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Determination of the geometric parameters κz and κr of a linear Paul trap
Chinese Journal of Physics 60 (2019) 61–67
Contents lists available at ScienceDirect
Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph
Determination of the geometric parameters κz and κr of a linear Paul trap
T
H.X. Lia,b, Y. Zhanga,b, S.G. Hea, X. Tonga,
⁎
a
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, China University of Chinese Academy of Sciences, Beijing, 100049, China
b
ARTICLE INFO
ABSTRACT
Keywords: Geometric parameters Linear Paul trap Secular motion excitation Coulomb crystal
An effective method for determining the geometric parameters of a linear Paul trap is provided and carried out in this paper. The method is based on exciting the secular motion of laser-cooled Ca+ ions in a linear Paul trap with varied trapping potentials. Taking advantage of the segmented structure of the linear Paul trap, both the axial and the radial geometric parameters are obtained by applying external excitation fields to the trapping electrodes directly. The axial geometric parameter of our linear Paul trap is determined to be 0.374(3), and, for the first time, the radial geometric parameter is determined to be 0.928(2).
1. Introduction Experiments with trapped and laser-cooled ions have been motivated by the possibility for high-accuracy spectroscopy [1], improved frequency standards [2], phase transitions [3], quantum computing [4], quantum logic spectroscopy [5], and experiments in fundamental physics [6]. When all the trapped ions in an ion trap have been cooled to a few millikelvin by laser cooling, ordered structures of translationally cold ions, usually referred to as ion Coulomb crystals, are formed [7]. In a Coulomb crystal, all the ions are confined in their local potential minimum, and the nearest neighboring Coulomb potential energy is typically hundreds of times larger than the average kinetic energy of the ions [8]. So far, systematic studies of ion Coulomb crystals have been performed in Penning traps and Paul traps including their linear configurations [9-16]. In a Penning trap the charged particles are held in a combination of electrostatic and magnetic fields, while in a Paul trap a spatially varying time-dependent electric field, typically in the radio-frequency (RF) domain, and an electrostatic field are used to confine the charged particles in the central region of the trap [17]. In a Paul trap the electrodes are arranged to generate an almost ideal quadrupolar electric potential close to the trap center. The trajectory of a single ion in the trap can be described by the motion in a harmonic pseudo-potential [18]. The charged particles have a typically slower harmonic motion which is called “secular motion”. The vibrating frequency of the secular motion depends on the charge-to-mass ratio of the ions. Therefore, the excitation of the mass-dependent motional resonances of the trapped ions is often used for mass spectrometry of the trapped ions [19]. The studies of the resonant secular frequency also help to determine the geometric parameters and characteristics of the designed ion trap. This in turn can help in the design of the ion trap. For a linear configuration Paul trap, the total potential can be described as [20]
⁎ Corresponding author: Wuhan Institute of Physics and Mathematics (WIPM), Chinese Academy of Sciences (CAS), West No.30 Xiao Hong Shan Wuhan, Hubei, 430071 PR China. E-mail address: [email protected] (X. Tong).
https://doi.org/10.1016/j.cjph.2019.03.017 Received 23 February 2019; Accepted 15 March 2019 Available online 01 May 2019 0577-9073/ © 2019 The Authors. Published by Elsevier B.V. on behalf of The Physical Society of the Republic of China (Taiwan). This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Chinese Journal of Physics 60 (2019) 61–67
H.X. Li, et al.
r (r ,
z) =
1 M 2
2 2 r r
+
1 M 2
2 2 zz ,
(1)
where the radial and axial frequency of the ion secular motion, ωr and ωz, are given by r
=
2 2 2 r Q Urf 8M 2r04 2rf
z
=
2 z QUend . Mz 02
z QUend Mz 02
,
(2)
(3)
Here 2r0 is the distance between the two diagonal electrodes. 2z0 is the distance between the two endcap electrodes. κr and κz stand for the geometric parameters of the linear Paul trap in the radial and axial directions, respectively. Q and M stand for the charge and mass of the ion in the trap, respectively. The ion is confined by the combination of a radio frequency (Ωrf/2π) oscillating electric field with a peak to peak voltage Urf in the radial direction and a static potential Uend applied to the endcap electrodes of the trap. The geometric parameter in the axis direction κz has been extensively studied [21,22], while according to our knowledge the radial geometric parameter has not been experimentally studied and is usually treated as having the value 1 [23,24]. This assumption is sufficient for a small ensemble of ions, typically for a single or linear configuration of Coulomb crystals confined in the center of an ion trap, where the trapping potential along the radial direction is close to an ideal hyperboloid potential [25,26]. However, more accurate values of the geometric parameters are required when using the radial direction resonance excitation scheme in an ion trap for precision mass spectrometry to determine a large ensemble of Coulomb crystals with multi ionic species [27]. In this paper, we provide an experimental method for determining both the axial and radial geometric parameters in a linear Paul trap. More specifically, the motional resonances of the tapped and laser-cooled Calcium ions are determined nondestructively with a high mass-to-charge ratio resolution upon excitation using an additional oscillating electric field. By changing the trapping potential, both the axial and radial geometric parameters are derived according to Eq. (2) and Eq. (3) from the measured resonant frequencies of the Calcium ions. The experimental setup is introduced in Section 2. In a third section, the results and a discussion are presented. 2. Experimental setup The linear Paul trap used in our experiment is shown in Fig. 1. The trap consists of four parallel steel rods, each axially partitioned into three sections, the endcap electrodes are labeled as ‘A1∼A4’ and ‘C1∼C4’, and the center electrodes are labeled as ‘B1∼B4’ in Fig. 1(a). The diameter of each electrode rod is 8 mm, 2r0 is 7 mm, the lengths of the center (2z0) and end-cap electrodes (2ze) are 6.4 mm and 24 mm, respectively. A RF (radio frequency) field is applied to a pair of diagonal electrodes, and the other pair of diagonal electrodes are connected to ground, as shown in Fig. 1(b). Here, the peak to peak voltage Urf and frequency frf (Ωrf/2π) can be varied from 0 V to 400 V and 3.2 MHz to 4.2 MHz, respectively. The RF field provides the confinement for the ions in the radial plane (xy plane in Fig. 1). The eight endcap electrodes (A1-A4, C1eC4) are supplied with static electric voltages (Uend) from 0 V–10 V to confine the ions in the axial (z-axis) direction. The main experimental setup is shown in Fig. 2. Calcium atoms are evaporated from a Ca oven. The temperature of the oven can be heated to 350 °C by an electric current. Calcium atoms in an effusive skimmed beam are non-resonantly ionized by a 355 nm pulse laser in the central region of the ion trap to produce Calcium ions, as shown in Fig. 3(a). The trapped Ca+ ions are Doppler laser cooled on the 4s2S1/2 → 4p2P1/2 transition at 397 nm with a red detuning of ∆ω and addressed to the population lost to the 3d2D3/2 state by an 866 nm laser beam, as shown in Fig. 3(b). Both laser beams are reflected along the trap axis to achieve bi-directional laser cooling. A photo-multiplier tube (PMT) is used to monitor the fluorescence of the laser-cooled Ca+ ions. A color filter with center wavelength at 390 nm and full width at half maximum of 18 nm is located before the PMT for filtering out the 866 nm laser and scattered light from the background. The ions are also imaged onto a complementary metal-oxide semiconductor (CMOS) camera by a 10 times magnification lens system. The localized ion positions are shown as an
Fig. 1. Schematic views of the linear ion trap. (a) Side view of the trap. (b) Cross-sectional view of the trap and applied RF (radio frequency) potential. 62
Chinese Journal of Physics 60 (2019) 61–67
H.X. Li, et al.
Fig. 2. Schematic of the experimental setup, including the trap, laser beams and fluorescence collecting system.
Fig. 3. (a) Energy level diagram of the photo-ionizing Ca atom. (b) Energy level diagram of the Doppler laser cooling of the Ca+ ions. (c) An image of a Ca+ ion Coulomb crystal taken by the CMOS camera.
image of a Coulomb crystal in Fig. 3(c). In order to calibrate the magnification and mark the position of the lasers, a 250 μm diameter bare optical fiber fixed on a 3D manipulator (not shown in Fig. 2) is inserted into the trap center and imaged by the CMOS camera. To obtain the secular motion frequencies (ωr and ωz) of the trapped ions, AC (alternating current) voltages used for motion excitation are applied to a pair of diagonal endcap electrodes (A2, A3) for axial motion excitation and to a mid-electrode (B2) for radial motion excitation, as indicated in Fig. 4. Because of the segmented structure of our linear ion trap, the AC voltages can be applied directly to the trapping electrodes without affecting the RF trapping field. The scans of AC frequency ranging from 0 kHz to 400 kHz are performed with the fixed AC amplitude of 500 millivolt. The scan step is set to be 1 kHz. When the frequency of the excitation AC field is on resonance with the secular motion frequency of the ions in the trap, the ions in the Coulomb crystal are heated and set off the resonance of the cooling laser, resulting in the reduction of the emitted fluorescence from the Ca+ ions. Such changes of fluorescence intensity are record by CMOS and PMT and indicate the secular frequencies of the trapped ions. According to Eq. (2), considering 1/ 2rf , Urf2 , and Uend as variables for the linear function of r2 , the geometric parameters κr and κz can be obtained from the slope or intercept, respectively. Similarly, according to Eq. (3), considering Uend as a variable for the linear function of z2 , κz can be derived from the slope. 3. Results and discussion The resonant motion excitation frequency spectrum of calcium ions in both the axial and radial direction are shown in Fig. 5(a) and (b), respectively. The secular motion frequencies, under the experimental trapping conditions (shown in the caption of Fig. 5), are 89 kHz for the axial motion and 178 kHz for the radial motion.
Fig. 4. Scheme of electrodes for AC and RF. RF applied to the red electrodes is used to confine the Ca+ ions in the radial plane. AC applied to the yellow electrodes is used to excite the secular motion of Ca+ ions in the trap. 63
Chinese Journal of Physics 60 (2019) 61–67
H.X. Li, et al.
Fig. 5. The change of the fluorescence of Ca+ ions via scanning of the AC frequencies. (a) Excitation of the axial motion of Ca+ ions in the ion trap with Urf = 200 V, Ωrf = 3.20 MHz × 2π, Uend = 1.834 V. (b) Excitation of the radial motion of Ca+ ions in the ion trap with Urf = 185 V, Ωrf = 3.30 MHz × 2π, Uend = 2.295 V.
Fig. 6. The square of the secular motion frequencies of Ca+ ions in the axial direction ( z2 ) as a function of the endcap voltage Uend with the trapping potential Urf =180 V, Ωrf =3.50 MHz × 2π. Every date point is averaged over 10 measurements. The red line is a linear least square fit to the data. The uncertainty of z2 is derived from 2ωz × Δωz, where the Δωz is the frequency scan step of 1 kHz.
To obtain the geometric parameters, a series of secular motion frequencies in both the axial and radial directions are measured under different trapping potentials. In the axial direction, the static voltage Uend is varied to measure the corresponding axial secular motion frequency ωz. The square of the measured secular frequencies ωz are plotted as a function of Uend in Fig. 6 The geometric parameter of the axial direction κz is derived from the slope of the linear fitting line. In the radial direction, Ωrf is varied to measure the corresponding radial secular motion frequency ωr. The square of ωr is plotted as the function of 1/ 2rf with three different trapping potentials, shown as the data points A, B and C in Fig. 7(a). The geometric parameters of κr and κz can be derived from the slopes and intercepts of the linear fitting lines, respectively. In a similar way, in 64
Chinese Journal of Physics 60 (2019) 61–67
H.X. Li, et al.
Fig. 7. The square of the secular motion frequencies of Ca+ ions in the radial direction ( r2 ) as a function of (a) 1/ 2rf , (b) Urf2 and Uend (c). The color lines (black, red, blue) are the least square linear fit to the data in different trapping potentials. The uncertainty of r2 is derived from 2ωr × Δωr, where the Δωr is the frequency scan step of 1 kHz.
Fig. 7(b) and (c), the square of ωr is plotted as a function of Urf2 and Uend, respectively. The value of κr and κz can be obtained from the slopes or intercepts of the corresponding fitting lines. All the values of κr and κz obtained from the fitting lines of Figs. 6 and 7 are collected in Table 1. The column of “Group” indicates the experiments taken under the different trapping potentials shown in Fig. 7. In order to obtain a more credible and accurate value of κr and κz, the results of κr and κz in Table 1 are compared in Fig. 8(a) and (b), respectively. All the values of κr or κz agree within one standard derivation, indicating that the method used in our experiment to obtain the geometric parameters of a linear Paul trap is feasible and the results are credible. Further linear fittings with zero slope yield the results of κr and κz to be 0.928(2) and 0.374(3), respectively. The electric potential in the central region is the superposition of the electric potential produced by four electrodes in the linear Paul trap. Because the shape of the four electrodes are not an ideal hyperboloid, the potential generated in the central region is not an 65
Chinese Journal of Physics 60 (2019) 61–67
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Table 1 Values of κr and κz. Direction of excitation
Figure
Group
κr
Standard error of κr
κz
Standard error of κz
x y-plane
Fig. 7(a)
A B C A B C A B C –
0.93250 0.93438 0.93395 0.92676 0.93064 0.92262 0.92863 0.93044 0.92764 –
0.00886 0.00837 0.01009 0.00406 0.00481 0.00632 0.00338 0.00295 0.00313 –
0.39984 0.37832 0.40622 0.38870 0.38701 0.37414 0.36844 0.36774 0.36889 0.37448
0.03982 0.04408 0.04876 0.02102 0.02143 0.02507 0.01197 0.01154 0.00838 0.00227
Fig. 7(b) Fig. 7(c) z-axis
Fig. 6
Fig. 8. (a) The values of κr. (b) The values of κz.
absolute harmonic potential. Therefore, the geometric parameter in the radial direction, κr is close to, but not exactly, 1. In the z-axis direction, the endcap electrodes are separated from the middle electrodes, and the middle electrodes have a shielding effect, the axial potential near the central region of the trap generated by the endcap electrodes is significantly reduced. Thus, the geometric parameter in the axial direction κz is much less than 1. 4. Conclusions and outlook In this paper, we determined the geometric parameters of a linear Paul trap by performing resonant excitation on the trapped and laser-cooled Ca+ ions. The secular motion frequency of the Ca+ ions is obtained by the motion excitation using an AC electronic field. For the first time, the geometric parameter in the radial direction κr has been determined to be 0.928(2), which shows the effect of the non-ideal hyperboloid shaped electrodes used in the linear Paul trap. The geometric parameter in the axial direction κz is also obtained as the value of 0.374(3). Our linear Paul trap has a similar configuration compared to the traps used in Refs. [19,21,22], which have the values of κz as 0.244, 0.3 and 0.342, respectively. Laser-cooled fluorescence mass spectrometry relying on the excitation of secular ion motion in an ion trap is an important mass spectrometry technology in cold atomic and molecular physics [28,29]. The precise determination of the geometric parameters of a linear Paul trap is essential for the laser-cooled fluorescence mass spectrometry, particularly for determining the ion species during the sympathetic cooling and for the further high-resolution spectroscopy based on the photodissociation process [30-32]. Acknowledgments We acknowledge Pan Yong and Zhao Pei for the early work on this project and Du Li-Jun, Song Hong-Fang for the help of setting up the experiment. Funding This work was supported by the National Natural Science Foundation of China (Grant No. 91636216, 11504410 and 11474317) and Chinese Academy of Sciences (Grant No. XDB21020200). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.cjph.2019.03.017. 66
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Contents lists available at ScienceDirect
Chinese Journal of Physics journal homepage: www.elsevier.com/locate/cjph
Determination of the geometric parameters κz and κr of a linear Paul trap
T
H.X. Lia,b, Y. Zhanga,b, S.G. Hea, X. Tonga,
⁎
a
State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, China University of Chinese Academy of Sciences, Beijing, 100049, China
b
ARTICLE INFO
ABSTRACT
Keywords: Geometric parameters Linear Paul trap Secular motion excitation Coulomb crystal
An effective method for determining the geometric parameters of a linear Paul trap is provided and carried out in this paper. The method is based on exciting the secular motion of laser-cooled Ca+ ions in a linear Paul trap with varied trapping potentials. Taking advantage of the segmented structure of the linear Paul trap, both the axial and the radial geometric parameters are obtained by applying external excitation fields to the trapping electrodes directly. The axial geometric parameter of our linear Paul trap is determined to be 0.374(3), and, for the first time, the radial geometric parameter is determined to be 0.928(2).
1. Introduction Experiments with trapped and laser-cooled ions have been motivated by the possibility for high-accuracy spectroscopy [1], improved frequency standards [2], phase transitions [3], quantum computing [4], quantum logic spectroscopy [5], and experiments in fundamental physics [6]. When all the trapped ions in an ion trap have been cooled to a few millikelvin by laser cooling, ordered structures of translationally cold ions, usually referred to as ion Coulomb crystals, are formed [7]. In a Coulomb crystal, all the ions are confined in their local potential minimum, and the nearest neighboring Coulomb potential energy is typically hundreds of times larger than the average kinetic energy of the ions [8]. So far, systematic studies of ion Coulomb crystals have been performed in Penning traps and Paul traps including their linear configurations [9-16]. In a Penning trap the charged particles are held in a combination of electrostatic and magnetic fields, while in a Paul trap a spatially varying time-dependent electric field, typically in the radio-frequency (RF) domain, and an electrostatic field are used to confine the charged particles in the central region of the trap [17]. In a Paul trap the electrodes are arranged to generate an almost ideal quadrupolar electric potential close to the trap center. The trajectory of a single ion in the trap can be described by the motion in a harmonic pseudo-potential [18]. The charged particles have a typically slower harmonic motion which is called “secular motion”. The vibrating frequency of the secular motion depends on the charge-to-mass ratio of the ions. Therefore, the excitation of the mass-dependent motional resonances of the trapped ions is often used for mass spectrometry of the trapped ions [19]. The studies of the resonant secular frequency also help to determine the geometric parameters and characteristics of the designed ion trap. This in turn can help in the design of the ion trap. For a linear configuration Paul trap, the total potential can be described as [20]
⁎ Corresponding author: Wuhan Institute of Physics and Mathematics (WIPM), Chinese Academy of Sciences (CAS), West No.30 Xiao Hong Shan Wuhan, Hubei, 430071 PR China. E-mail address: [email protected] (X. Tong).
https://doi.org/10.1016/j.cjph.2019.03.017 Received 23 February 2019; Accepted 15 March 2019 Available online 01 May 2019 0577-9073/ © 2019 The Authors. Published by Elsevier B.V. on behalf of The Physical Society of the Republic of China (Taiwan). This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Chinese Journal of Physics 60 (2019) 61–67
H.X. Li, et al.
r (r ,
z) =
1 M 2
2 2 r r
+
1 M 2
2 2 zz ,
(1)
where the radial and axial frequency of the ion secular motion, ωr and ωz, are given by r
=
2 2 2 r Q Urf 8M 2r04 2rf
z
=
2 z QUend . Mz 02
z QUend Mz 02
,
(2)
(3)
Here 2r0 is the distance between the two diagonal electrodes. 2z0 is the distance between the two endcap electrodes. κr and κz stand for the geometric parameters of the linear Paul trap in the radial and axial directions, respectively. Q and M stand for the charge and mass of the ion in the trap, respectively. The ion is confined by the combination of a radio frequency (Ωrf/2π) oscillating electric field with a peak to peak voltage Urf in the radial direction and a static potential Uend applied to the endcap electrodes of the trap. The geometric parameter in the axis direction κz has been extensively studied [21,22], while according to our knowledge the radial geometric parameter has not been experimentally studied and is usually treated as having the value 1 [23,24]. This assumption is sufficient for a small ensemble of ions, typically for a single or linear configuration of Coulomb crystals confined in the center of an ion trap, where the trapping potential along the radial direction is close to an ideal hyperboloid potential [25,26]. However, more accurate values of the geometric parameters are required when using the radial direction resonance excitation scheme in an ion trap for precision mass spectrometry to determine a large ensemble of Coulomb crystals with multi ionic species [27]. In this paper, we provide an experimental method for determining both the axial and radial geometric parameters in a linear Paul trap. More specifically, the motional resonances of the tapped and laser-cooled Calcium ions are determined nondestructively with a high mass-to-charge ratio resolution upon excitation using an additional oscillating electric field. By changing the trapping potential, both the axial and radial geometric parameters are derived according to Eq. (2) and Eq. (3) from the measured resonant frequencies of the Calcium ions. The experimental setup is introduced in Section 2. In a third section, the results and a discussion are presented. 2. Experimental setup The linear Paul trap used in our experiment is shown in Fig. 1. The trap consists of four parallel steel rods, each axially partitioned into three sections, the endcap electrodes are labeled as ‘A1∼A4’ and ‘C1∼C4’, and the center electrodes are labeled as ‘B1∼B4’ in Fig. 1(a). The diameter of each electrode rod is 8 mm, 2r0 is 7 mm, the lengths of the center (2z0) and end-cap electrodes (2ze) are 6.4 mm and 24 mm, respectively. A RF (radio frequency) field is applied to a pair of diagonal electrodes, and the other pair of diagonal electrodes are connected to ground, as shown in Fig. 1(b). Here, the peak to peak voltage Urf and frequency frf (Ωrf/2π) can be varied from 0 V to 400 V and 3.2 MHz to 4.2 MHz, respectively. The RF field provides the confinement for the ions in the radial plane (xy plane in Fig. 1). The eight endcap electrodes (A1-A4, C1eC4) are supplied with static electric voltages (Uend) from 0 V–10 V to confine the ions in the axial (z-axis) direction. The main experimental setup is shown in Fig. 2. Calcium atoms are evaporated from a Ca oven. The temperature of the oven can be heated to 350 °C by an electric current. Calcium atoms in an effusive skimmed beam are non-resonantly ionized by a 355 nm pulse laser in the central region of the ion trap to produce Calcium ions, as shown in Fig. 3(a). The trapped Ca+ ions are Doppler laser cooled on the 4s2S1/2 → 4p2P1/2 transition at 397 nm with a red detuning of ∆ω and addressed to the population lost to the 3d2D3/2 state by an 866 nm laser beam, as shown in Fig. 3(b). Both laser beams are reflected along the trap axis to achieve bi-directional laser cooling. A photo-multiplier tube (PMT) is used to monitor the fluorescence of the laser-cooled Ca+ ions. A color filter with center wavelength at 390 nm and full width at half maximum of 18 nm is located before the PMT for filtering out the 866 nm laser and scattered light from the background. The ions are also imaged onto a complementary metal-oxide semiconductor (CMOS) camera by a 10 times magnification lens system. The localized ion positions are shown as an
Fig. 1. Schematic views of the linear ion trap. (a) Side view of the trap. (b) Cross-sectional view of the trap and applied RF (radio frequency) potential. 62
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H.X. Li, et al.
Fig. 2. Schematic of the experimental setup, including the trap, laser beams and fluorescence collecting system.
Fig. 3. (a) Energy level diagram of the photo-ionizing Ca atom. (b) Energy level diagram of the Doppler laser cooling of the Ca+ ions. (c) An image of a Ca+ ion Coulomb crystal taken by the CMOS camera.
image of a Coulomb crystal in Fig. 3(c). In order to calibrate the magnification and mark the position of the lasers, a 250 μm diameter bare optical fiber fixed on a 3D manipulator (not shown in Fig. 2) is inserted into the trap center and imaged by the CMOS camera. To obtain the secular motion frequencies (ωr and ωz) of the trapped ions, AC (alternating current) voltages used for motion excitation are applied to a pair of diagonal endcap electrodes (A2, A3) for axial motion excitation and to a mid-electrode (B2) for radial motion excitation, as indicated in Fig. 4. Because of the segmented structure of our linear ion trap, the AC voltages can be applied directly to the trapping electrodes without affecting the RF trapping field. The scans of AC frequency ranging from 0 kHz to 400 kHz are performed with the fixed AC amplitude of 500 millivolt. The scan step is set to be 1 kHz. When the frequency of the excitation AC field is on resonance with the secular motion frequency of the ions in the trap, the ions in the Coulomb crystal are heated and set off the resonance of the cooling laser, resulting in the reduction of the emitted fluorescence from the Ca+ ions. Such changes of fluorescence intensity are record by CMOS and PMT and indicate the secular frequencies of the trapped ions. According to Eq. (2), considering 1/ 2rf , Urf2 , and Uend as variables for the linear function of r2 , the geometric parameters κr and κz can be obtained from the slope or intercept, respectively. Similarly, according to Eq. (3), considering Uend as a variable for the linear function of z2 , κz can be derived from the slope. 3. Results and discussion The resonant motion excitation frequency spectrum of calcium ions in both the axial and radial direction are shown in Fig. 5(a) and (b), respectively. The secular motion frequencies, under the experimental trapping conditions (shown in the caption of Fig. 5), are 89 kHz for the axial motion and 178 kHz for the radial motion.
Fig. 4. Scheme of electrodes for AC and RF. RF applied to the red electrodes is used to confine the Ca+ ions in the radial plane. AC applied to the yellow electrodes is used to excite the secular motion of Ca+ ions in the trap. 63
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Fig. 5. The change of the fluorescence of Ca+ ions via scanning of the AC frequencies. (a) Excitation of the axial motion of Ca+ ions in the ion trap with Urf = 200 V, Ωrf = 3.20 MHz × 2π, Uend = 1.834 V. (b) Excitation of the radial motion of Ca+ ions in the ion trap with Urf = 185 V, Ωrf = 3.30 MHz × 2π, Uend = 2.295 V.
Fig. 6. The square of the secular motion frequencies of Ca+ ions in the axial direction ( z2 ) as a function of the endcap voltage Uend with the trapping potential Urf =180 V, Ωrf =3.50 MHz × 2π. Every date point is averaged over 10 measurements. The red line is a linear least square fit to the data. The uncertainty of z2 is derived from 2ωz × Δωz, where the Δωz is the frequency scan step of 1 kHz.
To obtain the geometric parameters, a series of secular motion frequencies in both the axial and radial directions are measured under different trapping potentials. In the axial direction, the static voltage Uend is varied to measure the corresponding axial secular motion frequency ωz. The square of the measured secular frequencies ωz are plotted as a function of Uend in Fig. 6 The geometric parameter of the axial direction κz is derived from the slope of the linear fitting line. In the radial direction, Ωrf is varied to measure the corresponding radial secular motion frequency ωr. The square of ωr is plotted as the function of 1/ 2rf with three different trapping potentials, shown as the data points A, B and C in Fig. 7(a). The geometric parameters of κr and κz can be derived from the slopes and intercepts of the linear fitting lines, respectively. In a similar way, in 64
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Fig. 7. The square of the secular motion frequencies of Ca+ ions in the radial direction ( r2 ) as a function of (a) 1/ 2rf , (b) Urf2 and Uend (c). The color lines (black, red, blue) are the least square linear fit to the data in different trapping potentials. The uncertainty of r2 is derived from 2ωr × Δωr, where the Δωr is the frequency scan step of 1 kHz.
Fig. 7(b) and (c), the square of ωr is plotted as a function of Urf2 and Uend, respectively. The value of κr and κz can be obtained from the slopes or intercepts of the corresponding fitting lines. All the values of κr and κz obtained from the fitting lines of Figs. 6 and 7 are collected in Table 1. The column of “Group” indicates the experiments taken under the different trapping potentials shown in Fig. 7. In order to obtain a more credible and accurate value of κr and κz, the results of κr and κz in Table 1 are compared in Fig. 8(a) and (b), respectively. All the values of κr or κz agree within one standard derivation, indicating that the method used in our experiment to obtain the geometric parameters of a linear Paul trap is feasible and the results are credible. Further linear fittings with zero slope yield the results of κr and κz to be 0.928(2) and 0.374(3), respectively. The electric potential in the central region is the superposition of the electric potential produced by four electrodes in the linear Paul trap. Because the shape of the four electrodes are not an ideal hyperboloid, the potential generated in the central region is not an 65
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Table 1 Values of κr and κz. Direction of excitation
Figure
Group
κr
Standard error of κr
κz
Standard error of κz
x y-plane
Fig. 7(a)
A B C A B C A B C –
0.93250 0.93438 0.93395 0.92676 0.93064 0.92262 0.92863 0.93044 0.92764 –
0.00886 0.00837 0.01009 0.00406 0.00481 0.00632 0.00338 0.00295 0.00313 –
0.39984 0.37832 0.40622 0.38870 0.38701 0.37414 0.36844 0.36774 0.36889 0.37448
0.03982 0.04408 0.04876 0.02102 0.02143 0.02507 0.01197 0.01154 0.00838 0.00227
Fig. 7(b) Fig. 7(c) z-axis
Fig. 6
Fig. 8. (a) The values of κr. (b) The values of κz.
absolute harmonic potential. Therefore, the geometric parameter in the radial direction, κr is close to, but not exactly, 1. In the z-axis direction, the endcap electrodes are separated from the middle electrodes, and the middle electrodes have a shielding effect, the axial potential near the central region of the trap generated by the endcap electrodes is significantly reduced. Thus, the geometric parameter in the axial direction κz is much less than 1. 4. Conclusions and outlook In this paper, we determined the geometric parameters of a linear Paul trap by performing resonant excitation on the trapped and laser-cooled Ca+ ions. The secular motion frequency of the Ca+ ions is obtained by the motion excitation using an AC electronic field. For the first time, the geometric parameter in the radial direction κr has been determined to be 0.928(2), which shows the effect of the non-ideal hyperboloid shaped electrodes used in the linear Paul trap. The geometric parameter in the axial direction κz is also obtained as the value of 0.374(3). Our linear Paul trap has a similar configuration compared to the traps used in Refs. [19,21,22], which have the values of κz as 0.244, 0.3 and 0.342, respectively. Laser-cooled fluorescence mass spectrometry relying on the excitation of secular ion motion in an ion trap is an important mass spectrometry technology in cold atomic and molecular physics [28,29]. The precise determination of the geometric parameters of a linear Paul trap is essential for the laser-cooled fluorescence mass spectrometry, particularly for determining the ion species during the sympathetic cooling and for the further high-resolution spectroscopy based on the photodissociation process [30-32]. Acknowledgments We acknowledge Pan Yong and Zhao Pei for the early work on this project and Du Li-Jun, Song Hong-Fang for the help of setting up the experiment. Funding This work was supported by the National Natural Science Foundation of China (Grant No. 91636216, 11504410 and 11474317) and Chinese Academy of Sciences (Grant No. XDB21020200). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.cjph.2019.03.017. 66
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