T. P. Dinneen, K-Y. N. Tan, T. T. Grove, and P. L. Gould, “Isotopic difference in trap loss collisions of laser cooled rubidium atoms,” Phys. Rev. Lett. 69897–900 (1992).
[Crossref] [PubMed]

K. Lindquist, M. Stephens, and C. E. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082–4090 (1992).
[Crossref] [PubMed]

1991 (2)

1990 (1)

C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571–1574 (1990).
[Crossref] [PubMed]

1989 (3)

1987 (1)

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[Crossref] [PubMed]

1985 (1)

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, “Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure,” Phys. Rev. Lett. 55, 48–51 (1985).
[Crossref] [PubMed]

Ackemann, T.

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,” Nature Phot. 8, 321–325 (2014).
[Crossref]

Aldous, M.

J. A. Rushton, M. Aldous, and M. D. Himsworth, “The feasibility of a fully miniaturized magneto-optical trap for portable ultracold quantum technology,” Rev. Sci. Instrum. 85, 121501 (2014).
[Crossref]

Ammar, M.

L. Huet, M. Ammar, E. Morvan, N. Sarazin, J. P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, “Experimental investigation of transparent silicon carbide for atom chips,” Appl. Phys. Lett. 100, 121114 (2012).
[Crossref]

Arnold, A. S.

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,” Nature Phot. 8, 321–325 (2014).
[Crossref]

C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, “Laser cooling with a single laser beam and a planar diffractor,” Opt. Lett. 35, 3453–3455 (2010).
[Crossref] [PubMed]

M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, “Single-laser, one beam, tetrahedral magneto-optical trap,” Opt. Express 17, 13601–13608 (2009).
[Crossref] [PubMed]

A. S. Arnold and P. J. Manson, “Atomic density and temperature distributions in magneto-optical traps,” J. Opt. Soc. Am. B 17, 497–506 (2000).
[Crossref]

J. P. McGilligan, P. F. Griffin, E. Riis, and A. S. Arnold, in preparation (2015).

A. S. Arnold, Fig. 4.12, (DPhil thesis, Sussex, 1999).

J. P. Cotter, P. F. Griffin, E. A. Hinds, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, in preparation.

Ashkin, A.

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, “Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure,” Phys. Rev. Lett. 55, 48–51 (1985).
[Crossref] [PubMed]

Barry, J. F.

J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. DeMille, “Magneto-optical trapping of a diatomic molecule,” Nature 512, 286–289 (2014).
[Crossref] [PubMed]

Bjorkholm, J. E.

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, “Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure,” Phys. Rev. Lett. 55, 48–51 (1985).
[Crossref] [PubMed]

Blanshan, E.

F.-X. Esnault, E. Blanshan, E. N. Ivanov, R. E. Scholten, J. Kitching, and E. A. Donley, “Cold-atom double-Λ coherent population trapping clock,” Phys. Rev. A 88, 042120 (2013).
[Crossref]

Booth, J. L.

K. Jooya, N. Musterer, K. W. Madison, and J. L. Booth, “Photon-scattering-rate measurement of atoms in a magneto-optical trap,” Phys. Rev. A 88, 063401 (2013).
[Crossref]

Cable, A.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[Crossref] [PubMed]

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, “Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure,” Phys. Rev. Lett. 55, 48–51 (1985).
[Crossref] [PubMed]

Camara, A.

A. Camara, R. Kaiser, and G. Labeyrie, “Behavior of a very large magneto-optical trap,” Phys. Rev. A 90, 063404 (2014).
[Crossref]

Castin, Y.

M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[Crossref]

Chu, S.

D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, and S. Chu, “Optical molasses and multilevel atoms: experiment,” J. Opt. Soc. Am. B 6, 2072–2083 (1989).
[Crossref]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[Crossref] [PubMed]

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, “Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure,” Phys. Rev. Lett. 55, 48–51 (1985).
[Crossref] [PubMed]

Clairon, A.

M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[Crossref]

Cohen-Tannoudji, C.

Cooper, C. J.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[Crossref] [PubMed]

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[Crossref]

Cotter, J. P.

C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

S. Pollock, J. P. Cotter, A. Laliotis, F. Ramirez-Martinez, and E. A. Hinds, “Characteristics of integrated magneto-optical traps for atom chips,” New J. Phys. 13, 043029 (2011).
[Crossref]

S. Pollock, J. P. Cotter, A. Laliotis, and E. A. Hinds, “Integrated magneto-optical traps on a chip using silicon pyramid structures,” Opt. Express 17, 14109–14114 (2009).
[Crossref] [PubMed]

J. P. Cotter, P. F. Griffin, E. A. Hinds, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, in preparation.

Dalibard, J.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[Crossref] [PubMed]

J. Dalibard and C. Cohen-Tannoudji, “Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,” J. Opt. Soc. Am. B 6, 2023–2045 (1989).
[Crossref]

Davis, K. B.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[Crossref] [PubMed]

DeMille, D.

J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. DeMille, “Magneto-optical trapping of a diatomic molecule,” Nature 512, 286–289 (2014).
[Crossref] [PubMed]

Dinneen, T. P.

C. D. Wallace, T. P. Dinneen, K-Y. N. Tan, T. T. Grove, and P. L. Gould, “Isotopic difference in trap loss collisions of laser cooled rubidium atoms,” Phys. Rev. Lett. 69897–900 (1992).
[Crossref] [PubMed]

Donley, E. A.

F.-X. Esnault, E. Blanshan, E. N. Ivanov, R. E. Scholten, J. Kitching, and E. A. Donley, “Cold-atom double-Λ coherent population trapping clock,” Phys. Rev. A 88, 042120 (2013).
[Crossref]

Drewsen, M.

M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[Crossref]

Edwards, N. H.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[Crossref] [PubMed]

Esnault, F.-X.

F.-X. Esnault, E. Blanshan, E. N. Ivanov, R. E. Scholten, J. Kitching, and E. A. Donley, “Cold-atom double-Λ coherent population trapping clock,” Phys. Rev. A 88, 042120 (2013).
[Crossref]

Falke, St.

N. Poli, M. Schioppo, S. Vogt, St. Falke, U. Sterr, Ch. Lisdat, and G. M. Tino, “A transportable strontium optical lattice clock,” Appl. Phys. B 117, 1107–1116 (2014).
[Crossref]

Firth, W. J.

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,” Nature Phot. 8, 321–325 (2014).
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Foot, C. J.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[Crossref] [PubMed]

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[Crossref]

Franke-Arnold, S.

N. Radwell, G. Walker, and S. Franke-Arnold, “Cold-atom densities of more than 1012 cm−3 in a holographically shaped dark spontaneous-force optical trap,” Phys. Rev. A 88, 043409 (2013).
[Crossref]

Gomes, P. M.

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,” Nature Phot. 8, 321–325 (2014).
[Crossref]

Gould, P. L.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1998).
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C. D. Wallace, T. P. Dinneen, K-Y. N. Tan, T. T. Grove, and P. L. Gould, “Isotopic difference in trap loss collisions of laser cooled rubidium atoms,” Phys. Rev. Lett. 69897–900 (1992).
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C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, “Laser cooling with a single laser beam and a planar diffractor,” Opt. Lett. 35, 3453–3455 (2010).
[Crossref] [PubMed]

M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, “Single-laser, one beam, tetrahedral magneto-optical trap,” Opt. Express 17, 13601–13608 (2009).
[Crossref] [PubMed]

J. P. Cotter, P. F. Griffin, E. A. Hinds, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, in preparation.

J. P. McGilligan, P. F. Griffin, E. Riis, and A. S. Arnold, in preparation (2015).

Grison, D.

M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[Crossref]

Grove, T. T.

C. D. Wallace, T. P. Dinneen, K-Y. N. Tan, T. T. Grove, and P. L. Gould, “Isotopic difference in trap loss collisions of laser cooled rubidium atoms,” Phys. Rev. Lett. 69897–900 (1992).
[Crossref] [PubMed]

Grover, J. A.

Guerlin, C.

L. Huet, M. Ammar, E. Morvan, N. Sarazin, J. P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, “Experimental investigation of transparent silicon carbide for atom chips,” Appl. Phys. Lett. 100, 121114 (2012).
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W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[Crossref] [PubMed]

J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
[Crossref]

Hänsel, W.

W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[Crossref] [PubMed]

J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
[Crossref]

Hillenbrand, G.

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[Crossref]

Himsworth, M. D.

J. A. Rushton, M. Aldous, and M. D. Himsworth, “The feasibility of a fully miniaturized magneto-optical trap for portable ultracold quantum technology,” Rev. Sci. Instrum. 85, 121501 (2014).
[Crossref]

Hinds, E. A.

C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

S. Pollock, J. P. Cotter, A. Laliotis, F. Ramirez-Martinez, and E. A. Hinds, “Characteristics of integrated magneto-optical traps for atom chips,” New J. Phys. 13, 043029 (2011).
[Crossref]

S. Pollock, J. P. Cotter, A. Laliotis, and E. A. Hinds, “Integrated magneto-optical traps on a chip using silicon pyramid structures,” Opt. Express 17, 14109–14114 (2009).
[Crossref] [PubMed]

J. P. Cotter, P. F. Griffin, E. A. Hinds, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, in preparation.

Hollberg, L.

S. Chu, L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin, “Three-dimensional viscous confinement and cooling of atoms by resonance radiation pressure,” Phys. Rev. Lett. 55, 48–51 (1985).
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Hommelhoff, P.

W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[Crossref] [PubMed]

Hostetter, J.

J. Miao, J. Hostetter, G. Stratis, and M. Saffman, “Magneto-optical trapping of holmium atoms,” Phys. Rev. A 89, 041401 (2014).
[Crossref]

Hu, Z.

Huet, L.

L. Huet, M. Ammar, E. Morvan, N. Sarazin, J. P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, “Experimental investigation of transparent silicon carbide for atom chips,” Appl. Phys. Lett. 100, 121114 (2012).
[Crossref]

Ironside, C. N.

C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

Ivanov, E. N.

F.-X. Esnault, E. Blanshan, E. N. Ivanov, R. E. Scholten, J. Kitching, and E. A. Donley, “Cold-atom double-Λ coherent population trapping clock,” Phys. Rev. A 88, 042120 (2013).
[Crossref]

Jhe, W.

Joffe, M. A.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[Crossref] [PubMed]

Jooya, K.

K. Jooya, N. Musterer, K. W. Madison, and J. L. Booth, “Photon-scattering-rate measurement of atoms in a magneto-optical trap,” Phys. Rev. A 88, 063401 (2013).
[Crossref]

Kaiser, R.

A. Camara, R. Kaiser, and G. Labeyrie, “Behavior of a very large magneto-optical trap,” Phys. Rev. A 90, 063404 (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,” Nature Phot. 8, 321–325 (2014).
[Crossref]

Ketterle, W.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[Crossref] [PubMed]

Kim, J. A.

Kimble, H. J.

Kitching, J.

F.-X. Esnault, E. Blanshan, E. N. Ivanov, R. E. Scholten, J. Kitching, and E. A. Donley, “Cold-atom double-Λ coherent population trapping clock,” Phys. Rev. A 88, 042120 (2013).
[Crossref]

Labeyrie, G.

A. Camara, R. Kaiser, and G. Labeyrie, “Behavior of a very large magneto-optical trap,” Phys. Rev. A 90, 063404 (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,” Nature Phot. 8, 321–325 (2014).
[Crossref]

Laliotis, A.

S. Pollock, J. P. Cotter, A. Laliotis, F. Ramirez-Martinez, and E. A. Hinds, “Characteristics of integrated magneto-optical traps for atom chips,” New J. Phys. 13, 043029 (2011).
[Crossref]

S. Pollock, J. P. Cotter, A. Laliotis, and E. A. Hinds, “Integrated magneto-optical traps on a chip using silicon pyramid structures,” Opt. Express 17, 14109–14114 (2009).
[Crossref] [PubMed]

Laurent, P. H.

M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[Crossref]

Lee, J.

Lee, K. I.

Lett, P. D.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1998).
[Crossref]

P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, “Optical molasses,” J. Opt. Soc. Am. B 6, 2084–2107 (1989).
[Crossref]

Lindquist, K.

K. Lindquist, M. Stephens, and C. E. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082–4090 (1992).
[Crossref] [PubMed]

Lisdat, Ch.

N. Poli, M. Schioppo, S. Vogt, St. Falke, U. Sterr, Ch. Lisdat, and G. M. Tino, “A transportable strontium optical lattice clock,” Appl. Phys. B 117, 1107–1116 (2014).
[Crossref]

Madison, K. W.

K. Jooya, N. Musterer, K. W. Madison, and J. L. Booth, “Photon-scattering-rate measurement of atoms in a magneto-optical trap,” Phys. Rev. A 88, 063401 (2013).
[Crossref]

Manson, P. J.

Martin, A.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[Crossref] [PubMed]

McCarron, D. J.

J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. DeMille, “Magneto-optical trapping of a diatomic molecule,” Nature 512, 286–289 (2014).
[Crossref] [PubMed]

McGilligan, J. P.

J. P. McGilligan, P. F. Griffin, E. Riis, and A. S. Arnold, in preparation (2015).

Metcalf, H. J.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1998).
[Crossref]

Miao, J.

J. Miao, J. Hostetter, G. Stratis, and M. Saffman, “Magneto-optical trapping of holmium atoms,” Phys. Rev. A 89, 041401 (2014).
[Crossref]

Monroe, C.

C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571–1574 (1990).
[Crossref] [PubMed]

Morvan, E.

L. Huet, M. Ammar, E. Morvan, N. Sarazin, J. P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, “Experimental investigation of transparent silicon carbide for atom chips,” Appl. Phys. Lett. 100, 121114 (2012).
[Crossref]

Musterer, N.

K. Jooya, N. Musterer, K. W. Madison, and J. L. Booth, “Photon-scattering-rate measurement of atoms in a magneto-optical trap,” Phys. Rev. A 88, 063401 (2013).
[Crossref]

Nadir, A.

M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[Crossref]

Noh, H. R.

Norrgard, E. B.

J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. DeMille, “Magneto-optical trapping of a diatomic molecule,” Nature 512, 286–289 (2014).
[Crossref] [PubMed]

Nshii, C. C.

C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

Oppo, G. L.

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,” Nature Phot. 8, 321–325 (2014).
[Crossref]

Orozco, L. A.

Perrin, H.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[Crossref] [PubMed]

Phillips, W. D.

P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, “Observation of atoms laser cooled below the Doppler limit,” Phys. Rev. Lett. 61, 169–172 (1998).
[Crossref]

P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, “Optical molasses,” J. Opt. Soc. Am. B 6, 2084–2107 (1989).
[Crossref]

Pocholle, J. P.

L. Huet, M. Ammar, E. Morvan, N. Sarazin, J. P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, “Experimental investigation of transparent silicon carbide for atom chips,” Appl. Phys. Lett. 100, 121114 (2012).
[Crossref]

Poli, N.

N. Poli, M. Schioppo, S. Vogt, St. Falke, U. Sterr, Ch. Lisdat, and G. M. Tino, “A transportable strontium optical lattice clock,” Appl. Phys. B 117, 1107–1116 (2014).
[Crossref]

Pollock, S.

S. Pollock, J. P. Cotter, A. Laliotis, F. Ramirez-Martinez, and E. A. Hinds, “Characteristics of integrated magneto-optical traps for atom chips,” New J. Phys. 13, 043029 (2011).
[Crossref]

S. Pollock, J. P. Cotter, A. Laliotis, and E. A. Hinds, “Integrated magneto-optical traps on a chip using silicon pyramid structures,” Opt. Express 17, 14109–14114 (2009).
[Crossref] [PubMed]

Prentiss, M.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[Crossref] [PubMed]

Pritchard, D. E.

W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, “High densities of cold atoms in a dark spontaneous force optical trap,” Phys. Rev. Lett. 70, 2253–2256 (1993).
[Crossref] [PubMed]

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[Crossref] [PubMed]

Raab, E. L.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, and D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987).
[Crossref] [PubMed]

Radwell, N.

N. Radwell, G. Walker, and S. Franke-Arnold, “Cold-atom densities of more than 1012 cm−3 in a holographically shaped dark spontaneous-force optical trap,” Phys. Rev. A 88, 043409 (2013).
[Crossref]

Ramirez-Martinez, F.

S. Pollock, J. P. Cotter, A. Laliotis, F. Ramirez-Martinez, and E. A. Hinds, “Characteristics of integrated magneto-optical traps for atom chips,” New J. Phys. 13, 043029 (2011).
[Crossref]

Reichel, J.

L. Huet, M. Ammar, E. Morvan, N. Sarazin, J. P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, “Experimental investigation of transparent silicon carbide for atom chips,” Appl. Phys. Lett. 100, 121114 (2012).
[Crossref]

W. Hänsel, P. Hommelhoff, T. W. Hänsch, and J. Reichel, “Bose-Einstein condensation on a microelectronic chip,” Nature 413, 498–501 (2001).
[Crossref] [PubMed]

J. Reichel, W. Hänsel, and T. W. Hänsch, “Atomic micromanipulation with magnetic surface traps,” Phys. Rev. Lett. 83, 3398–3401 (1999).
[Crossref]

Riis, E.

C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, “Laser cooling with a single laser beam and a planar diffractor,” Opt. Lett. 35, 3453–3455 (2010).
[Crossref] [PubMed]

M. Vangeleyn, P. F. Griffin, E. Riis, and A. S. Arnold, “Single-laser, one beam, tetrahedral magneto-optical trap,” Opt. Express 17, 13601–13608 (2009).
[Crossref] [PubMed]

D. S. Weiss, E. Riis, Y. Shevy, P. J. Ungar, and S. Chu, “Optical molasses and multilevel atoms: experiment,” J. Opt. Soc. Am. B 6, 2072–2083 (1989).
[Crossref]

J. P. McGilligan, P. F. Griffin, E. Riis, and A. S. Arnold, in preparation (2015).

J. P. Cotter, P. F. Griffin, E. A. Hinds, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, in preparation.

Rink, J.

C. J. Cooper, G. Hillenbrand, J. Rink, C. G. Townsend, K. Zetie, and C. J. Foot, “The temperature of atoms in a magneto-optical trap,” Europhys. Lett. 28, 397–402 (1994).
[Crossref]

Robb, G. R. M.

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,” Nature Phot. 8, 321–325 (2014).
[Crossref]

Robinson, H.

C. Monroe, W. Swann, H. Robinson, and C. Wieman, “Very cold trapped atoms in a vapor cell,” Phys. Rev. Lett. 65, 1571–1574 (1990).
[Crossref] [PubMed]

Rolston, S. L.

Rushton, J. A.

J. A. Rushton, M. Aldous, and M. D. Himsworth, “The feasibility of a fully miniaturized magneto-optical trap for portable ultracold quantum technology,” Rev. Sci. Instrum. 85, 121501 (2014).
[Crossref]

Saffman, M.

J. Miao, J. Hostetter, G. Stratis, and M. Saffman, “Magneto-optical trapping of holmium atoms,” Phys. Rev. A 89, 041401 (2014).
[Crossref]

Salomon, C.

M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[Crossref]

Santarelli, G.

M. Drewsen, P. H. Laurent, A. Nadir, G. Santarelli, A. Clairon, Y. Castin, D. Grison, and C. Salomon, “Investigation of sub-Doppler cooling effects in cesium magneto-optical trap,” Appl. Phys. B 59, 283–298 (1994).
[Crossref]

Sarazin, N.

L. Huet, M. Ammar, E. Morvan, N. Sarazin, J. P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, “Experimental investigation of transparent silicon carbide for atom chips,” Appl. Phys. Lett. 100, 121114 (2012).
[Crossref]

Schioppo, M.

N. Poli, M. Schioppo, S. Vogt, St. Falke, U. Sterr, Ch. Lisdat, and G. M. Tino, “A transportable strontium optical lattice clock,” Appl. Phys. B 117, 1107–1116 (2014).
[Crossref]

Scholten, R. E.

F.-X. Esnault, E. Blanshan, E. N. Ivanov, R. E. Scholten, J. Kitching, and E. A. Donley, “Cold-atom double-Λ coherent population trapping clock,” Phys. Rev. A 88, 042120 (2013).
[Crossref]

Schwartz, S.

L. Huet, M. Ammar, E. Morvan, N. Sarazin, J. P. Pocholle, J. Reichel, C. Guerlin, and S. Schwartz, “Experimental investigation of transparent silicon carbide for atom chips,” Appl. Phys. Lett. 100, 121114 (2012).
[Crossref]

See, P.

C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

J. P. Cotter, P. F. Griffin, E. A. Hinds, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, in preparation.

Sesko, D. W.

Shevy, Y.

Shimizu, F.

Shimizu, K.

Sinclair, A. G.

C. C. Nshii, M. Vangeleyn, J. P. Cotter, P. F. Griffin, E. A. Hinds, C. N. Ironside, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, “A surface-patterned chip as a strong source of ultracold atoms for quantum technologies,” Nature Nanotech. 8, 321–324 (2013).
[Crossref]

J. P. Cotter, P. F. Griffin, E. A. Hinds, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, in preparation.

Steane, A. M.

C. G. Townsend, N. H. Edwards, C. J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P. Szriftgiser, H. Perrin, and J. Dalibard, “Phase-space density in the magneto-optical trap,” Phys. Rev. A 52, 1423–1440 (1995).
[Crossref] [PubMed]

Steinecker, M. H.

J. F. Barry, D. J. McCarron, E. B. Norrgard, M. H. Steinecker, and D. DeMille, “Magneto-optical trapping of a diatomic molecule,” Nature 512, 286–289 (2014).
[Crossref] [PubMed]

Stephens, M.

K. Lindquist, M. Stephens, and C. E. Wieman, “Experimental and theoretical study of the vapor-cell Zeeman optical trap,” Phys. Rev. A 46, 4082–4090 (1992).
[Crossref] [PubMed]

Sterr, U.

N. Poli, M. Schioppo, S. Vogt, St. Falke, U. Sterr, Ch. Lisdat, and G. M. Tino, “A transportable strontium optical lattice clock,” Appl. Phys. B 117, 1107–1116 (2014).
[Crossref]

Stratis, G.

J. Miao, J. Hostetter, G. Stratis, and M. Saffman, “Magneto-optical trapping of holmium atoms,” Phys. Rev. A 89, 041401 (2014).
[Crossref]

Swann, W.

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J. Miao, J. Hostetter, G. Stratis, and M. Saffman, “Magneto-optical trapping of holmium atoms,” Phys. Rev. A 89, 041401 (2014).
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[Crossref] [PubMed]

Phys. Rev. Lett. (7)

C. D. Wallace, T. P. Dinneen, K-Y. N. Tan, T. T. Grove, and P. L. Gould, “Isotopic difference in trap loss collisions of laser cooled rubidium atoms,” Phys. Rev. Lett. 69897–900 (1992).
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Other (3)

J. P. McGilligan, P. F. Griffin, E. Riis, and A. S. Arnold, in preparation (2015).

A. S. Arnold, Fig. 4.12, (DPhil thesis, Sussex, 1999).

J. P. Cotter, P. F. Griffin, E. A. Hinds, P. See, A. G. Sinclair, E. Riis, and A. S. Arnold, in preparation.

Supplementary Material (2)

» Media 1: JPG (140 KB)     
» Media 2: JPG (141 KB)     

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Figures (6)

Fig. 1
Fig. 1

Magneto-optical trap geometries: six-beam MOT (a), reflection MOT (b) and grating MOT (c). The black cuboid indicates a vacuum cell, yellow fibers deliver cooling and repumping light (red) which is collimated by lenses (blue) and appropriately circularly-polarised with quarter waveplates (green). The vacuum pump, atom source, anti-Helmholtz magnetic coils etc. are omitted for clarity.

Fig. 2
Fig. 2

The ‘zoomed in’ microstructure on the different 2×2cm2 chips used in our studies. The black and white zones are separated by a height difference of ~200 nm, one quarter of the wavelength of the rubidium cooling light (780nm). Chips TRI15 and TRI12 have the pattern shown in (a), comprising three one-dimensional gratings with period d (blue line) of 1470 and 1200nm, respectively. The other chip, CIR, has the pattern shown in (b), with a grating unit cell (blue square) side length d = 1080nm. First order Bragg-diffracted beams have angles 32° (TRI15), 41° (TRI12), 46° (CIR) with respect to the chip surface normal.

Fig. 3
Fig. 3

The full theoretical acceleration above chip TRI12, shown both as vector streamlines and magnitude. The white arrow indicates the centre and direction of the beam incident on the grating (gold line at figure base), whereas the white cross indicates the quadrupole magnetic field centre. In figure a, a uniform beam intensity profile is used, whereas in b we use a Gaussian beam with 1/e2 radius of 2cm, like in the experiment. Black dashed lines indicate the boundaries of the beam overlap (capture) volume.

Fig. 4
Fig. 4

The full GMOT characterisation when using chips TRI15 and TRI12 with beam diffraction angles α = 32° and 41°, respectively. The labels a-f indicate theoretical atom number (a), experimental atom number (b) and spatial density (c), the theoretical temperature (d) and experimental temperatures in directions parallel (e) and perpendicular (f) to the grating. Light gray indicates indeterminate or out-of-range data. Media 1 shows the theoretical temperature (d) for IS = 1.67mW/cm2, which has clearer experimental agreement.

Fig. 5
Fig. 5

The full GMOT detuning and intensity characterisation when using chip CIR (standard Gaussian input beam a-f) and CIRND (CIR with a T = 0.74 ND filter in the central 3mm diameter of the input beam, a1-f1), both with beam diffraction angle α = 46°. Note the slight increase in atom number, and significant decrease in temperature using chip CIRND relative to chip CIR. Media 2 is the IS = 1.67mW/cm2 version of the theoretical temperature (images (d)).

Fig. 6
Fig. 6

Temperature measurements in directions both parallel (red) and perpendicular (blue) to the grating for chip TRI12 (a) and CIRND (b). Both chips give a 3D average temperature T = 2 3 T + 1 3 T = 50 μ K.

Equations (8)

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T U = T D 1 + β T + 4 Δ 2 / Γ 2 4 Δ / Γ ,
R i ( v ) = Γ 2 I i / I S 1 + β T + 4 ( Δ k i v ) 2 / Γ 2 , with R i ( 0 ) = I i I S R = β i R .
D = i = 0 N 2 β i R { k i x 2 + k 2 3 , k i y 2 + k 2 3 , k i z 2 + k 2 3 }
D = 2 k 2 I 0 6 I S R ( { 2 , 2 , 8 ) } + sec α { 3 sin 2 α + 2 , 3 sin 2 α + 2 , 6 cos 2 α + 2 } )
F ( v ) γ · v = { γ v x , γ v y , γ v z } { γ , γ } = γ 6 { sin α tan α 4 , 1 + cos α 2 } , where γ 6 = 16 I 0 k 2 R Δ I S Γ 2 ( 1 + β T + 4 Δ 2 Γ 2 )
{ T , T } = T U 6 { 3 + csc 2 ( α / 2 ) , 3 + sec α } ,
a = v d v d x = η h k Γ I 2 m I S ( 1 1 + I T I S + 4 ( Δ k v ) 2 Γ 2 1 1 + I T I S + 4 ( Δ + k v ) 2 Γ 2 ) .
N = 4 π r 2 8 σ ( v c v T ) 4 ,

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