atom chips (reichel:atom chips o-bk) || cryogenic atom chips

Part Four New Directions

311

10Cryogenic Atom ChipsGilles Nogues, Adrian Lupascu, Andreas Emmert, Michel Brune, Jean-Michel Raimond, andSerge Haroche

10.1Introduction

It is quite remarkable to observe that in 1995, when Weinstein and Libbrecht wrotetheir seminal article on microscopic atomic traps [1], the very same research grouppublished experimental results on long-lived neutral atom traps in a cryogenicenvironment [2]. They reported trapping lifetimes of nearly 10 minutes for mag-netostatic traps created by centimeter-size superconducting coils. Moreover theyclaimed that “with a cryogenic system one can use superconducting magnets andSQUID detectors to trap and non-destructively sense spin-polarized atoms”. Thissentence foresees some of the possibilities of cryogenic atom chips using supercon-ducting materials. Low temperatures and superconductivity do not only offer ultra-high vacuum conditions and high currents without dissipation. They also openthe way to a new class of experiments where ultra-cold atoms will be integratedon a chip together with solid-state devices whose coherent manipulation is onlypossible at low temperatures. A good example of such a device is a Superconduct-ing Quantum Interference Device (SQUID) coupled to the magnetic moment ofthe trapped atomic cloud [3]. Other interesting candidates like coplanar waveguideresonators or the nano-electro-mechanical system (NEMS) are also worth mention-ing.

It took some time after the initial proposal of [1] to develop and operate the firstatom chips (see Chapter 1) at room temperature and an even longer time to oper-ate them in cryogenic conditions. The first results on superconducting atom chipswere reported at ENS Paris in 2006 [4] and at NTT Basic research labs in 2007 [5].Many groups in the world are now building experiments along this line of researchand exciting developments are expected. The first part of this chapter (Section 10.2)will present a review of the existing experiments on cryogenic chips. We will tryto show by which aspects they may be similar to or differ from equivalent room-temperature setups. In Section 10.3 we will give a few examples of new experimentsthat could be carried out with cryogenic chips.

Atom Chips. Edited by Jakob Reichel and Vladan VuleticCopyright © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-40755-2

312 10 Cryogenic Atom Chips

10.2Superconducting Atom Chip Setup:Similarities and Differences with Conventional Atom Chips

10.2.1Experimental Considerations

Superconducting atom chips differ in many aspects from the usual atom chip ex-periments at room temperature. We focus in this section on the technical issuesassociated with building a setup that can be properly operated at cryogenic tem-peratures. Although they may appear technical, such differences are often a directillustration of the physics of superconducting materials.

10.2.1.1 Chip Fabrication and WiringCryogenic atom chips use superconducting films to carry the required trapping cur-rents. Indeed, the typical power dissipated by the Joule effect in normal metal chipsranges from 1 to 10 W. This is not compatible with cryogenic systems. Among allthe possible superconducting materials, present experimental efforts are focusingon niobium (Nb, Tc D 9.2 K) or niobium nitride (NbN, Tc D 15 K) [4]. More com-plicated alloys such as magnesium diboride (MgB2, Tc D 39 K) [5], or even high-Tc

oxides like YBCO (YBa2Cu3O7x , Tc D 87 K) [6] are also being investigated. It isimportant to note that the critical temperatures of Nb and MgB2 given here corre-spond to the bulk material properties. For thin films, especially if the thickness isbelow a few 10 nm, the presence of defects can significantly reduce Tc. The maxi-mum critical current in the smallest Nb wire (thickness 1 µm, width 40 µm) in [4]is 1.8 A at 4.2 K. It corresponds to a critical current density of 4.5 MA cm2. Currentdensities as high as 10 MA cm2 are reported for MgB2 at 4.2 K [5] and YBCO at50 K [6]. Those values are of the same order of magnitude as the current densitiestypically observed in normal metal atom chips at room temperature [7] and hencesuperconductors will not offer the possibility to reduce the size of the conductorsat similar currents. They will essentially differ in the fact that absolutely no heat isdissipated in the chip during the trapping.

Niobium-based chips are relatively easy to fabricate. They use standard tech-niques already applied to the case of normal metal chips (see Chapter 3). Super-conducting Nb films, with thicknesses ranging from a few nm to a few µm, canbe deposited on a Silicon substrate using cathodic sputtering. Standard optical orelectronic lithography techniques can then be used to define the trapping wire andcontact pads. Reactive ion etching (RIE) with fluor ions is used to remove unde-sired Niobium surfaces. Figure 10.1 shows the process for the fabrication of thechip of [4]. The superconducting current-carrying wires are complete after step 7.The complete process includes two additional steps for depositing a layer of goldon top of the wires in order to reflect the laser beams necessary for the operationof an on-chip mirror-MOT.

The technology involved in the fabrication of the MgB2 chip of [5] is more de-manding. A 1.6-µm-thick layer of superconductor is grown by molecular beam

10.2 Superconducting Atom Chip Setup: Similarities and Differences with Conventional Atom Chips 313

Figure 10.1 Complete fabrication processfor the niobium chip of [4]. Starting from asilicon wafer, the superconducting Nb film isdeposited by cathodic sputtering. The steps3 to 6 correspond to the optical lithographyused for creating a structured hard mask of

aluminum on top of the Nb layer. The hardmask protects the wires during the reactiveion etching (step 7). Steps 8 and 9 allow oneto deposit a gold mirror on top of the chip inorder to reflect the lasers arriving on it.

epitaxy (MBE) on a sapphire substrate. For MgB2 the patterning of the circuit can-not rely on lithography and etching techniques. In the case of [5] for example, itwas performed by removing unnecessary parts of the film by ion milling.

Similar film deposition methods as in [5] are used for the fabrication of the high-Tc chip described in [6]. A 600–800-nm film of YBCO is grown by epitaxy on ayttria-stabilized zirconia (YSZ) single-crystal substrate in order to match the latticeparameters. It is important to control carefully the substrate temperature duringthe process. In the case of YBCO, it is even important to match the temperatureexpansion coefficients of the film and the substrate. A difference between the twoexpansion coefficients could be the source of additional surface tensions duringthe cooling of the chip that can significantly degrade the superconducting prop-erties of the film. The patterning of the YBCO layer can be done by combiningstandard lithography techniques together with wet chemical etching or laser abla-tion.

A very important issue for all setups is the connection of the wire to the outside.It is relatively difficult to pass the required current in the structure and to avoidat the same time local heating due to the Joule effect in the connection wires orthe contact pad. Because of the very small heat capacity of the chip materials at lowtemperature, even a moderate dissipation can rapidly lead to a local temperature in-crease above Tc, in which case superconductivity is lost. A possible solution to thisproblem consists in using only superconducting materials both for the lead wiresand their contact on the film [8]. Another radical solution consist in suppressing


completely the connecting wires and to induce permanent currents in a supercon-ducting loop as shown in [5]. If a bias magnetic field is applied perpendicularly toa loop of superconductor before its transition to the superconducting state, the cir-cuit will then react in order to keep constant the magnetic flux through the loop.As a consequence persistent currents will exist in the loop after switching off theexternal field. It is possible to switch off or even tune the current by performing alocal transition to the dissipative normal state. This is done in [5] by shining a laseron a portion of the loop.

10.2.1.2 The Cryogenic CellThe first experimental question that arises when one starts a low-temperature ex-periment is the choice of the cryogenic system. It will greatly determine the kindof studies one is able to carry out. In the case of atom chips the cryostat should becompatible with standard conditions for ultra-cold atom manipulation, like ultra-high vacuum (UHV), good optical access, and large magnetic field gradients. Aswas pointed out in [2], trapping ultra-cold atoms in a cryostat can lead to a verylong lifetime because of the huge adsorption on the cold surface surrounding thecloud. UHV conditions can be reached without baking out the cold cell.

The problem of the optical access to the cloud is much more difficult to solve.It seems reasonable to keep the imaging system at room temperature. However, alarge numerical aperture for the collection lens outside the cryostat implies that thesuperconducting atom chip will be directly exposed to blackbody radiation at 300 Kwith a large solid angle. This is usually not admissible. The coldest parts of theexperiment are protected from the room-temperature radiation by thermal shieldsat intermediate temperatures. It is also necessary to bring some lasers inside thesetup in order to trap the atoms and to probe them.

In many experiments, the trapping of the ultra-cold cloud is achieved by combin-ing magnetic fields created by on-chip wires together with external bias coils. Thetypical size of the latter is of the order of a few cm and they must be placed at sim-ilar distances from the trapped cloud. Having the coils outside the cryostat limitsthe volume of the cryostat and might generate potentially harmful eddy currentsin the cold setup while changing the external fields. On the other hand it might betechnically difficult to arrange and thermalize bias coils inside the cryostat.

The first two experiments that have reported the operation of superconductingchips so far use radically different approaches:

In [4] (see Figure 10.2a) the cold cell is in direct thermal contact with a 4.2-K liquid He reservoir. The volume of the cell is large enough (a few dm3) toaccommodate the chip holder as well as the external bias coils. The latter need,therefore, to be superconducting. The first imaging lens is 20 cm away from thetrapped cloud. An intermediate thermal shield at liquid nitrogen temperature(77 K) screens most of the blackbody radiation. Cold viewports are mounted at4.2 and 77 K. A good thermalization of all the wires connected to the cold partof the setup allows one to run the experiment for hours without difficulties.As opposed to room-temperature experiments, one cannot rely on the atomic


4 K

77 K

300 K

DriftChamber

Rb Cell

TrappingBeam

Pushing Beam

Atomic Beam

2D-MOTOptical Windows

Front Observation Axis

Atom Chip

Mirror-MOTBeams

z

x y

60

cm

(a)

(b)

Figure 10.2 Scheme of the experimental setup for [4] (a) and [5] (b).

background vapor pressure to load the trap with atoms because it is dramaticallylow at cryogenic temperature. The problem can be circumvented by using a low-velocity beam of atoms which is recaptured in a magneto-optical trap (MOT) inthe cryostat [4]. An additional UHV chamber at room temperature is thereforerequired in order to prepare the cold atoms.

In [5] the experimental cell is much more compact and mounted at the end ofa cold finger (see Figure 10.2b). The chip can be cooled down to 3 K but itstemperature is significantly higher during normal operation. It is protected by ashield at 15 K. A better optical access is provided but thermalization of the lead


wires might be a problem. Atoms are magnetically trapped outside the cryostatin an external UHV chamber and conveyed inside the cryostat with movablecoils [5].

10.2.2Trapping and Cooling: First Results

10.2.2.1 Magnetic TrapAs we have seen previously in Section 10.2.1.1, typical current densities inside su-perconducting slabs are similar to what is generally used in normal metal atomchips. As a consequence the trapping parameters such as the trapping frequencies,the trap volume, the distance to the chip, and so on . . . do not differ significantlyfrom the values observed for room-temperature chips (at least as long as the cloudis not too close to the surface, see Section 10.3.1). A typical experimental sequencefor the ENS experiment is described in [4]. It was later refined in [9] where the firstobservation of a BEC (Bose–Einstein condensate) on a superconducting atom chipwas reported. All the usual steps of normal metal chip experiments [10], see Chap-ter 2) are present. About 5 107 atoms are loaded in 5 s in a mirror-MOT whosemagnetic field is created by the lower part of a rectangular coil placed 1.5 mm be-hind the chip. It mimics a centimeter-size U-wire whose magnetic field, combinedwith the homogeneous field created by the bias coils inside the cryostat, producesa quadrupole field a few mm away from the chip surface. Atoms are then trans-ferred in 20 ms with an efficiency of 86 % to an on-chip mirror-MOT that uses aU-shaped superconducting wire on the chip. The currents in the U-wire and theexternal bias fields are then changed in 20 ms in order to compress the MOT andbring the atomic cloud to a distance of about 500 µm from the surface. The lasersare detuned during the compression step as well as during the 2-ms-long opticalmolasses that follows it. About 107 are present at the end of the molasses, at atemperature of 20 µK. Finally, an optical pumping to a low-field seeking state isperformed before the transfer to the magnetic trap. The values of the experimentalparameters at each step (current, bias field, laser power, and detuning) are close tothose typical of room-temperature atom chips. Between 1 and 3 106 atoms aretrapped 400 µm away from the surface at a temperature between 60 and 20 µK.Those numbers were measured by observation of the atomic cloud using standardabsorption imaging 100 ms after the transfer to the magnetic trap.

The situation is slightly different for the permanent current magnetic trap pre-sented in [5]. As explained in Section 10.1, the trapping current is induced in a su-perconducting loop long before the atoms are brought into the system. Moreover,there is no intermediate mirror-MOT close to the chip surface and the atoms aremagnetically trapped and transferred inside the cryostat with the help of externalmovable coils (see Section 10.2.1.2 and Figure 10.2b). The center of the quadrupolemagnetic trap created by those coils is simply superposed with the location of thepermanent trap, then the coils are rapidly switched off. The number of atoms andthe cloud temperature 100 ms after this transfer are 3105 and 200 µK, respectively.


200

1000

0 20 40 60 80 100 120 140 160

Ato

m n

umbe

r (a.

u.)

Time (s)

0 5

10 15 20 25

0 50 100 150

T (µ

K)

Time (s)

Figure 10.3 Number of atoms (log scale) asa function of time measured in [11]. The sol-id line is a double exponential fit of the data.During the first part the trapping atoms arerapidly lost because of evaporation. The atom-

ic cloud temperature decreases during thisphase (see inset). In the last part of the curveatomic loss is significantly reduced (trappinglifetime τ D 350 s)

For both experiments corresponding to [4] and [5], relatively long trapping life-times have been observed. Figure 10.3 shows the number of atoms as a functionof time in the ENS setup at a distance of 250 µm from the surface [11]. It clearlydisplays a double exponential behavior. Atoms are rapidly lost at the beginning ofthe sequence. During this time interval the cloud temperature decreases with thesame characteristic time. The loss of atoms during that time interval is attributedto natural evaporation: because of the finite depth of the trap the hottest atoms pro-duced by collision can escape while the remaining cloud cools down after rether-malization. After 20 s, the temperature remains constant and the trapping lifetimedramatically increases to 350(70) s. This value, in the minute range, is comparableto the one observed in [2]. If one assumes that the lifetime is limited by collisionswith a residual background gas of He at 4 K, the corresponding pressure would be 1 1011 mbar. The actual pressure is probably much smaller than this value.From a systematic study of the trap lifetime as a function of the distance to the trap-ping wire, one concludes that the main source of loss here remains the technicalcurrent noise in the wire which induces magnetic Zeeman transitions towards un-trapped states. This phenomenon is removed with the use of permanent current.In [5], atoms were observed for a time as long as 15 s. Surface evaporation takesplace during the first second, followed by another regime where the atom numberremains constant within an accuracy of 10 % (corresponding lifetime 80 s). A de-tailed study of lifetime as a function of distance was recently published by the NTTgroup [12], that suggests that phenomena related to the superconducting nature ofthe surface, like magnetic field distortion resulting from flux penetration, can havea strong impact on the trap decay.

10.2.2.2 Forced Evaporation and Quantum DegeneracyFurther cooling of the atomic sample has been realized at ENS by using the stan-dard technique of forced evaporative cooling [9]. In order to speed up the evapora-


tion process, the elastic collision rate is first increased by an adiabatic compressionof the cloud. After the compression, the Zeeman transition frequency towards un-trapped states at the bottom of the trap potential is approximately 9 MHz. Thisvalue minimizes the influence of technical noise, present at lower frequencies, atthe expense of a reduced confinement of the cloud. With these settings, the atomsare located 85 µm away from the surface. Calculations of the magnetic field giveaxial and longitudinal trapping frequencies equal to 2π 6 kHz and 2π 100 Hz,respectively. A RF knife whose initial frequency corresponds to the energy of thehottest atoms in the trap is slowly ramped down. It is produced with the help ofa current that is fed in an extra superconducting wire, located on the chip about3.3 mm away from the atoms. Under these conditions, elastic collisions producehot atoms which are expelled, while the mean temperature of the remaining sam-

0

0.2

0.4

0.6

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0 50 100 150 200 250 300 350

Con

dens

ed fr

actio

n

Temperature (nK)

(a)

(b)

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0 10 20 30 40 50

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ical

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Figure 10.4 (a) Horizontal cross-section ofthe absorption image of the atomic cloud after17 ms of expansion, for a final evaporationramp frequency f D 9.871 MHz. The sol-id line is a Gaussian fit. (b) Same as (a) withf D 9.841 MHz. It shows the emergence of a

condensed fraction, fitted by a Thomas–Fermidistribution superimposed on a thermal back-

ground (solid line). (c) Condensed fractionof atoms as a function of the cloud temper-ature T. The latter is varied by adjusting thefinal frequency of the evaporation ramp. Thesolid lines are theoretical fractions in the ab-sence of interactions for the transition tem-peratures Tc D 70, 100, and 130 nK.

10.3 Perspectives for Cryogenic Atom Chips: A New Realm of Investigations 319

ple decreases. The frequency of the knife is ramped down in order to match thetemperature decrease, and thus force the process.

Figure 10.4a,b show two cross-sections in the absorption image of the atomiccloud for the final RF frequencies f D 9.871 and 9.841 MHz, respectively. Themagnetic trap is first adiabatically decompressed, then suddenly switched off. Bothimages are taken 17 ms after releasing the cloud. A sharp peak clearly appears onFigure 10.4b on top of a broader thermal Gaussian profile. This bimodal distribu-tion is strong evidence for the presence of a BEC. The condensed part of the samplecan be fitted by a 2D Thomas–Fermi distribution, whose amplitude is used to de-termine the fraction of condensed atoms for Figure 10.4c. The cloud temperature T

on the abscissa of the graph has been obtained by fitting the width of the thermalfraction with a Gaussian. The solid lines in Figure 10.4c correspond to the fit of thecondensed fraction in the absence of interaction between atoms for three differentcritical temperatures (Tc D 70, 100, and 130 nK), which are qualitatively compatiblewith the experimental points. The corresponding critical temperature and numberof atoms at the onset of BEC are Tc D 100(30) nK and N D 1.0(5) 104, respec-tively. Those results also prove that it is possible in present experiments to produceand observe almost pure condensates with about 1000 atoms.

10.3Perspectives for Cryogenic Atom Chips: A New Realm of Investigations

We present now some experiments that could be undertaken in a near future withcryogenic chips. In Section 10.3.1 we show that many properties of the supercon-ducting film can affect the magnetic trap. Hence atom chips offer a new toolboxfor testing the different theoretical models of current flow and electrodynamicresponse in superconductors. Section 10.3.2 will be devoted to the integration ofultra-cold trapped atoms with standard superconducting elements like SQUIDs orcoplanar resonators. Finally Section 10.3.3 will discuss the possibility to trap on achip a single circular Rydberg atom excited from the ultra-cold cloud. Indeed thosehighly sensitive quantum states require cryogenic conditions for coherent opera-tions over long times.

10.3.1Probing the Superconducting Film Current Distribution

The results of Section 10.2 demonstrate that it is possible to produce an ultra-coldatomic cloud close to superconducting films at low temperature. The propertiesof such a surface are dramatically different from those of a normal metal. For ex-ample it is well known that a superconducting material has the ability to generatepermanent currents in order to screen an applied magnetic field (Meissner effect).In the case of an atom chip, those currents are likely to affect the trapping poten-tial [13] and reduce its depth when the atoms are brought very close to the surface.In the experiment of [13] atoms are trapped close to a relatively large cylindrical su-


perconducting wire which is well described by the Meissner state. But most of thematerials used in present chips (see Section 10.2.1.1 are thin films of superconduc-tor in the mixed state, with a magnetic flux which partially penetrates through thematerial. This case is treated in [14] where the Bean Model of the superconductingmixed state is used to describe the current distribution. One of the most importantpredictions of this model is that the current flow maintains a memory of its previ-ous physical states and hence exhibits a hysteretic behavior. This is very differentfrom the Meissner case, where the current distribution corresponds to a thermalequilibrium and shows no memory effect. Evidence of this phenomenon has beenrecently observed with the ENS setup: permanent currents in the superconductingfilm strongly distort the trapping potential seen by the atoms as in [13]. However,the current distribution depends dramatically on the magnetic field applied to thechip at the time of its transition to the superconducting state. As a consequence itis possible to set completely different trap configurations by controlling the valueof this applied field [15]. The agreement with the Bean Model is very good.

The presence of vortices could also introduce a corrugation of the trapping poten-tial at distances of the order of the inter-vortex separation. The latter depends on thelocal applied magnetic field and ranges typically from 0.1 to a few µm [16]. It wouldbe fascinating to access the distribution and the dynamics of those structures.Present microscopic magnetic-field imaging experiments with trapped BEC [17]on atom chips (see Chapter 7) could provide a way to make those observations.

The response of the superconducting film to an AC electromagnetic field is alsoof importance. Chapters 4 and 5 explain in great detail the interaction of an ultra-cold cloud with a surface at micrometer distance. Thermal current fluctuations inthe film result in a random magnetic field at the cloud position. The latter can berelatively strong because the radiation takes place in the near-field regime. Somespectral components at RF frequency for the random field can induce atomic tran-sitions towards untrapped high-field-seeking Zeeman sublevels, in which case theatom is expelled from the trap. The fluctuation-dissipation theorem lets us guessthat the loss rate from the trap is directly related to the dissipation on the surfaceat the Zeeman transition frequency. The latter is in the kHz to MHz range, that is,much smaller than typical superconducting bandgap frequencies. Hence one ex-pects to observe very low dissipation and significantly longer trapping times for su-perconductors compared to normal metals (for a detailed review, see Section 4.3.2).Moreover, measuring the trapping time close to superconducting bodies will offera way to test the different theoretical models for the electrodynamics of supercon-ductors [18, 19]. For example, the presence of vortices in the superconducting filmsused so far could possibly affect the spin-flip rate of the atoms. As long as the filmis superconducting the vortices are pinned. However, the random hopping of thevortex lines from one pinning center to another at finite temperature generates anadditional flux noise [16]. Moreover, the response of the vortex lattice to an elec-tromagnetic wave at the Zeeman transition frequency adds an additional term ofdissipation. Recent calculations suggest that it could be the dominant source of lossfrom the trap [20].


10.3.2Integration of Atom Chips with Superconducting Circuit Elements

10.3.2.1 Coupling with a Superconducting QubitWe have just seen that the presence of vortices in the superconducting film willprobably play a crucial role in the atom–surface interaction. A clear imaging of thevortices as in [17] remains, however, very challenging. A possible way to overcomethis difficulty could be to design a larger piece of superconductor (with dimen-sions around 10 µm) through which magnetic flux is quantized. This can be veryeasily achieved by making a simple loop of superconducting material. The mostimportant point here is that, in the particular case where the loop is interrupted bya Josephson-junction (a very thin isolating barrier), it becomes possible to createquantum superpositions of different flux-states corresponding to different super-currents in the loop. The characteristic energy scale for this phenomenon is theJosephson energy J. It can be precisely controlled by replacing the single junctionby two split junctions, in which case the device is called a DC-SQUID. Such sys-tems are very powerful quantum circuit elements. They can be operated in a fullycoherent way if the temperature T is lower than J/ kB. They have proved to be verygood candidates for the physical realization of a quantum bit (qubit) [21].

Reference [22] studies the interaction of a BEC close to a DC-SQUID. The ex-perimental situation considered is presented in Figure 10.5. The diameter of theSQUID is 10 µm. A field B0 of 100 mG perpendicular to the loop is enough to ap-ply a half quantum of flux Φ0/2 D h/4e through it. It is, therefore, easy to prepareit in a superposition of two flux-states. One can consider without loss of generalitythat the flux inside the loop is either 0 or Φ0 (states j0i and j1i). Hence the SQUIDstates correspond to two different values of supercurrent in the loop with equal am-plitude and opposite directions. Let’s assume now that atoms in a BEC are trapped10 µm away from the loop. The magnetic field at the bottom of the trap can beadjusted with an external bias field along the x-direction. But the field created bythe loop adds a contribution along the x-direction and distorts the potential. Thedisplacement of the trap center depends on the supercurrent and one observes thatthe distance between the minima of the two resulting perturbed configurations as-sociated to j0i and j1i is of the order of the diameter of the superconducting loop.Moreover, the amplitude of the field perturbation is 5 mG which is much larg-er than the typical chemical potential for the BEC presented in Section 10.2.2.2.

µ

Figure 10.5 Experimental conditions of [22].


This means that the atoms of the BEC can be in either one of two distinguish-able modes corresponding to the fundamental level for two well-separated traps. Ifone adiabatically brings the cloud close to the chip, the final quantum state for theSQUID+BEC system is entangled and can be written as [22]:

jΨ i D 1p2

(j0i ˝ j0, Ni01 C j1i ˝ jN, 0i01) , (10.1)

where j0, Ni01 is the BEC state representing zero atoms in one perturbed config-uration of the trap and N atoms in the other configuration. The final SQUID-BECstate is an entangled state of both systems. Strong correlations between the inter-nal state of the SQUID and the position of the condensate are expected. However, ifone tries to observe the atomic cloud directly, this operation corresponds to a localoperation on one of the two subsystems. One needs to trace the different SQUIDstates and the BEC is then a mere statistical mixture of the two modes. It is, how-ever, possible to go beyond that limitation. For example, manipulating the SQUIDstate and measuring it makes it possible to perform a projective measurement forthe SQUID in the j0i ˙ j1i state basis [21]. Because of the strong correlations be-tween the two parts of the entangled state, the BEC is then projected onto:

jΨ iBEC D 1p2

(j0, Ni01 ˙ jN, 0i01) , (10.2)

which is a coherent superposition of two macroscopically distinguishable states, orin other words a “Schrödinger-cat state”. Such states are very important from thefundamental point of view as they provide a tool to explore the frontier betweenthe microscopic and macroscopic worlds. For example, the loss of a single atomfrom the trap is enough to destroy the quantum superposition. Very long trappinglifetimes are therefore necessary. Another problem is that the magnetic potentialseen by the atoms is changed as soon as the SQUID state is determined. As aconsequence, the two modes of the BEC in Eq. (10.2) are no longer stationary andwill evolve and interfere in a complex manner. Moreover, the coherent operation ofthe superconducting qubit requires that the performances of the cryogenic systemare pushed to temperatures no larger than 100 mK. This is a very challenging andstimulating proposal as it combines two emerging fields of physics: atom chips andsuperconducting qubit manipulation.

10.3.2.2 Coupling with a Superconducting Resonator: On-Chip CQEDMany recent experiments on superconducting qubits have been realized by cou-pling them to an on-chip microwave resonator [23] (see Figure 10.6). The latter ismade of a coplanar waveguide: a slab of superconducting material between twolateral ground planes which is interrupted at each end in order to create two “mir-rors.” It offers ideal conditions for observing the strong coupling regime: the modevolume is very small as the photon energy is “squeezed” in a 1D structure. TheQ factor at low temperature ranges from 104 to 106. It now becomes possible toreproduce with solid-state devices all the experiments which had been pioneeredwith atomic systems in the field of cavity quantum electrodynamics (CQED).


Figure 10.6 The circuit QED chip of [23] (a).A scheme of the principle of the chip is dis-played at the bottom right. It couples a copla-nar waveguide resonator with a supercon-ducting qubit. The resonator is made of a filmof superconducting niobium. A conductor isplaced centrally between two lateral ground

planes. In order to define a cavity, a “mirror”on both ends is made by cutting the line (b).The resonant frequency is 6 GHz (wavelength5 cm, which explains the large dimensionsof the chip). The Q factor is adjusted by tun-ing the capacitance of each mirror. It is of theorder of 104 in that case.

On the other hand, those circuit QED cavities could become very powerful toolsif one brings atomic systems into their vicinity. It appears quite natural to try tointegrate them with atom chips. As an example, it was recently proposed to use acoplanar waveguide as a means to couple two separate polar molecules [24]. Thelatter would be electrically trapped on the chip.

In order to evaluate the achievable coupling for this kind of experiment, let usconsider the case of an atom trapped at a distance z from the coplanar resonator ofFigure 10.7. The atom could be magnetically trapped with the help of an additionalon-chip wire (not shown in the figure) through which a DC current is flowing.Both electric and magnetic field amplitudes of the cavity mode decrease rapidlywith distance from the surface [24]. The characteristic length of this decay is ofthe order of the gap w between the central conductor and the ground planes. Asa consequence a significant coupling is possible only if z w . We assume thatthis is the case and limit ourselves to gap distances ranging from 1 to 10 µm. Inorder to account for the decay of the field we will assume that the electric field atthe cloud position E (resp. the magnetic field H) is only a fraction η D 0.1 of themaximum electric field in the chip plane E0 (resp. H0).

We will first study the hyperfine transition of 87Rb. This particular transitionhas been used for demonstrating the coherent manipulation of trapped atoms onan atom chip [25]. The cavity mode frequency ω0/2π is about 6.8 GHz. It is inthe frequency range in which circuit QED cavities have been realized so far, withQ factors as high as 106. We present here a simplified calculation of the atom-cavitycoupling. A more detailed study can be found in [26]. One can evaluate the vacuumRabi coupling g following the same procedure as in [24, 27]. The zero-point energy


(a)

(b)

(c)

Figure 10.7 Cross-section of the coplanar res-onator displaying the geometrical parametersof the transmission line (a). We consider thecase where the central conductor width s isof the order of the gap w. The corresponding

electric and magnetic field lines are shown in(b) and (c), respectively. Both field amplitudesdecrease rapidly with a characteristic length ofthe order of w.

of the resonator can be written as:

E0 D 12

„ω0 D 12

C V 20 C 1

2LI 2

0 , (10.3)

where V0 and I0 are the resonator voltage and current quantum fluctuations. C

and L are the resonator characteristic capacitance and inductance. If the resonatoris matched with standard microwave devices, its impedance is Z D 50 Ω and C Dπ/2ω0Z . For the considered frequency one gets V0 D p„ω0/2C D 1.75 µV andI0 D V0/Z D 35 nA. One can then calculate the magnetic induction at the cloudposition B D ηµ0H0 D ηµ0I0/w D 4.4 µG for w D 10 µm. Because the hyperfinetransition is only coupled through the magnetic dipole Hamiltonian the resultingvacuum Rabi frequency is g D µBB D 2π 6 Hz (µB is the Bohr magneton). Weare not in the strong coupling regime, g being 3 orders of magnitude smaller thanthe cavity linewidth D ω0/Q D 2π 6.8 kHz for Q D 106.

On the other hand the use of a hyperfine transition allows us to obtain very longatomic lifetime γ . In the case of present experiments, one can assume that γ islimited by transitions towards untrapped Zeeman levels or any other loss mecha-nism from the magnetic trap. In order to be conservative, we consider γ D 0.1 s1

at 10 µm. The corresponding cooperativity factor C0 D N g2/(2γ ), where N is theatom number coupled to the cavity, is equal to 0.03 for N D 1. This means that theonset of quantum effects is observable with about 30 atoms coupled to the cavity. ABEC with N D 104 atoms would clearly modify the cavity transmission spectrum.By reciprocity the atoms could act on the cavity field and shift its frequency, the


cloud playing in a way the role of a dielectric slab whose index of refraction de-pends on the atomic internal state. The measurement of the resulting phase-shiftby standard microwave homodyning techniques could then provide informationon the internal state of the BEC [26, 28].

Turning now to the case of an electric dipole transition, one expects the couplingto be much larger. In the case of a single dipolar molecule (transition frequencyaround 10 GHz), [24] evaluates a vacuum Rabi frequency of 40 kHz at 1 µm fromthe surface, which fulfills the conditions for strong coupling. It could be possiblealso to replace the molecules by highly excited atoms in Rydberg states [27]. Exper-iments on circular Rydberg atoms, highly excited quantum levels with maximumangular momentum, have proven them extremely sensitive systems to electric andmagnetic fields, both static and dynamic. As an illustration, the electric dipole onthe transition between two adjacent circular levels with principal quantum num-ber n D 50 and 51 (ω0/2π D 50 GHz, for more details see Section 10.3.3) isd D 1776 ea0, where e is the charge of the electron and a0 is the Bohr ra-dius. Its coupling frequency to the cavity described above is then g D ηdE0/„ DηdV0/„w D 2π 375 kHz for w D 10 µm. A strong coupling regime could beobserved if the Q factor does not degrade significantly at the transition frequency,which is still an open question.

10.3.3Atom Chips for Circular Rydberg States

Circular Rydberg states are ideal tools for the investigation of numerous quantumphenomena [29]. Its remarkable sensitivity, in conjunction with its relative ease ofdetection, makes a single circular Rydberg atom an excellent probe of microwavefield photon number ranging from one to a few tens [29] and a very good candi-date for implementation of simple quantum information algorithms. A circularstate with principal quantum number n can only emit a photon on a σ-polarizedtransition towards the lower circular state n 1. For n D 51 the correspondingfree-space radiative lifetime is 30 ms. However, it has been shown that cavity QEDeffects can dramatically inhibit the spontaneous emission and hence increase thestate lifetime.

A possible trapping geometry was proposed in [30]. It is presented in Figure 10.8.Trapping is achieved with the help of the strong quadratic Stark polarizability αof levels e and g [α 200 Hz/(V/m)2 for weak fields] in a field E approximatelyaligned with O z. Since circular states are high-field seekers, there can be no stat-ic field trap. Therefore, we turn to a dynamic scheme, reminiscent of the Paul iontraps (see Chapter 13), combining static and oscillating fields. Our design is compa-rable to that discussed in [31] for ground-state atoms. We note that atom chips havealready demonstrated the ability to trap ground-state atoms with electric fields [32].In both articles however, the required trapping fields are orders of magnitude largerthan that necessary for trapping highly polarizable Rydberg states.

The trap is composed of two chips facing each other made up of concentric elec-trodes. Different voltages can be applied to the electrodes. In Figure 10.8 the static


η

η

Figure 10.8 Section of the proposed circu-lar Rydberg trap in a vertical x Oz plane, withapplied potentials. The trap has a cylindricalsymmetry around Oz. The diameter of theinner electrode is 1 mm. The plate spacing,

1 mm, is appropriate for spontaneous emis-sion inhibition. The electrodes are shadedaccording to the phase of the oscillating po-tential U1.

potential U0 creates the homogeneous directing field E0u z . The potential U1 < U0,oscillating at frequency ω1, creates a smaller, AC, approximately hexapolar field E1.In order to cancel the amplitude of E1 at the origin O for all times, we apply thepotential ˙ηU1 to the outer electrodes. The factor η is determined by the elec-trode geometry. Calculations of [31] show that the resulting atomic Stark energy,α(E0u z C E1)2, has a roughly quadratic spatial dependence around the origin O.It looks, however, like a saddle-point potential with trapping and anti-trapping di-rections depending on the sign of U1. Stable trapping configurations are obtainedonly if one modulates U1. Finally, the yet smaller static potential U2 creates an ap-proximately quadrupolar field. This provides a force, nearly constant in the trapregion, compensating gravity (antiparallel to O z).

The main advantage of the above geometry is that radiative lifetime for bothlevels can be greatly increased by inhibiting their spontaneous emission. Let usassume that the atom is placed between two parallel infinite conducting plates ina geometry similar to the one of Figure 10.8. We apply a voltage across the plates(voltage ˙U0) in order to fix a quantization axis for the quantum levels. Hence bothlevels e and g only have a single decay channel: the emission of a millimeter-wavephoton circularly polarized with respect to the quantization axis. This emissiononly couples to the transverse electric (TE) modes of the cavity made by the twoplates. If the latter are separated by a distance d < λ/2 (λ 6 mm being theemitted photon wavelength), there exists no TE mode at the transition frequen-cy and the decay is inhibited. Levels e and g can simultaneously be made long-lived.

A detailed assessment of the trap performances is given in [33]. It relies on nu-merical simulations of the electric field created by the various voltages in the setup.Then we numerically integrate a set of atomic trajectories with randomly choseninitial conditions compatible with an ultra-cold cloud in a standard magnetic mi-crotrap at an adjustable temperature T0. For U0 D 0.2 V, U2 D 0.003 V and ahexapolar potential U1 D 0.155 V oscillating at ω1 D 2π 430 Hz, the resultingmotion is the combination of a fast micromotion, at frequency ω1/2π, with a slowoscillation whose longitudinal (along O z) and transverse (orthogonal to O z) fre-quencies are 175 and 64 Hz respectively. For such low frequencies the quantizationaxis for the atomic levels follows adiabatically the space- and time-dependent di-


rection of the local electric field. At T0 D 90 µK, the motion has an extension ofabout 100 µm. We have checked that for T0 > 100 nK, the atomic excursion in theRydberg trap is much larger than the de Broglie wavelength, allowing a classicaltreatment of the trajectory. The trap depth, Td, which we define as the T0 value forwhich half of the atoms remain within 400 µm of the origin, is Td D 180 µK, wellwithin the reach of standard ultra-cold atom experiments. Deeper traps, with trap-ping frequencies in the kHz range, are achievable with higher but still moderatevoltages (U0 around 10 V).

In order to take full advantage of a Rydberg atom microtrap for quantum infor-mation or high-precision spectroscopy, one needs to keep a superposition of states e

and g coherent over the longest possible time. The inhibition of spontaneous emis-sion limits the effect of radiative decoherence for both states. Taking into accountmany possible imperfections to the inhibition, [33] estimates a radiative decoher-ence time of about τr D 3 s in the particular case of a superposition of states e and g.The most important cause of decoherence is actually the absorption of blackbodyradiation photons. The value of τr is reached only if the trap is cooled down to 1 K,corresponding to 0.1 mean thermal photon at the Rydberg transition frequency.Very low temperatures are therefore required.

However, the main source of dephasing in the microtrap is due to the slightlydifferent Stark polarizabilities of levels e and g. Figure 10.9 shows the energy levelsas a function of the electric-field amplitude. The frequency ν eg of the transition hasa strong dependence on the electric-field amplitude E, δν eg D ν eg(E ) ν eg(0) D25.5 Hz/(V/m)2 E 2. If one considers the case of an atom prepared from a cloud

Electric field (V/m)0 200 400 600 800

–0.8

–0.4

0.050.0

50.4

50.8

51.2

×5

×5

|e⟩

|i⟩

|g⟩Freq

uenc

y (G

Hz)

Figure 10.9 Energies of levels g, e, and i as afunction of the electric field. Circular states gand e have a small, approximately quadraticStark effect, magnified in this figure by a factor5. Level i has a much larger, linear Stark effect.

An additional microwave field (vertical arrow)predominantly mixes g and i and reduces thedifference in Stark polarizability between e andthe resulting dressed level Qg.


at 0.3 µK under the trapping conditions previously presented, it experiences overits trajectory a mean electric-field amplitude Ea D 400 V/m, with an excursion of∆E D ˙1 V/m. The corresponding frequency broadening ∆ν eg D 2δν eg Ea ∆E isof the order of 20 kHz. This broadening is inhomogeneous as it differs from onetrajectory to another and could only be controlled by perfect control of the initialconditions of the atomic motion. The dephasing time associated to this motionalbroadening is of the order of a few tens of µs. In addition to this dephasing, thetrapping frequencies (ω, ωz ) for states e and g differ by about 10 %. The trajec-tories for the two states are, therefore, rapidly separated (‘Stern–Gerlach’ effect).Coherence would be lost should this separation exceed the wave-packet coherencelength, of the order of the de Broglie wavelength (about 0.5 µm for the conditionsconsidered).

Similar broadenings are observed in many traps for ground-state atoms, wherethe potential energy is also, in general, level-dependent. However, a proper choiceof external parameters like bias field or laser frequency can reduce significantly thiseffect. This possibility was clearly demonstrated in [25] in the context of magneticatom chips, paving the way for future atom chip clock devices (see Chapter 8). Weproposed the artificial re-creation of a similar situation by shining an additionalmicrowave field, linearly polarized, on the trapped atom [30]. To first approxima-tion, this field couples g to the Rydberg state i (n D 51, m D mg D 49), whichexperiences a large linear Stark effect [polarizability 1 MHz/(V/m)]. The Stark po-larizability of the ‘dressed’ eg level is thus significantly modified. The dressing fieldhas a smaller effect on e which is far off-resonance. The Rabi frequency Ω0 and thedetuning δ0 of the dressing microwave can therefore be tailored to cancel the first-and to minimize the second-order terms in the expansion of ν eg(E ) around Ea. Thecalculation of the energy shift for levelseg and e is given in [33]. The optimal param-eters Ω0 and δ0 have reasonable values and limit the variation of the ‘dressed’transition frequency to 10 Hz over the ˙1 V/m electric field range explored byan atom for T0 D 0.3 µK. The level dressing reduces the transition broadening byover 3 orders of magnitude. Combined with standard refocusing techniques, likespin-echoes, it allows one to preserve the coherence of a state superposition fortimes as long as one second.

10.4Conclusion

The operation of atom chips in a cryogenic environment has become a reality. Thepresent experimental observations are very similar to the results usually found forroom-temperature systems, although very long trapping times have already beenobserved. However, the understanding and control of permanent currents for trap-ping the cloud has already provided initial evidence that new phenomena, that onlyexist at low temperature, significantly affect the atomic ensemble. The coupling ofsuperconducting micro- or nanostructures like SQUIDs or coplanar waveguidesto the ultra-cold cloud will certainly lead to very interesting results in the coming

10.4 Conclusion 329

years. We also think that cryogenic conditions offer an ideal environment aroundthe atoms for preparation in exotic states like Rydberg levels. A new realm of inves-tigations, with important results for both atomic and solid-state physics, is open-ing.

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atom chips (reichel:atom chips o-bk) || cryogenic atom chips

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