Optimization of the current extracted from an ultracold ion source

During the loading phase, atoms from the background vapor are loaded into the MOT at a constant loading rate R_L [18]. Atoms are lost from the trap primarily because of collisions of trapped atoms with the background gas. The loading of the MOT is described by the rate equation

$$\begin{equation} \frac{{\rm d} N}{{\rm d} t} = R_{\mathrm{L}}- \frac{N}{\tau_{\mathrm{M}}}, \end{equation}\noindent \tag{ 2 }$$

with N being the number of trapped atoms and τ_M the lifetime of the MOT. In contrast, during the expansion phase R_L = 0 and the atoms can also more easily escape; the lifetime of the MOT during this phase τ_M' is shorter than τ_M. A set of two differential equations, one for the loading phase and the other for the expansion phase, with N(0) = N(t_cycle) as the periodic boundary condition, allows a unique solution for the steady state. In this case, the lifetimes of the MOT are chosen to be τ_M = 200 ms and τ'_M = 100 ms, values that are realistic for the setup. The sequence's timings, t_l = 1000 μs and t_e = 800 μs, are typically the highest used in the experiments, and even then, the number of atoms changes only less than 1% during the sequence. The average number 〈N〉 of atoms is therefore a good approximation for the number of atoms in the MOT and can be expressed as

$$\begin{equation} \langle N \rangle = \frac{N_{\infty}}{1 + \zeta~t_{\mathrm{e}}/t_{\mathrm{l}}} \approx N(t), \end{equation} \noindent \tag{ 3 }$$

with ζ = τ_M/τ'_M and N_∞ = R_Lτ_M as the number of atoms when the MOT is loaded (t ≫ τ_M). From equation (3), it follows that if t_i → 0 (only the loading phase is present), then 〈N〉 ≈ N_∞. Finally, the core atomic density is given by

$$\begin{equation} N_{\mathrm{c}} = \frac{\langle N \rangle}{(2 \pi)^{3/2} \sigma_{\mathrm{M}}^3}. \end{equation} \tag{ 4 }$$

Typical values for the parameters of the model can be found in table 1. These values are used for the figures and calculations found in this section.

An on-resonance excitation laser beam is used to excite the atoms in the core, which will be coincidentally ionized with an ionization laser. The excitation laser beam also accelerates the illuminated atoms in the direction of its propagation [8], so they can eventually reach the core region and be ionized. This effect is termed 'pushing effect' from now on. The atoms get accelerated by a force F_z in the z-direction (due to the radiation pressure) given by

$$\begin{equation} F_z(v_z) = \hbar k \frac{\Gamma}{2} \frac{s_0}{1+s_0+(2 \delta(v_z) / \Gamma)^2}, \end{equation}\noindent \tag{ 5 }$$

where ℏ is Dirac's constant, Γ = 5.98 × 2π Mrad s⁻¹ is the linewidth of the atomic transition, k = 2π/λ_e is the magnitude of the wave vector and δ(v_z) = −2πv_z(t)/λ_e is the Doppler shift of the laser beam. The velocity of the atoms in the z-direction v_z(t) can be determined from the Newtonian expression F_z = m dv_z/dt, where m is the atomic mass of rubidium. This leads to

$$\begin{equation} v_z(t) = \frac{V_{\mathrm{c}}}{2} \sqrt{1 +s_0} \left( \alpha^{1/3} - \alpha^{-1/3} \right), \end{equation} \tag{ 6 }$$

where V_c = 4.66 m s⁻¹ is the capture velocity and

$$\begin{equation} \alpha = \frac{t+\sqrt{\tau_{\mathrm{v}}^2 +t^2}}{\tau_{\mathrm{v}}} \end{equation} \tag{ 7 }$$

in which

$$\begin{equation} \tau_{\mathrm{v}} = \frac{1}{3} \frac{(1 + s_0)^{3/2}}{s_0} \frac{2\,m}{\hbar k^2} \end{equation} \tag{ 8 }$$

is the time constant for the increase in velocity. A typical value is τ_v = 39 μs for s₀ = 1. The distance traveled in a certain time t is numerically calculated from equation (6) as

$$\begin{equation} \mathrm{d}(t)=\int_0^t{v_z(t)\,\mathrm{d}t}. \end{equation} \tag{ 9 }$$

Due to the pushing effect, ionization is not limited to atoms present in the core region at the start of the ionization phase, as we assumed previously [7]. Instead, all the atoms in a cylindrically shaped region as shown in figure 3 can be pushed through the ionization region and contribute to the current. Hence, the atoms in the cylindrically shaped region N_cyl = N_p + N_c need to be considered to determine the number of atoms in the core region. Here, N_p is the number of atoms that will be pushed toward the core region and N_c is the number of atoms present in the core region. The relevant geometry is sketched in figure 3. In this figure, the center of the MOT is at z = 0. The core region is the part of the cylinder highlighted in gray matching $|z| < \sqrt {\frac {\pi }{2}} \sigma _{\mathrm {i}}$; σ_i is the rms radius of the ionization laser beam and σ_e is the rms radius of the excitation laser beam. The factors $\sqrt 2$ and $\sqrt {2 \pi }$ are two normalization constants: the effective radius of the cylinder $\sqrt 2 \sigma _{\mathrm {e}}$ is chosen such that the area A_e = 2πσ²_e is the same for the actual Gaussian laser beam and in this approximated geometry; the length of the core region $\sqrt {2 \pi } \sigma _{\mathrm {i}}$ is chosen such that the volume of the core V_c = (2π)^3/2σ²_eσ_i is the same in both descriptions (assuming σ_e = σ_i = σ_c).

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The number of atoms in the core is given by integrating the local cylinder atomic density n_cyl(z) as

$$\begin{equation} N_{\mathrm{c}} = \int_{-\sqrt{\frac{\pi}{2}} \sigma_{\mathrm{i}}}^{\sqrt{\frac{\pi}{2}} \sigma_{\mathrm{i}}}{n_{\rm cyl}(z) A_{\mathrm{e}} \mathrm{d}z}. \end{equation}\noindent \tag{ 10 }$$

Note that the integration over the direction perpendicular to the z-axis has been replaced with multiplying by the area A_e. This approximation holds since the size of the excitation laser beam is significantly smaller than the size of the MOT (σ_e ≪ σ_M). In reality, n_cyl(z) < n(z), because atoms have been pushed out of the volume during the previous ionization phase. However, we approximate n_cyl(z) ≈ n(z), since it holds in the case of long loading times and short ionization times. We are indeed interested in those times where the maximum of the current is expected to occur. For long ionization times and short loading times the current derived from the model will then be larger than in reality since the approximation not longer holds, but this is also a less interesting case for the current optimization. After a time t, the entire cylinder has been pushed by the excitation laser beam and all the atoms originally present in the volume have been displaced by a distance d(t). Hence, the atoms present in the core region at a time t have originated from a different part of the MOT. This can be incorporated into the model by replacing the density term n(z) with n(z − d(t)).

The atoms initially present inside the cylinder are not confined there. They can diffuse out through its surface and may not reach the core region. The outward diffusion rate of particles is given by

$$\begin{equation} -\frac{{\rm d} N_{\rm cyl}}{{\rm d} t}=\int{\frac{n_{\rm cyl}(z) \langle v \rangle dA_{\rm cyl}}{4}}=\frac{\langle v \rangle}{4} \frac{\sqrt2}{\sigma_{\mathrm{e}}} N_{\rm cyl}, \end{equation}\noindent \tag{ 11 }$$

where 〈v〉 is the thermal velocity. This equation can be written in terms of the diffusion time constant τ_d as

$$\begin{equation} -\frac{{\rm d} N_{\rm cyl}}{{\rm d} t}=\frac{N_{\rm cyl}}{\tau_{\mathrm{d}}}, \end{equation}\noindent \tag{ 12 }$$

where τ_d is given by

$$\begin{equation} \tau_{\mathrm{d}}=\frac{\sigma_{\mathrm{e}}}{2 \sqrt2 \langle v \rangle}. \end{equation}\noindent \tag{ 13 }$$

Using a typical value for the thermal velocity 〈v〉 = 0.3 m s⁻¹ and for the excitation laser beam size σ_e = 30 μm results in τ_d = 350 μs (see table 1).

Alltogether, the contribution of the atoms that started out in the cylinder to the number of excited atoms in the core region is given by

$$\begin{equation} N_{\rm core}^{\rm (cyl)}(t) = A_{\mathrm{e}} \,{\rm e}^{-\frac{t}{\tau_{\mathrm{d}}}} f(s_0,\delta(v_z)) \int_{-\sqrt{\frac{\pi}{2}} \sigma_{\mathrm{i}}}^{\sqrt{\frac{\pi}{2}} \sigma_{\mathrm{i}}}{n(z-d(t)) \mathrm{d}z}. \end{equation} \tag{ 14 }$$

In equation (14), f is the excited state fraction of atoms, which can be expressed as

$$\begin{equation} f(s_0,\delta(v_z)) = \frac{1}{2} \frac{s_0}{1+s_0+(2 \delta(v_z) / \Gamma)^2}. \end{equation} \tag{ 15 }$$

Equation (14) can be further simplified using the midpoint rule to

$$\begin{equation} N_{\rm core}^{\rm (cyl)}(t) \dot{=} (2 \pi)^{(3/2)} \sigma_{\mathrm{e}}^2 \sigma_{\mathrm{i}} n(-d(t)) \,{\rm e}^{-\frac{t}{\tau_{\mathrm{d}}}} f(s_0,\delta(v_z)). \end{equation} \tag{ 16 }$$

Figure 4 is a plot of N^(cyl)_core(t), normalized with its maximum N_init ≡ N^(cyl)_core(0), against time for three different values of s₀. This is always a monotonically decreasing function, mainly due to the lower density outside the core region. For a very low saturation parameter (s₀ = 0.1) the pushing effect become less important, and the dominant loss process of atoms is diffusion out of the cylinder, which means that the number of atoms decays exponentially with a time constant τ_d.

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Apart from the atoms present in the cylinder at t = 0, there is also a contribution of atoms that diffuse into the cylinder. The diffusion flux density into the cylinder is given by

$$\begin{equation} \phi_{\rm diff}^{\rm (in)}(z') = \textstyle\frac{1}{4} n(z') \langle v \rangle \end{equation} \tag{ 17 }$$

with z' the coordinate at which a particle enters the cylinder. This flux needs to be integrated over the surface area (${\rm d}A_{\mathrm{cyl}}=2 \pi \sqrt {2} \sigma _{{\rm e}} \,{\rm d}z'$) to obtain the inward diffusion flux

$$\begin{equation} \frac{{\rm d} N_{\rm cyl}}{{\rm d} t} = \int_{-\infty}^{\sqrt{\frac{\pi}{2}} \sigma_{\mathrm{i}}}{\phi_{\rm diff}^{(\mathrm{in})}(z') 2 \sqrt2 \pi \sigma_{\mathrm{e}} \mathrm{d}z'}. \end{equation} \tag{ 18 }$$

This rate should be integrated over time to obtain the number of atoms that have entered the cylinder. However, these atoms can also diffuse out of the cylinder again, and only the number of atoms that make it to the core region is of interest for this model.

A change of variable z' = z − d(t'), where z is the position of a particle that entered the core at a position z' after a time of flight t', is performed in order to integrate over the core region instead of over the cylinder,

$$\begin{equation} \frac{{\rm d} N_{\rm core}^{\rm (diff)}}{{\rm d} t} = \frac{\pi}{\sqrt2} \sigma_{\mathrm{e}} \langle v \rangle \int_{-\sqrt{\frac{\pi}{2}} \sigma_{\mathrm{i}}}^{\sqrt{\frac{\pi}{2}} \sigma_{\mathrm{i}}} {n(z-d(t)) \mathrm{d}z}. \end{equation} \tag{ 19 }$$

Therefore, the contribution of the atoms that diffused into the cylinder to the number of the excited atoms in the core region is given by

$$\begin{equation} N_{\rm core}^{\rm (diff)}(t)\ \dot{=}\ \pi^{3/2} \sigma_{\mathrm{e}} \sigma_{\mathrm{i}} \langle v \rangle \int_0^t{n(-d(t')) \,{\rm e}^{-\frac{t'}{\tau_{\mathrm{d}}}} f(s_0,\delta)\, \mathrm{d}t'}. \end{equation} \tag{ 20 }$$

This expression has also been approximated using the midpoint rule. Note that N^(diff)_core(t) is the time integral of equation (16) apart from some constants.

Figure 5 plots N^(diff)_core(t) for three values of the saturation parameter. It has been normalized with its maximum value N_final ≡ N^(diff)_core(∞). This is a monotonically increasing function because at large t, the atoms still diffuse into the part of the cylinder close to the core region at the same rate. For a large saturation parameter (s₀ = 10), the convergence to the final value is faster, as for a low saturation parameter the pushing effect is less important.

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Both figures 4 and 5 have been normalized and it is interesting to see how the normalization constants N_init and N_final relate to each other. This is shown in figure 6, where N_init and N_final are plotted versus s₀.

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Both of the functions increase for a larger saturation parameter. Hence, for the current optimization a larger value of the saturation parameter of the excitation laser beam is beneficial. For a finite s₀, the following relation holds: N_init > N_final.

The ionization rate r_i for excited atoms in the core region can be expressed as

$$\begin{equation} r_{\mathrm{i}} = \tau_{\mathrm{i}}^{-1} = \frac{\sigma_{\mathrm{PI}} P_{\mathrm{i}} \lambda_{\mathrm{i}}}{A_{\mathrm{i}} h c}, \end{equation}\noindent \tag{ 21 }$$

where σ_PI = 1.48 × 10⁻¹⁷ cm² is the photo-ionization cross section for ⁸⁵Rb [19], the ionization laser beam is focused to an area A_i = 2πσ²_i, h is Planck's constant and c is the speed of light. The quantity τ_i is the typical time it takes for an atom to be ionized. The available ionization laser beam power in the laboratory is currently such that the average time spent by the atoms in the core region is small compared to the ionization time constant. Therefore, the number of ionized atoms is small (less than 30%) compared to the total number of atoms present in the core region. The time-dependent ion current I(t), in this limit, is given by

$$\begin{equation} I(t) = e r_{\mathrm{i}} \{ N_{\rm core}^{\rm (diff)}(t) + N_{\rm core}^{\rm (cyl)}(t) \}. \end{equation}\noindent \tag{ 22 }$$

The typical temporal behavior of the ion current has been plotted in figure 7. In this plot, the current has been separated into these two contributions (similarly to what was discussed before for the number of atoms in the core region). The diffusion current I_diff(t) = er_iN^(diff)_core(t) increases with time, up to a final value. In contrast, the current due to atoms starting in the cylinder I_cyl(t) = er_iN^(diff)_cyl(t) decreases to zero with time. The total current I(t) = I_diff(t) + I_cyl(t) typically resembles an exponential decay with a time constant τ, with an initial maximum I_peak and a steady level I_final.

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It is also useful to define the average current $\skew3\overline {I}$, the peak current I_peak, the final current I_final and the charge per pulse Q:

$$\begin{equation} \skew3\overline{I} = \frac{1}{t_{\rm cycle}} \int_0^{t_{\rm i}}{I(t)\,\mathrm{d}t}, \end{equation} \tag{ 23 }$$

$$\begin{equation} I_{\rm peak} = \mbox{max} \left(I(t)\right), \end{equation} \tag{ 24 }$$

$$\begin{equation} I_{\rm final} = I(t_{\rm i}), \end{equation} \tag{ 25 }$$

$$\begin{equation} Q = \int_0^{t_{\rm i}}{I(t)\,\mathrm{d}t}. \end{equation} \tag{ 26 }$$

Figure 8 shows a typical two dimensional 2D plot of the average current as a function of the ionization time on the horizontal scale and of the loading time on the vertical scale. A clear maximum is $\skew3\overline {I}=2.5\,{\rm pA}$ obtained with t_i ≈ 50 μs and t_l ≈ 100 μs. The loading time is twice the ionization time, corresponding to 〈N〉 = N_∞/2, from equation (3). From the type of plot shown in figure 8, a few important quantities can be extracted. The first one is simply the maximum of the average current $\skew3\overline {I}_{\max }$. Also, for a given ionization time for which the average current is maximized there is an optimum. A curve is drawn as a dashed line in figure 8 that indicates the dependence. To first order, the curve is a straight line that can be characterized by its slope. We extract the slope by linear regression. The slope ρ = t_l/t_i represents the optimal ratio between the loading and the ionizing times, and is related to the optimal duty cycle D at which the highest current is obtained. The duty cycle is given by

$$\begin{equation} D=\frac{t_{\mathrm{i}}}{t_{\rm cycle}} \cong \frac{1}{\rho + 1}, \end{equation} \tag{ 27 }$$

where the 10 μs in the definition of t_cycle have been ignored since t_cycle ≫ 10 μs.

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4.1.1. Trap temperature

The trap temperature is an important parameter in our model, as it affects the diffusion between the core region and the rest of the MOT, as well as the expansion of the MOT itself while the trapping lasers are turned off. After the trapping laser beams are turned off at t = 0, the MOT expands according to the relation

$$\begin{equation} \sigma_{\mathrm{M}}^2(t_{\mathrm{e}}) = \sigma_{\mathrm{M}}^2(0) + \left(\frac{k_{\mathrm{b}} T_{\mathrm{s}}}{m}\right) t_{\mathrm{e}}^2, \end{equation}\noindent \tag{ 28 }$$

where k_b is Boltzmann's constant [20]. The MOT cameras are triggered to measure the fluorescence light emitted by the trapped atoms at a variable delay time up to 8 ms after the trapping lasers are turned off (at t = 0). The trapping laser beams are flashed on for 250 μs to produce the fluorescence light. The rms radii of the MOT are calculated by fitting the images with a 2D Gaussian. The square of the rms radius of the MOT is plotted against the square of the expansion time in figure 9(a). The trap temperature, determined from a fit of equation (28) to the data, is (230 ± 30) μK, not far from the Doppler temperature T_D = 143 μK of the ⁸⁵Rb transition. The uncertainty in the extracted temperature reflects the degree of agreement between the data sets.

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4.1.2. Loading rate and lifetime

4.1.3. The number of atoms

This section discusses the temporal behavior of the measured current pulses. Two measurements have been selected and are shown in figure 10: the first one for a large saturation parameter s₀ = 4 and t_i = 130 μs and the second one for a smaller saturation parameter s₀ = 0.2 and t_i = 770 μs. The loading time is t_l = 250 μs in both cases. The two plots show many differences. One of these lies in the magnitude of the peak current: I_peak = 62 pA in figure 10(a) versus I_peak = 1.1 pA in figure 10(b). This is partly due to the difference in the excited state fraction ratio f(4,0)/f(0.2,0) = 4.8, but the difference in the number of trapped atoms is more important. In figure 10(b), the ionization time is about 3 times longer than the loading time, whereas in figure 10(a) this is a factor of 2 shorter. From figure 9(d), this corresponds to 〈N〉 = 2.5 × 10⁸ atoms and 2.5 × 10⁷ atoms, respectively. The two effects combined change the peak current by a factor of 48, which is comparable with the difference in peak current in the two measurements (a factor ≈ 56).

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The second difference is the ratio between the peak and the final current I_peak/I_final. This ratio is 2.8 for s₀ = 4 since the final current is I_final = 22 and 1.4 for s₀ = 0.2 since I_final = 0.7 pA. This difference can be explained with the model and is caused by the different s₀ as shown in figure 6, where the ratios of 3.9 and 2.2 are respectively calculated at the same s₀. The measured values are both a factor of 1.5 lower, possibly due to local variation of the MOT density.

Finally, the currents resemble quite well an exponential decay, with time constants of τ = 45 μs in figure 10(a) and τ = 102 μs in figure 10(b), drawn as solid lines. The value for figure 10(a) is somewhat uncertain because the final current level is not yet reached due to the short t_i. Nevertheless, the time constant clearly depends on s₀ as predicted by the model (see figures 4 and 5).

The model predicts that the atoms that are ionized have been pushed into the core region by the excitation laser beam. Therefore the total charge created can be increased by moving the ionization laser beam away from the center of the MOT, in the positive z-direction (see figure 1). In this way the excitation laser cylinder (see figure 3) can be made longer, increasing the number of atoms it contains. However, for larger displacements the current will be reduced once outward diffusion becomes important. In addition, the initial current will be reduced because the ionization laser beam is no longer at the center of the MOT (where the density is maximum). In order to verify this picture, an experiment is performed where the current is measured for several z-positions of the ionization laser beam. This experiment uses a long ionization time t_i = 900 μs, so that the current will nearly reach its final value, a very long loading time of t_l = 5 ms and s₀ = 4. Here Δz > 0 corresponds to a displacement of the ionization laser spot in the same direction as the atoms are pushed.

For several values of Δz (in the units of σ_M ≈ 0.8 mm), calculated and measured current profiles have been plotted in figure 11. For the calculations, a shifted density term

$$\begin{equation} n(z)=N_{\mathrm{c}} \,{\rm e}^{\left( -(z-\Delta z)^2 / {2 \sigma_{\mathrm{M}}^2} \right)} \end{equation} \tag{ 29 }$$

is used instead of the density at z = 0 of equation (4). Comparing the two figures, a number of similarities are observed. In both cases, the initial current decreases when Δz is increased, because the density at the ionization laser position decreases. Also, in both cases, the current gets an extra 'bump': it no longer matches an exponential decay, but has a broad peak added to it. The time at which this peak occurs increases with Δz. It is approximately given by the time needed for an atom to be pushed over the distance Δz, the solution of equation (9). This gives t_peak = 111 μs for Δz = 0.5 σ_M, t_peak = 168 μs for Δz = σ_M and t_peak = 215 μs for Δz = 1.5 σ_M for the model. The first three peak positions based on this approach have been marked with colored arrows in figure 11. This simple approach does not take into account effects such as outward diffusion and increasing of the Doppler shifts, which both will make the real peak occur earlier. Also, because these peaks are superimposed on the exponential decay, the actual peak positions will seem to occur earlier. Nevertheless, it gives quite a good approximation of the behavior in both the model and the experiment.

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Next, both the model and the experiment show that shifting the ionization laser too far will result in a lower current for the whole time the ionization laser beam is turned on, compared to Δz = 0. This is the case for Δz ⩾ σ_M both in the modeled current and in the experiment. Clearly, the outward diffusion and decreasing excited state fraction for larger time of flight prevent the extra atoms in the cylinder from being ionized. For smaller shifts, Δz = 0.5 σ_M, both the model and the experiment show a slightly larger final current, which can be desirable for extremely long pulses, where only the final level of the current is relevant. For shifts 0 < Δz < 0.5 σ_M an optimum occurs, where also for shorter ionization times the obtained current is larger than without the shift.

One aspect where the model and experiment do not agree is how much the initial current decreases when Δz is increased. This is probably caused by deviations of the number density from strict Gaussian behavior, observable as asymmetries present in the shape of the MOT.

The main parameters of the model have been measured (see section 4.1) and the model has been verified (see section 4.3). The main goal of this paper was to optimize the extracted current. The average current, equation (23), is measured while the loading and the ionization times are varied. This results in a contour plot similar to that presented in figure 8. These measurements have been made at different values of the excitation laser beam saturation parameter and ionization laser beam power. Every set of measurements was carried out in a random order to prevent drift from influencing the results. The measurement time varied between 30 and 60 min depending on the number of steps. Typically, the ionization power energy was P_i = 46 mW, unless otherwise specified.

4.4.1. Saturation parameter dependence

The saturation parameter is changed by varying the power of the excitation laser beam with the use of an AOM. Figure 12 shows two scans performed at s₀ = 1 and s₀ = 12. In both the scans, t_l was incremented in steps of 100 μs between 50 and 1250 μs and t_i ranges between 50 and 850 μs in steps of 80 μs, a total of 143 points. These plots strongly resemble the one from the model shown in figure 8, especially the one at s₀ = 12. In general, a higher saturation parameter is beneficial for increasing the average current. The maximum average current is marked with a black dot in the plots. The average current is about a factor of two higher, $\skew3\overline {I} \approx 8.2$ pA at s₀ = 12 compared to $\skew3\overline {I} \approx 3.8\,{\rm pA}$ at s₀ = 1, and this is mainly caused by the higher excited state fraction: f(12,0)/f(1,0) ≈ 1.8. A stronger 'pushing effect' is also important, since it leaves less time for the atoms to diffuse out of the cylinder.

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It follows from the plot that it is better to ionize for a short time, going for a higher repetition rate rather than a low one. The optimum ratio ρ where the maximum of the average current occurs is drawn as a dashed line in the two figures: ρ = 2.3 for s₀ = 12 and ρ = 1.4 for s₀ = 1. Clearly, it becomes beneficial to use a shorter ionization phase for a higher saturation parameter and this is a direct consequence of the larger ratio between the peak and the final current already discussed in section 4.2. The maximum average current measured in all the experiments was (13 ± 1) pA, obtained at a duty cycle of 36%. The maximum peak current measured was (88 ± 5) pA. The uncertainty of the current is related to the calibration of the MCP.

The contour plot of the charge per pulse for s₀ = 12 is plotted in figure 13. The maximum in this graph is 8.8 fC, which corresponds to 5.5 × 10⁴ ions. In general, longer loading and ionization times increase the charge per pulse, but the graph shows that there is a limit: at a given loading time, increasing the ionization time will reduce the charge since too long an ionization time eventually reduces the available time to load atoms in the trap. The slope ρ can also be defined for charge plots in a similar way as discussed in section 3 for the average current. In this case, ρ = 1.9 (dashed line in the plot). The ratio is slightly smaller than that for the average current of figure 12(b). This is as expected since a longer ionization time will result in some extra charge, but the period t_cycle will be longer, reducing the average current. But since the slope is very similar, optimization of the average current coincides with optimization of the charge per pulse.

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4.4.2. Ionization power dependence

The power of the ionization laser beam used in the experiments was so small that removal of atoms by ionization can be neglected when determining the number of atoms in the core region. A change in the laser power therefore should not change the temporal behavior of the current pulses but just the number of ionized atoms. Figure 14 shows two contour plots from an experiment where the ionization laser power has been decreased by a factor of 3.8 with the use of a neutral density filter. Here, the size of the core region in the three dimensions is (39 × 63 × 71) μm³, resulting in a volume V_c = 2.8 × 10⁻¹² m³. Also, t_l was incremented in steps of 150 μs between 50 and 1250 μs and t_i ranges between 50 and 500 μs in steps of 75 μs; in total the scan consisted of 63 points. As predicted, the shape of the contour in the two figures is very similar and the average current decreased by a factor of 4.1, similar to the decrease in laser power.

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4.4.3. Summary plots

In this section, two summary plots for several measurements carried out with a core volume V_c = 2 × 10⁻¹³ m³ are presented. In figure 15, the slopes ρ at which the optimum of the average current occurs are plotted against the saturation parameter of the excitation laser beam. When more measurements are available for a specific s₀, they correspond to measurements made at different ionization power. The large uncertainty bars are related to the linear fit not matching the real behavior perfectly, as ρ is not a constant in practice. For this reason the plot is qualitative. It does appear, however, that the slope increases with increasing saturation parameter in agreement with the prediction of section 4.2 shown in figure 12. In general, with increasing saturation parameter, a larger increase in the peak current is expected compared to an increase in the final current. Hence, the gain in charge obtained by ionizing for a longer time is offset by a reduction in the number of atoms if the loading time is not adjusted, so that it is instead beneficial to reduce the ionization time. This then leads to a higher optimal slope ρ.

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In figure 16, the peak current is plotted against the saturation parameter. A higher saturation parameter of the excitation laser beam means a higher excited state fraction and therefore a higher initial peak current. Hence, the expected behavior of the peak current is to be proportional to the excited state fraction: I_peak∝f(s,δ → 0), from equation (15). This relation is also plotted in the figure as a solid line. The shape of the curve indeed corresponds to the trend of the data points.

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