Measurement of ion displacement via RF power variation for excess micromotion compensation
aa r X i v : . [ phy s i c s . a t o m - ph ] M a r Measurement of ion displacement via RF power variation for excess micromotion compensation
Measurement of ion displacement via RF power variation for excessmicromotion compensation
Ryoichi Saito , , a) Kota Saito , and Takashi Mukaiyama , Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531,Japan Quantum Information and Quantum Biology Division, Institute for Open and Transdisciplinary Research Institute,Osaka University, Osaka 560-8351, Japan. (Dated: 3 March 2021)
We demonstrate a method of micromotion minimization of a trapped ion in a linear Paul trap based on the precisionmeasurement of the ion trapping position displacement due to a stray electric field in the radial plane by ion fluores-cence imaging. The amount of displacement in the radial plane is proportional to the strength of a stray electric field.Therefore, we evaluated the micromotion compensation condition by measuring the ion displacements from the ionequilibrium position using two different radial trap frequencies with various combinations of the compensation voltage.The residual electric field uncertainty of this technique reached a few volts per meter. This compensation techniquedoes not depend on the orientation of the incident cooling laser or the detuning and imaging direction. Therefore, thismethod is suitable for a planar ion trap, a stylus ion trap, which limits the propagation angle of lasers, or miniaturizedion trap systems for sensing and metrological applications.
I. INTRODUCTION
A laser-cooled ion stored in an ion trap is a valuable plat-form for developing devices based on quantum phenomena.A single to several trapped ions in a vacuum, considered asa well-isolated system, is used in the development of quan-tum techniques, especially quantum computing, quantum sim-ulation, and quantum sensing. The design and architectureof an ion trap have been investigated to optimize scaling upthe number of entangling ions or enable miniaturization. Ex-amples of ingenious ion traps for quantum devices includethe segmented ion trap , microfabricated ion trap , planartrap , and stylus trap .An ion in a Paul trap has a motional mode directly drivenby the RF electric field that confines the ion in a harmonic po-tential in terms of pseudopotential approximation, so-calledmicromotion , regardless of the complexity of the trap archi-tecture. Because micromotion minimization has a strong re-lationship with the ion trap’s performance, micromotion com-pensation is a necessary and essential initial procedure for uti-lizing trapped ions for precise quantum level manipulation.Therefore, several compensation methods have been devel-oped. These compensation procedures are characterized by anindicator of excess micromotion, and include motional side-band spectroscopy , RF-photon correlation , parametricexcitation , atomic loss due to atom-ion collisions , andion trajectory .This paper demonstrates a micromotion minimizationmethod based on ion position detection with two different trapconfinement variations. In an ideal case, an ion in a Paul trapis stored at the RF null point due to the trap RF electric field,while in general, an unexpected stray electric field shifts theequilibrium position from the RF null point. The strength of a) Electronic mail: [email protected] b) Electronic mail: [email protected] the stray electric field can be evaluated from the measurementof the ion displacement due to two different radial confine-ments. Minimization of the ion displacement gives us the mi-cromotion well-compensated codition. Finally, we obtain aresidual electric field uncertainty of a few volts per meter byour compensation measurement.The method introduced in this study does not depend onthe orientation of the incident cooling laser or its detuningand imaging direction. Therefore this technique is suitablefor various ion trap architectures, including planar traps, andminiaturized ion trap systems for metrological applications,which have structural limitations for an incident laser. Fur-thermore, this method does not require any stable lasers toexcite the narrow transition for sideband spectroscopy or anyhigh-speed measuring instruments to detect photon emissiontiming for RF-photon correlation measurements. Only a con-ventional ion fluorescence imaging system is required to de-termine the ion trapping position. By comparison with thecompensation method of ion trajectory , which requires con-tinuous scanning of the RF amplitude, our method is simpleand convenient because we seek the RF null from two differ-ent RF amplitudes. While this compensation method relies onion fluorescence images, the residual electric field’s limitationis carried out by the resolution of ion images in principle. Wealso mention systematic error due to forces other than the elec-tric field; thus, our imaging resolution of ion position sensingreached a sensitivity of 125 zN. This is comparable to thescattering force of the cooling laser. II. THEORY AND METHOD
In this section, we introduce the theoretical treatment of mi-cromotion in a linear Paul trap and the concept of our micro-motion minimization technique.We provide an overview of the trapped single ion motion ina conventional linear Paul trap. Micromotion in a linear Paultrap arises in a radial plane in an ideal case. Therefore, weeasurement of ion displacement via RF power variation for excess micromotion compensation 2describe the ion motion only in a radial plane for brevity. Anequation of motion of a single ion that has mass m and charge Q in a linear Paul trap can be described as¨ r i + [ a i + q i cos ( Ω t )] Ω r i = QE stray , i m . (1)where t is time, r i is the ion position, and i = x ′ , y ′ indicates theCartesian coordinate in the radial plane. The a i and q i are usu-ally called the trapping parameters and can be determined bythe mass of the ion, the ion trap geometric condition, and theapplied voltage to the electrodes. E stray = (cid:0) E stray , x ′ , E stray , y ′ (cid:1) is the stray electric field and is assumed to be a static and uni-form electric field in the region of the trapped ion.The solution of Eq.(1), which is called the Mathieu equa-tion, in the case where | a i | , q i ≪ r i ( t ) ≈ [ r , i + r , i cos ( ω i t + φ i )] h + q i ( Ω t ) i , (2)where r , i ≈ QE stray , i m ω i . (3)Here, r , i and φ i are determined by the initial condition ofthe ion. The secular frequency ω i is described as ω i = Ω p a i + q i / ω i commonly called secular motion, and the other is the micro-motion following the RF frequency Ω .The term r , i in Eq.(2) that is dependent on the stray elec-tric field shows the equilibrium position of the ion from the RFnode. In the case of stray electric field E stray , i =
0, a trappedion is confined at the RF node. In such a well-compensatedcondition, the ion’s motional energy can be minimized by thelaser cooling technique. A non-negligible stray electric fieldshifts the ion trapping position given by Eq.(3) from the RFnode. This positional shift arises from a motion driven by theRF electric field, which is called excess micromotion. Thisexcess micromotion cannot be significantly reduced by lasercooling because it synchronizes with the RF period, and en-ergy transfer between the ion and RF electric field always ex-ists in a single RF period . Thus, to minimize the ion motionalenergy in the ion trap, the external electric field should be ap-plied to reduce the stray electric field and compensate for thetrapping position of the ion in two dimensions.Excess micromotion can arise not only from a finite strayelectric field, such as one caused by a charged insulator, forexample, but also from RF phase mismatching between theelectrodes or geometric distortion of the ion trap. However,we will not consider these factors here.Next, we discuss the trapping position shift of the ion fromthe RF node due to the stray electric field in relation to trapconfinement. From Eq.(3), the ion’s trapping position is de-scribed by an equilibrium of the spring force resulting fromthe pseudopotential and the electrostatic force of the strayfield. Therefore, the ion trapping position is close to the RFnode with asymptotically increasing RF confinement. In case of infinite trap confinement ω i → ∞ , the trapping position ofthe ion is coincident with the RF node under the condition ofa finite stray electric field E stray , i = ω i and ω ′ i .Then the ion displacement between the two different trappingpositions ∆ r , i can be written as ∆ r , i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) QE stray , i m ω i − ω ′ i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4)This equation shows that the stray electric field is detectableby measuring the ion displacement due to the two differenttrap frequencies. The ion displacement is proportional to thestray electric field; thus, the ion displacement becomes zerowhen the stray electric field is reduced by applying anotherexternal electric field for compensation.The ion displacement in Eq.(4) is represented in one dimen-sion. Therefore, ∆ r = q ∑ i ∆ r , i yields the ion displacementin the radial plane.Our compensation method concept is based on measuringthe ion displacement due to the difference in the ion confine-ment in the radial plane (see the left side of Fig1). Moreover,we determine the trap condition in which the trapping posi-tion does not change with RF power variation by applying anexternal electric field.An external force can also cause the trapping position toshift due to the light pressure of the Doppler cooling laser orgravity, for example. These external forces can cause system-atic errors in micromotion compensation. Therefore we haveto manage these forces carefully for the precise detection ofthe RF node. On the other hand, this fact shows that the de-tection of the trapping position can also be used as a smallforce sensor. III. EXPERIMENTAL SETUP
In this study, we used a conventional linear Paul trap thathas nine electrodes (two RF electrodes, two RF ground (GND)electrodes, four END electrodes, and a Comp electrode forcompensation of ion position). The schematic diagram of ourlinear ion trap seen from the axial direction is shown in theright side of Fig. 1. Radial confinement was realized by twoRF electrodes and two GND electrodes (shown as RF, GNDA, and GND B in Fig.1). DC voltage was applied to fourEND electrodes, which are attached to the front and back ofthe GND electrodes along the z direction to achieve axial trapconfinement.Neutral ytterbium vapor from an atomic oven was pho-toionized by 399 nm and 369 nm lasers and then a single ionwas trapped at the center of the ion trap. The trapped single Yb + ion was cooled by Doppler cooling of the S / – P / transition using a 369 nm cooling laser and a 935 nm repump-ing laser. We also irradiated 760 nm laser to pump out the Yb + ion in F / state to the cooling cycle. These lasers were over-lapped and co-propagated through the trap center along theaxial direction. Another set of cooling and repumping laserseasurement of ion displacement via RF power variation for excess micromotion compensation 3 FIG. 1. Schematic diagram of ion displacement in the radial plane(left side) and the linear Paul trap seen from the axial direction inour system (right side, not scaled). (left) An ion in a linear Paul traptrapped at the RF node without a stray electric field. The ion’s equi-librium position is shifted proportionally to a finite stray electric field(orange point). Furthermore, the shift amount of the ion also dependson its radial confinement. If the trap frequency is decreased by tun-ing the RF power, the equilibrium position shifts away from the RFnull (purple point). Excess micromotion can be detected by the iondisplacement between the two different confinements that rely on thestray electric field amplitude. (right) An ion is confined by RF volt-age applied to the RF electrodes and RF ground (GND) electrodes,in two dimensions, and axial confinement is achieved by DC voltageapplied to the END electrodes located on the front and back of theGND electrodes. Minimization of micromotion is carried out by ap-plying DC voltage to the GND A and Comp electrodes to cancel outthe stray electric field (GND B electrode is connected to laboratoryground). was incident through the trap center in the y - z plane to coolthe radial ion motion efficiently.In typical trapping conditions, we drove the RF electrodesusing a helical resonator with 570 V at a resonance frequencyof Ω = π × . ω r = π × ( .
90 MHz , . ) . We also applied 68 . z direction was ω z = π × .
17 MHz. We deliberately ap-plied modulation voltage superposing to the GND A or ENDelectrode and excited the secular ion motion to independentlymeasure the trap frequency. In our proposed micromotionminimization method, we trapped the ion in several radial trapconfinement conditions by changing the RF power applied tothe helical resonator.To compensate the excess micromotion in the radial plane,we applied additional compensation voltages V GND and V Comp to the electrodes GND A and Comp (colored blue and yel-low in Fig.1), respectively. The GND A and Comp electrodeswere aligned from the trap center at 144 . ◦ counterclockwisewith respect to y axis and at 16 . ◦ clockwise with respect tothe x axis, respectively. Each electrode and the trap centermade an angle of 109 . ◦ . Applying DC voltages to these elec-trodes created a static and approximately uniform electric fieldaround the trap center in each direction.The ion fluorescence from S / – P / transition due to thelaser cooling cycle was collected by an objective lens installedat the top of the ion trap in Fig.1. Because our imaging sys-tem captured the fluorescence image in the y – z plane, we es-timated the ion position in direction by imaging the lens po-sition when an ion image was in focus . Ion images were acquired around the focus position ± µ m typically whilescanning the imaging lens mounted on the motorized stage.The objective lens for imaging the ion, which had a numeri-cal aperture of 0.28, was mounted on a piezo motor stage andthe focusing lens was positioned along the x direction. Thegathered fluorescence light propagated along the x axis andreached an electron-multiplying charge-coupled device (EM-CCD) camera and a photomultiplier tube (PMT). The imagingmagnification of our imaging system was 27.0, and the esti-mated collection efficiency of emitted photons from the ionwas 1 . IV. RESULTA. Measurement of ion position in radial plane
To compensate for the excess micromotion caused by astray electric field based on ion displacement due to trap con-finement variation, we measured the two-dimensional ion po-sition in the radial plane, namely the x - y plane in our system.Figure 2 shows a typical result of ion position measurement.Each fluorescence image in Fig.2(a–c) was captured during400 ms accumulation of emitting photons from an ion; fur-thermore, it was fitted by Gaussian to estimate the position in y direction and width of the ion image. The width of the ionimage as a function of imaging lens position is illustrated inFig.2(d). We determined the focus position by fitting thesedata using an equation of the Gaussian beam waist and deduc-ing the position in the x direction. The absolute position of anion in the radial plane was measured using this procedure.The absolute position in the x direction had a few microm-eter systematic error in each scanning sequence, caused bylack of piezo motor stage position repeatability. To avoid thiscrucial error, we measured the ion position in different radialconfinements in a single scan of the stage. B. Two dimensional ion displacement
We measured the trapping position of an ion at two dif-ferent radial trap confinements using the method describedin Section II. These two equilibrium points yielded the iondisplacement at the compensation condition derived from thevoltage applied to electrodes GND A and Comp. To detect andminimize micromotion, we scanned the compensation voltagetwo-dimensionally and measured the ion displacement.Figure 3 shows ion displacement measurement results fordetermining the optimum trap condition. E GND , , E Comp axesshow the induced electric field around the small region of thetrap center derived by applying a voltage to the GND A andComp electrode, respectively. This electric field for compen-sation was calibrated from the other measurements by detect-ing the equilibrium position with the varying voltage appliedto the GND A or Comp electrode. Each point represents theobserved ion displacement at each compensation electric fieldcombination. This ion displacement was induced by varyingeasurement of ion displacement via RF power variation for excess micromotion compensation 4 !" ! & ’( ) * + ,-. ) / )3 + ! ;,< ;=<;@< FIG. 2. Ion position measurement in the radial plane by fluorescenceimage of an ion and its focus position. (a–c) Ion fluorescence imagesat different imaging lens positions around the focus position. We fitthis fluorescence image by Gaussian distribution and evaluated thefull width at half-maximum (FWHM) in each image. The coordi-nates indicated in (a) show the axes that we defined and illustratedin Fig.1. The white bar in (b) shows 5 µ m scale of the images. (d)Each plot shows the FWHM of the ion image at each objective lensposition. We determined the ion position along the imaging depth(namely x ) direction by Gaussian beam waist fitting (indicated bythe red line). the radial trap frequency with a finite stray electric field con-dition. Therefore, we realized a change in the trap frequencyby tuning the RF power applied to the RF electrodes.The ion displacement measurement with 3 .
71 dBmRF power variation is shown in Fig.3(a). This RFpower change corresponds to ( .
90 MHz , .
75 MHz ) → ( .
21 MHz , .
09 MHz ) variation of the radial trap frequen-cies. This result clearly shows the cone shape expected fromEq.4 and an intensity profile on the bottom of the graph dis-plays a fitted outcome by a cone function. We estimated theresidual stray electric field uncertainty of (cid:0) ∆ E GND , ∆ E Comp (cid:1) =( . / m , . / m ) from this measurement.To reveal the optimum trap condition more precisely, weinvestigated the same measurement with a larger RF powervariation of 4 .
75 dBm with smaller scanning of the elec-tric field for compensation, and the result is displayed inFig.3(b). In this measurement, we changed the radial con-finement of ( .
90 MHz , .
75 MHz ) → ( .
06 MHz , .
95 MHz ) to evaluate the ion displacement. (cid:0) ∆ E GND , ∆ E Comp (cid:1) =( . / m , . / m ) was obtained from this result. The resid-ual stray electric field uncertainty improved compared to theresult of the RF power variation 3 .
71 dBm because the iondisplacement ∆ r becomes sensitive by tuning the magnitudeof the RF power variation.This tunable sensitivity of micromotion compensation is anotable point of this method because we can seek a compen-sated condition step by step to suit the current trap situation.Therefore, we expect that a better residual electric field canbe obtained by decreasing the lower limit of radial confine- ment. However, this is restricted by the instability of an ionwith less than 1 MHz of radial trap confinement in our system.This trap instability could be caused by a radial trap frequencylower than the axial one, micromotion induced by other fac-tors (e.g., geometrical distortion of an ion trap), micromotionin the axial direction due to the breakdown of an ion trap trans-lational symmetry, or a combination of these elements.The optimum condition of the ion trap can be determinedinstinctively and simply from Fig.3 because the minimum iondisplacement condition is represented in the two-dimensionalparameter space that corresponds to electric fields cancelingout the stray electric field in the two-dimensional radial plane.This easy-to-recognize representation is one of the advantagesof this compensation technique compared to the other tech-niques based on ion position sensing. Moreover, only animaging system of ion fluorescence is required to introducethis compensation measurement. This convenience is anotheradvantage of this technique. From the point of view of sen-sitivity, the obtained uncertainty of the residual electric fieldis not smaller than but comparable to that of other reportedtechniques .Because the ion displacements were evaluated by two ab-solute equilibrium points of an ion in the radial plane, we cananalyze the ion displacements in the respective axes of ∆ r , x and ∆ r , y , which are shown in Fig.4(a) and (b), respectively.The measured data are indicated as the plots, and the intensityplot at the bottom represents fitting by a valley shape func-tion. The valley bottoms of the results in Fig.4(a) and (b)show the micromotion-minimized position in the x and y di-rections, respectively. This line represents the nodal line ofthe RF electric field near the trap center along the x or y direc-tion, and the geometry of the ion trap determines the slope ofthe line. Because the valley bottom of each direction only hasa sensitivity for the respective axis, the intersection of thesetwo lines coincides with the two-dimensional micromotion-minimized condition of Fig.3. A similar situation occurs inthe RF-photon correlation method, which is only sensitive tomicromotion in a projection direction to the radial plane of theincident cooling laser.The data of ∆ r , x are noisier than the data of ∆ r , y by com-parison between Fig.4(a) and (b). This is caused by the lowaccuracy of ion position determination in the imaging depth( x ) direction compared to that in the imaging plane ( y and z directions). Because an angle between E Comp near the trapcenter and the x axis is supposed to be small from the trapgeometry, the shifts of the equilibrium positions induced by E Comp are nearly along x direction. It is reasonable that thevalue of the residual stray electric field uncertainty in E Comp direction is larger than that in the E GND direction.
C. Limitation of detecting residual micromotion
Next, to reveal the performance of this measurementmethod, the sensitivity and the limitation of detecting residualmicromotion are discussed. We also mention the equilibriumposition shift due to forces other than the electric field. Theseunexpected shifts may induce systematic error for micromo-easurement of ion displacement via RF power variation for excess micromotion compensation 5
FIG. 3. Measured ion displacement with various combinations ofelectric fields derived by compensation voltage applied to the GNDA and Comp electrodes. Each point represents the measured data,and the intensity plot on the bottom shows the fitting of these databy the cone function. Results of tuning of the RF power variation(a) 3 .
71 dBm and (b) 4 .
75 dBm. The obtained residual stray electricfield uncertainty is (cid:0) ∆ E GND , ∆ E Comp (cid:1) = ( . / m , . / m ) from(b). The sensitivity of this compensation method depends on themagnitude of the RF power variation because an ion’s equilibriumposition can easily shift in the case of a small stray electric field witha low trap frequency. tion minimization in the case of the compensation methodbased on ion position detection.We evaluated the magnitude of the RF power variation de-pendence of the two-dimensional ion displacement of ∆ r with the various static electric fields in Fig.5. In this mea-surement, to change the magnitude of the RF power variation,the higher trap frequencies were fixed, and only the lower fre-quencies were tuned. We started measuring the ion displace-ments at RF null in the radial plane determined by the pro-cedure discussed above (data set of 0 V / m) and applied theelectric field by tuning E GND . The RF power variation cor-responding to the condition in Fig.3(a) is shown as the blackdashed line and the inset displays a small region of the verticalaxis.The ion displacements increased along the horizontal axis
FIG. 4. Measured ion displacement in one dimension with variouselectric field combinations of (cid:0) E GND , E Comp (cid:1) . The measured ion dis-placements induced by an RF power variation of 3 .
71 dBm is shownas the plotted points and the fitting result of these ion displacementdata by a valley shape function is indicated as the intensity plot atthe bottom. We investigated (a) ∆ r , x and (b) ∆ r , y , namely the iondisplacement in x and y directions. The valley bottoms of (a) and(b) compose lines in E GND - E Comp plane and the condition of two-dimensional micromotion compensation in Fig.3 is achieved by theintersection of these lines. because the large equilibrium position shift was derived bylow trap confinement. This result shows the sensitivity of thismethod in terms of distinguishable residual micromotion ateach RF power variation in our system. It is evident that abetter residual stray electric field uncertainty can be achievedby lowering the trap confinement for measuring the ion dis-placement. The red shaded area in the inset indicates theuncertainty of the ion displacement determination measure-ment. Therefore, the plots in the shaded area cannot be dis-tinguished from one another. The accuracy of the ion positionmeasurement is mainly determined by a focal point measure-ment (namely, a position measurement in the x direction) inthe current situation because the typical uncertainty of a focalpoint is more than one order of magnitude larger than that inthe imaging plane. The uncertainty of a focal point dependson the depth of the field. Thus, we can improve the accu-easurement of ion displacement via RF power variation for excess micromotion compensation 6racy of a focal point measurement by replacing the imaginglens with one having a smaller focal length and larger numer-ical aperture. In principle, this is restricted by the Rayleighlength. On the other hand, the limitation of ion position deter-mination in the imaging plane (namely, the y and z directions)depends on the imaging resolution. The calculated localizedarea of an ion in the ion trap is much smaller than the ion im-age width at the focal point (typical value is about 1 µ m; referto Fig.2(d)). Therefore, the current imaging resolution is ap-proximately 1 µ m, and it can be reduced approximately by afactor of 0 .
37 in terms of the diffraction limit. To sum up,measuring ion displacement with a considerable RF powervariation and accuracy of the ion position determination arecrucial for this micromotion minimization technique.Finally, we mention the equilibrium position shift of an iondue to forces other than the electric field. The detectableminimum ion displacements of ∆ r , ∆ r , x , and ∆ r , y by ourimaging system are approximately 0 . µ m, 0 . µ m, and0 . µ m, respectively (the minimum detectable ∆ r corre-sponds to the red shaded area in the inset of Fig.5). This de-tectable minimum ion displacement is equivalent to an ion dis-placement due to a force of 2 . . . We do nothave to consider the shift in ion equilibrium position due togravity because the gravitational force for a single ytterbiumion is 2 . S – P transition of Yb + ion,in which the cooling laser is half red detuned of the line widthfrom the resonance and the saturation parameter is equal toten. Therefore, we can ignore the ion position shift due to thecooling laser along the x direction. In contrast, the shift in the y direction is comparable to the accuracy of the ion positiondetermination in the imaging plane. This can cause a system-atic error in the determination of ion position if the coolinglaser intensity or detuning fluctuates. V. CONCLUSION
In conclusion, we have developed and demonstrated a sim-ple method for minimizing the excess micromotion in a linearPaul trap based on measuring the ion displacement due to astray electric field. The magnitude of the stray electric fieldwas evaluated by its ion displacement, which was measuredby determining the equilibrium position with two different ra-dial trap frequencies detected by fluorescence images of anion. We found the micromotion-compensated condition bysearching for an ion displacement of zero, and the uncertaintyof the residual electric field reached a few volts per meter.The sensitivity achieved is not superior but comparable to thatof existing compensation techniques . Because the equilib-rium position shift increases with decreasing trap frequency,the sensitivity of micromotion detection can easily be tunedusing this method. Furthermore, this compensation techniquedoes not depend on the orientation and detunig of the incidentcooling laser or imaging direction. Therefore, this method is !" $ %& ’ ( ) * + , - . / /&
56 "! 78’+%9/:’;-:)-1)%&’2(<4 ’!’=>0’ 5?@’=>0’@"?@’=>0’" 6’=>0’6@A’=>0’B@ ’=>0’"A! ’=>0 ?!"?
FIG. 5. RF power variation dependence of the ion displacements. Wemeasured the ion displacements with decreasing trap frequency ofthe lower side in various static electric fields by tuning E GND . Eachpoint represents the average of three independent measurements, andthe error bar shows the standard error. The black dashed line indi-cates the same measurement conditions as Fig.3(a). The inset dis-plays an enlarged vertical axis, and the red shaded area representsthe typical uncertainty of an ion displacement measurement. applicable to planar ion traps, stylus ion traps, which limit thepropagation angle of lasers, or miniaturized ion traps and sys-tems. Moreover, complex laser systems or instruments are notrequired in this measurement because this method is based onthe detection of ion positions with trap potential modulation.Thus, this technique can simplify the ion trap system and itsexperiments. This can lead to the development of quantuminformation and sensing techniques using ion traps.
VI. ACKNOWLEDGEMENT
This work was supported by JST-Mirai Program GrantNumber JPMJMI17A3, Japan.
VII. DATA AVAILABILITY
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