A Comprehensive Study on A Tapered Paul Trap: From Design to Potential Applications (2024)

1 Introduction

Trapped ions possess unrivaled features covering a broad range of fields such as quantum metrology and sensing [1, 2, 3, 4], quantum information [5, 6, 7], quantum simulation [8] and quantum thermodynamics [9, 10], to name a few. The geometry of ion traps varies from macroscopic three-dimensional linear Paul traps [11], needle traps [12], segmented micro-traps [13, 14], to surface traps [15, 16] tailored to different applications. Here, we present and characterize a macroscopic tapered Paul trap that possesses distinct properties that have been exploited for studying nonlinear mechanical oscillators with applications for force sensing and amplification [17]. The special features due to the tapered design have also been used to implement a single atom heat engine [18] with the potential to implement a thermal machine in the quantum regime [19].
This article begins with an introduction to the design and construction of this tapered ion trap. We then proceed to a numerical field simulation and a 3D trajectory simulation. These simulations allow us to evaluate the performance of our trap. Following this, we provide a detailed description of the experimental setup and present characterizations of this system. Our measurements characterize the axial position dependent radial confinement as a special feature of this system. We also present sideband spectroscopy performed on the $4^{2}S_{1/2}\leftrightarrow 3^{2}D_{5/2}$ quadrupole transition using the electron shelving technique. Finally, we discuss the potential applications of this system in various fields, including quantum thermodynamics, quantum metrology and sensing, and quantum information.

2 Trap design and field simulation

Conventional linear Paul traps, as used in many state-of-the-art quantum information experiments [11, 20, 8], are designed and constructed to minimize couplings between motional degrees of freedom. However, such couplings can be beneficial in several research areas, such as studies with single-atom heat engines [21, 18] in the quantum regime, quantum metrology experiments, and exploring new methods for quantum information. To facilitate this coupling, we deviate from the conventional design of the Paul trap by inclining the four blade-shaped radiofrequency (RF) electrodes to the axial symmetry axis by an angle $\vartheta=10\,^{\circ}$, which is chosen such that adequate radial-axial coupling is achieved while maintaining stable trapping conditions. Special care is taken to avoid axial micromotion by driving the four blades with RF voltages of opposite sign between each pair of blades (see Fig. 1). Furthermore, micromotion is minimized by a proper alignment of the RF electrodes with respect to the axial $z$ axis of the trap, as shown in Fig. 1. This is achieved by using four silicon nitride insulating balls and corresponding recesses to position the fork-shaped radio frequency electrodes precisely.The endcap electrodes are inserted into the RF electrodes and insulated from the latter by ceramic rings. Additionally, four compensation electrodes are inserted and insulated by ceramic sleeves.

The RF blades have a length of 4 mm, and the endcap electrodes are spaced 4.8 mm apart. This yields a factor of 16 smaller trapping volume compared to the tapered Paul trap used in our earlier work [18]. The endcap electrodes are narrowed towards the center of the trap and have a through-hole of 0.8 mm diameter, allowing for optical access along the axial axis. The electric fields generated by this geometry are simulated using a boundary element method [22, 23]. In order to solve the potential of the trap, the computer-aided design files, including the blade-electrode pairs, the endcap electrodes, and the compensation electrodes, are imported. The polygon mesh in Fig. 2 (a) used for this numerical simulation is generated by the finite element mesh generator Gmsh [24]. Based on the imported geometry, the electrostatic potential is solved using the boundary element solver package Bempp [23]. By applying the pseudopotential approximation [25, 22], the trapping potential can be described by $\Phi_{\text{eff}}(x,y,z)=q\left|\nabla\Phi(x,y,z)\right|^{2}/\left(4m\omega_{$ [26] with $\Phi(x,y,z)$ the electrostatic potential, $q$ the charge of the particle, $m$ its mass and $\omega_{\text{RF}}$ the RF frequency. In Fig. 2(b), a cross-section of the pseudopotential in the $x$-$z$-plane is depicted. The confinement in the radial direction is illustrated by the black solid and cyan dashed lines, which are obtained by fitting a quadratic function to the potential. The confinement is getting stronger towards positive $z$. To understand the behavior of a single ion inside the potential, we perform a trajectory simulation of a single calcium ion using the velocity Verlet propagator [22]. We set the voltages on the two pairs of the blade electrodes as $\tilde{V}_{\text{RF1}}=\hat{V}_{\text{RF1}}\cos(\omega_{\text{RF}}t)$ and $\tilde{V}_{\text{RF2}}=-\hat{V}_{\text{RF2}}\cos(\omega_{\text{RF}}t)$ respectively, where $\hat{V}_{\text{RF1,2}}$ denotes the amplitude of the RF voltages. We deviate $\hat{V}_{\text{RF1}}$ and $\hat{V}_{\text{RF2}}$ by 1.8 % to mimic the imperfect symmetry of the experimental setup, which lifts the degeneracy of the two radial modes. By varying the voltages on the endcaps, we can shift the potential minimum along the axial direction. We repeat the simulation with different axial potential minimum positions and perform a Fourier transformation to the simulated trajectories in $x$ and $y$ to determine the corresponding trapping frequencies. The simulation result is compared to measurements in Sec. 4.

3 Experimental setup

Figure 3 (a) shows the electric connections to the Paul trap. The dynamic confinement in the radial directions ($x$, $y$) is provided by RF voltages ($+\widetilde{V}_{\mathrm{RF}}$, $-\widetilde{V}_{\mathrm{RF}}$) applied to the two opposite-positioned pairs of blades, respectively. An RF signal at 11.17 MHz with a power of -17 dBm is amplified by a 5-watt power amplifier with a gain of 46 dB. The amplified signal is then sent to a helical resonator for further amplification and impedance matching to the trap. This helical resonator is specifically designed with a $\lambda/2$ geometry [27], whose middle point is grounded with two open ends connected to the two pairs of RF blades. This configuration provides two RF voltages with a $\pi$ phase difference that is necessary to avoid excessive axial micromotion. The RF voltages are probed by voltage dividers composed of two capacitors for ease of characterization and adjustment. Through careful adjustment of the sliding contact on the coil and the tunable capacitors on the helical resonator, we tune to an impedance-matched frequency of 11.17 MHz. The phase difference of the two RF signals is measured to be 179.51($\pm 0.28$) degrees with a relative amplitude difference of 1.29%. The axial confinement is achieved by applying DC voltages ($V_{\mathrm{D1}}$ and $V_{\mathrm{D2}}$) to the endcap electrodes. Additionally, four auxiliary electrodes are implemented around the Paul trap (see Fig. 1 in yellow). These electrodes are supplied with four independent DC voltage sources, which allow for rotating radial principal axes and compensation of the micromotion [28].
The trap is contained in an ultrahigh-vacuum chamber. In our experiment, ${}^{40}\text{Ca}^{+}$ ions are used. The calcium sample is sublimated in an oven to form a collimated atomic beam shooting through the center of the trap. The atoms are ionized via a two-photon process by the combination of a 422 nm and a 375 nm laser [29]. A 397 nm laser red-detuned from the $4^{2}S_{1/2}\leftrightarrow 4^{2}P_{1/2}$ transition of ${}^{40}\text{Ca}^{+}$ is focused into the trap for Doppler cooling. The short lifetime of the $4^{2}P_{1/2}$ state leads to a natural linewidth of 22.1 MHz. As the ion at state $4^{2}P_{1/2}$ has a probability of 0.064 to decay to the metastable state $3^{2}D_{3/2}$ [30], an 866 nm laser is used to repump the ion back to the $4^{2}P_{1/2}$ state to close the laser cooling cycle. A second repumper laser at 854 nm is used for pumping the $3^{2}D_{5/2}\leftrightarrow 4^{2}P_{3/2}$ transition. The resonantly scattered photons at 397 nm are used for fluorescence imaging. In addition, a 729 nm laser is locked to a high finesse cavity. It is used to address the $4^{2}S_{1/2}\leftrightarrow 3^{2}D_{5/2}$ quadrupole transition to implement resolved sideband cooling. To define a quantization axis, permanent magnets are mounted outside the vacuum chamber to generate a magnetic field as shown in Fig. 3(b), which also lifts the degeneracy of magnetic quantum states.

4 System characterization

Next, we measure the trapping frequency of a single ${}^{40}\text{Ca}^{+}$ ion by exciting its motion with a periodic force. To do so, we red-detune the 397 nm laser by 33.7 MHz from the $4^{2}S_{1/2}\leftrightarrow 4^{2}P_{1/2}$ transition and apply a sinusoidal modulation to its intensity. We monitor the fluorescence from the ion as a function of the modulation frequency. Resonant scattering of photons excites the mechanical oscillation when the intensity modulation frequency matches one of the harmonic trapping frequencies. Figure 4(a) displays the fluorescence signal from the ion as a function of intensity modulation frequency during an axial trapping frequency measurement. Here, the fluorescence signal is imaged by a camera with an exposure time of 400 ms at each frequency. Each column of the displayed image represents a summation of a fluorescence image over the radial directions. As the intensity modulation frequency is tuned close to the axial trapping frequency at 99.8 kHz, a coherent oscillation along the axial direction is excited. Since the exposure time of the camera is much longer than one oscillation period, the recorded images integrate the fluorescence signal over many oscillation cycles, which can be formulated as the position distribution function of sinusoidal oscillation convoluted with the Gaussian point spread function of the imaging system. The oscillation amplitude (red circles) is obtained by fitting each column to this model. Figure 4(b) shows the fluorescence signal from the ion when one of the radial oscillation modes is excited. Here, each column represents an integration of the image over the axial direction. The oscillation amplitude (red circles) in Fig. 4(b) is obtained following the same procedure as in Fig. 4(a).
The red and blue circles in Fig. 5 show the measured trapping frequencies of the $x$ and $y$ radial modes as a function of the axial position of the ion. To obtain the data, we shift the axial position of the trapped ${}^{40}\text{Ca}^{+}$ ion by changing the voltages applied to the endcap electrodes ($V_{\mathrm{D1}}$,$V_{\mathrm{D2}}$), and keeping the axial trapping frequency constant. The radial frequency measurement is performed at each axial position by a descending sweep of the modulation frequency across the radial modes. The red and blue dotted lines display the results from trajectory simulations mentioned in Sec. 2, showing good agreement with the measured data. The solid lines are from data fitting to the following analytical formula

To obtain the amplitude of the coherent oscillation in the radial directions, we need to determine the angle between the imaging plane and the two radial principal axes. As shown in Fig. 5, these two radial modes are non-degenerate by 8 kHz. The orientation of principal axes can be controlled by voltages on the compensation electrodes (see Fig. 1) to compress or stretch the trapping potential. The tapered geometry leads to an asymmetric shielding effect to the compensation electrodes. In order to maintain the axial position of the ion, the voltages of the four compensation electrodes are applied in a manner as shown in the inset of Fig. 6(a). We show the radial frequency response projected to the imaging plane at different values of the compensation voltage $V_{C}$ in Fig. 6(a). By increasing $V_{C}$ from 0 to 150 Volt, we can observe the radial principal axes rotating clockwise when looking along the $z$-axis.The two radial principal axes both have an angle to the imaging plane of 45$\,{}^{\circ}$ when setting $V_{C}=62.5\,V$.

After the principal axes are set, micromotions of the ion can be compensated by shifting the ion’s position in the radial plane by applying voltages on the compensation electrodes. To keep the axial position of the trapped ion, the changes in the voltages applied to the opposite-positioned electrodes need to be of opposite sign, as shown in the inset of Fig. 6(b). As the frequency of the micromotion is 11.17 MHz, which is smaller than the natural linewidth of the $4^{2}S_{1/2}\leftrightarrow 4^{2}P_{1/2}$ transition, the spectrum of this transition will be broadened due to excess micromotions [28]. In this measurement, the laser is red detuned by 33 MHz from the resonance. The micromotion is reduced by minimizing the fluorescence signal of the $4^{2}S_{1/2}\leftrightarrow 4^{2}P_{1/2}$ transition [28]. As shown in Fig. 6(b), the fluorescence signal is plotted against the voltages $\Delta V_{C1,3}$ and $\Delta V_{C2,4}$, which are set to 18.5 V and -60 V to minimize the micromotion.

5 Sideband spectroscopy

We have also measured the sideband-resolved spectrum of the $4^{2}S_{1/2}\leftrightarrow 3^{2}D_{5/2}$ quadrupole transition. The long lifetime of 1.05 s [31] of the $3^{2}D_{5/2}$ state leads to a narrow bandwidth. A frequency-stabilized laser at 729 nm with a nominal linewidth of 1 Hz is used to address this transition. At the trap, 1.3 mW of 729 nm laser power is available, with a focal diameter at the ion of around 50$\,\mu$m. The laser frequency can be swept across a range of 20 MHz with a double-pass acousto-optic modulator (AOM) setup, which is driven at a center frequency of 330 MHz. The driving signal of the AOM is controlled by a fast RF switch. There are in total ten transitions between the different sublevels of $4^{2}S_{1/2}$ and $3^{2}D_{5/2}$, spanning 20 MHz under the 3-Gauss magnetic field inside the trap. The 729 nm laser orientation is chosen parallel to the quantization axis defined by the magnetic field [see Fig. 3(b)], selecting transitions with $|\Delta m_{J}|\,=\,$1 [32].

The quadrupole excitation spectrum shown in Fig. 7 is obtained by the electron shelving method [33]. The inset shows the pulse sequence of the measurement. The ion is initially laser-cooled using a linearly polarized 397 nm laser pulse. It is then pumped by a circularly polarized 397 nm pulse to the $|4^{2}S_{1/2},+1/2\rangle$ Zeeman state. Afterwards, a 729 nm laser pulse is sent to the ion for 70$\,\mu$s. Finally, a 397 nm pulse is turned on along with the camera to measure the fluorescence of the ion. The ion appears dark when the 729 nm pulse excites the ion to the $D$-state. Otherwise, the ion appears bright on the camera. The spectrum of the $|4^{2}S_{1/2},+1/2\rangle\leftrightarrow|3^{2}D_{5/2},-1/2\rangle$ transition is obtained by sweeping the driving frequency of the AOM. At each frequency, the measurement is repeated 300 times. The 866 nm laser beam is continuously on during the measurement. The 854 nm laser beam is only used before the 729 nm laser manipulation to initialize the ion to its $4^{2}S_{1/2}$ state. The strongest peak in the center of the spectrum is the carrier transition. To confirm that the remaining peaks are motional sidebands, the spectrum is taken with different axial trapping frequencies. Higher-order sidebands are also visible in the spectrum, as our system is not operating in the Lamb-Dicke regime. There are also hybrid motional sidebands where both the radial and axial motional modes are accessed.

6 Discussion and Outlook

With this paper, we want to draw attention to our tapered Paul-trap platform that allows utilizing its special features, such as mechanical nonlinearities, to implement novel schemes for quantum information, to advance metrology and sensing methods, and to push forward quantum thermal machines for probing thermodynamics in the quantum regime. A detailed description of the setup presented in this article allows analytical and numerical investigations of the trap and provides a sound basis for designing, optimizing, and tailoring properties of the system for new applications and basic research.
Motional modes of trapped ions are used as a quantum bus for quantum information processing [34]. In the tapered trap, radial modes for neighboring ions have different frequencies, allowing local addressing in frequency domain. This enables new ways of implementing quantum simulation and gate operations by selectively addressing the modes in frequency space, which allows for global laser irradiation of an ion crystal.
The mechanical nonlinearity opens up new possibilities to advance quantum sensing and metrology through a range of effects, such as multistability, stochastic and vibrational resonances [35]. Similar to demonstrations with nanomechanical oscillators  [36], where a broadband noise has led to significant signal amplification, our system has shown similar behavior [17] through vibrational resonance.Additionally, combining quantum resources such as squeezing with nonlinearities could further increase the sensitivity similar to effects predicted with quantum thermal machines based on trapped ions [37].
Thermodynamics in the quantum regime is a new active research field [38, 39]. Trapping ions in tapered traps have shown pioneering results with the first single ion heat engine being realized [18] and parameters for work extraction at the Curzon-Ahlborn efficiency limit being predicted [21]. Quantum resources such as squeezing are expected to increase the efficiency of heat extraction [37] and transient non-confining potentials can be used for speeding up a single ion heat pump through the use of shortcuts to adiabaticity [40]. The single-ion heat engine can also be used as a sensitive thermal probe [41] and the sensitivity can be enhanced through the application of squeezing. The tapered trap design also potentially allows for implementing local non-classical baths combining dissipative state preparation [42] with frequency space addressing of the local radial modes, which is of fundamental interest in quantum resource theory [43].

Acknowledgement

We thank Florian Elsen and David Zionski for early-stage work on the ion trap setup and simulation. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projects No. 499241080, No. 384846402, No. 328961117, No 510794108 - through the QuantERA grant ExTRaQT, the Research Unit Thermal Machines in the Quantum World (FOR 2724), and the Collaborative Research Center ELCH (SFB 1319), and by the federal state of Hesse, Germany, through project SMolBits within the LOEWE program. M.M. and D.W. acknowledge support from Germany’s Excellence Strategy - Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1-390534769.

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