Heating rate and spin flip lifetime due to near field noise in layered superconducting atom chips
Rachele Fermani, Tobias Mueller, Bo Zhang, Michael J. Lim, Rainer Dumke
aa r X i v : . [ phy s i c s . a t o m - ph ] D ec Heating rate and spin flip lifetime due to near field noisein layered superconducting atom chips
R. Fermani,
1, 2, ∗ T. M¨uller,
1, 2
B. Zhang, M. J. Lim, † and R. Dumke Nanyang Technological University, Division of Physics and Applied Physics, 21 Nanyang Link, Singapore 637371 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 (Dated: August 9, 2018)We theoretically investigate the heating rate and spin flip lifetimes due to near field noise foratoms trapped close to layered superconducting structures. In particular, we compare the case of agold layer deposited above a superconductor with the case of a bare superconductor. We study aniobium-based and a YBa Cu O − x (YBCO)-based chip. For both niobium and YBCO chips at atemperature of 4.2 K, we find that the deposition of the gold layer can have a significant impact onthe heating rate and spin flip lifetime, as a result of the increase of the near field noise. At a chiptemperature of 77 K, this effect is less pronounced for the YBCO chip. PACS numbers: 34.35.+a, 37.10.Gh, 42.50.Ct
I. Introduction
In the area of magnetic trapping of ultracold atoms,considerable attention has been recently devoted to theinteraction of atomic clouds with the surfaces of bothsuperconducting atom chips [1, 2, 3, 4, 5, 6, 7, 8,9, 10, 11, 12] and superconducting solid state devices[13, 14, 15, 16]. Technological advances will allow a newgeneration of fundamental experiments and applicationsinvolving the control of the interface between atomicsystems and quantum solid state devices. The imple-mentation of such technologies depends on the ability tocontrol and efficiently manipulate atoms close to super-conducting surfaces. However, below a certain separa-tion the atom-surface coupling is strong enough that thetrapping potential is modified by the interaction of theatomic magnetic moment with the near-field magneticnoise. This leads to atom heating and spin flip inducedatomic loss, which is the basic limitation in metal-basedatom chips [17, 18, 19, 20, 21]. The origin of near fieldnoise lies in the thermally-induced fluctuating currentsand in the resistivity of the chip materials in accordancewith the fluctuation-dissipation theorem. The magneticfield noise is significantly smaller in the vicinity of a su-perconductor [2, 3], however for the small distances in-volved, heating and thermally-induced spin flip may be-come relevant.The reduction of the spin flip rate has been shown ex-perimentally for different chip types: with and withouta gold layer above the superconductor [10, 11, 12]. Thepurpose of the top gold layer in atom chips is to use themin mirror–MOTs [22]. This can be achieved by coatingthe chip surface with a reflective metal film. The metalfilm can also protect the superconductor when it is op- ∗ Electronic address: [email protected] † Permanent address: Department of Physics and Astronomy,Rowan University, 201 Mullica Hill Road, Glassboro, New Jersey,USA erated near the critical current density, to avoid possiblequenching [7]. However, the presence of the metal filmabove the superconducting layer may increase the nearfield noise significantly. chip base z z d d d chip basesuperconductor s u p e r c o n d u c t o r gold (a)(b) Bare Superconductor chip structureGold + Superconductor chip structure z conductor chip z perconductor c FIG. 1: Schematic representation of the chip structure: (a)the bare superconductor chip structure consists of a layer ofsuperconducting material of thickness d deposited on a chipbase (usually a crystal); (b) the superconductor+gold chipstructure consists of a thin film of gold of thickness d ontop of the superconducting layer of thickness d < d , whichis deposited on the chip base. The superconducting layer iseither niobium or YBCO. We report our investigation of the heating rate andthe thermally-induced spin flip rate of a neutral atomheld close to two different superconducting structures:one with and one without a metal film above the su-perconducting layer. We consider the case of two dif-ferent superconductors in the Meissner state: niobium(a conventional s -wave superconductor) and YBCO (ahigh temperature d –wave superconductor). We presenta first-principles derivation of the heating rate, adoptinga model for the heating mechanism similar to [19, 23].For the derivation of the spin flip rate the reader mayrefer to earlier work [21]. The evaluation of the rates isdone within the framework of the electromagnetic fieldquantization in absorbing dielectric media [24, 25]. De-spite the quantum electrodynamics formalism being orig-inally developed for dielectric and metallic media, it canbe appropriately adopted to account for the electromag-netic field dissipation in superconducting materials in theMeissner state [5].The paper is organized as follows. In Sec. II, we de-scribe the model we adopt for the heating rate and wegive its general formulation. We present the quantummechanical derivation of the heating rate in AppendixA. In Sec. III, we report the formulas and simulationsfor the heating rate for two superconducting chip struc-tures: with and without a metal layer. The comparisonis done by considering niobium and YBCO superconduc-tors. For completeness, we extend the comparison of thetwo materials by studying the spin flip lifetime in Sec. IV.Conclusions are given in Sec. V. II. Model and Formulas
The heating of an atom harmonically trapped canbe understood as a transition from a motional state ofthe atomic trap to a higher motional state of the sametrap. This is due to fluctuations of the trap center x ,as schematically represented in Fig. 2. We consider athree-dimensional harmonic oscillator with Hamiltonian H = p / mω x − x · F where F is a force causingthe trap centre to fluctuate. The force is proportionalto the gradient of the magnetic potential experienced bythe atom, which is a consequence of the Zeeman couplingof the atomic magnetic moment µ to a spatially varyingmagnetic field B ( r ). Fluctuations of the magnetic field B ( r A ) at the atom’s position r A will induce fluctuationsof the trap center and this process is described by theHamiltonian ˆ H int = − ˆ x · ∇ (cid:16) µ · ˆ B ( r A ) (cid:17) , (1)where ˆ x = P j x j ˆ X j , with ˆ X j = (ˆ a † j + ˆ a j ) / x j the av-erage position of the trap center along the j th direction.In the following, a concise notation will be adopted bydropping the sum over j and assuming that j = x, y, z .For an atom initially in the state | n i , the average trapcenter for a transition to a state | m = n i is given by x = h m | ˆ x | n i which is nonzero only for m = n ± x j, ± = p ~ /M ω v,j q n + ± , where ω v,j is thetrapping frequency along the j th direction and M is theatomic mass.We obtain the rate γ j,n → n ± for the transition n → n ± j th trapping direction as derived in Ap- Γ → nd excited motional state1 st excited motional stateground motional state Γ → surface FIG. 2: Schematic representation of the heating rate model.The heating rate is modelled as a transition from a givenmotional state of the trap to higher motional states. Thetrapping potential is considered harmonic. pendix A γ j,n → n ± = πx j, ± ~ S F ( r A , ω v,j ) , (2)where S F ( r A , ω v,j ) is the spectrum of the fluctuatingforce causing the shift of the trap centre. In order toobtain the heating rate, we consider the average energyof the system h E ( t ) i = P n P ( n, t ) ~ ω ( n + ) where thesum is performed over the motional trap states | n i , and P ( n, t ) is the probability that at time t the system is inthe state | n i . The average heating rate Γ ǫ,j ≡ h ˙ E ( t ) i j isgiven by [23]Γ ǫ,j = X n P ( n, t ) ~ ω v,j ( γ j,n → n +1 − γ j,n → n − ) , (3)and by substituting Eq. (2) into the previous equationwe obtain Γ ǫ,j = πM S F ( r A , ω v,j ) , (4)where P n P ( n, t ) = 1. The expression of Eq. (4) is themost general formula we obtain for the heating rate andmust be evaluated for each separate structure. In par-ticular, the spectrum of the fluctuating force has beenobtained in Appendix A as S F ( r A , ω ) = ~ πǫ c ( n th + 1) −→ ∇ (cid:16) µ · Im h −→ ∇ × G ( r A , r A , ω ) × ←− ∇ i · µ (cid:17) ←− ∇ , (5)where n th = 1 / ( e ~ ω/k B T −
1) is the mean thermal pho-ton number and G ( r A , r A , ω ) is the Green function. TheGreen function accounts for the spectrum of the fluctuat-ing magnetic field according to the following expression S B ( r , r ′ , ω ) = h ˆ B ( r , ω ) ˆ B ( r ′ , ω ′ ) i (6)= ~ πǫ c ( n th + 1)Im h −→ ∇ × G (r , r ′ , ω ) × ←−∇ i δ ( ω − ω ′ ) . All the information regarding the trapping parametersand the geometry of the system are contained in theGreen function.
III. Heating rate
We present our numerical evaluations for the heatingrate of Eq. (4) for two different chip structures as shownin Fig. 1(a)-(b). Both structures contain a superconduct-ing layer, either niobium or YBCO, in the Meissner state.We evaluate the heating rate for a Rb atom held at adistance z above the chip. We consider the followingtrapping frequencies along the three directions in space: ω v,x = 0 . ω v,y = ω v,z = 1 KHz.The heating rate along the j th trapping direction forthe niobium-based chip structure can be obtained fromEq. (4) by substituting the Green functions for isotropicmedia as reported in [26], leading toΓ ǫ,j = α j µ B ~ M ǫ c ( n th + 1) (7) Z dη η π e − ηz " R T M ω v,j c + 2 η R T E where α x = α y = 1 and α z = 2, and µ B is the Bohrmagneton. The Fresnel coefficients R T E and R T M , for
T E and
T M waves, are computed according to the na-ture and number of layers constituting the structure. Inparticular, the Fresnel coefficients for the n th layer followthe recursive formula R qn,m = r qn,m + R qm,m +1 e iβ m d m r qn,m R qm,m +1 e iβ m d m , (8)where m = n + 1 denotes the consecutive layer to n ,with thickness d m , β m = p k m − η , k m = ǫ m ω /c , and ǫ m is the relative permittivity of the m th layer. Thelabel q denotes either the TE or the TM components ofthe electromagnetic field. The r qn,m terms represent theFresnel coefficients for the simplest geometry, two halfspaces, and are given by r T En,m = β n − β m β n + β m , r T Mn,m = ǫ m β n − ǫ n β m ǫ m β n + ǫ n β m . (9)In Figure 3 we plot the heating rate Γ ǫ,z along the z direction versus the surface distance for three differentchip structures at liquid helium temperature T = 4 . µ m, we estimate the heating rate forthe gold substrate to be of the order of Γ ǫ, Au ≃ − K/s which is a measurable effect. At the same distance,
100 nm 1 Μ m 10 Μ m 100 Μ m10 - - - - - Surface distance z H ea t i ng r a t e G Ε , z @ K (cid:144) s D FIG. 3: Heating rate for a Nb-superconducting chip at 4.2K, the thickness of the Nb layer is 1 µ m. Each of the threecurves represents a different structure: the bare niobium chip(solid line), the niobium+gold chip with gold thickness of 50nm (dashed line) and a simple gold surface (dotted line). the heating for the niobium-based chip is reduced but inprinciple still detectable with a top gold layer of 50 nmas Γ ǫ, Nb+Au ≃ − . However, the rate for the bare nio-bium chip is too small to be of experimental relevancebecause Γ ǫ, Nb ≃ − K/s. Trapping distances below1 µ m are challenging because of the van der Waals at-tractive potential. However, for z ∼
100 nm, the heatingrate of the niobium+gold surface approaches the heatingrate of the gold substrate. This indicates that at verysmall distances only the effect of the metal is relevant.For the YBCO-based chip structure, the Green func-tion for anisotropic media as reported in [27] is adoptedand the heating rate along the z -direction (perpendicularto the chip surface) is given byΓ ǫ,z = ~ µ B M ǫ c ( n th + 1) (10) ∞ Z dη η π e − ηz Im (cid:2) B N k z + 2 η B M (cid:3) , with k z = ω z /c , while the heating rate Γ ǫ, k for the planeparallel to the chip surface is given by the sum of thefollowing expressionsΓ ǫ,r = ~ µ B M ǫ c ( n th + 1) (11) ∞ Z dη η π e − ηz (cid:2) B N k r + 3 η B M (cid:3) , Γ ǫ,φ = ~ µ B M ǫ c ( n th + 1) (12) ∞ Z dη π e − ηz Im (cid:2) B N k φ + η B M (cid:3) , with k r,φ = ω r,φ /c . The scattering coefficients B M and B N play the same role of the Fresnel coefficients ofEq. (8) but account for the anisotropic properties of theYBCO layer. These coefficients can be obtained by fol-lowing [27, 28].
100 nm 1 Μ m 10 Μ m 100 Μ m10 - - - - Surface distance z H ea t i ng r a t e G Ε , z @ K (cid:144) s D FIG. 4: Heating rate for a YBCO-based chip at 77 K, thethickness of the YBCO layer is 1 µ m. Each of the threecurves represents a different structure: the bare YBCO chip(solid line), the YBCO+gold chip with gold thickness of 50nm (dashed line) and a simple gold surface (dotted line). Our simulations for the heating rate of the YBCO chipare presented in Fig. 4. We plot the heating rate alongonly the z trapping direction because this term domi-nates. We observe that the bare YBCO surface yields thelowest heating rate, compared to the other two structureswith the gold layer. However, the difference between thebare YBCO surface and the YBCO+gold chip structureis dramatically reduced at liquid nitrogen temperature T = 77 K, and is less than one order of magnitude fora wide range of surface distances. The heating rate isdetectable for distances around 1 µ m where Γ ǫ, YBCO ≃ Γ ǫ, YBCO+Au ≃ − K/s. At smaller temperatures, therate corresponding to the two structures differs by a feworders of magnitude, i. e. Γ ǫ, YBCO+Au / Γ ǫ, YBCO ≃ − at T = 4 . µ m does not result ina measurable difference at 77 K. Moreover, the typicalvalues of magnetic bias fields and temperatures used forsuperconducting chip experiments suggest that a high-temperature superconductor is likely to be operated inthe Shubnikov phase. This involves the partial penetra-tion of magnetic field in the form of vortices but this isnot considered in the present work. IV. Spin flip lifetime
In the following section the comparison between thebare superconducting chip and the superconductor+goldstructure is extended by considering the spin flip lifetime τ of a neutral atom. The transition rate for thermallyinduced spin flips Γ SF = 1 /τ between an initial hyperfinestate | i i and a final hyperfine state | f i is given in its mostgeneral formulation as [21]Γ SF = 2 cµ B g S ~ h f | ˆ S q | i ih i | ˆ S p | f i S B ( r A , ω SF ) , (13)where ω SF is the spin transition frequency and h i | ˆ S p | f i the spin matrix element for | i i → | f i . We restrict thecalculation to a two-level system evaluating the spin fliplifetime for the transition | F = 1 , m F = − i → | F =1 , m F = 0 i . We choose a typical experimental transitionfrequency ω SF = 2 π
560 KHz [17], and we note that thenear field noise spectrum does not change significantlyin the rf frequency range. We evaluate the spin matrixelements via the Clebsch-Gordan coefficients and obtainfor the non-zero matrix elements |h i | ˆ S x | f i| = |h i | ˆ S z | f i| =1 / y is the spin quantization direction).The spin flip lifetime close to the niobium-based chip isobtained as the inverse of the following spin flip rate [28]Γ SF, Nb = 3 π ( µ B g S ) c ǫ ~ Z dη η (2 π ) e − ηz (cid:2) R T E (cid:3) ( n th + 1) , (14)where R T E is the Fresnel coefficient as in Eq. (8). Thespin flip lifetime as a function of the surface distanceis plotted in Fig. 5 for three different structures. Theshortest spin flip lifetime is obtained for the gold surface.The longest lifetime is found close to the bare niobiumsurface and is two orders of magnitude longer than thelifetime near the niobium+gold chip.
100 nm 1 Μ m 10 Μ m 100 Μ m0.11010 Surface distance z L i f e t i m e Τ @ s D FIG. 5: Spin flip lifetime for a niobium-based chip at 4.2 K,the thickness of the Nb layer is 1 µ m. Each of the three curvesrepresents the lifetime corresponding to a different structure:the bare niobium chip (solid line), the niobium+gold chipwith gold thickness of 50 nm (dashed line) and a simple goldsurface (dotted line). The spin flip rate above the YBCO-based chip can bewritten as [28]Γ
SF,
YBCO = ( µ B g S ) ~ ǫ c ( n th + 1) (15) ∞ Z dη e − ηz π Im (cid:20) η B M + B N ω SF c (cid:21) , where B M and B N are the scattering coefficients ofEqs. (10-12). The spin flip lifetime τ = 1 / Γ SF,
YBCO isplotted in Fig. 6 for the three different structures: goldsurface, YBCO surface, YBCO+gold structure. Simi-larly to the niobium-based chip structures, the spin fliplifetime close to the YBCO+gold structure is shorterthan the one obtained near the bare superconductor,however the difference is less than an order of magni-tude. A complete accounting of the total lifetime mustinclude other experimental restrictions, such as collisionswith background gas [8].
100 nm 1 Μ m 10 Μ m 100 Μ m0.11010 Surface distance z L i f e t i m e Τ @ s D FIG. 6: Spin flip lifetime for a YBCO-based chip at 77 K, thethickness of the YBCO layer is 1 µ m. The two upper linesrepresent the case with a bare YBCO wire (solid line) andwith a gold layer of thickness 50 nm on top of YBCO (dashedline). The lifetime for a simple gold surface is reported forcomparison (dotted line). We conclude this section by plotting in Fig. 7 the heat-ing rate and spin flip lifetime of the niobium+gold andYBCO+gold chip as a function of the gold thickness. Theniobium+gold chip has the lowest heating rate. The rateincreases for both structures with increasing the thick-ness of the gold until the layer thickness reaches 1 µ m.The presence of the superconductor is then irrelevantand the heating rate is the same as what one would ob-tain close to a gold substrate. Typical values for thegold thickness are around 50 −
100 nm. For such valuesthe heating rate for the YBCO+gold chip is one orderof magnitude larger than the niobium+gold chip at 4.2K. Similarly, the niobium+gold chip exhibits the longestlifetime. For both structures the lifetime decreases with increasing gold thickness until it approaches the lifetimeobtained close to the single gold layer. This happens fora layer thickness of the order of 10 µ m.
100 nm 1 Μ m 10 Μ m10 nm1 nm10 - - - - Gold thickness d H ea t i ng r a t e G Ε , z @ K (cid:144) s D
100 nm 1 Μ m 10 Μ m10 nm1 nm110100100010 Gold thickness d L i f e t i m e Τ @ s D FIG. 7: Heating rate and spin flip lifetime of an atom 1 µ mabove a superconducting chip with a top gold layer. Bothquantities are plotted as a function of the gold thickness d (as in Fig. 1 (b)), for a niobium+gold chip (solid curve),YBCO+gold chip (dashed curve) and for a gold substrate(dotted line). Both chips have a temperature of 4.2 K andthe thickness of the superconducting layer is 1 µ m. V. Conclusions
We have investigated the heating rate and spin fliplifetime for a superconducting atom chip considering thecases with and without a gold layer deposited above thesuperconductor. All the results presented are valid for asuperconductor in the Meissner state. The heating ratefor trapping distances around 1 µ m is measurable andmay become significant for a sufficiently cold atom cloud.However, by adding a top gold layer, this rate increasessuch that for layers thicker than 1 µ m, the presence ofthe superconductor becomes irrelevant. Even for a sub-micron gold layer, an atom held at a close distance of 100nm is influenced only by the metal, and both the heatingrate and spin flip lifetime approach the values obtainedin the proximity of a infinitely thick gold substrate.In particular, we have considered a niobium-based anda YBCO-based chip. At a chip temperature of 4.2 K, thedifference between the case with the gold layer and with-out is marked both for heating rate and spin flip lifetime.Such difference is less pronounced when the YBCO chipis operated at 77 K. In summary, our study shows thatthe deposition of a metal layer above a superconductinglayer can diminish the advantages of using a supercon-ductor to overcome the limitations given by a metal. Thepresented results are of interest for future development oftechnologies involving superconducting atom chips. Fur-ther investigations will include the treatment of type IIsuperconductors in the Shubnikov phase. Acknowledgments
We acknowledge financial support from Nanyang Tech-nological University (grant no. WBS M58110036), A-Star (grant no. SERC 072 101 0035 and WBS R-144-000-189-305) and the Centre for Quantum Technologies,Singapore. MJL acknowledges travel support from NSFPHY-0613659 and the Rowan University NSFG program.
A. Quantum mechanical derivation of the heatingrate
We outline the quantum mechanical derivation of theheating rate based on the model introduced in Sec. II.The total Hamiltonian of a harmonically trapped atomheld close to a substrate (the conducting properties of thesubstrate are not yet specified at this stage), is given bythe sum of three different Hamiltonians as ˆ H = ˆ H f +ˆ H a + ˆ H int . The Hamiltonian for the atom in the unper-turbed harmonic trap reads ˆ H a = ~ ω v,j (cid:16) ˆ a † j ˆ a j + (cid:17) . TheHamiltonian of the electromagnetic field arising from anabsorbing and dispersing medium is given within the for-malism of quantum electrodynamics for dielectric media[24, 25] as ˆ H f = R d r dω ~ ω ˆ f † ( r , ω )ˆ f ( r , ω ), with ˆ f ( r , ω )and ˆ f † ( r , ω ) being the bosonic operators accounting forthe collective excitation of the medium and the electro-magnetic field. They satisfy the usual equal-time com-mutation relations h ˆ f ( r , ω ) , ˆ f † ( r ′ , ω ′ ) i = δ ( r − r ′ ) δ ( ω − ω ′ )and their correlation function at temperature T reads h ˆ f ( r , ω )ˆ f † ( r ′ , ω ′ ) i = ( n th + 1) δ ( r − r ′ ) δ ( ω − ω ′ ), where n th is the mean thermal photon number.The interaction Hamiltonian of Eq. (1) can be writtenin the rotating wave approximation asˆ H int = − x j (cid:16) ˆ a † j ∂ j µ k ˆ B + k ( r A ) + H . c . (cid:17) , (A1)where x j denotes either x j, + for n → n +1 or x j, − for n → n −
1. The magnetic field B + k ( r A ) can be obtained viaˆ B ( r ) = ˆ B + ( r )+ ˆ B − ( r ) where ˆ B +( − ) ( r ) = ∞ R dω ˆ B ( † ) ( r , ω ) is the positive-frequency part, and the single componentis given byˆ B k ( r , ω ) = r ~ πǫ ωc Z d r ′ p ǫ I ( r ′ , ω ) ǫ kjs ∂ j G sn ( r , r ′ , ω ) ˆ f n ( r ′ , ω ) . (A2)In the Heisenberg picture, the equation of motion forcreation operator ˆ a + j is given by˙ˆ a † j ( t ) = iω v,j ˆ a † j − i ~ x j ∂ j µ k ˆ B − k ( r A ) , (A3)and the bosonic field operator becomes˙ˆ f n ( r , ω, t ) = − iω ˆ f n (A4)+ i ˆ a j x j µ k √ ~ πǫ ωc p ǫ I ( r , ω ) ∂ j ǫ kqs ∂ q G ∗ sn ( r A , r , ω ) , where the time integral in the Markov approximation be-comesˆ f n ( r , t ) = ˆ f n,free ( r , t ) + i ˆ a j ( t ) ζ ( ω v,j − ω ) (A5) x j µ k √ ~ πǫ ωc p ǫ I ( r , ω ) ∂ j ǫ kqs ∂ q G ∗ sn ( r A , r , ω ) , and ζ ( x ) = πδ ( x )+ i P x − ( P denotes the principal part).This formal solution is going to be substituted into themagnetic field of Eq. (A2) and making use of the follow-ing integral relation for the Green function [25] Z d s ω c ǫ I ( s , ω ) G ( r , s , ω ) G ∗ ( r ′ , s , ω ) = Im G ( r , r ′ , ω ) , (A6)we obtainˆ B i ( r A , ω ) = ˆ B i,free ( r A , ω ) + i ˆ a j ( t ) x j πǫ c (A7) ζ ( ω v,j − ω ) ∂ j Im h −→ ∇ × G ( r A , r A , ω ) × ←− ∇ i ik µ k , where the gradient ←− ∇ = P j = x,y,z ∂ j is acting on thesecond argument of the Green function as denoted bythe left pointing arrow. After integrating Eq. (A7) over ω and substituting it into Eq. (A3), we can write theequation of motion for the creation operator as˙ˆ a † j ( t ) = [ − γ j / i ( ω v,j − δω j )] ˆ a † j − i ~ x j ∂ j µ p ˆ B − p,free ( r A ) . (A8)where γ j = γ j,n → n ± is the rate associated with the n → n ± j th direction. The rate arisesfrom the δ ( x ) of the ζ ( x ) function appearing in Eq. (A5),and is given by γ j = x j ~ ǫ c (A9) −→ ∇ (cid:16) µ · Im h −→ ∇ × G ( r A , r A , ω v,j ) × ←− ∇ i · µ (cid:17) ←− ∇ , while the term δω j of Eq. (A8) arises from the principal-part of the ζ ( x ) function and denotes the frequency shiftwhich we assume to be negligible.If the dielectric body is in thermal equilibrium withits surroundings, then the magnetic field is in a thermalstate with temperature T , and the magnetic field spec-trum has to be multiplied by the mean thermal photonnumber n th + 1. The spectrum of the magnetic field canbe expressed as S B ( r A , ω ) = ~ πǫ c ( n th + 1) (A10)Im h −→ ∇ × G ( r A , r A , ω ) × ←− ∇ i , and hence the spectrum of the fluctuating force causingthe shift of the center of mass is S F ( r A , ω ) = ~ πǫ c ( n th + 1) (A11) −→ ∇ (cid:16) µ · Im h −→ ∇ × G ( r A , r A , ω ) × ←− ∇ i · µ (cid:17) ←− ∇ , such that the transition rate for the j th trapping direc-tion can be written as γ j,n → n ± = πx j, ± ~ S F ( r A , ω v,j ) . (A12) [1] T. Nirrengarten, A. Qarry, C. Roux, A. Emmert,G. Nogues, M. Brune, J.-M. Raimond, and S. Haroche,Phys. Rev. Lett. , 200405 (2006).[2] S. Scheel, P.K. Rekdal, P.L. Knight, and E.A. Hinds,Phys. Rev. A , 042901 (2005).[3] Bo-Sture K. Skagerstam and P.K. Rekdal, Phys. Rev. A , 052901 (2007).[4] S. Scheel, R. Fermani, and E.A. Hinds, Phys. Rev. A ,064901 (2007).[5] U. Hohenester, A. Eiguren, S. Scheel, and E.A. Hinds,Phys. Rev. A , 033618 (2007).[6] T. Mukai, C. Hufnagel, A. Kasper, T. Meno, A. Tsukada,K. Semba, and F. Shimizu, Phys. Rev. Lett. , 260407(2007).[7] T. M¨uller, X. Wu, A. Mohan, A. Eyvazov, Y. Wu andR. Dumke, New J. Phys. , 073006 (2008).[8] B. Kasch, H. Hatterman, D. Cano, T. E. Judd, S. Scheel,C. Zimmermann, R. Kleiner, D. K¨olle, and J. Fort´agh,arXiv:0906.1369 (2009).[9] A. Emmert, A. Lupascu, G. Nogues, M. Brune, J. M. Rai-mond, and S. Haroche, Eur. J. D , 173 (2009).[10] G. Nogues, C. Roux, T. Nirrengarten, A. Lupascu, A.Emmert, M. Brune, J.-M. Raimond, S. Haroche, B.Plaais and J.-J. Greffet, Eur. Phys. Lett. , 13002(2009).[11] C. Hufnagel, T. Mukai, and F. Shimizu, Phys. Rev. A , 053641 (2009).[12] T. M¨uller, B. Zhang, R. Fermani, K.S. Chan, Z.W. Wang,C.B. Zhang, M.J. Lim, R. Dumke, arXiv:0910.2332(2009).[13] L. Tian, P. Rabl, R. Blatt, and P. Zoller, Phys. Rev. Lett. , 247902 (2004). [14] A. S. Sørensen, C. H. van der Wal, L. I. Childress, andM. D. Lukin, Phys. Rev. Lett. , 063601 (2004).[15] D. Petrosyan and M. Fleischhauer, Phys. Rev. Lett. ,170501 (2008).[16] J. Verd´u, H. Zoubi, Ch. Koller, J. Majer, H. Ritsch, andJ. Schmiedmayer, Phys. Rev. Lett. , 043603 (2009).[17] M.P.A. Jones, C.J. Vale, D. Sahagun, B.V. Hall, andE.A. Hinds, Phys. Rev. Lett. , 080401 (2003).[18] D.M. Harber, J.M. McGuirk, J.M. Obrecht, andE.A. Cornell, J. Low Temp. Phys. , 229 (2003).[19] C. Henkel, S. P¨otting, and M. Wilkens, Appl. Phys. B , 379 (1999).[20] R. Folman, P. Kr¨uger, J. Schmiedmayer, J. Denschlag,and C. Henkel, Adv. At. Mol. Opt. Phys. , 263 (2002).[21] P.K. Rekdal, S. Scheel, P.L. Knight, and E.A. Hinds,Phys. Rev. A , 013811 (2004).[22] J. Reichel, W. H¨ansel, T.W. H¨ansch, Phys. Rev. Lett. , 3398 (1999).[23] M.E. Gehm, K.M. O’Hara, T.A. Savard, andJ.E. Thomas, Phys. Rev. A, , 3914 (1998).[24] W. Vogel, D.-G. Welsch, and S. Wallentowitz, QuantumOptics, An Introduction (Wiley-VCH, Weinheim, 2001).[25] L. Kn¨oll, S. Scheel, and D.-G. Welsch, in
Coherence andStatistics of Photons and Atoms , edited by J. Peˇrina (Wi-ley, New York, 2001).[26] H.T. Dung, L. Kn¨oll, and D.-G. Welsch, Phys. Rev. A , 3931 (1998).[27] L.-W. Li, J.-H. Koh, T.-S. Yeo, M.-S. Leong, and P.-S. Kooi, IEEE Trans. Antennas Propagat.52