# Electro-Optical Ion Trap for Experiments with Atom-Ion Quantum Hybrid Systems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## Featured Application

**This work reports the novel design of a micromotion-free electro-optical trap for ions, integrated with a standard linear Paul trap. The ion trap is developed for experiments with atom-ion quantum mixtures and allows the ions to reach ultracold temperatures by sympathetic cooling with the neutral atoms, eventually leading to atom-ion collisions in the s-wave scattering regime.**

## Abstract

## 1. Introduction

## 2. The Electro-Optical Trap

- $0>\varphi >{\varphi}_{\mathrm{iso}}$: The ions can be trapped in a single minima of the optical lattice, thus forming a disk-shaped crystal.
- $\varphi ={\varphi}_{\mathrm{iso}}$: The three trapping frequencies are equal, so the potential is isotropic.
- ${\varphi}_{\mathrm{iso}}>\varphi >{\varphi}_{\mathrm{trap}}$: The confinement along the interference direction is weaker than the other two. Considering the typical depth of the optical potentials and the Coulomb repulsion, the ions might lie in different minima of the optical lattice.

#### Loading Ions into the Electro-Optical Trap

## 3. Trap Design

#### 3.1. Electro-Optical Trap Design

#### 3.2. Paul Trap Design

#### 3.3. The Atomic Source

#### 3.4. Machining Tolerances

#### 3.5. Materials

## 4. Simulations on the Trapping System

#### 4.1. Electrical Simulations

^{th}superficial charge densities of the surfaces ${S}_{i}$ and N is the panel number. Equation (3) can be reformulated in a compact way as $A\xb7\overrightarrow{q}=\overrightarrow{p}$, where the $N\times N$ matrix A, which depends only on the mesh geometry, connects the vector $\overrightarrow{q}$ containing the superficial charge distributions ${\sigma}_{i}$ to the vector $\overrightarrow{p}$ representing the applied voltage on the i

^{th}panel. BEMSOLVER employs two techniques for solving Equation (3). The first is the generalized minimum residual (GMRES) method [28], which aims at computing an approximate solution by running an iterative algorithm until a certain tolerance fixed by the user is reached. This method starts with an initial guess ${\overrightarrow{q}}_{0}$ for the solution, then evaluates the first residual ${\overrightarrow{r}}_{0}=\overrightarrow{p}-A{\overrightarrow{q}}_{0}$, on which the convergence of the algorithm is checked. If another iteration is needed, the new solution vector is computed as:

#### 4.1.1. Paul Trap Stability Diagram

#### 4.1.2. Residual Axial Radiofrequency

^{th}direction, ${x}_{0,i}$ is the position shift from ${x}_{1,i}$ due to the stray electric field, ${\omega}_{RF}$ is the RF frequency, and ${\omega}_{i}$ is the secular motion frequency. Consequently, the micromotion amplitude along the i

^{th}direction can be estimated as $\left({x}_{0,i}\phantom{\rule{0.166667em}{0ex}}{q}_{i}\right)/2$. Ideally, along the RF electrodes’ axis, the parameter ${q}_{z}$ is null near the trap center; instead, it assumes nonzero values if the finite electrodes’ length and misalignment effects are introduced. In order to evaluate ${q}_{z}$ and the residual axial micromotion amplitude, a modified version of the trap assembly, having the blade electrodes displaced from their exact position, was simulated. Since the mechanical tolerance on the blades’ position was $5\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m, the angular deviations were neglected in the simulation, while the linear displacements were considered equal to the tolerance for taking into account the worst possible case. This evaluation showed a residual axial micromotion amplitude of about $0.5$ nm, which corresponded to a micromotion energy of about $1\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$K.

#### 4.2. Thermal Simulations

#### 4.2.1. RF Power Dissipation

#### 4.2.2. Ions Loading from Paul Trap to Electro-Optical Trap

#### 4.2.3. Ovens’ Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Tomza, M.; Jachymski, K.; Gerritsma, R.; Negretti, A.; Calarco, T.; Idziaszek, Z.; Julienne, P.S. Cold hybrid ion-atom systems. arXiv
**2017**, arXiv:1708.07832. [Google Scholar] [CrossRef] [Green Version] - Idziaszek, Z.; Simoni, A.; Calarco, T.; Julienne, P.S. Multichannel quantum-defect theory for ultracold atom-ion collisions. New J. Phys.
**2011**, 13, 083005. [Google Scholar] [CrossRef] - Sias, C.; Köhl, M. Hybrid quantum systems of ions and atoms. arXiv
**2014**, arXiv:1401.3188. [Google Scholar] - Kollath, C.; Köhl, M.; Giamarchi, T. Scanning tunneling microscopy for ultracold atoms. Phys. Rev. A
**2007**, 76, 063602. [Google Scholar] [CrossRef] [Green Version] - Zipkes, C.; Palzer, S.; Sias, C.; Köhl, M. A trapped single ion inside a Bose-Einstein condensate. Nature
**2010**, 464, 388–391. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ratschbacher, L.; Zipkes, C.; Sias, C.; Köhl, M. Controlling chemical reactions of a single particle. Nat. Phys.
**2012**, 8, 649–652. [Google Scholar] [CrossRef] - Hall, F.H.J.; Aymar, M.; Bouloufa-Maafa, N.; Dulieu, O.; Willitsch, S. Light-Assisted Ion-Neutral Reactive Processes in the Cold Regime: Radiative Molecule Formation versus Charge Exchange. Phys. Rev. Lett.
**2011**, 107, 243202. [Google Scholar] [CrossRef] - Härter, A.; Hecker Denschlag, J. Cold atom-ion experiments in hybrid traps. Contemp. Phys.
**2014**, 55, 33–45. [Google Scholar] [CrossRef] [Green Version] - Cetina, M.; Grier, A.T.; Vuletić, V. Micromotion-Induced Limit to Atom-Ion Sympathetic Cooling in Paul Traps. Phys. Rev. Lett.
**2012**, 109, 253201. [Google Scholar] [CrossRef] - Feldker, T.; Fürst, H.; Hirzler, H.; Ewald, N.V.; Mazzanti, M.; Wiater, D.; Tomza, M.; Gerritsma, R. Buffer gas cooling of a trapped ion to the quantum regime. arXiv
**2019**, arXiv:quant-ph/1907.10926. [Google Scholar] [CrossRef] - Schneider, C.; Enderlein, M.; Huber, T.; Schaetz, T. Optical trapping of an ion. Nat. Photonics
**2010**, 4, 772–775. [Google Scholar] [CrossRef] [Green Version] - Huber, T.; Lambrecht, A.; Schmidt, J.; Karpa, L.; Schaetz, T. A far-off-resonance optical trap for a Ba
^{+}ion. Nat. Commun.**2014**, 5, 5587. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lambrecht, A.; Schmidt, J.; Weckesser, P.; Debatin, M.; Karpa, L.; Schaetz, T. Long lifetimes in optical ion traps. arXiv
**2016**, arXiv:1609.06429. [Google Scholar] - Enderlein, M.; Huber, T.; Schneider, C.; Schaetz, T. Single Ions Trapped in a One-Dimensional Optical Lattice. Phys. Rev. Lett.
**2012**, 109, 233004. [Google Scholar] [CrossRef] [Green Version] - Schmidt, J.; Lambrecht, A.; Weckesser, P.; Debatin, M.; Karpa, L.; Schaetz, T. Optical Trapping of Ion Coulomb Crystals. Phys. Rev. X
**2018**, 8, 021028. [Google Scholar] [CrossRef] [Green Version] - Perego, E. A Novel Setup for Trapping and Cooling Barium ions for Atom-ion Experiments. Ph.D. Thesis, Politecnico di Torino, Torino, Italy, 2019. [Google Scholar]
- Kaur, J.; Singh, S.; Arora, B.; Sahoo, B.K. Magic wavelengths in the alkaline-earth-metal ions. Phys. Rev. A
**2015**, 92, 031402. [Google Scholar] [CrossRef] [Green Version] - Sage, J.M.; Kerman, A.J.; Chiaverini, J. Loading of a surface-electrode ion trap from a remote, precooled source. Phys. Rev. A
**2012**, 86, 013417. [Google Scholar] [CrossRef] [Green Version] - Cetina, M.; Grier, A.; Campbell, J.; Chuang, I.; Vuletić, V. Bright source of cold ions for surface-electrode traps. Phys. Rev. A
**2007**, 76, 041401. [Google Scholar] [CrossRef] [Green Version] - De, S.; Dammalapati, U.; Jungmann, K.; Willmann, L. Magneto-optical trapping of barium. Phys. Rev. A
**2009**, 79, 041402. [Google Scholar] [CrossRef] [Green Version] - Naik, D.S.; Kuyumjyan, G.; Pandey, D.; Bouyer, P.; Bertoldi, A. Bose-Einstein condensate array in a malleable optical trap formed in a traveling wave cavity. Quantum Sci. Technol.
**2018**, 3, 045009. [Google Scholar] [CrossRef] [Green Version] - Nagourney, W. Quantum Electronics for Atomic Physics; Oxford University Press: New York, NY, USA, 2010. [Google Scholar]
- Herskind, P.F.; Dantan, A.; Albert, M.; Marler, J.P.; Drewsen, M. Positioning of the rf potential minimum line of a linear Paul trap with micrometer precision. J. Phys. B At. Mol. Phys.
**2009**, 42, 154008. [Google Scholar] [CrossRef] [Green Version] - Roos, C.F. Controlling the Quantum State of Trapped Ions. Ph.D. Thesis, Leopold-Franzens-Universitat Innsbruck, Innsbruck, Austria, 2000. [Google Scholar]
- Doležal, M.; Balling, P.; Nisbet-Jones, P.B.R.; King, S.A.; Jones, J.M.; Klein, H.A.; Gill, P.; Lindvall, T.; Wallin, A.E.; Merimaa, M.; et al. Analysis of thermal radiation in ion traps for optical frequency standards. Metrologia
**2015**, 52, 842. [Google Scholar] [CrossRef] [Green Version] - Brun, R.; Rademakers, F. ROOT—An Object Oriented Data Analysis Framework. Nucl. Inst. Methods Phys. Res. Sect. A
**1997**, 389, 81–86. [Google Scholar] [CrossRef] - Singer, K.; Poschinger, U.; Murphy, M.; Ivanov, P.; Ziesel, F.; Calarco, T.; Schmidt-Kaler, F. Colloquium: Trapped ions as quantum bits: Essential numerical tools. Rev. Mod. Phys.
**2010**, 82, 2609–2632. [Google Scholar] [CrossRef] [Green Version] - Saad, Y.; Schultz, M. GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. SIAM J. Sci. Stat. Comput.
**1986**, 7, 856–869. [Google Scholar] [CrossRef] [Green Version] - Greengard, L.; Rokhlin, V. A fast algorithm for particle simulations. J. Comput. Phys.
**1987**, 73, 325–348. [Google Scholar] [CrossRef] - Landa, H.; Drewsen, M.; Reznik, B.; Retzker, A. Classical and quantum modes of coupled Mathieu equations. J. Phys. A Math. Theor.
**2012**, 45, 455305. [Google Scholar] [CrossRef] - Likins, P.W.; Lindh, K.G. Infinite determinant methods for stability analysis of periodic-coefficient differential equations. AIAA J.
**1970**, 8, 680–686. [Google Scholar] [CrossRef] - Ozakin, A.; Shaikh, F. Stability analysis of surface ion traps. arXiv
**2011**, arXiv:quant-ph/1109.2160, arXiv:quant–ph/11092160. [Google Scholar] - Detti, A.; De Pas, M.; Duca, L.; Perego, E.; Sias, C. A compact radiofrequency drive based on interdependent resonant circuits for precise control of ion traps. Rev. Sci. Instrum.
**2019**, 90, 023201. [Google Scholar] [CrossRef] [Green Version] - Berkeland, D.J.; Miller, J.D.; Bergquist, J.C.; Itano, W.M.; Wineland, D.J. Minimization of ion micromotion in a Paul trap. J. Appl. Phys.
**1998**, 83, 5025–5033. [Google Scholar] [CrossRef] [Green Version] - Leschhorn, G.; Hasegawa, T.; Schaetz, T. Efficient photo-ionization for barium ion trapping using a dipole-allowed resonant two-photon transition. Appl. Phys. B
**2012**, 108, 159–165. [Google Scholar] [CrossRef] [Green Version] - Berto, F.; Sias, C. Prospects for single photon sideband cooling in fermionic Lithium. arXiv
**2019**, arXiv:physics.atom-ph/1912.08104. [Google Scholar] - Perego, E.; Pomponio, M.; Detti, A.; Duca, L.; Sias, C.; Calosso, C.E. A scalable hardware and software control apparatus for experiments with hybrid quantum systems. Rev. Sci. Instrum.
**2018**, 89, 113116. [Google Scholar] [CrossRef]

**Figure 1.**Sketch of the electro-optical trap, in which a static electric quadrupole potential (red arrows) is used for trapping ions along two orthogonal directions (y and z axes), while the confinement along the antitrapping direction (x axis) is provided by the interference pattern of two crossed Gaussian laser beams (drawn in blue). For positive ions, the electrodes generating the quadrupole potential must be negatively charged.

**Figure 2.**Electro-optical trap “stability diagram”. The shaded region shows all the values $\{P,\varphi \}$ for which the trapping frequencies in the three directions are simultaneously positive, thus indicating a stable trapping potential. The laser wavelength, the beam waists, and the crossing angle chosen for trapping Ba${}^{+}$ are $\lambda =451.7$ nm, ${w}_{0x}={w}_{0y}=40\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}$m, and $2{\theta}_{c}=10\xb0$, respectively. The red line indicates the parameters $\{{P}_{\mathrm{iso}},{\varphi}_{\mathrm{iso}}\}$ for which the confinement is equal in each direction. The region above this line corresponds to a disk-like-shaped potential.

**Figure 3.**CAD assembly of the electro-optical trap, formed by the cone electrodes and the bow-tie cavity mirror mounts. The two mirror mounts on the left side have been removed for a better view.

**Figure 4.**On the left, the front view of the ceramic lateral support. Each hole is used to hold a specific electrode. Extrusions for optical access are indicated with the label “lasers”. On the right, a CAD assembly of some of the electrodes (two RF blades and four endcap electrodes) inserted in the lateral support. The distance between the endcaps’ tips is 3 mm along the y axis and 1 mm along the z axis.

**Figure 5.**CAD section of the oven system for producing atomic beams of neutral barium. (1) Copper heat-sink. (2) Venting holes. (3) Screw passing hole for fixing the heat-sink on the flange. (4) Cold upper part of the (5) tubes. (6) Tube stripes for the electric connections. (7) Threaded hole for the ground connecting the heat-sink. (8) Screw passing holes for fixing the base on the flange. (9) Pipes and (10) screw passing holes of the cold ceramic (11) skimmer. (12) Inspection viewports of the (13) cylindrical oven cavities. (14) Anti-sputter wall for protecting the vertical optical access. (15) and (16) Output holes.

**Figure 6.**Conceptual sketch of the Paul trap electrodes’ position in relation to the electro-optical trap cone-shaped hollow electrodes. The RF electrodes are rotated 45° with respect to the endcaps’ frame, and the confinement along the z axis is provided by the negatively charged endcaps.

**Figure 7.**Stability diagram of the Paul trap, characterized by a coupling angle of $45\xb0$ between the RF electrodes and endcaps’ axes. The black thick dots represent the $\{q,a\}$ pairs for which the Paul trap (described by a system of coupled Mathieu equations) is stable, whereas the small gray dots indicate values of parameters for which the trap is unstable. The blue curves, calculated with the multiple-scale perturbation theory, enclose a stability region that is wider than the numerically estimated one, yet well describing the stability behavior for small values of q. The green area is the stability region of the corresponding uncoupled system.

**Figure 8.**Average counts of photons emitted by neutral barium atoms decaying from the 5d6p ${}^{3}$D${}_{1}$ level are plotted as a function of time. The neutrals are emitted from one of the two ovens by applying a train of 20 current pulses 400 ms long with a duty cycle of 75%, whereas the amplitude is linearly decreased from 100 A to 70 A to keep the temperature from exponentially rising. Time $t=0$ corresponds to the moment at which the current pulses’ train starts.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Perego, E.; Duca, L.; Sias, C.
Electro-Optical Ion Trap for Experiments with Atom-Ion Quantum Hybrid Systems. *Appl. Sci.* **2020**, *10*, 2222.
https://doi.org/10.3390/app10072222

**AMA Style**

Perego E, Duca L, Sias C.
Electro-Optical Ion Trap for Experiments with Atom-Ion Quantum Hybrid Systems. *Applied Sciences*. 2020; 10(7):2222.
https://doi.org/10.3390/app10072222

**Chicago/Turabian Style**

Perego, Elia, Lucia Duca, and Carlo Sias.
2020. "Electro-Optical Ion Trap for Experiments with Atom-Ion Quantum Hybrid Systems" *Applied Sciences* 10, no. 7: 2222.
https://doi.org/10.3390/app10072222