Influence of static electric fields on an optical ion trap

We recently reported on a proof-of-principle experiment demonstrating optical trapping of an ion in a single-beam dipole trap superimposed by a static electric potential [Ch. Schneider, M. Enderlein, T. Huber, and T. Schaetz, Nat. Photon. 4, 772 (2010)]. Here, we discuss, first, the experimental procedures focusing on the influence and consequences of the static electric potential. This potential can easily prevent successful optical trapping, if its configuration is not chosen carefully. Afterward, we analyze the dipole trap experiments with different analytic models, in which different approximations are applied. According to these models the experimental results agree with recoil heating as the relevant heating effect. In addition, a Monte Carlo simulation has been developed to refine the analysis. It reveals a large impact of the static electric potential on the dipole trap experiments in general. While it supports the results of the analytic models for the parameters used in the experiments, the analytic models are no longer valid for significantly different parameters. Finally, we propose technical improvements for future realizations of experiments with optically trapped ions.

Figure 1

Scheme of the trapping setup. The linear RF trap consists of gold-plated, bladelike electrodes, which are mounted using alumina insulators and stainless steel holders. The DC electrodes are divided into six segments each: three belong to a zone with greater electrode–ion distance and three to a zone with smaller electrode–ion distance. (The latter zone is not used in the current dipole trap experiments.) The atomic oven can evaporate a beam of neutral magnesium atoms mainly in direction of the zone with large electrode–ion distance. One or more atoms are photoionized and Doppler cooled in the RF potential using laser beams in the $y$-$z$ plane, which intersect the $z$ axis at an angle of 22.5${}^{\circ }$. The dipole trap beam propagates in the $x$-$z$-plane and crosses the $z$ axis at an angle of 45${}^{\circ }$. The fluorescence light from the ion(s) is detected with a CCD camera through a sapphire window above the trap (not shown). Parts of an insulator and a holder are omitted to offer a view on the trapping zone. The arrows of the coordinate system indicate a length scale of 10 mm; the distance between the atomic oven and its aperture is not to scale.

Figure 2

Fluorescence images of one, two, and three ions in the RF trap. The radial frequencies are set to ${\omega }_{x,y}\approx 2\pi \times 870\mathrm{kHz}$, while the axial trap frequency of a single ion amounts to (a) ${\omega }_z=2\pi \times (114.0\pm 0.5)\mathrm{kHz}$ and (b) ${\omega }_z=2\pi \times (40\pm 2)\mathrm{kHz}$, respectively. The dashed lines indicate the position of a single ion. The objective lens used to image the ions is intentionally defocused to gain additional information about the ions' positions perpendicular to the imaging plane: The ion images will become larger (smaller) if the ion moves toward (away from) the CCD camera. Panel (b) reveals anharmonicities of the axial potential.

Figure 3

Axial potential of the RF trap in cubic approximation $U_{\mathrm{DC},\mathrm{cub}}$ and harmonic approximation $U_{\mathrm{DC},\mathrm{harm}}$. The cubic approximation is derived from the positions of one, two, and three ions and the frequency of the axial oscillation of a single ion [Fig. 2b]. The light pressure of the Doppler cooling beam is subtracted. The axial potential has a maximum at a distance of 135 $\mu$m from the minimum and a depth of approximately $k_B\times 0.5$ K.

Figure 4

Lowest-order stability region of Mathieu's equations of an infinitely long linear RF trap (without considering an axial confinement). The motion is stable for pairs $(q,a)$ inside the gray shaded region. Outside this region the motion is stable only in the $x^{\prime }$ direction or the $y^{\prime }$ direction or unstable in both radial directions. The arrow indicates a path in the stability diagram, when the RF voltage is decreased. It starts from a typical operational point of the RF trap. The motion of the ion will become unstable for any $a\ne 0$ before zero RF voltage is reached.

Figure 5

Example for lowest-order stability region of Mathieu's equations of a linear RF trap considering a specific defocusing due to an axial confinement. We assumed some $V_{\mathrm{DC},\mathrm{out}}\ne 0$ and different geometry factors ${\kappa }_{x^{\prime }}\ne {\kappa }_{y^{\prime }}$. It demonstrates that a decrease of the RF voltage will eventually lead to an ion loss for any $a$ parameter (even $a=0$) before zero RF voltage is reached. Additionally, it shows that $V_{\mathrm{DC},\mathrm{out}}=0$ need not be the optimal value to stay inside the stable region as long as possible.

Figure 6

RF voltage of the RF trap during the experimental sequence. The amplitude is measured with a pick-up antenna inside the helical resonator, followed by filters to suppress extraneous RF signals from other sources in the laboratory. It is recorded by an oscilloscope, which is the reason for the noise. The final ramp-up of the RF voltage is slightly nonlinear as the slope of the ramp does not meet the specifications of the RF synthesizer. The magnification (inset) shows the effect of the TTL-controlled RF switch for times $t\le T_{\mathrm{optical}}$. The confinement due to the residual RF voltage during $T_{\mathrm{optical}}$ can be neglected.

Figure 7

Relevant part of the electronic energy level scheme of ${}^{24}{\mathrm{Mg}}^{+}$. As ${}^{24}{\mathrm{Mg}}^{+}$ has no nuclear spin ($I=0$), there are only one ground level (${\mathit{S}}_{1/2}$) with angular momentum $J=1/2$ and two exited levels (${\mathit{P}}_{1/2}$ and ${\mathit{P}}_{3/2}$) with angular momenta $J=1/2$ and $J=3/2$, respectively. The $M_J$ sublevels are shown. The dashed line indicates the ${\sigma }^{+}$-polarized dipole trap beam with a detuning of $\Delta$ from the cycling transition.

Figure 8

Theoretically expected maximal total energy gain ${(d\overline{E}/dt)}_{\mathrm{tot},\max }$ and maximal energy gain due to recoil heating ${(d\overline{E}/dt)}_{\mathrm{s},\max }$ as a function of the detuning $\Delta$ for a Gaussian beam dipole trap (according to Ref. [20]). The graphs are calculated for a waist radius $w_0=7$ $\mu$m, a potential depth $U_0=-k_B\times 38$ mK, a position in the direction of the beam propagation $z_k=0$ of the ion, and the radial position $r$ with maximal heating rate. The gray shaded area is magnified in the inset, which allows us to extract the energy gain for the experimental parameters. The two curves are almost identical such that for the given parameters recoil heating is expected to be dominant.

Figure 9

Theoretically expected maximal total energy gain ${(d\overline{E}/dt)}_{\mathrm{tot},\max }$ and maximal energy gain due to recoil heating ${(d\overline{E}/dt)}_{s,\max }$ as a function of the detuning $\Delta$ for a standing-wave dipole trap created by retroreflecting the Gaussian beam (according to Ref. [20]). The graphs are calculated for a waist radius $w_0=7$ $\mu$m, a potential depth $U_0=-k_B\times 38$ mK, a radial position $r=0$ of the ion, and the position in direction of the beam propagation $z_k$ with maximal heating rate. The power $P_{\mathrm{optical}}\left(\Delta \right)$ of the dipole trap beam corresponds to a quarter of the power of Fig. 8 to account for the retroreflection and interference in the standing wave. The gray shaded area is magnified in the inset. The total heating rate approximately doubles at the detunings used in the single-beam dipole trap experiments.

Figure 10

Comparison of the energy-cutoff and radial-cutoff model: The optical trapping probabilities $P_{\mathrm{en}}$ according to the energy-cutoff model and $P_{\mathrm{rad}}$ according to the radial-cutoff model are shown as functions of the dipole trap duration $T_{\mathrm{optical}}$. The points represent the measured optical trapping probabilities and the error bars the standard deviation of the mean [8]. The parameters for which the curves of the analytic models are shown originate from a joint fit to the experimental data of this figure and Fig. 11. While the curves are almost identical, the best fit values for the initial temperature and the heating rate are approximately 20$\%$ greater for the radial-cutoff model. Experimental parameters: $w_0=(7\pm 1)$ $\mu$m, $\Delta =-2\pi \times 275$ GHz, $P_{\mathrm{optical}}=190$ mW, and ${\omega }_z=2\pi \times 47$ kHz.

Figure 11

Comparison of the energy-cutoff and radial-cutoff model (analogously to Fig. 10): The optical trapping probabilities $P_{\mathrm{en}}$ and $P_{\mathrm{rad}}$ according to the models are shown as functions of the power $P_{\mathrm{optical}}$ of the dipole trap beam. Experimental parameters: $w_0=(6.5\pm 1.0)$ $\mu$m, $\Delta =-2\pi \times 300$ GHz, $T_{\mathrm{optical}}=0.5$ ms, and ${\omega }_z=2\pi \times 41$ kHz.

Figure 12

Equipotential surfaces $U\left(\vec{r}\right)=U_{\mathrm{mpe},\max }$ of the total potential. The parameters of the graphs correspond to the experimental parameters $U_0=-k_B\times 38$ mK, ${\omega }_z=2\pi \times 47$ kHz, and $w_0=7$ $\mu$m. The frequency in the $y$ direction is (a) ${\omega }_y^2={\omega }_{y,\max }^2$, (b) ${\omega }_y^2={\left(1.2{\omega }_{y,\max }\right)}^2$, and (c) ${\omega }_y^2={\left(0.95{\omega }_{y,\max }\right)}^2$, respectively (see text). The points defining $U_{\mathrm{mpe},\max }$ in (a) appear as “cut” trough the egglike surface enclosing the trap center (indicated by the arrow). Note that ${\omega }_{y,\max }^2$ is negative, i.e., an increase of the absolute value of ${\omega }_y^2$ leads to a stronger defocusing effect in the $y$ direction (b), while a decrease leads to a stronger defocusing in a direction in the $x$-$z$ plane (c).

Figure 13

Probability distributions $f\left(E\right)$ of the particle energies $E$ during Doppler cooling in the RF trap. The curves show the results from the Monte Carlo simulation and related fits of the Boltzmann distribution of the three-dimensional harmonic potential. The frequency splitting is set to ${\omega }_{x^{\prime }}-{\omega }_{y^{\prime }}=2\pi \times 2.5$ kHz and the tilting angle to $\alpha ={35}^{\circ }$. The simulation is performed for a cooling duration of 2 s with time steps of $\Delta t=0.1/\Gamma$. “(i)” denotes the parameters used during Doppler cooling corresponding to Fig. 10: ${\omega }_z=2\pi \times 47$ kHz, ${\omega }_{x^{\prime }}=2\pi \times 950$ kHz, $P_{\mathrm{DL}}=8.6$ $\mu$W, and $P_{\mathrm{DLdet}}=270$ $\mu$W. The axial displacement of the ion due to the light pressure of the Doppler cooling beams amounts to $\Delta z=8.8$ $\mu$m. The radial displacement is of the order of few nanometers and can be neglected. The energies $E$ are referred to the harmonic oscillator shifted by $\Delta z$. The curves “(ii)” correspond to Fig. 11: ${\omega }_z=2\pi \times 42$ kHz, ${\omega }_{x^{\prime }}=2\pi \times 880$ kHz, $P_{\mathrm{DL}}=2.1$ $\mu$W, and $P_{\mathrm{DLdet}}=138$ $\mu$W. These parameters yield a displacement due to light pressure of $\Delta z=2.8$ $\mu$m. The curves of the corresponding Boltzmann distributions fit well to the simulated curves. The temperatures according to the fits of the Boltzmann distribution are $T^{\left(\mathrm{i}\right)}=3.1$ mK for the parameters of (i) and $T^{\left(\mathrm{ii}\right)}=2.7$ mK for the parameters of (ii).

Figure 14

Comparison of the probability distributions $f\left(E\right)$ of the particle energies $E$ during Doppler cooling in the RF trap for (a) different splittings ${\omega }_{x^{\prime }}-{\omega }_{y^{\prime }}$ and (b) angles $\alpha$. The curves show the results from the Monte Carlo simulation with the same parameters as used in Fig. 13 for parameters (ii). For small frequency splitting ${\omega }_{x^{\prime }}-{\omega }_{y^{\prime }}$ or small angles $\alpha$ Doppler cooling becomes inefficient.

Figure 15

Monte Carlo simulations of the dipole trap experiments (Figs. 10 and 11) for an angle $\alpha ={35}^{\circ }$ and different frequencies ${\omega }_{y^{\prime }}^2$. The initial values for the position, velocity, and internal state are obtained from the respective results of the Doppler cooling simulations depicted in Fig. 13. The points represent the measured optical trapping probabilities and the error bars the standard deviation of the mean. The Monte Carlo simulations agree with the measured trapping probabilities for the depicted parameters.

Figure 16

Monte Carlo simulations of one dipole trap experiment (Fig. 10) for a frequency ${\omega }_{y^{\prime }}^2=-(2\pi \times 50$ kHz)${}^2$ and different angles $\alpha$ between principal axes of the static electric potential and the axes of the coordinate system. The initial values for the position, velocity, and internal state are obtained from the same Doppler cooling simulation of Fig. 13 for all three curves. Note that this allows for a better comparison of the influence of $\alpha$ on the dipole trap experiment, but assumes a better Doppler cooling result for small angles than predicted by the simulations in Fig. 14. The trapping probability shows a strong dependence on the angle $\alpha$.