Abstract

An atom placed in a small high-finesse optical cavity will dominantly emit into modes sustained by the cavity. If the cavity supports many frequency-degenerate modes, the radiation pattern depends strongly on the position of the atom. These patterns can be used to detect the position of the atom with high sensitivity.

© 2003 Optical Society of America

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References

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  1. C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science 287, 1447 (2000).
    [Crossref] [PubMed]
  2. P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, Nature 404, 365 (2000).
    [Crossref] [PubMed]
  3. T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, Phys. Rev. Lett. 88, 163002 (2002).
    [Crossref]
  4. E. M. Purcell, Phys. Rev. 69, 681 (1946).
    [Crossref]
  5. P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
    [Crossref]
  6. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  7. S. E. Morin, C. C. Yu, and T. W. Mossberg, Phys. Rev. Lett. 73, 1489 (1994).
    [Crossref] [PubMed]
  8. T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, New J. Phys. 3, 11.111.20 (2001).
  9. The expression for ψeff can be derived by induction. We start by taking the first two basis modes ψ1,ψ2, with ψ1ra+ψ2ra>0. In the two-dimensional subspace spanned by ψ1,ψ2 a unitary transformation can be applied, forming the following two orthonormal superpositions: χ1r=ψ1*raψ1r+ψ2*raψ2r/ψ1ra2+ψ2ra21/2 and χ˜1r=-ψ2raψ1r+ψ1raψ2r/ψ1ra2+ψ2ra21/2. It is easily verified that χ1ra≥ψ1ra,ψ2ra and χ˜1ra=0. Hence, in this two-dimensional subspace, χ1 is the mode with the largest value at ra and χ˜1 does not couple to the atom. Now, basis mode ψ3 is combined with χ1, and new superpositions χ2 and χ˜2 are constructed that, respectively, maximize and zero the mode value at ra in the subspace spanned by χ1,ψ3. This procedure is repeated for all ψi, so that the last constructed mode is the effective mode, ψeff=χN-1, given in Eq. (2), and the χ˜i form a noncoupling basis. By construction, ψeff is unique apart from a phase factor.
  10. G. Hechenblaikner, M. Gangl, P. Horak, and H. Ritsch, Phys. Rev. A 58, 3030 (1998).
    [Crossref]
  11. H. J. Carmichel, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).
  12. P. Horak and H. Ritsch, Phys. Rev. A 64, 033422 (2001).
    [Crossref]

2002 (2)

T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, Phys. Rev. Lett. 88, 163002 (2002).
[Crossref]

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[Crossref]

2001 (2)

T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, New J. Phys. 3, 11.111.20 (2001).

P. Horak and H. Ritsch, Phys. Rev. A 64, 033422 (2001).
[Crossref]

2000 (2)

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science 287, 1447 (2000).
[Crossref] [PubMed]

P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, Nature 404, 365 (2000).
[Crossref] [PubMed]

1998 (1)

G. Hechenblaikner, M. Gangl, P. Horak, and H. Ritsch, Phys. Rev. A 58, 3030 (1998).
[Crossref]

1994 (1)

S. E. Morin, C. C. Yu, and T. W. Mossberg, Phys. Rev. Lett. 73, 1489 (1994).
[Crossref] [PubMed]

1946 (1)

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[Crossref]

Carmichel, H. J.

H. J. Carmichel, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).

Doherty, A. C.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science 287, 1447 (2000).
[Crossref] [PubMed]

Fischer, T.

T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, Phys. Rev. Lett. 88, 163002 (2002).
[Crossref]

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[Crossref]

T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, New J. Phys. 3, 11.111.20 (2001).

P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, Nature 404, 365 (2000).
[Crossref] [PubMed]

Gangl, M.

G. Hechenblaikner, M. Gangl, P. Horak, and H. Ritsch, Phys. Rev. A 58, 3030 (1998).
[Crossref]

Hechenblaikner, G.

G. Hechenblaikner, M. Gangl, P. Horak, and H. Ritsch, Phys. Rev. A 58, 3030 (1998).
[Crossref]

Hood, C. J.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science 287, 1447 (2000).
[Crossref] [PubMed]

Horak, P.

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[Crossref]

P. Horak and H. Ritsch, Phys. Rev. A 64, 033422 (2001).
[Crossref]

G. Hechenblaikner, M. Gangl, P. Horak, and H. Ritsch, Phys. Rev. A 58, 3030 (1998).
[Crossref]

Kimble, H. J.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science 287, 1447 (2000).
[Crossref] [PubMed]

Lynn, T. W.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science 287, 1447 (2000).
[Crossref] [PubMed]

Maunz, P.

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[Crossref]

T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, Phys. Rev. Lett. 88, 163002 (2002).
[Crossref]

T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, New J. Phys. 3, 11.111.20 (2001).

P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, Nature 404, 365 (2000).
[Crossref] [PubMed]

Morin, S. E.

S. E. Morin, C. C. Yu, and T. W. Mossberg, Phys. Rev. Lett. 73, 1489 (1994).
[Crossref] [PubMed]

Mossberg, T. W.

S. E. Morin, C. C. Yu, and T. W. Mossberg, Phys. Rev. Lett. 73, 1489 (1994).
[Crossref] [PubMed]

Parkins, A. S.

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science 287, 1447 (2000).
[Crossref] [PubMed]

Pinkse, P. W. H.

T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, Phys. Rev. Lett. 88, 163002 (2002).
[Crossref]

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[Crossref]

T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, New J. Phys. 3, 11.111.20 (2001).

P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, Nature 404, 365 (2000).
[Crossref] [PubMed]

Puppe, T.

T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, Phys. Rev. Lett. 88, 163002 (2002).
[Crossref]

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[Crossref]

T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, New J. Phys. 3, 11.111.20 (2001).

Purcell, E. M.

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[Crossref]

Rempe, G.

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[Crossref]

T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, Phys. Rev. Lett. 88, 163002 (2002).
[Crossref]

T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, New J. Phys. 3, 11.111.20 (2001).

P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, Nature 404, 365 (2000).
[Crossref] [PubMed]

Ritsch, H.

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[Crossref]

P. Horak and H. Ritsch, Phys. Rev. A 64, 033422 (2001).
[Crossref]

G. Hechenblaikner, M. Gangl, P. Horak, and H. Ritsch, Phys. Rev. A 58, 3030 (1998).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

Yu, C. C.

S. E. Morin, C. C. Yu, and T. W. Mossberg, Phys. Rev. Lett. 73, 1489 (1994).
[Crossref] [PubMed]

Nature (1)

P. W. H. Pinkse, T. Fischer, P. Maunz, and G. Rempe, Nature 404, 365 (2000).
[Crossref] [PubMed]

New J. Phys. (1)

T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, New J. Phys. 3, 11.111.20 (2001).

Phys. Rev. (1)

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[Crossref]

Phys. Rev. A (2)

G. Hechenblaikner, M. Gangl, P. Horak, and H. Ritsch, Phys. Rev. A 58, 3030 (1998).
[Crossref]

P. Horak and H. Ritsch, Phys. Rev. A 64, 033422 (2001).
[Crossref]

Phys. Rev. Lett. (3)

P. Horak, H. Ritsch, T. Fischer, P. Maunz, T. Puppe, P. W. H. Pinkse, and G. Rempe, Phys. Rev. Lett. 88, 043601 (2002).
[Crossref]

T. Fischer, P. Maunz, P. W. H. Pinkse, T. Puppe, and G. Rempe, Phys. Rev. Lett. 88, 163002 (2002).
[Crossref]

S. E. Morin, C. C. Yu, and T. W. Mossberg, Phys. Rev. Lett. 73, 1489 (1994).
[Crossref] [PubMed]

Science (1)

C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science 287, 1447 (2000).
[Crossref] [PubMed]

Other (3)

The expression for ψeff can be derived by induction. We start by taking the first two basis modes ψ1,ψ2, with ψ1ra+ψ2ra>0. In the two-dimensional subspace spanned by ψ1,ψ2 a unitary transformation can be applied, forming the following two orthonormal superpositions: χ1r=ψ1*raψ1r+ψ2*raψ2r/ψ1ra2+ψ2ra21/2 and χ˜1r=-ψ2raψ1r+ψ1raψ2r/ψ1ra2+ψ2ra21/2. It is easily verified that χ1ra≥ψ1ra,ψ2ra and χ˜1ra=0. Hence, in this two-dimensional subspace, χ1 is the mode with the largest value at ra and χ˜1 does not couple to the atom. Now, basis mode ψ3 is combined with χ1, and new superpositions χ2 and χ˜2 are constructed that, respectively, maximize and zero the mode value at ra in the subspace spanned by χ1,ψ3. This procedure is repeated for all ψi, so that the last constructed mode is the effective mode, ψeff=χN-1, given in Eq. (2), and the χ˜i form a noncoupling basis. By construction, ψeff is unique apart from a phase factor.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

H. J. Carmichel, An Open Systems Approach to Quantum Optics (Springer-Verlag, Berlin, 1993).

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Figures (2)

Fig. 1
Fig. 1

(a) Hermite–Gaussian basis modes, HG0,2,HG1,1, and HG2,0, respectively, for the frequency-degenerate N=2 mode family. (b)–(d) Example showing another orthonormal basis set of modes where the effective mode (b) couples maximally to the atom (cross) and all noncoupling modes are given by superpositions of (c) and (d).

Fig. 2
Fig. 2

Examples of cavity emission patterns with an atom (cross) at (a)–(c) r=1.6w0 and (d)–(f) r=2.8w0 near resonant with the N=10 mode family. The plots on the left are normalized with respect to the maximal value of the HG0,0 mode, ψ0,00,0. (a) and (c) and (b) and (d) are radial cuts and two-dimensional plots for side pumping, respectively. The emission patterns equal the effective mode. (c), (f) A cavity pumped through the input mirror with a plane-wave pump. These patterns result from the interference of the noncoupling modes and the effective mode whose amplitude is influenced by the atom.

Equations (4)

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ψm,nx,y=Cm,nexp-x2+y2w02Hm2xw0Hn2yw0,
ψeffr,ra=i=0Nψi*raϕeffraψir,
Enc+r=0ηnc1κ-iΔcψncr,
Eeff+r=0ηeffγ-iΔageff2-ΔaΔc+γκ-iΔcγ+Δaκ×ψeffr.

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