Could dark energy be measured in the lab?

The experimentally measured spectral density of current noise in Josephson junctions provides direct evidence for the existence of zero-point fluctuations. Assuming that the total vacuum energy associated with these fluctuations cannot exceed the presently measured dark energy of the universe, we predict an upper cutoff frequency of nu_c=(1.69 +- 0.05) x 10^12 Hz for the measured frequency spectrum of zero-point fluctuations in the Josephson junction. The largest frequencies that have been reached in the experiments are of the same order of magnitude as nu_c and provide a lower bound on the dark energy density of the universe. It is shown that suppressed zero-point fluctuations above a given cutoff frequency can lead to 1/f noise. We propose an experiment which may help to measure some of the properties of dark energy in the lab.


I. INTRODUCTION
In his "second theory" of black-body radiation, Planck [1] (cf. also [2]) found the average energy of a collection of oscillators at temperature T and frequency ν to bē U (ν, T ) = 1 2 hν + hν exp(hν/kT ) − 1 . (1) The first (temperature independent) term is now referred to as the zero-point energy and commonly related to vacuum fluctuations. The second term gives rise [1,3] to the Planck black body spectrum ρ(ν, T ) = 8πhν 3 c 3 1 exp(hν/kT ) − 1 (2) that is relatively flat for hν << kT and which approaches zero for hν >> kT . In spite of early convictions by some investigators that the zero-point energy term in Equation (1) would not have any experimental correlate, this has not been the case. Indeed, the zero-point term has proved important in explaining X-ray scattering in solids [4]; understanding of the Lamb shift between the s and p levels in hydrogen [5,6]; predicting the Casimir effect [7,8,9]; understanding the origin of Van der Waals forces [7] [10,11]; explaining Compton scattering [5]; and predicting the spectrum of noise in electrical circuits [12,13,14,15]. It is this latter effect that concerns us here.
Koch et al. [13] measured the frequency spectrum of current fluctuations in Josephson junctions. At low temperatures and high frequencies the experimental spectrum is dominated by zero-point fluctuations, confirming the physical relevance of the zero-point term in Equation (1) up to frequencies of the order ν max = 6 × 10 11 Hz.
Here we re-analyze their experimental results in light of recent astronomical estimates of dark energy density in the universe [16,17,18,19].
Our hypothesis is that the signature of zero-point fluctuations measured by Koch et al. imply a non-vanishing vacuum energy density in the universe. This vacuum energy would have large scale gravitational effects, and cannot exceed the measured dark energy density of the universe as determined in astronomical measurements [16,17]. On this basis we predict a cutoff frequency (ν c ) for the zero-point fluctuations in Josephson junction experiments, which is only slightly larger than the maximum frequency ν max reached in Koch et al.'s 1982 experiment. Future experiments, based on Josephson junctions that operate in the THz region [20,21], could thus help to clarify whether this cutoff exists and whether the dark energy of the universe is related to the vacuum fluctuations that play a role in the Josephson junction experiments.

II. ESTIMATING A CUTOFF FREQUENCY FOR ZERO-POINT FLUCTUATIONS
If Planck [1] and Nernst [3] had used the relation ρ(ν, T ) = 8πhν 2Ū (ν, T )/c 3 , then instead of Equation (2) they would have obtained Equation (3), which is correct from the perspective of quantum electrodynamics [22], predicts that if all frequencies ν are taken into account then there should be an infinite energy per unit volume since lim νc→∞ νc 0 ρ(ν, T )dν diverges. To avoid this one could introduce a cutoff frequency ν c < ∞. Split the total energy density into where is due to zero-point fluctuations, and corresponds to the radiation energy density generated by photons of energy hν. Integration of (5) up to ν c yields while integration of (6) over all frequencies yields the well-known Stefan-Boltzmann law Suppose Equation (5) is valid only up to a cutoff frequency ν c , due to new but as yet unknown physics. How might we determine ν c ? We propose using estimates of the dark energy density to place an upper limit on the value calculated from Equation (7). Current estimates [16,17] indicate that dark energy constitutes 73% of all energy in the universe. To calculate the dark energy density ρ dark we need the critical energy density ρ c of a flat universe (the data of [17] indicate that the universe is flat), which is ρ c = 10.539h 2 Hubble GeV/m 3 = 10.539 × (0.71 ± 0.04) 2 GeV/m 3 . Finally, we have If we set then

III. MEASUREMENTS OF ZERO-POINT FLUCTUATIONS IN JOSEPHSON JUNCTIONS
The behavior of a resistively shunted Josephson junction is modeled as a particle that moves in a tilted periodic potential, and the effect of the noise current is to produce random fluctuations of the tilt angle [12]. This situation is captured by the stochastic differential equation Here δ is the phase difference across the junction, R is the shunt resistor, C the capacitance of the junction, I is the mean current, I 0 the noise-free critical current, and I N is the noise current. As shown in [12,23], the junction noise current should have a spectral density given by The first term in Equation (13) is due to vacuum fluctuations, and the second one is due to ordinary Bose-Einstein statistics. This predicted spectral behaviour has been experimentally verified in the work of [13] measuring the current noise in a resistively shunted Josephson junction at two different temperatures. Further, the computed cutoff frequency (11) is less than one order of magnitude larger than the highest frequency used in these experiments. Fig. 1 shows how well the predicted form of the power spectrum (13) is experimentally verified up to frequencies of order 6 × 10 11 Hz (note that no fitting parameters are used in this figure). For more recent theoretical work on the quantum noise theory of Josephson junctions, see [24,25].
Zero-point fluctuations thus have theoretically predicted and experimentally measured effects in Josephson junctions. We therefore expect that the energy density associated with these fluctuations has physical meaning as well: It is a prime candidate for dark energy, being isotropically distributed and temperature independent. Note that the experimentally measured fluctuations in  [13] for two different temperatures. The solid line is the prediction of Equation (13), whereas the dashed line is given by (4hν/R)(exp(hν/kT ) − 1) −1 . Fig. 1 are physical reality and have to be distinguished from 'theoretical' zero-point fluctuations that just formally enter into QED calculations without any cutoff. The vacuum energy associated with the measured data in Fig. 1 cannot be easily discussed away.
Assuming that the vacuum energy associated with the measured fluctuations in Fig. 1 is physically relevant, we predict that the measured spectrum in Josephson junction experiments must exhibit a cutoff at the critical frequency ν c . If not, the corresponding vacuum energy density would exceed the currently measured dark energy density of the universe. In future experimental measurements that may reach higher frequencies one would have to carefully distinguish between intrinsic cutoffs (due to experimental constraints) and fundamental cutoffs (due to new physics).

A. Lower bound on dark energy density
The largest frequency reached in the Koch et al. [13] experiment was ν max ≃ 6 × 10 11 Hz ≈ 1 3 ν c . From (7) this implies a minimum value of dark energy density in the universe: If larger frequencies ν max could be reached in a similar experiment, they would provide a better lower bound.

B. hνc and neutrino masses
The energy associated with the computed cutoff frequency ν c E c = hν c = (7.00 ± 0.17) × 10 −3 eV (15) coincides with current experimental estimates of neutrino masses. The LMA (large-mixing angle) solution of the solar neutrino problem yields a mass square difference of roughly ∆m 2 sun ≃ 7 × 10 −5 eV 2 between two neutrino species [26]. Assuming a hierarchy of neutrino masses, this gives a neutrino mass of the order of magnitude m ν ≃ 8 × 10 −3 eV.
If this coincidence is confirmed in future experiments, one might try to develop a theory that links the cutoff frequency of the zero-point fluctuations to an as yet unknown property of the neutrino sector of the standard model. For previous work that relates the dark energy scale to the mass of neutrinos, see [27]. Generally, in quantum field theory bosons are associated with positive vacuum energy and fermions with negative energies [28]. In supersymmetric models both contributions cancel exactly. To explain a coincidence of the type hν c ≃ m ν c 2 , a possible idea would be that negative vacuum energy associated with neutrinos (or neutrino-like particles) might cancel positive vacuum energy associated with photons as soon as the energy E = hν exceeds the neutrino rest mass. A toy model of this type is worked out in section V.

C. Effective degrees of freedom contributing to dark energy
Photons and other particles contribute to the total vacuum energy density of the universe. General quantum field theoretical considerations imply that a particle of mass m and spin j makes a contribution [28] in units where = c = 1. Here k represents the momentum and the energy is given by E = √ k 2 + m 2 . The integral is divergent and the actual contribution depends on the regularization scheme chosen.
It is likely that the Josephson junction experiment only measures vacuum fluctuations that couple to electric charge, since this experiment is purely based on electromagnetic interaction (see also [29] on possible interactions of mesoscopic quantum systems with gravity). Thus this experiment is likely to see only a fraction κ < 1 of the total dark energy of the universe. This would modify the expected cutoff frequency as In particular, a small κ can significantly lower the cutoff frequency. A measurement of κ would thus give information on the effective number of degrees of freedom that produce the entire dark energy density of the universe.

V. DARK ENERGY AND 1/F NOISE
In the following we consider a simple model where a bosonic contribution to vacuum energy is suppressed by a fermionic contribution as soon as the energy exceeds hν c = mc 2 , where m is the mass of the fermion under consideration.
Assume j = 1/2. From Equation (16) we obtain the fermionic contribution to the vacuum energy as where k = |k| and k max is a suitable upper cutoff. Transforming from k to E = √ k 2 + m 2 this can be written as Additionally the massless boson contributes with in agreement with Equation (7), setting E = hν and = c = 1. Adding the two contributions, one obtains The integrand in Equation (21), divided by E 2 /π 2 , represents the effective zero-point energy of this problem. Correlated vacuum fluctuations of this type would thus produce in Josephson junctions the power spectrum There is a rapid decrease of spectral power above the critical frequency hν c = mc 2 . For frequencies hν > mc 2 Equation (23) implies For large hν and we have Thus, asymptotically, the vacuum fluctuation spectrum is inversely proportional to ν so suppressed vacuum fluctuations produce 1/f noise. 1/f noise is commonly observed in many electric circuits, and was also observed in Koch et al.'s experiment but was subtracted from the data [13]. Our simple theoretical considerations show that high-frequency 1/f noise can arise naturally if bosonic vacuum fluctuations are suppressed by fermionic ones. If the coefficient multiplying 1/ν in Equation (26) is measured in the experiment, then it can be used to determine the cutoff scale hν c = mc 2 .

VI. CONCLUSION
We propose a repeat of the experiments of Koch et al. with new generations of Josephson junctions at higher frequencies. If it is possible to increase the maximum frequency by a factor of about 3, then this experiment could provide valuable information on the nature of dark energy. If the vacuum energy associated with the fluctuations measured in Fig. 1 is physically relevant, then we predict that a deviation from linear growth of S(ν) will be seen at higher frequencies, and in fact a rapid decrease of zero-point power near the critical frequency ν c is expected. If this is not seen in the experiment, then we must conclude that the dark energy of the universe probably has nothing to do with vacuum fluctuations at all but is purely classical. Alternately, if this decrease is not observed another interpretation would be that the Josephson junction experiment is insensitive to the process which cancels the photonic vacuum energy at large frequencies. If the frequency cutoff is observed, it could be used to determine the fraction κ of dark energy density that is produced by electromagnetic processes. Moreover, if the Josephson junction experiment is repeated at different temperatures, then a possible temperature dependence of ν c could provide information on whether the dark energy density is really independent of the expansion of the universe (i.e. its temperature) or whether it changes slightly with the expansion (as in the models [30]). Finally, we think that it could be interesting to analyze experimentally observed high-frequency 1/f noise in electrical circuits under the hypothesis that it could be a possible manifestation of suppressed zero-point fluctuations.