On the Casimir effect between superconductors

TL;DR

This paper calculates the Casimir effect between superconductors, showing the transition effect is too small to detect with current experiments but could be observed with improved sensitivity and different material configurations.

Contribution

It provides theoretical predictions for the Casimir force change during superconducting transition and suggests experimental setups to observe this effect.

Findings

Transition effect is over 250 times smaller than current experimental sensitivity.

Using a Au-NbTiN cavity could enhance the effect to detectable levels.

An order of magnitude increase in sensitivity could enable observation of the transition effect.

Abstract

A recent experiment [R. A. Norte et al. Phys. Rev. Lett. {\bf 121}, 030405 (2018)] probed the variation of the Casimir force between two closely spaced thin Al films, as they transition into a superconducting state, observing a null result. We present here computations of the Casimir effect for superconductors, based on the Mattis-Bardeen formula for their optical response. We show that for the Al cavity used in the experiment the effect of the transition is over two hundred and fifty times smaller than the experimental sensitivity, in agreement with the observed null result. We demonstrate that a large enhancement of the effect can be achieved by using a cavity consisting of a Au mirror and a superconducting NbTiN film. We estimate that the effect of the superconducting transition would be observable with the proposed Au-NbTiN configuration, if the sensitivity of the apparatus could be increased by an order of magnitude.

Equations

$$P(a,T)=-\frac{k_{B}T}{\pi }\sum _{l=0}^{\infty }{}^{\prime }{\int }_0^{\infty }d{k}_{\perp }{k}_{\perp }{q}_l$$

$$\times \sum _{\alpha }{\left[\frac{e^{2a{q}_l}}{r_{\alpha }^{(1)}(i{\xi }_l,k_{\perp })r_{\alpha }^{(2)}(i{\xi }_l,k_{\perp })}-1\right]}^{-1},$$

$$r_{\mathrm{TE}}^{(k)}(i{\xi }_l,k_{\perp })=\frac{q_l-s_l^{(k)}}{q_l+s_l^{(k)}},$$

$$r_{\mathrm{TM}}^{(k)}(i{\xi }_l,k_{\perp })=\frac{{ϵ}_l^{(k)}{q}_l-s_l^{(k)}}{{ϵ}_l^{(k)}{q}_l+s_l^{(k)}},$$

$$\Delta P(a;T)=P(a,T)-P(a,T_{\mathrm{c}}),$$

$$ϵ(i\xi )={ϵ}_0+\frac{{\Omega }^2}{\xi (\xi +\gamma )},$$

$${\sigma }_{\mathrm{BCS}}(i\xi )=\frac{{\Omega }^2}{4\pi }\left[\frac{1}{(\xi +\gamma )}+\frac{g(\xi ;T)}{\xi }\right],$$

$$g(\xi ;T)={\omega }_s^2(T)+B(T)\xi \log (\Delta /\hslash \xi )+o(\xi ),$$

$${ϵ}_{\mathrm{BCS}}(i\xi )={ϵ}_0+4\pi \frac{{\sigma }_{\mathrm{BCS}}(i\xi )}{\xi }={ϵ}_0+\frac{{\Omega }^2}{\xi }\left[\frac{1}{\xi +\gamma }+\frac{g(\xi ;T)}{\xi }\right].$$

$$ϵ(i\xi )={ϵ}_0+(1-n_{\mathrm{s}}(T))\frac{{\Omega }^2}{\xi (\xi +\gamma )}+n_{\mathrm{s}}(T)\frac{{\Omega }^2}{{\xi }^2}.$$

$$n_{\mathrm{s}}(T)=\left[1-{\left(\frac{T}{T_{\mathrm{c}}}\right)}^4\right]\Theta (T_{\mathrm{c}}-T),$$

$$g(\xi )=\Theta (T_{\mathrm{c}}-T){\int }_{-\infty }^{\infty }\frac{dϵ}{E}\tanh \left(\frac{E}{2k_{B}T}\right)\mathrm{Re}[G_{+}(i\xi ,ϵ)],$$

$$G_{+}(z,ϵ)=\frac{{ϵ}^2{Q}_{+}(z,E)+(Q_{+}(z,E)+i\hslash \gamma )A_{+}(z,E)}{Q_{+}(z,E)[{ϵ}^2-(Q_{+}(z,E)+i\hslash \gamma {)}^2]},$$

$$E=\sqrt{{ϵ}^2+{\Delta }^2},$$

$$Q_{+}(z,E)=\sqrt{(E+\hslash z{)}^2-{\Delta }^2},$$

$$A_{+}(z,E)=E(E+\hslash z)+{\Delta }^2.$$

$$\Delta =c_1{k}_B{T}_{\mathrm{c}}\sqrt{1-\frac{T}{T_{\mathrm{c}}}}\left(c_2+c_3\frac{T}{T_{\mathrm{c}}}\right).$$

Full text

I Introduction

One of the most spectacular manifestations of vacuum fluctuations of quantum fields is provided by the Casimir effect Casimir48 . This is the tiny force acting between two discharged dielectric bodies, which results from the modification of the spectrum of quantum and thermal fluctuations of the electromagnetic field in the region of space bounded by the two bodies. In his pioneering work, Casimir studied this phenomenon for the idealized case of two perfectly conducting plane-parallel mirrors at zero temperature. The investigation of the Casimir effect in real material media started with the fundamental paper of Lifshitz lifs , which presented a derivation of the force between two plane-parallel dielectric slabs in vacuum, at finite temperature. In recent years, intense experimental and theoretical efforts have been made to probe the dependence of the Casimir force on the shapes and material properties of the test bodies. For a review of the Casimir effect, and its perspective applications to nanotechnology the reader may consult several recent books and review articles book1 ; parse ; book2 ; RMP ; capasso ; buh ; woods ; mehran .

Many experiments have now probed the Casimir effect with test bodies made of diverse materials, embedded in different media. Apart from two metallic conductors in vacuum, which still constitute the standard configuration, experiments have been carried out with semiconductors umar2006 ; umar2006bis ; umar1 , conductive oxides iannuzzi ; umar2 ; umar3 , magnetic materials bani2 ; ricardomag and liquid crystals munday . Experiments exist as well in which the bodies are immersed in gases or in liquids liq ; palas .

Another interesting class of candidate materials for Casimir experiments is represented by superconductors ala1 ; ala2 . The study of the Casimir effect in superconductors is indeed very interesting, since these materials constitute an excellent arena bimontesuper to investigate yet unresolved fundamental problems book2 ; RMP about the influence of relaxation phenomena on the strength of the Casimir force between metallic bodies. Unfortunately, observing the influence of the superconducting transition on the Casimir effect is very difficult, because on theory grounds one expects that the effect is extremely small. This can be understood by considering that the transition modifies significantly the optical properties of a superconductor only for frequencies of the order of $k_{B}T_{c}/\hbar$, where $T_{c}$ is the critical temperature. This region represents only a very small fraction of the spectrum of frequencies that contribute to the Casimir interaction between two bodies at distance $a$. The latter spectrum is known to stretch up to the characteristic frequency $\omega_{c}=c/a$, which for typical submicron separations is tens of thousands times larger than the frequency $k_{B}T_{c}/\hbar$ for classical BCS superconductors. In view of the difficulty of a direct force measurement, in ala1 ; ala2 we proposed an indirect approach, based on observation of the Casimir-induced shift of the critical magnetic field $H_{\rm c}$ of a thin superconducting film, constituting one of the two plates of a rigid Casimir cavity. An experiment with an Al film based on this scheme, placed an upper bound on the shift of the critical field not far from theoretical predictions superc ; annalisa .

An alternative route to successful detection is represented by differential measurements, which offer the advantage of a far superior sensitivity in comparison to absolute force measurements. An experiment based on the observation of the differential Casimir force between a Au-coated sphere and the two sectors of a microfabricated plate, respectively made of superconducting Nb and Au, was indeed proposed in bimonteiso . The latter setup exploits the principle of isoelectronic differential measurements bimoiso1 ; bimoiso2 , whose power in precision Casimir measurements has been demonstrated by a room temperature experiment ricardomag with a microfabricated plate consisting of alternating Au and Ni sectors.

More recently, an unpublished experiment leiden measured the Casimir force between a Au-coated sphere with a radius $R=100\;\mu$m and a superconducting NbTiN film, with a critical temperature $T_{c}=13.6$ K. The experimental data for room temperature showed good agreement with theoretical predictions. The low-temperature data displayed however an anomalous behavior, due to an unexpected twenty percent increase in the measured force, for which no explanation could be found. Apart from this, the experiment did not detect any change in the strength of the Casimir force across the superconducting transition, and placed an upper bound of 2.6 % on its magnitude.

A promising on-chip platform for observing the Casimir force between superconductors has been described very recently in norte . The apparatus consists of two micro-fabricated Al-coated SiN parallel strings, having a length of 384 $\mu$m and a width of 926 nm. By application of a large tensile stress, the strings can be kept perfectly parallel, at litographically determined fixed separations. Several cavities of different widths were realized on the same chip, the minimum separation being of one hundred nm. One of the two strings is attached to the movable mirror of an optomechanical cavity, whose resonance frequency is monitored by a laser. The detection scheme is based on the idea that when the system transitions to superconductivity, the resulting variation of the Casimir force between the Al strings should affect the mutual distance between the strings, thus determining a change in the length of the the cavity and therefore in its resonance frequency. The experiment norte provides a nice implementation the differential measurement scheme, since the apparatus is sensitive to the variation $\Delta F(T)=F(T)-F(T_{\rm c})$ across the superconducting transition of the Casimir force $F(T)$ on the Al strings. Up to edge effects, the force $F(T)$ can be expressed as $F(T)=P(T)\times A$, where $P(T)$ is the unit-area Casimir force, i.e. the Casimir pressure, and $A$ is the area of the strings. The above relation shows that the measurement of $\Delta F(T)$ is directly related to the variation $\Delta P(T)=P(T)-P(T_{\rm c})$ of the Casimir pressure across the critical temperature of the superconducting transition ($T_{c}=1.2$ K for Al). The null result reported by the experiment sets on the magnitude of $\Delta P$ an upper bound of 6 mPa, which represents the sensitivity of the apparatus.

In this paper we work out a detailed theory of the Casimir effect in superconducting cavities. We compute the variation $\Delta P(T)$ of the Casimir pressure for two distinct configurations of a superconducting planar cavity. In the first configuration, similarly to the experiment norte , both plates are made of the same superconductor, while in the second configuration, similarly to the experiment leiden , one of the two superconducting plates is replaced by a Au mirror. We model the frequency-dependent permittivity of the superconductor by the Mattis-Bardeen formula mattis ; tinkham , which provides the best known description of the optical properties of superconductors. We present numerical results for NbTiN and Al which are the superconductors used in the experiments leiden and norte respectively. It is important to note that optical measurements performed on NbTiN superconducting films hong show excellent agreement with the local limit (so called dirty-limit) of the Mattis-Bardeen formula, providing strong support in favor of our theoretical model. Our computations show that for the Al cavity used in the experiment norte the magnitude of $\Delta P$ is over two hundred and fifty times smaller than the experimental sensitivity. Our results, while in agreement with the null result reported by the experiment, make it unlikely that the effect of the superconducting transition can be observed with an Al cavity. We find however that the magnitude of $\Delta P(T)$ can be enhanced by a factor of fifteen, by considering a cavity composed by a Au mirror and a NbTiN film, having a thickness larger than two hundred nm. The enhancement factor increases to thirty-four if the separation $a$ is decreased from 100 nm to 60 nm. This is an encouraging result, since it shows that the effect would be detectable with a Au-NbTiN cavity, if the sensitivity of the apparatus could be improved by one order of magnitude.

The plan of the paper is as follows: in Sec. II we review the general formalism for computing the Casimir pressure between two superconducting plates, and we present the models we use to describe their optical properties. In Sec. III we present the results of our numerical computations. In Sec. IV we present our conclusions. Finally, in the Appendix we provide the explicit formula for the analytic continuation to the imaginary frequency axis of the Mattis-Bardeen formula for the frequency dependent conductivity of BCS superconductors.

II General formalism for the Casimir pressure

We consider a Casimir cavity, formed by two plane-parallel homogeneous and isotropic dielectric plates at temperature $T$, separated by an empty gap of width $a$. We denote by $\epsilon^{(k)}(\omega)$, $k=1,2$ their respective (complex) permittivities (we only consider non magnetic materials, and thus we set $\mu^{(1)}=\mu^{(2)}\equiv 1$). According to Lifshitz formula lifs , the Casimir pressure $P(a,T)$ among the plates can be expressed as (negative pressures correspond to attraction):

[TABLE]

[TABLE]

where $k_{B}$ is Boltzmann constant, $k_{\perp}$ is the in-plane momentum, the prime in the sum indicates that the $l=0$ term is taken with weight one-half, $\xi_{l}=2\pi lk_{B}T/\hbar$ are the imaginary Matsubara frequencies, $q_{l}=\sqrt{\xi_{l}^{2}/c^{2}+k_{\perp}^{2}}$, and the sum over $\alpha={\rm TE,TM}$ is taken over the independent states of polarization of the electromagnetic field, i.e. transverse magnetic ($\rm TM$) and transverse electric ($\rm TE$). Finally, the symbols $r^{(k)}_{\alpha}(i\xi_{l},k_{\perp})$ denote the Fresnel reflection coefficients of the $k$-th slab:

[TABLE]

[TABLE]

where $s^{(k)}_{l}=\sqrt{\epsilon^{(k)}_{l}\xi_{l}^{2}/c^{2}+k_{\perp}^{2}}$, and $\epsilon^{(k)}_{l}\equiv\epsilon^{(k)}(i\xi_{l})$. If instead of thick homogeneous slabs, one considers more complex mirrors constituted by plane-parallel metallic films deposited on some substrate, the corresponding Casimir pressure can still be computed by the general Lifshitz formula Eq. (1), provided that the Fresnel reflection coefficients Eqs. (2-3) are replaced by the reflection coefficients of the layered mirrors book2 . We shall consider two distinct confìgurations for our system: in the first one, both plates are made of the same superconducting material. Concretely, we shall consider two superconductors, i.e. Al (which is the superconductor used in the experiment norte ), and NbTiN (which is the superconductor used in the experiment leiden ). The corresponding configurations shall be denoted as Al-Al and NbTiN-NbTiN, respectively. The respective Casimir pressures are obtained by substituting into Lifshitz formula the permittivities of Al or NbTiN, respectively: $\epsilon^{(1)}_{l}=\epsilon^{(2)}_{l}=\epsilon^{(\rm Al/NbTiN)}(i\xi_{l})$. In the second configuration, one of the two superconducting plates is replaced by a Au mirror. This second configuration shall be analyzed in detail only for the case of NbTiN, and we shall denote it as the Au-NbTiN configuration. The corresponding Casimir pressure is obtained by setting into Eq. (1) $\epsilon^{(1)}_{l}=\epsilon^{(\rm Au)}(i\xi_{l})$ and $\epsilon^{(2)}_{l}=\epsilon^{(\rm NbTiN)}(i\xi_{l})$.

To compute the Casimir pressure, one needs the permittivities $\epsilon^{(k)}_{l}$ of the materials constituting the plates. In a concrete experimental situation, one would ideally like to measure the optical data of the used samples, for the experimental values of the temperature. The permittivities $\epsilon^{(k)}_{l}$ for the physically inaccessible imaginary frequencies $i\xi_{l}$ would then be computed on the basis of the optical data, using Kramers-Kronig dispersion relations book2 . In order to obtain a precise theoretical estimate of the Casimir pressure $P(a,T)$ for a separation $a$, it is in principle necessary to know the optical data for all frequencies lower than ten or twenty times the characteristic cavity frequency $\omega_{c}=c/2a$ book2 . For $a=100$ nm, $\omega_{c}=1.5\times 10^{15}$ rad/s.

It is fortunate that in the problem at hand we do not really need this much information about the optical properties of the materials. Indeed, the quantity that interests us is not the Casimir pressure $P(a,T)$ at a single temperature, but rather its variation $\Delta P(a;T)$ across the critical temperature $T_{\rm c}$:

[TABLE]

where $T<T_{\rm c}$. Now, it is known tinkham that the superconductive transition affects significantly the optical properties of a superconductor only for frequencies corresponding to photon energies smaller than (a few times) the BCS gap $\Delta(0)$. From BCS theory tinkham one knows that $\Delta(0)=1.76k_{B}T_{\rm c}$. For Al ($T_{\rm c}=1.2$ K) this gives $\Delta(0)=1.8\times 10^{-4}$ eV, while for NbTiN ($T_{\rm c}=13.6$ K) $\Delta(0)=2.1\times 10^{-3}$ eV. For these small photon energies the optical response of a normal metal is dominated by intraband transitions. The latter can be phenomenologically described by a Drude-model dielectric function of the form

[TABLE]

where the contribution from core interband transitions has been included in $\epsilon_{0}$. Here $\Omega$ is the plasma frequency for intraband transitions, and $\gamma$ is the relaxation frequency. To compute $\Delta P$ we have used the simple Drude model in Eq. (5) to describe the permittivity of Au, as well as the permittivity of the superconductors in the normal state. In our computations we have neglected the temperature dependence of both the plasma frequency $\Omega$ and of the core-electron permittivity $\epsilon_{0}$, and thus we used their room temperature values. The relaxation frequency is instead temperature dependent, and in general it decreases as the temperature is decreased. At cryogenic temperatures $\gamma$ approaches a constant sample dependent residual value. Following the standard convention, we express the residual relaxation frequency in terms of the corresponding room temperature frequency $\gamma_{0}$ by the formula $\gamma=\gamma_{0}/RRR$ where RRR is the residual resistance ratio. The values of the parameters were chosen as follows. For Au, we used the standard values $\Omega=9$ eV/$\hbar$ and $\gamma_{0}=35$ meV/$\hbar$ book2 , while from the tabulated optical data Palik we obtained $\epsilon_{0}=6.3$. For Al, we used $\Omega=13$ eV/$\hbar$, $\gamma_{0}=100$ meV/$\hbar$, and $\epsilon_{0}=1.03$ Palik . Finally, for NbTiN we used the values quoted in leiden i.e. $\Omega=5.33$ eV/$\hbar$ and $\gamma_{0}=0.465$ eV/$\hbar$. In leiden the optical data of the used NbTiN films were determined by ellipsometry in the frequency range from 1.89$\times 10^{11}$ rad/s to 1.13$\times 10^{16}$ rad/s, both at room temperature and at 16 K. The optical data were afterwards fitted by a Lorentz-Drude model with four oscillator terms. Unfortunately the values of the corresponding parameters were not reported explicitly. We are thus unable to provide a value for the contribution of core electrons for this material. We have verified however that the pressure variation $\Delta P(a;T)$ remains practically unchanged when the value $\epsilon_{0}$ for NbTiN is varied in the interval from one to ten. The value of the RRR parameter depends on the sample preparation procedure, and therefore it cannot be fixed a priori. For the NbTiN sample used in the experiment leiden , the fit to the optical data at 16 K gave $\gamma=0.415$, which corresponds to RRR=1.12. To probe the sensitivity of the pressure variation $\Delta P$ on this parameter, in our computations we varied its value in the interval from one to ten.

Next, we describe the model for the permittivity of the superconductors. For this we rely on the Mattis-Bardeen formula mattis for the conductivity $\sigma$, which is known to provide an accurate representation of the optical response of BCS superconductors tinkham . In its general form, the Mattis Bardeen formula depends both on the frequency $\omega$ and the wavevector $q$, since superconductors display spatial dispersion. However, the $q$-dependence is negligible in the so-called dirty limit $\ell/\xi_{0}\ll 1$, where $\ell=v_{F}/\gamma$ is the mean free path, and $\xi_{0}=\hbar v_{F}/\pi\Delta(0)$ is the correlation length, with $v_{F}$ the Fermi velocity. The dirty limit condition is well satisfied by both Al ($\ell/\xi_{0}=5.7\times 10^{-3}$) and NbTiN ($\ell/\xi_{0}=1.4\times 10^{-2}$). This is confirmed by optical measurements of NbTiN films in the THz region, that are in excellent agreement with the local dirty-limit of the Mattis-Bardeen formula hong . The analytic continuation of the Mattis-Bardeen formula to the imaginary frequency axis has been worked out in bimonteBCS , where it is shown that $\sigma_{\rm BCS}(i\xi)$ can be conveniently decomposed as:

[TABLE]

The first term between the square brackets on the r.h.s. of the above Equation coincides with the familiar Drude contribution to the conductivity of a normal metal, while the second term represents the BCS correction. The explicit expression of the function $g(\xi)$ is given in the Appendix. Here is a brief summary of its main properties. The function $g(\xi)$ is different from zero only for $T<T_{\rm c}$, and vanishes identically for $T\rightarrow T_{\rm c}$. For $T<T_{\rm c}$, it is a positive and monotonically decreasing function of $\xi>0$, approaching a finite value $g(0)<1$ for $\xi\rightarrow 0$, and going to zero for $\xi\rightarrow\infty$. Its value depends parametrically on the temperature-dependent BCS gap $\Delta$ as well as on the relaxation frequency $\gamma$. In addition to that, $g(\xi)$ has an explicit dependence on the temperature. For small $\xi$ the function $g(\xi)$ has the expansion:

[TABLE]

where $\omega_{s}(T)$ represents the (normalized) effective superfluid plasma frequency. A plot of the function $g(\xi;T)$ for NbTiN (RRR=1.12) is shown in Fig. 1 for $T/T_{\rm c}=0.9$ (blue line) and for $T/T_{\rm c}=0.1$ (red line). By adding the contribution of core electrons, we thus arrive at the following formula for the permittivity of the superconductor:

[TABLE]

The BCS term proportional to $g(\xi;T)$ in the expression of $\epsilon_{\rm BCS}$ can be interpreted as a plasma-model contribution, with an effective $\xi$-dependent plasma frequency $\Omega_{\rm eff}(\xi)=\Omega\sqrt{g(\xi;T)}$. In Fig. 2 we show logarithmic plots of the BCS permittivity of NbTiN as a function of $\xi/2\Delta(T)$, for $T/T_{\rm c}=0.9$ (blue line) and for $T/T_{\rm c}=0.1$ (red line). The dashed line corresponds to the Drude permittivity Eq. (5). The figure shows that the BCS permittivity approaches the Drude permittivity for $\xi/2\Delta(0)\gg 1$.

It is interesting to compare the BCS formula for the permittivity with the old-fashioned Casimir-Gorter two fluid-model gorter ; tinkham . According to this model a fraction $n_{\rm s}(T)$ of the conduction electrons contributes to the supercurrent, while the remaining fraction $n_{\rm n}(T)=1-n_{\rm s}(T)$ remains normal. Superconducting electrons behave as a dissipationless plasma, while normal electrons are described by the usual dissipative Drude model. Core electron remain unaltered. According to this simple physical picture, the permittivity of the two-fluid model is written as:

[TABLE]

The fraction $n_{\rm s}(T)$ of superconducting electrons follows the Casimir-Gorter law:

[TABLE]

where $\Theta(x)$ is the Heaviside step-function: $\Theta(x)=1$ for $x>0$, and $\Theta(x)=0$ for $x\leq 0$. In Fig. 2 we show plots of the two-fluid model for NbTiN, for $T/T_{\rm c}=0.9$ (blue dot-dashed line) and for $T/T_{\rm c}=0.1$ (red dot-dashed line). Comparison with the BCS permittivity (solid lines) shows that the two-fluid model overestimates the permittivity of a superconductor by a very large factor. We note that the two fluid model was used in leiden to compute the Casimir force between superconductors.

III Numerical computation of $\Delta P$

In this Section we present our numerical computations of the pressure variation $\Delta P(a;T)$, based on the expressions of the permittivity described in the previous Section.

We consider first a Casimir cavity constituted by two thick plates made of Al, which is the superconductor used in the experiment norte . In Fig. 3 the corresponding $\Delta P(a;T)$ is plotted versus the temperature $T$ (in K), for the separation $a=100$ nm which was the minimum separation probed in the experiment. We took RRR=1. For comparison, we show in the same Figure the variation of the pressure that would obtain in the absence of the transition (dashed line). We see that the solid curve lies below the dashed one, in accordance with one’s expectation that the superconducting transition determines an increase in the Casimir attraction with respect to the normal state, since superconductors are better reflectors than normal metals. The magnitude of $\Delta P$ is seen to be smaller than 0.05 mPa at all temperatures below $T_{\rm c}$. To get a feeling of how small an effect this represents, we note that the magnitude of the Casimir force $P(T_{\rm c})$ at the critical temperature is estimated to be of 6.8 Pa (this value was computed using the simple representation Eq. (5) for the permittivity of Al, and must be just considered as an approximate estimate. A more accurate estimate would require a better description of core electrons). Using this estimate, we obtain $\Delta P/|P(T_{\rm c})|<$7$\times 10^{-6}$ across the transition.

The Casimir cavity used the experiment norte consisted of two identical layered plates, each consisting of an Al film with a thickness $w=18$ nm, deposited on a SiN substrate.

To determine how the thickness $w$ of the Al films influences $\Delta P$, it is necessary to replace in Lifshitz formula the Fresnel reflection coefficients for a thick Al slab Eqs. (2-3) by those for the layered Al-SiN plate book2 . The resuls of this computation are shown in Fig. 4. We see that the thick-plate limiting value of 0.041 mPa is nearly reached for a film thickness of 250 nm, but for a thickness of 18 nm the magnitude of $\Delta P$ decreases to 0.023 mPa. Recalling that the experiment norte has an estimated sensitivity of 6 mPa, we see that the theoretical pressure variation for the Al cavity is over two hundred and fifty times smaller than the sensitivity. While this is consistent with the null result reported by the experiment, it makes one think that observation of the effect with the Al cavity is hardly possible in the near future.

Our computations predict that a significant increase in the magnitude of the pressure variation can be achieved by using NbTiN in the place of Al. This is demonstrated by Fig. 5, which displays the pressure variation for two thick NbTiN plates at a separation $a=100$ nm, versus the temperature $T$ (RRR=1.12). As we said earlier, we could not find in the literature enough information on the optical properties of NbTiN, to fix the value of $\epsilon_{0}$ in Eq. (8). For this reason, we repeated the computations using two widely different values for $\epsilon_{0}$. It is fortunate that the pressure variation is insensitive to the contribution of core electrons, as it can be seen from Fig. 5 where the solid and dotted lines correspond to $\epsilon_{0}=1$ and $\epsilon_{0}=10$, respectively. The weak dependence of $\Delta P$ on $\epsilon_{0}$ is explained by the fact that the pressure variation is determined by the optical response of the materials at frequencies of the order the thermal frequency $k_{B}T_{\rm c}/\hbar$, for which the Drude term is overwhelmingly large compared to $\epsilon_{0}$. Comparison of Fig 5 with Fig. 3 shows that the variation of the Casimir pressure for a NbTiN cavity is five times larger than the corresponding variation for an Al cavity.

A further increase of the pressure variation can be achieved by replacing one of the NbTiN plates by a Au mirror. This was the combination of materials adopted in the unpublished experiment leiden . We assume in what follows that the thickness of the Au coating of the first mirror is larger than 200 nm. This ensures that for the purposes of the Casimir effect that mirror can be considered as equivalent to an infinitely thick Au slab book2 . We note that Ref. leiden does not provide data for the RRR of Au at 16 K. In our computations we take $RRR_{\rm Au}=1$. It can be seen from Fig. 6 that for $a=100$ nm the maximum variation pressure for the Au-NbTiN cavity has a magnitude of 0.42 mPa, which is nine times larger than the corresponding maximum pressure variation of the Al cavity (see Fig. 3). In Fig. 7 we show the pressure variation of the Au-NbTiN cavity as a function of the separation $a$ (in nm), for $T/T_{\rm c}=0.5$. The red and blue curves correspond to RRRNbTiN=1.12 and RRRNbTiN=5, respectively. In Fig. 8 the pressure variation is displayed versus the residual resistance ratio RRR of the NbTiN film, for the two separations $a=100$ nm (red curve) and $a=60$ nm (blue curve). In Fig. 9 the pressure variation is displayed versus the residual resistance ratio RRR of the Au film, for the two separations $a=100$ nm (red curve) and $a=60$ nm (blue curve). For both curves, the RRR of the NbTiN film has the fixed value RRRNbTiN=1.12. We note that by increasing the value of RRR for the Au plate, it is possible to obtain a significant increase in the magnitude of $\Delta P$. This indicates that it would be beneficial to realize a Au mirror having a long mean free path for the electrons.

Finally, in Fig. 10 we display the pressure variation of the cavity formed by a thick Au mirror and a NbTiN film of thickness $w$, as a function of the film thickness $w$ (in $\mu$m), for a fixed separation $a=100$ nm. The pressure variation is moderately dependent on the properties of the substrate of the superconducting film. We verified this by comparing the results for a free-standing film (solid line of Fig. 10) with those for a substrate having a static permittivity equal to ten (dashed line in Fig. 10). The influence of the substrate of course decreases for thicker films. The plot shows that NbTiN films with a thickness larger than two hundred nm are essentially undistinguishable from an infinitely thick slab.

The important conclusion that can be drawn from the computations described above, is that a large enhancement of the pressure variation $\Delta P$ can be achieved by replacing the thin Al plates used in the experiment norte , by a cavity composed by a thick Au mirror and a NbTiN film having a thickness larger than two hundred nm. To get a quantitative idea of the magnitude of the enhancement that can be achieved in this way, consider as an example a cavity with a width $a=100$ nm, at a temperature $T=0.5\,T_{\rm c}$. For the Al cavity of norte , one gets $\Delta P=-0.023$ mPa while for the Au-NbTiN cavity (with RRRAu=1 and RRRNbTiN=1.12) one finds $\Delta P=-0.36$ mPa. While this figure represents a 15.8-fold enhancement with respect to the Al cavity, it is still 16.5 times smaller than the sensitivity of 6 mPa. One can get closer to the sensitivity threshold by decreasing the separation $a$. For example, going down to a=60 nm, one gets $\Delta P=-0.77$ mPa, which is 7.8 times smaller than the sensitivity. The remaining gap can be partly filled by improving the mean free path $\ell$ of the Au mirror. If Au mirrors with RRR=3 can be made, that would give $\Delta P=-0.98$ which is just 6.1 times smaller than the sensitivity. This shows that the effect of the transition would be observable with the Au-NbTiN cavity if the sensitivity of the apparatus could be improved by only one order of magnitude.

As a final remark, we note that in the unpublished experiment using a superconducting Au-NbTiN cavity leiden , the Casimir pressure was computed using the Casimir-Gorter two-fluid model Eq.(9) for the permittivity of the superconductor. Unfortunately, the results obtained in this way are not quite correct. Using this model, the authors estimated that for $a=100$ nm, the variation of the Casimir pressure $\Delta P(a;T)$ for $T\ll T_{\rm c}$ was of -265 mPa, corresponding to a 5.1 % fractional change $\Delta P(T)/|P(T_{\rm c})|$ of the Casimir pressure. The corresponding variation $\Delta P(a;T)$ obtained by us using the BCS permittivity is of -0.42 mPa (see Fig. 6), which amounts to a pression fractional change $\Delta P(T)/|P(T_{\rm c})|=8\times 10^{-5}$. We thus see that the two-fluid model overestimates the magnitude of the pressure variation by a factor larger than 600. We note also that the prediction of a 5.1 % change in the pressure is in disagreement with the experimental bound of 2.5 %, while of course the prediction of the BCS model is consistent with it.

IV Conclusions

A much debated problem in the theory of the Casimir effect is the role of relaxation phenomena of free charge carriers in Lifshitz theory book2 ; RMP . Different prescriptions have been proposed in the literature to compute the Casimir force between conducting test bodies, that go by the names of Drude and plasma prescriptions book2 ; RMP . Superconductors offer a unique possibility to investigate this problem bimontesuper . Unfortunately, it is very difficult to probe the influence of the superconducting transition on the Casimir force, because the effect of the transition is expected to be very small bimontesuper . A recent experiment with thin superconducting Al films reported a null result norte .

In this paper we have developed a detailed theory for the Casimir effect with superconducting plates. Our analysis relies on the Mattis-Bardeen formula for the frequency-dependent conductivity of BCS superconductors, which represents the best known theoretical description of the optical properties of superconductors. We performed numerical computations for Al and for NbTiN, which are the superconductors used in the experiments norte and leiden , respectively. The excellent agreement with the Mattis-Bardeen formula demonstrated by recent optical measurements on superconducting NbTiN hong , lends strong support to the validity of our theoretical analysis. We estimate that for the Al cavity used in the experiment norte , the magnitude of the variation of the Casimir pressure across the transition is over two hundred and fifty times smaller than the sensitivity of the experiment. This result, while consistent with the observed null result, makes it unlikely that the effect of the superconducting transition can be observed with an Al cavity. We find however that the expected signal can be enhanced by a factor of fifteen by substituting the thin Al films used in norte with a Casimir cavity constituted by a Au mirror and a NbTiN superconducting film, having a thickness larger than two hundred nm. The enhancement factor increases to thirtyfour times, if the width of the cavity is decreased from 100 nm to 60 nm. According to our computations, a further improvement is possible by using a Au mirror with a long mean free path for the electrons. Our analysis shows that the effect of the transition to superconductivity would be observable with the Au-NbTiN cavity, if the sensitivity of the apparatus used in norte could be increased by one order of magnitude.

Acknowledgements.

The author thanks R. A. Norte for useful discussions on the experiment norte . . *

Appendix A Expression of the function $g(\xi)$

In this Appendix we display the explicit expression of the function $g(\xi)$ that enters in Eq. (6), providing the analytic continuation to the imaginary frequency axis of the Mattis-Bardeen formula for the conductivity of a superconductor. Details on its derivation can be found in bimonteBCS . The function $g(\xi)$ can be expressed as:

[TABLE]

where $\Theta(x)$ is the Heaviside step-function: $\Theta(x)=1$ for $x>0$, and $\Theta(x)=0$ for $x\leq 0$ and

[TABLE]

with

[TABLE]

[TABLE]

and

[TABLE]

Here, $\Delta$ is the temperature-dependent gap. From BCS theory tinkham one knows that

[TABLE]

where $c_{1}=1.764$, $c_{2}=0.9963$ and $c_{3}=0.7735$