Manipulation and coherence of ultra-cold atoms on a superconducting atom chip
S. Bernon, H. Hattermann, D. Bothner, M. Knufinke, P. Weiss, F. Jessen, D. Cano, M. Kemmler, R. Kleiner, D. Koelle, J. Fortágh
MManipulation and coherence of ultra-cold atoms on a superconducting atom chip
S. Bernon ∗ , H. Hattermann ∗ , D. Bothner, M. Knufinke, P. Weiss,F. Jessen, D. Cano, M. Kemmler, R. Kleiner, D. Koelle and J. Fortágh CQ Center for Collective Quantum Phenomena and their Applications,Physikalisches Institut, Eberhard-Karls-Universität Tübingen,Auf der Morgenstelle 14, D-72076 Tübingen, Germany and ∗ These authors contributed equally to this work
The coherence of quantum systems is crucial to quantum information processing.While it has been demonstrated that superconducting qubits can process quantuminformation at microelectronics rates, it remains a challenge to preserve the coher-ence and therefore the quantum character of the information in these systems. Analternative is to share the tasks between different quantum platforms, e.g. cold atomsstoring the quantum information processed by superconducting circuits. In our experi-ment, we characterize the coherence of superposition states of Rb atoms magneticallytrapped on a superconducting atom-chip. We load atoms into a persistent-current trapengineered in the vicinity of an off-resonance coplanar resonator, and observe that thecoherence of hyperfine ground states is preserved for several seconds. We show thatlarge ensembles of a million of thermal atoms below nK temperature and pureBose-Einstein condensates with . × atoms can be prepared and manipulated atthe superconducting interface. This opens the path towards the rich dynamics of strongcollective coupling regimes. The quantum physics of interfaces is attracting greatinterest because quantum state transfer between sub-systems is required for quantum measurements, quan-tum information processing and quantum communication[1]. To overcome the fast decoherence of superconduct-ing qubits, the engineering of various hybrid quantumsystems recently became a subject of intensive research[2–7]. The success of strong coupling between supercon-ducting two-level systems and microwave cavities [8], theimplementation of quantum algorithms with supercon-ducting circuits [9–11] and the successful realization ofsuperconducting surface traps for ultracold atoms [12–14]encourage the development of superconductor/cold atomhybrids. So far, some fundamental interactions betweenthe two systems have been observed [14–16]. However, co-herent coupling between the systems remains a scientificand technological challenge. Although theoretical pro-posals suggest using atomic ensembles as quantum mem-ories in a hybrid quantum computer [17–19], the trappingof atoms in the vicinity of a superconducting coplanar mi-crowave waveguide resonator (CPR) is still required.Long coherence times and state transfer are central is-sues for quantum information processing. In cold atomicensembles, the fine control of inhomogeneous dephasingsources [20, 21] allow long storage times of a single col-lective excitation [22]. A similar control in chip-basedtrapped atomic clocks [23], allowed to preserve coher-ent states of rubidium hyperfine levels over tens of sec-onds [24]. In addition, the energy spectrum of rubid-ium atoms can be used to convert the quantum infor-mation to the near infrared, in the telecom band [21],where long distance quantum communication can be real-ized. Hybrid systems of cold atoms and superconductorsare therefore very appealing for a solid state, atomic andphotonic quantum interface. Nevertheless, the questionhow to preserve such coherence and optical properties inthe complex environment of a hybrid system needs to besolved.Here, we report on the preparation of coherent atomic samples at a superconducting interface. We load ultra-cold Rb atoms into a magnetic trap generated by asuperconducting niobium thin film structure. We mea-sure exceptionally long lifetimes of fully spin polarizedstates ( > s). In either of the hyperfine ground statesof rubidium, we reach the critical temperature of Bose-Einstein condensation with more than one million atoms.The coherence of the superpositions of these groundstates is measured for various positions on the supercon-ducting interface. In a self-centered persistent-currenttrap engineered in the vicinity of a CPR we observe acoherence time T ∼ s. This demonstrates that coldatom trap inhomogeneities can be controlled in this com-plex environment to a metrological level, paving the waytowards long living single excitations.The experimental apparatus combines a cryostat (JanisST-400, W cooling power) holding a superconductingatom-chip and a cold atom setup integrated in a singleultra high vacuum (UHV) chamber (Fig. 1(a)). The vac-uum ( ∼ − mbar) provides excellent heat isolation be-tween the superconducting chip surface at a temperatureof T = 4 . K and the room temperature electromagnetsthat are used for the preparation of cold atomic samples[25]. A copper radiation shield at ∼ K protects thechip from the room temperature thermal radiation. Aslit of mm height on the shield gives optical access toan optical tweezer that transports atom clouds from theroom temperature environment to the superconductingatom-chip.The superconducting chip with Z -shaped wires (red line)and a quarter-wave CPR structure is shown in Fig.1(b)and (c). The niobium film structures ( nm thickness)were fabricated on monocrystalline sapphire by mag-netron sputtering, optical lithography and reactive ionetching (SF ). Niobium is, in our experimental condi-tions, a type II superconductor with a transition tem-perature of . K. At . K, the wires carry mean cur-rent densities of up to × A/cm , corresponding toa current of A for a wire of µm width. Supply wires a r X i v : . [ phy s i c s . a t o m - ph ] F e b Magnetic traps at room temperatureCryogenic set-up20 K Radiation shieldSuperconducting chip (4.2 K)Optical dipole trap Transport with dipole trapAtom cloud preparationSingle vacuum chamber p = 10 -11 mbar (a) mm (b) (c) -1000100 horizontal position (μm) v e r ti ca l po s iti on ( μ m ) OD I n t e g r a t e d d e n s it y ( a . u . ) (d)(e)(f ) m F = F = 2F = 1 Δ g F =1/2 Energy-2 -1 0 1 2 g F =-1/2 xy z FIG. 1:
Hybrid system of ultra-cold atoms andsuperconductors. (a) In-vacuo setup (to scale): On the right side, the atoms aretrapped and cooled in a room temperature environment. The leftpart shows the superconducting chip attached to the cryostat at . K and surrounded by a gold-plated radiation shield at ∼ K.The atoms are transported from one environment to the other( mm distance) by an optical tweezer. (b) Photograph of thesuperconducting atom-chip mounted onto an oxygen free copperholder. (c) Microscope image of the superconducting trappingstructure. Visible are four Z -wires for trapping and aquarter-wave coplanar microwave waveguide resonatorcapacitively coupled to the feedline. (d) Absorption image of aBose-Einstein condensate in state | (cid:105) with N BEC = 3 × atomsafter 15 ms time of flight. Color scale corresponds to opticaldensity (OD). (e) Normalized integrated density showing thebi-modal structure of a BEC (black points). In dashed blue a fitto the thermal background and in dotted red a fit to the centralThomas Fermi profile. (f) Energy diagram of Rb in a magneticfield. In dark are the three magnetically trappable states. Thecoupling of | F = 1 , m F = − (cid:105) ( | (cid:105) ) and | F = 2 , m F = 1 (cid:105) ( | (cid:105) ) isrealized by a two photon transition. (normal conducting copper) are connected by ultrasonicsoldering to the niobium. The sapphire substrate is sim-ilarly soldered to the copper mount of the cryostat. Twosuperconducting wires of µm diameter pass below thechip and help to maintain the longitudinal confinementof the trap (Fig. 1(b)).The preparation of atomic clouds follows standard techniques of magneto-optical (MOT) and magnetic trap-ping (see Methods for details) and leads, in the roomtemperature environment, to a cold cloud of × atoms at T < µK. After transfer to an optical dipoletrap, the cloud is transported into the cryogenic envi-ronment to the loading position at ∼ µm from thesuperconducting chip surface. The atoms are then trans-ferred into a harmonic magnetic trap formed by super-posing the field generated by the current I Trap = 0 . Adriven through the largest Z -shaped trapping wire withan homogeneous magnetic bias field B bias = 4 G ap-plied along y and a magnetic offset field B off = 0 . Gapplied along x [28]. The atoms loaded are then adiabat-ically compressed in a trap with oscillation frequencies { ω x , ω y , ω z } / π = { , , } Hz and subsequentlycooled by radio-frequency forced evaporation to temper-atures of ∼ nK. By changing the length of the initialMOT phase (1-10 s), the atom number in the supercon-ducting trap ( − ) and the final state (thermal cloudor BEC) can be conveniently controlled without affectingthe temperature. Due to the strong cryopumping andthe suppression of thermally driven magnetic field fluc-tuations [16], the lifetimes of the atoms in such a sur-face trap are predicted to be exceptionally long [29–31].For a cloud of N at = 10 atoms polarized in state | (cid:105) ( | F = 1 , m F = − (cid:105) ) at a density of at/cm heldin the compressed trap with offset field B off = 0 . G, wemeasure a lifetime of more than min. Together withthe recently measured lifetime of min for an experi-ment entirely shielded at nitrogen temperature [32], weexpect that the sub-kelvin cryoshields of dilution refrig-erators will provide excellent vacuum and thermal noiseconditions for cold atom setups.We form pure Bose-Einstein condensates (BEC) withup to . × atoms in either of the spin states | (cid:105) or | F = 2 , m F = 2 (cid:105) . The lifetime of a condensate of × atoms in a trap with frequencies { , , } Hzis s and is density limited by three-body collisionallosses [33]. Such high atom numbers are of special interestfor quantum information protocols in which the collectiveenhancement allows one to reach an effective strong cou-pling regime [19], for superradiance experiments and forthe realization of an on-chip maser [34].In the following, we explore the properties of the coldcloud in the vicinity of the superconducting CPR. Wefirst consider the positioning of atoms in the close vicinityof the CPR where we additionally engineer a persistent-current trap. As a further step, we study the coherenceof atomic superposition states in different positions ofthe CPR mode volume. In this report, the resonator is,by design, off resonance from the atomic transition andshould therefore not affect the internal state dynamics ofthe sample.
Positioning of atomic clouds into a CPR
In aCPR, the electromagnetic fields are concentrated in thegaps between the ground planes and the central conduc-tor. To maximize the atom-cavity coupling, atoms needto be positioned in the close vicinity of these gaps. Nev-ertheless, for trap-wire distances smaller than the wirewidth, the trapping parameters are significantly affectedby the Meissner-Ochsenfeld effect (MOE) [15]. To limit g C h i p d i s t a n ce ( μ m ) -20 0 20 40 6001020304050 Freezingfield C u rr e n t d e n s it y ( a . u . ) Horizontal position (µm) E n e r gy / k B ( μ K ) Bias field
28° 41° 50° 56° 61° 65° 68° I Trap I Scr - I Scr
Horizontal position (μm) C h i p d i s t a n ce ( μ m ) g,z α α Bias field y Ground plane Trapping wire Atomic ensemble I Trap
MagneticpotentialCurrent density Central conductorGap: 10 μmBiasfield I Scr -I Scr
FeedlineSuperconducting loop zx y(a) (b)(c)
FIG. 2:
Positioning atoms close to a superconducting coplanar microwave resonator (CPR). (a) Scheme of the trapping of atoms in the gap of a superconducting quarter wavelength CPR. The trap inside the gap is a result of themagnetic fields generated by the current of the trapping wire I Trap and by the screening current I Scr in the ground plane. These fieldscancel with an externally applied bias field. The embedded plot (black dots) is the simulated distribution of the screening currents in thesuperconductor. These currents keep the flux in the superconducting loop constant and the interior of the films field free. The transverseprofile of the magnetic potential is shown in color. The dark blue corresponds to the potential minimum. (b) Position of the trap fordifferent currents in the trapping wire and different angles α between the bias field and the surface of the chip. α = B bias z /B bias y is variedby changing B bias z with B bias y = 2 . G constant. The position of the atoms has been measured by in-situ absorption imaging (SeeMethods). For small angles, the trap behaves as for a normal conducting chip, i.e., when the current is reduced in the trapping wire( I Trap ), the trap moves towards it. For large angles, this behavior is modified and the trap is focussed into the gap between the centerconductor and the ground plane of the CPR. The agreement between measurement (circles) and simulations (dotted and solid line withdots) proves that positioning of the atoms in the gap of the CPR can be facilitated by screening currents I Scr in the ground planes. Thesimulations [26, 27] are performed with no adjustable parameter and assume a Meissner state for the superconductor. Gravity, g , isoriented upwards. (c) Top: Potential energy landscape of a persistent-current trap above the central conductor (red) of the CPR. Thistrap is generated by the superposition of a vertical homogeneous bias field B bias ≈ . G and the field induced by the screening currents.To enhance the screening currents at a given bias field, a non-zero flux is trapped in the gap of the CPR during the cool down of thecryostat (Freezing field, B freezing ≈ . G). Isolines are separated by nK. Bottom: Screening current density distribution induced bythe combination of bias field and freezing field. such deformations in the vicinity of the CPR, the de-sign shown in Fig.1(c) includes four Z -shaped wires withwidths ranging from µm to µm. Starting from thelargest Z -wire trap, the atoms are first horizontally trans-ferred to the third Z -wire trap. The fourth and last wire( µm width) revealed to be unnecessary. At µmbelow the third trapping wire, the trap needs to be ro-tated towards the CPR gap. Two parameters are classi-cally used to manipulate atoms on a chip; the value ofthe trapping wire current modifies the trap-wire distanceand the direction of the bias field rotates the trap aroundthe trapping wire. Because of the MOE, positioning thatis straight forward for a normal conducting chip, is moresubtle for a superconducting atom-chip. In the presentcase, we take advantage of the conservation of magneticflux in superconducting loops to directly guide the atomsbelow the gap of the CPR.The design of the quarter-wave CPR includes a su-perconducting loop formed by the ground planes of the resonator (blue in Fig. 2(a)). When a field perpendic-ular to the substrate is applied, such as the bias field,a screening current is induced in the ground planes andensures the conservation of the magnetic flux in this su-perconducting loop. Such a current, that circulates justnext to the gap, generates a magnetic field profile thatguides the atoms into the gap. At distances comparableto the width of the ground planes, such guiding is furtherenhanced by the MOE that focusses magnetic field linesand generates magnetic gradients that center the cloud inthe gap. The guiding of atoms into the gap is observedby in-situ measurements of the position of the atomiccloud for different bias field orientations (angle α in Fig.2(b)) and different currents in the wire (Fig. 2(b)). Foreach experimental point, the cloud is first brought to atrap-wire distance of ∼ µm, rotated to the angle α and then moved to the desired trapping current. Themeasured position ( y, z ) agrees well with a 2D simula-tion of the London equations [26], that includes gravity P r ob a b ilit y ( F = ) P r ob a b ilit y ( F = ) Ramsey time (s) P r ob a b ilit y ( F = ) (a)
25 μm14 μm60 μm
Unused GP GPCC
Microwave resonatorSapphire substrate
Trapping wire
Reference Hybrid Persistent
Trap frequencies (Hz)
Atomic density (x10 at.cm -3 ) Coherence time (s)
Mean frequency (Hz) (b)(c)
Expected τ inh (s)
FIG. 3:
Atomic coherence in a superconducting coplanar microwave resonator (CPR). (a) Ramsey fringes measured in the time domain for different trapping positions on the superconducting atom-chip. From top to bottom,the trapping positions correspond to the reference trap (green star in (b)), the hybrid trap (blue square in (b)) and the persistent-currenttrap (red disk in (b)). The coherence time obtained for the three sets are respectively . s, s and . s that indicate collision inducedspin self rephasing effects [24] (see main text and Methods) (b) Positions of the different traps below the superconducting chip. Thereference trap (green star) is situated µm straight below the trapping wire. The hybrid trap, that is generated by applied and inducedcurrents, stands µm below the µm wide gap of the CPR. The persistent-current trap is µm below the central conductor (CC) ofthe CPR that has a width of µm. GP stands for ground planes. Horizontal and vertical axes are to scale. (c) Table of the experimentalparameters for the three measurements shown in (a). The density quoted is the mean atomic density. The mean frequency is thegeometrical average. τ inh = √ / ∆ is the coherence time expected for the corresponding residual frequency inhomogeneity ∆ (seeMethods). and the conservation of flux in the resonator loop (seeMethods). The simulations are performed without freeparameters. As experimentally observed and consistentwith our model, we note that the two gaps of the CPRare not equivalent. Due to the opposite direction of thecurrent in the ground planes and the orientation of thebias field, only the closest gap to the trapping wire canbe accessed. We call this trap, resulting from applied andinduced currents, a hybrid trap.A further advantage in using superconductors to ma-nipulate cold atoms, is the possibility to engineer trapsinduced by persistent currents. These persistent-currenttraps [13] render the direct injection of currents onthe chip unnecessary and therefore suppress the relatedsource of noise. In our geometry, such a trap can be gen-erated directly below the CPR. Here we demonstrate aself-centered trap generated by a single homogeneous biasfield and its induced screening currents. By trapping fluxin the resonator loop with a freezing field applied duringthe cooling of the cryostat, the magnitude of the screeningcurrents can be controlled independently from the appliedbias field. This experimentally realized trap is simulatedin Fig. 2(c) where both the resulting potential and thecurrent densities are depicted. The simulation presentedincludes the distortion due to the MOE, the conserva-tion of magnetic flux in the loop and the effect of vorticestrapped inside the film during the cooling of the cryo-stat [35]. The agreement of position and trap frequenciesbetween the experiment and our simple model requiresan adjustment of the input parameters by less than .For a small BEC in the trap of Fig. 2(c) with oscillationfrequencies { , , } Hz, we measure a lifetime of s, probably limited by the residual technical noiseand the shallow trap depth of ∼ nK. Coherence in a superconducting CPR
To studythe coherence lifetime of atomic ensembles in the closevicinity of a CPR, we perform Ramsey measurements atdifferent positions below the superconducting chip. Thesemeasurements compare the coherence of a trapped atomicensemble in a superposition of the states | (cid:105) and | (cid:105) to ahigh stability MHz reference oscillator that has a shortand long term frequency stability ∆ f /f < × − .These two states are chosen for the low sensitivity oftheir transition frequency to magnetic inhomogeneities ata magnetic offset field of 3.228 G (see Methods) [36]. Theatomic cloud is first prepared in a thermal and pure stateof | (cid:105) . The initialization of the coherent superposition isrealized by a π/ two-photon excitation that starts the in-terferometric sequence. After a variable waiting time T R ,the interferometer is closed by a second π/ pulse. Thepopulations N ( T R ) and N ( T R ) in respectively | F = 1 (cid:105) and | F = 2 (cid:105) are consecutively read out by state selec-tive absorption imaging (See Methods), and the resultingprobability of | F = 2 (cid:105) : N ( T R ) / ( N ( T R )+ N ( T R )) is dis-played in Fig. 3(a).To study the capabilities of our setup, we first describea reference measurement that is performed far from theCPR: µm below the third trapping wire. This measure-ment is conducted with . × atoms at a temper-ature of nK in a magnetic trap with oscillationfrequencies { , , } Hz, corresponding to a meandensity (in units of at/cm ) n ≈ . (green starin Fig. 3(a)). In this configuration, the coherence timeis T = 20 . . s ( /e exponential decay time). Due tomagnetic noise in this non-shielded apparatus and driftsover the length of the scan (several hours), the phasestarts to be lost for T R > s. This decay time T ex-ceeds the time of s predicted by the residual trap inho-mogeneities ∆ / π ≈ . Hz (see Methods, [37]). Thisindicates that we entered the spin self-rephasing regime[24], where the identical spin rotation rate ω ex / π = 3 Hzdominates both ∆ and the rate of lateral elastic col-lisions γ c = 1 s − . This regime can be understood as acontinuous spin-echo process triggered by forward atomiccollisions.The second trap under study is the hybrid trap, situated µm below the gap of the CPR. As mentioned before,this position is particularly privileged for the perspec-tive of strong atom-cavity coupling. The Ramsey fringesshown by blue squares in Fig. 3(a) were obtained for . × atoms at a temperature of nK heldin a confined trap { . , , } Hz, corresponding to n ≈ . . In the present situation, the coherencetime is T = 3 . s and is mainly limited by the atomloss during T R that has a decay time of . s. Thisloss is probably due to spin exchanging collisions fromatoms in state | (cid:105) towards the states | F = 2 , m F = 0 (cid:105) (untrapped) and | F = 2 , m F = 2 (cid:105) (trapped) [38]. Onetypical signature of this loss channel is the asymmetry ofthe probability for large T R . These large collisional lossesare due to the strong confinement of this trap. This is thedrawback of the guiding mechanism that involves strongmagnetic gradients generated by the close-by screeningcurrents. The fringes presented in figure 3(a) result froman optimization of the temperature that is high enoughto minimize collisional losses and low enough to enter thespin-self rephasing regime ( ω ex / π = 9 . Hz, γ c = 3 . s − , ∆ / π = 0 . Hz).The last position studied corresponds to the persistent-current trap previously mentioned. In this trap the mea-surement was performed with . × atoms at nK, corresponding to n ≈ . and to therelated spin self rephasing parameters ω ex / π = 6 Hz, γ c = 1 . s − , ∆ / π = 0 . Hz. The coherence lifetimeobtained is T = 7 . s and is mainly limited, in thisshallow trap ( ∼ nK), by the atom loss decay time . s.The three measurements are compared in Fig. 3(c). Itshows that the trap deformation induced by the supercon-ducting CPR results in an increase of the trap frequencieswhich strongly impact the coherence of the atomic cloud.It also proves that this influence can be engineered toreach performances comparable to the state of the art ofmetrological experiments [24].In conclusion, we have demonstrated that hybrid sys-tems of cold atoms and superconductors are now suffi-ciently mature to produce excellent experimental condi-tions for the study of the quantum properties of suchan interface. The production of large thermal clouds andBECs of a million atoms opens the path to the strong col-lective coupling to a CPR, and to the transfer of quantuminformation between the two systems. We also demon-strated that, in the vicinity of a superconducting CPR,magnetic traps could be efficiently engineered to producerobust and controllable conditions. The measurement of coherence lifetimes of more than s in this new type ofenvironment gives strong hope that cold atoms could beused as a quantum memory for superconducting devices.We also point out that the motion of particles plays aninteresting role here. While this is usually considered tobe a source of decoherence and prevents, for example, theuse of spin-echo techniques, it might actually be turnedto preserve the coherence of a quantum memory. Acknowledgements:
We would like to thankThomas Udem from the MPQ Munich as well as MaxKahmann and Ekkehard Peik from the PTB Braun-schweig for the loan of the reference oscillators. This workwas supported by the Deutsche Forschungsgemeinschaft(SFB TR 21) and ERC (Socathes). H.H. and D.B. ac-knowledge support from the Evangelisches Studienwerke.V.Villigst. M.Kn. and M.Ke. acknowledge supportfrom the Carl Zeiss Stiftung.
Author contributions
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Methods:
Atom cloud preparation:
The magneto-optical trap(MOT) is loaded from a 2D-MOT. For a loading time of s , theMOT contains ∼ Rb atoms at a temperature of ∼ µK.With this method, we do not observe perturbation of the back-ground pressure. After an optical molasses, the atomic cloud isoptically pumped into one of the two hyperfine ground states | (cid:105) or | F = 2 , m F = 2 (cid:105) (total angular momentum states) and istransferred through a magnetic quadrupole into a harmonic Ioffe-Pritchard type trap. It is further cooled by forced RF evaporationto a temperature of ∼ . µK, slightly above the BEC transition.The remaining × atoms are loaded into an optical tweezer( λ = 1064 nm laser, P = 500 mW, focused to w = 25 µm beamwaist) and transported without significant loss or heating over a distance of mm to the superconducting chip. During the opti-cal transfer, a quantization field of 350 mG along x is applied tomaintain the polarization of the sample. Magnetic field calibration:
At the position of theatomic cloud, below the superconducting chip, the magnetic field iscontrolled by 3 orthogonal pairs of coils that allow to independentlycontrol the 3 directions of space. The calibration of the residualfield and the field generated by the coils is realized in-situ by mi-crowave spectroscopy. To that purpose, the atoms are prepared in | (cid:105) and the magnetic field is set to the desired field of study. Afterall Eddy currents have damped out, the atoms are released fromthe optical dipole trap. The microwave transition considered is | (cid:105) to | F = 2 , m F = 0 (cid:105) . In the absence of a magnetic field it has afrequency of .
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GHz and a first order Zeeman sensitivityof kHz/G. In a first coarse step, the absolute field is reducedto below 5 mG. In a second fine step, each pair of coils is switchedon in a row to generate a frequency shift of approximately kHz.This shift is then measured with an accuracy of ± Hz. This way,each pair of coils is in-situ and independently calibrated (residualcontribution from other pairs below 1 (cid:104) ) and the residual field ismeasured to below 1 mG while the coils are calibrated with an ac-curacy better than 3 (cid:104) .To avoid perturbation by the MOE of the nearby superconductor,this calibration is done with the cryostat at T ≈ K. Imaging and measurement of position:
Theatoms are observed by absorption imaging with a variable timeof flight. Due to Eddy currents in the mechanical system, the mea-sured atom number is not absolute and depends on the time offlight (TOF). For TOF < ms, the calibration of the detection ofthe state | (cid:105) with respect to the state | (cid:105) is obtained by minimizingthe variance of the total atom number detected over the length ofthe scan. For TOF> ms, atom numbers stay constant, showingthat Eddy currents are no more an issue at the corresponding dis-tances. The absolute calibration of the atom number is obtainedfrom the critical temperature of Bose-Einstein condensation. Thiscalibration is the main source of uncertainty of the atom numbers.Therefore, the atom number uncertainty quoted in the text do notrepresent shot to shot fluctuations. In-situ , the position of theatomic cloud is measured along the three directions of space by tworeflection imaging systems that are aligned along x and y . The cal-ibration of distances is realized by time of flight of the magneticallyinsensitive state | F = 2 , m F = 0 (cid:105) . To avoid spurious effect, thechip is uncoated. Magnetic field calculations:
To calculate the currentdensities in the superconductor and the subsequent atomic trap de-formation, we solved the London equations using the 2D algorithmdescribed in [26]. This treatment is particularly valid in the trap-ping region where thin films are parallel to each other. The conser-vation of flux in the superconducting loop of the resonator is furthertaken into account by imposing net currents in the grounds of theresonator. The influence of an homogeneous density of vorticespointing along u z is modeled by the superposition of an homoge-neous field along u z (undeformed) and the opposite field deformedby the superconducting structure in a pure Meissner state. In allsimulations, the quantization of flux is neglected ( Φ / Φ ≈ × − ).The simulations presented in Fig. 2 have been confirmed using a3D simulator (3D-MLSI [39]). The effect of gravity is included inall simulations. Details on the measurement of coherence life-time:
The measurement of the T time of the atomic qubitformed by the state | (cid:105) and | (cid:105) is realized by a Ramsey typeexperiment. As shown in Fig. 1(f), the two states are cou-pled via a two photon transition involving a microwave photon at f MW = 6 .
833 378 16
GHz and an RF photon at f RF = 1 . MHz.Both frequencies are generated by commercial synthesizer phase locked to a high stability MHz quartz oscillator (Oscilloquartz,8607-BHM15), and their sum is frequency detuned form the atomictransition by ∆ R / π . For all the measurement presented, the mi-crowave with power P MW is radiated by an helicoidal antenna sit-uated outside the vacuum chamber at a distance from the atomsof 20 cm. The radio frequency with power P RF is coupled on thechip to the largest Z -wire. The Rabi frequency Ω R obtained ineach situation is summarized in table I. Figure 4 shows the dif-ferential frequency shift of the qubit transition that is well ap-proximated by ∆ ν ( r ) = ν | (cid:105) − ν | (cid:105) = ∆ ν + β ( B ( r ) − B ) , with ∆ ν = 4 . kHz, B = 3 .
228 917(3)
G[24] and β = 431 . Hz/G [36]. At the magic offset field B , this shift is first order insensitive F r e qu e n c y s h i f t ( k H z ) Offset field (G)0-1-2-3-5-40 1 2 3 4
FIG. 4:
Magnetic field dependence of thedifferential frequency shift of the transition | (cid:105) to | (cid:105) . The black points are experimental data. The red dashed line isthe prediction given by the Breit-Rabi formula [40].to the magnetic field. The sensitivity of the coherence time to themagnetic inhomogeneities of the trap is therefore highly reduced.The measurements presented in Fig. 3a are performed with an off-set field B off slightly lower than B . This configuration is knownas the mutual compensation scheme [36, 37], which allows to com-pensate the negative collisional shift ∆ c ( r ) / π = − . n ( r ) / Hz by the positive magnetic shift ∆ B ( r ) = 2 π ∆ ν ( r ) . In suchconditions, an optimum residual radial frequency homogeneity ∆ = (cid:113) (cid:104) (∆ c ( r ) + ∆ B ( r )) (cid:105) − (cid:104) ∆ c ( r ) + ∆ B ( r ) (cid:105) is obtained foran optimal offset field B opt1 that depends on the number of par-ticles, the temperature and the geometry of the trap. In table I,we give the optimum value of ∆ for each experimental configu-ration. In the abscence of spin-rephasing, such inhomogeneitiesshould result in a decay of the Ramsey contrast with a time con-stant τ inh = √ / ∆ [37]. TABLE I: Experimental parameters of the measurementshown in Fig. 3a.
Properties Reference Hybrid Persistent P MW (dBm) 19 -10 -14 P RF (dBm) -2 11.5 11.5 ∆ R / π (Hz) 5.7 4.4 6.5 Ω R / π (Hz) 432 416 179 B (G) 3.193(4) 3.086(8) 3.20(3) B opt1 (G) 3.197 3.17 3.17 ∆ / π (Hz) 0.04 0.12 0.1 τ inhinh