# «NORTHWESTERN UNIVERSITY Toward Rotational Cooling of Trapped SiO+ by Optical Pumping A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL ...»

## NORTHWESTERN UNIVERSITY

Toward Rotational Cooling of Trapped SiO+ by Optical Pumping

## A DISSERTATION

## SUBMITTED TO THE GRADUATE SCHOOL

## IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

for the degree

## DOCTOR OF PHILOSOPHY

Field of Physics and Astronomy By David Tabor

## EVANSTON, ILLINOIS

June 2014 UMI Number: 3627205 All rights reserved## INFORMATION TO ALL USERS

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789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106 - 1346 c Copyright by David Tabor 2014 All Rights Reserved ABSTRACT Toward Rotational Cooling of Trapped SiO+ by Optical Pumping David Tabor This thesis presents a scheme for preparation of trapped molecular ions with a high degree of internal state purity by optical pumping with a broadband pulse-shaped femtosecond laser; the internal structure of SiO+ permits fast stepwise pumping through the tens of rotational levels populated in a room-temperature distribution. Two analyses, which guided the experimental implementation, are presented: (1) a novel method of quantifying anharmonicity in the trapping potentials, which limits the number of ions that can be trapped, and (2) a rate-equation simulation of the quantum state evolution during pumping. Experimental implementation of pulse shaping and its characterization are discussed, as is the molecular spectroscopy used to reference this light to the rotational cooling transitions. Internal state analysis can be performed using resonance enhanced multiphoton dissociation.

## Acknowledgements

The work reported here beneﬁted from the eﬀorts and contributions of many more people than myself.Each of my labmates came to my aid many times. Particularly deserving of thanks are Jason Nguyen and Yen-Wei Lin, who collaborated directly with me on this experiment.Jason deserves much of the credit for initially identifying SiO+ as a promising candidate and working out the details of the optical pumping scheme. Yen-Wei and Jason both spent a great deal of time and eﬀort working on the experimental implementation.

This project beneﬁted from the contributions of many able undergraduates. In particular, Marc Bourgeois, Scott Williams and Daniel Villalon made major contributions to the results reported here.

Selim Shahriar and John Ketterson, in addition to serving on the committee for this thesis, each guided my development as a graduate student and became a reliable source of aid. The neighboring Ketterson lab often lent equipment and advice without which I would have been stuck. Selim’s quantum optics class was particularly enlightening.

I am deeply indebted to Brian Odom for his support and guidance. He worked tirelessly to provide the best possible environment for his students to develop. He also ensured we had paychecks.

In this experiment, 138 Ba+ is cooled and trapped as a diagnostic tool for the apparatus as well as to permit SiO+ loading to be visually conﬁrmed. The ﬁrst two sections of this chapter summarize some basics of Ba+ structure and the formalism of radiofrequency (rf) Paul traps. Much greater detail on these topics is available in the literature and not repeated here. The ﬁnal section of this chapter summarizes the sources of heating experienced by a trapped ion in some detail; it is foundational to the discussion in Ch. 2.

ground 6S1/2 state, 493 nm light drives excitations to the 6P1/2 state. This excited state may decay back to the 6S1/2 state, or alternatively it can decay to the 5D3/2 state. The relative branching ratio between these two channels is Γ1 /Γ2 ≈ 12 MHz / 4 MHz ≈ 3.

(A decay to 5D5/2 is also possible, but the rate is negligible in comparison to the other two channels. Other than these three, no other channels are present.) The line widths reported here are measured by [26].

The 493 nm 6S1/2 → 6P1/2 transition is referred to as the cooling transition and in this experiment is provided by a frequency-doubled Ti:sapphire laser (Toptica DLPro 123).

Repumping along the 5D3/2 → 6P1/2 transition at 650 nm is produced by an extended cavity diode laser (ECDL) (Toptica DLPro 123).

Repumping of the Zeeman substructure of these transitions requires both σ + and σ − polarizations to be present in the 650 nm repumping beam. Experimentally, a Helmholtz coil is used to deﬁne a quantization axis, and linearly polarized 650 nm light is oriented at 45◦ to this axis. The decomposition of this orientation along the quantization axis contains both σ + and σ − components.

Ion trapping by radiofrequency (rf) electric ﬁelds was ﬁrst demonstrated in 1954 by Paul in a trap with hyperbolic electrodes [29]. Attempts to maximize the ﬁeld-free trapping volume led to the development of the linear Paul trap by Prestage in 1989 [31].

Although less harmonic than hyperbolic traps, the ﬁeld-free central axis of linear traps often make theirs the preferred geometry for experiments involving multiple laser-cooled ions, including precision spectroscopy [34, 21], quantum information processing [13, 19], and cavity quantum electrodynamics applications [14].

A linear quadrupole ion trap (Fig. 1.2) is formed from four parallel cylindrical electrodes of radius re, held at a separation r0, with an rf voltage signal applied to the electrodes. The resulting electric ﬁeld conﬁnes the ion motion radially; axial conﬁnement (in the z -direction) is provided by endcap electrodes (not shown in Fig. 1.2), to which a ˆ static voltage is applied.

Near the trap center, the electric potential is given by

where Vrf and Ωrf are the voltage and frequency of the applied rf drive, Vec is the static voltage applied to the endcap electrodes, z0 is half the distance between the endcap electrodes, and κ is a geometric factor. While this description is accurate near the trap center and along the z-axis, signiﬁcant deviations can occur elsewhere in the trap interior.

** Figure 1.2.**

End view of a linear quadrupole trap. A sinusoidally varying voltage is applied to the rod electrodes, with one pair held 180◦ out of phase with the other.

The radial motion of an ion in the potential of Eq. (1.1) is described by the Mathieu diﬀerential equations, and the stability of this motion is expressed using Mathieu parameters which depend only on Vrf, Vec, Ωrf, κ, and the charge-to-mass ratio q/m. Solutions to the Mathieu equations contain regions of parameter space where ion motion is stable [27], and trap parameters are chosen to operate in one such stable region.

The motion of a trapped ion can be described approximately as a superposition of

frequencies ωx, ωy, and ωz. An analytic relationship exists between the time-dependent

**rf potential and the time-independent pseudopotential [7]:**

When the thermal energy of a trapped ion cloud is cooled suﬃciently below the repulsive Coulomb interaction energy, the ions settle into a Coulomb crystal with geometry dened by the trapping potential [49, 20]. Crystallization of certain atomic ion species with closed-cycle electronic transitions can be accomplished directly by laser cooling [9, 32];

crystallization of other species (e.g., molecular ions) can be accomplished sympathetically by laser cooling of a co-trapped atomic species [3] or in situ formation from a pre-cooled reactant [28]. Crystallization can be prevented by various heating mechanisms.

Although radial and axial micromotion vanish, respectively, along the trap axis and at z = 0, a large crystal necessarily extends beyond these regions. Micromotion heating results from the transfer of micromotion energy into secular energy, which leads to an elevated secular temperature. Although the scaling is non-trivial, simulations show that the micromotion heating rate strongly increases with micromotion amplitude and with secular temperature [37, 36]. As a crystal grows, additional ions are held at locations with increasingly strong rf ﬁelds, causing them to experience increasingly large micromotion.

The resulting heating limits the minimum obtainable temperature for a crystal of a given size, and eventually prevents further crystal growth.

Nonlinear resonance heating can occur if a resonance condition is satisﬁed between secular frequencies and the rf drive frequency. For linear Paul traps, the condition is

where nx, ny, and nz are integers [8]. The corresponding condition for a single particle in a hyperbolic trap has been derived [44], with the resonances weakening with larger n. This behavior has been conﬁrmed in both hyperbolic traps [2] and linear traps [8].

For suﬃciently cold trapped samples, it is understood that the frequency of the center of mass (COM) modes, or other normal modes, should be used in Eq. (1.4) rather than the single-particle frequencies [50]. For an inﬁnitely long linear Paul trap, or for a small crystal in a ﬁnite trap, where the axial rf drive vanishes, a non-deforming crystal is only expected to be heated on resonances with nz = 0. However, a crystal which is large enough to sample axial fringing ﬁelds near the endcaps is expected to be excited also by resonances with nz 0.

Anharmonicity in the trapping potential causes the mode frequencies to shift as a crystal grows and samples less harmonic regions of the trap potential. Eventually, the anharmonicity-shifted mode frequencies of a growing crystal will meet the resonance condition of Eq. (1.4), and some level of heating will occur, potentially halting further crystal growth.

Even for a crystal of deﬁnite size, it is non-trivial to predict the precise response when it is swept through a given nonlinear heating resonance. Predicting the heating response in the non-equilibrium scenario of a growing crystal, as anharmonic frequency shifts cause it to cross a resonance, is yet more complicated. Modeling of the heating rates and whether crossing a given subharmonic resonance will actually melt a crystal or prevent further growth is beyond the scope of this analysis.

A trap capable of growing large Coulomb crystals is desirable for many types of experiment; in particular, for precision spectroscopy, large crystals improve the rate at which statistics are collected, often directly improving the achieved precision. This chapter describes an original method [42] for quantifying the susceptibility of a trap geometry to the two heating eﬀects (micromotion heating and nonlinear resonance heating) described in Ch. 2. The axial component of micromotion, which leads to ﬁrst-order Doppler shifts along the preferred spectroscopy axis in precision experiments, is also compared.

Four diﬀerent linear trap implementations are compared (Fig. 2.1, with corresponding voltage given in Table 2.1): a conventional in-line endcap design with two diﬀerent voltage conﬁgurations (designs A and B), a plate endcap design (design C, similar to a design analyzed in Ref. [30]), and an original rotated endcap design (design D). In all trap designs, z0 = 8.5 mm, r0 = 4 mm, and re = 4.5 mm, with r0 and re chosen to agree with the optimal ratio re = 1.14511r0 for minimizing the leading-order contribution to anharmonicity in the rf potential [33]. Design C uses thin circular plates of radius 2re +r0 as endcap electrodes, with a 2 mm radius hole left for axial optical access. Design D has four small cylindrical endcap electrodes of radius 1.75 mm, rotated 45◦ around the z-axis;

the central axes of these endcap electrodes are on the same radius as the central axes

of the rf electrodes. All numerical simulations presented here use values corresponding Ba+ (q = 1.602 × 10−19 C, m = 2.292 × 10−25 kg) and our operating rf frequency to (Ωrf = 2π × 3.00 MHz).

The potential resulting from grounding all electrodes except the endcaps is denoted φec and φrf is likewise deﬁned for the rf electrodes. Both are found using the ﬁnite element method to solve the Laplace equation, with the trap geometries discretized by the meshing software Gmsh [12].

To allow evaluation of the potentials at locations not on the mesh, the numerical solutions are ﬁtted to an expansion in associated Legendre polynomials

Terms with odd values of l and m are excluded from our ﬁtting due to symmetry.

An analytic form of the gradient of Eq. (2.1) is straightforward to ﬁnd using

which can be derived from a more general identity given by [1]. Fitting the Laplace solutions to the above expansion thus permits eﬃcient evaluation of φf it or its gradient φf it at arbitrary location from the coeﬃcients {Clm }. In this way, the pseudopotential ˜ φrf and φec are determined up to the scaling set by choosing Vrf and Vec. Choices of these voltages, shown in Table 2.2, are made to equalize the single particle secular frequencies ωz and ωr across all designs at experimentally reasonable values ωx = ωy = 2π × 418 kHz and ωz = 2π × 18.9 kHz.

** Table 2.2.**

Vrf and Vec used in comparisons and calculated value of κ for each trap.