1. Introduction

Polarization gradient cooling has been proven as a useful method to achieve sub-Doppler temperatures [1,2,3]. Although it is widely used there are still unrevealed phenomena such as the transport characteristics and the dynamic aspects associated with the cooling mechanism. In a polarization gradient field, atomic samples are produced with kinetic energies of less than half the optical potential depth, leading to a localization of atoms at the minima of the potential corresponding to maxima of the laser intensity [4]. The lifetime of an atom in such a well is limited since energy fluctuations owing to spontaneous emission occasionally allow the atom to accumulate enough energy to overcome the potential barrier, leading to random walks of the atoms between lattice sites. For shallow optical potentials, it has been predicted that these energy fluctuations lead to long flights of an atom that can reach over many wavelengths before it is trapped again, since the Sisyphus damping force is negligible for an atom with a larger momentum. The resultant atomic trajectory corresponds to a random walk phenomenon called Levy-flights [5,6,7]. In this paper we present recent results obtained with a new experimental technique [8] allowing to perform trajectory measurements in a one dimensional polarization gradient field. The atomic trajectories can be traced via spontaneously emitted photons in the course of the energy dissipation without disturbing the cooling process. This setup is ideally suitable to measure a spatial diffusion coefficient of an optical lattice. In the past this has been extremely difficult in standard 3D optical lattices with free atoms [9,10].

In our experiment a single ion is radially confined in a radio frequency trap. It interacts with a polarization gradient field which is irradiated into the trap in tangential direction using the lin⊥lin configuration for a 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

An rf quadrupole ring trap was used to confine a single ²⁴Mg⁺ ion in the radial direction [12]. The trap was operated with an 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 200μrad, 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_ext/2π = 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 ²S_1/2 → ²P_3/2 atomic transition of ²⁴Mg⁺ at λ =280 nm. The natural linewidth of the excited state is Γ/2π = 42.7 MHz. The schematics of this 1D potential is depicted in Fig. 1.

In the experiment the position of the ion was measured by its displacement x(t_i) 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 μs. The effective position resolution of the imaging system was estimated experimentally to be 3 μm, 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 10² ~ 10³, limited mainly by the presence of fluorescence light scattered from the surface of the trap electrodes.

The real-time position information was transferred to a personal computer to record the time history of the single ion displacement, {x(t_i)}, (i = 1, 2,…, 2¹⁶), 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 μm symmetrically to the minimum of the external electric potential.

 

Figure 1. Sketch of the ring trap. The shape of the electrostatic and optical potentials is depicted in the inset. The electrostatic potential is formed by two additional electrodes which are not shown in the figure. The lower picture shows a trajectory measurement. In this special case another ion enters the observation region defined by the electrostatic potential from the left (see text for details). The kinetic energy of the ions corresponds to a temperature of roughly 1 mK.

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A demonstration of trapped and laser-cooled ions in the observation region of the quadrupole ring trap is also given in the video. The original number of four ions is constantly reduced until a single ion is obtained. The ions are thrown out one by one by operating the ring trap for a few seconds at the edge of the stability region of the trap. This period can be recognized by the fact, that the ions are heated and pushed out of the observation region. On two occasions an additional ion is trapped with some time delay, after the previous ones had already settled in an equilibrium position. These “collisions” are similar to the one shown in the lower part of Fig. 1.

3. Analysis of the Data

In order to extract macroscopic quantities such as spatial diffusion constants and kinetic energies from the measured atomic trajectories x(t_i), we employed an autocorrelation function analysis. The x(t_i) were used to calculate

which leads to the position autocorrelation function ⟨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.

In a next step the spatial diffusion coefficient has to be evaluated. For this purpose we start with the Langevin equation for an ion confined in an harmonic potential:

 

Figure 2. Results for the position autocorrelation functions for different optical potential depths. The oscillation at the U ₀ = 37E_R is due to the motion of the ion in the external electric potential. For all the curves the laser intensity is I = 4I_Sat.

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Due to photon recoils and dipole fluctuations the ion is subjected to the Langevin force ⟨F(t + τ)F(t)⟩ = 2γk_BTδ(τ), where k_B is the Boltzmann constant, 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:

This limit applies to deep optical potentials, where the damping is much faster than one oscillation period of the ion in the harmonic potential and the atomic transport can be considered as diffusive. The time derivative of Eq. (3) at τ = 0 gives the spatial diffusion coefficient:

Fig. 2 shows the shape ofϕ(τ) for different optical potential depths $U_0=\frac{2}{3}\mathit{ħ\delta }{s}_0$, where $s_0=\frac{\frac{I}{I_{\mathit{Sat}}}}{1+{\left(\frac{2\delta }{\Gamma }\right)}^2},I_{\mathit{Sat}}=\frac{2{\pi }^2\mathit{ħc}\Gamma }{3{\lambda }^3},$ 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 $E_R=\frac{{\mathit{ħ}}^2{k}^2}{2m}\mathrm{with}\frac{E_R}{h}=106\mathrm{kHz}$. The slow rise in the autocorrelation function for the two highest optical potentials indicates, that the ion is trapped in the optical potential and therefore moving in the high friction limit A spatial diffusion coefficient can be defined.

For the two lowest optical potential depths a sinusoidal oscillation with the frequency ω_ext 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 ϕ(τ) is

The ion’s secular- and micromotion are not seen, owing to the time resolution of the detection system of 10 μs. 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

Fig. 3 shows the spatial diffusion coefficient D_x as a function of optical potential depth for a fixed laser intensity of I = 15I_sat. The optical potential depth was varied by changing the laser detuning 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)).

 

Figure 3. The spatial diffusion coefficient D_x as a function of optical potential depth U ₀. The symbols mark measurements taken on different days.

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A comparison between some of the experimental results of Fig. 3 with theoretical calculations is shown in Fig. 4. The Monte-Carlo simulations have been carried out for two cases: one with and one without the external electric potential. Details of these calculations will be described elsewhere [13].

From this comparison of theory and experiment we can draw the following conclusions. For U ₀ > 200E_R 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 U ₀. In this region the two Monte-Carlo simulations show a growing disagreement with decreasing U ₀.

For U ₀ < 200E_R the experimental data agree well with the results from a semiclassical Fokker-Planck-Kramers (FPK) treatment [9], which does not include the localization due to the periodic optical potential. In this domain we suppose in our experiment a relative large contribution of micromotion to the ion dynamics since there, the influence of the optical lattice is small and easily washed out. This fact makes the agreement between the FPK theory and our experiment not unreasonable.

Our results from the simulations can also be compared with the quantum calculations done by Marksteiner et. at. [7] and Pax et. al. [11]. Marksteiner et. al. predicted a minimum spatial diffusion constant of 40ħ/m, which is more than twice as large as our value of 15ħ/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.

 

Figure 4. Comparison between theory and experiment. The experimental data are identical to the ones plotted in Fig. 3. The theory from Ref. [9] is marked with FPK. This treatment neglects localization by an optical potential. The difference between the two Monte-Carlo simulations is that in the ion case the superimposed external electric potential is included.

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5. Conclusions

In this paper we have described a new method to investigate the spatial diffusion of a single ion in an optical lattice. The results obtained so far show that the method will be suitable to perform detailed studies of the processes involved. In the moment there are still some experimental drawbacks such as e.g. the influence of micromotion being in principle avoidable in future experiments.

So far our study reflects only classical aspects of polarization gradient cooling. However, it also seems possible to reach the quantum limit if the laser beams used for fluorescence detection and grating formation are at different wavelengths so that the optical resolution can be brought into a domain, which is better than the period of the optical lattice. Experiments in this direction are in principle possible and under preparation in the moment.