A pulsed ion microscope to probe quantum gases
C. Veit, N. Zuber, O. A. Herrera-Sancho, V. S. V. Anasuri, T. Schmid, F. Meinert, R. Löw, T. Pfau
AA pulsed ion microscope to probe quantum gases
C. Veit, N. Zuber, O. A. Herrera-Sancho,
1, 2, 3, 4
V. S. V. Anasuri, T. Schmid, F. Meinert, R. Löw, and T. Pfau
5. Physikalisches Institut and Center for Integrated Quantum Science and Technology,Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany Escuela de Física, Universidad de Costa Rica, 2060 San Pedro, San José, Costa Rica Centro de Investigación en Ciencia e Ingeniería de Materiales,Universidad de Costa Rica, 2060 San Pedro, San José, Costa Rica Centro de Investigación en Ciencias Atómicas, Nucleares y Moleculares,Universidad de Costa Rica, San José, Costa Rica (Dated: August 20, 2020)The advent of the quantum gas microscope allowed for the in situ probing of ultracold gaseousmatter on an unprecedented level of spatial resolution. The study of phenomena on ever smallerlength scales as well as the probing of three-dimensional systems is, however, fundamentally limitedby the wavelength of the imaging light, for all techniques based on linear optics. Here we report on ahigh-resolution ion microscope as a versatile and powerful experimental tool to investigate quantumgases. The instrument clearly resolves atoms in an optical lattice with a spacing of 532 nm over afield of view of 50 sites and offers an extremely large depth of field on the order of at least 70 µm.With a simple model, we extract an upper limit for the achievable resolution of approximately200 nm from our data. We demonstrate a pulsed operation mode which in the future will enable 3Dimaging and allow for the study of ionic impurities and Rydberg physics.
The ability to observe natural phenomena on the sin-gle particle level has led to major breakthroughs in mod-ern physics. Prominent milestones in this context werethe pioneering experiments in cloud chambers [1], thediscovery of Rutherford scattering [2, 3], the first imag-ing of a solid surface with atomic resolution [4] and theobservation of quantum jumps for single trapped ions[5, 6]. In the realm of quantum gases, time and positionsensitive single atom detection enabled Hanbury Brown-Twiss type experiments for both bosons and fermions[7, 8] and a scanning electron microscope was used for thefirst high-resolution in situ detection of individual atoms[9, 10]. Nowadays, optical quantum gas microscopes offerthe possibility to image strongly interacting atoms loadedinto optical lattices with single-site resolution [11, 12] andhave been used to investigate Bose- and Fermi-Hubbardphysics [13–16]. More recently, ensemble-averaged imag-ing with super-resolution was demonstrated [17, 18]. Allmicroscopes based on linear optics, however, are funda-mentally limited in their resolution and depth of field bythe wavelength of the imaging light. A high-resolutionimaging method capable of extracting three-dimensionalinformation over an extended volume is therefore highlydesirable and would allow for the study of quantum cor-relations, impurity physics and transport phenomena inunprecedented ways.Here we present a high-resolution imaging systembased on ion optics, and offering a multitude of intriguingfeatures. Charged-particle optics is commonly employedin cold atom experiments in the form of momentum-spacespectrometers [19, 20], but has also been used to imageneutral ground-state atoms [9, 21, 22] and Rydberg atoms[22–26]. Our instrument is especially inspired by the ionmicroscope presented in Ref. [21], for which a spatial res-olution of smaller than 2 . Lens 1Lens 2Lens 3MCP -+t t t - + ++ FIG. 1. Concept of the ion microscope. Neutral atoms in aquantum gas (shown in a 1D periodic potential) are convertedinto ions at time t = t via a suitable ionization process (e.g.near-threshold photo-ionization). After a variable and op-tional wait time, an extraction field is pulsed on ( t = t ) andthe ions are imaged by means of an ion microscope consistingof three electro-static lenses onto a spatially resolving detec-tor ( t = t ). The high time resolution of the multi-channelplate detector (MCP) allows for the study of dynamical pro-cesses (e.g. in strongly interacting many-body systems) and3D imaging via the time-of-flight information. The versatilenature of the concept enables the study of ground-state en-sembles, Rydberg systems and ionic impurities. based on the following simple concept (Fig. 1). Neutralatoms are converted into ions and are subsequently im-aged by means of a magnification system onto a spatiallyand temporally resolving detector. This allows for theinvestigation of ground-state ensembles, Rydberg excita- a r X i v : . [ phy s i c s . a t o m - ph ] A ug tions and ionic impurities with the very same apparatus.Owing to the excellent time resolution of commerciallyavailable ion detectors [27, 28], both time-dependentmeasurements and 3D imaging via the time-of-flight in-formation can be realized [29]. The choice of the ion-ization process constitutes a powerful degree of freedom,enabling the combination of the high spatial resolutionwith spectroscopic techniques and permitting weak mea-surement schemes. Furthermore, near-threshold photo-ionization offers the possibility to produce ultracold ions[30] and opens the door to the spatially resolved studyof ionic impurities and ion-atom scattering in the quan-tum regime [31–35]. In this context, an exciting prospectis the observation of polaron formation and transportdynamics from the two-body collisional timescale to thefew- and many-body timescales. We foresee further ap-plications of our approach in the imaging of spatial or-dering in three-dimensional Rydberg-blockaded ensem-bles [36], the detection of fine spatial structures in bulkFermi gases (e.g. Friedel oscillations [37] and spatial cor-relations) and the probing of dynamic many-body pro-cesses with a high time resolution, just to name a fewexamples. In the following, we briefly describe our ionmicroscope and then focus on its performance and imag-ing characteristics. We begin by discussing a continuousoperation mode, in which the object plane is permanentlyimmersed in an extraction field, before demonstrating apulsed extraction scheme especially suited for the studyof ions and Rydberg atoms. THE ION MICROSCOPE
The design of our ion microscope is depicted in Fig.2a. The whole microscope column has a length of 135 cmand consists of three electro-static three-cylinder lenses[38] connected via field-free drift sections. In order tocompensate for mechanical tolerances, each lens is fol-lowed by a quadrupole deflector. The object plane liesbetween a repeller and an extractor electrode, the latterof which also acts as the lower lens electrode of the firstlens. To allow for the study of Rydberg atoms [39] andions, the extraction field can be pulsed on a time scale of30 ns. The observation of free ions serves as an excellentprobe for stray electric fields, which we cancel by apply-ing small compensation voltages to extractor and repelleras well as to four radially arranged field plates (see Meth-ods). A schematic of the electrode configuration is shownin Fig. 2b. The ions are detected with a multi-channelplate assembly (MCP) in combination with a delay-linedetector [28], offering a high spatial and temporal resolu-tion on the order of 100 µm and 200 ps, respectively [40].We expect the detection efficiency of our MCP to be lim-ited to a value close to the open-area ratio of ∼
70 % [41]and plan to improve this efficiency in the future by em-ploying a funnel-type MCP offering an open-area ratio of c m M = x100 to x1500V D2 V V D1 V D1 V V V E V R a b xy z FIG. 2. Ion microscope. (a) The microscope column consistsof three electro-static lenses (main electrodes marked in red,drift sections in gray) and a spatially resolving delay-line de-tector on top of an MCP stack. The object plane lies betweena repeller electrode (hosting an ITO-coated aspheric opticallens) and an extractor electrode which at the same time arepart of a six plate electric field control and render the first lensan immersion lens. Each lens is followed by an electro-staticquadrupole deflector (blue). (b) Schematic representation ofthe electrode configuration. Imaging properties and magni-fication are determined by the extraction voltages ( V E and V R ), the lens voltages ( V to V ) and the drift tube voltage( V D1 and V D2 ). The latter are equal for all measurementspresented and referred to as V D .
90 % [42]. As detailed in the Methods section, we aim fora further enhancement of the detection efficiency by ex-ploiting facilitated Rydberg excitation around ionic seeds(see Ref. [43] and references within). More informationon our instrument can be found in Ref. [29].In contrast to most conventional optical systems,electro-static lens systems offer easy tunablility of imag-ing properties via the electrode voltages. We explore theparameter space of our ion microscope by means of nu-merical trajectory simulations performed with a commer-cially available program [44] and find that our system al-lows for both 2D and 3D imaging [29]. In a first step, wefocus here on the performance of our microscope regard-ing 2D imaging. In this mode, we keep all drift tubes atthe potential of the MCP front plate ( − . V E = − V R = −
500 V.Owing to their small initial velocity, the cold ions areextracted on trajectories almost parallel to the extrac-tion field. Consequently, a sharp projection of the object
L 479 nm780 nm 6.9 μm479 nm780 nm 1064 nm532 nm a b c
FIG. 3. Tuning of the magnification. A large range of M is achievable by tuning V and V ( V E = − V R = −
500 V, V = − . V D = − . L = 100 nm (see inset)and parameters resembling the experimental situation. Shown are only data points for which the line spacing is clearly resolvedand low imaging distortion is present (see Methods for details). Most values of M are accessible by tuning only V . (b, c)Comparison between experiment and simulation along the red lines ( V = −
250 V, V = −
110 V) shown in (a). (b) Theatoms were loaded into a retro-reflected 1064 nm lattice and photo-ionized in a crossed beam configuration ( w ≈
26 µm, w ≈ M was inferred from the period of thedetected patterns. All experimental error bars (see Methods) are significantly smaller than the marker size. plane can be observed in the detector plane even for volt-age configurations for which no actual image formationoccurs. CHARACTERISTICS OF THE IMAGINGSYSTEM
In the following, we present experimental results onthe characteristics of the previously described imagingsystem. For most of the measurements, Rb atoms wereloaded into a one-dimensional optical lattice with a spac-ing of 532 nm and photo-ionized in the crossing volume oftwo laser beams with wavelengths of 780 nm and 479 nm.The 780 nm beam was blue-detuned by 78 MHz from the | P / , F = 3 i intermediate state and aligned parallel tothe optical axis of the microscope (z-direction, see Fig.2a). The 479 nm beam was aligned perpendicular to theoptical axis (y-direction) and tuned such as to realize anear-threshold ionization with typical excess energies onthe order of h ×
100 GHz or below, where h corresponds tothe Planck constant. The above described setup was usedto characterize the magnification, field of view (FOV),depth of field (DOF) and resolution of our microscope.The waists of the ionization beams were shaped to suitthe specific measurement. For magnifications too smallto resolve the optical lattice, we held the atoms in anoptical dipole trap and induced a spatial structure of the 780 nm light field by projecting the diffraction pattern ofa double slit onto the atoms. The period of the resultingionization pattern was measured at large magnificationsusing the optical lattice as a ruler. All measurementspresented are integrated over several experimental cyclesand, as detailed later, compensated for phase drifts ofthe optical lattice as well as for minor distortions of thedetected images. Additional information on the experi-ment and the data evaluation are given in the Methodssection. Magnification
To explore the exceptional tunability of the magnifica-tion M of our instrument, we fixed all electrode voltagesbut V and V (see caption of Fig. 3) and used the lattertwo to tune the focal lengths of the second and third lens.Figure 3a shows a corresponding simulation of the totalmagnification revealing that high-quality imaging resultscan be obtained for magnifications ranging from below ×
200 to above × V (Fig. 3c) or V (Fig. 3b) for which the respective other voltage wasfixed (see red lines in Fig. 3a). As delineated above, the a b cde FIG. 4. Field of view. The FOV was investigated for the volt-age configuration corresponding to the largest magnificationshown in Fig. 3b ( M = 1441). (a) Measured data (sum over1200 experimental cycles, object coordinates), corrected fora global phase Φ g (drift of the lattice) and a local phase Φ l (distortion). The marked region is shown enlarged in (b) (rawdata) and (e) (phase-corrected data). (c) Φ g over time. (d)Φ l is a function of the spatial coordinates and was extractedfrom the local deviation of the experimental data from theexpected regular lattice structure. magnification was extracted by imaging either atoms inan optical lattice, or a diffraction pattern of known pe-riod (see insets). Evidently, the simulations describe ourimaging system fairly accurate with the largest deviationfrom the experiment being smaller than 10 %. For ourdetector diameter (40 mm), the demonstrated magnifica-tion range maps to a FOV between approximately 30 µmand 300 µm. This permits both high-resolution studiesand the observation of large atomic ensembles. Field of view
In order to confirm that the large magnifications suit-able for high-resolution imaging can be made use of with-out any sacrifices concerning the FOV, we utilized ouroptical lattice as a test pattern. To this end, we em-ployed a large 780 nm beam ( w ≈
36 µm) and shapedthe 479 nm beam to a horizontal light sheet with a waistof w x, ≈
40 µm. The measurement shown in Fig. 4aillustrates that the lattice can be clearly resolved overthe whole detector area, corresponding to a FOV of 50lattice sites. The data shown is post-processed in twoways (see also Methods). First, we make use of the reg-ular structure of the lattice to compensate for a localphase Φ l caused by small distortions common to all mea-surements (Fig. 4d). Second, we compensate for a time dependent global phase Φ g caused by a thermal drift ofthe lattice over the measurement time of almost six hours(Fig. 4c). We find that the local phase Φ l stays below2 π over the whole detector and use the data shown inFig. 4d to compensate for the observed distortion in allour measurements. A comparison between the raw dataand the post-processed data is shown in Figs. 4b,e forthe marked region in Fig. 4a. Even the raw data showsa good contrast. Depth of field
Field of view and DOF of an imaging system deter-mine the maximum dimensions of the objects that canbe imaged. In the case of high-resolution optical micro-scopes, the imaging volume is typically restricted by asmall DOF originating from the large numerical apertureof the systems. In contrast, the DOF of our ion micro-scope is remarkably large as demonstrated in the follow-ing. The DOF was probed for the same voltage config-uration as used for the FOV measurement by employinga large 780 nm beam ( w ≈
36 µm) and by shaping the479 nm beam to a vertical light sheet ( w x, ≈ w z, ≈
100 µm). Due to the large extent of the lightsheet in z-direction, the height of the ionization regionwas primarily limited by the diameter of the atomic cloud(1 /e -diameter of approximately 70 µm). Careful align-ment of the lattice planes was required in order to at-tain a good imaging contrast. A resulting measurementis shown in Fig. 5a along with the normalized magni-tude of the FFT corresponding to the integrated latticeprofile (Fig. 5b). As expected from simulations, we findthat the imaging result does not suffer from the increasedDOF and is very comparable to the results obtained fromthe magnification measurements (see inset of Fig. 3b forcomparison). Given the diameter of the atomic cloud,we can quantify the DOF to be on the order of at least70 µm. In comparison with quantum gas microscopes,typically featuring a DOF smaller than a few microme-ters, the DOF or our ion optics is exceptionally large andallows for the study of three-dimensional bulk gases. Resolution
From the measurements presented above it is evidentthat, for large magnifications, the resolution of the mi-croscope is significantly smaller than the period of ourtest target given by the optical lattice. In order to stillcharacterize the resolution, we take advantage of the lo-calization of the atoms caused by the tight confinementin the optical lattice. The signature of this localization isfound in the FFT of our measured data, in which a signalat twice the lattice frequency is apparent (see e.g. Fig.5b). We use the amplitude of this second-order peak as a st nd ca b FIG. 5. Depth of field and resolution. (a) Imaging resultfor an increased depth of field limited only by the cloud size(1 /e -diameter of 70 µm). (b) Absolute value of the FFT cor-responding to the integrated lattice profile shown in (a). (c)The magnitudes of the first and second-order FFT peaks (hereas a function of V and for a tightly confined ionization region)serve as a qualitative measure of the resolution. All experi-mental error bars (see Methods) are smaller than the markersize. The insets show the experimental data for V = −
150 V.Solid line and shaded area: theoretical prediction for the up-per bound of the second-order peak amplitude for a resolutionof 200 ±
20 nm. Experimental circumstances are detailed inthe main text. qualitative measure of the resolution r of our microscopeand use a simple model to give a quantitative upper limitof r . Due to the finite detector resolution, we expect ourimaging system to perform best at large magnifications.Consequently, the voltage configuration corresponding tothe data shown in Fig. 3b was chosen for our resolu-tion studies. In order to minimize effects of residual dis-tortions, a tightly confined photo-ionization volume wasused for the measurement ( w ≈ w ≈ V . The significant amplitude of the second-order peak indicates an excellent resolution over almostthe whole measurement range. The drop of contrast atsmall magnitudes of V is in accordance with our sim-ulations and results from both imaging aberrations anda decrease of magnification. To extract an upper limitof the resolution, we assume all atoms to occupy thelowest band of the optical lattice and employ a Gaus-sian approximation for the single-site wave function (seeMethods for details). We then convolve the density pro-file with a Gaussian point spread function (PSF), scaleit according to the magnification of our microscope andbin the profile with the bin size used for our experimen- a b FIG. 6. Pulsed operation mode. (a) Imaging result for t tof = 0 µs and magnitude of the FFT corresponding to theintegrated lattice profile. The clear signature of a second-order Fourier peak indicates a resolution on the order of halfthe lattice spacing (266 nm). (b) The decay of the first-orderFFT peak serves as an indicator for the blurring of the latticestructure as a function of t tof . The timescale of the decayreveals the cold temperature of the produced ions in the fewtens of microkelvin range. Error bars correspond to a conser-vative estimate of the statistical error (see Methods). tal data (100 µm). Expectedly, the FFT of the modelprofile shows a second-order peak at twice the lattice fre-quency, enabling comparison with the experiment. Weidentify the full width at half maximum of the PSF asthe resolution r and find that, for the highest magnifi-cations, we get good agreement with our experimentaldata for r ≈
200 nm (see solid line in Fig. 5c). In con-sideration of the non-optimized loading procedure of theoptical lattice, the atoms most certainly occupy severalbands instead of only the lowest band. Since this leads toa less confined wave function (and therefore to a smalleramplitude of the second-order FFT peak), the actual res-olution of our microscope is probably significantly smallerthan 200 nm. Indeed, numerical simulations accountingfor realistic mechanical tolerances and voltage noise sug-gest that a resolution on the order of 100 nm is achievable[29]. This permits the in situ observation of phenomenataking place at the length scale of the healing length ofBose-Einstein condensates [45] or the Fermi wavelengthof ultracold Fermi gases [37].
Pulsed operation
A powerful feature of our ion microscope is the abilityto directly image ions and field-ionized Rydberg atoms.Both the study of Rydberg and ion physics, however,would be hindered by a constant extraction field. There-fore, our instrument is designed such as to allow for apulsed extraction. In the measurements presented inthe following, a fast high-voltage switch was employedto toggle between small compensation voltages and largeextraction voltages being applied to extractor and re-peller (Fig. 2). By the additional use of four radialfield plates, this procedure enables us to precisely cancelstray electric fields while the extraction field is switchedoff. The pulsed operation mode was tested for the sameoptical configuration as was used for the measurementsdiscussed in the previous paragraph and slightly differentvoltage settings resulting in a magnification of M = 1467(see Methods). In every experiment, 6000 ionization cy-cles were realized, each consisting of a 1 µs long photo-ionization pulse followed by a variable wait time t tof , afterwhich the extraction voltage was switched on. A mea-surement result for t tof = 0 µs is shown in the inset ofFig. 6a together with the corresponding FFT. As forthe data measured in the continuous operation mode,second-order Fourier peaks are clearly observable and in-dicate a resolution on the order of half the lattice spac-ing. For increasing wait time t tof , the visibility of thelattice structure decreases and a decay of the first-orderFourier peak can be observed (Fig. 6b). This approx-imately Gaussian-shaped decay with a 1 /e -timescale of2 . ∼ k B ×
13 µK,with k B being the Boltzmann constant) and the occu-pancy of several excited bands in the optical lattice. Areduction of the ionization energy in combination with anultracold atomic sample will in the future allow for thecreation of much colder ions and enable the investigationof ion-atom hybrid systems in the quantum regime. CONCLUSION AND OUTLOOK
We have presented a high-resolution ion microscopeallowing for the time-resolved probing of quantum gaseson a single atom level. The magnification of the imag-ing system was shown to be highly tunable, enabling theinvestigation of both isolated microscopic few-body pro-cesses and extended many-body systems. With a resolu-tion better than 200 nm and an exceptionally large DOFof more than 70 µm, our microscope is excellently suitedto study bosonic and fermionic bulk quantum gases onthe length scale of the healing length and the Fermi wave-length, respectively. A pulsed operation mode enablesthe spatially resolved study of ion-atom hybrid systemsand Rydberg ensembles and permits the ultra-precisemeasurement and subsequent compensation of stray elec-tric fields.We believe that charged-particle optics holds greatpromise for the field of ultracold quantum gases and willallow for a whole range of new experimental techniques aswell as an unprecedented level of precision. Key aspectsin this respect are the exceptional spatial and temporalresolution, the possibility of 3D imaging and the free-dom to combine the spatial resolution with spectroscopictechniques (e.g. enabling spin-resolved detection).As a next step, we plan to use the 3D imaging capabili- ties of our apparatus in combination with near-thresholdphoto-ionization to create and study ultracold ionic im-purities in a degenerate quantum gas [31, 33]. Using thesame experimental tools, we also aim for the observationof individual ion-atom collisions in the quantum regime[35].
ACKNOWLEDGMENTS
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Trajectory simulations.
The imaging properties ofthe ion optics were simulated by propagating test pat-terns through the electro-static potential correspondingto a given electrode voltage configuration. For the sim-ulation shown in Fig. 3a, two test patterns with a lat-eral line spacing of L = 100 nm and an extent of 10 µmalong the optical axis were employed (see inset for theshape of the patterns). Each pattern consisted of 800ions given an isotropic velocity distribution correspond-ing to an initial kinetic energy of k B ×
10 µK. One ofthe patterns was centered on the optical axis, whereasthe other one was offset by a distance corresponding to42 . Preparation of the atomic cloud.
Each experi-mental cycle started with the loading of approximately4 × Rb atoms from an effusive oven into a magneto-optical trap (MOT). For this, a double-species Zeemanslower was employed which in the future will allow usto perform experiments with Lithium [46]. The atomswere then optically transported along the x-axis (Fig. 2)from the MOT chamber into a separate science cham-ber, above which the ion microscope is located. Thiswas accomplished by the use of a transport trap consist-ing of two 1064 nm laser beams passing through a lensmounted on an air-bearing translation stage [47]. Thebeams cross under a small angle of 2 . ◦ , producing atransverse intensity profile with a 1 /e -waist of approx-imately 70 µm at the crossing point. For the measure-ments performed at low magnifications (Fig. 3c), weheld typically 2 × to 3 × atoms at a temperatureof approximately 8 µK in the transport trap. For theexperiments performed in an optical lattice, we rampeddown the power of the transport trap on a time scale of1 . /e -waist of w ≈
110 µm. After the ramping process,the atoms were solely held by the optical lattice poten-tial with a depth of ∼ E r , where the recoil energy is defined as E r = (cid:126) k / m . Here m corresponds to theatomic mass, (cid:126) is the reduced Planck constant and k isthe wave vector of the lattice light. All experiments wereperformed in unpolarized samples with the atoms beingpumped to the F = 2 hyperfine state. Depending onthe specific measurement, a complete experimental cyclelasted between 10 and 13 s. Photo-ionization sequence.
After the preparationof the atomic cloud, a photo-ionization phase followed.For the measurements performed in the continuous op-eration mode of the microscope, the ionization sequencelasted 200 ms, during which time the 780 nm laser wasswitched on continuously. Due to technical reasons, the479 nm laser was pulsed with a duty cycle of 95 % and afrequency of 50 kHz. For the measurements conducted inthe pulsed operation mode, the ionization sequence lasted1 . t tof (see main text), before the extrac-tion field was pulsed on. In order to minimize the effect ofdark counts which occur at a typical rate of a few Hertz,the detector was gated such as to register only signalscompatible with the employed ionization sequence. Thenumber of detected ions per experimental cycle (i.e. forone atomic cloud) depended on the specific measurementand ranged typically between a few tens and a few hun-dreds of ions. As an example, the measurement shown inFig. 5a consists of 18400 ion counts which were acquiredover the course of one hour. Ion detection.
The employed delay-line detector of-fers a spatial and temporal resolution on the order of100 µm and 200 ps, respectively, and features a maximumcontinuous detection rate of 1 MHz with a multi-hit deadtime on the order of 10 to 20 ns [40]. The delay-line iscombined with two stacked MCPs with an active diam-eter slightly larger than 40 mm and an open-area ratio(OAR) of 70 % (F1217-01 MOD7, Hamamatsu). Sincethe detection efficiency η of MCPs is typically limited tovalues close to the OAR [41], we plan to exchange thefront MCP with a funnel-type model offering an OAR of90 % [42]. In addition, we envision to enhance the over-all quantum efficiency to detect a single ion to higherthan 99 % using a Rydberg antiblockade effect (see Ref.[43] and references therein) which allows for the reso-nant laser excitation of Rydberg atoms on a shell aroundthe ion. Due to the Stark effect of Rydberg atoms [39],this shell can have a radius on the order of 1 µm. Asthe extraction field of the ion microscope is pulsed on,the Rydberg atoms are field ionized and the initially sin-gle ion is converted into n ions (typically n = 2 − − η ) n . Since the time scale for the Rydberg excitationcan be very fast ( ∼
10 to 100 ns), the proposed schemecan be repeated several times in a single atomic cloud.The secondary ions responsible for the described pream-plification process share on average a common center ofgravity with the initial ion. However, depending on theamplification protocol, a small position noise as well as aloss of temporal resolution is introduced. For the specificapplication, a trade-off between spatial resolution, timeresolution and quantum efficiency will have to be found.
Determination of the magnification.
For the mea-surements performed in an optical lattice (see Fig. 3b),the FFT corresponding to the integrated lattice profilewas fitted assuming a Gaussian shape of the first- andsecond order peaks evoked by the periodicity of the lat-tice. The magnification was then determined from thefitted peak positions and the known wavelength of thelattice laser λ lat = (1063 . ± .
01) nm. Error bars weredetermined by considering the conservative estimate ofthe systematic error of λ lat and the standard error ofthe fitted peak positions. For the characterization ofsmall magnifications for which the lattice could not beresolved (see Fig. 3c), a spatial structure was inducedinto the 780 nm photo-ionization beam by passing thebeam through a double slit (slit separation and width of3 mm and 1 mm, respectively), before focusing the beamonto the atoms. To this end, an in-vacuum asphericlens with an effective focal length of 26 mm was used.The described setup resulted in an image of the far-fielddiffraction pattern of the double slit in the focal planeof the lens. The period of the diffraction pattern λ diff was measured at a known magnification previously cali-brated with the optical lattice. In comparison with themeasurements performed in an optical lattice, the 479 nmbeam was enlarged in x-direction to w x ≈
100 µm andthe 780 nm beam measured approximately w x ≈ λ diff andthe standard error of the fitted pattern period. All errorbars in Fig. 3 are significantly smaller than the markersize. Data processing.
The employed detector delivers atime stamp and a position for each detected ion. Wecompensated for minor distortions of the detector imagein y-direction (Fig. 2) by applying a spatially dependentshift to the detected y-position of all ions. The spatialdistribution of this shift was found by using the regu-lar structure of the optical lattice as a test pattern (Fig.4). The local distortion could be interpolated from thediscrepancy between the observed lattice phase and thephase expected for an ideal lattice structure (Fig. 4d).We have checked that the influence of wave-front aberra-tions of the lattice beams is small by shifting the imagealong the x-direction of the detector using the electro-static deflector after the third lens (Fig. 2). To compen-sate for thermal drifts of the optical lattice, we sorted thedetection events into groups of 100 ions and calculatedthe FFT corresponding to the integrated lattice profilefor each of the groups. We then extracted the global phase of the lattice from the complex amplitude of thefirst-order FFT peak and corrected for the drift accord-ingly. Subsequently, the data was binned with a bin sizeof 100 µm. For measurements for which the magnitudesof the FFT peaks were extracted, we determined a sta-tistical error of these by considering again groups of 100ions. For each of the groups, the complex amplitudes ofthe first- and second-order FFT peaks were determined.An estimate for the error of the peak magnitudes wasthen found by dividing the standard deviation of the am-plitudes by √ n g , where n g is the number of groups. Pulsed operation mode.
The measurements shownin Fig. 6 were acquired using the following electrode volt-ages: [ V R , V E , V , V , V , V D ] = [400, − − − − − . For the pulsed measurements shown inFig. 6b, the magnitude of the first-order FFT peak de-creased with increasing t tof . In order to still accuratelycompensate for the thermal drift of the lattice, a mea-surement for t tof = 0 µs was performed in every secondexperiment cycle. The thermal drift of the lattice wasthen deduced only from these additional measurementsand corrected for according to the procedure detailed inthe previous paragraph. Approximation of the single-site wave function.
For atoms occupying the lowest energy band of our deepoptical lattice, we approximate the single-site wave func-tion by a ground-state harmonic oscillator wave function.To this end, we consider the Taylor expansion of the lat-tice potential around an energy minimum V ( x ) ≈ V k x and find the probability density of the lowest harmonicoscillator state to be | ψ ( x ) | = (2 mV k ) / ( π (cid:126) ) / exp (cid:18) − √ mV k (cid:126) x (cid:19) . (1)Here V is the depth of the lattice potential, m is theatomic mass and k is the wave vector of the 1064 nmlattice light. Compensation of stray electric fields.
For thepresented measurements performed in the pulsed opera-tion mode, stray electric fields within the ionization vol-ume were compensated by applying suitable voltages tothe six compensation electrodes. In order to calibratethese voltages on the order of a few tens of millivolts,free ions were observed for up to 70 µs. The compensationvoltages were adjusted such as to minimize the displace-ment of the detected ion distribution as the observationtime was increased. For the field component along theoptical axis, the time-of-flight information from the iondetection was employed. From the observed spread ofthe ion cloud, the typical magnitude of residual fields inthe transverse direction can be quantified to be on theorder of 100 µV //