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

Casimir energy for two

Keywords: numbers: 12.20.Ds, 12.20.-m, 74.25.-q, 74.78.Fk

published or not. The documents may come

I. INTRODUCTION

The ARCHIMEDES experiment [START_REF] Calloni | Towards weighing the condensation energy to ascertain the Archimedes force of vacuum[END_REF] is designed for testing whether the energy of vacuum fluctuations, foreseen by quantum electrodynamics, contributes to gravity, through the coupling demanded by quantum field theory in curved spacetime [START_REF] Bimonte | Energy-momentum tensor for a Casimir apparatus in a weak gravitational field[END_REF][START_REF] Fulling | How does Casimir energy fall?[END_REF][START_REF] Bimonte | From Rindler space to the electromagnetic energymomentum tensor of a Casimir apparatus in a weak gravitational field[END_REF][START_REF] Esposito | Energy-momentum tensor of a Casimir apparatus in a weak gravitational field: Scalar case[END_REF], where the Einstein tensor is taken to be proportional to the expectation value of the regularized and renormalized energy-momentum tensor of matter fields. The idea is to weigh the vacuum energy stored in a rigid Casimir cavity made by parallel conducting plates, by modulating the reflectivity of the plates upon inducing a transition from the metallic to the superconducting phase [START_REF] Calloni | Towards weighing the condensation energy to ascertain the Archimedes force of vacuum[END_REF]. In order to enhance the effect, a multilayer cavity is considered, obtained by superimposing many 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 superconducting transition, separated by nonconducting planes. A crucial aspect to be tested is thus the behavior of the Casimir energy [START_REF] Casimir | Introductory remarks on quantum electrodynamics[END_REF] for a multi-cavity when the layers undergo the phase transition from the metallic to the superconducting phase. Until now only the case of a cavity having a single layer that undergoes the superconducting transition was considered, the other reflecting plate being just metallic (not superconducting), in Refs. [START_REF] Bimonte | Towards measuring variations of Casimir energy by a superconducting cavity[END_REF][START_REF] Bimonte | Variations of Casimir energy from a superconducting transition[END_REF]. The generalization to the case of a system of coupled superconducting layered cavities is still lacking. With respect to the ARCHIMEDES project the main goal is to study the possibility of enhancing the modulation factor η = ∆Ecas Ecas were ∆E cas is the difference of Casimir energy in normal and superconducting states. The value obtained in Ref. [START_REF] Bimonte | Towards measuring variations of Casimir energy by a superconducting cavity[END_REF][START_REF] Bimonte | Variations of Casimir energy from a superconducting transition[END_REF], considering a cavity with a single superconducting layer and a transition temperature of about 1 K is η l ≈ 10 -8 . This value was compliant with a previous experiment devoted to ascertain the vacuum energy contribution to the total condensation energy [START_REF] Bimonte | Low noise cryogenic system for the measurement of the Casimir energy in rigid cavities[END_REF][START_REF] Allocca | Results of measuring the influence of Casimir energy on superconducting phase transitions[END_REF], but it is not sufficient to prove the weight of the vacuum, because it is in absolute too small. It is therefore necessary to consider high-T c superconductors where condensation energy is much higher and also 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 superconductor, a factor as high as η h = 4 ⋅ 10 -4 has been estimated, under the approximation of flat plasma sheets at zero temperature, no conduction in normal state (here E cas is the energy of the ideal cavity) and charge density of n = 10 14 cm -2 . The ARCHIMEDES sensitivity is expected to be capable of ascertain the interaction of gravity and vacuum energy also for values lower than η h = 4 ⋅ 10 -4 , up to 1 100 of this value [START_REF] Calloni | Towards weighing the condensation energy to ascertain the Archimedes force of vacuum[END_REF]. Clearly it is important to understand more firmly if dealing with layered superconducting structures the modulation depth can be sufficiently high. This is the study of the present paper. Considering in particular the multi-layer cavity, the general assumption adopted so far has been that the Casimir energy obtained 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 the thickness of each metallic layer separating the various cavities is very large with respect to the penetration depth of the radiation field). Of course, this is no longer true if the thickness of these metallic inter-cavity layers gets thinner and thinner. The evaluation of the Casimir energy for such a configuration is the subject of the present study. It is worth stressing that this is only a first step because in the final version ARCHIMEDES experiment will make use of high-T c superconducting oxides with a built-in layered structure, like YBa 2 Cu 3 O 7-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 Casimir cavities, made by traditional BCS (low-T c ) superconducting material (niobium), so as to deal with relatively manageable and well established formulas. On trying to preserve a macroscopic approach, we limited our study to thicknesses between 10 and 100 nm. In the following, referring to Fig. 1,d i is the distance of the ith cavity from the (i -1) -th, (thickness of the ith cavity), within the slabs 1, 3 and 5 there is vacuum while the zones 0, 2, 4 and 6 are made of niobium. The thicknesses of the zones 0 and 6 are assumed to be infinite. Although this choice is dictated by simplicity and by the well-established superconducting properties of Nb, this is a first necessary step to prepare future studies aimed at considering the more elaborate case of high-T c superconductors, as required by the roadmap of the ARCHIMEDES experiment.

Section II studies the Casimir energy of a multilayer cavity, while Sec. III evaluates the Casimir energy in the normal and superconducting phases. Variation of the energy in the transition is obtained in Sec. IV, including a detailed numerical analysis of the Matsubara zero-mode contribution. Section V extends this scheme to the three-layer configuration, and concluding 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 [START_REF] Bimonte | Towards measuring variations of Casimir energy by a superconducting cavity[END_REF][START_REF] Bimonte | Variations of Casimir energy from a superconducting transition[END_REF], at finite temperature, the Casimir variation across the transition from a metallic to a superconducting phase is obtained as the difference between the free Casimir energy in the metallic state and the same after the transition to superconducting state takes place: δE(T ) = E n -E s . The energy per unit area of a single cavity, (012) in Fig. 1, can be written, at finite temperature T , as the sum of the contributions of the transverse electric (T E) and transverse magnetic (T M ) modes (see, for example, [START_REF] Bordag | Advances in the Casimir Effect[END_REF]) :

E[d 1 , d 2 ] = k B T ∞ ′ l=0 dk (2π) 2 log ∆ T E (ξ l ) + log ∆ T M (ξ l ) =∶ ∞ l=0 E[l, d 1 , d 2 ] (1)
where ξ l = 2πlk B T are the Matsubara frequencies, k B is the Boltzmann constant, l = 0, 1, 2, . . ., the superscript ′ on the sum means that the zero mode must be multiplied by a factor 1 2 ,

∆ T M (ξ l ) = r 0,1 T M (ξ l )r 1,2 T M (ξ l )e -2d 1 K 1 + 1 , ∆ T E (ξ l ) = r 0,1 T E (ξ l )r 1,2 T E (ξ l )e -2d 1 K 1 + 1
and the reflection coefficients r i,j (T M,T E) (ξ l ) are given by (see [START_REF] Bordag | Advances in the Casimir Effect[END_REF]):

r i,j T M (ξ l ) = i (ξ l )K j (ξ l ) -j (ξ l )K i (ξ l ) i (ξ l )K j (ξ l ) + j (ξ l )K i (ξ l ) , r i,j T 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 2 ⊥ + i (ξ l )ξ 2 l .
We point out that our approach captures the relevant length scale of a superconductor, the London penetration depth λ L , through the expression of i (ξ l ) in terms of the correction to the optical conductivity when entering the superconducting state, δσ BCS (iξ), see below and Appendix B. In particular, for ξ l → 0, we have i (ξ l )ξ 2 l → λ -2 L . We characterize the properties of the ith material trough the dielectric function i (ξ l ) and the change in the Casimir energy is given simply by the modification of the (ξ l ) due to the transition [START_REF] Bimonte | Towards measuring variations of Casimir energy by a superconducting cavity[END_REF][START_REF] Bimonte | Variations of Casimir energy from a superconducting transition[END_REF]. As we said, in the following we report calculations for the case in which 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 ith slab the material is Nb and the spacer is vacuum (the modifications introduced by a dielectric spacer deserve a separate study).

To obtain the formulas for two and three cavities we solve the problem by imposing the continuity of the tangential component of the ⃗ E and ⃗ H fields (non-magnetic media) and the normal component of the ⃗ D and ⃗ B at the interface [START_REF] Bordag | New developments in the Casimir effect[END_REF][START_REF] Jackson | Classical Electrodynamics[END_REF]. Thus, for example, in the case 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 report the expression for the T M -modes in appendix A) from which it is possible to extract the case of one, two, and three cavities by taking (i, j) = 1 . . . 4, (i, j) = 1 . . . 8, (i, j) = 1 . . . 12, respectively:

∆ (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 the formulas for the two cases can be obtained using respectively the T M or T E reflection coefficients. Defining (no summation over repeated indices) E ijl = r i,j r j,l e -2d j K j + 1, F ijl = r i,j e -2d j K j + r j,l , G ijl = r i,j + e -2d j K j r j,l , H ijl = e -2d j K j + r i,j r j,l , we have for the single cavity (012) in Fig. 1 ∆ (1) = E 012 ;

(

for two cavities (012 -234):

∆ (2) = E 012 E 234 + e -2(d 2 k 2 ) F 012 G 234 =∶ E 012 E 234 + I (2) 012 and (3) log ∆ (2) = log E 012 E 234 + log ⎛ ⎝ 1 + I (2) 012 E 012 E 234 ⎞ ⎠ , (4)
and for the three cavities:

∆ (3) = E 012 E 234 E 456 + e -2(d 2 k 2 +d 4 k 4 ) F 012 H 234 G 456 +e -2d 2 k 2 E 456 F 012 G 234 + e -2d 4 k 4 E 012 F 234 G 456 =∶ E 012 E 234 E 456 + I (3) + E 456 I (2) 012 + E 012 I (2) 234 , (5) log ∆
(3) = log E 012 E 234 E 456 + log 1 + I (3) E 012 E 234 E 456 + log ⎛ ⎝ 1 + E 456 I (2) 012 + E 012 I (2) 234 E 012 E 234 E 456 + I (3) ⎞ ⎠ . (6)
In this way, when

d 2 → ∞ [see Eq. ( 3
)] I (2)
012 → 0 and log ∆ (2) = log

E 012 E 234 = log E 012 + log E 234 .
That is to say, when the two cavities are far away their energy is simply the sum of the individual contributions. In this respect the second term on the right of Eq. ( 3), I (2) , can be seen as the energy due to the coupling of the two cavities (012) -(234).

When

d 1 = d 3 = d 5 , d 2 = d 4 , 0 = 2 = 4 , 1 = 3 = 5 we obtain E 012 T M = E 234 , F 012 = F 234 so that we can omit the subscripts: log ∆ (2) = log ∆ (1) 2 + log 1 + I (2) (∆ (1) ) 2 . ( 7
)
For the three cavities (012 -234 -456), formulas are written so as to make evident the 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 (012 -234), (234 -456), and the one coming from the coupling of the three: I (3) . Thus, under the previous hypothesis, ∆ (3) = ∆ (1) 3 + 2I (2) + I (3) and we can write log ∆ (3) = log ∆ (1) 3 + log 1 + 2

I (2) (∆ (1) ) 3 + I (3) + log 1 + I (3) (∆ (1) ) 3 . (8)
In a sense, we are writing the energy as a sum of the energy of the single cavity plus the coupling energy between the nearest neighbor, plus the coupling energy among the second nearest neighbor and so on. In this way we will have a clear indication of the strength of the coupling between the cavities at the various orders. As far as we know this way of displaying the various contribution to the Casimir energy has been obtained for the first time in [START_REF] Kenneth | [END_REF] 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 recovered by observing that the determinant of a N × N complex block matrix can be obtained in terms of the determinants of its constituent blocks [18].

The Casimir energy in the superconducting phase is obtained by replacing, in the reflection coefficients, the expression of the dielectric function with the corresponding obtained using the BCS theory [START_REF] Zimmermann | Optical conductivity of BCS superconductors with arbitrary purity[END_REF][START_REF] Bimonte | Optical BCS conductivity at imaginary frequencies and dispersion energies of superconductors[END_REF], see Appendix B. In the following we will characterize the dielectric properties of the material by means of the Drude model (but see conclusions):

(iξ) = 1 + σ(iξ) ξ , with σ k (iξ) = σ 0k γ + ξ ,
for conducting materials and

σ k (iξ) = σ 0k γ + ξ + δσ BCS (iξ), for superconducting materials ,
where the expression of δσ BCS (iξ) is given in Appendix B (see [START_REF] Bimonte | Optical BCS conductivity at imaginary frequencies and dispersion energies of superconductors[END_REF]).

Thus

δE(T ) = E n (T ) -E s (T ) = k T 2 ∞ ′ l=0 dk (2π) 2 ⎛ ⎝ log ∆ (k) n,T E ∆ (k) s,T E + log ∆ (k) n,T M ∆ (k) s,T M ⎞ ⎠ =∶ ∞ l=0 δE[l, d 1 , d 2 ].
where ∆

(k) n,T E,T M , ∆ (k)
s,T E,T M are the generating functions (this nomenclature denotes here just 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 are considering (1, 2, or 3) we must take k = 1, 2, 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 for the critical temperature and plasma frequency T c = 9.25 K, ̵ hω nio = 9.268 eV and work at the temperature T = 9.157 K. We start by choosing d 1 = 300 nm, and d 2 = 600 nm, so as to have results that can be compared with standard formulas.

We find for the energy in the normal phase E n , for fixed d 1 , d Thus, at least 500 Matsubara modes are necessary to obtain a result stable at the second decimal digit.

These results can be compared with the approximate result obtained by Bordag et al. [START_REF] Bordag | Advances in the Casimir Effect[END_REF]

(t 0 = c ̵ h 2 a k B T , D 0 = c a ωp ): E = c ̵ h 8πa 3 D 0 2 3ζ(3) t 0 - 4π 3 25 + D 0 2π 3 45 - ζ(3) t 0 + ζ(3) 2 t 0 - π 3 90 = -1.258 ⋅ 10 -8 .
As expected, because of the strong suppression of the exponential for large d 2 , the contribution of the coupling term between the two cavities (012)-( 123) is about ten orders of magnitude smaller than the energy obtained from each cavity. With

E[d 1 , d 2 ] = k T 2 ∞ ′ l=0 dk (2π) 2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ log E 012 E 234 + log ⎛ ⎝ 1 + I (2) 012 E 012 E 234 ⎞ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦TM + [T E] (9)
=∶ E

(2)

T M [d 1 , d 2 ] + C (2) T M [d 1 , d 2 ] + E (2) T E [d 1 , d 2 ] + C (2) T E [d 1 , d 2 ] ( 10
) =∶ E (2) [d 1 , d 2 ] + C (2) [d 1 , d 2 ]
we obtain (n mod = 500):

E n (2)
[300, 600] = -2.42035 ⋅ 10 -8 ; C n

[300, 600] = -2.30743 ⋅ 10 -17 .

Indeed, having d 1 ≪ d 2 , the total energy is simply the sum of the energies of the two cavities:

E n (2) [300, 600] 2 = -2.42035 2 ⋅ 10 -8 = -1.2102 ⋅ 10 -8 ≈ -1.258 ⋅ 10 -8
In the superconducting phase we have, more or less, the same behavior: The Casimir energy is always greater than the corresponding energy in the normal phase so that, as expected, the difference is always positive. Once more, the contribution from the energy of the two cavities is much larger than the contribution from the coupling, but "only" by about four orders of magnitude:

n mod E s n mod [
E s 500 [300, 600] = -2.4210⋅10 -8 , E s (2) [300, 600] = -2.4208⋅10 -8 , C s (2)
[300, 600] = -2.0979⋅10 -12 .

IV. VARIATION OF THE ENERGY ACROSS THE TRANSITION

In computing the difference in energy between the two phases, we find that a few tens (50) of modes are sufficient to obtain good values. This is a consequence of the fact that the high-energy part of the spectrum is essentially the same in the metal and in the superconductor, making the energy difference a quantity that converges much more rapidly than the individual terms, as a function of the upper cutoff in the Matsubara frequency, n mod .

On defining δE (2) + δC (2) , i.e.

δE n mod [d 1 , d 2 ] = E n (2) [d 1 , d 2 ] -E s (2) [d 1 , d 2 ] + C n (2) -C s (2)
=∶ δE (2) + δC (2) as the difference between the terms coming from the energy of the two cavities in the normal and superconducting phase, plus the difference between the values of the coupling in the two phases respectively, we have: We note that δE (2) is of the same order of magnitude of δC (2) , and when

n mod δE n mod [
d 1 = d 2 δC (2)
is about two orders of magnitude larger: (12)

IV.1. The Matsubara zero-mode contribution

It turns out that this unexpected behavior is due to the contribution from the Matsubara zero mode. This is evident in the following table where we report, for the nth Matsubaramode, the values of the Casimir energy in the normal and superconducting phase, and their difference ( A close look at the table makes it evident that the result is almost completely due to the contribution of the coupling term of the zero mode. Indeed, C

d 1 = d 2 = 100 nm): n E[n, 100, 100] = {E (2) + C (2) } T E {E (2) + C (2) } T M {E (2) + C (2) } T E+T M E n 0 + 0 -6.
s,T M is about 3 orders of magnitude larger than the corresponding in the normal case C

(2) n,T M while all the other terms are of the same order of magnitude (in some case egual) so that in the difference they cancel each other.

V. ENERGY OF THE THREE-LAYER CONFIGURATION

The behavior discussed in the previous section is confirmed for the three-layer configuration:

E[d 1 , d 2 , d 4 ] = k B T 2 ∞ ′ l=0 dk (2π) 2 log E 012 E 234 E 456 + log 1 + I (3) E 012 E 234 E 456 + log ⎛ ⎝ 1 + E 456 I (2) 012 + E 012 I (2) 234 E 012 E 234 E 456 + I (3) ⎞ ⎠ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦TM + [T E], (13)
=∶ E (3) [d 1 , d 2 , d 4 ] + C (3) [d 1 , d 2 , d 4 ] + C (2) [d 1 , d 2 , d 4 ] E n mod [d 1 , d 2 , d 4 ] ∶= n mod l=0 E[l, d 1 , d 2 , d 4 ],
To have a comparison between the formulae for two and three cavities let us compute the Casimir energy for the three layer when d 4 is very large. In this case, since the third cavity is distant from the other two, it decouples and the result would be the sum of the energy of a double cavity plus the energy of a third one. Indeed we find:

E 500 n (100, 100, 300) E

(3) n

C

(2) n

C

(3) n -6.2636 ⋅ 10 -7 -6.2574 ⋅ 10 -7 -6.2390 ⋅ 10 -10 6.73512 ⋅ 10 -15

this is exactly three halves the energy of a double cavity:

E 500 n (100, 100) = -4.17785 ⋅ 10 -7 ≈ -

2 3 6.26365 ⋅ 10 -7 = 4.17577 ⋅ 10 -7 , (14)
as expected. Of course, this is a consequence of the strong exponential suppression present in this term, see the expression of I (3) in Eq.( 5). Taking thus the contribution due to the coupling of the three cavities is three orders of magnitude larger than in the previous case but still, for the normal Casimir energy, very much smaller than the sum of the energies of the three individual cavities.

d 4 = d 1 = d 2 =
Once again, things are different when computing the difference between the energy in the normal and superconducting phase. Indeed, in this case the contribution from the nth mode is, with obvious significance for the indicated symbols: n δE(n, 100, 100, 100) δE We immediately realize that even in this case the energy is due almost completely to the coupling of nearest cavities δC (2) . Note that the δC (3) term is about one order of magnitude smaller than the corresponding δC (2) . Summing on the first n modes we find n mod δE n mod (100, 100, 100) For layers 10 nm thick we find: with a = 1.73 ⋅ 10 -9 J m 2 , b = 3.69 ⋅ 10 -8 J m 2 , x 0 = 12.63 nm. Note that the red dots are completely covered by the fitting curve. The only term that substantially depends on d 1 is δE 3 , i.e. the sum of the energies of the single cavity whose thickness is d 1 . On the contrary the other terms almost depend on d 2 , d 4 exclusively. Being δE 3 very much smaller than δC 2 and δC 3 , this fit is very stable with respect the variation of d 1 see FIG. 3 where the same fitting curve is overimposed on the data relative to d 1 = 500 nm We conclude that, the contribution from the coupling of the three cavities being so large: δC 3 can turn out to be only one order of magnitude smaller than δC 2 , it will be therefore necessary 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 generalization to three is straightforward). In the ξ ↦ 0 limit, assuming vacuum between the two superconducting layers (see appendix B), one obtains r i,j T M = 1 for both the normal and superconducting case, so that: where d i are measured in nm, K i in nm -1 , and K i,n s = k 2 ⊥ + α 2 i,n s . We immediately realize that the contribution of the energies of the two cavities E

δE (3) δC (2) δC (3
E 500 n (10, 10, 10) E (3) n C (2) n C (3)
E (2) T M,n = E (2) T M,s = k B T 4 dk (2π) 2 log 1 -e -2d 1 K 1 C (2) T M,n = k B T 4 dk (2π) 2 log 1 -e -2d 2 K 0,n C (2)
T M,s = k B T 4 dk (2π) 2 log 1 -e -2d 2 K 0,s 20
T M is exactly the same in the normal and in the superconducting phase so that they cancel in the difference. On the contrary the contribution of the interaction terms C

(2) T M in the two phases are different thanks to the presence of K 0,n and K 0,s respectively. (Note that the dependence of these two terms on d 1 cancels). Naturally, the strong suppression due to the exponential ensures that the main contribution to the integral comes from small wave numbers k ⊥ (hereafter measured in nm -1 ), taking

d 1 = d 2 = 100 nm and α 2 1 ∶= k 2 ⊥ + ω nio c 2 = k 2 ⊥ + 0.00221 nm -1 , α 2 0,n ∶= k 2 ⊥ ,
T M,n = 1.78 ⋅ 10 -8 J and C

(2) T M,s = -3.011 ⋅ 10 -8 J. However in this case, even though the two terms are very much closer, 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 [START_REF] Bordag | New developments in the Casimir effect[END_REF]. We expect that this behavior could change 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 of the Casimir energy in the normal and superconducting phase for a multilayered cavity. This is of particular interest for the ARCHIMEDES experiment aimed at weighing the vacuum energy of a multi-cavity by modulating the reflectivity of the constituting plates from the metallic to the superconducting phase. As pointed out in [START_REF] Calloni | Towards weighing the condensation energy to ascertain the Archimedes force of vacuum[END_REF] with a single cavity and with a standard BCS superconductor a ratio η = ∆Ecas Ecas ∼ 10 -8 is expected. For this value there would be 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 a term 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 least two) structure and it can be attributed to the strong contribution of the T M Matsubara zero 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 static electric field in a superconductor (and in a metal as well) is rapidly screened on short length-scales, the magnetic field parallel to the vacuum-Nb interface can penetrate over a substantial distance, set by the London penetration depth. This length is shortest in clean Nb, but is still of the order of tens of nm, and increases in the presence of impurities. It is not surprising therefore, that the zero-frequency TM mode links the various adjacent cavities, 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 necessary to take into account the contribution from the coupling of pairs of cavities that can lead to a strong enhancement of the effect. This behavior is confirmed in the case of a three-layer configuration where, in addition, the contribution of the coupling of the three cavities turns out to be about one order of magnitude smaller. At this stage we plan to obtain in a future work an estimate of the contribution of (at least) four coupled cavities. Because of the strong contribution of the zero mode we expect to be able to discriminate between the Drude or plasma model in computing the zero-mode contribution for the Casimir energy. We wish to point out that, even though these results are encouraging, the shift in energy is still not as large as needed. Indeed (see [START_REF] Calloni | Towards weighing the condensation energy to ascertain the Archimedes force of vacuum[END_REF], and Refs. therein), to extract the signal we need an energy shift 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 100 (1, 1, 1) = 6.371⋅10 -8 This is a consequence of the smooth dependence of δE on d 2 . Indeed, for d 2 ≤ 10 nm, it can be fitted as (see [START_REF] Bimonte | Variations of Casimir energy from a superconducting transition[END_REF]):

δE 100 (10, d 2 , d 2 ) = δE 0 1 + d 2 D s
with δE 0 = 9.29 ⋅ 10 -8 , s = 0.92, D = 2.30 nm. Thus, in this way we can gain at most one order of magnitude. This result strongly support our idea of obtaining such an improvement by using high-temperature superconducting oxides, like YBa 2 Cu 3 O 7-x . In this case, in fact, larger areas can be used (two orders of magnitude), a larger number of layer, ∼ 10 6 , can be assembled together, relying on the fine built-in layered structure of cuprates, with thickness of the order of 1 nm. It is possible to work at high temperature, ∼ 100 K (gaining here a factor 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 present analysis to such a situation is underway.

To obtain the reflection coefficients for the zero mode we have to compute the limit ξ → 0. In this way, considering that in the 1, 3, and 5 regions there is vacuum, we find lim ξ→0 r i,j T M (iξ) = 1, lim ξ→0 r i,j T E (iξ) = 0 in the conducting phase. Noting that, when ξ → 0, δσ BCS (z) can be approximated by [START_REF] Bimonte | Optical BCS conductivity at imaginary frequencies and dispersion energies of superconductors[END_REF] δσ BCS (iξ) ≈ ω 2 s ξ, with

ω 2 s = ω 2 p γ ⎛ ⎜ ⎝ π∆ tanh ∆ 2k b T -4γ∆ 2 ∞ 0 tanh √ ∆ 2 +x 2 2k b T √ ∆ 2 + x 2 (γ 2 + x 2 ) dx ⎞ ⎟ ⎠ ,
we obtain for r i,j T M,T E in the superconducting phase:

lim ξ→0 r i,j T M (iξ) = 1, lim ξ→0 r i,j T E (iξ) = k ⊥ -k 2 ⊥ + ω 2 si k ⊥ + k 2 ⊥ + ω 2 si .

  δE 50 [50, 50] = 1.11259 ⋅ 10 -9 = 1.24592 ⋅ 10 -11 + 1.10013 ⋅ 10 -9 (11) δE 50 [10, 10] = 9.3812 ⋅ 10 -9 = 2.34485 ⋅ 10 -11 + 9.35775 ⋅ 10 -9 .

) 10 5 .

 5 47634 ⋅ 10 -10 1.2644 ⋅ 10 -11 5.81547 ⋅ 10 -10 -4.6557 ⋅ 10 -11 50 5.47789 ⋅ 10 -10 1.28122 ⋅ 10 -11 5.81533 ⋅ 10 -10 -4.6557 ⋅ 10 -11 100 5.47800 ⋅ 10 -10 1.28236 ⋅ 10 -11 5.81533 ⋅ 10 -10 -4.6557 ⋅ 10 -11

n - 1 .- 3 . × 10 - 9 - 2 . 5 × 10 - 9 - 2 . × 10 - 9 - 1 . 5 × 10 - 9 - 1 . × 10 - 9 - 5 . × 10 -10 0 δC 3 FIG. 2 :

 13925109291510919532 FIG.2:The behavior with respect to d 2 = d 4 = d ∈ [10, 50] nm of the Casimir energy δE and of the various components δE 3 , δC 2 , and δC 3 for the three-layer configuration with d 1 = 10 nm. In the plot of δE it is shown, also, the fitting curve.

5 . × 10 - 9 1. × 10 - 8 1. 5 × 10 - 8 δEFIG. 3 :

 510910851083 FIG. 3: The behavior with respect to d 2 = d 4 = d ∈ [10, 50] nm of the Casimir energy δE for the three-layer configuration with d 1 = 500 nm and the fitting curve with the parameters obtained for the case d 1 = 10 nm.

  300, 600] ⋅ 10 12 δE (2) ⋅ 10 12 δC (2) ⋅ 10 12

	10	6.54447	4.44655	2.09792
	30	6.55295	4.45504	2.09792
	50	6.55406	4.45614	2.09792
	100	6.55461	4.45669	2.09792

  039 ⋅ 10 -10 -2.175 ⋅ 10 -13 -6.041 ⋅ 10 -10 10 -10 -8.5958 ⋅ 10 -13 -1.19878 ⋅ 10 -9 -5.58814 ⋅ 10 -12 -1.67018 ⋅ 10 -9 δE 1.80615 ⋅ 10 -14 -2.07135 ⋅ 10 -16 2.31737 ⋅ 10 -16 -1.64377 ⋅ 10 -15 1.64423 ⋅ 10 -14 E n -5.11734 ⋅ 10 -10 -1.87681 ⋅ 10 -13 -1.1032 ⋅ 10 -9 -9.34075 ⋅ 10 -13 -1.61606 ⋅ 10 -9 50 E s -5.11735 ⋅ 10 -10 -1.8768 ⋅ 10 -13 -1.1032 ⋅ 10 -9 -9.34068 ⋅ 10 -13 -1.61606 ⋅ 10 -9 δE 3.31224 ⋅ 10 -16 -1.22071 ⋅ 10 -18 3.90994 ⋅ 10 -17 -6.84111 ⋅ 10 -18 3.62262 ⋅ 10 -16

	0	E s δE	-6.284 ⋅ 10 -12 -1.647 ⋅ 10 -13 6.284 ⋅ 10 -12 + 1.647 ⋅ 10 -13	-6.039 ⋅ 10 -10 -2.692 ⋅ 10 -10 0.0 + 2.690 ⋅ 10 -10	-8.795 ⋅ 10 -10 2.754 ⋅ 10 -10
		E n	-2.076 ⋅ 10 -10 -6.060 ⋅ 10 -12	-1.207 ⋅ 10 -9 -1.235 ⋅ 10 -10	-1.545 ⋅ 10 -9
	1	E s δE	-2.087 ⋅ 10 -10 -6.048 ⋅ 10 -12 1.161 ⋅ 10 -12 -1.228 ⋅ 10 -14	-1.207 ⋅ 10 -9 -1.223 ⋅ 10 -10 9.480 ⋅ 10 -16 -1.235 ⋅ 10 -12	-1.545 ⋅ 10 -9 -8.536 ⋅ 10 -14
		E n	-4.64934 ⋅ 10 -10 -8.59787 ⋅ 10 -13 -1.19878 ⋅ 10 -9 , -5.58978 ⋅ 10 -12	-1.67017 ⋅ 10 -9
	10	E s	-4.64952 ⋅

TABLE I :

 I Contributions

of the T E and T M modes for different values of n and summing the first 50 modes: δE 50 [100, 100] = 2.76004 ⋅ 10 -10 = 8.54145 ⋅ 10 -12 + 2.67463 ⋅ 10 -10 .

  10 -7 -6.25742 ⋅ 10 -7 -1.25193 ⋅ 10 -9 4.39881 ⋅ 10 -12

					100 nm we find
	E 500 n (100, 100, 100)	E n (3)	C	(2) n	C	(3) n
	-6.26989 ⋅

  ⊥ + 4.306 ⋅ 10 -6 nm -1 , we recover the numbers in TableI. Of course the huge difference reduce drastically for smaller values of d 2 because smaller values of d 2 allow for the contribution to the integral from larger values of k ⊥ so that the dependence on α 2 i,n s is less evident. For example with d 2 = 10 nm we get C

	and α 2 0,s ∶= k 2 ⊥ + ωs c	2	= k 2

ACKNOWLEDGMENTS

G.E. and C.S. are grateful to the Department of Physics "Ettore Pancini" of Federico II University, Naples, for hospitality and support.

Appendix A:

For the case of the T M -modes the matching conditions give the following 12 × 12 matrix of coefficients:

Computing the determinant of the minors of dimensions 4, 8, and 12 respectively we obtain Eqs. [START_REF] Bimonte | Energy-momentum tensor for a Casimir apparatus in a weak gravitational field[END_REF][START_REF] Fulling | How does Casimir energy fall?[END_REF][START_REF] Esposito | Energy-momentum tensor of a Casimir apparatus in a weak gravitational field: Scalar case[END_REF].

Appendix B:

On writing

where σ(iξ) is the conductivity along the imaginary frequencies, we will obtain the dielectric function in the Drude model for the normal case simply by taking

with ω 2 p = 4πne 2 m the plasma frequency and γ the relaxation parameter. While in the superconducting phase the conductivity can be written as [START_REF] Bimonte | Optical BCS conductivity at imaginary frequencies and dispersion energies of superconductors[END_REF]

where the correction within the BCS model is given by (in the following ̵ h = 1)