Optical clocks based on linear ion chains with high stability and accuracy

Optical clocks based on linear ion chains with high stability and accuracy

J. Keller Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany    D. Kalincev Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany    T. Burgermeister Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany    A. P. Kulosa Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany    A. Didier Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany    T. Nordmann Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany    J. Kiethe Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany    T. E. Mehlstäubler Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany
July 12, 2019

Trapped-ion optical clocks are capable of achieving systematic fractional frequency uncertainties of and possibly below. However, the stability of current ion clocks is fundamentally limited by the weak signal of single-ion interrogation. We present an operational, scalable platform for extending clock spectroscopy to arrays of Coulomb crystals consisting of several tens of ions, while allowing systematic shifts as low as . We observe 3D excess micromotion amplitudes inside a Coulomb crystal with atomic resolution and prove that the related frequency shifts of individual ions can be controlled simultaneously at the level. Using this method, we demonstrate regions of and length where Coulomb crystals can be trapped with time dilation shifts due to micromotion close to and below , respectively. By further characterizing the trapping environment and calculating cooling dynamics for mixed In/Yb crystals, we show that achievable clock uncertainties due to multi-ion operation in our setup are below . This work opens up the path to precision spectroscopy in many-body ion systems, enabling entanglement-enhanced ion clocks and providing a well-controlled, strongly coupled quantum system.

37.10.Ty, 06.30.Ft

Optical atomic clocks Ludlow et al. (2015) are currently the most accurate man-made devices, capable of providing frequency references with fractional uncertainties approaching one part in . The two most successful approaches thus far are ensembles of to neutral atoms stored in an optical lattice (e.g. Beloy et al. (2014); Lisdat et al. (2016); Nemitz et al. (2016); Campbell et al. (2017)), and single trapped ions (e.g. Chou et al. (2010); Dubé et al. (2014); Huntemann et al. (2016)). The latter benefit from the ability to be strongly confined due to their charge, which, to first order, does not affect the electronic energy structure (by definition, the time-averaged electric field vanishes at the equilibrium ion position). The resulting strong localization and excellent control over the trapping environment for single ions are an ideal premise for high-accuracy spectroscopy. Besides being excellent candidates for optical clocks, ions also provide transitions with high sensitivities for tests of general relativity Pruttivarasin et al. (2015); Dzuba et al. (2016), or the search for physics beyond the standard model Rosenband et al. (2008); Huntemann et al. (2014); Godun et al. (2014). Stable and reproducible optical clocks also give rise to novel applications like chronometric leveling, which uses relativistic time dilation to determine differences in gravitational potential Bjerhammar (1985); Vermeer (1983). With fractional frequency resolutions of , corresponding to of height difference on Earth’s surface, optical clocks become competitive sensors for geodesy Lion et al. (2017); Müller et al. (2018); Mehlstäubler et al. (2018).

Current single-ion clocks are fundamentally limited by their statistical uncertainty due to the low signal-to-noise ratio of a single quantum absorber Itano et al. (1993). The corresponding instabilities of a few necessitate measurement times of tens of days before resolving the atomic transition frequency to within . These long integration times complicate comparisons between clocks and significantly impede practical applications. So far, it has been unclear whether precision spectroscopy can be extended to many ions without compromising accuracy. Here, we demonstrate the ability to scale the high level of control from a single ion to spatially extended and strongly coupled many-body systems and argue, based on experimental observations, that corresponding systematic uncertainties as low as can be reached. Interrogating ions in parallel will reach a given instability times faster Herschbach et al. (2012). By enabling multi-ion clock operation Herschbach et al. (2012); Arnold et al. (2015), our platform also allows the implementation of proposed novel clock schemes, such as cascaded clock interrogation of separate ensembles Rosenband and Leibrandt (2013); Borregaard and Sørensen (2013) or exploiting quantum correlations Leroux et al. (2010); Kessler et al. (2014); Lebedev et al. (2014). The unperturbed storage of extended multi-ion ensembles demonstrated here is also a key requirement for scaling ion based quantum simulation Schneider et al. (2012); Blatt and Roos (2012); Hess et al. (2017) and quantum information processing Blatt and Wineland (2008) systems to sizes and computation times at which they outperform classical computations.

In this work, we demonstrate the operation of a new scalable platform for precision spectroscopy and resolve the open question whether micromotion within an ion chain can be controlled sufficiently well to allow fractional frequency uncertainties at the level or below. Our measurements show that by exploiting symmetries in trap design, micromotion, together with all further systematic frequency shifts related to multi-ion operation, contributes less than to the fractional uncertainty of multi-ion clocks based on In ions. Our approach supports ion numbers on the order of at this level of accuracy and, besides In Becker et al. (2001); Ohtsubo et al. (2017), can also benefit clocks based on other ion species providing transitions with low sensitivity to electric field gradients, such as Al Beloy et al. (2017), Yb (SF) Huntemann et al. (2016), Lu Barrett (2015) or Th Peik and Tamm (2003).

Figure 1: Precision scalable ion trap array. (a) Electrode geometry: the trap consists of one -long and seven -long segments. (b) Trap assembly from a stack of four wafers. (c) Photograph of the trap, showing the onboard low-pass filters and one thermistor. (d) Large Coulomb crystal (here: fluorescing Yb ions), which can serve as a high-stability frequency reference. (e) Sketch of the crystal configuration considered in the uncertainty budget of Table 1. It corresponds to the lower of the three displayed crystal examples of linear In/Yb crystals. The spatial extension is () for ().

Our platform consists of a linear rf trap array (shown in Fig 1a-c), developed for precision spectroscopy on separate ion ensembles. In this type of trap, large Coulomb crystals can be stored to serve as high-stability frequency references, while short chains on the order of ten ions allow clock operation with high accuracy (see Fig. 1d-e). Separation into relatively small linear crystals limits the complexity of the motional spectrum and supports internal state readout with single-ion resolution. Four laser-cut and structured AlN wafers with gold electrodes form the trap geometry. The semi-monolithic design ensures scalability and symmetrically shaped electrodes with manufacturing tolerances below . Integrated Pt100 sensors allow for in-situ temperature measurements during clock operation. Further trap features include 3D optical access, dedicated electrodes for 3D stray field compensation in each segment, and integrated RC filters close to each electrode to prevent pickup electronic noise from exciting eigenmodes of the Coulomb crystals. The electrical design has previously been tested in a simple prototype trap Pyka et al. (2014).

The strongest concern for precision spectroscopy in extended ion many-body systems is the time dilation due to fast motion of individual ions, driven by the confining rf field at (excess micromotion, EMM): while a single ion can be placed at the point of lowest (ideally, vanishing) rf field amplitude by the application of static electric fields (see, e.g. Huber et al. (2014); Keller et al. (2015); Gloger et al. (2015); Meir et al. (2017)), doing so with a chain of ions relies on the existence of a nodal line in the trapping field. Axial rf field components prevent the formation of such a line when the translation symmetry of the trap is broken by its finite length, segmentation, and possible manufacturing imperfections. Secondly, the line of minimal rf amplitude is not necessarily identical to the line of minimal radial potential and it is not clear that the available compensation fields can correct for this over the whole extent of the crystal. Finally, collective effects exist within the crystal, in which ions are affected by the fluctuating Coulomb repulsion from neighboring ions undergoing micromotion Landa et al. (2012); Arnold et al. (2015). We show here that this last effect scales as in linear ion chains at an axial trapping frequency and is negligible for our parameters Sup (). Due to the storage of all ions at positions of minimal rf amplitude, our concept is not restricted to atomic species that allow for a cancellation of micromotion induced AC Stark and time dilation shifts Madej et al. (2012). It also avoids ion heating by an rf enabled coupling of noise at frequencies of to motional modes at Blakestad et al. (2009).

Figure 2: Excess micromotion in Coulomb crystals. (a) Axial rf field component in two trap segments (of and length, respectively), measured by moving a single ion along the trap axis. The shaded areas mark amplitudes corresponding to time dilation shifts below and for In, respectively. The confining rf voltage has an amplitude of and frequency . (b) Setup for stroboscopic micromotion measurements across Coulomb crystals with atomic resolution (see Sup () for details). (c) Visualization of 3D micromotion amplitudes of individual ions inside a Coulomb crystal. The motional amplitudes are scaled by a factor of with respect to the ion distances. (d) Corresponding time dilation shifts, assuming the mass of In. (e) Axial rf fields experienced by each ion in two separate a 14-ion chains, measured at two different crystal positions in the segment, compared to single-ion PMT measurements. Integration times were () per observed phase and laser direction for crystal 1 (2). “Crystal 1” is the same data as shown in (c) and (d).

Using a photomultiplier tube (PMT) and the photon-correlation technique Berkeland et al. (1998), we have previously demonstrated the capability of quantifying the frequency shifts resulting from EMM with uncertainties below for a single ion Keller et al. (2015). Here, using this technique, we first benchmark the new trap array by measuring the motion (mean velocity ) of a single Yb ion at different positions along the trap axis, as shown in Fig. 2a. The observed rf electric field amplitudes are below () over an entire -long (-long) segment, corresponding to fractional time dilation shifts below () for In. In these relations, denotes the ion mass, the speed of light and the elementary charge. The change in time dilation shift over a characteristic ion distance of is below everywhere in these segments, three orders of magnitude smaller than has previously been reported Chou et al. (2010). The resulting axial range with shifts below allows for the storage of more than ions in each segment.

The question remains whether a single-ion measurement can be extrapolated to predict the rf driven motion of multiple simultaneously trapped ions. We have therefore developed a method to simultaneously observe EMM in individual ions within a Coulomb crystal. The experimental setup is shown in Fig. 2b: ion fluorescence is imaged onto an image intensifier, the cathode of which is gated at the rf drive frequency of the trap for a fraction (here, ) of the rf period. The resulting stroboscopic image is observed on an EMCCD camera, and accumulated photon counts within a region of interest constitute the signal for each ion. By varying the phase of the gating pulses with respect to the trap voltage, we obtain a sinusoidal signal, which is evaluated according to the model in Keller et al. (2015). 3D information about the ion velocities is obtained by switching the direction of the illuminating laser. Further information on the experimental setup and data evaluation can be found in Sup (). Details on the electronic circuit for generating the short high voltage pulses at a repetition rate of and the signal calibration are published elsewhere Beev et al. (2017).

Figure 2c shows the 3D micromotion amplitudes observed with this method in a crystal consisting of 14 Yb ions, confined at . In this figure, the motional amplitudes of the ions have been exaggerated by a factor of with respect to the ion distances, as the largest value is . The signals were averaged over for each of the four phases and three laser directions, and all resulting amplitude uncertainties are below . The time dilation shifts due to the mean velocities of for In ions are shown in Fig. 2d. Those due to axial micromotion are close to for crystals up to in length, which, depending on the axial confinement, corresponds to ca.  ions (at ). The radial components are caused by a drift of stray fields during the measurement time of more than 9 hours for that crystal. As they are uniform across the crystal, more careful suppression below is possible during clock operation, e.g. by imaging the fluorescence of a single ion onto a dedicated PMT during the Doppler cooling phases of the clock cycle Keller et al. (2015). A comparison of the new method with single-ion PMT measurements, shown in Fig. 2e for two different crystal positions, shows matching results. This proves that in the chosen symmetric electrode geometry, the observation of a single ion is sufficient to predict the EMM amplitudes across a full Coulomb crystal at this level of accuracy.

narrow line cooled sympathetically cooled
values in max. shift spread uncertainty max. shift spread uncertainty comment
Time dilation (thermal) assuming
Heating (per second)
Time dilation (EMM)
AC Stark (thermal MM)
AC Stark (EMM)
El. quadrupole shift without P Beloy et al. (2017)
BBR at without 111The contribution from can be experimentally reduced to below Keller et al. (2018)
Table 1: Multi-ion operation related uncertainty contributions for linear ion chains as shown in Fig. 1e, with trap parameters , () and (), respectively. All values are given as fractional frequency deviations in units of . “Max. shift” corresponds to the most affected ion, “spread” denotes the difference between the most and least affected ion.

In the following we show that all further relevant uncertainty contributions related to multi-ion operation can be controlled at the level, for the example of an In clock. Our findings, listed in Table 1 and summarized below, are discussed in more detail in Keller et al. (2018).

Additional time dilation shifts arise from thermal motion of the laser cooled ions and motional heating during the dark clock interrogation time Brownnutt et al. (2015). The heating rates due to fluctuating charges on the trap electrodes determine the achievable equilibrium temperatures of the crystal eigenmodes. They are of particular importance for multi-ion clock operation, where the phonon distribution at each clock ion needs to be determined precisely. We observe a dependence of electric field noise Keller et al. (2018), with a heating rate of less than 1 phonon per second for Yb ions trapped at , which is low in comparison with other ion traps of similar size Brownnutt et al. (2015). Based on these heating rate measurements, we calculate the cooling dynamics of In/Yb Coulomb crystal configurations and determine the attainable frequency uncertainties. We consider two scenarios: sympathetic Doppler cooling using Yb ions alone or followed by a second cooling stage on the narrow SP intercombination line in In. The heavier Yb ions are placed in the center of the crystal in both cases (see Fig. 1e). For sympathetic cooling on Yb, a trap aspect ratio close to the phase transition from a linear chain to a two-dimensional crystal provides strong Coulomb coupling between the ions. The trap frequencies are in the radial and in the axial direction, yielding a chain length of for the crystal shown in Fig. 1e. Our calculations show that in the sympathetically cooled case, all motional modes reach temperatures below Keller et al. (2018), with a corresponding time dilation shift of for the most affected ion. The phonon distributions in the individual modes can be determined from coherent sideband excitation Chen et al. (2017). Second-stage cooling on the narrow intercombination line in In allows for lower temperatures and a weaker axial confinement, which also reduces other systematic frequency shifts, such as thermally induced micromotion Wübbena et al. (2012); Keller et al. (2018). With trap frequencies of and , the chain length is for the 13 ions. Temperatures of have been demonstrated for direct cooling of In Peik et al. (1999). For both scenarios, we assume temperature uncertainty for all modes Poulsen and Drewsen (2012).

The electric quadrupole shift is of particular importance in spectroscopy with Coulomb crystals, where large electric field gradients can be caused by neighboring ions. A clock transition as in In is favorable due to its low differential electric quadrupole moment Beloy et al. (2017). The resulting frequency shifts are () for the innermost In ions in the sympathetically (narrow line) cooled configuration if the quantization field is chosen parallel to the trap axis Keller et al. (2018).

We determine the black-body radiation (BBR) environment of the ions during trap operation via in-situ trap temperature measurements using integrated Pt100 thermistors, which are calibrated to within . An rf voltage amplitude of () at , resulting in a confinement of (), leads to a mean temperature increase of () Keller et al. (2018). According to our thermal model of the trap Doležal et al. (2015), this corresponds to an effective temperature increase of () at the ion position, with an overall uncertainty of (0.08 ), which gives rise to a () fractional frequency uncertainty. To avoid an increased uncertainty due to contributions from the vacuum chamber, its temperature also needs to be stabilized at this level. Presently, the uncertainty in the theoretical value of the differential static polarizability contributes Safronova et al. (2011), which is the largest contribution to the uncertainty budget. In the future, this can be reduced to below by measurements of laser-induced light shifts Keller et al. (2018).

Magnetic field deviations over the trapping region are required to be small for operation with multiple ion ensembles. With typical magnetic shielding and accounting for manufacturing tolerances of our magnetic field coils, the maximal B field variation across a crystal (the whole trap array) is () Keller et al. (2018). The corresponding spread of first-order Zeeman shifts on the transition of 6.5  () is well below the natural linewidth of the transition. Single-ion resolution allows the clock servo to account for known inhomogeneities, which can be pre-determined using a more sensitive transition (e.g. in Yb).

Collisions with the background gas can shift electronic potentials and introduce a phase error in clock interrogation. For a known chemical composition, the magnitude of this shift can be estimated from molecular potentials Vutha et al. (2017). To obtain a worst-case estimate without any assumptions regarding the species involved in the collisions, we extend the treatment in Rosenband et al. (2008) to Ramsey spectroscopy and assume a maximally detrimental phase shift of in each collision Keller et al. (2018). From an experimentally observed collision rate of per ion, we obtain an upper bound of for the absolute shift in the considered 10-In ion crystals Keller et al. (2018). The actual value and uncertainty require further experimental investigation, but a reduction below seems achievable.

Lastly, we consider the Debye-Waller effect Wineland et al. (1998), as it can reduce clock stability in the presence of highly excited or low-frequency modes. Relative Rabi frequency fluctuations of less than () are expected for the sympathetically (narrow line) cooled case, which result in instability limits of () when using a Ramsey interrogation scheme Keller et al. (2018).

In summary, we have demonstrated an operational scalable platform for precision spectroscopy on ion ensembles with fractional frequency uncertainties at the level, and resolved the question whether time dilation shifts due to rf-driven motion can be controlled in such extended many-body systems. Our findings open up the possibility to overcome the fundamental statistical uncertainty limitation of single-ion optical clocks and implement novel clock schemes, while allowing systematic uncertainties at this level. The concept can be applied to any clock transition with low differential electric quadrupole moment, enabling multi-ion clocks based on, e.g. Al Schulte et al. (2016), Lu Barrett (2015) or Th Peik and Tamm (2003), and possibly even further species where insensitive transitions can be designed using collective states Roos et al. (2006). In Yb, the quadrupole moment of the F state Huntemann et al. (2012) is low enough to yield absolute shifts below for arrays of 10-ion chains confined with , allowing electric quadrupole shift uncertainties below as well. In the absence of a narrow cooling line, lower frequency uncertainties can be achieved by relaxing the axial confinement for interrogation or employing advanced cooling methods Lechner et al. (2016); Ejtemaee and Haljan (2017); Scharnhorst et al. (2017).

The authors thank M. Drewsen for helpful discussions on intensifier gating electronics, N. Beev for developing the circuit, the PTB departments 5.3 and 5.5 for the collaboration on trap fabrication, and S. A. King for helpful comments on the manuscript. This work was supported by DFG through grant ME3648/1-1 and SFB 1227 (DQ-mat), project B03.


  • Ludlow et al. (2015) A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik,  and P. O. Schmidt, Rev. Mod. Phys 87, 637 (2015).
  • Beloy et al. (2014) K. Beloy, N. Hinkley, N. B. Phillips, J. A. Sherman, M. Schioppo, J. Lehman, A. Feldman, L. M. Hanssen, C. W. Oates,  and A. D. Ludlow, Phys. Rev. Lett. 113, 260801 (2014).
  • Lisdat et al. (2016) C. Lisdat, G. Grosche, N. Quintin, C. Shi, S. M. F. Raupach, C. Grebing, D. Nicolodi, F. Stefani, A. Al-Masoudi, S. Dörscher, S. Häfner, J.-L. Robyr, N. Chiodo, S. Bilicki, E. Bookjans, A. Koczwara, S. Koke, A. Kuhl, F. Wiotte, F. Meynadier, E. Camisard, M. Abgrall, M. Lours, T. Legero, H. Schnatz, U. Sterr, H. Denker, C. Chardonnet, Y. Le Coq, G. Santarelli, A. Amy-Klein, R. Le Targat, J. Lodewyck, O. Lopez,  and P.-E. Pottie, Nat. Commun. 7, 12443 (2016).
  • Nemitz et al. (2016) N. Nemitz, T. Ohkubo, M. Takamoto, I. Ushijima, M. Das, N. Ohmae,  and H. Katori, Nat. Photonics 10, 258 (2016).
  • Campbell et al. (2017) S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. Darkwah Oppong, R. L. McNally, L. Sonderhouse, J. M. Robinson, W. Zhang, B. J. Bloom,  and J. Ye, Science 358, 90 (2017).
  • Chou et al. (2010) C. W. Chou, D. B. Hume, J. C. J. Koelemeij, D. J. Wineland,  and T. Rosenband, Phys. Rev. Lett. 104, 070802 (2010).
  • Dubé et al. (2014) P. Dubé, A. A. Madej, M. Tibbo,  and J. E. Bernard, Phys. Rev. Lett. 112, 173002 (2014).
  • Huntemann et al. (2016) N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm,  and E. Peik, Phys. Rev. Lett. 116, 063001 (2016).
  • Pruttivarasin et al. (2015) T. Pruttivarasin, M. Ramm, S. G. Porsev, I. I. Tupitsyn, M. S. Safronova, M. A. Hohensee,  and H. Häffner, Nature 517, 592 (2015).
  • Dzuba et al. (2016) V. A. Dzuba, V. V. Flambaum, M. S. Safronova, S. G. Porsev, T. Pruttivarasin, M. A. Hohensee,  and H. Häffner, Nat. Phys. 12, 465 (2016).
  • Rosenband et al. (2008) T. Rosenband, D. B. Hume, P. O. Schmidt, C. W. Chou, A. Brusch, L. Lorini, W. H. Oskay, R. E. Drullinger, T. M. Fortier, J. E. Stalnaker, S. A. Diddams, W. C. Swann, N. R. Newbury, W. M. Itano, D. J. Wineland,  and J. C. Bergquist, Science 319, 1808 (2008).
  • Huntemann et al. (2014) N. Huntemann, B. Lipphardt, C. Tamm, V. Gerginov, S. Weyers,  and E. Peik, Phys. Rev. Lett. 113, 210802 (2014).
  • Godun et al. (2014) R. M. Godun, P. B. R. Nisbet-Jones, J. M. Jones, S. A. King, L. A. M. Johnson, H. S. Margolis, K. Szymaniec, S. N. Lea, K. Bongs,  and P. Gill, Phys. Rev. Lett. 113, 210801 (2014).
  • Bjerhammar (1985) A. Bjerhammar, B. Geod. 59, 207 (1985).
  • Vermeer (1983) M. Vermeer, Report of the Finnish Geodetic Institute 83, 1 (1983).
  • Lion et al. (2017) G. Lion, I. Panet, P. Wolf, C. Guerlin, S. Bize,  and P. Delva, J. Geod. 91, 597 (2017).
  • Müller et al. (2018) J. Müller, D. Dirkx, S. M. Kopeikin, G. Lion, I. Panet, G. Petit,  and P. N. A. M. Visser, Space Sci. Rev. 214, 5 (2018).
  • Mehlstäubler et al. (2018) T. E. Mehlstäubler, G. Grosche, C. Lisdat, P. O. Schmidt,  and H. Denker, to appear in Rep. Prog. Phys.  (2018).
  • Itano et al. (1993) W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L. Moore, M. G. Raizen,  and D. J. Wineland, Phys. Rev. A 47, 3554 (1993).
  • Herschbach et al. (2012) N. Herschbach, K. Pyka, J. Keller,  and T. E. Mehlstäubler, Appl. Phys. B 107, 891 (2012).
  • Arnold et al. (2015) K. Arnold, E. Hajiyev, E. Paez, C. H. Lee, M. D. Barrett,  and J. Bollinger, Phys. Rev. A 92, 032108 (2015).
  • Rosenband and Leibrandt (2013) T. Rosenband and D. R. Leibrandt, ArXiv e-prints  (2013), arXiv:1303.6357 .
  • Borregaard and Sørensen (2013) J. Borregaard and A. S. Sørensen, Phys. Rev. Lett. 111, 090802 (2013).
  • Leroux et al. (2010) I. D. Leroux, M. H. Schleier-Smith,  and V. Vuletić, Phys. Rev. Lett. 104, 250801 (2010).
  • Kessler et al. (2014) E. M. Kessler, P. Kómár, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye,  and M. D. Lukin, Phys. Rev. Lett. 112, 190403 (2014).
  • Lebedev et al. (2014) A. V. Lebedev, P. Treutlein,  and G. Blatter, Phys. Rev. A 89, 012118 (2014).
  • Schneider et al. (2012) C. Schneider, D. Porras,  and T. Schaetz, Rep. Prog. Phys. 75, 024401 (2012).
  • Blatt and Roos (2012) R. Blatt and C. F. Roos, Nat. Phys. 8, 277 (2012).
  • Hess et al. (2017) P. W. Hess, P. Becker, H. B. Kaplan, A. Kyprianidis, A. C. Lee, B. Neyenhuis, G. Pagano, P. Richerme, C. Senko, J. Smith, W. L. Tan, J. Zhang,  and C. Monroe, Phil. Trans. R. Soc. A 375, 20170107 (2017).
  • Blatt and Wineland (2008) R. Blatt and D. J. Wineland, Nature 453, 1008 (2008).
  • Becker et al. (2001) T. Becker, J. V. Zanthier, A. Y. Nevsky, C. Schwedes, M. N. Skvortsov, H. Walther,  and E. Peik, Phys. Rev. A 63, 051802 (2001).
  • Ohtsubo et al. (2017) N. Ohtsubo, Y. Li, K. Matsubara, T. Ido,  and K. Hayasaka, Opt. Express 25, 11725 (2017).
  • Beloy et al. (2017) K. Beloy, D. R. Leibrandt,  and W. M. Itano, Phys. Rev. A 95, 043405 (2017).
  • Barrett (2015) M. D. Barrett, New. J. Phys. 17, 053024 (2015).
  • Peik and Tamm (2003) E. Peik and C. Tamm, EPL 61, 181 (2003).
  • Pyka et al. (2014) K. Pyka, N. Herschbach, J. Keller,  and T. E. Mehlstäubler, Appl. Phys. B 114, 231 (2014).
  • Huber et al. (2014) T. Huber, A. Lambrecht, J. Schmidt, L. Karpa,  and T. Schaetz, Nat. Commun. 5, 5587 (2014).
  • Keller et al. (2015) J. Keller, H. L. Partner, T. Burgermeister,  and T. E. Mehlstäubler, J. Appl. Phys. 118, 104501 (2015).
  • Gloger et al. (2015) T. F. Gloger, P. Kaufmann, D. Kaufmann, M. Tanveer Baig, T. Collath, M. Johanning,  and C. Wunderlich, Phys. Rev. A 92, 043421 (2015).
  • Meir et al. (2017) Z. Meir, T. Sikorsky, R. Ben-shlomi, N. Akerman, M. Pinkas, Y. Dallal,  and R. Ozeri, J. Mod. Opt. 65, 387 (2017).
  • Landa et al. (2012) H. Landa, M. Drewsen, B. Reznik,  and A. Retzker, New. J. Phys. 14, 093023 (2012).
  • (42) See Supplemental Material below for details on the experimental setup and evaluation.
  • Madej et al. (2012) A. A. Madej, P. Dubé, Z. Zhou, J. E. Bernard,  and M. Gertsvolf, Phys. Rev. Lett. 109, 203002 (2012).
  • Blakestad et al. (2009) R. B. Blakestad, C. Ospelkaus, A. P. VanDevender, J. M. Amini, J. Britton, D. Leibfried,  and D. J. Wineland, Phys. Rev. Lett. 102, 153002 (2009).
  • Berkeland et al. (1998) D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano,  and D. J. Wineland, J. Appl. Phys. 83, 5025 (1998).
  • Beev et al. (2017) N. Beev, J. Keller,  and T. E. Mehlstäubler, Rev. Sci. Instrum. 88, 126105 (2017).
  • Keller et al. (2018) J. Keller, T. Burgermeister, D. Kalincev, A. Didier, A. P. Kulosa, T. Nordmann, J. Kiethe,  and T. E. Mehlstäubler, ArXiv e-prints  (2018), arXiv:1803.08248 .
  • Brownnutt et al. (2015) M. Brownnutt, M. Kumph, P. Rabl,  and R. Blatt, Rev. Mod. Phys. 87, 1419 (2015).
  • Chen et al. (2017) J.-S. Chen, S. M. Brewer, C. W. Chou, D. J. Wineland, D. R. Leibrandt,  and D. B. Hume, Phys. Rev. Lett. 118, 053002 (2017).
  • Wübbena et al. (2012) J. B. Wübbena, S. Amairi, O. Mandel,  and P. O. Schmidt, Phys. Rev. A 85, 043412 (2012).
  • Peik et al. (1999) E. Peik, J. Abel, T. Becker, J. von Zanthier,  and H. Walther, Phys. Rev. A 60, 439 (1999).
  • Poulsen and Drewsen (2012) G. Poulsen and M. Drewsen, ArXiv e-prints  (2012), arXiv:1210.4309 .
  • Doležal et al. (2015) M. Doležal, P. Balling, P. B. R. Nisbet-Jones, S. A. King, J. M. Jones, H. A. Klein, P. Gill, T. Lindvall, A. E. Wallin, M. Merimaa, C. Tamm, C. Sanner, N. Huntemann, N. Scharnhorst, I. D. Leroux, P. O. Schmidt, T. Burgermeister, T. E. Mehlstäubler,  and E. Peik, Metrologia 52, 842 (2015).
  • Safronova et al. (2011) M. S. Safronova, M. G. Kozlov,  and C. W. Clark, Phys. Rev. Lett. 107, 143006 (2011).
  • Vutha et al. (2017) A. C. Vutha, T. Kirchner,  and P. Dubé, Phys. Rev. A 96, 022704 (2017).
  • Wineland et al. (1998) D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King,  and D. M. Meekhof, J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998).
  • Schulte et al. (2016) M. Schulte, N. Lörch, I. D. Leroux, P. O. Schmidt,  and K. Hammerer, Phys. Rev. Lett. 116, 013002 (2016).
  • Roos et al. (2006) C. F. Roos, M. Chwalla, K. Kim, M. Riebe,  and R. Blatt, Nature 443, 316 (2006).
  • Huntemann et al. (2012) N. Huntemann, M. Okhapkin, B. Lipphardt, S. Weyers, C. Tamm,  and E. Peik, Phys. Rev. Lett. 108, 090801 (2012).
  • Lechner et al. (2016) R. Lechner, C. Maier, C. Hempel, P. Jurcevic, B. P. Lanyon, T. Monz, M. Brownnutt, R. Blatt,  and C. F. Roos, Phys. Rev. A 93, 053401 (2016).
  • Ejtemaee and Haljan (2017) S. Ejtemaee and P. C. Haljan, Phys. Rev. Lett 119, 043001 (2017).
  • Scharnhorst et al. (2017) N. Scharnhorst, J. Cerrillo, J. Kramer, I. D. Leroux, J. B. Wübbena, A. Retzker,  and P. O. Schmidt, ArXiv e-prints  (2017), arXiv:1711.00738 .
  • James (1998) D. F. V. James, Appl. Phys. B 66, 181 (1998).

Supplemental Material: Optical clocks based on linear ion chains with high stability and accuracy

Appendix A Stroboscopic imaging photon-correlation measurements on Coulomb crystals

a.1 Experimental setup

Our experimental setup for spatially resolved micromotion measurements in Coulomb crystals is shown in Fig. 2b in the main text. The fluorescence is imaged onto an image intensifier,222Proxivision BV 2581 TZ 5N the cathode of which is driven with square pulses, corresponding to about of the rf period. These pulses are triggered, with variable delay, by the trap drive. Details on the electronic circuit can be found in Ref. Beev et al. (2017). Photo-electrons are subsequently multiplied using a multi-channel plate assembly and converted back to photons by acceleration onto a fluorescent screen. Finally, the screen image is observed on an EMCCD camera and integrated for each setting of phase (in total 4) and laser direction (3). All 12 settings are interleaved and the collected images are added up in post-processing. Figure S1 shows an example of such an image for a total exposure of : The spatial resolution is slightly deteriorated by the image intensifier, as the applied cathode pulses of ca.  are half of the design value, allowing for more transverse electron motion during the time of flight. As the mean photon number during each gate pulse is about per ion, mutual electron repulsion at this stage is negligible. Individual ions can be clearly resolved in the final image, and circular regions of interest are used to separate their signals. No modulation was observed in stray light imaged with this setup, proving that there is no additional modulation due to, e.g. electronic noise at the trap drive frequency.

Figure S1: Stroboscopic Coulomb crystal image for one rf phase setting obtained with the image intensifier setup and an integration time of .

a.2 Data evaluation

We observe fluorescence on the transition in Yb, which is modulated by EMM as


The transfer function of the image intensifier setup can be written as


where is the fraction of the rf period for which the intensifier is gated, is the variable delay of the pulses with respect to the rf drive, accounts for a possible deviation from ideal rectangular pulses, and denotes convolution. The observed signal is the product of Eqns. S1 and S2, integrated over the detection time:


where the last step assumes , such that the integration limits can be replaced by , is the Fourier transform of ,


and the Fourier transform of . By adjusting , the magnitude of can be varied in order to distinguish the from the contribution:


The quantity , which is used to determine the micromotion amplitude Keller et al. (2015), is thus observed with a contrast of


Due to the unknown contribution of , needs to be determined experimentally, which for our setup is shown in Ref. Beev et al. (2017). The evaluation according to Keller et al. (2015) yields a micromotion modulation index , which can be converted into an rf electric field via


with ion mass , laser wave vector projection onto the micromotion velocity and elementary charge . The resulting time dilation shift depends on as


and for the radial directions, the equivalent DC electric field displacing the ions is


where denotes the secular frequency and the Mathieu parameter for the ponderomotive potential.

Appendix B Collective micromotion effects in an ion chain

Consider the situation depicted in Fig. S2: two ions at equilibrium positions and , of which ion 2 undergoes micromotion due to an rf electric field pointing along . The corresponding micromotion amplitude is . This displacement results in an electric field at the position of ion 1 of


since at equilibrium positions. Following James (1998), the equilibrium distances between ions and in a harmonically confined chain can be expressed as , where are of order 1 and independent of trapping parameters, and


The relative field experienced by ion 1 thus scales as


Of all the configurations considered in this work, the highest value for this expression by far is , which occurs for the center ions in the crystal of Fig. 1e () in the sympathetic cooling trap configuration (, ).

Figure S2: Geometry considered for the derivation of collective micromotion effects in a chain.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description