The PVLAS experiment: measuring vacuum magnetic birefringence and dichroism with a birefringent Fabry–Perot cavity

Abstract

Vacuum magnetic birefringence was predicted long time ago and is still lacking a direct experimental confirmation. Several experimental efforts are striving to reach this goal, and the sequence of results promises a success in the next few years. This measurement generally is accompanied by the search for hypothetical light particles that couple to two photons. The PVLAS experiment employs a sensitive polarimeter based on a high finesse Fabry–Perot cavity. In this paper we report on the latest experimental results of this experiment. The data are analysed taking into account the intrinsic birefringence of the dielectric mirrors of the cavity. Besides a new limit on the vacuum magnetic birefringence, the measurements also allow the model-independent exclusion of new regions in the parameter space of axion-like and milli-charged particles. In particular, these last limits hold also for all types of neutrinos, resulting in a laboratory limit on their charge.

1 Introduction

Vacuum magnetic birefringence is a very small macroscopic quantum effect stemming from the 1936 Euler–Heisenberg–Weisskopf effective Lagrangian density for slowly varying electromagnetic fields. To lowest order that Lagrangian density is Equation 1: the classical Maxwell term, quadratic in the fields, plus two quartic terms weighted by a parameter A-sub-e — the square of the difference between the squared electric field over the square of the speed of light and the squared magnetic field, and seven times the square of the scalar product of the electric and magnetic fields divided by the speed of light.

Here the parameter A-sub-e is given by Equation 2: two times the square of the fine structure constant times the cube of the reduced Compton wavelength of the electron, divided by forty-five times the magnetic permeability of free space times the electron rest energy — a value of 1.32 times ten to the minus twenty-four per tesla squared. The reduced Compton wavelength is the reduced Planck constant divided by the electron mass times the speed of light, the fine structure constant is the usual combination of the elementary charge, the permittivity of free space, the reduced Planck constant and the speed of light.

The first term in Equation 1, quadratic in the fields, is the classical Lagrangian corresponding to Maxwell’s equations in vacuum, for which the superposition principle holds and no light-by-light interaction is expected. The other terms, instead, imply that Electrodynamics is nonlinear even in vacuum, giving rise to a new class of observable effects.

The Quantum Electrodynamics (QED) representation of the simplest phenomena we are interested in is given by Feynman diagrams in which four photons interact through a virtual electron-positron pair. In one of them, two photons interact with an external field; this is the process that leads, in vacuum, to magnetic birefringence, namely to different indices of refraction for light polarised parallel and perpendicular to an external magnetic field. Let us consider the complex index of refraction, the real index plus an imaginary extinction coefficient. The relationship between the extinction coefficient and the absorption coefficient is that the absorption coefficient equals four pi times the extinction coefficient divided by the wavelength in vacuum. It can be shown that the magnetic birefringence derived from Equation 1 is Equation 3: the difference between the parallel and perpendicular indices of refraction equals three times A-sub-e times the square of the external field. This corresponds, at an external field of 2.5 tesla, to a birefringence of 2.5 times ten to the minus twenty-three — Equation 4. The calculations also show that the magnetic dichroism is instead negligible: no appreciable imaginary part of the index of refraction is predicted.

Magnetic birefringence accompanied by magnetic dichroism could, though, be generated in vacuum through the creation of so far hypothetical light bosonic spin-zero axion-like particles (ALPs), in an analog of the Primakoff effect. Two different Lagrangians describe the pseudoscalar and the scalar cases: for the pseudoscalar, the coupling constant times the field times the scalar product of the electric and magnetic fields; for the scalar, the coupling constant times the field times the difference of the squared electric and magnetic fields. Natural Heaviside-Lorentz units are used, so that one tesla is 195 electron-volts squared and one metre is 5.06 times ten to the sixth inverse electron-volts. One finds Equation 5: the induced birefringence is the square of the coupling times the square of the external field, divided by twice the square of the particle mass, multiplied by one minus the sine of twice the dimensionless parameter divided by twice that parameter; and the induced dichroism is the square of the coupling times the external field times the magnetic field length, divided by four times the photon energy times the length, multiplied by the square of the sine of the parameter over the parameter. Here the dimensionless parameter is the particle mass squared times the length divided by four times the photon energy. The last formula corrects an earlier published expression in which the factor of one over the photon energy times the length was missing.

Consider now the vacuum fluctuations of particles with a small fractional charge and mass. The photons traversing a uniform magnetic field may interact with such fluctuations, resulting in a phase delay and, for photon energy above twice the particle rest energy, in a pair production. We consider separately the cases of Dirac fermions and of scalar bosons. The indices of refraction of photons with polarisation respectively parallel and perpendicular to the external magnetic field have two different mass regimes defined by a dimensionless parameter — three halves times the photon energy over the particle rest energy, times the fractional charge times the external field times the reduced Planck constant over the square of the particle rest energy. Equation 6 gives the birefringence for Dirac fermions as a parameter A, defined in analogy to Equation 2 but with the fractional charge and the particle mass in place of the electron charge and mass, times the square of the external field, times three in the small-parameter regime and times a numerical coefficient built from Gamma functions multiplied by the parameter to the power minus four thirds in the large-parameter regime.

In the limit of large masses the expression reduces to Equation 3 with the substitution of the fractional charge for the electron charge and the particle mass for the electron mass. Note that for small masses the birefringence depends on the parameter to the power minus four thirds, resulting in a net dependence of the fermion birefringence on the external field to the two-thirds power rather than on its square as in Equation 3. For dichroism one finds a corresponding two-regime expression, exponentially suppressed in the small-parameter regime and going as the parameter to the power minus one third in the large-parameter regime.

The results for the case of milli-charged scalar particles are very similar to the case of Dirac fermions. Again there are two mass regimes defined by the same parameter. The magnetic birefringence and the dichroism take the same functional forms with different numerical coefficients. As can be seen, there is a sign difference with respect to the case of Dirac fermions, both for birefringence and for dichroism.

The PVLAS (Polarisation of Vacuum with LASer) experiment in Ferrara is the fourth generation of a measurement scheme that dates back to the end of the seventies. Previous experimental efforts were based at CERN, at BNL, and at Legnaro (Italy). The experiment aims at the direct measurement of the small polarisation changes undergone by a linearly polarised laser beam traversing a dipole magnetic field in vacuum. To this end, a pair of polarising prisms, two permanent magnets, an optical high-finesse Fabry–Perot cavity, and heterodyne detection are employed. A quarter-wave-plate placed after the Fabry–Perot switches the measurement from ellipticity to rotation (dichroism). The signal is detected in the extinguished beam with polarisation orthogonal to the input polarisation.

The Fabry–Perot cavity has the role of lengthening the optical path inside the magnetic field. It is realised with two dielectric mirrors with extremely high reflectivity. Unfortunately, the mirrors have a small intrinsic linear birefringence in reflection. A first consequence of this fact is that, if linearly polarised laser light is at maximum resonance inside the cavity, the orthogonal polarisation component is not. This means that the amplitude of the observed signal is reduced; this fact is evidenced during the calibration of the polarimeter with magnetic birefringence in gas (Cotton-Mouton – or Voigt – effect). Recent anomalously low Cotton-Mouton results could perhaps be explained in this way. As a second consequence, ellipticities and rotations are mixed, due to the birefringence of the mirrors. As we will see, both phenomena can be managed, in some cases even with profit. Moreover, the intrinsic birefringence of the mirrors may play a role in the excess noise currently observed in the PVLAS experiment.

In this article we present a detailed account of the polarimetric method employed by the PVLAS experiment, with a novel interpretation of the experimental data. What we describe here has consequences for all the experiments that use Fabry–Perot cavities for polarimetry, and in particular for those trying to measure vacuum magnetic birefringence. Section 2 analyses the experimental scheme, taking into account the intrinsic birefringence of the mirrors. Section 3 describes the experimental set-up with the calibration measurements. Then the measurement of the mirrors’ equivalent wave-plates and of the two resonance curves are presented. In Sect. 4 the ellipticity and rotation measurements in vacuum are discussed, together with the new limits on the existence of axion-like and milli-charged particles.

2 The PVLAS experimental method

Linearly polarised light of wavelength lambda is fed to a Fabry–Perot optical cavity. The cavity beam traverses the bore of a dipole magnet, with the magnetic field making an angle, variable in time, with respect to the polarisation direction. A variable ellipticity is then added to the polarisation of the beam transmitted by the cavity. For rotation measurements, a quarter-wave-plate is inserted at the exit of the cavity with one of its axes aligned to the input polarisation, transforming the rotation eventually acquired by the beam inside the magnetic field region into an ellipticity (and, at the same time, the ellipticity into a rotation). Finally a polariser, crossed with respect to the input prism, extinguishes the polarisation component of the beam parallel to the input polarisation. The residual intensity is then collected with a light detector and Fourier analysed.

In order to calculate the effect, we use Jones’ matrices to describe the beam and the optical elements. The most general optical element describing linear magnetic birefringence and dichroism can be written, in its own axes and neglecting an overall attenuation factor, as a diagonal matrix whose entries are the exponential of a small complex number and unity. That complex number is twice the ellipticity times the imaginary unit minus twice the rotation. Here twice the ellipticity is the phase difference between the two polarisation directions added by the optical element, and one minus the exponential of minus twice the rotation is the fraction of the absorbed electric field. Without loss of generality, one axis is considered as the absorbing as well as the slow axis. The ellipticity is the maximum ellipticity that the light can acquire due to the element, while the rotation is the maximum rotation. The ellipticity is the ratio of the minor to the major axis of the ellipse described by the electric field vector of the light.

In the case of the vacuum birefringence of Equation 3, the ellipticity for a length of 1.64 metres of a magnetic field of 2.5 tesla and light wavelength of 1.064 micrometres is Equation 7: pi times the birefringence times the length divided by the wavelength, which is 1.2 times ten to the minus sixteen.

Placing the element at an angle with respect to the polarisation direction gives the rotated Jones matrix. To show the salient features of our polarimetric method, we begin with neglecting the effect of the Fabry–Perot cavity. The electric field after the analyser is then the product, from left to right, of the Jones matrices of the analyser, of the ellipticity modulator, and of the quarter-wave-plate, acting on the rotated element and the input polarisation vector. In the quarter-wave-plate matrix, the coefficient is unity for ellipticity measurements, when the wave-plate is out of the optical path and the matrix coincides with the identity, whereas it is one plus the imaginary unit over the square root of two for rotation measurements.

For ellipticity measurements, with the quarter-wave-plate not inserted, the intensity collected at the extinction photodiode is Equation 8: the input intensity times the sum of the square of the modulation amplitude and twice the modulation amplitude times the ellipticity times the sine of twice the angle, plus higher order terms. For rotation measurements, with the quarter-wave-plate inserted, Equation 9 is the same expression with the rotation in place of the ellipticity. The light having the same polarisation as the input is collected at the transmission photodiode and has an intensity equal to the vacuum permittivity times the speed of light times half the square of the input field amplitude.

The heterodyne method is employed to measure the ellipticity and the rotation: the angle of the birefringent element is varied linearly in time at the magnet rotation frequency, and the modulator ellipticity is varied sinusoidally at a much higher modulation frequency. The sought for value of each of the two quantities can be extracted from the measurement of the transmitted intensity and from the amplitude and phase of three components in a Fourier transform of the extinguished intensity: the component at twice the modulation frequency, and the components at the modulation frequency plus and minus twice the magnet rotation frequency. By using a lock-in amplifier to demodulate the residual intensity at the modulation frequency, instead of the two sidebands there is a single component at twice the magnet rotation frequency, and the resulting ellipticity and rotation signals are given by Equation 10: the component at twice the rotation frequency divided by two times the square root of twice the product of the transmitted intensity and the component at twice the modulation frequency, which equals that component at twice the rotation frequency times the modulation amplitude divided by four times the component at twice the modulation frequency.

The ellipticity and rotation signals come with a well defined phase equal to twice the reference angle of the birefringent element. That reference angle is minus the angle between a laboratory reference direction and the polarisation direction. With this position, the axes of the element coincide with the laboratory axes and the ellipticity is a maximum at a determined time within each magnet rotation. We will return to this topic in the calibration section. In the absence of signals due to magnetic birefringence or dichroism, the noise level at the signal frequency translates into an upper limit for the measured quantity.

2.1 The Fabry–Perot cavity as an optical path multiplier

To take into account the multiple reflections of the Fabry–Perot cavity, we consider the physical parameters of the mirrors, namely the reflectivity, transmissivity, and losses, assumed equal for both mirrors and summing to unity. If the distance between the two mirrors is given, the phase acquired by the light in a round trip is four pi times that distance divided by the wavelength. Then one can write, for the electric field after the cavity, Equation 11: the sum over all round trips of the reflectivity times the round-trip phase factor times the square of the rotated element matrix, applied to the transmitted input field — which resums to the inverse of the identity minus that same quantity. The electric field after the analyser is then given by Equation 12, the product of the analyser, modulator and quarter-wave-plate matrices acting on the cavity output.

In the case of ellipticity measurements, since at resonance the round-trip phase is a multiple of two pi, and given that the reflectivity is close to one, the intensity collected by the extinction photodiode is, at lowest order, Equation 13: the input intensity times the sum of the square of the modulation amplitude and four times the modulation amplitude times the ellipticity divided by one minus the reflectivity, times the sine of twice the angle. Analogously, in the case of rotation measurements, one has Equation 14 with the rotation in place of the ellipticity, while the transmitted intensity is given by Equation 15.

By comparing these formulas with the corresponding ones calculated above without the Fabry–Perot cavity, one sees that the expressions are very similar, with the latter ones having the signals of Equation 10 amplified by a factor equal to two over one minus the reflectivity, which is approximately twice the finesse of the cavity divided by pi. The finesse can be up to about a million. This can be interpreted as a lengthening of the optical path by that same factor, as the very form of Equation 11 suggests. Besides heterodyne detection, high amplification is another key feature of the polarimetric technique adopted by the PVLAS experiment. In this way, the ellipticity of Equation 7 becomes of order ten to the minus ten.

We now introduce another issue of the Fabry–Perot cavity that will be fully discussed in the next paragraph. Let us suppose that the resonance condition is not fully matched, namely that the Fabry–Perot cavity is not exactly locked to the top of the resonance curve. Equations 13 and 14 then become Equations 16 and 17, in which the numerator of the signal term becomes twice the amplification times the ellipticity minus the square of the amplification times the rotation times the sine of the detuning, over one plus the square of the amplification times the square of the sine of half the detuning, and correspondingly with ellipticity and rotation exchanged and the sign reversed for the rotation channel. The transmitted intensity acquires the same Airy denominator.

One can see that, in a cavity locked away from resonance, there is a cross talk between the birefringence and dichroism signals as defined by Equation 10: a rotation is measured even in the case of pure birefringence. Conversely, in the case of pure dichroism, a signal mimicking a birefringence is observed.

2.2 Mirror birefringence

Let us now tackle the problem of dealing with birefringent mirrors. If the two mirrors impose small phase differences on light in just one reflection, one must introduce in the above calculations two further wave-plate matrices, each diagonal with entries the exponential of plus and minus half the respective phase difference; both phase differences can be thought of as positive quantities, without loss of generality. Assuming, for simplicity, that the slow axes of the mirror wave-plates are both aligned to the input polarisation, the polarisation auto-states of the Fabry–Perot cavity are given by two Airy denominators, one for each polarisation, in which the round-trip phase is shifted by plus and minus half the sum of the two mirror phase differences.

The above equations show that the resonance curves of the two polarisation modes are no longer centred at resonance, and are separated by the sum of the two mirror phase differences. In other words, the resonance frequencies of the two polarisations are slightly different.

In the PVLAS experiment, the emission frequency of the laser is locked to the resonance frequency of the cavity by means of a feedback electronic circuit based on the Pound and Drever locking scheme, in which the error signal is carried by the light reflected from the cavity through the input polariser. As a consequence, while the light having the input polarisation is at the top of the resonance curve, the orthogonal component is not. As the frequency width of the cavity is a few tens of hertz, for a frequency difference of this order of magnitude the orthogonal component may be filtered significantly. Hence, as a first issue, when analysing the extinguished beam one has to necessarily take into account the fact that its intensity is reduced with respect to the other polarisation by the factor of Equation 18: one over one plus the square of the amplification factor times the square of the sine of half the mirror phase difference, a quantity never greater than one.

By varying the input polarisation direction and the relative angular position of the two mirrors, it is possible to minimise the effect of the wave-plates of the mirrors by aligning the slow axis of one mirror against the fast axis of the other. This ensures that the two curves are as near as possible, in which case the effective phase difference is equal to the difference of the two mirror phase differences rather than their sum.

As a second issue, analogously to Equations 16 and 17, a symmetrical mixing appears between rotations and ellipticities. In fact, the electric field at the exit of the cavity acquires the two mirror wave-plates inside the resummation, and the intensity at the detector for small phase differences and reflectivity close to one is given by Equations 19 and 20, which have the same form as Equations 16 and 17 with the detuning replaced by the detuning minus half the mirror phase difference. It can be shown that any small static ellipticity or rotation acquired before or after the cavity does not interfere with the signal at twice the magnet rotation frequency and can thus be neglected.

If the laser is locked to the maximum value of the transmitted intensity, one has, for an ellipticity measurement, Equation 21: the intensity times the square of the modulation amplitude plus the modulation amplitude times the attenuation factor times the quantity twice the amplification times the ellipticity plus the square of the amplification times the rotation times the mirror phase difference, all times the sine of twice the angle. For a rotation measurement one has Equation 22, the same with ellipticity and rotation exchanged and the sign of the second term reversed.

With respect to Equations 13 and 14, the expected signals of ellipticity and rotation are attenuated by the factor of Equation 18. Moreover, a cross talk between the two measurement channels appears: even with no true rotation, a spurious rotation is observed. The ratio of the spurious rotation and of the true ellipticity is Equation 23: minus half the amplification factor times the mirror phase difference — hence allowing a direct determination of the sum of the birefringences of the two mirrors. Analogously, even with no true ellipticity, an apparent ellipticity appears. In the absence of both signals, an upper limit coming from the measurement of one of the two quantities, ellipticity or rotation, translates in an upper limit also on the other one.

2.3 Intrinsic noise of the polarimeter

We now calculate the limit sensitivity of the apparatus. Starting from Equation 10, if the noise at the lower sideband is uncorrelated to the noise at the upper sideband, one must take into account a factor of the square root of two due to the folding of the spectrum around the modulation frequency. If the root-mean-square noise spectral density of the light intensity at the signal frequency is given, the expected peak sensitivity of the polarimeter is that noise density divided by the transmitted intensity and by the modulation amplitude.

Several intrinsic effects contribute to the sensitivity, all of which can be expressed as a noise in the light intensity impinging on the detector. We consider first the intrinsic root-mean-square shot noise due to the direct current in the detector, the square root of twice the elementary charge times the direct current times the bandwidth. According to Equations 8 or 9, the direct current inside the photodiode is given by the detection efficiency times the transmitted intensity times half the square of the modulation amplitude. However, any pair of crossed polarising prisms has a nonzero minimum extinction coefficient for intensity. For the best polarisers, the extinction coefficient can be as low as ten to the minus eight. This effect introduces an additional term in the detected intensity, the transmitted intensity times the extinction coefficient. This leads to the shot noise intensity and the shot-noise-limited sensitivity given in the text.

Other effects contributing to the noise are the Johnson noise of the transimpedance of the photodiode, given by the square root of four times Boltzmann’s constant times the temperature divided by the square of the efficiency and the transimpedance; the photodiode dark noise, the dark current divided by the efficiency; and the relative intensity noise of the light emerging from the cavity, the transmitted intensity times the relative intensity noise spectral density. For the last of these we consider that the contributions of all the peaks in the Fourier spectrum add incoherently to the intensity noise at the modulation frequency, and that the magnet rotation frequency is far below the modulation frequency.

All the intrinsic contributions were evaluated as functions of the modulation amplitude in typical operating conditions, with an efficiency of about 0.7 amperes per watt, a transmitted intensity of 8 milliwatts, an extinction coefficient of 2 times ten to the minus seven, a transimpedance of a million, a dark current of 25 femtoamperes root-mean-square per root hertz, and a relative intensity noise of about 3 times ten to the minus seven per root hertz. The expected noise has a minimum for a modulation amplitude of about ten to the minus two, which is the value normally used.

3 Experimental setup

The experiment is hosted inside a class 10,000 clean room. All the optics lay upon a single 4.5 tonne granite honeycomb table measuring 4.8 by 1.5 by 0.5 metres. The optical table is seismically isolated from the ground by means of actively operated pneumatic supports. All the mechanical components of the apparatus are made of nonmagnetic materials.

The light source is a 2 watt Non Planar Ring Oscillator Nd:YAG laser at a wavelength of 1064 nanometres, having tuneable emission frequency. The tuning capabilities of the laser are used to lock the emission frequency of the laser to the resonance frequency of the cavity. Laser light is mode matched to the Fabry–Perot cavity with a single lens and is linearly polarised immediately before the first mirror. The cavity length is 3.303 metres, corresponding to a free spectral range of 45.4 megahertz. The dielectric mirrors, 6 millimetres thick and 25.4 millimetres in diameter, have fused silica substrates with a radius of curvature of minus 2 metres, and are mounted on three-axis mirror mounts. The Gaussian cavity mode is the fundamental transverse mode, with a beam radius on the mirrors of 1.2 millimetres. The decay time of the cavity has been measured to be 2.45 plus or minus 0.05 milliseconds, corresponding to a finesse of about 700,000, hence to a path amplification factor of 445,000, and to a reflection coefficient of 0.9999955. The frequency width of the resonance is 65 hertz, corresponding to a phase interval of less than ten to the minus five radians.

The laser frequency is matched to the resonance frequency of the cavity by means of a modified Pound–Drever–Hall feedback system. The electronic feedback circuit has the unique feature of allowing the adjustment of the reference point of the loop, equivalent to varying the detuning in Equations 19 and 20. This allows the scanning of the Airy curve of the intensity transmitted by the cavity around its maximum. The amplitude of this interval is in principle limited to the linear range of the error function, but is in practice slightly less. The feedback circuit parameters are controlled by a microprocessor that, in the case the feedback unlocks, re-locks automatically. In a measurement run lasting several days this normally results in a dead time of less then 5 percent.

After the cavity, the light crosses the photoelastic ellipticity modulator, that adds a small ellipticity variable at the modulation frequency. In the case of rotation measurements, the quarter-wave-plate is inserted. Finally, the light leaves the polarimeter through the analyser, that separates the two polarisations. The two beams are collected by two 1 square millimetre InGaAs photodiodes. The photocurrents are amplified by two low noise transimpedance amplifiers. The extinguished signal is demodulated by two lock-in amplifiers, at the modulation frequency and at its second harmonic. All the relevant signals are properly filtered, digitised, and stored for data analysis.

The magnetic field region is provided by two 94 centimetre long, 28 centimetre diameter dipole magnets in Halbach configuration, placed between the mirrors and having a central bore of 20 millimetres. Each magnet weighs 450 kilograms. The magnets are sustained by an aluminium structure mechanically decoupled from the rest of the optical table. Overall, the magnets provide an integral of the squared field along the path of 10.25 plus or minus 0.06 tesla squared metres. As for the effective length of each magnet and the value of the magnetic field, in the following we will use the full width at half maximum of the squared field profile, 0.82 metres, and hence a field of 2.50 tesla. The centres of the two magnetic regions are separated by about 150 centimetres. Stray field on the axis at a position 20 centimetres outside the magnets is less than 1 gauss. The magnets can rotate around their axes at a frequency up to 10 hertz, so that the magnetic field vectors of the two magnets rotate in planes normal to the path of the light stored in the cavity. Two magnetometers, measuring the small stray field of the two magnets, monitor the magnetic field directions.

The synchronous motors driving the two magnets are controlled by two phase-locked signal generators. The same signal generators trigger the data acquisition. The two magnets can rotate at the same frequency with the two magnetic fields making an arbitrary angle, but normally each magnet rotates at its own frequency. In this way the results of one magnet are a countercheck for the results of the other. The two rotation frequencies are chosen so to have a common subharmonic whose frequency is used to start data acquisition: at the beginning of each acquisition run, the two magnets have the fields in the same direction. The sampling rate is normally 16 samples per turn for the faster magnet. The rotation frequency of the other magnet is then chosen in such a way that its number of samples per turn contains only factors 2 and 5. A practical example: the first magnet at 8 hertz, sampling rate 128 hertz, the second magnet at 6.4 hertz, acquisition start trigger 1.6 hertz, giving 20 samples per turn for the second magnet. We have verified that the phase relations between all the generators and the magnets rotation never change during data acquisition.

Two analyses are performed in parallel on the intensity collected by the extinction diode. In both cases, this signal is first demodulated for the modulation frequency and then the ellipticity or the rotation is calculated through Equation 10 by using the values of the intensity measured by the transmission diode and of the modulation amplitude determined from the component at twice the modulation frequency. An online analysis is performed by means of a fast-Fourier-transform spectrum analyser. Normally, an integration time of 32 seconds is chosen and vector averaging is performed between subsequent spectra. The start trigger ensures that the phases of all the partial spectra are referred to the same angular position of the magnets. This analysis produces visual results in real time, but is not fully exploiting one of the main advantages of the experimental method, namely the frequency selection. In the offline analysis, since all the phases are under control, data acquired in separate time blocks, but with the same experimental conditions, are joined in a single long time series called run. As the time base lengthens, the frequency resolution of the Fourier transform becomes better and better. When doing this, one has to ensure that the magnet-rotation component of the Fourier transform of the signal from the magnetometer occupies a single frequency bin. This was verified to be true even for the longest runs, having a bin size of about a microhertz. Time intervals containing anomalous features are expunged from the data. The results of runs differing in the rotation frequency of the magnets or for any other relevant experimental parameter are averaged by using a weighted vector average procedure.

The polarimeter, from the entrance polariser to the analyser, is housed inside a high-vacuum enclosure consisting of five chambers aligned along the light beam path and connected by metallic bellows and by two glass tubes with 12 millimetre inner diameter traversing the bores of the two magnets. The entrance chamber hosts the polariser, whereas the exit chamber contains the quarter-wave-plate, the photoelastic modulator, and the analyser. Each mirror is placed inside a separate chamber, preceded and followed by 10 millimetre diameter iris diaphragms carved from strongly absorbing glass. The light enters and exits the vacuum through two anti-reflection-coated optical glass windows. A system of baffles is placed inside the glass tubes. The central vacuum chamber serves as a pumping station and also contains a central 5 millimetre diameter diaphragm.

The vacuum system is pumped by turbo-molecular and non-evaporable getter pumps, and has a base pressure of less than ten to the minus seven millibar; the residual atmosphere, monitored by two Residual Gas Analysers, is mainly composed of water vapour, hydrogen and a small amount of methane produced by the getter pumps. This guarantees that no magnetic birefringence signal from the Cotton–Mouton effect on residual gases in the vacuum chamber can interfere with the vacuum measurements. To reduce mechanical vibrations, during measurements in vacuum, only the turbo pump of the central chamber is kept on to pump methane produced by the getter pumps and the noble gases. The system can be filled with high purity gases through a leak valve; in this case, the gas pressure is measured with a capacitive transducer. To ensure gas purity, the all-metal gas line is pumped by a turbo pump before gas filling. When the chamber is dosed with noble gases, the getter pumps are not shut off.

3.1 Calibration

The apparatus is calibrated measuring the magnetic linear birefringence of gases, the Cotton–Mouton or Voigt effect. This effect is perfectly analogous to the vacuum magnetic birefringence described by Equation 3, but is far more intense already at low gas pressures. The birefringence generated in an atmosphere of gas at a given pressure by a magnetic field is the unit birefringence times the square of the field in tesla times the pressure in atmospheres, where the unit birefringence is the value generated in one atmosphere of gas by a field of one tesla. Typical values of the unit birefringence range from a minimum of about 2 times ten to the minus sixteen per tesla squared per atmosphere for helium, to about minus 2.3 times ten to the minus twelve for oxygen, and to about ten to the minus eleven for a few other simple molecules.

These measurements give two calibration parameters: the amplitude and the phase of the ellipticity signal. The amplitude can be compared to theoretical calculations as well as to other experimental results, and calibrates the linear response of the polarimeter; the second parameter is the phase of the ellipticity signal, which is determined by the geometry and the electronic response of the apparatus. The phase of the signal directly depends on the angle of the polariser; this parameter has not a single value during the experiment, but is adjusted from time to time. Electronic components such as lock-ins and filters introduce a phase which depends on the frequency of the signal. The phase of the Cotton–Mouton signals defines what we call the physical phase of the measurements; we expect that the vacuum magnetic birefringence comes with the same phase as the Cotton–Mouton measurement of the noble gases. Any signal in quadrature with respect to the physical phase has to be considered as spurious. As a general principle, all the measured signals are projected onto the physical axis. We explicitly note that the gas measurements are interpreted in terms of a pure birefringence. In fact, for gases, no dichroism is associated to a transverse magnetic field; however, a Faraday rotation, due to the time variation of an eventual small longitudinal component of the rotating magnetic field at the position of the mirrors, comes at the magnet rotation frequency and not at twice that frequency.

With the vacuum chamber filled with 230 microbar of argon gas, the Cotton–Mouton ellipticity signal is observed in the demodulated spectrum of the residual intensity after the analyser, and so is a rotation signal. This indicates that the Fabry–Perot resonances of the two orthogonal polarisations are separated, and the calculations of Sect. 2.2 apply. Taking the ratio of the amplitudes of the two peaks, using Equations 21 and 22, one finds a mirror phase difference of 3.7 microradians, corresponding to an attenuation factor of 0.59. The frequency distance of the two Airy curves is 27 hertz. From these data one can extract a value for the unitary birefringence of argon gas at room temperature of 7.5 plus or minus 0.5 times ten to the minus fifteen per tesla squared per atmosphere.

3.2 Studies of the mirrors’ wave-plates

In Sect. 2.2 we assumed that the axes of the birefringent wave-plates of the two mirrors were always aligned to the input polarisation. Here we use a full description of the wave-plates of the two mirrors, placing the second one at an azimuthal angle with respect to the first one. We recall that the effect of two birefringent wave-plates is equivalent to that of a single wave-plate with a phase difference given by Equation 24: the square root of the sum of the square of the difference of the two mirror phase differences and four times their product times the square of the cosine of the azimuthal angle between them. It is placed at an angle with respect to the slow axis of the first mirror given by Equation 25: the cosine of twice that angle equals the ratio of the two mirror phase differences plus the cosine of twice the azimuthal angle, all divided by the square root of the sum of the square of that ratio minus one and four times the ratio times the square of the cosine of the azimuthal angle.

As noted before, the ratio of Equation 23 is exactly the phase difference of the equivalent wave-plate of the mirrors, amplified by minus half the amplification factor. By varying two of the three quantities — the direction of the mirror axes and the input polarisation direction — one is able to change the phase difference of the equivalent wave-plate of the mirrors while keeping the polarimeter at extinction, namely with the input polarisation aligned with the axis of the equivalent wave-plate. As this procedure changes the equivalent wave-plate, it also changes the ratio of rotation to ellipticity. One is then able to align the fast axis of one mirror wave-plate to the slow axis of the other. In this configuration, if the two mirror phase differences were equal, the resonance curves of the two polarisation auto-states would appear superimposed. If they are unequal, the two resonance curves are as near as possible given their difference.

The ratio of the values of rotation to ellipticity in a Cotton–Mouton measurement was plotted as a function of the azimuthal angle of the first mirror. Each rotation step, of about 15 degrees, has been followed by cavity realignment through the adjustment of the two tilt stages of the mirror, by optimisation and measurement of the extinction ratio, and by measurement of the finesse. The experimental points are fitted with Equation 23, where the phase difference is given by Equation 24. The best fit produces values for the two amplified mirror phase differences and for the angular position of the maxima with respect to the initial angular position of the input mirror. With half the amplification factor equal to about 2.2 times ten to the fifth, the phase differences of the two mirrors are calculated to be 2.4 plus or minus 0.1 microradians and 1.9 plus or minus 0.1 microradians. From this fit only it is not possible to label each mirror with its phase difference for reflection. According to the relative angular position of the two mirrors, the value of the equivalent phase difference can be found between 0.6 microradians and 4.3 microradians, which is equivalent to saying that the Airy curve of the ellipticity resonance is 5 to 31 hertz away from the resonance of the input polarisation.

The values taken by the polariser angle while tracking the best extinction ratio in the process described above were plotted against the input mirror angle and fitted with Equation 25. The best fit produces a ratio of the two mirror phase differences of 0.62 plus or minus 0.08, allowing the assignment of the phase delay of each mirror. This value is slightly different from the one obtained by the previous fit, but is compatible within the fit uncertainties. However, the zero references of the azimuthal angle in the two fits appear to be different by about 10 degrees, well beyond the fit uncertainty. This might be due to the presence of other birefringent elements — the mirror substrates and the photoelastic modulator — between the two crossed polarisers. As these elements are fixed during the measurement, while the equivalent wave-plate of the mirrors is varying, their contribution to the total anisotropy varies from one measurement to the other. The position of the polariser tracks the position of the equivalent wave-plate of all the wave-plates of the system, and not only of that of the mirrors. On the contrary, the ratio data, being the ratio of signals at twice the magnet rotation frequency, do not suffer from the same problem. Anyway, the smallness of the difference of the two determinations of the reference angle indicates that the importance of birefringent elements other than the reflecting surface of the mirrors is very limited.

A unique feature of our apparatus is the possibility of varying the set point of the feedback electronic circuit that locks the laser frequency to the resonance frequency of the cavity. This allows to perform polarimetric measurements with arbitrary values of the detuning, in this way fully testing the mathematics presented in Sect. 2.2. Measuring the ellipticity, the transmitted intensity and the rotation for 230 microbar of argon as functions of the feedback set point gives an experimental realisation of the predicted three-curve pattern. The continuous lines are the fits of the data obtained with Equations 15, 21 and 22. In the three fits, a single value of the resonance width has been used. Ellipticity and rotation curves are forced to have the same centre of resonance and the same amplitude coefficient. The fit determines the scale factor between the feedback set point and the detuning phase. The distance between the two Airy curves is found to be 1.5 microradians, with negative sign, corresponding to a frequency difference of the two resonance frequencies of about 11 hertz.

4 Vacuum measurements results and discussion

In this section we present the polarimetric measurements carried out on vacuum in the attempt to test its opto-magnetic properties. Differently from what we used to do before, we now normally rotate the two magnets at different frequencies, as a strategy to beat the systematics. Two of the ellipticity runs make use of the same data, analysed at two different frequencies corresponding to the second harmonic of the rotation frequencies of the two magnets; the same holds for another pair. Since the measurements have been taken making use of a birefringent cavity, the ellipticity data can be interpreted also in terms of rotation; the converse is also true. The integrated noise level in the ellipticity measurement allows to cast upper limits on the magnetic birefringence predicted by QED, and also on the existence of hypothetical particles coupling to two photons, axion-like and milli-charged. The runs listed run at signal frequencies from 8 to 12.5 hertz with integration times from 1.4 times ten to the fifth to ten to the sixth seconds, at a finesse of 7.0 times ten to the fifth and an attenuation factor of 0.65, together with an earlier run taken at a finesse of 6.7 times ten to the fifth and an attenuation factor of 0.50.

Two ellipticity runs, one at a magnet rotation frequency of 4 hertz with an integration time of ten to the sixth seconds and one at 6.25 hertz with an integration time of 8.9 times ten to the fifth seconds, have been discarded due to the presence of spurious structures in the Fourier transform of the signals around the signal frequency. In fact, a signal coming from a magnetic birefringence cannot occupy more than a single bin. These structures are the consequence of a misalignment of the glass tubes traversing the rotating magnets. We have developed an alignment procedure for the tubes that prevents the appearance of systematic peaks in the spectra, but this does not prevent a small drift of their positions during the long runs.

For each run, the amplitudes of the complex Fourier transform of the signal in a narrow interval around the signal frequency show the absence of any structure due to spurious signals. The values at the signal frequency, projected along the physical axis, represent the results of the measurement. The histograms of the ellipticity noise amplitude values are fitted with the Rayleigh distribution of a two-dimensional variable whose two components are independent Gaussian variables with the same standard deviation. In our case, those two components are the projections of the complex Fourier components of the signal onto the physical and the quadrature axes. The values obtained for the standard deviation define the noise level of the measurement for a given integration time. The measured sensitivity of the apparatus at the frequency of interest is then the square root of the integration time times that standard deviation.

Each measurement line can be interpreted also in terms of the reciprocal quantity, because of the mixing of ellipticity and rotation, using Equations 21 and 22. The ellipticity lines give four determinations of the magnetic birefringence of vacuum; as many determinations of the dichroism are given by the rotation lines. The measured ellipticities and rotations lie in the range of a few times ten to the minus eleven to a few times ten to the minus ten in phase, with standard deviations from about 5 times ten to the minus ten to 2.6 times ten to the minus nine, and measured sensitivities from about 3 times ten to the minus seven to 2.1 times ten to the minus six per root hertz. The weighted averages of the in-phase determinations are Equation 26, a vacuum magnetic birefringence of minus 1.5 plus or minus 3.0 times ten to the minus twenty-two at a field of 2.5 tesla, and Equation 27, a vacuum magnetic dichroism of minus 1.6 plus or minus 3.5 times ten to the minus twenty-two at the same field.

One must note that the measured ellipticity and rotation are intrinsically integral quantities. As a consequence, the values of the birefringence and dichroism are not point functions, but average quantities. Moreover, they are calculated with the length of the magnets defined for convenience as the full width at half maximum of the squared field profile. Hence, they have a precise meaning only in the cases in which their expression is proportional to the square of the external field, namely the QED vacuum and the birefringence of axion-like and milli-charged particles in the limit of large mass.

The quadrature value of the birefringence results to be plus 5.2 plus or minus 3.2 times ten to the minus twenty-two. All the numbers found are compatible with zero. The value of the measured birefringence is an order of magnitude larger than the birefringence predicted by QED in Equation 4 and serves only as an upper limit.

The time evolution of the measurement of the QED magnetic birefringence of vacuum, with the measured values normalised to the square of the external field so that different experiments can be compared, runs from the BFRT experiment through Legnaro, the Ferrara test setup, BMV and PVLAS 2014 to the present result. By extrapolation, one could predict that it should not take too long before the measurement is performed successfully. Anyway, this will not happen if the sensitivity of the polarimeter will not improve by an order of magnitude. The next section briefly discusses the noise issue.

4.1 Noise considerations

The values found for the sensitivity of the polarimeter are a factor four better than the values obtained in previous versions of the experiment, but are still far from the theoretical value of 6 times ten to the minus nine per root hertz that is computed by adding all the known noise sources. With respect to the 2014 version of the experiment, a few minor changes have been made: the input polariser was substituted and a few iris diaphragms have been inserted along the beam. It is not clear which of these changes determined the improvement.

It is not clear either which could be the sources of the excess noise. A few things are known, though: first of all, the noise comes from the cavity; in fact, when the mirrors are removed, the polarimeter performance is limited only by intrinsic noise; this would exclude the laser as a source of noise. Since we are talking of noise in ellipticity and rotation, one must find a mechanism that produces noise in these two quantities.

A possible source of noise is the intrinsic birefringence of the mirrors. One could imagine a few mechanisms for a wide band modulation of this parameter. One of them could be mechanical movement of the mirrors induced by seismic noise: as the surface of such mirrors has a birefringence pattern both in amplitude and in axis direction, one could imagine that environmental mechanical noise moves the beam spot on the surface of the mirror, modulating the birefringence in a wide frequency range. However, this mechanism can be excluded: the amplitude of the ellipticity signal generated by forcing the optical bench to oscillate at a single frequency with known amplitude was measured and compared to the observed mechanical noise floor at the signal frequency. The measurement was repeated for the three spatial directions; in all cases the observed noise floor was found much too weak to account for the observed sensitivity of the polarimeter. Moreover, no improvement of the sensitivity was observed when the polarimeter was running in the quietest situations, during nights, with air conditioning switched off, and so on.

Considering again the intrinsic birefringence, another mechanism that could be invoked to explain the sensitivity is the insufficient thermal stability of the mirrors. This mechanism would imply a dependence of the sensitivity upon the light power inside the cavity. Such a dependence is observed only for frequencies below about 1 hertz. Nonetheless, we are planning to cool the mirrors down to the liquid nitrogen temperature.

A notable aspect of the observed noise is that it is quite independent from the value of the attenuation coefficient, as was observed during the rotation of the mirrors reported in the previous section. This seems to indicate that the noise may originate from diffused light inside the polarimeter and have nothing to do with intrinsic birefringence of the mirrors. However, the system of optical baffles and diaphragms that was installed along the beam path was able to get rid of the spurious signals at the signal frequency that haunted the measurements in the past, but seems not to have benefited the wide band noise. Further studies are ongoing.

4.2 Limits on hypothetical particles

The measurements of ellipticity and rotation can be used to draw an exclusion plot in the plane of mass and coupling for axion-like particles. One must note, however, that it is not possible to average together measurements taken with different magnet lengths, as Equation 5 shows. The best limits we can provide derive from the ellipticity measurements taken with one rotating magnet and from the rotation measurements taken with two magnets. The limits hold for both scalar and pseudoscalar axion-like particles. Below 0.5 milli-electron-volts, the most stringent results are given by a recent measurement by the OSQAR experiment, whereas our ellipticity measurement dominates the region above one milli-electron-volt. Between these two values, our rotation measurement almost coincides with the 2010 ALPS result. One must obviously remind that the whole region has already been excluded by the CAST solar helioscope down to a coupling of about ten to the minus ten per giga-electron-volt. The interest for the laboratory experiments resides in the fact that their results are model independent.

For milli-charged particles, two independent limits are derived from the birefringence and the dichroism measurements of Equations 26 and 27, the latter being more stringent in the low-mass range, below about 0.1 electron-volts, whereas the former is dominating the high-mass range. We explicitly note that the fermion exclusion plot applies also to all types of neutrinos, limiting their charge to be less than about 3 times ten to the minus eight of the elementary charge for mass smaller than 10 milli-electron-volts.

5 Conclusions

We have presented a detailed report of the status of the PVLAS experiment, which strives to push further the frontier of the opto-magnetic polarimetry of small signals. As for the magnetic birefringence of vacuum, the new measurements are approaching the goal of the experiment. The measurements have given new limits also on the existence of hypothetical particles which couple to two photons, both axion-like and milli-charged. The sensitivity, although improved with respect to the past, has not yet reached the level that would guarantee the capability to perform the measurement in a reasonable time. The challenge of the experiment is now to lower the wide band noise. A few tests are ongoing, which should reduce the noise or at least shade light on its nature. Among them, we plan to rotate the magnets faster to reduce the incidence of the one-over-frequency noise, to further reduce the scattered light, to search for mirrors with even higher reflectivity and lower losses and with smaller intrinsic birefringence, and to test the possibility of significantly lowering the temperature of the mirrors.

Acknowledgments

We gratefully acknowledge the invaluable technical help and infinite patience of Luca Landi from the University of Ferrara.

(The fifteen figures, the three data tables and the forty-five-item reference list are omitted for length; the complete text is at the source. The collaboration’s twenty-five-year review of the whole programme is on this site at /library/stm-08601764f6.)