The PVLAS experiment: A 25 year effort to measure vacuum magnetic birefringence

Abstract

This paper describes the 25 year effort to measure vacuum magnetic birefringence and dichroism with the PVLAS experiment. The experiment went through two main phases: the first using a rotating superconducting magnet and the second using two rotating permanent magnets. The experiment was not able to reach the predicted value from QED. Nonetheless the experiment set the current best limits on vacuum magnetic birefringence and dichroism for a field of 2.5 T, namely, a birefringence of (12 ± 17) × 10⁻²³ and a dichroism whose absolute value is (10 ± 28) × 10⁻²³. The uncertainty on the birefringence is about a factor 7 above the predicted value of 2.5 × 10⁻²³ at 2.5 T.

1. Introduction

The velocity of light in vacuum is considered today to be a universal constant and is defined as c = 299 792 458 m/s in the International System of Units, as one over the square root of the product of the vacuum magnetic permeability and the vacuum permittivity, which describe the properties of classical electromagnetic vacuum. Classically this relation derives directly from Maxwell's equations in vacuum. Due to their linearity c does not depend on the presence of other electromagnetic fields (photons, static fields).

Today Quantum Electrodynamics (QED) describes electrodynamics to an incredibly accurate level having been tested in many different systems at a microscopic level: the anomalous magnetic moments of the electron and the muon, the Lamb shift, Delbrück scattering and so on. One fundamental process predicted since 1935, before the formulation of QED, namely 4-field interactions with only photons present in both the initial and final states, still needs attention. In the above mentioned measurements either the accuracy is such that the 4-field interaction must be taken into account as a correction to the first order effect being observed or, as is the case of Delbrück scattering, this contribution must be distinguished from a series of other effects. The 4-field interaction considered to first order will lead to two effects: light-by-light (LbL) scattering and vacuum magnetic (or electric) linear birefringence (VMB) due to low energy coherent Delbrück scattering. This first effect occurs at a microscopic level whereas VMB describes a macroscopic effect related to the index of refraction and is a direct manifestation of quantum vacuum. In recent years, with the ATLAS experiment at the LHC accelerator, LbL scattering at high energies has been observed via photon pair emission during Pb-Pb peripheral collisions for photon energies much smaller than the electron rest energy. Furthermore, optical polarimetry of an isolated neutron star has led Mignani et al. to publish evidence of VMB.

It remains that this purely quantum mechanical effect still needs a direct laboratory verification in the low energy regime, for photon energies much smaller than the electron rest energy, at the macroscopic level.

As will be discussed in the following sections, not only does the index of refraction depend on the presence of external fields but it depends also on the polarisation direction of the propagating light. In the presence of an external magnetic field perpendicular to the propagation direction of a beam of light, one finds that the index of refraction for light polarised parallel to the field is one plus seven times the parameter A-sub-e times the square of the external field, and for light polarised perpendicular to the field it is one plus four times that parameter times the square of the external field. Similarly, in the presence of an electric field the roles of the factors seven and four are exchanged, with the square of the external electric field divided by the square of the speed of light in place of the square of the magnetic field.

As will be discussed in Section 2, the parameter A-sub-e describes the non linearity of Maxwell's equations due to vacuum fluctuations, and equals 1.32 × 10⁻²⁴ T⁻², being built from the reduced Planck constant, the electron mass, the speed of light, the vacuum permeability and the square of the fine structure constant.

The first modern proposal to measure QED non-linearities due to vacuum polarisation at very low energies dates back to 1979 and a first attempt was performed at CERN during the beginning of the '80s to study the feasibility of such a measurement. Since then the idea to detect the induced birefringence due to an external magnetic field using optical techniques has been an experimental challenge. Optical elements and lasers have since improved tremendously but as of today, in spite of the unceasing efforts, the direct measurement of VMB is still lacking. Another ongoing attempt to tackle the same physics is detecting the refraction of light-by-light in vacuum. Searches for direct photon-photon elastic scattering are also reported in the literature. Note that heuristic approaches to this very same matter started before and prescinded from any theoretical justification. More literature and a general treatment of non linear vacuum properties can be found in the references.

Following two precursor experiments, the one at CERN and the other at the Brookhaven National Laboratories (BNL) briefly described in Sections 4.1 and 4.2, the PVLAS (Polarizzazione del Vuoto con LAser) experiment, financed by INFN (Istituto Nazionale di Fisica Nucleare, Italy), performed a long lasting attempt starting from 1993. This experiment went through two major phases: the first with a rotating superconducting magnet and the second with two rotating permanent magnets.

In this paper we will describe at length the 25 year development of the PVLAS experiment and present the final results which represent today the best limit on VMB, closest to the expected value. In the description of the various phases of the experiment many details are given which generally are excluded in scientific papers. We believe this is an opportunity to gather all this information together in a single publication. The last three years of activity of PVLAS coincided with the PhD thesis of one of the authors. More details on the experiment can be found in that work.

In Section 2 we will present the physics related to PVLAS including the possibility of searching for physics beyond the Standard Model. In Section 3 we will describe the general experimental method of the experiment including systematic effects. In Section 4 each attempt with its peculiarities will be described: there we will discuss the limitations of each effort and the results obtained. Finally in Sections 6, 7 and 8 we will present the calibration method, systematics-hunting, noise issues and results of the last phase of the PVLAS experiment.

2. Theoretical considerations

(Section 2.1, on classical electromagnetism and the constitutive relations in a medium, is omitted for length; the complete text is at the source.)

2.2. Light-by-light interaction at low energies

This classical scenario changed drastically with the introduction of three new facts at the beginning of the 20th century:

• Einstein's energy-mass relation, energy equals mass times the square of the speed of light;
• Heisenberg's uncertainty principle, the product of an energy uncertainty and a time uncertainty being at least half the reduced Planck constant;
• Dirac's relativistic equation of the electron admitting negative energy states today identified as anti-matter.

These three facts together allow vacuum to fluctuate changing completely the idea of vacuum and allowing for non linear electrodynamic effects in vacuum. Today vacuum is considered as a minimum energy state. Citing from O. Halpern's letter (1933):

"…. Here purely radiation phenomena are of particular interest inasmuch as they might serve in an attempt to formulate observed effects as consequences of hitherto unknown properties of corrected electromagnetic equations. We are seeking, then, scattering properties of the 'vacuum'."

In 1935, soon after Halpern's intuition, two of Heisenberg's students H. Euler and B. Kockel determined a relativistically, parity-conserving effective Lagrangian density which, to second order in the invariants of the electromagnetic field tensor, takes into account electron-positron vacuum fluctuations. This Lagrangian was derived in the approximation of low energy photons, for photon energies much smaller than the electron rest energy.

The effective Euler-Kockel Lagrangian density leads to non linear effects even in the absence of matter thereby violating the superposition principle, one of the building blocks of Maxwell's theory in vacuum.

2.2.1. Leading order vacuum birefringence and dichroism in Electrodynamics

In general, the index of refraction of a medium is a complex quantity, a real part plus an imaginary part. The real part (known as the index of refraction tout court) determines the velocity of propagation of light in the medium, whereas the imaginary part, known as the index of absorption, describes the absorption of the medium.

A medium is said to be birefringent if the index of refraction depends on the polarisation state of the propagating light. Both linear and circular birefringences exist: the first is a birefringence for linearly polarised light whereas the second is a birefringence for circularly polarised light (also known as optical activity). Similarly a medium is said to be dichroic if the index of absorption depends on the polarisation (both linear and circular).

Consider a linearly polarised beam of light propagating through an external field perpendicular to its direction of travel. Indicating with the subscripts parallel and perpendicular the polarisation direction, that is the electric field direction of the light, with respect to the external magnetic field, one finds that the index of refraction for the parallel polarisation is one plus seven times A-sub-e times the square of the external field, and for the perpendicular polarisation one plus four times A-sub-e times the square of the external field. Both are greater than unity and a birefringence is apparent:

(37) The difference between the parallel and perpendicular indices of refraction, the Euler-Kockel vacuum magnetic birefringence, equals three times A-sub-e times the square of the external magnetic field.

A measurement of the induced birefringence of vacuum due to an external magnetic field would therefore allow a direct verification of the Euler-Kockel Lagrangian.

(Sections 2.2.2 and 2.2.3, on higher order corrections and on the Born-Infeld Lagrangian, and Sections 2.3 and 2.4, on axion like particles and millicharged particles, are omitted for length; the complete text is at the source.)

2.5. Chameleons and Dark energy

An open issue of modern cosmology is the understanding of the cosmic acceleration. The presence of a scalar field sourcing the dark energy responsible for this acceleration is envisaged in several theories. To comply with experimental bounds, a screening mechanism preventing the scalar field to act as a fifth force is however necessary. The chameleon mechanism provides a way for this suppression via nonlinear field self-interactions and interactions with the ambient matter. It can be seen that the chameleon fields behave as Axion Like Particles with respect to photons in an experiment of PVLAS type, with a coupling constant to photons. Brax and co-workers have calculated the effect on the rotation and ellipticity measurements in the presence of a chameleon field. One feature of the chameleon model is that the ellipticity is predicted to be much larger than the rotation. This can be viewed as a generic prediction of chameleon theories and it is due to the fact that chameleons could be reflected off the cavity mirrors. The difficulty in calculating the expected effects is that these are related to the geometrical size of the cavity, the magnetic field and the density of matter in the laboratory vacuum. For these reasons in this paper we will not try to extract chameleon information from the PVLAS data.

3. The experimental method

3.1. Polarimetric scheme

Birefringence and dichroism are local properties of a medium and can be determined by detecting their effect on the propagation of light. Here we will discuss the polarimetric scheme adopted by PVLAS in the attempt to measure magnetically induced vacuum birefringence and dichroism.

Consider a monochromatic linearly polarised beam of light propagating along an axis. Let us also assume that the polarisation, that is the electric field, is directed vertically, and let this beam propagate through a uniformly birefringent medium of thickness L whose slow and fast axes are perpendicular to the direction of travel. Finally let the slow axis of the medium form an angle with the polarisation direction. The components of the electric field along the slow and fast axes of the propagating beam will acquire a phase difference at the output of the medium equal to two pi over the wavelength, times the difference of the two indices of refraction, times the thickness L. More generally, the total optical path difference between the two components of the electric field is the integral of the birefringence along the path.

(The remainder of Chapter 3 — the Fabry-Perot interferometer as an optical path multiplier, heterodyne detection, systematic effects, calibration and the survey of comparable apparatus — and the whole of Chapter 4, on the PVLAS forerunners at CERN, at Brookhaven and at the Legnaro National Laboratories, are omitted for length; the complete text is at the source.)

5. The PVLAS-FE experiment

5.1. Summary

The many years of experience led to the final PVLAS-FE setup in which all of the previous experience was put together. Four general features were implemented:

1. the polarimeter was to be mounted on a single vibration isolated optical bench to reduce seismic noise coming from the ground;
2. the rotation frequency of the magnets was to be as high as possible;
3. the finesse was to be as high as possible supposedly to increase the signal-to-noise ratio;
4. all components of the polarimeter were to be non magnetic to avoid magnetic coupling between the optics and the rotating stray magnetic field.

Magnet rotations up to about 10 Hz were imagined during the design phase allowing a further reduction of the expected cavity noise contribution: at the time it was clear that the higher the signal frequency, the lower was the noise. What was still not clear was the proportionality of the ellipticity sensitivity with the number of passes, or in other words, the independence of the optical path difference noise on that number for large numbers. For this reason the highest achievable finesse was still a goal. At the time the experiment was designed, fields up to 2.5 T were available with permanent magnets over a diameter of about 2 cm. Due to the available space, a total magnetic field length longer than about 2 m was difficult. The two magnet scheme was implemented to take advantage of the effective signal cancellation with perpendicular fields tested in the PVLAS-Test setup. Each of the PVLAS-FE magnets had a field length of 0.82 m.

The final sensitivity in optical path difference of the PVLAS-FE apparatus was 3.5 × 10⁻¹⁹ m per root hertz at 16 Hz, a factor of about 10 worse than the required one to reach VMB detection in about a million seconds. Furthermore measurements showed the independence on the number of passes, for large numbers, of the optical path difference noise, meaning that increasing the finesse would have been useless to gain in signal to noise ratio. An increase in frequency to compensate this missing factor of about 10 was beyond our possibility.

An integrated noise floor in optical path difference of (1.0 ± 1.4) × 10⁻²² m, limited by statistics and not by systematics, was our final value after a run time of about 5 × 10⁶ s. Translated to vacuum magnetic birefringence, with a field length of 0.82 m, this leads to a birefringence of (12 ± 17) × 10⁻²³ at 2.5 T. The one sigma uncertainty is a factor of about 7 from the predicted value of 2.5 × 10⁻²³ at 2.5 T. Similarly for the dichroism limit, the PVLAS-FE final value was (10 ± 28) × 10⁻²³ at 2.5 T.

5.2. General description of the apparatus

The experiment was located on the ground floor of an experimental hall at the Department of Physics and Earth Sciences of the University of Ferrara, Ferrara, Italy, inside a temperature controlled (23 degrees plus or minus 1 degree) and relative humidity controlled (about 56%) clean room of ISO-4 class.

A Nd:YAG laser (Innolight Mephisto, 2 W power) emitted at a wavelength of 1064 nm. The beam first passed through a quarter-wave plate reducing the initial ellipticity of the laser beam. A first half-wave plate placed before a two stage Faraday isolator allowed the adjustment of the power being injected into the Fabry-Perot cavity. The beam then passed through a lens to match the laser waist with the cavity waist for optimal mode matching. Two steering mirrors followed by a second half-wave plate brought the beam to the entrance of the vacuum system with the desired alignment and polarisation direction. Between the second steering mirror and this second half-wave plate a glass window allowed the sampling of the reflected power from the cavity for phase locking the laser to the cavity via the Pound-Drever-Hall technique. The same glass plate was also used to sample the beam power at the Fabry-Perot input. The sidebands for the Pound-Drever-Hall locking circuit were generated directly in the laser rather than with an external phase modulator. An automatic locking servo-circuit allowed operation of the apparatus with an almost unitary duty-cycle.

The second half-wave plate together with the rotatable polariser allowed the alignment of the light polarisation with one of the axes of the equivalent wave plate of the cavity. The light path between the two mirrors passed through the bores of the two dipole magnets. At the cavity output an extractable quarter-wave plate was used to transform, when necessary, a polarisation rotation into an ellipticity and vice versa. The light then passed through the resonant photo-elastic ellipticity modulator and the analyser, normally set to maximum extinction. Both the extraordinary and ordinary beams from the analyser exited the vacuum enclosure: the former measured the power transmitted by the cavity, whereas the extinguished beam power contained information on the ellipticity and rotation acquired by the light polarisation.

The length of the PVLAS-FE high-finesse Fabry-Perot cavity was 3.303 ± 0.005 m, and the measured finesse was 770 000 ± 6 000, with the cavity running during the vacuum measurements at a finesse of about 700 000.

(The remainder of Chapter 5, and the whole of Chapter 6 on commissioning, calibration, systematics-hunting and the noise budget, and Chapter 7 on the measurements themselves, are omitted for length; the complete text is at the source.)

8. Vacuum measurement results and time evolution

8.1. Limits on vacuum magnetic birefringence and dichroism

The results can be averaged to give the final limits on vacuum magnetic birefringence and dichroism of the PVLAS-FE experiment, for a total run time of about 5 × 10⁶ s:

(214) The vacuum magnetic birefringence is (12 ± 17) × 10⁻²³ at 2.5 T.

(215) The absolute value of the vacuum magnetic dichroism is (10 ± 28) × 10⁻²³ at 2.5 T.

These values represent the current best limits on these quantities obtained by optical means. The value for the dichroism is reported as an absolute value because its sign was never determined but was common to all measurements. Therefore the relative signs of all the dichroism values are consistent.

We note that the vacuum magnetic birefringence predicted by the Euler and Kockel Lagrangian is 2.5 × 10⁻²³ at 2.5 T, i.e. the integrated noise level of the PVLAS-FE measurement is a factor seven larger than the predicted effect. The practical impossibility to beat the noise in the actual scheme was a show stopper and spurred for new ideas.

The historical time evolution of the measurement of vacuum magnetic birefringence normalised to the square of the external field, from different experiments, allows the comparison of the limits of the different experiments trying to measure VMB. In such a comparison, the PVLAS-FE experiment appears with three points, representing the integrated progression of this measurement by the experiment. The first two points correspond to already published papers whereas the third, including the 2016 data, represents the final result of the experiment concerning VMB. The global 2016 value is a birefringence normalised to the square of the field of (+19 ± 27) × 10⁻²⁴ T⁻².

8.2. Limits on hypothetical particles

8.2.1. Axion Like Particles. The results of the polarimetric measurements of the PVLAS-FE experiment can be used to draw exclusion plots in the plane of axion mass against coupling constant for Axion Like Particles, at 95% confidence level, alongside the measurements by the OSQAR and the ALPS collaborations.

8.2.2. Millicharged Particles. Exclusion plots for millicharged particles at 95% confidence level derive from the birefringence and dichroism values above, for fermion and for scalar millicharged particles. Two independent limits are derived from the birefringence and the dichroism values, the latter being more stringent in the low-mass range at or below 0.1 eV, whereas the former is dominating the high-mass range. We explicitly note that the fermion exclusion plot also applies to all types of neutrinos, limiting their charge to be less than about 3 × 10⁻⁸ of the electron charge for masses smaller than 10 meV.

9. Conclusions

The PVLAS-FE experiment officially ended on December 31st 2017 after 25 years. The present paper represents the final results of the experiment with the magnetically induced birefringence and dichroism limits summarised above. Unfortunately implementing further improvements to the PVLAS-FE setup of at least a factor ten to reach the predicted VMB value was not possible given the wide band noise intrinsic to the presence of the Fabry-Perot cavity which we believe to be of thermal origin. Nor was it conceivable to integrate a factor fifty times longer to only reach a signal-to-noise ratio of one.

A total run time of 5 × 10⁶ s was made possible thanks to the use of permanent magnets which, for the first time, allowed a detailed debugging of the setup. One of the key issues was the coupling between the diffused light inside the vacuum tube passing through the magnets and the induced movement of the tubes due to the small transverse gradient of the rotating magnetic field. The reduction of the diffused light with the use of baffles and a careful monitoring of the acceleration of the tubes allowed their centering thereby reducing systematic signals to below the achieved noise floor.

The phase coherence of the rotating magnets was another key factor allowing to distinguish between a physical signal and a mechanically induced disturbance which appeared at times. Indeed one of the requirements of a physical signal was that it occupy a single bin in the demodulated Fourier spectrum even after a very long integration time, with a bin width of about one microhertz or less.

From the experience presented in this paper and assuming to have all systematics under control, a new experiment to measure VMB using a Fabry-Perot based polarimeter will need to take into account the intrinsic optical path difference noise. The key factor, we believe, is to improve the optical path difference source, that is the integral of the square of the external field along the light path. A few possible ideas could be:

• use a relatively low magnetic field of about one to two tesla but increase the magnetic field length. This would allow the use of permanent or normal conducting magnets. Such a solution could then be applied to a gravitational wave antenna detector with long arms where the optics is in continuous development. In this situation an integral of about 200 tesla squared metres would be necessary considering an integration time of a million seconds. One extremely interesting feature of this solution is the independent measurement of the two indices of refraction allowing the direct determination of the two parameters in the general Lagrangian;
• continue to develop pulsed magnets as in BMV and OVAL. Given a pulse width of about 10 ms, the integrated peak noise per pulse will be about 3 × 10⁻¹⁹ m per pulse. Assuming a pulse rate of about 0.1 Hz and a running time of about a month — a longer running time would not allow debugging in the presence of unexpected peaks — results in a required integral of about 150 tesla squared metres;
• use a constant field superconducting magnet and modulate the induced ellipticity using the polarisation. One possibility to achieve this is to insert two co-rotating half-wave plates inside the Fabry-Perot, one at each end of the magnetic field. Due to the losses introduced by these wave plates, clearly this would reduce the maximum finesse to about one to five thousand. The intrinsic optical path difference noise of 10⁻¹⁸ m per root hertz at 4 Hz could still be reached provided the intensity exiting the cavity is about 50 mW. Assuming the usual integration time of a million seconds results in a requirement for the magnetic field integral of about 250 tesla squared metres.

This last configuration seems to be the most attractive in that such superconducting magnets already exist at CERN. In particular the use of a spare LHC dipole magnet with a maximum field of 9 T and a field length of 14.3 m, resulting in an integral of about 1158 tesla squared metres, would in principle allow a signal-to-noise ratio of one in less than a day. Following this line a new collaboration is coalescing and a Letter of Intent has been submitted to CERN and initial testing to demonstrate the feasibility is underway.

Acknowledgements

This paper is dedicated to the memory of Emilio (Mimmo) Zavattini, Erseo Polacco and Guido Petrucci. They were colleagues, teachers, friends, and more.

The PVLAS experiment was financed by the Istituto Nazionale di Fisica Nucleare (INFN), Italy and by the Italian Ministry of Research (MIUR).

(The 175-item reference list is omitted for length; the complete text is at the source.)