Effect of vortices on the spin-flip lifetime of atoms in superconducting atom-chips
Gilles Nogues, Cédric Roux, Thomas Nirrengarten, Adrian Lupascu, Andreas Emmert, Michel Brune, Jean-Michel Raimond, Serge Haroche, Bernard Placais, Jean-Jacques Greffet
aa r X i v : . [ qu a n t - ph ] J u l epl draft Effect of vortices on the spin-flip lifetime of atoms in supercon-ducting atom-chips
G. Nogues , C. Roux , T. Nirrengarten , A. Lupas¸cu , A. Emmert , M. Brune , J.-M. Raimond , S.Haroche , , B. Plac¸ais and J.-J. Greffet Laboratoire Kastler Brossel, ENS, UPMC, CNRS - 24 rue Lhomond, 75005 Paris, France, EU Coll`ege de France - 11 place Marcelin Berthelot, 75005 Paris, France, EU Laboratoire Pierre Aigrain, ENS, UPMC, CNRS - 24 rue Lhomond, F-75231 Paris Cedex 05, France, EU Laboratoire EM2C, Ecole Centrale Paris, CNRS, Grande Voie des Vignes, 92295 Chˆatenay-Malabry, France, EU
PACS – Atom traps and guides
PACS – Quantum description of interaction of light and matter; related experiments
PACS – Vortex dynamics
Abstract. - We study theoretically the lifetime of magnetically trapped atoms in the close vicinityof a type-II superconducting surface, in the context of superconducting atom-chips. We accountfor the magnetic noise created at the cloud position by the vortices present in the superconductorand give specific results for our experiment which uses a niobium film. Our main result is thatatom losses are dominated by the presence of vortices. They remain however dramatically smallerthan in equivalent room-temperature atom-chip setups using normal metals.
Atom chips allow to trap ultracold atomic gases in thevicinity of micron-sized current carrying wires [1] or per-manent magnetic structures [2]. Microfabrication tech-niques allow to design complex trapping potentials and torealize versatile manipulation of atoms thanks to the con-trol of currents or radiofrequency fields in the vicinity ofthe trapped cloud [3, 4]. Atoms chips are now consideredas a powerful toolbox that can be used for fundamentalstudies [5,6], atomic interferometry [7,8] or quantum gateimplementation [9].In many such experiments, atoms are required to bevery close to the surface of the trapping structures. Un-fortunately, additional losses from the trap are experi-mentally observed in these conditions [10, 11]. Johnson-Nyquist current noise in the trapping metallic wires pro-duce magnetic field fluctuations at the position of theatoms, which can induce Zeeman transitions towards un-trapped magnetic sublevels. This phenomenon is stronglyenhanced in the near-field of conductors for typical spin-flip radiofrequencies (in the MHz range) [12,13]. The typ-ical geometry of these experiments is presented in Fig. 1.A possibility to overcome these difficulties consist in usingcryogenic atom-chips made of superconducting materials,for which dissipation at RF frequencies, and hence fluctua- (a)
E-mail: [email protected] tions, are dramatically reduced [14]. Successfull operationof superconducting atom-chips has been reported [15–17],with the aim of developping new hybrid atomic–solid-statesystems. Concurrently, theoretical studies have madequantitative prediction for the lifetime increase with re-spect to normal metals. However they strongly dependon the model of superconductivity which is used in thecalculation. Most recent articles agree to predict an en-hancement of at least 6 orders of magnitude [18–20].In this Letter, we emphasize the importance of thevortex dynamics in the superconducting material on theatomic losses. In current atom-chip experiments, DC mag-netic fields of the order of 10-100 G are applied orthogo-nally to thin layers of type-II superconductors. Hence, weexpect the thin film to be in the mixed-state phase, withvortices present in the superconducting material. Ref. [21]showed that the random hopping of a vortex line from apinning site to another could affect the trap lifetime. Westress here that, in addition, the motion of the vortex linesubmitted to a RF field is in itself responsible for dissi-pation, a phenomenon known as “flux-flow”. However,this motion is partially suppressed because of the pin-ning of the vortices on material defects. Typical spin-flipfrequencies are significantly smaller than vortex pinningcharacteristic energies (in the GHz range). Many theoret-p-1. Nogues et al. (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:5)(cid:8)(cid:9) (cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:6)(cid:7)(cid:13)(cid:13)(cid:8)(cid:15)(cid:16)(cid:7)(cid:5)(cid:17)(cid:18)(cid:15)(cid:11) (cid:19) (cid:20)
Fig. 1: An atom in an initial state | i i is trapped at position(0 , , d ) in vacuum near a film of metal or superconductor ofthickness h . We assume that the substrate under the film isdielectric and does not affect the atomic lifetime. At the levelof the atomic cloud, an external magnetic field sets the quan-tization axis along the x direction. ical and experimental studies have already been carriedout on the dissipation of type-II superconductors in thislow-frequency regime. We note that observations are ac-counted for only if one assumes a non-local response of thematerial to an applied electromagnetic field. We presentin this letter an adaptation of the theoretical frameworkdeveloped in Ref. [12] to this particular non-local situa-tion. We have adapted the theory describing the vortex-dynamics in niobium [22–24] to the situation of atom-chips. On the basis of the measurements of the previousreferences, we evaluate quantitatively the influence of vor-tex dissipation on atomic lifetime for the particular caseof our superconducting atom-chip experiment [15, 17]. Spin-flip lifetime calculation. –
Let us first con-sider the simple case of an infinite thickness conductingslab ( h → ∞ ). The atom can decay towards an untrappedstate | f i , ω being the frequency of the i → f transition.We define k = ω/c . The contribution of a semi-infinitespace to the spin-flip rate can be calculated in term of thefield Green’s functions [12] which is equivalent to evalu-ating the field radiated by the atom onto itself [25]. Thisfield can be decomposed into propagating and evanescentplane waves (Weyl decomposition). Each of these wavesis reflected by the surface according to Fresnel laws beforegoing back to the atom. One obtains [18]:Γ slab if ( ω ) = Γ if ( ω )( n th + 1) 38 Re (cid:18)Z ∞ dq qη ( q ) (1) × e iη ( q ) k d (cid:2) r p ( q ) − η ( q ) r s ( q ) + 2 q r s ( q ) (cid:3)(cid:17) , where Γ if ( ω ) = µ ( µ B g S ) k / (24 π ~ ) is the spin-flip ratein vacuum, n th = 1 / ( e ~ ωkBT −
1) is the mean photon numberat frequency ω , µ B is the Bohr magneton and g S the gy-romagnetic factor of the electron. The integration factor q is such that qk is the modulus of the wave vector compo-nent parallel to the surface. Evanescent waves correspondto q >
1. If we now consider the case a material described bya local dielectric permittivity ε ( ω ), the polarization-dependent Fresnel coefficients are: r s ( q ) = η ( q ) − η ( ω, q ) η ( q ) + η ( ω, q ) , r p ( q ) = ε ( ω ) η ( q ) − η ( ω, q ) ε ( ω ) η ( q ) + η ( ω, q )(2)where η ( q ) = p − q and η ( q, ω ) = p ε ( ω ) − q . Fora metal described by the Drude model, the permittivity,much larger than 1, is related to the local conductivity σ : ε ( ω ) = 1+ iσ/ ( ε ω ) ≈ iσ/ ( ε ω ). The characteristic lengthassociated with the material response is the skin-depth δ = p / ( µ σω ), typically in the µ m range for good conductorsat rf frequencies. The semi-infinite slab assumption holdsthen for h ≫ δ .In the near field regime, we have 1 /k ≫ d . Moreover ifwe make the experimentally reasonable assumption d ≫ δ ,the main contribution to the integral of Eq. (1) is for q values such that 1 ≪ q ≪ p | ε ( ω ) | . One thus obtains anasymptotic expression of Eq. (1) with η ≈ iq and η ( ω ) ≈ p ε ( ω ) leading to an analytical expression for Γ if [14]:Γ if ( ω ) ≈ Γ if ( n th + 1) Re " p ε ( ω ) k d (3) ≈ Γ if ( n th + 1) (cid:18) δk d (cid:19) , (4)For the case of superconductors, different theoreticalmodels for the material response have been used in orderto evaluate the spin-flip rate [18–20]: the phenomenologi-cal two-fluid model, the BCS microscopic model, and theEliashberg theory which takes into account the scatteringof Cooper pairs by the phonons. All those models predicta local complex conductivity σ = σ + iσ . As soon as thetemperature T is significantly below the critical tempera-ture T c , one has σ ≫ σ . Equation (3) then becomes [18]:Γ if ( ω ) ≈ Γ if ( ω )( n th + 1) √ ωµ k d σ σ / ! . (5)The spin-flip rate is then reduced by more than 6 ordersof magnitude as compared to the case of normal metal atsimilar temperature. Adaptation to the case of a type-II superconduc-tor. –
We turn now to the situation where vortices arepresent in the film. In order to find a relation betweenthe vortex dynamics and the electromagnetic radiation,we assume that the atom-surface distance is much largerthan the intervortex distance a . In this situation, theresponse of the vortex lattice to an electromagnetic fieldcan be treated like that of a continuous complex hydro-dynamic system, taking into account vortex pinning aswell as vortex interactions [26–28]. The theoretical mod-els that consider a local response of the mixed-state top-2ortices and spin-flip lifetime in superconducting atom-chipsthe electromagnetic field [29,30] underestimate the vortexdissipation at low frequency. It is necessary to considera non-local response of the superconductor [22, 23] whichmakes it impossible to define a local dielectric constantor a local conductivity for describing the material. HenceEqs. (2-5) do not apply directly.In the case of a non-local response of the supercon-ductor, dissipation is well described in term of surfaceimpedance Z S = µ E S /B S at frequency ω , where E S and B S are the tangential electric and magnetic fields onthe surface respectively. The use of Z S allows to includethe detailed microscopic response of the superconductingmedium into a linear local relation between the tangen-tial electric field at the surface and the surface current −→ K = −→ E S /Z S . The rate Γ if being related to the dissipa-tion in the material, we expect it to be proportional to Re ( Z S ).In order to find the relation between Γ if and Z S ,we express the Fresnel coefficients in terms of surfaceimpedance: r s ( q ) = η ( q ) Z S − Z η ( q ) Z S + Z , r p ( q ) = Z η ( q ) − Z S Z η ( q ) + Z S (6)where Z = µ c is the vacuum impedance. For the radia-tion of a dipole at a distance d the wave vector amplitude k q values mainly contributing to the integral in Eq. 1 areof the order of d − . For all reasonable superconductormodels and for the range of distances considered above( d ≫ µ m), these wave vectors are much smaller than theinverse of the field penetration depth at the frequency ω .The surface impedance Z S is then independent of q andequal to its value at normal incidence. Substituting Eq. 6into Eq. 1 and performing the integration is equivalent tothe substitution p ε ( ω ) = Z /Z S in Eq. 3:Γ if ( ω ) ≈ Γ if ( ω )( n th + 1) (cid:20) ωµ k d Re ( Z S ) (cid:21) . (7)Note that that if we replace the surface impedance byits standard value in the case of a normal metal Z metS =(1 − i ) / ( σδ ), we exactly recover Eq. (4).Let us stress that, within our approximations, the sur-face impedance Z S allows to calculate exactly the fieldradiated by the atom in the z > Two-mode non-local response of the vortex lat-tice. –
In order to derive Z S for a type-II supercon-ducting material, we evaluate the vortex response to anexternal oscillating magnetic field. As we restrict our-selves to the situation where the distance d is larger thanthe intervortex distance, it is possible to consider aver-aged macroscopic quantities for local electric and magneticfields as well as for supercurrent densities. In this frame,the response of the superconductor can be derived froman equivalent of the Ginzburg-Landau free energy relat-ing the macroscopic quantities [26]. For this purpose, it is necessary to introduce the vortex field −→ B = n V ϕ −→ ν ,where −→ ν is the local direction of the vortex lines, n V the density of vortices per unit area and ϕ = h/ e isthe quantum of flux. −→ B is related to the macroscopicmagnetic field −→ B by the generalized London equation −→ B + µ λ L −→∇ × −→ V s = −→ B , where −→ V s is the averaged macro-scopic velocity of the Cooper-pairs and λ L is the Londonlength. In absence of vortices ( B = 0), one recovers theLondon equation which leads to the Meissner effect. Inthe mixed state, a fraction −→ B of the applied magneticfield penetrates the film. In the presence of a non-uniformsupercurrent ( −→∇ × V s = 0) vortex and magnetic field linesdo not coincide.We consider now the case of an electromagnetic fieldarriving at normal incidence on the superconductingmedium according to the geometry of Fig. (2). The vor-tex lattice is initially in its equilibrium position −→ B = −→ B = B −→ e z . The magnetic and vortex fields experiencea small perturbation −→ B ( z ) = B −→ e z + b exp ( ikz − iωt ) −→ e x , −→ ν ( z ) = −→ e z + ν exp ( ikz − iωt ) −→ e x , where | ν | is the an-gle of the vortex line with the z -direction on the surfaceand | ν ( z ) | the same angle at the position z . Refs. [27, 28]show that the field in the medium is a superposition oftwo propagation modes with wave vector k f and k V . Thecharacteristic length of penetration δ f associated to thefirst mode is related to the flux-flow dissipation when avortex line is moving: δ f = s µ σ f ω ≈ s ρ N µ ω BB c . (8) B c is the second critical magnetic field of the material,and ρ N is the resistivity of normal electrons in the mate-rial at temperature T . As a consequence δ f is of the or-der of typical skin-depths for metals at low temperature.We define the complex penetration length λ f = 1 /ik f = δ f (1 + i ) / λ V = 1 ik V = λ L r µ ǫ l B + µ ǫ l . (9)where ǫ l is the vortex line potential [26]. The quantity ϕ ǫ l is the energy that is required to enter one unit oflength of vortex into the medium. The penetration length λ V is of the order of λ L . In usual superconductors one has λ V ≪ δ f . The second mode describes the non dissipative-screening of the magnetic field by supercurrents locatedclose to the surface.The incoming field must be decomposed onto thesetwo modes of propagation. For each mode we can de-fine the displacement of the vortex line at the surface u j =( f,V ) = R −∞ ν j ( z ) dz . The relative weight of each modeis determined by the boundary conditions at the surfacefor the magnetic and vortex fields. It strongly depends onthe surface geometry of the sample. We present in Fig. 2p-3. Nogues et al. ν (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:1)(cid:1)(cid:6)(cid:3)(cid:4)(cid:5)(cid:1) νλ (cid:1) λ (cid:1) δ (cid:2) δ (cid:2) (cid:2) (cid:7) (cid:2) (cid:7) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:1)(cid:1)(cid:6)(cid:3)(cid:4)(cid:5)(cid:1)(cid:4) (cid:4)(cid:6) (cid:7) (cid:6) (cid:7) (cid:8) (cid:3)(cid:4)(cid:5) (cid:3)(cid:6)(cid:5) (cid:9) Fig. 2: Amplitude of the transverse magnetic field b (solid line)and displacement of a vortex line u (dashed line): (a) in thecase of a flat surface, (b) in the case of a rough surface withcharacteristic length l . For both situations, the vortex direction ν must end perpendicularly to the surface. two different cases of propagation in the superconduct-ing medium for a perfectly flat (a) and a rough surface(b). In both cases the vortex field must end perpendicu-larly to the surface. In the second case its displacementis prevented by its pinning on surface defects. It was pro-posed in Refs. [22, 27] to link the vortex displacement atthe surface u = u f + u V to the angle ν by the relation u + lν = 0, where l is the phenomenological slippagelength that characterizes the material. Following Fig 2(b),one clearly sees that l is related to the roughness of thesurface but it also takes into account the interaction be-tween vortices that forces a collective response of the wholelattice [24].The complete determination of the surface magnetic andelectric fields, together with the vortex displacement ispresented in Refs. [27, 28] for a infinite half-space. Theresulting surface impedance is: Z ∞ S = − iµ ω B ( l + λ V ) λ f ( B + µ ǫ l )( l + λ V ) + µ ǫ l λ f (10) Finite-thickness effects. –
As shown below, in ourexperimental conditions, the slab thickness h =1 µ m isof the order of or smaller than δ f . The superconductingmedium thus cannot be described as a half-space and finitesize effects have to be taken into account. In the case ofa type-II superconductor, they have been studied boththeoretically and experimentally [23], in a regime wherethe magnetic field is the same on each side of the slab. Itdoes not correspond to our experimental situation becausethe presence of the magnetic dipole breaks the symmetrybetween the two sides of the slab.We have calculated the surface impedance Z S in thecase of the reflection of an incident electromagnetic wavearriving at normal incidence on a superconducting slab offinite thickness. It requires to take into account 7 modes ofpropagation: the incident, reflected and transmitted fieldsas well as the evanescent propagation modes in the film E / E i B / B i z (nm) 0 50 100 150 200 250 0 0.5 1 1.5 2 2.5 3 R e [ Z s ] ( n W ) h/ d f
10 Gauss20 Gauss50 Gauss100 Gauss
Fig. 3: (a) Electric (solid line) and magnetic (dashed line) fieldamplitudes in the superconducting slab of thickness h =1 µ m.The superconductor characteristic parameters δ f , λ V and l cor-respond to the niobium layer used in our experiment (see sec-tion “Numerical results”). The bias field applied perpendicu-larly to the slab is B = 100 G. The fields are expressed in unitsof the incident electromagnetic field amplitude. (b) Real partof the surface impedance as a function of h (in units of δ f ) fordifferent external bias field B with wave vectors ± k f and ± k V . Figure 3(a) presents theanalytical results for the variation of the electric and mag-netic fields inside the superconducting film. We observethat only a very small part, of the order of 10 − , of the in-cident field penetrates the slab, as expected for any metal.Moreover, most of the magnetic field is screened by super-conducting currents carried by the the modes ± k V whichdo not dissipate. Hence an even smaller fraction of the in-coming wave is dissipated by the vortex displacement andcontributes to Re ( Z S ).We present in Fig. 3(b) the calculated real part of Z S as a function of the slab thickness for different externalfield values B . We recover the value Re ( Z ∞ S ) of Eq. (10)as soon as h & δ f . In our experiment, we are in theopposite limit h ≪ δ f , for which we fit the results withthe phenomenological formula: Re [ Z S ( h )] ≈ hδ f Re ( Z ∞ S ) (11) Numerical results. –
In the specific case of our su-perconducting atom-chip [15, 17], we can evaluate all rel-p-4ortices and spin-flip lifetime in superconducting atom-chipsevant parameters and determine Z S . Resistance mea-surements of the sputtered Nb film give a normal re-sistivity above transition to the superconducting state ρ N =15 µ Ω cm. The comparison with measurements atroom temperature gives a residual resistance ratio (RRR)of 4.6, which indicates that the film is in the so-called“dirty limit”. Measurements of the critical magnetic fieldfor similar films in this limit gives B c = 4 . · G, a fac-tor 15 larger than the pure case value [31]. To first orderthe product B c B c is a quantity that weakly depends onthe quality of the film and remains almost constant [32].Hence, we expect B c to be a factor 15 smaller than thepure case value [33], i.e. B c = 80 G. The vortex po-tential ǫ l depends on the applied magnetic field B . Fornormal operation of an atom-chip, we fulfill the condition B ≈ B c ≪ B c and we have µ ǫ l ≈ . B c ≈
70 G [34].A microscopic calculation of l is out of the scope of thisLetter. It can be found in [24]. Nevertheless, it is possi-ble to evaluate this quantity by linking it to the criticalcurrent in the superconducting slab I c = RR j s d r . Onthe one hand, we have I c ≈ wǫ l ν c [23], where w ≫ h is the width of the slab and ν c is the maximum angle atwhich the vortex line can bend before their pinning on thesurface breaks. On the other hand, the critical angle isreached when the displacement u is of the order of the in-tervortex separation a = p ϕ /B . Combining those twoequations with the boundary condition between u and ν ,we obtain: l = ǫ l wI c r ϕ B . (12)We will now consider that the atoms are trapped abovethe h =1 µ m thick superconducting Z-wire of Ref. [15].We consider this finite width conductor ( w =40 µ m) asan infinite film, leading to a worst case estimate for thespin-flip rate. We assume that an homogeneous externalbias field B = 100 G is applied perpendicularly to theslab. This crude assumption corresponds to a worst casesituation. Another external magnetic field parallel to thesurface determines the spin-flip frequency ω = 2 π × δ f =9 . µ m and using Eq. (9) we obtain λ V = 29 nm (with λ L = 45 nm [33]). In presence of a cur-rent in the slab, the value l can be obtained by replacing I c by I c − I in Eq. (12) where I = 1 . I c = 1 .
76 A).We get l = 250 nm. Figure 4 compares the spin-fliplifetime τ = Γ − if derived from Eqs. (11) and (7) as afunction of the distance d . These predictions are com-pared to our measurements for a gold layer of thickness h = 200 nm [35]. We also present the results of the BCSmodel of Refs. [18–20].Our predictions stand in a regime where the distance d ≫ δ f , λ V , a . The results of Fig. 4 are therefore validfor d & µ m. In this regime the vortex dissipation is thelimiting factor for the spin-flip lifetime. We find a reduc-tion of about 3 orders of magnitude of the lifetime withrespect to the two-fluid model. However, superconducting t ( s ) d ( m m) two-fluid modelReal Nb film, finite thicknessReal Nb film, infinite thicknessIdeal Nb clean-limit, finite thicknessAu at 4.2K, infinite thickness Fig. 4: Spin-flip lifetime as a function of atom-surface dis-tance as predicted by Eq. (7) (solid line). It takes into ac-count the measurements of our real superconducting film prop-erties and its finite thickness h =1 µ m. The dashed lines cor-respond to two ideal cases where the thickness of the film is in-finite (long dashes), or its superconducting properties are ideal(clean limit, short dashes). The dotted line corresponds tothe superconducting BCS model [19]. The experimental pointscorrespond to our measurements for a Gold layer of thickness h = 200 nm [35]. They are in good agreement with the pre-dictions of Eq. (4), with σ Au, . K = 6 . × Ω m(dot-dashedline)
Nb remains significantly better than normal metals.We have also represented in Fig. 4 the predicted life-times in the case of an semi-infinite superconducting film,where the surface impedance is given by Eq. (10), or fora film with ideal purity and surface quality (clean limit).It is interesting to note that these two situations corre-spond to a degraded lifetime compared to the dirty film.This might seem paradoxical. In this clean superconductorregime however, surface pinning is significantly reduced,the vortices can move with a larger amplitude. In addi-tion the change of the normal fluid conductivity increasesthe viscous drag of the vortex lattice. Two phenomenonstherefore contribute to the increase of the dissipation.We also note that we have considered here a worst caselimit where we assume that all the surface is subjectedto a an external field of 100 G and a homogeneously dis-tributed current 1.4 A. A more precise calculation of thethe atomic losses should include the inhomogeneous cur-rent and vortices distribution on the surface. Neverthe-less, our results show that vortices play a crucial role inthe spin-flip lifetime of atoms in the close vicinity of atype-II atom-chip.In conclusion we have adapted the formalism of atomicspin-flip lifetime of an atom close to a metallic surface top-5. Nogues et al. the non-local electrodynamic response of the vortex latticein type-II superconductors. Note that this model shouldalso described type-I superconductors, as a thin film ofsuch a material will contain vortices. On the other hand,lifetime close to superconducting materials remains signif-icantly better than close to normal metals. Our resultspredict a lifetime of 10000 s at 20 µ m from the surfaceopening new perspectives for the coherent manipulationof ultracold atoms in the vicinity of superconductors. ∗ ∗ ∗ Laboratoire Kastler Brossel and Laboratoire PierreAigrain are joint research Laboratories of CNRS with´Ecole normale sup´erieure and Universit´e Pierre et MarieCurie. EM2C is a laboratory of CNRS associated to ´Ecolecentrale de Paris. We acknowledge support of the Eu-ropean Union (CONQUEST and SCALA projects, MarieCurie Fellowship program), of the Japan Science and Tech-nology corporation (International Cooperative ResearchProject : “Quantum Entanglement”), of the ANR, DGAand of the R´egion Ile de France (IFRAF and Cnano Idfconsortiums).
REFERENCES[1]
Fort´agh J. and
Zimmermann C. , Rev. Mod. Phys. , (2007) 235.[2] Hinds E. A. and
Hughes I. G. , J. Phys. D , (1999)R119.[3] H¨ansel W., Reichel J., Hommelhoff P. and
H¨anschT. W. , Phys. Rev. Lett. , (2001) 608.[4] Shin Y., Sanner C., Jo G. B., Pasquini T. A., SabaM., Ketterle W., Pritchard D. E., Vengalat-tore M. and
Prentiss M. , Phys. Rev. A , (2005)021604(R).[5] Schumm T., Hofferberth S., Andersson L. M., Wil-dermuth S., Groth S., Bar-Joseph I., Schmied-mayer J. and
Kruger P. , Nat. Physics , (2005) 57.[6] G¨unther A., Kraft S., Kemmier M., Koelle D.,Kleiner R., Zimmermann C. and
Fort´agh J. , Phys.Rev. Lett. , (2005) 170405.[7] Treutlein P., Hommelhoff P., Steinmetz T.,H¨ansch T. W. and
Reichel J. , Phys. Rev. Lett. , (2004) 203005.[8] Jo G. B., Shin Y., Will S., Pasquini T. A., Saba M.,Ketterle W., Pritchard D. E., Vengalattore M. and
Prentiss M. , Phys. Rev. Lett. , (2007) 030407.[9] Calarco T., Hinds E. A., Jaksch D., SchmiedmayerJ., Cirac J. I. and
Zoller P. , Phys. Rev. A , (2000)022304.[10] Jones M. P. A., Vale C. J., Sahagun D., Hall B. V. and
Hinds E. A. , Phys. Rev. Lett. , (2003) 080401.[11] Lin Y., Teper I., Chin C. and
Vuletic V. , Phys. Rev.Lett. , (2004) 050404.[12] Rekdal P. K., Scheel S., Knight P. L. and
HindsE. A. , Phys. Rev. A , (2004) 013811.[13] Henkel C. , Eur. Phys. J D , (2005) 59.[14] Scheel S., Rekdal P. K., Knight P. L. and
HindsE. A. , Phys. Rev. A , (2005) 042901. [15] Nirrengarten T., Qarry A., Roux C., Emmert A.,Nogues G., Brune M., Raimond J.-M. and
HarocheS. , Phys. Rev. Lett. , (2006) 200405.[16] Mukai T., Hufnagel C., Kasper A., Meno T.,Tsukada A., Semba K. and
Shimizu F. , Phys. Rev. Lett. , (2007) 260407.[17] Roux C., Emmert A., Lupas¸u A., Nirrengarten T.,Nogues G., Brune M., Raimond J.-M. and
HarocheS. , EPL , (2008) 56004.[18] Skagerstam B., Hohenester U., Eiguren A. and
Rekdal P. , Phys. Rev. Lett. , (2006) 070401.[19] Hohenester U., Eiguren A., Scheel S. and
HindsE. A. , Phys. Rev. A , (2007) 033618.[20] Skagerstam B. S. K. and
Rekdal P. K. , Phys. Rev. A , (2007) 052901.[21] Scheel S., Fermani R. and
Hinds E. A. , Phys. Rev. A , (2007) 064901.[22] L¨utke-Entrup N., Plac¸ais B., Mathieu P. and
SimonY. , Phys. Rev. Lett. , (1997) 2538.[23] L¨utke-Entrup N., Plac¸ais B., Mathieu P. and
SimonY. , Physica B , (1998) 75.[24] Plac¸ais B., L¨utke-Entrup N., Belessa J., MathieuP., Simon Y. and
Sonin E. B. , Europhys. Lett. , (2004) 655.[25] Haroche S. , Cavity quantum electrodynamics , in
Funda-mental Systems in Quantum Optics, Les Houches SummerSchool, Session LIII , edited by
Dalibard J., RaimondJ.-M. and
Zinn-Justin J. , (North Holland, Amsterdam)1992 p. 767.[26]
Mathieu P. and
Simon Y. , Europhys. Lett. , (1988)67.[27] Sonin E. B., Tagantsev A. K. and
Traito K. B. , Phys. Rev. B , (1992) 5830.[28] Plac¸ais B., Mathieu P., Simon Y., Sonin E. B. and
Traito K. B. , Phys. Rev. B , (1996) 13083.[29] Coffey M. W. and
Clem J. R. , Phys. Rev. Lett. , (1991) 386.[30] Brandt E. H. , Phys. Rev. Lett. , (1991) 2219.[31] Peroz C. and
Cillard C. , Phys. Rev. B , (2005)014515.[32] de Gennes P. and Pincus P. , Superconductivity of Met-als and Alloys (W. A. Benjamin) 1966.[33]
Finnemore D. K., Stromberg T. F. and
SwensonC. A. , Phys. Rev. , (1966) 231.[34] Abrikosov A. , Sov. Phys. JETP , (1957) 1174.[35] Emmert A., Lupas¸cu A., Nogues G., Brune M., Rai-mond J.-M. and
Haroche S. , Eur. Phys. J D , (2009)173.(2009)173.