Casimir energy for two and three superconducting coupled cavities
L. Rosa, S. Avino, E. Calloni, S. Caprara, M. De Laurentis, R. De Rosa, Giampiero Esposito, M. Grilli, E. Majorana, G. P. Pepe, S. Petrarca, P. Puppo, P. Rapagani, F. Ricci, C. Rovelli, P. Ruggi, N. L. Saini, C. Stornaiolo, F. Tafuri
CCasimir energy for two and three superconductingcoupled cavities
L. Rosa , , S. Avino , , E. Calloni , , S. Caprara , , M. De Laurentis , R. De Rosa , ,Giampiero Esposito , M. Grilli , , E. Majorana , G. P. Pepe , S. Petrarca , , P. Puppo , P.Rapagnani , , F. Ricci , , C. Rovelli , P. Ruggi , N. L. Saini , C. Stornaiolo , F. Tafuri Universit`a di Napoli Federico II, Dipartimento di Fisica “Ettore Pancini”,Complesso Universitario di Monte S. Angelo,Via Cintia Edificio 6, 80126 Napoli, Italy INFN Sezione di Napoli, Complesso Universitario di Monte S. Angelo,Via Cintia Edificio 6, 80126 Napoli, Italy INO-CNR, Comprensorio Olivetti,Via Campi Flegrei 34-80078 Pozzuoli (NA), Italy Universit`a di Roma “La Sapienza”,P.le A. Moro 2, I-00185, Roma, Italy ISC-CNR and Consorzio Nazionale Interuniversitario perle Scienze Fisiche della Materia (CNISM),Unit`a di Roma “La Sapienza”, P.le A. Moro 2, I-00185, Roma, Italy INFN Sezione di Roma, P.le A. Moro 2, I-00185, Roma, Italy Universit`a di Napoli Federico II, Dipartimento di Fisica “Ettore Pancini”,piazzale Tecchio 80, 80126 Napoli, Italy Aix Marseille Universit´e CNRS, CPT,UMR 7332, 13288 Marseille, FranceUniversit`e de Toulon, CNRS, CPT,UMR 7332, 83957 La Garde, France and European Gravitational Observatory (EGO), I-56021 Cascina (Pi), Italy a r X i v : . [ h e p - t h ] J u l bstract In this paper we study the behavior of the Casimir energy of a “multi-cavity” across the transitionfrom the metallic to the superconducting phase of the constituting plates. Our analysis is carriedout in the framework of the ARCHIMEDES experiment, aiming at measuring the interaction ofthe electromagnetic vacuum energy with a gravitational field. For this purpose it is foreseen tomodulate the Casimir energy of a layered structure composing a multi-cavity coupled system byinducing a transition from the metallic to the superconducting phase. This implies a thorough studyof the behavior of the cavity, where normal metallic layers are alternated with superconductinglayers, across the transition. Our study finds that, because of the coupling between the cavities,mainly mediated by the transverse magnetic modes of the radiation field, the variation of energyacross the transition can be very large.
PACS numbers: 12.20.Ds, 12.20.-m, 74.25.-q, 74.78.Fk . INTRODUCTION The ARCHIMEDES experiment [1] is designed for testing whether the energy of vacuumfluctuations, foreseen by quantum electrodynamics, contributes to gravity, through the cou-pling demanded by quantum field theory in curved spacetime [2–5], where the Einstein ten-sor is taken to be proportional to the expectation value of the regularized and renormalizedenergy-momentum tensor of matter fields. The idea is to weigh the vacuum energy storedin a rigid Casimir cavity made by parallel conducting plates, by modulating the reflectivityof the plates upon inducing a transition from the metallic to the superconducting phase [1].In order to enhance the effect, a multilayer cavity is considered, obtained by superimposingmany cavities. This structure is natural in the case of crystals of type-II superconductors,particularly cuprates, being composed by Cu-O planes, that undergo the superconductingtransition, separated by nonconducting planes. A crucial aspect to be tested is thus thebehavior of the Casimir energy [6] for a multi-cavity when the layers undergo the phasetransition from the metallic to the superconducting phase. Until now only the case of acavity having a single layer that undergoes the superconducting transition was considered,the other reflecting plate being just metallic (not superconducting), in Refs. [7, 8]. Thegeneralization to the case of a system of coupled superconducting layered cavities is stilllacking. With respect to the ARCHIMEDES project the main goal is to study the possi-bility of enhancing the modulation factor η = ∆ E cas E cas were ∆ E cas is the difference of Casimirenergy in normal and superconducting states. The value obtained in Ref. [7, 8], consideringa cavity with a single superconducting layer and a transition temperature of about 1 K is η l ≈ − . This value was compliant with a previous experiment devoted to ascertain thevacuum energy contribution to the total condensation energy [9, 10], but it is not sufficientto prove the weight of the vacuum, because it is in absolute too small. It is therefore nec-essary to consider high- T c superconductors where condensation energy is much higher andalso the absolute value of vacuum energy variation is expected to be correspondingly larger.On the other hand, in Ref. [11], considering a cavity based on a high- T c layered supercon-ductor, a factor as high as η h = ⋅ − has been estimated, under the approximation of flatplasma sheets at zero temperature, no conduction in normal state (here E cas is the energyof the ideal cavity) and charge density of n = cm − . The ARCHIMEDES sensitivity isexpected to be capable of ascertain the interaction of gravity and vacuum energy also forvalues lower than η h = ⋅ − , up to 1 /
100 of this value [1]. Clearly it is important to under-stand more firmly if dealing with layered superconducting structures the modulation depthcan be sufficiently high. This is the study of the present paper. Considering in particular themulti-layer cavity, the general assumption adopted so far has been that the Casimir energyobtained by overlapping many cavities is the sum of the energies of each individual cavity.This is true if the distances between neighboring cavities are large (in the sense that thethickness of each metallic layer separating the various cavities is very large with respect tothe penetration depth of the radiation field). Of course, this is no longer true if the thicknessof these metallic inter-cavity layers gets thinner and thinner. The evaluation of the Casimirenergy for such a configuration is the subject of the present study. It is worth stressing thatthis is only a first step because in the final version ARCHIMEDES experiment will makeuse of high- T c superconducting oxides with a built-in layered structure, like YBa Cu O − x ,for which a complete theory is as yet unavailable.Having this in mind, we start with a thorough analysis of two and three coupled Casimircavities, made by traditional BCS (low- T c ) superconducting material (niobium), so as to3eal with relatively manageable and well established formulas. On trying to preserve amacroscopic approach, we limited our study to thicknesses between 10 and 100 nm. In thefollowing, referring to Fig. 1, d i is the distance of the i − th cavity from the ( i − ) − th ,(thickness of the i − th cavity), within the slabs 1 , , , T c superconductors, as required by theroadmap of the ARCHIMEDES experiment.Section II studies the Casimir energy of a multilayer cavity, while Sec. III evaluates theCasimir energy in the normal and superconducting phases. Variation of the energy in thetransition is obtained in Sec. IV, including a detailed numerical analysis of the Matsubarazero-mode contribution. Section V extends this scheme to the three-layer configuration, andconcluding remarks are made in Sec. VI, while relevant details are given in the Appendices. II. THE CASIMIR ENERGY OF A MULTILAYER CAVITY
As it is customary [7, 8], at finite temperature, the Casimir variation across the transitionfrom a metallic to a superconducting phase is obtained as the difference between the freeCasimir energy in the metallic state and the same after the transition to superconductingstate takes place: δE ( T ) = E n − E s . The energy per unit area of a single cavity, ( ) in Fig. 1, can be written, at finite temperature T , as the sum of the contributions of thetransverse electric ( T E ) and transverse magnetic (
T M ) modes (see, for example, [12]) : E [ d , d ] = k B T ∞ ′ ∑ l = ∫ d k (cid:150) ( π ) ( log ∆ T E ( ξ l ) + log ∆ T M ( ξ l )) =∶ ∞ ∑ l = E [ l, d , d ] (1)where ξ l = πlk B T are the Matsubara frequencies, k B is the Boltzmann constant, l = , , , . . . , the superscript ′ on the sum means that the zero mode must be multiplied by afactor ,∆ T M ( ξ l ) = ( r , T M ( ξ l ) r , T M ( ξ l ) e − d K + ) , ∆ T E ( ξ l ) = ( r , T E ( ξ l ) r , T E ( ξ l ) e − d K + ) and the reflection coefficients r i,j ( T M,T E ) ( ξ l ) are given by (see [12]): r i,jT M ( ξ l ) = (cid:15) i ( ξ l ) K j ( ξ l ) − (cid:15) j ( ξ l ) K i ( ξ l ) (cid:15) i ( ξ l ) K j ( ξ l ) + (cid:15) j ( ξ l ) K i ( ξ l ) , r i,jT E ( ξ l ) = K j ( ξ l ) − K i ( ξ l ) K j ( ξ l ) + K i ( ξ l ) , with K i ( ξ l ) = √ k ⊥ + (cid:15) i ( ξ l ) ξ l . We point out that our approach captures the relevant lengthscale of a superconductor, the London penetration depth λ L , through the expression of (cid:15) i ( ξ l ) in terms of the correction to the optical conductivity when entering the superconductingstate, δσ BCS ( iξ ) , see below and Appendix B. In particular, for ξ l →
0, we have (cid:15) i ( ξ l ) ξ l → λ − L .We characterize the properties of the i − th material trough the dielectric function (cid:15) i ( ξ l ) and the change in the Casimir energy is given simply by the modification of the (cid:15) ( ξ l ) due tothe transition [7, 8]. As we said, in the following we report calculations for the case in which4 d d d d d
0 1 2 3 4 5 6
FIG. 1: A three layer cavity. In the 0,2,4, and 6 zone there is Nb; in the 1,3,5 vacuum. d i is the thickness of the i − th slabthe material is Nb and the spacer is vacuum (the modifications introduced by a dielectricspacer deserve a separate study).To obtain the formulas for two and three cavities we solve the problem by imposing thecontinuity of the tangential component of the ⃗ E and ⃗ H fields (non-magnetic media) andthe normal component of the ⃗ D and ⃗ B at the interface [13, 14]. Thus, for example, in thecase of the three cavities (012-234-456) in Fig. 1 we have that the ∆ functions appearing in(1) are the determinant of the matrix of the coefficients M ij (just to give an idea we reportthe expression for the T M - modes in appendix A) from which it is possible to extract thecase of one, two, and three cavities by taking ( i, j ) = . . . ( i, j ) = . . . ( i, j ) = . . . ( T M,T E ) = det ( M ij ( T M,T E ) ) . In the following we will omit the subscript
T M, T E if no ambiguity is generated. All the5ormulas for the two cases can be obtained using respectively the
T M or T E reflectioncoefficients. Defining (no summation over repeated indices) E ijl = r i,j r j,l e − d j K j + , F ijl = r i,j e − d j K j + r j,l , G ijl = r i,j + e − d j K j r j,l , H ijl = e − d j K j + r i,j r j,l , we have for the single cavity ( ) in Fig. 1∆ ( ) = E ; (2)for two cavities ( − ) :∆ ( ) = E E + e − ( d k ) F G =∶ E E + I ( ) and (3)log ∆ ( ) = log ( E E ) + log ⎛⎝ + I ( ) E E ⎞⎠ , (4)and for the three cavities:∆ ( ) = E E E + e − ( d k + d k ) F H G + e − d k E F G + e − d k E F G =∶ E E E + I ( ) + E I ( ) + E I ( ) , (5)log ∆ ( ) = log ( E E E ) + log ( + I ( ) E E E )+ log ⎛⎝ + E I ( ) + E I ( ) E E E + I ( ) ⎞⎠ . (6)In this way, when d → ∞ [see Eq. (3)] I ( ) → ( ) = log E E = log E + log E . That is to say, when the two cavities are far away their energy is simply the sum of theindividual contributions. In this respect the second term on the right of Eq. (3), I ( ) , canbe seen as the energy due to the coupling of the two cavities ( ) − ( ) .When d = d = d , d = d , (cid:15) = (cid:15) = (cid:15) , (cid:15) = (cid:15) = (cid:15) we obtain E T M = E , F = F sothat we can omit the subscripts:log ∆ ( ) = log ( ∆ ( ) ) + log ( + I ( ) ( ∆ ( ) ) ) . (7)For the three cavities ( − − ) , formulas are written so as to make evidentthe contribution to the energy resulting from the sum of the energies of the single cavity,with respect to the one coming from the coupling of the two possible pairs of cavities ( − ) , ( − ) , and the one coming from the coupling of the three: I ( ) . Thus,under the previous hypothesis,∆ ( ) = ( ∆ ( ) ) + I ( ) + I ( ) and we can writelog ∆ ( ) = log ( ∆ ( ) ) + log ( + I ( ) ( ∆ ( ) ) + I ( ) ) + log ( + I ( ) ( ∆ ( ) ) ) . (8)6n a sense, we are writing the energy as a sum of the energy of the single cavity plus thecoupling energy between the nearest neighbor, plus the coupling energy among the secondnearest neighbor and so on. In this way we will have a clear indication of the strength ofthe coupling between the cavities at the various orders. As far as we know this way ofdisplaying the various contribution to the Casimir energy has been obtained for the first timein [15] where the so called T GT G formula ( T being the Lippmann-Schwinger T operator an G the translation matrix), is used, see also [16, 17]. In our case it can be simply recoveredby observing that the determinant of a N × N complex block matrix can be obtained interms of the determinants of its constituent blocks [18].The Casimir energy in the superconducting phase is obtained by replacing, in the reflec-tion coefficients, the expression of the dielectric function with the corresponding obtainedusing the BCS theory [19, 20], see Appendix B. In the following we will characterize thedielectric properties of the material by means of the Drude model (but see conclusions): (cid:15) ( iξ ) = + σ ( iξ ) ξ , with σ k ( iξ ) = σ k γ + ξ , for conducting materials and σ k ( iξ ) = σ k γ + ξ + δσ BCS ( iξ ) , for superconducting materials , where the expression of δσ BCS ( iξ ) is given in Appendix B (see [20]).Thus δE ( T ) = E n ( T ) − E s ( T )= k T ∞ ′ ∑ l = ∫ d k (cid:150) ( π ) ⎛⎝ log ∆ ( k ) n,T E ∆ ( k ) s,T E + log ∆ ( k ) n,T M ∆ ( k ) s,T M ⎞⎠ =∶ ∞ ∑ l = δE [ l, d , d ] . where ∆ ( k ) n,T E,T M , ∆ ( k ) s,T E,T M are the generating functions (this nomenclature denotes herejust the determinant of the matrix whose zeros provide, implicitly, the allowed energies)of the normal and superconducting phases, and, depending on how many cavities we areconsidering ( , , or 3 ) we must take k = , , or 3 respectively. III. CASIMIR ENERGY IN THE NORMAL AND SUPERCONDUCTING PHASES
All results described hereafter are obtained for Nb, and we use the following values forthe critical temperature and plasma frequency T c = .
25 K, ̵ hω nio = . eV and work atthe temperature T = .
157 K. We start by choosing d =
300 nm, and d =
600 nm, so as tohave results that can be compared with standard formulas.We find for the energy in the normal phase E n , for fixed d , d , and different valuesof number of Matsubara modes ( n mod ) : E nn mod [ d , d ] = ∑ n mod j = E n [ j, d , d ] (N.B. in thefollowing all the quoted numbers that concern energy or difference of energy are in Jm ):7 mod E nn mod [ , ] ⋅ − . − . − . − . − . − . t = c ̵ h a k B T , D = ca ω p ): E = c ̵ h πa [ D ( ζ ( ) t − π ) + D ( π − ζ ( ) t ) + ζ ( ) t − π ]= − . ⋅ − . As expected, because of the strong suppression of the exponential for large d , the con-tribution of the coupling term between the two cavities (012)-(123) is about ten orders ofmagnitude smaller than the energy obtained from each cavity. With E [ d , d ] = k T ∞ ′ ∑ l = ∫ d k (cid:150) ( π ) ⎡⎢⎢⎢⎢⎣ log ( E E ) + log ⎛⎝ + I ( ) E E ⎞⎠⎤⎥⎥⎥⎥⎦ T M + [
T E ] (9) =∶ E ( ) T M [ d , d ] + C ( ) T M [ d , d ] + E ( ) T E [ d , d ] + C ( ) T E [ d , d ] (10) =∶ E ( ) [ d , d ] + C ( ) [ d , d ] we obtain ( n mod = E n ( ) [ , ] = − . ⋅ − ; C n ( ) [ , ] = − . ⋅ − . Indeed, having d ≪ d , the total energy is simply the sum of the energies of the twocavities: E n ( ) [ , ] = − . ⋅ − = − . ⋅ − ≈ − . ⋅ − In the superconducting phase we have, more or less, the same behavior: n mod E sn mod [ , ] ⋅ − . − . − . − . − . − . E s [ , ] = − . ⋅ − , E s ( ) [ , ] = − . ⋅ − , C s ( ) [ , ] = − . ⋅ − . IV. VARIATION OF THE ENERGY ACROSS THE TRANSITION
In computing the difference in energy between the two phases, we find that a few tens ( ) of modes are sufficient to obtain good values. This is a consequence of the fact thatthe high-energy part of the spectrum is essentially the same in the metal and in the super-conductor, making the energy difference a quantity that converges much more rapidly thanthe individual terms, as a function of the upper cutoff in the Matsubara frequency, n mod .On defining δE ( ) + δC ( ) , i.e. δE n mod [ d , d ] = E n ( ) [ d , d ] − E s ( ) [ d , d ] + C n ( ) − C s ( ) =∶ δE ( ) + δC ( ) as the difference between the terms coming from the energy of the two cavities in the normaland superconducting phase, plus the difference between the values of the coupling in thetwo phases respectively, we have: n mod δE n mod [ , ] ⋅ δE ( ) ⋅ δC ( ) ⋅
10 6 . . . . . . . . . . . . δE ( ) is of the same order of magnitude of δC ( ) , and when d = d δC ( ) is about two orders of magnitude larger: δE [ , ] = . ⋅ − = . ⋅ − + . ⋅ − (11) δE [ , ] = . ⋅ − = . ⋅ − + . ⋅ − . (12) IV.1. The Matsubara zero-mode contribution
It turns out that this unexpected behavior is due to the contribution from the Matsubarazero mode. This is evident in the following table where we report, for the n − th Matsubara-mode, the values of the Casimir energy in the normal and superconducting phase, and theirdifference ( d = d =
100 nm): 9 E [ n, , ] = { E ( ) + C ( ) } T E { E ( ) + C ( ) } T M { E ( ) + C ( ) } T E + T M E n + − . ⋅ − − . ⋅ − − . ⋅ − E s − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − δE . ⋅ − + . ⋅ − . + . ⋅ − . ⋅ − E n − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − E s − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − δE . ⋅ − − . ⋅ − . ⋅ − − . ⋅ − − . ⋅ − E n − . ⋅ − − . ⋅ − − . ⋅ − , − . ⋅ − − . ⋅ − E s − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − δE . ⋅ − − . ⋅ − . ⋅ − − . ⋅ − . ⋅ − E n − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − E s − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − − . ⋅ − δE . ⋅ − − . ⋅ − . ⋅ − − . ⋅ − . ⋅ − TABLE I: Contributions of the
T E and
T M modes for different values of n and summing the first 50 modes: δE [ , ] = . ⋅ − = . ⋅ − + . ⋅ − . A close look at the table makes it evident that the result is almost completely due to thecontribution of the coupling term of the zero mode. Indeed, C ( ) s,T M is about 3 orders ofmagnitude larger than the corresponding in the normal case C ( ) n,T M while all the other termsare of the same order of magnitude (in some case egual) so that in the difference they canceleach other. V. ENERGY OF THE THREE-LAYER CONFIGURATION
The behavior discussed in the previous section is confirmed for the three-layer configura-tion: E [ d , d , d ] = k B T ∞ ′ ∑ l = ∫ d k (cid:150) ( π ) [ log ( E E E ) + log ( + I ( ) E E E )+ log ⎛⎝ + E I ( ) + E I ( ) E E E + I ( ) ⎞⎠⎤⎥⎥⎥⎥⎦ T M + [
T E ] , (13) =∶ E ( ) [ d , d , d ] + C ( ) [ d , d , d ] + C ( ) [ d , d , d ] E n mod [ d , d , d ] ∶= n mod ∑ l = E [ l, d , d , d ] , To have a comparison between the formulae for two and three cavities let us compute theCasimir energy for the three layer when d is very large. In this case, since the third cavityis distant from the other two, it decouples and the result would be the sum of the energy of10 double cavity plus the energy of a third one. Indeed we find: E n ( , , ) E ( ) n C ( ) n C ( ) n − . ⋅ − − . ⋅ − − . ⋅ − . ⋅ − this is exactly three halves the energy of a double cavity: E n ( , ) = − . ⋅ − ≈ −
23 6 . ⋅ − = . ⋅ − , (14)as expected. Of course, this is a consequence of the strong exponential suppression presentin this term, see the expression of I ( ) in Eq.(5). Taking d = d = d =
100 nm we find E n ( , , ) E ( ) n C ( ) n C ( ) n − . ⋅ − − . ⋅ − − . ⋅ − . ⋅ − thus the contribution due to the coupling of the three cavities is three orders of magnitudelarger than in the previous case but still, for the normal Casimir energy, very much smallerthan the sum of the energies of the three individual cavities.Once again, things are different when computing the difference between the energy in thenormal and superconducting phase. Indeed, in this case the contribution from the n − th mode is, with obvious significance for the indicated symbols: n δE ( n, , , ) δE ( ) δC ( ) δC ( ) . ⋅ − . ⋅ − . ⋅ − − . ⋅ − − . ⋅ − . ⋅ − − . ⋅ − . ⋅ −
10 2 . ⋅ − . ⋅ − − . ⋅ − . ⋅ −
100 8 . ⋅ − . ⋅ − − . ⋅ − . ⋅ − We immediately realize that even in this case the energy is due almost completely to thecoupling of nearest cavities δC ( ) . Note that the δC ( ) term is about one order of magnitudesmaller than the corresponding δC ( ) . Summing on the first n modes we find n mod δE n mod ( , , ) δE ( ) δC ( ) δC ( )
10 5 . ⋅ − . ⋅ − . ⋅ − − . ⋅ −
50 5 . ⋅ − . ⋅ − . ⋅ − − . ⋅ −
100 5 . ⋅ − . ⋅ − . ⋅ − − . ⋅ − For layers 10 nm thick we find: E n ( , , ) E ( ) n C ( ) n C ( ) n − . ⋅ − − . ⋅ − − . ⋅ − . ⋅ − and δE ( , , ) δE ( ) δC ( ) δC ( ) . ⋅ − . ⋅ − . ⋅ − − . ⋅ − d × - × - × - δ E 20 30 40 50 d × - × - × - × - × - × - × - δ E
20 30 40 50 d × - × - × - × - δ C
20 30 40 50 d - × - - × - - × - - × - - × - - × - δ C FIG. 2: The behavior with respect to d = d = d ∈ [ , ] nm of the Casimir energy δE and of the various components δE , δC , and δC for the three-layer configuration with d = nm . In the plot of δE it is shown, also, the fitting curve.To give an idea of the dependence of the Casimir energy on the parameters d , d , d , weshow in the figure 2 the contribution of the three terms δE , δC , and δC to the energydifference between the normal and the superconducting phase, δE , with respect to d = d = d ∈ [ , ] nm with d = nm . In blue it is shown a fit of δE obtained by means of thefunction δE = a + b e −( xx ) with a = . ⋅ − J / m , b = . ⋅ − J / m , x = . nm .Note that the red dots are completely covered by the fitting curve. The only term thatsubstantially depends on d is δE , i.e. the sum of the energies of the single cavity whosethickness is d . On the contrary the other terms almost depend on d , d exclusively. Being δE very much smaller than δC and δC , this fit is very stable with respect the variation of d see FIG. 3 where the same fitting curve is overimposed on the data relative to d = nm We conclude that, the contribution from the coupling of the three cavities being so large: δC can turn out to be only one order of magnitude smaller than δC , it will be thereforenecessary to analyze the situation of four coupled cavities.Some comments about the contribution of the T M zero mode are in order at this point(in the following we will analyze the configuration of two coupled cavities but the gener-alization to three is straightforward). In the ξ ↦ r i,jT M = E ( ) T M,n = E ( ) T M,s = k B T ∫ d k (cid:150) ( π ) log ( − e − d K ) C ( ) T M,n = k B T ∫ d k (cid:150) ( π ) log ( − e − d K ,n ) C ( ) T M,s = k B T ∫ d k (cid:150) ( π ) log ( − e − d K ,s ) d × - × - × - δ E FIG. 3: The behavior with respect to d = d = d ∈ [ , ] nm of the Casimir energy δE forthe three-layer configuration with d = nm and the fitting curve with the parametersobtained for the case d = nm .where d i are measured in nm, K i in nm − , and K i,n / s = √ k ⊥ + α i,n / s . We immediatelyrealize that the contribution of the energies of the two cavities E ( ) T M is exactly the samein the normal and in the superconducting phase so that they cancel in the difference. Onthe contrary the contribution of the interaction terms C ( ) T M in the two phases are differentthanks to the presence of K ,n and K ,s respectively. (Note that the dependence of thesetwo terms on d cancels). Naturally, the strong suppression due to the exponential ensuresthat the main contribution to the integral comes from small wave numbers k ⊥ (hereaftermeasured in nm − ), taking d = d =
100 nm and α ∶= √ k ⊥ + ( ω nio c ) = √ k ⊥ + . − , α ,n ∶= k ⊥ , and α ,s ∶= √ k ⊥ + ( ω s c ) = √ k ⊥ + . ⋅ − nm − , we recover the numbers in TableI. Of course the huge difference reduce drastically for smaller values of d because smallervalues of d allow for the contribution to the integral from larger values of k ⊥ so that thedependence on α i,n / s is less evident. For example with d =
10 nm we get C ( ) T M,n = . ⋅ − Jand C ( ) T M,s = − . ⋅ − J. However in this case, even though the two terms are very muchcloser, their absolute value is larger so that they still give a strong contribution to the energy,thus the values of δE is large, see Eqs. (12) and (13). We expect that this behavior couldchange when a dielectric is inserted between the two layers. VI. CONCLUDING REMARKS
In this paper we performed a series of numerical calculations aimed at the computation ofthe Casimir energy in the normal and superconducting phase for a multilayered cavity. Thisis of particular interest for the ARCHIMEDES experiment aimed at weighing the vacuumenergy of a multi-cavity by modulating the reflectivity of the constituting plates from themetallic to the superconducting phase. As pointed out in [1] with a single cavity and with astandard BCS superconductor a ratio η = ∆ E cas E cas ∼ − is expected. For this value there would13e no possibility for the experiment to detect the signal. However, and quite surprisingly,our results are orders of magnitude larger: We obtained a very large contribution from aterm resulting from the coupling of nearest neighbor cavities in the superconducting phase.This strong enhancement of η results from the use of a superconducting multi-layer (at leasttwo) structure and it can be attributed to the strong contribution of the T M
Matsubarazero mode. From the point of view of the experiment these results are quite promising.The important role played by the static TM physically arises because, while a staticelectric field in a superconductor (and in a metal as well) is rapidly screened on shortlength-scales, the magnetic field parallel to the vacuum-Nb interface can penetrate over asubstantial distance, set by the London penetration depth. This length is shortest in cleanNb, but is still of the order of tens of nm, and increases in the presence of impurities. Itis not surprising therefore, that the zero-frequency TM mode links the various adjacentcavities, providing a substantial inter-cavity contribution to the Casimir energy. Therefore,in computing the Casimir energy of a large number of overlapping cavities, it is necessaryto take into account the contribution from the coupling of pairs of cavities that can lead toa strong enhancement of the effect. This behavior is confirmed in the case of a three-layerconfiguration where, in addition, the contribution of the coupling of the three cavities turnsout to be about one order of magnitude smaller. At this stage we plan to obtain in a futurework an estimate of the contribution of (at least) four coupled cavities. Because of the strongcontribution of the zero mode we expect to be able to discriminate between the Drude orplasma model in computing the zero-mode contribution for the Casimir energy. We wish topoint out that, even though these results are encouraging, the shift in energy is still not aslarge as needed. Indeed (see [1], and Refs. therein), to extract the signal we need an energyshift of the order of few joules. With this kind of configuration, even using a very thin layer,of the order of few nanometers, the energy shift is relatively small: δE ( , , ) = . ⋅ − This is a consequence of the smooth dependence of δE on d . Indeed, for d ≤
10 nm, it canbe fitted as (see [8]): δE ( , d , d ) = δE + ( d D ) s with δE = . ⋅ − , s = . , D = . nm . Thus, in this way we can gain at most oneorder of magnitude. This result strongly support our idea of obtaining such an improvementby using high-temperature superconducting oxides, like YBa Cu O − x . In this case, in fact,larger areas can be used (two orders of magnitude), a larger number of layer, ∼ , can beassembled together, relying on the fine built-in layered structure of cuprates, with thicknessof the order of 1 nm. It is possible to work at high temperature, ∼
100 K (gaining here afactor ten), and, possibly, other two order of magnitude can be gained from ∆ T . Of course,this prevision can prove to be too optimistic and for this reason the extension of the presentanalysis to such a situation is underway. ACKNOWLEDGMENTS
G.E. and C.S. are grateful to the Department of Physics “Ettore Pancini” of Federico IIUniversity, Naples, for hospitality and support.14 ppendix A:
For the case of the
T M -modes the matching conditions give the following 12 ×
12 matrixof coefficients: M = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ − (cid:15) (cid:15) (cid:15) − K K − K e d K (cid:15) e − d K (cid:15) − e − d K (cid:15) − e d K (cid:15) e d K K − e − d K K e − d K K − e d K K e − K x (cid:15) e K x (cid:15) − e K x (cid:15) − e − K x (cid:15) − e − K x K e K x K − e K x K e − K x K e K x (cid:15) e − K x (cid:15) − e − K x (cid:15) − e K x (cid:15) e K x K − e − K x K e − K x K − e K x K e − K x (cid:15) e K x (cid:15) − e K x (cid:15) − e − K x (cid:15)
00 0 0 0 0 0 0 − e − K x K e K x K − e K x K e − K x K
00 0 0 0 0 0 0 0 0 e K x (cid:15) e − K x (cid:15) − e − K x (cid:15) e K x K − e − K x K e − K x K ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . Computing the determinant of the minors of dimensions 4 ,
8, and 12 respectively we obtainEqs. (2,3,5).
Appendix B:
On writing (cid:15) ( iξ ) = + σ ( iξ ) ξ where σ ( iξ ) is the conductivity along the imaginary frequencies, we will obtain the dielectricfunction in the Drude model for the normal case simply by taking σ ( iξ ) = ω p / πγ + ξ with ω p = πne / m the plasma frequency and γ the relaxation parameter. While in thesuperconducting phase the conductivity can be written as [20] σ ( iξ ) = ω p γ + ξ + δσ BCS ( iξ ) where the correction within the BCS model is given by (in the following ̵ h = δσ BCS ( iξ ) = σ γξ ∫ +∞−∞ tanh ( E T ) Re [ G + ( iξ, η )] dηE ,G + ( z, η ) = η Q + ( z, E ) + A + ( z, E )( Q + ( z, E ) + iγ ) Q + ( z, E )[ η − ( Q + ( z, E ) + iγ ) ,A + ( z, E ) = E ( E + z ) + ∆ ,Q + ( z, E ) = ( E + z ) − ∆ ,E = √ η + ∆ .
15o obtain the reflection coefficients for the zero mode we have to compute the limit ξ → ,
3, and 5 regions there is vacuum, we findlim ξ → r i,jT M ( iξ ) = , lim ξ → r i,jT E ( iξ ) = ξ → δσ BCS ( z ) can be approximated by [20] δσ BCS ( iξ ) ≈ ω s / ξ, with ω s = ω p γ ⎛⎜⎝ π ∆ tanh ∆2 k b T − γ ∆ ∫ ∞ tanh √ ∆ + x k b T √ ∆ + x ( γ + x ) dx ⎞⎟⎠ , we obtain for r i,jT M,T E in the superconducting phase:lim ξ → r i,jT M ( iξ ) = , lim ξ → r i,jT E ( iξ ) = k ⊥ − √ k ⊥ + ω si k ⊥ + √ k ⊥ + ω si . [1] E. Calloni, M. De Laurentis, R. De Rosa, F. Garufi, L. Rosa, L. Di Fiore, G. Esposito, C.Rovelli, P. Ruggi, and F. Tafuri, Towards weighing the condensation energy to ascertain theArchimedes force of vacuum, Phys. Rev. D (2014) no.2, 022002[2] G. Bimonte, E. Calloni, G. Esposito and L. Rosa, Energy-momentum tensor for a Casimirapparatus in a weak gravitational field, Phys. Rev. D (2006) 085011 Erratum: Phys. Rev.D (2007) 049904 Erratum: Phys. Rev. D (2007) 089901 Erratum: Phys. Rev. D (2008) 109903[3] S. A. Fulling, K. A. Milton, P. Parashar, A. Romeo, K. V. Shajesh, and J. Wagner, How doesCasimir energy fall?, Phys. Rev. D , 025004 (2007).[4] G. Bimonte, G. Esposito and L. Rosa, From Rindler space to the electromagnetic energy-momentum tensor of a Casimir apparatus in a weak gravitational field, Phys. Rev. D (2008) 024010[5] G. Esposito, G. M. Napolitano and L. Rosa, Energy-momentum tensor of a Casimir apparatusin a weak gravitational field: Scalar case, Phys. Rev. D (2008) 105011[6] H. B. G. Casimir, Introductory remarks on quantum electrodynamics, Physica , 846 (1953).[7] G. Bimonte, E. Calloni, G. Esposito, L. Milano and L. Rosa, Towards measuring variationsof Casimir energy by a superconducting cavity, Phys. Rev. Lett. (2005) 180402[8] G. Bimonte, E. Calloni, G. Esposito and L. Rosa, Variations of Casimir energy from a super-conducting transition, Nucl. Phys. B (2005) 441[9] G. Bimonte, D. Born, E. Calloni, G. Esposito, U. Huebner, E. Il’ichev, L. Rosa, F. Tafuri, andR. Vaglio: ”Low noise cryogenic system for the measurement of the Casimir energy in rigidcavities ” J. Phys. A , 164023 (2008).[10] A. Allocca, G. Bimonte, D. Born, E. Calloni, G. Esposito, U. Huebner, E. Il’ichev, L. Rosa, nd F. Tafuri: ”Results of measuring the influence of Casimir energy on superconductingphase transitions” Jour. Super. and Novel Mag. , 2557 (2012).[11] A. Kempf, J. Phys. A , 164038 (2008).[12] M. Bordag, G.L. Klimtchisktaya, U. Mohideen, and V.M. Mostepanenko, Advances in theCasimir Effect (Oxford University Press, Oxford, 2009).[13] M. Bordag, U. Mohideen and V.M. Mostepanenko, New developments in the Casimir effect,Phys. Rep. , 1 (2001).[14] J. D. Jackson,
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