application/xmlCould dark energy be measured in the lab?Christian BeckMichael C. MackeyZero-point fluctuationsDark energyJosephson junctionsPhysics Letters B 605 (2005) 295-300. doi:10.1016/j.physletb.2004.11.060journalPhysics Letters BCopyright © 2004 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26936053-413 January 20052005-01-13295-30029530010.1016/j.physletb.2004.11.060http://dx.doi.org/10.1016/j.physletb.2004.11.060doi:10.1016/j.physletb.2004.11.060http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.2PLB21549S0370-2693(04)01626-010.1016/j.physletb.2004.11.060Elsevier B.V.PhenomenologyFig. 1Spectral density of current noise as measured in Koch et al.'s experiment [13] for two different temperatures. The solid line is the prediction of Eq. (13), whereas the dashed line is given by (4hν/R)(exp(hν/kT)−1)−1.Could dark energy be measured in the lab?ChristianBeckac.beck@qmul.ac.ukhttp://www.maths.qmul.ac.uk/~beckMichael C.Mackeyb1mackey@cnd.mcgill.cahttp://www.cnd.mcgill.ca/people_mackey.htmlaSchool of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UKbCentre for Nonlinear Dynamics in Physiology and Medicine, Departments of Physiology, Physics and Mathematics, McGill University, Montreal, PQ, Canada1Also: Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK.Editor: P.V. LandshoffAbstractThe 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 νc=(1.69±0.05)×1012 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 νc 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.PACS74.81.Fa98.80.-k03.70.+kKeywordsZero-point fluctuationsDark energyJosephson junctions1IntroductionIn 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 be (1)U¯(ν,T)=12hν+hνexp(hν/kT)−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 (2)ρ(ν,T)=8πhν3c31exp(hν/kT)−1 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 Eq. (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 [7–9]; understanding the origin of van der Waals forces [7]; interpretation of the Aharonov–Bohm effect [10,11]; explaining Compton scattering [5]; and predicting the spectrum of noise in electrical circuits [12–15]. 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 Eq. (1) up to frequencies of the order νmax=6×1011 Hz. Here we reanalyze their experimental results in light of recent astronomical estimates of dark energy density in the universe [16–19].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 (νc) for the zero-point fluctuations in Josephson junction experiments, which is only slightly larger than the maximum frequency νmax 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.2Estimating a cutoff frequency for zero-point fluctuationsIf Planck [1] and Nernst [3] had used the relation ρ(ν,T)=8πhν2U¯(ν,T)/c3, then instead of Eq. (2) they would have obtained (3)ρ(ν,T)=8πν2c3[12hν+hνexp(hν/kT)−1]=4πhν3c3[1+2exp(hν/kT)−1]=4πhν3c3coth(hνkT). Eq. (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 limνc→∞∫0νcρ(ν,T)dν diverges. To avoid this one could introduce a cutoff frequency νc<∞.Split the total energy density into (4)ρ(ν,T)=ρvac(ν)+ρrad(ν,T), where (5)ρvac(ν)=4πhν3c3 is due to zero-point fluctuations, and (6)ρrad(ν,T)=8πhν3c31exp(hν/kT)−1 corresponds to the radiation energy density generated by photons of energy hν. Integration of (5) up to νc yields (7)∫0νcρvac(ν)dν=4πhc3∫0νcν3dν=πhc3νc4, while integration of (6) over all frequencies yields the well-known Stefan–Boltzmann law (8)∫0∞ρrad(ν,T)dν=π2k415ℏ3c3T4. Suppose Eq. (5) is valid only up to a cutoff frequency νc, due to new but as yet unknown physics. How might we determine νc? We propose using estimates of the dark energy density to place an upper limit on the value calculated from Eq. (7).Current estimates [16,17] indicate that dark energy constitutes 73% of all energy in the universe. To calculate the dark energy density ρdark we need the critical energy density ρc of a flat universe (the data of [17] indicate that the universe is flat), which is ρc=10.539hHubble2 GeV/m3=10.539×(0.71±0.04)2 GeV/m3. Finally, we have (9)ρdark=0.73ρc=(3.9±0.4) GeV/m3. If we set (10)πhc3νc4≃ρdark then (11)νc≃(1.69±0.05)×1012 Hz.3Measurements of zero-point fluctuations in Josephson junctionsThe 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 (12)ℏC2eδ¨+ℏ2eRδ˙+I0sinδ=I+IN. 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, I0 the noise-free critical current, and IN is the noise current. As shown in [12,23], the junction noise current should have a spectral density given by (13)S(ν)=2hνRcoth(hνkT)=4hνR(12+1exp(hν/kT)−1).The first term in Eq. (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 6×1011 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 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 νc. 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).4Implications for dark energy from present and future experiments4.1Lower bound on dark energy densityThe largest frequency reached in the Koch et al. [13] experiment was νmax≃6×1011 Hz≈13νc. From (7) this implies a minimum value of dark energy density in the universe: (14)ρdark⩾πhc3νmax4=0.062 GeV/m3. If larger frequencies νmax could be reached in a similar experiment, they would provide a better lower bound.4.2hνc and neutrino massesThe energy associated with the computed cutoff frequency νc(15)Ec=hνc=(7.00±0.17)×10−3 eV 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 Δmsun2≃7×10−5 eV2 between two neutrino species [26]. Assuming a hierarchy of neutrino masses, this gives a neutrino mass of the order of magnitude mν≃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 hνc≃mνc2, 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 5.4.3Effective degrees of freedom contributing to dark energyPhotons 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](16)ρvac=12(−1)2j(2j+1)∫d3k(2π)3k2+m2 in units where ℏ=c=1. Here k represents the momentum and the energy is given by E=k2+m2. 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 (17)νc=(κρdarktotal)1/4(c3πh)1/4. 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.5Dark energy and 1/f noiseIn 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 hνc=mc2, where m is the mass of the fermion under consideration.Assume j=1/2. From Eq. (16) we obtain the fermionic contribution to the vacuum energy as (18)ρvacferm=−∫d3k(2π)3k2+m2=−12π2∫0kmaxk2k2+m2dk, where k=|k| and kmax is a suitable upper cutoff. Transforming from k to E=k2+m2 this can be written as (19)ρvacferm=−12π2∫mEmaxE2−m2E2dE. Additionally the massless boson contributes with (20)ρvacbos=+12π2∫0EmaxE3dE, in agreement with Eq. (7), setting E=hν and ℏ=c=1. Adding the two contributions, one obtains (21)ρvac=12π2∫0Emax(E3−E2−m2E2θ(E−m))dE, where the θ-function is defined by (22)θ(x)={1,x⩾0,0,x<0. The integrand in Eq. (21), divided by E2/π2, represents the effective zero-point energy of this problem. Correlated vacuum fluctuations of this type would thus produce in Josephson junctions the power spectrum (23)S(ν)=4R{12hν,hν⩽mc2,12(hν−h2ν2−m2c4),hν>mc2. There is a rapid decrease of spectral power above the critical frequency hνc=mc2. For frequencies hν>mc2 Eq. (23) implies (24)S(ν)=2Rhν(1−1−m2c4h2ν2). For large hν(25)1−m2c4h2ν2≈1−m2c42h2ν2, and we have (26)S(ν)=1Rm2c41hν. 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 Eq. (26) is measured in the experiment, then it can be used to determine the cutoff scale hνc=mc2.6ConclusionWe 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 νc 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 νc 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]). 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