## Diffusion of a single ion in a one-dimensional optical lattice

Optics Express, Vol. 3, Issue 2, pp. 97-103 (1998)

http://dx.doi.org/10.1364/OE.3.000097

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### Abstract

We present an experimental study of the spatial diffusion of a single ion in a polarization gradient field. A ^{24}Mg^{+} ion was radially confined in a two-dimensional radio-frequency (*rf*) trap, while an optical lattice superimposed on a weak electric potential was applied along the free axis. With the help of a statistical analysis of single ion trajectories, a spatial diffusion constant was obtained as a function of optical potential depth. The results are compared to semiclassical theoretical models for trapped ions and neutral atoms.

© Optical Society of America

## 1. Introduction

1. P.D. Lett, R.N. Watts, C.I. Westbrook, W.D. Phillips, P.L. Gould, and H.J. Metcalf, Phys. Rev. Lett. **61**, 169 (1988) [CrossRef] [PubMed]

5. J.-P. Bouchaud and A. Georges, Phys. Rep. **195**, 127 (1990) [CrossRef]

6. M.F. Shlesinger, G.M. Zaslavsky, and J. Klafter, Nature **363**, 31 (1993). [CrossRef]

8. H. Katori, S. Schlipf, and H. Walther, Phys. Rev. Lett. **79**, 2221 (1997) [CrossRef]

9. T.W. Hodapp, C. Gerz, C. Furtlehner, C.I. Westbrook, W.D. Phillips, and J. Dalibard, Appl. Phys. B **60**, 135 (1995) [CrossRef]

10. C. Jurczak, B. Desruelle, K. Sengstock, J.-Y. Courtois, C.I. Westbrook, and A. Aspect, Phys. Rev. Lett. **77**, 1727 (1996) [CrossRef] [PubMed]

*J*

_{1/2}→

*J*

_{3/2}transition. We compare our experimental results to simulations performed with the semiclassical Monte-Carlo technique, which has been proven to be in good agreement with quantum mechanical simulations over a wide parameter range [7,11].

## 2. Experimental Setup

*rf*quadrupole ring trap was used to confine a single

^{24}Mg

^{+}ion in the radial direction [12

12. I. Waki, S. Kassner, G. Birkl, and H. Walther, Phys. Rev. Lett. **68**2007 (1992) [CrossRef] [PubMed]

*rf*frequency of 6.5 MHz, with a resultant secular frequency for the ion’s radial motion of 900 kHz. As we used only a small part of the ring with an angle of 200

*μ*rad, the trap actually functioned practically as a linear trap. The ion was confined along the free axis by a shallow electric potential with an oscillation frequency of

*ω*/2

_{ext}*π*= 13kHz, which enabled us to observe a single ion for long time periods. As mentioned above a periodic optical potential was produced tangential to the trap axis with a pair of counterpropagating, crossed-linear-polarized laser beams, which were slightly red detuned from the

^{2}S

_{1/2}→

^{2}P

_{3/2}atomic transition of

^{24}Mg

^{+}at

*λ*=280 nm. The natural linewidth of the excited state is Γ/2

*π*= 42.7 MHz. The schematics of this 1D potential is depicted in Fig. 1.

*x*(

*t*) from the minimum of a superimposed external electric potential. The ion position was determined by detecting its fluorescent photons which pass through a microscope lens with a numerical aperture (NA) of NA = 0.28. The ion image was projected onto a resistive anode element based single photon counting position analyzer with a time resolution of 10

_{i}*μ*s. The effective position resolution of the imaging system was estimated experimentally to be 3

*μ*m, due mainly to the thermal motion of the ion in the focal direction. The total photon detection efficiency was roughly 5 × 10

^{-5}. The signal to noise ratio was 10

^{2}~ 10

^{3}, limited mainly by the presence of fluorescence light scattered from the surface of the trap electrodes.

*x*(

*t*)}, (

_{i}*i*= 1, 2,…, 2

^{16}), which corresponds to an observation time of several seconds to a few minutes depending on the lattice parameters. An example of an ion trajectory is depicted in the lower half of Fig. 1. This part of the figure shows in addition the collision with another ion in the quadrupole ring trap. First a single ion was localized at the center of the electric potential. After 8 seconds a second ion enters the observation region from the left and forces the first ion into a new equilibrium position. They settle at a distance of 150

*μ*m symmetrically to the minimum of the external electric potential.

## 3. Analysis of the Data

*x*(

*t*), we employed an autocorrelation function analysis. The

_{i}*x*(

*t*) were used to calculate

_{i}*x*(0)

*x*(

*τ*)⟩. The angle brackets denote averaging over time

*t*. In this analysis the contribution of stray photons and any position uncertainty in the optical imaging result in a constant offset only.

*F*(

*t*+

*τ*)

*F*(

*t*)⟩ = 2

*γk*(

_{B}Tδ*τ*), where

*k*is the Boltzmann constant,

_{B}*γ*the friction coefficient,

*m*the mass, and

*T*the temperature of the ion. The friction coefficient

*γ*was phenomenologically introduced to describe the spatial diffusion between the lattice sites. This is valid as long as we are interested in a time scale, which is much longer than the ion’s localization time in the optical potential. Inserting Eq. (2) into Eq. (1) we are able to derive for the high friction limit:

*τ*= 0 gives the spatial diffusion coefficient:

*ϕ*(

*τ*) for different optical potential depths

*δ*, and

*I*are the saturation parameter, the saturation intensity of the transition, the laser detuning, and the intensity, respectively. The optical potential depth is given in units of photon recoil energy

*ω*of the external potential appears. This clearly shows that the atomic transport in the lattice changed from a slow diffusive regime to one where the fast oscillation in the external potential is dominant. Sisyphus cooling is unable to cool the ion’s kinetic energy below the optical potential depth so that the ion is no longer localized. In this low friction limit, the result for

_{ext}*ϕ*(

*τ*) is

*μ*s. The sinusoidal oscillation in the position correlation function also infers an oscillation in the momentum correlation function. This fact is also important for the interpretation of anomalous diffusion but cannot be discussed here (for details see [13]).

## 4. Spatial Diffusion Coefficient

*D*as a function of optical potential depth for a fixed laser intensity of

_{x}*I*= 15

*I*. The optical potential depth was varied by changing the laser detuning

_{sat}*δ*. The values where obtained by fitting Eq. (3) to the experimental autocorrelation function

*ϕ*(

*τ*) in the high friction limit and by calculating

*ϕ′*(0) (see Eq. (4)).

*U*

_{0}> 200

*E*there exists a reasonable agreement between the Monte-Carlo simulations and the experimental data. In this region the Monte-Carlo simulations for a free atom and a trapped ion also agree rather well. This parameter range is dominated by the confinement of the ion in the optical potential. Therefore, the influence of the external potential disappears. A different situation shows up at low

_{R}*U*

_{0}. In this region the two Monte-Carlo simulations show a growing disagreement with decreasing

*U*

_{0}.

*U*

_{0}< 200

*E*the experimental data agree well with the results from a semiclassical Fokker-Planck-Kramers (FPK) treatment [9

_{R}9. T.W. Hodapp, C. Gerz, C. Furtlehner, C.I. Westbrook, W.D. Phillips, and J. Dalibard, Appl. Phys. B **60**, 135 (1995) [CrossRef]

*et. at*. [7] and Pax

*et. al*. [11]. Marksteiner

*et. al*. predicted a minimum spatial diffusion constant of 40

*ħ*/

*m*, which is more than twice as large as our value of 15

*ħ*/

*m*being in agreement with Ref. [11]. The discrepancy to Ref. [7] can be attributed to a difference in the chosen saturation parameter. We and Ref. [11] used a laser detuning of

*δ*/Γ ≈ 10 while their value was

*δ*/Γ ≈ 100, which would explain the smaller spatial diffusion constant.

## 5. Conclusions

## 6. Acknowledgement

## References

1. | P.D. Lett, R.N. Watts, C.I. Westbrook, W.D. Phillips, P.L. Gould, and H.J. Metcalf, Phys. Rev. Lett. |

2. | J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. |

3. | G. Grynberg and S. Triché, in |

4. | Y. Castin, J. Dalibard, and C. Cohen-Tannoudji, in |

5. | J.-P. Bouchaud and A. Georges, Phys. Rep. |

6. | M.F. Shlesinger, G.M. Zaslavsky, and J. Klafter, Nature |

7. | S. Marksteiner, K. Ellinger, and P. Zoller, Phys. Rev. |

8. | H. Katori, S. Schlipf, and H. Walther, Phys. Rev. Lett. |

9. | T.W. Hodapp, C. Gerz, C. Furtlehner, C.I. Westbrook, W.D. Phillips, and J. Dalibard, Appl. Phys. B |

10. | C. Jurczak, B. Desruelle, K. Sengstock, J.-Y. Courtois, C.I. Westbrook, and A. Aspect, Phys. Rev. Lett. |

11. | P. Pax, W. Greenwood, and P. Meystre, Phys. Rev. |

12. | I. Waki, S. Kassner, G. Birkl, and H. Walther, Phys. Rev. Lett. |

13. | H. Katori, S. Schlipf, L. Perotti, and H. Walther, to be published |

**OCIS Codes**

(020.7010) Atomic and molecular physics : Laser trapping

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Focus Issue: Quantum structures in nonlinear optics and atomic physics

**History**

Original Manuscript: May 11, 1998

Published: July 20, 1998

**Citation**

S. Schlipf, H. Katori, L. Perotti, and Herbert Walther, "Diffusion of a Single Ion in a
One-Dimensional Optical Lattice," Opt. Express **3**, 97-103 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-2-97

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### References

- P. D. Lett, R. N. Watts, C. I. Westbrook, W. D. Phillips, P. L. Gould, and H. J. Metcalf, Phys. Rev. Lett. 61, 169 (1988). [CrossRef] [PubMed]
- J. Dalibard and C. Cohen{Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
- G. Grynberg and S. Triche, in Proceedings of the International School of Physics Enrico Fermi, Course CXXXI, edited by A. Aspect et al. IOS Press, Amsterdam (1996).
- Y. Castin, J. Dalibard and C. Cohen-Tannoudji, in Proceedings of the LIKE workshop, edited by L. Moi et.al. ETS Editrice, Paris (1991).
- J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 127 (1990). [CrossRef]
- M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature 363, 31 (1993). [CrossRef]
- S. Marksteiner, K. Ellinger, and P. Zoller, Phys. Rev. A53, 3409 (1996)
- H. Katori, S. Schlipf, and H. Walther, Phys. Rev. Lett. 79, 2221 (1997). [CrossRef]
- T. W. Hodapp, C. Gerz, C. Furtlehner, C. I. Westbrook, W. D. Phillips, and J. Dalibard, Appl. Phys. B 60, 135 (1995). [CrossRef]
- C. Jurczak, B. Desruelle, K. Sengstock, J.-Y. Courtois, C. I. Westbrook, and A. Aspect, Phys. Rev. Lett. 77, 1727 (1996). [CrossRef] [PubMed]
- P. Pax, W. Greenwood, and P. Meystre, Phys. Rev. A56, 2109 (1997).
- I. Waki, S. Kassner, G. Birkl, and H. Walther, Phys. Rev. Lett. 68 2007 (1992). [CrossRef] [PubMed]
- H. Katori, S. Schlipf, L. Perotti, and H. Walther, to be published.

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