aa r X i v : . [ phy s i c s . a t o m - ph ] J un Dynamics of ion cloud in a linear Paul trap
Pintu Mandal ∗ , Manas Mukherjee † Raman Center for Atomic, Molecular and Optical SciencesIndian Association for the Cultivation of Science2A & 2B Raja S. C. Mullick Road, Kolkata 700 032 † Present address:
Centre for Quantum TechnologiesNational University of Singapore, Singapore - 117543 ∗ Email: [email protected]
October 17, 2018
Abstract
A linear ion trap setup has been developed for studying the dynamics of trapped ion cloudand thereby realizing possible systematics of a high precision measurement on a single ionwithin it. The dynamics of molecular nitrogen ion cloud has been investigated to extractthe characteristics of the trap setup. The stability of trap operation has been studied withobservation of narrow nonlinear resonances pointing out the region of instabilities withinthe broad stability region. The secular frequency has been measured and the motionalspectra of trapped ion oscillation have been obtained by using electric dipole excitation.It is applied to study the space charge effect and the axial coupling in the radial plane.1
Introduction “A single atomic particle forever floating at rest in free space” [1] is an ideal system forprecision measurement and a single trapped ion provides the closest realization to thisideal. A single or few ions can be trapped within a small region of space in an ion trap andthey are free from external perturbations. Such a system has been used for the precisionmeasurement of electron’s g - factor [2], various atomic properties like the lifetime ofatomic states [3], the quadrupole moment [4, 5, 6] etc . Precision table-top experiments offundamental physics in the low energy sector like the atomic parity violation measurement,nuclear anapole moment measurement, electron’s electric dipole moment measurement areeither in progress in different laboratories worldwide or proposed [7, 8, 9, 10]. Any high-precision experiment appears with systematics which are required to be tracked or removedand hence a systematic investigation on the system itself is essential at the initial stage.In order to prepare for measuring atomic parity violation with trapped ions, a series ofexperiments have been performed in a linear ion trap to fully understand its behaviourand associated systematics. In this colloquium, the results of some experiments will bepresented that are of preeminent interest to an audience coming from a variety of physicsdisciplines. It is organised with a brief overview on the physics of ion trapping in a linearPaul trap, description of the experimental setup and followed by results.
An electrostatic field can not produce a potential minimum in three dimensional space asis required for trapping the charged particles. It is therefore, either a combination of staticmagnetic field and an electric field is used (Penning trap) or a combination of a time-varying and an electrostatic field is used (Paul trap). In Paul trap a radio-frequency (rf)potential superposed with a dc potential is applied on electrodes of hyperbolic geometryto develop quadrupolar potential in space. The geometry of the electrodes evolved overthe decades for ease in machining, smooth optical access to the trapped ions etc . Figure 1shows one of the most frequently used trap geometries with four three-segmented rodsplaced symmetrically at four corners of a square and is commonly called a linear Paultrap. The four rods at each end are connected together and a common dc potential ( V e ) isapplied so as to produce an axial trapping potential. The diagonally opposite rods at themiddle are connected and a rf ( V cos Ω t ) in addition to a dc potential ( U ) is applied onone pair with respect to the other pair for providing a dynamic radial confinement. Theradial potential inside the trap isΦ( x, y, t ) = ( U − V cos Ω t ) x − y r ! , (1)where 2 r is the separation between the surfaces of the diagonal electrodes as depictedin figure 1(b). The equipotential lines are rectangular hyperbolae in the xy plane havingfour-fold symmetry about the z axis. The equation of motion of an ion of charge e andmass m under the potential Φ( x, y, t ) (eqn. 1) can be represented as d xdt = − emr ( U − V cos Ω t ) x (2) d ydt = emr ( U − V cos Ω t ) y. XY U – V cos Ω t V e XY (U – V cos Ω t) l V e (a) (b) l m l e l e Figure 1: (a) Schematic of the linear ion trap used in the experiment. (b) End view of thefour middle electrodes with relevant electrical connections. Various dimensions as markedby l e , l m , l , r e and r are described in section 3.These equations (eqn. 2) can be rewritten as d udζ + ( a u − q u cos 2 ζ ) u = 0 , (3)with u = x, y , where a x = − a y = 4 eUmr Ω ,q x = − q y = 2 eV mr Ω , (4)and ζ = Ω t/
2. Eqn. 3 is standard Matheiu differential equation and its solution providesstability or instability of the ion motion [11] depending on the values of the parameters a and q as defined in eqn. 4. There exists a region in a vs. q diagram for which the ion-motionis stable along a particular direction, for example along x . A similar stability region existsfor the motion along y direction. An intersection between these two stability regions thussignifies a stable motion in xy plane. For stable ion motion the trap should be operated at q < . ω n = (2 n ± β )Ω2 , n = 0 , , , ... (5)Here β is a function of the trap operating parameters a , q and for their small values, β = q a + q /
2. The fundamental frequency ω (that corresponds to n = 0) of secularmotion and other micromotion frequencies are given by ω = β Ω2 , (6) ω ± = Ω ± ω ,ω ± = Ω ± ω n x ω x + n y ω y = Ω , n x , n y = 0 , , , ... (7)where ω x and ω y are the secular frequencies for the motion along x and y respectively.Here n x + n y = k is the order of the multipole. If one of the trap parameters is varied,a parametric resonance appears at a definite value subjected to the condition defined byeqn. 7 and it gives rise to instabilities called “black canyons” [14] within the stabilitydiagram. -200 V – 2.5 KV Signal outAmplifierTTL converterMCSPC Channel electron multiplier VacuumRF @ 1.4 MHz20 V Extraction cylinderFilament Trap20 V to -45 V DAQ Discriminator Figure 2: Schematic of the experimental setup. The trap, filament and the CEM withother ion optics (extraction cylinder) are housed in a vacuum chamber. The functioningand control of the signal processing devices are explained in the text.
The schematic of the whole experimental setup is presented in figure 2. It consists of alinear Paul trap as shown in figure 1, an ionization setup, extraction and detection setup.The linear trap is assembled from four three-segmented electrodes each placed at fourcorners of a square of side ( l ) 12 .
73 mm [figure 1](b). Each of twelve rods are of diameter(2 r e ) 10 mm. The four middle rods are of length ( l m ) 25 mm while all others are 15 mmlong ( l e ) [figure 1(a)]. The separation between the surfaces of the diagonally opposite rods42 r ) is 8 mm. The middle electrode is separated from the end electrodes by a gap of2 mm. The molecular nitrogen ions (N +2 ) are created by electron impact ionization. Theions are dynamically trapped for few hundreds ms before they are extracted by loweringthe axial potential in one direction. The extracted ions are detected by a channel electronmultiplier (CEM). The CEM produces one pulse corresponding to each ion and the pulseis successively processed through an amplifier, a discriminator, a TTL converter beforeit is fed into a multichannel scalar (MCS) card which ultimately counts the number ofions reaching the CEM. This time-of-flight (TOF) technique provides a detection efficiencyaround 10%. The time sequences are generated by National Instruments’ Data Acquisition(DAQ) hardware which is controlled by Labview and monitored by a personal computer(PC).The trap is operated at a rf frequency of 1 .
415 MHz and no dc potential is appliedto the middle electrodes ( U = 0, a u = 0). The end electrodes are kept at +20 V whiletrapping. At the time of extraction, the end electrodes at the ion-exit-side are switchedfast (within 75 ns ) from +20 V to −
45 V. N q Figure 3: Number of trapped ions ( N ) as a function of q ( a = 0). Sudden fall of N aboutsome specific values of q corresponds to nonlinear resonances as explained in the text. Thenumbers 6, 7, 8 describe the order of the multipoles to which the resonances are assigned. The stability behaviour of the trap is studied by varying the trap-operating-parameter q while keeping the other parameter a at zero. The q is varied in steps of 0 . .
35 V while the number of trapped ions ( N ) is plottedin figure 3 as a function of q . It shows that N grows with q initially but decreases above q ≈ .
5. It remains almost constant and shows a plateau for 0 . < q < .
5. The q scanningis restricted to 0 . q . These are due to the existence ofhigher order multipoles within the trap potential as explained in section 2. The resonancesappear at q = 0 . , . . q = 0 . q = 0 . V cos Ω t v i cos ω t I IIIV III C
Figure 4: Schematic of the circuit used for dipole excitation of trapped ions. The dipoleexcitation signal v i cos ωt is applied between the electrodes marked as I and III.While operating the trap for a single ion, the region of instabilities should be avoidedas the ion gains energy from the time varying trapping field corresponding to these oper-ating regions and its motional amplitude increases. It can add to systematics in precisionmeasurement on the ion. Electric dipole excitation of the trapped ions has been employed to measure their secularfrequency and to obtain motional spectra. An electric dipole field has been applied onone of the middle electrodes as shown schematically in figure 4. The amplitude of theexcitation potential ( v i ) is kept fixed while its frequency is tuned so as to match withthe secular frequency of the trapped ions. The trap operating parameters are kept fixedduring the experiment. After the ions are loaded into the trap, the dipole excitation field6s applied for few hundreds of ms. After a short waiting time, the ions are released anddetected. The frequency of the excitation signal ( ω ) is varied and the total number of ionsis detected in each step.
164 168 172 176 180 184 188 192 196 200 2040.30.40.50.60.70.80.91.0 N o r m a li ze d i on c oun t Frequency (kHz)
Figure 5: Dipole excitation resonance of trapped ions. Solid line shows a fit to the datawith model function described in eqn. 8.
The experimentally obtained ion counts ( N ) have been normalised after dividing by themaximum ion count ( N max ) during a particular experiment. The normalised ion count( N n = N/N max ) with associated uncertainty, has been plotted as a function of the frequency( ω/ π ) of the dipole excitation signal. Figure 5 shows such a dipole excitation resonanceplot obtained with an excitation amplitude v i = 50 mV. The frequency is scanned from165 kHz to 205 kHz in steps of 500 Hz. The excitation signal is applied during 150 ms ineach step. The experimental data points have been fitted with the following function, N n = N + A exp [ − exp( − ω ′ ) − ω ′ + 1] , (8)with ω ′ = ( ω − ω ) /σ . Here ω is the resonant frequency and is equal to the secularfrequency of the trapped ions. N is an offset, A is a scaling factor and σ is the full-widthat half-maxima (FWHM) of the resonance. The secular frequency of the trapped ionsobtained from the fit is 182 . The motional spectra of the trapped ions as described in section 2 have been measured byvarying the dipole excitation signal frequency over long range. Figure 6 shows the motional7
00 400 600 800 1000 1200 1400 1600 18000.40.50.60.70.80.91.0 w /2 pw /2 pw /2 pw /2 p3w /2 pw / pw /2 p N Frequency (kHz)
Figure 6: The normalized ion count N plotted as a function of the dipole excitationfrequency (in kHz) presenting the motional spectra of the trapped ion cloud. The amplitudeof the excitation voltage is v i = 100 mV and the trap operating parameters are set at a = 0, q = 0 .
39 for N +2 . The frequency of the trap supply voltage is Ω = 2 π × . ω = 2 π ×
184 kHz that corresponds to the trap operating parameter a = 0, q = 0 .
39 and it is the strongest one. The second and third harmonics are observed at386 kHz and 577 kHz respectively. The other motional spectra as described in eqn. 6 areobserved at ω − = 2 π ×
915 kHz, ω − = 2 π × .
109 MHz, ω = 2 π × .
492 kHz and ω = 2 π × .
685 MHz.
The accurate measurement of the motional frequency of the trapped ions is essential fordifferent studies on them [16]. In a real linear Paul trap the radial motion is coupled withthe axial motion and hence a variation in the axial potential affects the secular frequency ofthe ions [15]. The motional frequency of the trapped ions for different axial potentials hasbeen measured with the technique described in section 4.2.1 and from this measurementthe geometrical radial-axial coupling constant has been determined. This is importantfor any precision spectroscopic study on a single ion confined in this setup. The dipoleexcitation technique is also applied to study the shift in the motional frequency due tospace charge created by the trapped ions. It is observed that the frequency decreases whilethey oscillate collectively with increasing space charge [17]. Detailed discussion on thesetopics can be found elsewhere [16, 17]. 8
Conclusions
This colloquium paper describes the development of an ion trap facility at IACS andthe results of some experiments fundamentally based on the dynamics of a trapped ioncloud. It presents a demonstration of some first principles of ion-trap-physics that are ofcommon interest to an audience coming from wide variety of physics and participating inthis colloquium. The results are also some significant feeds to the precision measurementbased on a single ion in a linear Paul trap.
The authors thank S. Das, D. De Munshi and T. Dutta, presently at the Centre for Quan-tum Technologies, National University of Singapore, for their support in developing theexperimental setup at IACS, and beyond it. The machining support from Max-PlanckInstitute, Germany is gratefully acknowledged.
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