Cavity enhanced dipole forces for dark field seeking atoms and molecules
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L 20. L 00. Cavity-enhanced dipole forces for dark-field seeking atoms and molecules. David McGloin, Kishan Dholakia. Tim Freegarde. Dipartimento di Fisica, Università di Trento 38050 Povo (TN), Italy. J F Allen Physics Research Laboratories, University of St Andrews,

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Cavity-enhanced dipole forces for dark-field seeking atoms and molecules

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Cavity enhanced dipole forces for dark field seeking atoms and molecules

L20

L00

Cavity-enhanced dipole forces for

dark-field seeking atoms and molecules

David McGloin, Kishan Dholakia

Tim Freegarde

Dipartimento di Fisica,

Università di Trento

38050 Povo (TN), Italy

J F Allen Physics Research Laboratories,

University of St Andrews,

Fife KY16 9SS, Scotland

OPTICAL BOTTLE BEAM

Dipole force traps for dark-field seeking states of atoms and molecules require regions of low intensity that are completely surrounded by a bright optical field. Confocal cavities allow the resonant enhancement of these interesting transverse mode superpositions, and put deep off-resonant dark-field seeking dipole traps within reach of low-power diode lasers.

COAXIAL RING ARRAY

  • Freegarde & Dholakia, Phys Rev A, in press

  • see Arlt & Padgett, Opt. Lett. 25 (2000) 191-193

  • Freegarde & Dholakia, Opt. Commun. 201 99 (2002)

  • see Zemánek & Foot, Opt. Commun. 146 119 (1998)

  • use single Gaussian beam of waist w1 larger than that of the fundamental cavity mode (w0 = aw1)

  • counterpropagating beam smaller by same factor (w2 = aw0)

  • beams of equal power cancel where nodal surfaces intersect

  • Laguerre-Gaussian superposition:

  • cancellation at cavity centre

  • constructive interference elsewhere thanks to different radial dependences and Gouy shifts

OPTICAL DIPOLE FORCE

(

)

Jdipole traps eliminate the magnetic fields needed for MOTs1

  • return beam larger than forward beam to avoid nodal surfaces

æ

ö

w

z

1

1

(

)

ç

÷

2

=

-

r

z

ln

1

(

)

ç

÷

(

)

(

)

0

2

2

FAR OFF RESONANCE2-5

Jbroadband interaction and

Jminimal scattering, hence suitable for spectrally complex atoms and molecules

Lintense laser beam needed to compensate for interaction weakness

BLUE-DETUNED6-10

Jdark-field seeking to minimize residual perturbations

Lneed isolated islands of low intensity for closed trapping region

RESONANT CAVITIES11-13

Jcan greatly increase circulating intensity, as optical absorption is low

Loptical field not a single cavity mode

w

z

w

z

w

z

è

ø

  • with  = 0.5, the maximum modulation depth is 7%.

2

2

1

  • intensity minima form a series of coaxial rings spaced by l/2

  • traps deepest when a = 0.492

  • r0 ~ 0.7 w0(z)

COMPOSITION

  • five component superposition optimizes trap depth for given radius:

towards

high intensity

  • high

=

+

-

-

-

0

.

691

0

.

332

0

.

165

0

.

332

0

.

525

E

L

L

L

L

L

00

10

20

30

40

towards

low intensity

  • trap intensity nearly half that at centre of simple Gaussian beam with same waist and power as forward beam

  • 99.99% mirrors with 100 mW at 780 nm would give 5 K trap depth for 85Rb at 0.2 nm detuning

  • low

  • with a = 0.492, 99% of power in first 5 modes

CONFOCAL CAVITIES

L/R2

  • transverse mode degeneracy allows enhancement of mode superpositions for complex field geometries

1

confocal

Intensity distribution within a perfectly confocal resonator.

Above left: mean intensity shown for central 40% of the cavity. The solid lines show where the forward beam has fallen to e-2 of its on-axis intensity.

Above: viewed on a wavelength scale around the cavity centre, the modulation due to interference between the counter-propagating beams is apparent. Here,

l = 100 mm,  = 780nm,  = 2.

Left: depth of modulation due to interference between forward and return beams. Black=0, white=100%.

0

1

L/R1

Intensity distribution around the centre of a confocal cavity.Dashed and solid lines indicate the nodal and antinodal planes; the dotted line shows where the lowest part of the trap wall is maximum. Logarithmic contours (four per decade) refer to the peak intensity on axis. l = 100 mm, l = 780 nm, a = 0.492.

Intensity distribution when the cavity mirrors are 0.1 mm from their confocal separation (Dl/l = 0.001), for r2 = 0.99,

t2 = 0.01. The nodal surfaces, shown dashed, are now curved, reflecting the increase in Gouy phase with mode number. Central and trapping intensities are reduced by about a third.

  • three different views of physics:

RAY OPTICS

  • 2 round trips before repeating

  • inverted image after 1 round trip

  • returning beam  forward beam

Applications:

  • trapping of spectrally complex atoms and molecules

  • investigation of vortices in quantum degenerate gases14

  • coupling between adjacent microtraps15

  • cooling via coupling to cavity radiation field16-18

R2

R1

GAUSSIAN BEAMS

HALF TRIP

ROUND TRIP

even

odd

even

CAVITY MODES

Cartesian

i + j

Hermite-Gaussian

  • half modes simultaneously resonant

  • (anti-)symmetric image =

  • superposition of even(odd) modes

cylindrical

2p + |m|

Laguerre-Gaussian

Amplitudes ap0 of mode components forming the complete five-component optical bottle beam with =2.

LAGUERRE-GAUSSIAN BEAMS

MECHANICAL AMPLIFIER

  • the Laguerre-Gaussian cavity modes are solutions to the paraxial wave equation in cylindrical polar coordinates,

col intensity

  • moving the mirrors from their confocal separation causes an amplified displacement of the trap centre

  • amplification by same factor as intensity enhancement

(

)

(

)

(

)

m

æ

ö

-

+

+

æ

ö

æ

ö

1

exp

i

2

p

m

1

tan

z

z

2

2

2

4

p

!

2

r

2

r

r

i

kr

(

)

trap centre

intensity

ç

÷

ç

÷

ç

÷

J

=

-

-

+

J

-

R

m

r

,

z

,

L

exp

i

m

i

kz

L

(

)

ç

÷

ç

÷

(

)

(

)

(

)

ç

÷

(

)

(

)

(

)

pm

p

+

d

p

+

2

2

2

1

p

m

!

w

z

2

R

z

w

z

w

z

w

z

è

ø

è

ø

è

ø

0

m

trap centre position

(

)

2

2

p

z

w

0

(

)

(

)

(

)

(

)

(

)

2

where are Laguerre polynomials and , , .

=

+

m

=

+

=

L

x

w

z

w

0

1

z

z

R

R

z

z

z

p

R

R

l

z

  • an arbitrary field may be written as a superposition

Variation of trap col (dotted) and trap centre (dashed) intensities – in units of the well depth at zero mirror displacement – and trap centre position (right hand scale) as mirrors are displaced from their confocal separation.

(

)

(

)

å

e

=

r

,

z

a

L

r

,

z

(

)

-

+

w

w

w

w

pm

pm

p

s

!

j

=

+

=

j

j

0

1

1

0

sin

s

1

p

pm

a

cos

sin

+

ps

0

w

w

w

w

p

!

s

!

(

1

)

L

  • Laguerre-Gaussian beams , of non-resonant waist radius w1, correspond to superpositions of resonant L-G beams with the same azimuthal index m = s. The first three coefficients are:

0

1

1

0

qm

(

)

  • amplification mechanism may be compared to Vernier scale between Gouy phases of different Laguerre-Gaussian components

[

]

+

p

s

!

(

)

+

-

=

j

j

j

-

+

j

s

1

p

1

2

2

a

cos

sin

p

cos

s

1

sin

(

)

ps

1

+

p

!

s

1

!

a

(

)

pmq

[

]

[

]

+

{

}

p

s

!

(

)

(

)

+

-

=

j

j

j

-

+

j

j

-

+

j

-

j

s

1

p

2

2

2

2

2

2

a

cos

sin

p

cos

s

1

sin

p

cos

s

2

sin

p

cos

(

)

ps

2

+

2

!

p

!

s

2

!

SINGLE TOROID

LARGE PERIOD STANDING WAVE

  • REFERENCES

  • 1R. Grimm, M. Weidemüller, Y. B. Ovchinnikov, Adv. At. Mol. Opt. Phys. 42 (2000) 95-170

  • 2S. L. Rolston, C. Gerz, K. Helmerson, P. S. Jessen, P. D. Lett, W. D. Phillips, R. J. Spreeuw, C. I. Westbrook, Proc. SPIE 1726 (1992) 205-211

  • 3J. D. Miller, R. A. Cline, D. J. Heinzen, Phys. Rev. A 47 (1993) R4567-4570

  • 4M. D. Barrett, J. A. Sauer, M. S. Chapman, Phys. Rev. Lett. 87 (2001) 010404

  • 5T. Takekoshi, B. M. Patterson, R. J. Knize, Phys. Rev. Lett. 81 (1998) 5105-5108

  • 6N. Davidson, H. J. Lee, C. S. Adams, M. Kasevich, S. Chu, Phys. Rev. Lett. 74 (1995) 1311-1314

  • 7P. Rudy, R. Ejnisman, A. Rahman, S. Lee, N. P. Bigelow, Optics Express 8 (2001) 159-165

  • 8S. A. Webster, G. Hechenblaikner, S. A. Hopkins, J. Arlt, C. J. Foot, J. Phys. B 33 (2000) 4149-4155

  • 9T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimuzu, H. Sasada, Phys. Rev. Lett. 78 (1997) 4713-4716

  • R. Ozeri, L. Khaykovich, N. Davidson, Phys. Rev. A 59 (1999) R1759-1753

  • 11J. Ye, D. W. Vernooy, H. J. Kimble, Phys. Rev. Lett. 83 (1999) 4987-4990

  • 12S. Jochim, Th. Elsässer, A. Mosk, M. Weidemüller, R. Grimm, Int. Conf. on At. Phys., Firenze, Italy, poster G.11 (2000)

  • 13P. W. H. Pinkse, T. Fischer, P. Maunz, T. Puppe, G. Rempe, J. Mod. Opt. 47 (2000) 2769-2787

  • 14E. M. Wright, J. Arlt, K. Dholakia, Phys. Rev. A 63 (2000) 013608

  • 15P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, G. Rempe, Phys. Rev. Lett. 84 (2000) 4068-4071

  • 16T. Zaugg, M. Wilkens, P. Meystre, G. Lenz, Opt. Commun. 97 (1993) 189-193

  • 17M. Gangl, H. Ritsch, Phys. Rev. A 61 (1999) 011402

  • 18V. Vuletic, S. Chu, Phys. Rev. Lett. 84 (2000) 3787-3790

  • in preparation

  • in preparation

  • see D M Giltner et al, Opt. Commun. 107 227 (1994)

  • pattern period = l/sinq

  • 2-D Hermite-Gaussian analysis; astigmatism renders out-of-plane direction non-confocal

  • high Q:all (odd) even modes

  • give (anti-)symmetric

  • field pattern

  • finite Q:half-axial modes

  • contribute

  • dissimilar forward/return waist sizes eliminate nodal planes

  • magnetic field free toroidal trap for study of vortices in condensates14


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