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

Abstract

In this paper we study the Casimir energy of a sample made by N cavities, with $N\gg 1$, 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.

1 Introduction

The principal goal of the Archimedes experiment [1] is to measure the coupling of the vacuum fluctuations of Quantum Electrodynamics (QED) to the gravitational field of the Earth. The coupling is obtained, as usual in Quantum Field Theory in Curved spacetime [2,3,4,5], assuming the Einstein tensor to be proportional to the expectation value of the regularized and renormalized energy-momentum tensor of matter fields, in particular, for the Archimedes experiment, of the electromagnetic field. The idea is to weigh the vacuum energy stored in a rigid Casimir cavity [6], made by parallel conducting plates, by modulating the reflectivity of the plates upon inducing a transition from the metallic to the superconducting state [1]. The “modulation factor” is defined as $\eta =\frac{\Delta E}{E_0}$ were $\frac{\Delta E}{A}$ is the difference of Casimir energy (per square meter) in the normal and in the superconducting state, and $\frac{E_0}{A}$ is the (absolute value) of the Casimir energy (per square meter), at zero temperature, of an ideal cavity of the same thickness d: $\frac{E_0}{A}=\frac{\pi ^2 \hbar c}{720 d^3}$.

In Ref. [1] it was shown that, in order to measure such an effect, $\eta$ must be of the order $\eta \sim 10^{-5}$ and that, to this purpose, a multi-cavity, obtained by superimposing many cavities must be used. 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 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 of structure has been carried out for a sample made by up to three “relatively thick” (of the order of ten nanometer) dielectric layers. In the present paper we extend the analysis to any number of cavities for both situations: layers consisting of “thick” dielectric slabs and layers consisting of “thin” plasma sheets.

Indeed, in Ref. [8], considering a cavity based on a high-$T_{\mathrm c}$ layered superconductor, a factor as high as $\eta =4\times 10^{-4}$ has been estimated (for flat plasma sheets at zero temperature and no conduction in the normal state, so that $\Delta E$ corresponds to the energy of the ideal cavity, and charge density $n = 10^{14}$ cm$^{-2}$). The Archimedes sensitivity is expected to be capable of assessing the interaction of gravity and vacuum energy also for values lower than $\eta =4\times 10^{-4}$, up to 1/100 of this value [1]. It is then crucial to understand the level of modulation achievable with layered superconducting structures. This is the scope of the present paper.

Considering in particular the multi-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.

Section II studies the Casimir energy of a multilayered cavity, assuming either dielectric or plasma sheet matching conditions at each interface between the layers. In Sec. III, numerical calculations are carried out and an analytic model capable of describing the Casimir energy at finite temperature is given. Finally, in Sec. IV, a possible model for describing the variation (and the modulation) of the Casimir energy across the transition is introduced. Our concluding remarks are found in Sec. V.

2 The Casimir energy of N coupled cavities

In this section we deduce the Casimir Energy of n coupled cavities, even though in the present paper we are interested in applying our results to plasma sheets, we will discuss the case of dielectrics first and then recover the plasma sheets results as a suitable limiting case.

In the following, referring to Fig. 1, $d_i$ is the distance of the i-th cavity from the $(i-1)$-th, (thickness of the i-th cavity), within the slabs 1, 3, 5 and N there is vacuum while within the regions 0, 2, 4, 6 and $N+1$ there is dielectric (in the case of dielectric model) or vacuum (for the plasma-sheet model). The thickness of the regions 0 and $N+1$ is assumed to be infinite.

Fig. 1

The layered structure considered in this paper. For the dielectric case all even-numbered regions include a dielectric material and all odd-numbered regions include vacuum. For plasma-sheet model the layers marked by bold lines are simple interfaces of zero thickness and $d_i$ is the thickness of the i-th slab

The general expression for the Casimir energy (per unit area), at finite temperature, will be written in the usual manner [9,10,11]

$$\begin{aligned} E=\, {k_B \,T}\sum _{l=0}^{~\infty \,~\prime }\int \frac{ \mathrm d \mathbf{k_{\bot }}}{(2 \pi )^2} \,\left[ \log {\Delta ^{\mathrm{TE}}(\zeta _l)}+\log {\Delta ^{\mathrm{TM}}(\zeta _l)}\right] \end{aligned}$$

(1)

where the $\Delta$ are the so called generating functions (in the following we will omit the subscript $\mathrm TM(TE)$ if no ambiguity is generated), $\zeta _l=2\pi l k_B T$ are the Matsubara frequencies, $k_B$ is the Boltzmann constant, $l={0,1,2,\ldots }$, and the superscript $^\prime$ on the sum means that the zero mode must be multiplied by a factor $\frac{1}{2}$. The generating functions are obtained by computing the determinant of the most general boundary conditions at each singular layer located at $d_0, d_0+d_1,d_0+d_1+d_2$...etc. (see Fig. 1; see also the appendix) [12].

For the sake of clarity, we only give here the general argument about the procedure for obtaining the generating functions, referring the reader to the appendix for the complete computation. In the appendix we show that the $\Delta$ functions can be written in terms of a sort of generalised reflection coefficients:

$$\begin{aligned} R_\mathrm{TM}^{i,j} & = \frac{\epsilon _j(\mathrm i\zeta _l)K_i-\epsilon _i(\mathrm i\zeta _l)K_j - {2\frac{\Omega }{\zeta _l^2}} K_iK_j}{\epsilon _j(\mathrm i\zeta _l)K_i+\epsilon _i(\mathrm i\zeta _l)K_j + {2\frac{\Omega }{\zeta _l^2}} K_iK_j},~~~~~R_\mathrm{TE}^{i,j} = \frac{K_i-K_j +2 \Omega }{K_i+K_j +2 \Omega }, \\ S_\mathrm{TM}^{i,j} & = \frac{\epsilon _j(\mathrm i\zeta _l)K_i-\epsilon _i(\mathrm i\zeta _l)K_j + {2\frac{\Omega }{\zeta _l^2}} K_iK_j}{\epsilon _j(\mathrm i\zeta _l)K_i+\epsilon _i(\mathrm i\zeta _l)K_j + {2\frac{\Omega }{\zeta _l^2}} K_iK_j},~~~~~S_\mathrm{TE}^{i,j} = \frac{K_i-K_j -2 \Omega }{K_i+K_j +2 \Omega }, \\ T_\mathrm{TM}^{i,j} & = \frac{\epsilon _j(\mathrm i\zeta _l)K_i+\epsilon _i(\mathrm i\zeta _l)K_j - {2\frac{\Omega }{\zeta _l^2}} K_iK_j}{\epsilon _j(\mathrm i\zeta _l)K_i+\epsilon _i(\mathrm i\zeta _l)K_j + {2\frac{\Omega }{\zeta _l^2}} K_iK_j}, ~~~~~T_\mathrm{TE}^{i,j} = \frac{K_i+K_j -2 \Omega }{K_i+K_j +2 \Omega }, \end{aligned}$$

where $K_i=\sqrt{k_\perp ^2+\epsilon _i(\mathrm i\zeta _l) \zeta ^2_l}$, $k_\perp =(k_x,k_y)$, $\Omega =\frac{\mu _0 n_{2D} q^{*2}}{m^{*} }$, $\mu _0$ is the magnetic permeability of vacuum, $n_{2D}$ is the two dimensional carrier density in the layer, and $q^*$ and $m^*$, respectively, their charge and mass. The standard dielectric boundary conditions (dbc) will be recovered by imposing $\Omega =0$ and the plasma sheet boundary conditions (psbc) by requiring $\epsilon _i(\mathrm i\zeta _l)=1,~~\forall i$ (in this case, $K_i=K_j$).

After introducing the auxiliary functions

$$\begin{aligned} {E}^{ijk}\,=\,& {} {\mathrm e^{-2 d_j K_j} \text {S}^{j,k} \text {R}^{i,j}}+1, \\ F^{ijk}\,=\, & {} {\mathrm e^{-2 d_j K_j} \text {R}^{i,j}\text {T}^{j,k}}+\text {R}^{j,k}, \\ G^{ijk}\,=\, & {} {\mathrm e^{-2 d_j K_j} \text {S}^{j,k} \text {T}^{i,j}}+{\text {S}^{i,j}}, \\ H^{ijk}=\, & {} {\text {S}^{i,j} \text {R}^{j,k}}+{\mathrm e^{-2 d_j K_j} \text {T}^{i,j} \text {T}^{j,k}} \end{aligned}$$

and (henceforth, we will assume all the cavities to be equal and consider only the indices $\{ijk\}= \{012\}$), on defining:

$$\begin{aligned} I_1\,=\, & {} E^{012};~~I_2=\,F^{012}\mathrm e^{-2 d_2 K_2}G^{012},\\ I_n=\, & {} F^{012}\mathrm e^{-2 d_2 K_2}\left( H^{012}\mathrm e^{-2 d_2 K_2}\right) ^{n-2}G^{012}, ~~~ \text{ for } n\ge 3, \end{aligned}$$

we can proceed to compute the generating functions.

2.1 The dielectric case

Let us consider Casimir cavities made of dielectric layers (of thickness $d_i$). To obtain the general expression for the $\Delta$ functions we can proceed inductively (a very detailed discussion up to three cavities can be found in Ref. [7]). For the cavity characterised by the numbers (012) in Fig. 1, with $\epsilon _{0}=\epsilon _{2}$, the generating function, for $\mathrm TM$ and $\mathrm TE$ modes, respectively, is obtained in the usual manner [7, 10] (see appendix). After regularization, i.e., setting to zero the Casimir energy when the two cavities are infinitely far away, the result can be written as $\Delta _1=E^{012}=I_1$. Let us now consider two cavities [(012), (234) in Fig. 1]. In this case, the generating function is the determinant of the $8\times 8$ matrix made by the first rows and columns of the matrix given in the appendix [7]. It can be written as a $2\times 2$ block matrix, thus [17, 18]

$$\begin{aligned} \Delta =\mathrm {det} \left( \begin{array}{cc} A&{} B \\ C &{} D \\ \end{array} \right) =\mathrm {det}(A)\,\mathrm {det}({1}-A^{-1}BD^{-1}C)\,\mathrm {det}(D), \end{aligned}$$

where $\{A,B,C,D\}$ are $4\times 4$ matrices, with $\mathrm {det}(A)=\mathrm {det}(D)=\Delta _1$.

When the two cavities are infinitely far away from each other ($d_2\rightarrow \infty$), $C=0$, $\Delta =\mathrm {det}(A)\,\mathrm {det}(D)=:\Delta _2$ and the Casimir energy will be simply the sum of the energies of the two cavities, $\log {(\Delta _2)}=\log {(\Delta _1^2)}=2 \log {(\Delta _1)}$. When they are brought at a distance $d_2$ from each other, in addition to the previous energy, there is the interaction energy accounted for by the term $\mathrm {det}({1}-A^{-1}BD^{-1}C)$. In this case $\Delta _2=\mathrm {det}(A)\,\mathrm {det}(D)\,\mathrm {det}({1}-A^{-1}BD^{-1}C)$ and, after regularization, it can be written (see appendix) as $\Delta _2=:I_1^2+I_2$, which defines $I_1$ and $I_2$, so that the corresponding Casimir energy depends on $\log {\Delta _2}=\log {(I_1^2+I_2)}=\log {(I_1^2)}+\log {(1+I_2/I_1^2)}$. The first term is simply the sum of the energies of the two cavities taken independently, the second term is the interaction energy between the two [7]. Therefore we can always reduce ourselves to the computation of determinants of products of $4\times 4$ matrix. The interaction in the case of $n\ge 3$ cavities is accounted for by the term $I_n$.

In this manner, using the inductive principle, it is not difficult to convince oneself that the generic $\Delta _N$ functions for the case of N dielectric cavities can be obtained in the following manner (a sort of Feynman diagram for the generating functions): let us define $\{k_1,k_2,\ldots ,k_J\}$ to be the J-th integer partition of N and $Q_J$ its multiplicity (the number of combinations that contain the same type of $I_k$ but in a different position) then

$$\begin{aligned} \Delta _N=\sum _{J} Q_J\left( I_{k_1} I_{k_2}\ldots I_{k_J}\right) . \end{aligned}$$

So, for example,

$$\begin{aligned} \Delta _{1} & = I_1, \\ \Delta _{2} & = (I_1)^2+I_2 , \\ \Delta _{3} & = (I_1)^3+I_1I_2+I_2I_1+I_3= (I_1)^3+2I_1I_2+I_3 ,\\ \Delta _{4} & = I_4+I_1I_3+I_3I_1+I_2^2+I_1I_1I_2+I_1I_2I_1+I_2I_1I_1+I_1^4 \\ & = I_4+2I_1I_3+I_2^2+3I_1^2I_2+I_1^4 , \end{aligned}$$

and, e.g.,

$$\begin{aligned} \Delta _{10} & = I_{1}^{10} + 9 I_{1}^8 I_{2} + 28 I_{1}^6 I_{2}^2 + 35 I_{1}^4 I_{2}^3 + 15 I_{1}^2 I_{2}^4 + I_{2}^5 + 8 I_{1}^7 I_{3} + 42 I_{1}^5 I_{2} I_{3} + 60 I_{1}^3 I_{2}^2 I_{3} \\ &\quad + 20 I_{1} I_{2}^3 I_{3} + 15 I_{1}^4 I_{3}^2 +30 I_{1}^2 I_{2} I_{3}^2 + 6 I_{2}^2 I_{3}^2 + 4 I_{1} I_{3}^3 + 7 I_{1}^6 I_{4} + 30 I_{1}^4 I_{2} I_{4} + 30 I_{1}^2 I_{2}^2 I_{4} \\ &\quad + 4 I_{2}^3 I_{4} + 20 I_{1}^3 I_{3} I_{4} +24 I_{1} I_{2} I_{3} I_{4} + 3 I_{3}^2 I_{4}+ 6 I_{1}^2 I_{4}^2 + 3 I_{2} I_{4}^2 + 6 I_{1}^5 I_{5} + 20 I_{1}^3 I_{2} I_{5} \\ &\quad + 12 I_{1} I_{2}^2 I_{5} + 12 I_{1}^2 I_{3} I_{5} + 6 I_{2} I_{3} I_{5} +6 I_{1} I_{4} I_{5} + I_{5}^2 + 5 I_{1}^4 I_{6} +12 I_{1}^2 I_{2} I_{6} + 3 I_{2}^2 I_{6} \\ &\quad + 6 I_{1} I_{3} I_{6} + 2 I_{4} I_{6} + 4 I_{1}^3 I_{7} + 6 I_{1} I_{2} I_{7} +2 I_{3} I_{7} + 3 I_{1}^2 I_{8} + 2 I_{2} I_{8} + 2 I_{1} I_{9} + I_{10}, \end{aligned}$$

for ten cavities.

2.2 The plasma sheets case

These formulae can be extended to the case in which the layers are characterised as plasma sheets. For example, the two dielectric cavities (012) and (234) can describe three plasma-sheet cavities, (012),  (123),  (234), by imposing $\epsilon _i=1$, and $\Omega \ne 0$. In other words, two dielectric cavities needs four layers located at $0,\,d_1,\,d_1+d_2,\,d_1+d_2+d_3$ but the same four layers correspond to three cavities having plasma sheet as boundaries. Consequently $N_{ps}$ (odd) plasma sheets can be obtained by $n=\frac{N_{ps}+1}{2}$ standard dielectrics by simply imposing $\epsilon _i(\mathrm i\zeta )=1$, and the extension of the previous formulae to the case of an odd number of plasma sheets is straightforward.

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 moving the last layer to infinity. From the mathematical point of view, this procedure corresponds to introducing a term $I'_n$ (which describes the interaction of the last interface with all the others), defined like as

$$\begin{aligned} I'_1=1;~~~~ I'_n=\lim _{G\rightarrow G'}I_n,~~\text{ if }~~n\ge 2 ; \text{ having } \text{ defined } G'^{~ijk}:=S^{ij}. \end{aligned}$$

(2)

In this manner, we have for two and four plasma sheet (please note that it is necessary to perform the limiting procedure first and then to group together the various terms)

$$\begin{aligned} \Delta _2^{ps}& = {} \lim _{G\rightarrow G'}\Delta _{2+1}^{ps} =\lim _{G\rightarrow G'}\Delta _2=\lim _{G\rightarrow G'} \left[ (I_1)^2+I_2 \right] =I_1I_1'+I_2'=I_1+I_2'; \end{aligned}$$

(3)

$$\begin{aligned} \Delta _4^{ps}& = {} \lim _{G\rightarrow G'}\Delta _{4+1}^{ps} =\lim _{G\rightarrow G'}\Delta _3=\lim _{G\rightarrow G'} \left[ (I_1)^3+I_1I_2+I_2I_1+I_3 \right] \\&= {} I_1I_1I'_1+I_1I'_2+I_2I'_1+I'_3=I_1^2+I_2+I_1I'_2+I'_3. \end{aligned}$$

(4)

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. $d_N\rightarrow \infty$ while leaving all the remaining $d_i, ~i\ne N$ finite).

3 Numerical results

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 the fact that the contribution of TE modes results various order of magnitude less than the one from TM modes. For this reason, in the following, it will be simply omitted.

We start by considering the variation of the Casimir energy as a function of the number of cavities for fixed thickness $d_i=2$ nm and $\Omega =\frac{\mu _0 n_{2D} q^{*2}}{m^{*} } = 49593.3$ m$^{-1}$ (see Refs. [16, 19]). We get $\frac{E[1]}{A}=-0.000197\,\mathrm{J}{\mathrm{m}^{-2}}$ and, for the ratio $\frac{E[N]}{N E[1]}$ between the Casimir Energy of N cavities E[N], and the product NE[1] between the number of cavities and the energy of a single cavity E[1], we find the values quoted in the following Table 1.

Table 1 The ratio $\frac{E[N]}{N E[1]}$ as a function of the number of cavities

The best fit is given by

$$\begin{aligned} \frac{1}{N}\frac{E[N]}{E[1]}=1.034\, -\frac{0.034}{N^{0.71}}, \end{aligned}$$

(5)

that gives a clear indication of the presence of an asymptote for $N\rightarrow \infty$. In Fig. 2 a comparison between the exact numerical result and the analytical fitted behaviour up to $N=19$ [Eq. (5)] is shown.

Thus, we obtained an asymptotic expression for the Casimir energy for large N,

$$\begin{aligned} E[N]\simeq (1.034\, E[1]) N \end{aligned}$$

(6)

and deduced that the coupling of the various cavities resulted in an increase of the Casimir energy of $3.4\%$. This result is very different from the result for dielectric layers, in which a strong coupling between the two and the three nearest cavities is found (see [7]). Indeed, considering (for giving an idea) a cavity made by two dielectric slabs (for example made by Niobium) 2 nm thick and separated by 2 nm of vacuum, we find $\frac{E[2]-2E[1]}{E[2]}\simeq 30 \%$ to be compared with the $1.2 \%$ obtained for plasma-sheet.

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

Fig. 2

The exact numerical result (dots) and the fitted results given by Eq. (5) (green line) of the function E[N]/(NE[1]) for $d=2$ nm

In order to have further confirmation of eq.(6), which is, after all, obtained at fixed $\Omega$ and d, we can use the Casimir energy functional dependence of a single cavity on these two parameters as reported in [15]: $E[1]=5 \times 10^{-3}\hbar c \sqrt{\frac{\Omega }{d^5}}$. With this in mind, we assume for E[N]/A the following functional form $E[N]/A=-(1.034 \,N)K\hbar c \frac{\Omega ^\alpha }{d^\beta }$, with arbitrary $K,~\alpha$ and $\beta$, and find their best estimate, using the method of least-squares, with respect to the exact results obtained numerically. We found $K=5.0\times 10^{-3}$, $\beta =2.4998$ and $\alpha =0.4998$, in perfect agreement with Ref. [15]. The comparison, shown in Fig. 3 (Casimir energy as a function of d) and Fig. 4 (Casimir energy as a function of $\Omega$), are a clear indication of the validity of the expression (6).

Fig. 3

A comparison between exact numerical values of the Casimir energy (dots) and the approximated formula, Eq. (7) (lines), with d expressed in nm and E/A in $J/m^2$

Fig. 4

A comparison between exact values of the Casimir energy (dots) and the approximated formula (lines), for $d=2$ nm, with $\Omega$ expressed in nm$^{-1}$ E/A in $J/m^2$

In conclusion, a good approximation for the Casimir Energy (at fixed temperature) for N plasma-sheet cavities can be written as

$$\begin{aligned} \frac{E[N]}{A}=-\left( 1.034\, K \hbar c \frac{\sqrt{\Omega }}{d^{5/2}}\right) N=\left( -1.63 \times 10^{-28}(\mathrm{J m})\right) \left( N\frac{\sqrt{\Omega }}{d^{5/2}}(\mathrm{m}^{-3})\right) \end{aligned}$$

(7)

with E[N]/A measured in $\mathrm{J m}^{-2}$.

Based on the above formulae, in the following section we give an estimate for the variation of the Casimir energy across the metal-superconductor transition.

4 The variation of the Casimir energy in the YBCO

The typical structure of a YBCO cell (a very well studied material for its numerous fundamental applications [20, 21]) is represented in Fig. 5, in which $\delta =4.25 {\text{\AA} }$ is the thickness of our plasma sheet and $d=3.18 {\text{\AA} }$ is the distance between the layers.

Fig. 5

Typical layered structure of $YBa_2Cu_3O_7$ [22]

By observing that [16] $\Omega (0)=\frac{\delta }{2 \lambda _{ab}^2(0)}$, at $T=0$ K, we can write for the Casimir energy of one cavity in the superconducting state as

$$\begin{aligned} \frac{E (0)}{A}=-1.63\times 10^{-28}{\sqrt{\frac{\delta }{2d^5}}\frac{1}{\lambda _{ab}(0)}}. \end{aligned}$$

(8)

Using the BCS relation $\lambda (T)=\frac{\lambda (0)}{\sqrt{1-(T/T_{\mathrm c})^{4/3}}}$ [22], corresponding to the case of d-wave pairing, as it is suitable for cuprates, for $T<T_{\mathrm c}$ and for one cavity we get

$$\begin{aligned} \frac{E (T)}{A}= & {} -\frac{1.63\times 10^{-28}}{\lambda _{ab}(T)}{\sqrt{\frac{\delta }{2d^5}} =-{1.63\times 10^{-28}}{\sqrt{\frac{\delta }{2d^5}}}\frac{\sqrt{1-(T/T_{\mathrm c})^{4/3}}}{\lambda _{ab}(0)}} \end{aligned}$$

(9)

Thus, using for YBa$_2$Cu$_3$O$_7$ [19], $T_{\mathrm c}=92$ K, $\lambda _{ab}(0)=1415\,{\text{\AA} }$, $d=3.36\,{\text{\AA} }$, and $\delta =5.84\,{\text{\AA} }$, we have

$$\begin{aligned} \frac{E (90)}{A}= & {} -9.51\times 10^{-3}\sqrt{1-(90/92)^{4/3}} =-0.001616~~ \mathrm{J m}^{-2}. \end{aligned}$$

(10)

For the normal phase, $T>T_{\mathrm c}$, we will use the data (and formulae) of Ref. [23].

At $T=100$ K, we have $n_{3D}=3.1\times 10^{25}$ m$^{-3}$, which implies $n_{2D}=n_{3D}~ \delta = (3.1\times 10^{25})(5.84\times 10^{-10}) =1.810\times 10^{16}\,m^{-2}$. In the Archimedes experiment a transition of a few degrees around the critical temperature is required to induce a continuous superconducting-normal transition. For this reason, in the following, we will choose, to fix the ideas, a four degree interval around $T_c$, thus for $T=94$ K, we have $n_{2D}= 1.317\times 10^{16} \frac{94}{100}=1.702 \times 10^{16}$ m$^{-2}$. Consequently, $\Omega =\frac{\mu _0 n_{2D} e^2}{2m^*}=300.505$ m$^{-1}$, and

$$\begin{aligned} \frac{ E (94)}{A}=-1.63\times 10^{-28}\sqrt{\frac{\Omega }{d^5}}=-0.001365~~\mathrm{J m}^{-2}. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{\Delta E}{A}=\frac{E (94)-E (90)}{A}=0.000251~~\mathrm{J m}^{-2}. \end{aligned}$$

As revealed by an inspection of the table in Ref. [19], it is clear that the previous results depend in a crucial way on the sample of YBCO used. A typical Resistance vs. Temperature curve of the YBCO crystals we are planning to use in the Archimedes experiment is reported in Fig. 6. Our reference values are $T_\mathrm{c}=89~K$ and $\lambda _{ab}(0)=1030\,{\text{\AA} }$ [24], thus, assuming all the other parameters unaltered, we have $\frac{E (87)}{A}=-0.002258~ Jm^{-2}$ and, in the normal phase, $T=91$ K, $\frac{E (91)}{A}=-0.001343~ Jm^{-2}$ so that $\frac{\Delta E}{A}=\frac{E (91)-E (87)}{A}=0.0009142~\mathrm{J m}^{-2}$.

Fig. 6

The transition of the sample of YBCO we are using

5 Conclusions

We have proposed a model for computing the variation of the Casimir energy of a YBCO sample across the metal-superconductor transition.

We have constructed a powerful procedure to compute the renormalised Casimir energy both in the case of cavities made of a large number of thick dielectric layers and in the case of cavities made by a large number of thin plasma sheet layers.

Our main assumption is that the last case can be used to describe the Casimir energy in YBCO and, more generally in cuprates (GdBCO), because of their natural built-in layered structure, both in the normal and in the superconducting phase. While the approach used here rests on a marked microscopic layer structure of the superconductor, the model uses phenomenological macroscopic parameters assuming which include mediated properties of the layer structure itself. The analysis conducted in [16] of the correlation between Casimir energy and experimental parameters such as penetration length in the plane $\lambda _{ab}$, effective mass $m^*$ and critical temperature $T_c$ is rather reassuring. Indeed, after their phenomenological analysis, the authors explicitly state in the conclusions that the plasma sheet model provides a good description for the behaviour of copper oxide HTSCs superconductors.

We suggested a possible way of characterising the variation of the Casimir energy at the metal-superconductor transition, giving a numerical estimate for the specific YBCO sample that we are using in the Archimedes experiment (at this time both YBCO and GdBCO superconductors are under consideration).

The computed value for the “modulation factor” $\eta =\frac{\Delta E}{E_0}$, for one cavity (since the total number of cavities within a sample will depend on the ratio between its total thickness and the thickness of the single layer, it is better to use the modulation factor referred to one cavity only), is thus in the range $\left( \frac{0.00025}{0.27}\sim \right) 0.0009\le \frac{\Delta E}{E_0}\le 0.003\left( \sim \frac{0.0009}{0.27}\right)$ ($E_0/A$ being the energy (per square meter) of an ideal cavity $11.68\;\text{\AA}$ thick) see Fig. 5, which is quite reassuring for the Archimedes experiment [1] . Of course, although encouraging, these results must be considered as an estimate of the orders of magnitude involved that reinforces the hypothesis that the variation of Casimir energy is not negligible compared to the condensation energy in a type II superconductor.

Appendix A

The generating functions are obtained by imposing the most general boundary conditions at each singular layer located at $d_0,\, d_0+d_1,\,d_0+d_1+d_2$...etc. (see Fig. 1). These are obtained, as usual, by integrating the Maxwell equations

$$\begin{aligned} \varvec{\nabla }\cdot \varvec{D}=\rho ,~~&~~ \varvec{\nabla }\times \varvec{E}+\frac{\partial \varvec{B}}{\partial t}=0, \\ \varvec{\nabla }\cdot \varvec{B}=0,~~&~~ \varvec{\nabla }\times \varvec{H}-\frac{\partial \varvec{D}}{\partial t}=\varvec{J}, \\ \varvec{D}= \epsilon \varvec{E},~~&~~ \varvec{B}=\mu \varvec{H},~~~(\epsilon =\epsilon _0,~\mu =\mu _0 \text{ in } \text{ vacuum}), \end{aligned}$$

across the discontinuity layers [12],

$$\begin{aligned} (\varvec{D}_i-\varvec{D}_{i-1})\cdot \mathbf {n}=\sigma ,~~&~~ (\varvec{B}_i-\varvec{B}_{i-1})\cdot \mathbf {n}=0, \\ \mathbf {n}\times (\varvec{E}_i-\varvec{E}_{i-1})=0, ~~&~~ \mathbf {n}\times (\varvec{H}_i-\varvec{H}_{i-1})=\mathbf {J}, \end{aligned}$$

where $\mathbf {n}=\hat{z}$ is the normal to the layers (parallel to the z-axis, going from the i-th to the $i+1$-th layer), $\sigma$ is the surface charge density, and $\mathbf {J}$ is the surface current density respectively (in principle they could be different at each layer, but we will not consider this situation). By virtue of the translational invariance in the (x, y) plane we can set

$$\begin{aligned} \varvec{E}={\varvec{f}}(z)\mathrm e^{\mathrm i(\varvec{k}_{||}\cdot \varvec{x}_{||}-\omega t)},~~~ \varvec{B}=\varvec{g}(z)\mathrm e^{\mathrm i(\varvec{k}_{||}\cdot \varvec{x}_{||}-\omega t)}. \end{aligned}$$

In the following, when discussing the plasma sheet model, we will consider the so called hydrodynamic model [13, 14], in which a continuous fluid with mass $m^*$ and charge $q^*$ is uniformly distributed in the layer with an overall-neutralizing background charge. The fluid displacement $\varvec{\xi }$ is purely tangential, $\varvec{\xi }\equiv (\xi _x,\xi _y)$ with surface charge and current densities related to the tangential component of the electric field $\varvec{E}_\parallel$ by

$$\begin{aligned} \ddot{\varvec{\xi }}=\frac{q^*}{m^*}{} \mathbf{{E}}_{\parallel }=: {q_0} \mathbf{{E}}_{\parallel },~~\mathbf{{J}}= n_{2D}\, q^* \dot{\varvec{\xi }}=: \sigma _0 \dot{\varvec{\xi }},~~\sigma =-n_{2D} \, q^*{\varvec{\nabla }}_{\parallel }\cdot {\varvec{\xi }}=- \sigma _0{\varvec{\nabla }}_{\parallel }\cdot {\varvec{\xi }}, \end{aligned}$$

$n_{2D}$ being the two dimensional carrier density in the layer, and $q^*$ and $m^*$ being their charge and mass, respectively. Under these assumptions, the most general boundary conditions at the $i=1,2,3 ...$-th boundaries are

$$\begin{aligned} \epsilon _i(\omega ) f_i^z-\epsilon _{i-1}(\omega ) f_i^z &\,=\, -\frac{\sigma _0 q_0}{\omega ^2} \frac{\partial f^z_i}{\partial z},\nonumber \\ \frac{\partial f^z_i}{\partial z}-\frac{\partial f^z_{i-1}}{\partial z} &\, =\, 0, ~~~~~\text{ for } \text{ the } \text{ TM } \text{ modes, } \text{ and } \end{aligned}$$

(A1)

$$\begin{aligned} \frac{\partial g^z_i}{\partial z}-\frac{\partial g^z_{i-1}}{\partial z}\,=\, & {} \Omega \; g^z_i,\nonumber \\ g_i^z-g_{i-1}^z\,=\, & {} 0, ~~~~~\text{ for } \text{ the } \text{ TE } \text{ modes }, \end{aligned}$$

(A2)

with $\Omega ={\mu _0\sigma _0 q_0}$. With these boundary conditions, the generating functions for the $\mathrm TM$ and $\mathrm TE$ modes, respectively, can be written in terms of the auxiliary functions

$$\begin{aligned} R_\mathrm{TM}^{i,j}\,=\, & {} \frac{\epsilon _j(\mathrm i\zeta _l)K_i-\epsilon _i(\mathrm i\zeta _l)K_j - {2\frac{\Omega }{\zeta _l^2}} K_iK_j}{\epsilon _j(\mathrm i\zeta _l)K_i+\epsilon _i(\mathrm i\zeta _l)K_j + {2\frac{\Omega }{\zeta _l^2}} K_iK_j},~~~R_\mathrm{TE}^{i,j} = \frac{K_i-K_j +2 \Omega }{K_i+K_j +2 \Omega } \\ S_\mathrm{TM}^{i,j}\,=\, & {} \frac{\epsilon _j(\mathrm i\zeta _l)K_i-\epsilon _i(\mathrm i\zeta _l)K_j + {2\frac{\Omega }{\zeta _l^2}} K_iK_j}{\epsilon _j(\mathrm i\zeta _l)K_i+\epsilon _i(\mathrm i\zeta _l)K_j + {2\frac{\Omega }{\zeta _l^2}} K_iK_j},~~~ S_\mathrm{TE}^{i,j} \,=\, \frac{K_i-K_j -2 \Omega }{K_i+K_j +2 \Omega } \\ T_\mathrm{TM}^{i,j}\,=\, & {} \frac{\epsilon _j(\mathrm i\zeta _l)K_i+\epsilon _i(\mathrm i\zeta _l)K_j - {2\frac{\Omega }{\zeta _l^2}} K_iK_j}{\epsilon _j(\mathrm i\zeta _l)K_i+\epsilon _i(\mathrm i\zeta _l)K_j + {2\frac{\Omega }{\zeta _l^2}} K_iK_j}, ~~~T_\mathrm{TE}^{i,j} = \frac{K_i+K_j -2 \Omega }{K_i+K_j +2 \Omega } \end{aligned}$$

with $K_i=\sqrt{k_\perp ^2+\epsilon _i(\mathrm i\zeta _l) \zeta ^2_l}$, and $k_\perp =(k_x,k_y)$. Standard dielectric boundary conditions (dbc) are recovered by imposing $\Omega =0$ and the plasma sheet boundary conditions (psbc) by requiring $\epsilon _j(\mathrm i\zeta _l)=1,~~\forall j$ (in this case $K_i=K_j$).

$$\begin{aligned} {E}^{ijk} & = {\mathrm e^{-2 d_j K_j} \text {S}^{j,k} \text {R}^{i,j}}+1, \\ F^{ijk} & = {\mathrm e^{-2 d_j K_j} \text {R}^{i,j}\text {T}^{j,k}}+\text {R}^{j,k}, \\ G^{ijk} & = {\mathrm e^{-2 d_j K_j} \text {S}^{j,k} \text {T}^{i,j}}+{\text {S}^{i,j}}, \\ H^{ijk}& = {\text {S}^{i,j} \text {R}^{j,k}}+{\mathrm e^{-2 d_j K_j} \text {T}^{i,j} \text {T}^{j,k}}. \end{aligned}$$

Considering henceforth all the cavities to be equal, we consider only the indices $\{ijk\}= \{012\}$, so that

$$\begin{aligned} I_1\,=\, & {} E^{012};~~I_2=F^{012}e^{-2 d_2 K_2}G^{012},\\ I_n\,=\, & {} F^{012}e^{-2 d_2 K_2}\left( H^{012}e^{-2 d_2 K_2}\right) ^{n-2}G^{012},~~ \text{ for } n\ge 3. \end{aligned}$$

Let us consider, for the case of the TM modes, three cavities. Letting $x_i=\sum _{n=0}^i d_n$, the matching conditions give rise to the $12 \times 12$ matrix of coefficients

$$\begin{aligned}& M =\\ &\quad\times\left( {\begin{array}{cccccccccccc} { - e^{{x_{0} K_{0} }} \epsilon _{0} } & { - e^{{ - x_{0} K_{1} }} \epsilon _{1} } & {e^{{x_{0} K_{1} }} \epsilon _{1} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ { - e^{{ - x_{0} K_{0} }} K_{0} } & {e^{{ - x_{0} K_{1} }} K_{1} } & { - e^{{x_{0} K_{1} }} K_{1} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {e^{{x_{1} K_{1} }} \epsilon _{1} } & {e^{{ - x_{1} K_{1} }} \epsilon _{1} } & { - e^{{ - x_{1} K_{2} }} \epsilon _{2} } & { - e^{{x_{1} K_{2} }} \epsilon _{2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {e^{{x_{1} K_{1} }} K_{1} } & { - e^{{ - x_{1} K_{1} }} K_{1} } & {e^{{ - x_{1} K_{2} }} K_{2} } & { - e^{{x_{1} K_{2} }} K_{2} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {e^{{ - K_{2} x_{2} }} \epsilon _{2} } & {e^{{K_{2} x_{2} }} \epsilon _{2} } & { - e^{{K_{3} x_{2} }} \epsilon _{3} } & { - e^{{ - K_{3} x_{2} }} \epsilon _{3} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - e^{{ - K_{2} x_{2} }} K_{2} } & {e^{{K_{2} x_{2} }} K_{2} } & { - e^{{K_{3} x_{2} }} K_{3} } & {e^{{ - K_{3} x_{2} }} K_{3} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {e^{{K_{3} x_{3} }} \epsilon _{3} } & {e^{{ - K_{3} x_{3} }} \epsilon _{3} } & { - e^{{ - K_{4} x_{3} }} \epsilon _{4} } & { - e^{{K_{4} x_{3} }} \epsilon _{4} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {e^{{K_{3} x_{3} }} K_{3} } & { - e^{{ - K_{3} x_{3} }} K_{3} } & {e^{{ - K_{4} x_{3} }} K_{4} } & { - e^{{K_{4} x_{3} }} K_{4} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & {e^{{ - K_{4} x_{4} }} \epsilon _{4} } & {e^{{K_{4} x_{4} }} \epsilon _{4} } & { - e^{{K_{5} x_{4} }} \epsilon _{5} } & { - e^{{ - K_{5} x_{4} }} \epsilon _{5} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & { - e^{{ - K_{4} x_{4} }} K_{4} } & {e^{{K_{4} x_{4} }} K_{4} } & { - e^{{K_{5} x_{4} }} K_{5} } & {e^{{ - K_{5} x_{4} }} K_{5} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {e^{{K_{5} x_{5} }} \epsilon _{5} } & {e^{{ - K_{5} x_{5} }} \epsilon _{5} } & { - e^{{ - K_{6} x_{5} }} \epsilon _{6} } \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {e^{{K_{5} x_{5} }} K_{5} } & { - e^{{ - K_{5} x_{5} }} K_{5} } & {e^{{ - K_{6} x_{5} }} K_{6} ,} \\ \end{array} } \right) \end{aligned}$$

Computing the determinant of the minors of dimensions $4,\,8$, and 12, we obtain the energy of one, two, and three cavities, respectively. After regularization, for the single cavity (012) in Fig. 1, we find

$$\begin{aligned} \Delta _{1} = {E}^{012}=I_1. \end{aligned}$$

(A3)

For two cavities $(012-234)$, we find

$$\begin{aligned} \Delta _{2}\,=\, & {} {E}^{012}{E}^{234}+\mathrm e^{-2(d_2k_2)}F^{012} G^{234}=:(I_1)^2 +I_{2}, \text{ and } \nonumber \\ \log \Delta _{2}\,=\, & {} \log (I_1^2) +\log \left( 1+\frac{I_{2}}{I_1^2} \right) . \end{aligned}$$

(A4)

Finally, for the three cavities, we find

$$\begin{aligned} \Delta _{3}\,&=\, {} E^{012}E^{234}E^{456}+ \mathrm e^{-2(d_2k_2+d_4k_4)}F^{012} H^{234}G^{456} \nonumber \\& \;\;+\mathrm e^{-2 d_2k_2}E^{456}F^{012} G^{234}+ \mathrm e^{-2 d_4k_4}E^{012}F^{234} G^{456} \nonumber \\&=:I_1^3+I_{3}+I_1I_{2}+I_1I_{2}, \end{aligned}$$

(A5)

$$\begin{aligned} \log \Delta _{3}= & {} \log \left( I_1^3\right) + \log \left( 1+\frac{2I_1I_{2}}{I_1^3}\right) + \log \left( 1+\frac{I_{3}}{I_1^3+2I_1I_2} \right) . \end{aligned}$$

(A6)

When $d_2\rightarrow \infty$, $I_{2}\rightarrow 0$ and

$$\begin{aligned} \log { \Delta _{2}} = \log {I_1^2}=2\log {I_1}. \end{aligned}$$

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 energy due to the coupling of the two cavities (012)–(234). For the three cavities (012–234–456), the formulas are written so as to make evident the contribution to the energy resulting from the sum of the energies of the single cavities, 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, $I^{(3)}$.