Casimir energy for N superconducting cavities: a model for the YBCO (GdBCO) sample to be used in the Archimedes experiment

In this paper we study the Casimir energy of a sample made by N cavities, with N≫1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\gg 1$$\end{document}, across the transition from the metallic to the superconducting phase of the constituting plates. After having characterised the energy for the configuration in which the layers constituting the cavities are made by dielectric and for the configuration in which the layers are made by plasma sheets, we concentrate our analysis on the latter. It represents the final step towards the macroscopical characterisation of a “multi-cavity” (with N large) necessary to fully understand the behaviour of the Casimir energy of a YBCO (or a GdBCO) sample across the transition. Our analysis is especially useful to the Archimedes experiment, aimed at measuring the interaction of the electromagnetic vacuum energy with a gravitational field. To this purpose, we aim at modulating the Casimir energy of a layered structure, the multi-cavity, by inducing a transition from the metallic to the superconducting phase. After having characterised the Casimir energy of such a structure for both the metallic and the superconducting phase, we give an estimate of the modulation of the energy across the transition.


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The principal goal of the Archimedes experiment [1] is to measure the coupling of the vacuum fluctuations of Quantum Electro-36 dynamics (QED) to the gravitational field of the Earth. The coupling is obtained, as usual in Quantum Field Theory in Curved 37 spacetime [2][3][4][5], assuming the Einstein tensor to be proportional to the expectation value of the regularized and renormalized 38 energy-momentum tensor of matter fields, in particular, for the Archimedes experiment, of the electromagnetic field. The idea is 39 to weigh the vacuum energy stored in a rigid Casimir cavity [6], made by parallel conducting plates, by modulating the reflectivity 40 of the plates upon inducing a transition from the metallic to the superconducting state [1]. The "modulation factor" is defined as 41 η = E E 0 were E A is the difference of Casimir energy (per square meter) in the normal and in the superconducting state, and E 0 A is the 42 (absolute value) of the Casimir energy (per square meter), at zero temperature, of an ideal cavity of the same thickness d: E 0 A = π 2 c 720d 3 . 43 In Ref. [1] it was shown that, in order to measure such an effect, η must be of the order η ∼ 10 −5 and that, to this purpose, 44 a multi-cavity, obtained by superimposing many cavities must be used. This structure is natural in the case of crystals of type-II 45 superconductors, particularly cuprates, being composed by Cu-O planes, that undergo the superconducting transition, separated by 46 nonconducting planes. A crucial aspect to be tested is the behavior of the Casimir energy [6] for a multi-cavity when the layers undergo the phase transition from the metallic to the superconducting phase. In a previous paper [7] a careful study for such a type 48 of structure has been carried out for a sample made by up to three "relatively thick" (of the order of ten nanometer) dielectric layers. 49 In the present paper we extend the analysis to any number of cavities for both situations: layers consisting of "thick" dielectric slabs 50 and layers consisting of "thin" plasma sheets. 51 Indeed, in Ref. [8], considering a cavity based on a high-T c layered superconductor, a factor as high as η = 4 × 10 −4 has been 52 estimated (for flat plasma sheets at zero temperature and no conduction in the normal state, so that E corresponds to the energy 53 of the ideal cavity, and charge density n = 10 14 cm −2 ). The Archimedes sensitivity is expected to be capable of assessing the 54 interaction of gravity and vacuum energy also for values lower than η = 4 × 10 −4 , up to 1/100 of this value [1]. It is then crucial to 55 understand the level of modulation achievable with layered superconducting structures. This is the scope of the present paper.

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Considering in particular the multi-cavity, the general assumption adopted so far has been that the Casimir  In this section we deduce the Casimir Energy of n coupled cavities, even though in the present paper we are interested in applying 67 our results to plasma sheets, we will discuss the case of dielectrics first and then recover the plasma sheets results as a suitable 68 limiting case.

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In the following, referring to  The general expression for the Casimir energy (per unit area), at finite temperature, will be written in the usual manner [9-11]

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where the are the so called generating functions (in the following we will omit the subscript TM(T E) if no ambiguity is generated),  Fig. 1; see also the appendix) 79 [12].

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For the sake of clarity, we only give here the general argument about the procedure for obtaining the generating functions, 81 referring the reader to the appendix for the complete computation. In the appendix we show that the functions can be written in 82 terms of a sort of generalised reflection coefficients: , μ 0 is the magnetic permeability of vacuum, n 2D is the two dimensional 88 carrier density in the layer, and q * and m * , respectively, their charge and mass. The standard dielectric boundary conditions (dbc) 89 will be recovered by imposing = 0 and the plasma sheet boundary conditions (psbc) by requiring i (iζ l ) = 1, ∀i (in this case, After introducing the auxiliary functions and (henceforth, we will assume all the cavities to be equal and consider only the indices {i jk} = {012}), on defining: we can proceed to compute the generating functions. be written as a 2 × 2 block matrix, thus [17,18]  Casimir energy will be simply the sum of the energies of the two cavities, log ( 2 ) = log ( 2 1 ) = 2 log ( 1 ). When they are 115 brought at a distance d 2 from each other, in addition to the previous energy, there is the interaction energy accounted for by the and, after regularization, it can be written (see 117 appendix) as 2 =: I 2 1 + I 2 , which defines I 1 and I 2 , so that the corresponding Casimir energy depends on log 2 = log ( ). The first term is simply the sum of the energies of the two cavities taken independently, the second term 119 is the interaction energy between the two [7]. Therefore we can always reduce ourselves to the computation of determinants of 120 products of 4 × 4 matrix. The interaction in the case of n ≥ 3 cavities is accounted for by the term I n .

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In this manner, using the inductive principle, it is not difficult to convince oneself that the generic N functions for the case of N So, for example,  The case of an even number of plasma sheets is more involved. It can be obtained starting with N ps + 1 (N ps even) cavities and 153 154 In this manner, we have for two and four plasma sheet (please note that it is necessary to perform the limiting procedure first and = I 1 I 1 I 1 + I 1 I 2 + I 2 I 1 + I 3 = I 2 1 + I 2 + I 1 I 2 + I 3 . (4)

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The fact that only one term at a time takes the prime corresponds to the fact that the last cavity only must be sent to infinity (i.e.  3 Numerical results 164 We are now in the position to discuss the dependence of the Casimir Energy of a N-cavity made of N −plasma sheets. We underline 165 the fact that the contribution of TE modes results various order of magnitude less than the one from TM modes. For this reason, in 166 the following, it will be simply omitted. 167 We start by considering the variation of the Casimir energy as a function of the number of cavities for fixed thickness d i = 2 168 nm and = μ 0 n 2D q * 2 we find the values quoted in the following Table 1.

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The best fit is given by E [2] 30% to be compared with the 1.2% obtained for plasma-sheet.

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Needless to say this result depends on the thickness of the cavity. For example in the same situation but with (more realistic) 184 thickness of the dielectric cavities (and of the vacuum) d = 50 nm, the same ratio turns out to be 3%. The same behaviour is 185 found for the case of three cavities, see discussion in [7] sec. 5.

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In order to have further confirmation of eq.(6), which is, after all, obtained at fixed and d, we can use the Casimir energy  In conclusion, a good approximation for the Casimir Energy (at fixed temperature) for N plasma-sheet cavities can be written as For the normal phase, T > T c , we will use the data (and formulae) of Ref. [23].

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Thus, 220 Fig. 6 The transition of the sample of YBCO we are using temperature T c is rather reassuring. Indeed, after their phenomenological analysis, the authors explicitly state in the conclusions that 238 the plasma sheet model provides a good description for the behaviour of copper oxide HTSCs superconductors. 239 We suggested a possible way of characterising the variation of the Casimir energy at the metal-superconductor transition, giving 240 a numerical estimate for the specific YBCO sample that we are using in the Archimedes experiment (at this time both YBCO and 241 GdBCO superconductors are under consideration).

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The computed value for the "modulation factor" η = E E 0 , for one cavity (since the total number of cavities within a sample will 243 depend on the ratio between its total thickness and the thickness of the single layer, it is better to use the modulation factor referred across the discontinuity layers [12], where n =ẑ is the normal to the layers (parallel to the z-axis, going from the i-th to the i + 1-th layer), σ is the surface charge 270 density, and J is the surface current density respectively (in principle they could be different at each layer, but we will not consider 271 this situation). By virtue of the translational invariance in the (x, y) plane we can set

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In the following, when discussing the plasma sheet model, we will consider the so called hydrodynamic model [13,14], in which a 275 continuous fluid with mass m * and charge q * is uniformly distributed in the layer with an overall-neutralizing background charge. The 276 fluid displacement ξ is purely tangential, ξ ≡ (ξ x , ξ y ) with surface charge and current densities related to the tangential component 277 of the electric field E by  with = μ 0 σ 0 q 0 . With these boundary conditions, the generating functions for the TM and TE modes, respectively, can be written 289 in terms of the auxiliary functions with K i = k 2 ⊥ + i (iζ l )ζ 2 l , and k ⊥ = (k x , k y ). Standard dielectric boundary conditions (dbc) are recovered by imposing = 0 295 and the plasma sheet boundary conditions (psbc) by requiring j (iζ l ) = 1, ∀ j (in this case K i = K j ).
Computing the determinant of the minors of dimensions 4, 8, and 12, we obtain the energy of one, two, and three cavities, respectively.

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Thus, when the two cavities are far away, their energy is simply the sum of the individual contributions and I (2) can be seen as the