[****]{}\

Vincenzo Branchina[^1]\[one\]

Department of Physics, University of Catania and\
INFN, Sezione di Catania, Via Santa Sofia 64, I-95123, Catania, Italy

Marco Di Liberto[^2]\[two\]

Scuola Superiore di Catania, Via S. Nullo 5/i, Catania, Italy

Ivano Lodato[^3]\[three\]

Scuola Superiore di Catania, Via S. Nullo 5/i, Catania, Italy and\
INFN, Sezione di Catania, Via Santa Sofia 64, I-95123, Catania, Italy

[Abstract]{}\

It has been recently claimed that dark energy can be (and has been) observed in laboratory experiments by measuring the power spectrum $S_I(\omega)$ of the noise current in a resistively shunted Josephson junction and that in new dedicated experiments, which will soon test a higher frequency range, $S_I(\omega)$ should show a deviation from the linear rising observed in the lower frequency region because higher frequencies should not contribute to dark energy. Based on previous work on theoretical aspects of the fluctuation-dissipation theorem, we carefully investigate these issues and show that these claims are based on a misunderstanding of the physical origin of the spectral function $S_I(\omega)$. According to our analysis, dark energy has never been (and will never be) observed in Josephson junctions experiments. We also predict that no deviation from the linear rising behavior of $S_I(\omega)$ will be observed in forthcoming experiments. Our findings provide new (we believe definite) arguments which strongly support previous criticisms.

Introduction
============

The origin of dark energy is one of the greatest mysteries confronting theoretical and experimental physics. Different proposals for the solution of this so called cosmological constant problem are put forward and many review articles nowadays discuss and compare these alternative approaches (see for instance [@peeb],[@padam],[@cope],[@nobe]).

Among many other issues, Copeland et al.[@cope] discuss the suggestion of Beck and Mackey[@bema1; @bema2] according to which dark energy can be (and has been) observed in laboratory experiments [@koch] by measuring the power spectrum of the noise current in a resistively shunted Josephson junction. If true, this would mark a dramatic progress in our understanding of the origin of dark energy.

According to[@bema1; @bema2], this power spectrum is due to thermal and vacuum fluctuations of the electromagnetic field in the resistor and these experiments[@koch] provide a measurement of the electromagnetic zero point energies, which they consider as being at the origin of dark energy.

These ideas have generated a certain debate and some authors[@jetz1; @doran; @maha] have argued against them. Beck and Mackey have rebutted these criticisms[@bema3] but they were again criticized in [@jetz2]. Copeland et al.close the section of their review devoted to this issue by saying that “time will tell who (if either) are correct”[@cope]. Needless to say, this issue is of the greatest importance and deserves further investigation.

Scope of this work is to bring additional elements to the analysis of this problem in the hope that the question posed in[@cope] could finally find an answer. To this end, it is necessary to review in some detail the Beck and Mackey proposal[@bema1; @bema2].

These authors begin by considering the work of Koch et al.[@koch], where the spectral density $S_I(\omega)$ of the noise current in the resistor of a resistively shunted Josephson junction was measured and confronted against the theoretical prediction, \[spectr\] S\_I() = (+ ), and good agreement was found between experimental results and theory ($T$ is the temperature and $R$ the resistance of the resistive shunt).

Eq.(\[spectr\]) comes from an application of the fluctuation-dissipation theorem (FDT)[@cawe] and we immediately recognize the term in parenthesis as the mean energy of a quantum harmonic oscillator of frequency $\omega$ in a thermal bath. Where does this Bose-Einstein (BE) distribution factor come from? Does it reflect an underlying “harmonic oscillator structure” of the system[@taylor]? If yes, which harmonic oscillators are involved in Eq.(\[spectr\])? A correct answer to these questions will turn out to be crucial in understanding the status of the Beck and Mackey proposal[@bema1; @bema2] and, we believe, in settling the controversy.

Beck and Mackey interpret this factor as coming from the modes of the electromagnetic field in interaction with the charged particles [@bema2] and claim that this experiment provides a direct measurement of vacuum fluctuations of the electromagnetic field (the $\frac{\hbar\omega}{2}$ term). Moreover, they assume that dark energy originates from vacuum fluctuations of fundamental quantum fields and conjecture that only those fluctuations which can be measured in terms of a physical power spectrum are gravitationally active, i.e. contribute to dark energy. Then, by observing that for strong and electroweak interactions it is unlikely that a suitable macroscopic detector exists that can measure the corresponding vacuum spectra, they conclude that the only candidate where we know that a suitable macroscopic detector exists is the electromagnetic interaction[@bema2].

Accordingly, by noting that astrophysical measurements give $\varrho_{dark} \sim 10^{-47} GeV^4$ (in natural units), they argue that there should be a physical cut-off frequency $\nu_c =\frac{\omega_c}{2\pi}\sim 1.7\, {\rm THz}$ such that, for frequencies above this cut-off, the spectral function of the noise current in the Josephson junction should behave differently than in Eq.(\[spectr\]). According to their hypothesis, in fact, for $\omega \geq \omega_c $ the $\frac{\hbar\omega}{2}$ term should be absent.

Coming back to the BE distribution factor in Eq.(\[spectr\]), Jetzer and Straumann[@jetz1] (see also [@kubo]) observed that this term simply comes from ratios of Boltzmann factors which appear in the derivation of the FDT and stressed that the $\frac{\hbar\omega}{2}$ term has nothing to do with zero point energies, while Beck and Mackey reply that this is contrary to the view commonly expressed in the literature[@bema3].

A simple look to the derivation of the FDT (see Section 2) shows that, as for the ratios of Boltzmann factors, Jetzer and Straumann[@jetz1] are definitely right. Nevertheless, in a sense that we are going to make clear in the following, there is an element of truth in the common lore according to which this factor can be regarded as due to a sort of underlying harmonic oscillator structure of the system (the resistive shunt in the case of the Koch et al. experiment[@koch]).

In a recent paper[@noi] we have shown that whenever linear response theory applies, which is the main hypothesis under which the FDT is derived, any generic bosonic and/or fermionic system can be mapped onto a fictitious system of harmonic oscillators in such a manner that the quantities appearing in the FDT coincide with the corresponding quantities of the fictitious one (for completeness, in sections 2 and 3 we briefly review these results. For a comprehensive exposition, however, see[@noi]).

This allows us to understand [*in which sense*]{} the harmonic oscillator interpretation can be put forward so that we shall be able to say whether the Beck and Mackey’s proposal is tenable or not. We shall see that it is not.

The rest of the paper is organized as follows. In Section 2 we briefly review the derivation of the FDT and consider two convenient expressions for the power spectrum of the fluctuating observable and for the imaginary part of the corresponding generalized susceptibility respectively. In Section 3 we consider the special case of a system of harmonic oscillators in interaction with an external field and show how the above mentioned mapping is constructed. In Section 4 we apply the results of the two previous sections to our problem, namely the dark energy interpretation[@bema1; @bema2] of the measured power spectrum $S_I(\omega)$[@koch] and show that this interpretation is untenable. Section 5 is for our conclusions.

The fluctuation-dissipation theorem
===================================

In the present section we briefly review the derivation of the FDT (see[@kubo] for more details) and provide expressions for the spectral function and the imaginary part of the generalized susceptibility which will be useful for our following considerations.

Consider a macroscopic system with unperturbed hamiltonian $\hat{H}_0$ under the influence of the perturbation \[inter\] = - f(t)(t), where $\hat{A}(t)$ is an observable (a bosonic operator) of the system and $f(t)$ an external generalized force[^4]. Let $|E_n\rangle$ be the $\hat{H}_0$ eigenstates (with eigenvalues $E_n$) and $\langle E_n|\hat{A}(t)|E_n \rangle =0$. Within the framework of linear response theory, the quantum-statistical average $\langle\hat{A}(t)\rangle_f$ of the observable $\hat{A}(t)$ in the presence of $\hat{V}$ is given by \[resp2\] (t)\_f = \_[-]{}\^t d t’ \_[\_[A]{}]{}(t-t’) f(t’) where $\chi_{_{A}}(t - t')$ is the generalized susceptibility, \[chi\] \_[\_[A]{}]{}(t - t’)=(t-t’) = -G\_R(t - t’) , with $\langle ... \rangle = 
\sum_{n} \varrho_n \langle E_n| ... |E_n \rangle$, $\varrho_n= e^{-\beta E_n}/Z$ , $Z=\sum_n e^{- \beta E_n}$, $G_R(t-t')$ being the retarded Green’s function and $\hat{A}(t)=e^{i\hat{H_0}t/\hbar}\hat{A}e^{-i\hat{H_0}t/\hbar}$.

If we now consider the mean square of the observable $\hat A(t)$ and write the generalized susceptibility $\chi_{_{A}}$ as $\chi_{_{A}} = \chi^{'}_{_{A}} + i\,\chi^{''}_{_{A}}$ (with $\chi^{'}_{_{A}}$ the real part and $\chi^{''}_{_{A}}$ the imaginary part of $\chi_{_{A}}$), it is not difficult to show (see [@kubo] and [@noi]) that the Fourier transform $\langle \hat{A}^2(\omega)\rangle$ of $\langle \hat{A}^2(t)\rangle$ is related to the Fourier transform $\chi_{_{A}}^{\,''}(\omega)$ through the relation \[fddt\] \^2()= \_[\_[A]{}]{}\^[”]{}() = \_[\_[A]{}]{}\^[”]{}() [coth]{}( ) =2\_[\_[A]{}]{}\^[”]{}() (1 2 + ), which is the celebrated FDT.

In Eq.(\[fddt\]) we recognize the ratio of Boltzmann factors alluded by Jetzer and Straumann[@jetz1]. Actually, it was already observed by Kubo et al.[@kubo] that the BE factor in Eq.(\[fddt\]) is simply due to a peculiar combination of Boltzmann weights and that there is no reference to physical harmonic oscillators of the system whatsoever. However, as we already said in the Introduction, there is an element of truth in the common lore which considers this term as due to a sort of harmonic oscillator structure of the system (the resistive shunt in the case of the Koch et al.experiment[@koch]).

In order to show that, we now refer to our recent work[@noi], where we have derived the following useful expressions for $\langle \hat{A}^2(\omega)\rangle$ and $\chi_{_A}^{\,''}(\omega)$ : \^2()&=& \_[j &gt; i]{}(\_i - \_j) |A\_[ij]{}|\^2 [coth]{}( ) \[o5\]\
&=& [coth]{}()\_[j &gt; i]{} (\_i - \_j) |A\_[ij]{}|\^2 ,\[o6\]\
”()&=&\_[j &gt; i]{}(\_i - \_j) |A\_[i j]{}|\^2 \[chii3\]. Clearly, from Eqs.(\[o6\]) and (\[chii3\]) the FDT (Eq.(\[fddt\])) is immediately recovered. However, what matters for our scopes are the explicit expressions in Eqs.(\[o5\]) and (\[chii3\]). Starting from these equations, in fact, we can easily show that it is possible to build up a mapping between the real system and a fictitious system of harmonic oscillators[@noi] in such a manner that $\chi_{_A}^{\,''}(\omega)$ and $\langle \hat{A}^2(\omega)\rangle$ are exactly reproduced by considering the corresponding quantities of the fictitious system. In the following section we outline the main steps for this construction (see [@noi] for details).

The Mapping
===========

In order to build up this mapping, we consider first a system ${\cal S}_{osc}$ of harmonic oscillators (each of which is labeled below by the double index $\{ji\}$ for reasons that will become clear in the following) whose free hamiltonian is: \[armonico\] H\_[osc]{} = \_[j &gt; i]{}( + q\_[ji]{}\^[2]{}), where $\omega_{ji}$ are the proper frequencies of the individual harmonic oscillators and $M_{ji}$ their masses. Let $| n_{j i}\rangle$ ($n_{ji}=0, 1,2,...$) be the occupation number states of the $\{ji\}$ oscillator out of which the Fock space of ${\cal S}_{osc}$ is built up. Let us consider also ${\cal S}_{osc}$ in interaction with an external system through the one-particle operator: \[armint\] V\_[osc]{} = - f(t) A\_[osc]{}, with \[onepart\] [A]{}\_[osc]{} = \_[j &gt; i]{} (\_[j i]{} [q]{}\_[ji]{} ). Obviously, the FDT applied to ${\cal S}_{osc}$ gives $\langle {\hat A}_{osc}^2(\omega)\rangle = 
\hbar \chi_{osc}^{\,''}(\omega) \,{\rm coth}\left( \frac{\beta\hbar\omega}{2}\right)$, but this is not what matters to us.

What is important for our purposes is that, as shown in[@noi], for ${\cal S}_{osc}$ we can exactly compute $\langle {\hat A}_{osc}^2(\omega) \rangle$ and $\chi_{osc}^{\,''}(\omega)$. The reason is that for this system, differently from any other generic system, we can explicitly compute the matrix elements of ${\hat A}_{osc}$. The result is (compare with Eqs.(\[o5\]), (\[o6\]) and (\[chii3\])): \_[osc]{}\^2()&=&\_[j&gt;i]{} \_[ji]{}\^2 [coth]{}() \[(-\_[ji]{}) +(+\_[ji]{})\]\[osc5\]\
&=& [coth]{}()\_[j&gt;i]{} \_[ji]{}\^2 \[(-\_[ji]{}) - (+\_[ji]{})\]\[oscc5\];\
\_[osc]{}\^[”]{} ()&=& \_[j&gt;i]{} \_[ji]{}\^2 \[(-\_[ji]{}) - (+\_[ji]{})\]\[chi3\].

Naturally, comparing Eq.(\[oscc5\]) with Eq.(\[chi3\]) we see that for ${\cal S}_{osc}$ the FDT holds true, as it should. However, for our scopes it is important to note the following. For this system, the ${\rm coth}\left(\frac{\beta \hbar\omega}{2}\right)$ factor of the FDT originates from the [*individual contributions*]{} ${\rm coth}\left(\frac{\beta \hbar\omega_{ji}}{2}\right)$ of each of the harmonic oscillators of ${\cal S}_{osc}$.

We can now build up our mapping. Let us consider the original system ${\cal S}$, described by the unperturbed hamiltonian $\hat H_0$, in interaction with an external field $f(t)$ through the interaction term $\hat V = - f (t)\,\hat A$ (see Eq.(\[inter\])), and construct a fictitious system of harmonic oscillators ${\cal S}_{osc}$, described by the free hamiltonian ${\hat H}_{osc}$ of Eq.(\[armonico\]), in interaction with the same external field $f(t)$ through the interaction term ${\hat V}_{osc}$ of Eq.(\[armint\]), with $\hat A_{osc}$ given by Eq.(\[onepart\]), where for $\alpha_{j i}$ we choose \[alfa\] \_[j i]{} = ()\^[12]{} (\_i - \_j)\^[12]{} |A\_[ij]{}| and for the proper frequencies $\omega_{ji}$ of the oscillators \[omega\] \_[ji]{}= (E\_j-E\_i)/&gt; 0, with $E_i$ the eigenvalues of the hamiltonian ${\hat H}_0$ of the real system.

By comparing Eq.(\[oscc5\]) with Eq.(\[o6\]) and Eq.(\[chi3\]) with Eq.(\[chii3\]), it is immediate to see that with the above choices of $\alpha_{ji}$ and $\omega_{ji}$ we have: \^2()&=& \_[osc]{}\^2()\[cen1\]\
\_[\_A]{}\^[”]{} () &=& \_[osc]{}\^[”]{} ()\[cen2\]. Eqs.(\[cen1\]) and (\[cen2\]) define the mapping we are looking for. They show that it is possible to map the real system ${\cal S}$ onto a fictitious system of harmonic oscillators ${\cal S}_{osc}$, [S]{}\_[osc]{}, in such a manner that $\chi_{_A}^{''} (\omega)$ and $\langle \hat{A}^2(\omega)\rangle$ of the real system are equivalently obtained by computing the corresponding quantities of the fictitious one. The key ingredient to construct such a mapping is the hypothesis that linear response theory is applicable (which is the central hypothesis under which the FDT is established).

Now, by considering the “equivalent” harmonic oscillators system ${\cal S}_{osc}$ rather than the real one, we can somehow regard the BE distribution factor ${\rm coth}\left( \frac{\beta\hbar\omega}{2}\right)$ of the FDT in Eq.(\[fddt\]) as originating from the individual contributions ${\rm coth}\left( \frac{\beta\hbar\omega_{ji}}{2}\right)$ of each of the oscillators of the equivalent fictitious system (see above, Eqs.(\[osc5\]), (\[oscc5\]) and (\[chi3\])). In this sense, this mapping allows for an oscillator interpretation of the BE term in the FDT.

At the same time, however, the above findings clearly teach us that the BE distribution term in the FDT [*does not describe the physical nature of the system*]{}. It rather encodes a fundamental property of any bosonic and/or fermionic system: whenever linear response theory is applicable, any generic system is equivalent (in the sense defined above) to a system of quantum harmonic oscillators.

Dark energy and laboratory experiments
======================================

We are now in the position to apply the results of the two previous sections to our problem. As we said in the Introduction, Beck and Mackey[@bema1; @bema2] interpret the Koch et al.experimental results[@koch] for the spectral density $S_I(\omega)$ of the noise current in a resistively shunted Josephson junction as a direct measurement of [*vacuum fluctuations of the electromagnetic field*]{} in the shunt resistor. Moreover, according to their ideas, these zero-point energies are nothing but the dark energy of the universe.

In view of our results, however, this interpretation seems to be untenable. Eq.(\[spectr\]) for $S_I(\omega)$ comes from an application of the FDT to the case of the noise current in the shunt resistor. Therefore, according to our findings, the BE distribution factor which appears in $S_I(\omega)$ has nothing to do with thermal and vacuum fluctuations of the electromagnetic field in the resistor. Our analysis shows that this factor rather reflects a general property of any quantum system valid whenever linear response theory applies. The resistor (as well as any other generic system) can be mapped onto a system of fictitious harmonic oscillators in such a manner that the the power spectrum of the noise current and the related susceptibility can be reproduced by considering the equivalent quantities for the fictitious oscillators.

It is in this sense, and [*only in this sense*]{}, that the BE factor can be interpreted in terms of harmonic oscillators, no other physical meaning can be superimposed on it. According to these considerations, we conclude that the claim that dark energy is observed in laboratory experiments[@bema1; @bema2] is based on an incorrect interpretation of the origin of the BE factor in the FDT.

We believe that this should help in solving the controversy, which is left open in the Copeland et al.review[@cope], between the proponents[@bema1; @bema2] of the dark energy interpretation of the Koch et al.experiments[@koch] and the opponents[@jetz1; @doran; @maha; @jetz2]. In this respect, it is worth to stress that our analysis provides new arguments which strongly support the conclusions of these latter works[@jetz1; @doran; @maha; @jetz2].

A distinctive new element of our work, which in our opinion should greatly help in settling the question, concerns the interpretation of the FDT presented in section 5 of[@bema2]. These authors note that, although the FDT is valid for arbitrary hamiltonians $H$, where $H$ need not to describe harmonic oscillators, in the FDT appears a [*universal function*]{} $H_{uni}$, $H_{uni}= \frac12\hbar\omega +\hbar\omega/(exp(\hbar\omega/kT)-1)$, which can always be interpreted as the mean energy of a harmonic oscillator. Then, they identify the $\frac12\hbar\omega$ in $H_{uni}$ as the source of dark energy.

The distinctive feature of our analysis is that, with the help of the formal mapping discussed in the previous section, which is valid for [*any generic system*]{}, the reason for the appearance of this [*universal*]{} function is immediately apparent. At the same time, however, this clearly shows that it cannot be claimed that the Koch et al.[@koch] experimental device is measuring zero point energies. As already noted in[@jetz1; @jetz2], these experiments simply measure a general quantum property of the system, the $\frac12\hbar\omega$ in $S_I(\omega)$ has nothing to do with zero point energies.

Another very important point related to these issues concerns future measurements[@barb; @warb] of the power spectrum $S_I(\omega)$ for values of the frequency higher than those measured by Koch et al.[@koch]. In fact, according to Beck and Mackey[@bema1; @bema2], in forthcoming experiments[@barb; @warb], which are purposely designed to test a higher frequency range of $S_I(\omega)$, we should observe a dramatic change in the behavior of the spectral function $S_I(\omega)$ for these higher values of the frequency due to the presence of a cut-off which separates the gravitationally active modes from those which are not gravitationally active (see the Introduction). In view of our findings, however, we do not expect to observe in these experiments[@barb; @warb] any change in the behavior of $S_I(\omega)$. We simply state that such a cut-off does not exist.

In this respect, we note that Beck and Mackey have recently proposed a new model for dark energy which should naturally incorporate such a cut-off[@bm3]. According to our analysis, this model seems to be deprived of any experimental and theoretical support.

As a consequence of our results, a deviation of $S_I(\omega)$ from the behavior given in Eq.(\[spectr\]) could be observed only if the central hypothesis on which the derivation of the FDT is based, namely the applicability of linear response theory, no longer holds true in this higher frequency region.

Summary and Conclusions
=======================

With the help of a general theorem, which shows that (under the assumption that linear response theory is applicable) any bosonic and/or fermionic fermionic system can be mapped onto a fictitious system of harmonic oscillators, we have shown that the appearance of a Bose-Einstein distribution factor in the power spectrum of the noise current of a resistively shunted Josephson junction[@koch] has nothing to do with a real (physical) harmonic oscillator structure of the shunt resistor. We then conclude that, contrary to recent claims[@bema1; @bema2], experiments where this power spectrum was measured[@koch] do not provide any direct measurement of zero point energies and, as a consequence, no dark energy has ever been measured in these laboratory experiments.

A direct consequence of our analysis is that, contrary to what is predicted in[@bema1; @bema2], we do not expect any deviation from the linear rising behavior of $S_I(\omega)$ with $\omega$. According to our analysis, in fact, the $\frac12\hbar\omega$ term in $S_I(\omega)$ has nothing to do with the dark energy in the universe, therefore we do not expect any cut-off which separates the gravitationally active zero point energies from the gravitationally non-active ones.

Finally, our analysis suggests that the theory which should naturally incorporate such a cut-off[@bm3] is deprived of any experimental and theoretical foundation.

We believe that our work provides a satisfactory answer to the intriguing and important question left open by Copeland et al. in the section of their review devoted to the possibility of measuring dark energy in laboratory experiments[@cope]: “time will tell who (if either) are correct”. According to our analysis, the opponents to the dark energy interpretation of the Koch et al.experiments[@koch] are correct.

We would like to thank Luigi Amico, Marcello Baldo, Pino Falci and Dario Zappalà for many useful discussions.

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[^1]: vincenzo.branchina@ct.infn.it

[^2]: madiliberto@ssc.unict.it

[^3]: ivlodato@ssc.unict.it

[^4]: More generally, we could consider a local observable and a local generalized force, in which case we would have $\hat{V} = -\int d^3\,\vec r \hat{A}(\vec{r})f(\vec{r},t)$, and successively define a local susceptibility $\chi(\vec{r},t;\vec{r'},t')$ (see Eq.(\[chi\]) below). As this would add nothing to our argument, we shall restrict ourselves to $\vec r$-independent quantities. The extension to include local operators is immediate.
