Single Ion Thermal Wave Packet Analyzed Via Time-Of-Flight Detection
Felix Stopp, Luis Ortiz-Gutiérrez, Henri Lehec, Ferdinand Schmidt-Kaler
SS INGLE ION THERMAL WAVE PACKET ANALYZED VIATIME - OF - FLIGHT DETECTION
Felix Stopp, Luis Ortiz-Gutiérrez, Henri Lehec and Ferdinand Schmidt-Kaler
QUANTUM, Institut für Physik, Universität Mainz, Staudingerweg 7, 55128 Mainz, Germany [email protected]
February 26, 2021 A BSTRACT
A single Ca ion is confine in the harmonic potential of a Paul trap and cooled to a temperature of afew mK, with a wave packet of sub- µ m spatial and sub- m / s velocity uncertainty. Deterministicallyextracted from the Paul trap, the single ion is propagating over a distance of 0.27 m and detected.By engineering the ion extraction process on the initial wave packet, theoretically modeling the iontrajectories, and studying experimentally the time-of-flight distribution, we directly infer the state ofthe previously trapped ion. This analysis allows for accurate remote sensing of the previous motionalexcitation in the trap potential, both coherently or incoherently. Our method paves a way to extract,manipulate and design quantum wave packets also outside of the Paul trap. Trapped charged particles in Paul or Penning traps, or equivalently neutral atoms in magnetic or optical potentials, allowtoday for outstanding control of the wave packet and its motional state in the harmonic confinement potential [1, 2].For ions and for trapped atoms, even the ground state of motion can be reached, where the wave packet is shaped as aGaussian and where the momentum and position uncertainty are limited by the Heisenberg uncertainty relation. Singleelectrons, protons, antiprotons and highly charged ions also have been transported within segmented Penning trapsbetween spatially separated potential wells to allow for ultra-precise measurement, e.g. measuring g-factors, nuclearproperties or testing fundamental symmetries of nature [3, 4, 5]. Propagation of electron beams in guiding electricoscillating fields [6] and also cold atom matter wave shuttles in magnetic guides are investigated for novel sensing andinterferometric applications [7, 8, 9]. Likewise, the fast transport of cold trapped ions in the guiding radio-frequencyfield of a segmented Paul trap has been successfully demonstrated [10, 11] and is currently used for scalable quantumprocessing [12, 13]. In all those cases, a guiding field and a confining three-dimensional potential was employed forthe transport. On the other hand, freely propagating ion and electron beams have many applications in technologyand science. This includes microscopy, material science, nano-fabrication by focused ion beam milling and electronlithography. The beams are accelerated, steered and focused by electric and magnetic fields. For specific setups withelectron beams [14], also for ultra cold atoms [15, 16, 17], even genuine matter-wave properties such as interferencehave been demonstrated. The release of trapped atoms and observation of arrival times after a free fall is since thebeginning used to reveal their temperature [18, 19]. Neutral atoms may be ionized and subsequently acceleratedby electrical fields [20], or electrostatic multi-reflection devices are used for accurate mass spectroscopy or lifetimemeasurements [21, 22, 23, 24]. Extraction of ions from Paul traps was demonstrated for single ions [25, 26, 27, 28].It is the focus of our work here to extend the excellent control over trapped ion wave packets into a situation offree-space propagation. Using a single ion in a Paul trap and optimizing the extraction event out of the potential, we areinvestigating the wave packet emission and transport in free space, actually over a distance more than six orders ofmagnitude larger as compared to its wave packet size. From the time-of-flight signal at the arrival we analyze an initiallyprepared motional state. In this publication, we start with a description of the experimental setup, we theoreticallydescribe the properties of wave packets, of their motional excitation and free-space propagation. Then, we move over tothe extraction mechanism for coherent states and thermal states and discuss how to find their characteristic signature in a r X i v : . [ phy s i c s . a t o m - ph ] F e b PREPRINT - F
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26, 2021Figure 1: (a)
Scheme of the experimental setup. Ion trap electrodes serve to hold a single ion, laser cooled and emittingfluorescence that is observed by a electron-multiplier-charged-coupled-device (EMCCD) camera. The distance betweenthe center of the trap and the detector is 0.27 m. (b)
Time sequence for ion extraction: after a chosen time delay τ trig ,triggered on a fixed phase of the RF drive (upper plot), the ion is extracted by applying a square voltage pulse ofamplitude U E and duration t puls (middle plot) to the endcap electrode E . Synchronously the RF drive starts shuttingdown. After a delay t tof from the beginning of the U E pulse, the extracted ion arrives at the detector and is counted(lower plot). (c) TOF histogram measured after N = 1000 single ion extraction events at U E = − V. ∆t tof denotesthe standard deviation of the Gaussian distribution that we fit to the data.the time-of-flight measurement. Finally, we sketch applications when extending the method for wave packet propagationin the quantum regime. The experimental setup is based on a X-shape segmented Paul trap, see Fig. 1(a), with geometry similar to that inRef. [29]. The trap features two chips with RF electrodes for a radial confinement of ω rad / π ≈ . MHz and two chipswith DC electrodes, as well as two endcaps E and E , used for extraction and ion state preparation, respectively. Theaxial trap frequency ω z / π is controlled between kHz and kHz by applying a DC control voltage on the middlesegment of the DC chips. Ca ions are produced by photoionization from a neutral atomic beam, trapped and laser cooledon the S / - P / dipole transition near 397 nm and a laser near 866 nm is used to pump out of the metastable D / level. In this way, continuous laser-induced fluorescence is emitted near nm. We image this light through anobjective on an EMCCD camera, with a . magnification, which allows us to resolve the number of ions in thecrystal. All experiments in this work have been performed with a single ion.The thermal state after Doppler-cooling can be excited to a higher average thermal phonon number by controlledheating of the ion. In such a way, the Gaussian width of the distribution increases both in position and momentum.Alternatively, we can engineer a coherently-excited thermal state by applying an oscillating electrical potential at ω z onthe endcap E , with the magnitude of the coherent excitation being controlled by the amplitude of the applied voltage.After initialization, the single ion is extracted by switching on a voltage of − V to endcap E which is pierced with a µ m hole and located 1.45 mm from the trap center. Extracted by the electric field, the ion is passing through theendcap hole, further steered by several deflection electrodes and finally detected by a secondary electron multiplier, 272mm downstream. We repeat this procedure of loading, initialization and extraction, for a number of times N and buildup a time-of-flight (TOF) histogram, see Fig. 1(c). 2 PREPRINT - F
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26, 2021Figure 2: Dynamic control of the trap drive RF power: (a)
Examples of traces obtained with the oscilloscope are shown(red, blue, green). After a fast initial down-ramping, the level of remaining RF power is chosen by a proper timing ofthe RF-switching. (b)
The RF setup consists of a signal generator Keithley-3390, whose output passes a series of twoRF switches Sw(1,2), each controlled by one of the TTL outputs of a SRS-DG535 delay generator. The top RF-branchhas a fixed attenuation, while second RF-branch has zero attenuation and a relative phase shift of π . Initially using thetop branch we switch with Sw1 to the lower branch and use the destructive interference from the strong RF-branch toquickly reduce the power in the helical resonator, much faster as compared to its decay time of ∼ µ s. Near the timewhen the helical resonator is almost empty, we employ switching Sw2 to control the level of the remaining RF.In order to optimize the single ion beam performance at low extraction energy, i.e. switching on a DC voltage of U E = − V for t puls = 1600 ns, special attention has been paid to the extraction time sequence, see Fig. 1(b).Depending on the amplitude and phase of the trap RF drive fringe field component along the trap axis, the ion can eitherbe accelerated or slowed down in the vicinity of the endcap E , causing in turn a modification of the TOF distribution.In order to have full control over these effects, the extraction DC voltage trigger is synchronized, using a delay generatorwith arbitrary phase with respect to the phase of the RF drive. Also, the RF drive amplitude is switched synchronouslywith the extraction event. With a typical switch-off time of ∼
350 ns, the remaining RF amplitude has already beenreduced to the residual level and a corresponding electric fringe field is present in vicinity of the endcap when the ionreaches this position. Through a fine tuning of the temporal shaping of the RF amplitude and phase, ion wave packets atarbitrary extraction conditions can be realized. The control of the RF drive is described in Fig. 2(a-b): choosing the RFswitching phase around ϕ RF = 0 and optimizing the residual RF amplitude during extraction, we can tailor the electricforces due to the RF fringe field, when the wave packet passes the endcap E . Note, that the initial ramping-down phaseis fast as compared to the ion motion at low extraction DC voltages, such that the forces on the wave packet are indeedcontrolled by the freely adjustable residual and constant RF amplitude. We will show, that by tuning its amplitude andphase, the wave packet is either narrowed down or stretched out during the extraction event. The latter setting allowsfor an improved sensitivity of the TOF distribution on details of the previously prepared motional state of the wavepacket, i.e. the RF residual amplitude may act as a magnification tool.3 PREPRINT - F
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26, 2021Figure 3: (a)
Measured TOF histogram: The average flight time (cid:104) t tof (cid:105) and the width of the distribution ∆t tof aredepending on the initial wave packet with ω z = 2 π · kHz and the RF drive phase ϕ RF at which the ion was extractedwith U E = − V. The average is modulated by a sine-function (blue) with a frequency of Ω RF = 2 π · . MHz dueto the electrical forces from the residual RF field. The width ∆t tof follows the squared sum (red) of a technical noisecontribution ∆t tech (red dotted) and of the effects due to the RF forces ∆t mod (red dashed), see eq. 6 and text for details.Shaded regions indicate the 1- σ margin of the fits. (b-c) Examples for measured TOF distributions demonstratingthe variation of the average value and width, using different extraction phases ϕ RF . Each histogram is taken with 100single ion extractions. At phase ϕ , see (b) , the width is close to a maximum and at ϕ , see (c) , the width is close to aminimum.For a better understanding of the ion dynamics inside the trap and during the extraction process, we will revisit in thenext section some theoretical elements for driven harmonic oscillator wave packets. We will provide tools to calculateTOF distributions associated with the extraction of coherently or incoherently excited states. A thermal wave packet in a harmonic potential is a statistical mixture of the oscillator eigenstates | n (cid:105) with populationprobabilities P n , given by the distribution of axial phonons P th n = (cid:104) n (cid:105) n ( (cid:104) n (cid:105) +1) n +1 where (cid:104) n (cid:105) = 1exp (cid:16) (cid:126) ω z k B T (cid:17) − (1)4 PREPRINT - F
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26, 2021denotes the temperature dependent mean phonon number. The time-averaged momentum distribution of the ion takesthen the form of a Gaussian [30]: Ψ th ( p z ) = 1 √ π ∆p th z exp (cid:16) − p z ∆p th z ) (cid:17) , (2)with a root mean square (RMS) width that reads ∆p th z ≈ √ mk B T , assuming the weak binding approximation k B T (cid:29) (cid:126) ω z , which is a good approximation for our operation conditions. Note, that this approximation is well justifiedfor Doppler-cooled ions with axial trap frequencies ω z / π of the order of 100 kHz, which are the typical conditions inthis work. Similarly, in position space, the distribution takes a Gaussian form with a RMS width of ∆z th ≈ (cid:115) k B Tmω z . (3)After the extraction the initial momentum distribution can be mapped onto the TOF distribution. We approximate thetheoretical TOF distribution by a Gaussian probability distribution, defined by its mean value (cid:104) t tof (cid:105) and its standarddeviation ∆t tof , see Fig. 1(c). The average (cid:104) t tof (cid:105) of the TOF can be decomposed into two parts: (cid:104) t tof (cid:105) = t phys + t tech , (4)where t tech accounts for fixed technical delays (cables, switches, detector, ect.) and t phys results from the physical effectsexpected for an ion, initially at rest at the trap center and submitted to an electric extraction potential U E , followed by afree flight into the detector. In our setup, a significant axial field contribution from the residual RF drive is present infront of the endcap and adds to the extraction potential U E . The physical TOF t phys can be modeled as t phys = (cid:115) md eκU E + t mod sin (Ω RF τ trig + φ ) (cid:124) (cid:123)(cid:122) (cid:125) := ϕ RF , (5)where the first term is the physical TOF from the trap into the detector at distance d , without the contribution of the RFdrive. For the acceleration process we take the geometry factor κ into account. The second term is the contribution bythe RF drive field. The time t mod denotes the amplitude of the TOF modulation and depends on the magnitude of the RFresidual field. The behaviour of eq. 5 is experimentally very well verified by our measurements as shown in Fig. 3(a-c),where we find a sinusoidal modulation.In order to understand the wave packet-stretching effect, we will now investigate the ion velocity during the accelerationphase between the center of the trap and endcap: Before application of the extraction potential, the velocity distributionfollows a Gaussian form, outlined in Fig. 4(a). As soon the extraction potential Φ E ( z ) is applied, ions with differentinitial velocities, marked with red ( v = 0 ), green ( v > ) and blue ( v < ), are accelerated. The remaining RF fieldcauses an additional periodic potential Φ RF ( z, t ) that increases or decreases the gradient along the z axis. The overallpotential can thus be written as Φ( z, t ) = Φ E ( z ) + Φ RF ( z ) sin(Ω RF t ) . The potential Φ( z, t ) in front of endcap E isshown in Fig. 4(b-d). Here the bare extraction potential, caused by the voltage U E = − V, is shown in black andthe total potential which contains the periodic contribution is represented in grey. The strongest force acting on theion, given by the potential gradient, is at z grad (cid:39) . mm. We investigate the interaction on the ions at this positionmore closely and distinguish three cases. Fig. 4(b): The extraction is started at phase ϕ RF = 0 . Ions without an initialmomentum (red) pass through the unmodified potential at z grad . In the ion momentum distribution, events with highervelocity (green) are more accelerated by the higher gradient, which was there a short moment before, while slowervelocities (blue) are less accelerated. The initial Gaussian distribution is thereby stretched further symmetrically. Thesituation is inverted for the setting ϕ RF = π , described in Fig. 4(d). Ions initially at higher velocities of the distributionare decelerated by the RF potential, while slower ions are accelerated. Depending on the potential strengths Φ E and Φ RF , initially slower ions may overtake the faster ones and reach the detector first and cause again a wider flight timedistribution. This process is acts symmetrical, such that a Gaussian shape is conserved. Adjusting parameters, thedistribution may be focused on any position z foc . This point can be shifted to the detector position z foc = z det bylowering the RF potential. If the point is still between the trap and the detector, < z foc < z det , then there is a phase < ϕ RF < π focusing is achieved approximately on the detector, see Fig. 4(c). Under typical experimental conditions,however, the spread of the wave packet is small as compared to the RF period, such that we can approximate the TOFdistribution by a Gaussian and only in cases of extreme compression becomes asymmetric. This is due to the fact, thatthe maximum modulation is t mod .To fully model the experimental data, we include an additional broadening ∆t tech into account, independent from ϕ RF .This constant term accounts for jitter in the switches, in the delay generator triggers and also jitter in the SEM ion5 PREPRINT - F
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26, 2021Figure 4: (a)
Schematic representation of a thermal Gaussian wave packet (blue curve, not to scale) in the trap potentialbetween both endcaps E and E (black curve). Ions have different initial velocities within the momentum distribution: v = 0 (red), v > (green) and v < (blue). As soon as the extraction starts, the ions pass through differentpotentials, caused by the periodic RF potential additional to the extraction potential in the red dashed area. (b) Temporalpotential curve and final TOF distribution for ϕ RF = 0 : Initial ion with v = 0 (red) is not influenced by the RFpotential, initial ion with v > (green) experiences a higher gradient, ion with v < (blue) a lower gradient. Thisleads to a broadening of the distribution. (c) Temporal potential curve and final TOF distribution for < ϕ RF < π : Thephase is precisely chosen such that initially slower ions perceive a higher gradient. All ions reach the detector at thesame time. (d) Temporal potential curve and final TOF distribution for ϕ RF = π : Initially slower ions experience astronger gradient and overtake the initially faster ions. (e) Numerically determined Gaussian distribution for T = 3 mKfor an exponentially decreasing RF field with U RF = 10 V (blue, focus between trap and detector) and U RF = 1 V (red,focus at the detector’s position). Each simulation point performed with 300 ions.6
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26, 2021detector and counting. As a result, the width of the TOF distribution ∆t tof now reads ∆t tof = (cid:113) ∆t phys + ∆t tech . (6)The predicted periodic change of the width ∆t tof as a function of the extraction phase ϕ RF is fully confirmed by ourmeasurements, see Fig. 3(a). For these experimental runs, technical limitations set a lower limit of the TOF widths,fitted here to ∆t tech = 2 . ns, dominated by the intrinsic width of the detected ion signal. Indeed, the typical widthof a single ion voltage drop pulse in our setup is ns. Additionally, a trigger jitter of around . ns leads to furthertechnical broadening. However, we observe a phase difference between maximum average shift at π / and maximumwave packet compression of about . rad. We conjecture that this is caused by deviations of the DC extraction voltage U E from a perfect square pulse. Indeed, we observe a ramping in the order of ns, as the switch and the supply haveto load the endcap, which acts like a capacitor. This unsymmetrical DC waveform may cause additional compressingeffects in the ion acceleration.Since the width of the TOF distribution ∆t phys is proportional to the initial momentum distribution width, we modelthe TOF distribution at ϕ RF = 0 for a thermal state as ∆t thphys ∼ √ T , where T denotes the temperature and theproportionality factor depends on a combination of d , q , m , U RF and U E . It is the goal of this work to optimize the wavepacket stretching such that TOF measurements are sufficiently sensitive to reveal the temperature of an ion in the trap,see subsect. 5.1. Also, we will employ TOF measurements to determine the heating of the ion, see subsect. 5.2. To coherently excite a wave packet, we apply an electric sinusoidal drive resonant with the trapping frequency ω z / π ,coined tickling , thus generating a momentum modulation. Therefore, the timing at which the ion arrives in the vicinityof E will be modulated. The phase of the coherent oscillation φ coh and its amplitude ϕ mod can be set at arbitrary valuesby the experimental parameters of the tickling. We describe the effect as an additional oscillating phase inside themodulation of eq. 5. The overall phase is then transformed to ϕ cohRF = ϕ RF + ϕ mod sin (cid:0) φ coh (cid:1) . (7)Similarly to the previously discussed case for thermal wave packets, we find the average of the TOF distribution and itswidth modified by the coherent excitation. For a phase of ϕ RF = 0 , the linear magnification as compared to the initial t mod results in a TOF characterized by an average value (cid:104) t tof (cid:105) = (cid:115) md eκU E + t mod sin[ ϕ mod sin (cid:0) φ coh (cid:1) ] + t tech . (8)Depending on the chosen setting of the phase φ coh of the coherent amplitude, the minimum and maximum average TOFare realized by (cid:104) t max/mintof (cid:105) = (cid:115) md eκU E ± t mod sin( ϕ mod ) + t tech . (9)Again, we included the technical contribution to fully model the experimental situation. Experiments with single ionextraction in coherent states will be further explored in sect. 4. We extract coherent states to study the stretching mechanism experimentally. Herefore, we excite the single trapped ionby the sinusoidal drive resonant to the axial oscillation frequency ω z / π = 247 kHz [31, 32, 33], in our case applied toendcap E , see Fig. 1(a). Note, that before and during the excitation, the ion is exposed to Doppler-cooling. We thenmake sure that a steady state is reached. We denote this convoluted wave packet as thermal-coherent state, however,under typical experimental conditions the coherent amplitude exceeds the thermal one by far.Single ion extraction events are characterized by the residual RF amplitude and the extraction phase ϕ mod , which iscontrolled by the time delay of the DC square pulse. Additionally, the phase of the coherent drive φ coh and its amplitude U tickle can be adjusted to determine the average position and momentum of the wave packet at the extraction instant.As seen from eq. 8 we have a sine function as the argument of another sine function, therefore we could, in principle,7 PREPRINT - F
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26, 2021Figure 5: Measured TOF shift (cid:104) t tof (cid:105) ∗ for corresponding t mod of ns (red), ns (blue) and ns (green).Depending on the residual RF amplitude the resonant coherent excitation characterized by U tickle results in a TOFvariation. We fit the prediction of eq. 10. Shaded areas show 1- σ margins.have very rich structures on the TOF mapping. Hereafter, we will treat (cid:104) t tof (cid:105) as a function of one variable, the coherentexcitation phase φ coh . In the low ϕ mod regime, (cid:104) t tof (cid:105) is a small amplitude sinusoid. However for higher coherentexcitation amplitudes it turns into a sine-enveloped sine function, where the maxima and minima start to approach thecentral value after reaching ± t mod .Experimentally, we set this t mod value with the remaining RF amplitude after the partial cancellation in front of thepierced endcap E . It corresponds to our magnifying factor and can be tuned by means of a delay generator on thetrigger time τ trig when switching the RF signal off. This mechanism works by taking advantage of an extra momentumobtained by to the residual RF at the extraction moment. It is of crucial importance in order to be able to time-resolvethe relatively small in-trap momenta via TOF detection method.In order to characterize and quantify this magnification, we identify the two coherent phases φ coh at which a maximumaverage TOF and a minimum average TOF take place, (cid:104) t maxtof (cid:105) and (cid:104) t mintof (cid:105) respectively. Once these coherent phases areidentified and kept fixed, through multiple extractions we can plot the difference between these two extreme averageTOF values as a function of the coherent excitation voltage amplitude.The magnifying mechanism is therefore characterized by repeating this procedure for three different RF-cancellationsshown in Fig. 2(a). Scanning the excitation voltage from mV pp to mV pp , at a certain intermediate voltage, (cid:104) t maxtof (cid:105) and (cid:104) t mintof (cid:105) are maximally delayed, however for lower and higher coherent amplitudes this difference starts to decreaseas shown in Fig. 5. For a very high voltage of the driving coherent excitation, an inverted difference is expected. Theprediction (solid lines in Fig.) are given by (cid:104) t tof (cid:105) ∗ ≡ (cid:104) t maxtof (cid:105) − (cid:104) t mintof (cid:105) t mod · sin( ϕ mod ) . (10)All the three cases show the same periodic behaviour ϕ mod = αU tickle with α = 8 . mrad / mV pp , as expected. Thismeasurement gives us a more accurate way of determining the magnification factor of our system, in contrast to thecoarse estimate via the degree of cancellation of the RF field.8 PREPRINT - F
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26, 2021Figure 6: (a)
Ion size-based temperatures T against the applied white noise amplitude U Noise . (b) These temperaturesare deduced from the ion size on the camera picture ∆z cam at an axial trap frequency of ω z = 2 π · kHz applied fordifferent noise amplitudes. We will first characterize our TOF based thermometry technique using ions with arbitrary temperatures. In order totune the initial temperature of the ions, we apply white noise of arbitrary amplitude U Noise on the endcap E . For thisgeometry, the axial degree of freedom is addressed. Thus, we can tune the temperature associated with the axial motion,typically between about 1 mK up to 200 mK. In order to relate the applied noise amplitude with temperature, we usethe spatial thermometry technique [30, 34]. We determine the RMS width of the ion ∆z th , see Fig. 6(b), and taking intoaccount the point spread function ∆z PSF = 1 . µ m of the imaging system, we infer the temperature, see eq. 3. Thisdata are shown in Fig. 6(a), featuring the expected quadratic behaviour of the temperature with the noise amplitude.Let us now focus on the TOF measurement after extraction. After the initialization in the thermal state, the appliednoise is stopped simultaneously with the extraction event. The choice of the RF drive phase ϕ RF at extraction turns outof particular importance. In order to get a good mapping of the initial velocities on the TOF signal, we chose a RFdrive phase centered at the linear regime at ϕ RF (cid:39) of the average TOF sinusoidal modulation, see eq. 5. Second, wechose a phase corresponding to an ascending slope for maximal stretching of the initial momentum distribution, seeexplanations in sect. 3. For the residual RF amplitude, we test the effect in two different settings: at an intermediatestretching (blue curve in Fig. 5) at minimal stretching (red). As expected, the latter does not feature a measurablelarge TOF modification depending on initial temperatures. However, at the intermediate RF amplitude we observe asufficiently large stretching for temperatures up to mK. For high amplitudes, the TOF stretching acts in non-linearway in case of large initial momenta.Results are shown in Fig. 7(a-c). In order to analyse the experimental results, we fit the measured points with theexpected function ∆t tof = (cid:112) b · T + ∆t tech where the technical noise t tech had been already determinated in previousmeasurements, see Fig. 3(a). We can see a good agreement of this model with the experimental data. From the fit,we extract a TOF width dependance in the temperature of b = 24(4) ( ns ) / mK . It is remarkable that such sensitivity is9 PREPRINT - F
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26, 2021Figure 7: Mapping the momentum distribution of the thermal states. (a)
Black points: width of the TOF distribution.Blue line: fit using a square root function. (b) and (c)
Histograms of the TOF distribution at respectively low temperature T ≈ and high temperature T ≈
65 mK . From the fits (red curve) we deduce a respective TOF broadening of ∆t tof = 5 . ns and ∆t tof = 43(2) ns.sufficient to resolve inital momenta of the trapped ion, corresponding to an axial temperature of mK. As discussedbefore, the sensitivity can be increased when using a higher residual RF amplitude, if lower thermal states should beanalized from the TOF signal. The ultimate limit on the wavepacket resolution is resulting from technical noise, and fora realistic value of 1 ns, temperatures of µ K become accessible. This corresponds to sideband cooling of the ionnear to the ground state. On the other hand, this technique can be adapted for high ion temperatures, which are hard todetermine from sideband spectroscopy [35] and even by dark resonance spectroscopy [36].
A first application of the thermometry is the determination of the heating rate. For this, we keep the ion in the dark afterlaser cooling for a variable time t h . Only then, we apply the extraction and determine from the TOF distribution the iontemperature increase with t h . We use the calibration from the previous section for converting TOF distributions intotemperatures. Conviniently, we have chose parameters for a linear stretching. The heating rate of ˙ (cid:104) n (cid:105) = 1 . ph / ms is within the error margins in agreement with similar traps in our lab. We emphasize that this new temperaturemeasurement technique does not require laser spectroscopy, even no laser excitation and may be favourable e.g. to studythe motional state and the cooling rates for ions of other species or of different isotopes, which are sympatheticallycooled by Coulomb interaction from a laser-cooled ion or ion crystal. Using a single ion wave packet, we have experimentally demonstrated how to control thermal and coherent excitationof the motion inside the trap, and how a designed extraction out of the harmonic oscillator potential conserves, or evenmagnifies the characteristic traits of the wave packet. From recording a TOF distribution we unambiguously detect thecoherent and the thermal excitation. 10
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26, 2021Figure 8: (a)
Axial broadening of the TOF distribution depending of the time in which the ion is heated up. (b)
Corresponding graph, which compares the measurement of the heating time with the ion temperature and the deducedphonon number.In the future, we intend to cool the ion to near its quantum ground state and extend our methods and investigations intothe quantum regime. We aim for expanding and propagating a ground state wave packet [37] in free space and exploreits TOF distribution. Such experiments will contribute to a better understanding of quantum wave packets propagationand may be of fundamental interest, e.g. for investigating the TOF detection statistics as a novel test of quantummechanics [38], possibly also testing Bohmian predictions [39]. Potentially, one may also extract the ionic wave packetthrough a tunneling barrier which could be controlled by tightly focused laser beams that induce optical potentials andexperimentally record the time that the wave packet has spent in the barrier region [40]. Such fundamental tests ofquantum mechanics have attracted much discussion recently. The extremely well-controlled extracted single ion wavepacket may complement experiments with tunneling cold atoms [41] or laser ionization [42]. The high control oversingle extracted ions at low energies may be utilized for experiments on friction near surfaces [43] or for injecting ionsinto surface structures with strong magnetic gradients to realize the Stern Gerlach effect with charged beams [44].On the technological side, the long-distance transport of ions in free space may open alternatives for scalable ion-qubitarchitectures, where ion traps are interconnected via free space segments [13], and ion-qubit Bell pairs are distributedfor later using this resource via gate teleportation or entanglement swapping [45].
This work has been supported by the Deutsche Forschungsgemeinschaft through the DIP program (Schm 1049/7-1) andby the VW Stiftung. We thank Uli Poschinger and Daniel Pijn for careful reading and helpful comments. We rememberin this work the vital encouragement by Prof. Dr. Detlef Dürr, who passed away January 2021, much too early.
References [1] Leibfried D, Blatt R, Monroe C and Wineland D 2003
Rev. Mod. Phys. (1) 281–324[2] Wineland D J and Itano W M 1979 Phys. Rev. A (4) 1521–154011 PREPRINT - F
EBRUARY
26, 2021[3] Häffner H, Beier T, Djeki´c S, Hermanspahn N, Kluge H J, Quint W, Stahl S, Verdú J, Valenzuela T and Werth G2003
Eur. Phys. J. D Nature
Phys. Rev. Lett. (17) 173001[6] Hoffrogge J and Hommelhoff P 2011
New J. Phys. Nature Physics Phys.Rev. A (6) 063620[9] Pandey S, Mas H, Drougakis G, Thekkeppatt P, Bolpasi V, Vasilakis G, Poulios K and von Klitzing W 2019 Nature
Phys. Rev. Lett. (8) 080501[11] Bowler R, Gaebler J, Lin Y, Tan T R, Hanneke D, Jost J D, Home J P, Leibfried D and Wineland D J 2012
Phys.Rev. Lett. (8) 080502[12] Kaushal V, Lekitsch B, Stahl A, Hilder J, Pijn D, Schmiegelow C, Bermudez A, Müller M, Schmidt-Kaler F andPoschinger U 2020
AVS Quantum Science Science Advances [14] Hasselbach F 2009 Rep. Prog. Phys. Phys. Rev. Lett. (15) 3008–3011[16] Bolpasi V, Efremidis N K, Morrissey M J, Condylis P C, Sahagun D, Baker M and von Klitzing W 2014 New J.Phys. Phys. Rev. Lett. (4)582–585[18] Lett P D, Watts R N, Westbrook C I, Phillips W D, Gould P L and Metcalf H J 1988 Phys. Rev. Lett. J. Opt. B: Quantum Semiclass. Opt Phys. Rev. Applied (6) 064049[21] Zajfman D, Strasser D, Heber O, Goldberg S, Diner A and Rappaport M 2004 Nuc. Instr. and Meth. in Phys.Research Sect. B
Phys. Rev. C (1) 011306[23] Ito Y, Schury P, Wada M, Naimi S, Smorra C, Sonoda T, Mita H, Takamine A, Okada K, Ozawa A and Wollnik H2013 Nuc. Instr. and Meth. in Phys. Research Sect. B
International Journal of Mass Spectrometry
Phys. Rev. Lett.
J. Phys. Soc. Jpn. Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. Lett. (18) 183002[30] Knünz S, Herrmann M, Batteiger V, Saathoff G, Hänsch T W and Udem T 2012
Phys. Rev. A (2) 023427[31] Carruthers P and Nieto M M 1965 American Journal of Physics Phys. Rev. A (4) 3112–311612 PREPRINT - F
EBRUARY
26, 2021[33] Leibfried D, Meekhof D M, Monroe C, King B E, Itano W M and Wineland D J 1997
Journal of Modern Optics New J.Phys. Phys. Rev. Lett. (4) 403–406[36] Roßnagel J, Tolazzi K N, Schmidt-Kaler F and Singer K 2015 New J. Phys. New J. Phys. Phys. Rev. Lett.
Scientific Reports J. Phys. Photonics Nature
Phys. Rev. Lett. (2) 023201[43] Intravaia F, Oelschläger M, Reiche D, Dalvit D and Busch K 2019
Phys. Rev. Lett.
New J. Phys. Nature Physics4