- Email: [email protected]
A precise measurement of the Cotton–Mouton effect in neon
Chemical Physics Letters 410 (2005) 288–292 www.elsevier.com/locate/cplett
A precise measurement of the Cotton–Mouton effect in neon M. Bregant a, G. Cantatore a, S. Carusotto b, R. Cimino c, F. Della Valle a, G. Di Domenico d, U. Gastaldi e, M. Karuza a, E. Milotti f, E. Polacco b, G. Ruoso E. Zavattini a, G. Zavattini d (PVLAS Collaboration)
e,* ,
a
Dipartimento di Fisica and INFN, Sez. di Trieste, Via Valerio, 2, 34127 Trieste, Italy Dipartimento di Fisica and INFN, Sez. di Pisa, Largo Pontecorvo, 3, 56126 Pisa, Italy c INFN, Laboratori Nazionali di Frascati, Via E. Fermi, 40, 00044 Frascati, Italy d Dipartimento di Fisica and INFN, Sez. di Ferrara, Via Paradiso, 12, 44100 Ferrara, Italy e INFN, Laboratori Nazionali di Legnaro, Viale dellÕUniversita`, 2, 35020 Legnaro, Italy Dipartimento di Fisica di Udine and INFN, Sez. di Trieste, Via Valerio, 2, 34127 Trieste, Italy b
f
Received 15 March 2005; in final form 13 May 2005 Available online 16 June 2005
Abstract In this Letter, we report a novel measurement of the magnetically induced birefringence (Cotton–Mouton effect) in neon. Using a highly sensitive apparatus we were able to precisely measure the specific birefringence value of Dnu = (5.9 ± 0.2) · 1016 at the wavelength of 1064 nm (for B = 1 T and atmospheric pressure) and T 290 K. The results reported here are in agreement with theory, while the only previous precise measurement differs significantly. 2005 Elsevier B.V. All rights reserved.
1. Introduction Linearly polarised light passing through a medium in the presence of a transverse magnetic field acquires an elliptical polarisation. This is known as the Cotton– Mouton effect (CME) since it was first investigated in detail by Cotton and Mouton [1]. The CME in gases is very small and systematic measurements have been performed only during the last decades [2]. The CME of neon has been measured precisely by Cameron et al. [3] in 1991 at k = 514.5 nm, and recently by Muroo et al. [4], but with a large error. Theoretical values have been calculated by Jaszunsky et al. [5], by Bishop and Cybulsky [6] and very recently by Rizzo et al. [7]. They show a large discrepancy from the value measured in [3].
*
Corresponding author. Fax: +39 049 641925. E-mail address: [email protected] (G. Ruoso).
0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.05.087
On the other hand, the measurement of [4] agrees with the calculated values, but the precision is rather poor. In this Letter, we report a novel precise measurements of the CME of neon at k = 1064 nm and T 290 K. Routine measurements of CME in gases, typically nitrogen, are normally performed to test the apparatus of the PVLAS experiment [8]. This experiment is searching for the small anisotropy induced on the vacuum by an external dipole magnetic field, as predicted by Quantum Electrodynamics (vacuum magnetic birefringence). In the PVLAS apparatus a superconducting dipole magnet is coupled to a very sensitive ellipsometer, and a high finesse Fabry–Perot cavity is used to increase the optical path in the magnetic field region. A peculiar feature of this set-up is that, in order to apply the heterodyne detection scheme, the whole magnet rotates around a vertical axis coincident with the optical axis. Recently, we also performed a measurement of the CME in krypton and xenon [9].
M. Bregant et al. / Chemical Physics Letters 410 (2005) 288–292
2. Experimental The apparatus and its layout are discussed in greater detail in [8], thus only a short description will be given here. Fig. 1 shows a schematic drawing of the experimental set-up. A linearly polarised 100 mW laser beam (k = 1064 nm) is frequency locked, using a modified Pound– Drever–Hall technique [10], to a very high finesse Fabry–Perot cavity. The cavity is placed in the warm bore of a cryostat containing a superconducting dipole magnet which provides a magnetic field up to 6 T over a length of 1 m [11,12]. A pair of crossed Glan–Laser polarisers are used to detect small changes in the polarisation state of the light beam after passing through the cavity. To enhance the sensitivity of the system, the magnetic field harmonically changes its direction, driven by a rotation of the cryostat holding the magnet, and a polarisation (ellipticity) modulator is used to effect heterodyne detection. This modulator is an in-house developed device [13] based on the stress birefringence of a BK7 glass window. It gives a controlled ellipticity of the order of 3 · 103, with a working frequency which can be as high as 1 kHz. The magnet rotation frequency is around 0.3 Hz, producing a beat signal with the much higher frequency of the polarization modulator. Polarisers, modulator and cavity are placed inside a high vacuum chamber, normally evacuated down to pressures p < 107 mbar when looking for magnetic vacuum birefringence. The chamber volume can be filled with gases at controlled pressure values from a fraction of a millibar
289
up to atmospheric pressure, measured by a set of capacitive transducers with accuracy <1%. A residual gas analyser (RGA) measures the composition of the residual gas inside the chamber when the pumping system is set off. This RGA has a Channeltron and a Faraday cup as detectors, and can be used for pressures up to 105 mbar. The extincted beam exiting from the second polariser is collected onto a InGaAs photodiode and the resulting photocurrent is then amplified using a low noise current amplifier. Its output is frequency downconverted using a lock-in amplifier which has as a reference oscillator the driving signal of the polarisation modulator. The output low frequency signal is sampled, by using a computer controlled slow ADC, at time intervals corresponding to 11.25 of rotation of the cryostat (32 equally spaced tags on the circumference of the rotating table that supports the cryostat, for a sampling frequency 10 Hz). A fast ADC [14] (sampling frequency 8.2 kHz) is also used to sample the direct signal from the amplifier. Data are saved for offline analysis. The ellipticity w induced on the beam by the Cotton– Mouton effect is related to the anisotropy of the index of refraction Dn = ni n^, where the i and ^ axes are defined with respect to the magnetic field direction. Following [2], we define a birefringence for unit field (1 T) and atmospheric pressure patm as 2 1 patm Dnu ¼ Dn ; ð1Þ B½T p where p is the pressure and B the magnetic field amplitude in Tesla. We have then 0L 1 Z F p w ¼ 2 sin 2h@ B2 ½T dzADnu ; ð2Þ k patm 0
Fig. 1. Schematic drawing of the apparatus: M1, M2, Fabry–Perot cavity mirrors; P1, P2, crossed polarisers; PM, polarisation modulator; RGA, residual gas analyser. The magnet rotates around the cavity axis.
where we have already taken into account the 2F/p amplification factor resulting from a cavity with finesse F (see [2]). The angle h is time-dependent and it is measured between the initial polarisation of the light and the magnetic field B, which lies in the plane perpendicular to the light propagation direction (dipole field). L is the length of the magnet bore. Due to the dependence of the ellipticity on sin 2h, the low frequency ADC signal has a spectral component at twice the magnet rotation frequency Xm, whose amplitude is used to measure w. Since the data sampling frequency is an integer multiple of the magnet rotation frequency, the relevant peak in the spectrum of the sampled signal does not suffer from spectral leakage and is contained in a single frequency bin (see Fig. 2). This frequency has the numeric value of 2 if measured in units of magnet rotation frequency. It is also possible to measure the relative sign of the ellipticity, since acquisition is triggered so that its starting time always corresponds to the same angular position of the rotating magnet, and absolute phase information can also be extracted.
290
M. Bregant et al. / Chemical Physics Letters 410 (2005) 288–292 10
Ellipticity
10
10
10
10
10
-4
-5
-6
-7
-8
-9
0
2
4 6 8 Frequency (units of magnet rotation frequency)
10
Fig. 2. Ellipticity spectrum of the demodulated photodiode signal with 8 mbar of neon and 4.53 T magnetic field. The peak amplitude at 2 units of magnet rotation frequency is 1.12 · 105. The peak at frequency 1 is probably due to the mechanical action of the stray magnetic field on the optical elements. No correlation has been found with the peak at frequency 2. In the spectrum the 1/f noise has been suppressed during the data analysis by using a software filter.
The finesse F in each run is deduced from the measurement of the cavity decay time s, from the relation pcs ; ð3Þ F ¼ d where c is the speed of light and d = 6.4 m is the cavity length. The measurements are repeated for each pressure of neon inside the chamber. Typical s values are around 400 ls, with an average absolute error of 10 ls. The temperature, measured with a probe placed on the outside of the vacuum pipe housing the high finesse cavity, is 289 K. Due to the presence of a localised cold source, given by the cryostat, a temperature gradient exists in the vacuum chamber containing the neon gas. Rough measurements indicate gas temperature values ranging between 287 and 291 K. The Cotton–Mouton anisotropy is dependent on the absolute temperature T: in particular a 1/T dependence is expected for monoatomic gases [2]. Uncertainties in the temperature measurements will not be considered in the global error reported in this Letter, but should be taken into account when comparing our results with other experiments. The magnet is energised at a given field intensity at the beginning of each measurement session and subsequently disconnected from the power supply to allow rotation. Due to the residual ohmic resistance of the high current switch, a current decay is observed. This is monitored using Hall probes and for each data taking run the effective field is considered. The error in the determination of the magnetic field is 0.075 T for field values around 5 T. Measurements have been performed in several runs and with different pressures. The neon bottle we used
was of type 5.0 (purity better than 99.999%). The chamber and the gas line were always evacuated to high vacuum before filling them with the desired gas. The chamber itself has a low static pressure: with no pumping the pressure is below 105 mbar for at least an hour, which is the time necessary to perform the CME measurement. Using the RGA it was also possible to measure that the main components of the residual gas are hydrogen and water vapour. A pressure of neon in the mbar range was normally sufficient to obtain a signal to noise ratio in ellipticity much larger than 100.
3. Results Table 1 lists the different data taking runs performed. Quantities of physical interest were obtained Table 1 The PVLAS data taking runs with neon discussed in this Letter Progressive #
Pressure (mbar)
Cavity decay time s (ls)
Magnetic field (T)
Measured ellipticity (·106)
1 2 3 4 5 6 7 8 9 10
5 5 5 10 15 20 7 10 3 8
375 375 375 375 375 375 420 420 450 440
4.80 4.80 4.80 4.77 4.74 4.73 4.96 4.77 4.77 4.53
7.17 6.88 6.87 13.1 19.6 26.0 11.0 14.6 4.47 11.2
The values of the finesse and of the magnetic field are measured for each run.
M. Bregant et al. / Chemical Physics Letters 410 (2005) 288–292 1.4 10
Reduced birefringence (mbar)
1.2 10
1 10
8 10
6 10
4 10
2 10
291
-14
-14
-14
-15
-15
-15
-15
0 0
5
10
15
20
25
Pressure (mbar) Fig. 3. Reduced birefringence values obtained as a function of pressure. The error bars are obtained by applying standard propagation formula to Eq. (5). The superimposed solid line is a least-squares fit to a linear function (see text).
from a Fourier analysis of the ADC sampled signal (see Fig. 1). For each run i we calculated the ellipticity wi, with its phase ui. To compare the different data sets it is necessary to transform each value into the reduced birefringence Dnri ¼ Dnu pi ;
ð4Þ
which is expressed for a 1 T magnetic field, using Eq. (3) for F: Dnri ¼ wi
k d patm ; 2p si c B2i ½TL
ð5Þ
where k = 1064 nm is the light wavelength, d is the cavity length and c the speed of light. For the field we have considered an effective length L with a mean field B: these values were obtained from a numerical model of the magnet (calculated using the finite elements software package TOSCA [15]) which has been verified by direct measurement of the magnetic field. By plotting each value Dnri against pressure one can fit the data to a linear function, where the angular coefficient is the birefringence Dnu at normal pressure as defined in Eq. (1). The values wi and their associated statistical uncertainties are obtained from a spectral analysis of the recorded fast ADC samples [14]: a typical spectrum of the demodulated photodiode signal, obtained for 8 mbar of pure neon in the chamber, can be seen in Fig. 2. This corresponds to an ellipticity w = 1.12 · 105, i.e., Dn = 9.2 · 1017 with a signal to noise ratio of the order of 103, for an integration time of 360 s. This shows the very high sensitivity of the PVLAS apparatus.
Fig. 3 shows a plot of the reduced birefringence versus pressure for all the runs, together with the best fit of a linear function of the type Dnr = a + Dnup. A leastsquares analysis gives for the angular coefficient the value Dnu = (5.9 ± 0.2) · 1016. The intercept at zero pressure is compatible with zero, being a = (6 ± 9) · 1017. The reduced v2, v2/n has a value of 0.85, corresponding to a probability of 64%. From the resulting phase of the signal (all ui values are equal within ±1), knowing the direction of the polarization of the light with respect to the direction of the magnetic field, we determined the sign of the effect to be positive.
4. Conclusions We summarize in Table 2 the different results available for the CME of Ne.
Table 2 Comparison between experimental data and theoretical calculations for Dnu Experiments This work (k = 1064 nm, T . 290 K) Ref. [3] (k = 514.5 nm, T = 298.15 K) Ref. [4] (k = 790 nm, T = 285 K)
5.9 ± 0.2 2.83 ± 0.15 6.9 ± 2.7
Theory Ref. [5] Ref. [6] Ref. [6] Ref. [7]
6.04 6.869 6.858 6.54 ± 0.23
(k = 514.5 nm, T = 273.15 K) (k = 514.5 nm, T = 273.15 K) (k = 632.8 nm, T = 273.15 K) (k = 1, T = 273.15 K)
All values are multiplied by 1016.
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M. Bregant et al. / Chemical Physics Letters 410 (2005) 288–292
The theoretical values of Dnu are reported following Rizzo et al. [2]. The values of [5] use the quadratic response method, those of [6] use perturbation theory and finite magnetic field method and those of [7] use highly correlated coupled cluster wave functions.
Acknowledgements We are grateful to all the people at the Laboratori Nazionali di Legnaro who helped us in building the apparatus. In particular we thank Selvino Marigo, of INFN Legnaro, and Aldo Zanetti, of INFN Trieste, for the precious collaboration also during data taking.
References [1] A. Cotton, H. Mouton, Ct. r. hebd. Seanc. Acad. Sci., Paris 141 (1905) 317. [2] C. Rizzo, A. Rizzo, D.M. Bishop, Int. Rev. Phys. Chem. 16 (1997) 81. [3] R. Cameron, G. Cantatore, A.C. Melissinos, J. Rogers, Y. Semertzidis, H. Halama, A. Prodell, F.A. Nezrick, C. Rizzo, E. Zavattini, J. Opt. Soc. Am. B 8 (1991) 520.
[4] K. Muroo, N. Ninomiya, M. Yoshino, Y. Takubo, J. Opt. Soc. Am. B 20 (2003) 2249. [5] M. Jaszunski, H.J.A. Jensen, P. Jorgensen, A. Rizzo, T. Helgaker, K. Ruud, Chem. Phys. Lett. 191 (1992) 599. [6] D.M. Bishop, S.M. Cybulski, Chem. Phys. Lett. 200 (1992) 153. [7] A. Rizzo, M. Kallay, J. Gauss, F. Pawlowski, P. Jorgensen, C. Hattig, J. Chem. Phys. 121 (2004) 9461. [8] D. Bakalov, F. Brandi, G. Cantatore, G. Carugno, S. Carusotto, F. Della Valle, A.M. De Riva, U. Gastaldi, E. Iacopini, P. Micossi, E. Milotti, R. Onofrio, R. Pengo, F. Perrone, G. Petrucci, E. Polacco, C. Rizzo, G. Ruoso, E. Zavattini, G. Zavattini, Quant. Semiclass. Opt. 10 (1998) 239. [9] M. Bregant, G. Cantatore, S. Carusotto, R. Cimino, F. Della Valle, G. Di Domenico, U. Gastaldi, M. Karuza, E. Milotti, E. Polacco, G. Ruoso, E. Zavattini, G. Zavattini, Chem. Phys. Lett. 392 (2004) 276. [10] G. Cantatore, F. Della Valle, E. Milotti, P. Pace, E. Zavattini, E. Polacco, F. Perrone, C. Rizzo, G. Zavattini, G. Ruoso, Rev. Sci. Instrum. 66 (1995) 2785. [11] M. Morpurgo, CERN Bulletin 28 (12 July 1982). [12] D. Bakalov, F. Brandi, G. Cantatore, G. Carugno, S. Carusotto, F. Della Valle, A.M. De Riva, U. Gastaldi, E. Iacopini, P. Micossi, E. Milotti, R. Onofrio, R. Pengo, F. Perrone, G. Petrucci, E. Polacco, C. Rizzo, G. Ruoso, E. Zavattini, G. Zavattini, Hyperfine Int. 114 (1998) 103. [13] F. Brandi, E. Polacco, G. Ruoso, Meas. Sci. Technol. 12 (2001) 1503. [14] E. Milotti, J. Comput. Phys. 202 (2005) 134. [15] Developed by Vector Fields Ltd., 24 Bankside Kidlington, Oxford OX5 1JE, UK.
A precise measurement of the Cotton–Mouton effect in neon M. Bregant a, G. Cantatore a, S. Carusotto b, R. Cimino c, F. Della Valle a, G. Di Domenico d, U. Gastaldi e, M. Karuza a, E. Milotti f, E. Polacco b, G. Ruoso E. Zavattini a, G. Zavattini d (PVLAS Collaboration)
e,* ,
a
Dipartimento di Fisica and INFN, Sez. di Trieste, Via Valerio, 2, 34127 Trieste, Italy Dipartimento di Fisica and INFN, Sez. di Pisa, Largo Pontecorvo, 3, 56126 Pisa, Italy c INFN, Laboratori Nazionali di Frascati, Via E. Fermi, 40, 00044 Frascati, Italy d Dipartimento di Fisica and INFN, Sez. di Ferrara, Via Paradiso, 12, 44100 Ferrara, Italy e INFN, Laboratori Nazionali di Legnaro, Viale dellÕUniversita`, 2, 35020 Legnaro, Italy Dipartimento di Fisica di Udine and INFN, Sez. di Trieste, Via Valerio, 2, 34127 Trieste, Italy b
f
Received 15 March 2005; in final form 13 May 2005 Available online 16 June 2005
Abstract In this Letter, we report a novel measurement of the magnetically induced birefringence (Cotton–Mouton effect) in neon. Using a highly sensitive apparatus we were able to precisely measure the specific birefringence value of Dnu = (5.9 ± 0.2) · 1016 at the wavelength of 1064 nm (for B = 1 T and atmospheric pressure) and T 290 K. The results reported here are in agreement with theory, while the only previous precise measurement differs significantly. 2005 Elsevier B.V. All rights reserved.
1. Introduction Linearly polarised light passing through a medium in the presence of a transverse magnetic field acquires an elliptical polarisation. This is known as the Cotton– Mouton effect (CME) since it was first investigated in detail by Cotton and Mouton [1]. The CME in gases is very small and systematic measurements have been performed only during the last decades [2]. The CME of neon has been measured precisely by Cameron et al. [3] in 1991 at k = 514.5 nm, and recently by Muroo et al. [4], but with a large error. Theoretical values have been calculated by Jaszunsky et al. [5], by Bishop and Cybulsky [6] and very recently by Rizzo et al. [7]. They show a large discrepancy from the value measured in [3].
*
Corresponding author. Fax: +39 049 641925. E-mail address: [email protected] (G. Ruoso).
0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.05.087
On the other hand, the measurement of [4] agrees with the calculated values, but the precision is rather poor. In this Letter, we report a novel precise measurements of the CME of neon at k = 1064 nm and T 290 K. Routine measurements of CME in gases, typically nitrogen, are normally performed to test the apparatus of the PVLAS experiment [8]. This experiment is searching for the small anisotropy induced on the vacuum by an external dipole magnetic field, as predicted by Quantum Electrodynamics (vacuum magnetic birefringence). In the PVLAS apparatus a superconducting dipole magnet is coupled to a very sensitive ellipsometer, and a high finesse Fabry–Perot cavity is used to increase the optical path in the magnetic field region. A peculiar feature of this set-up is that, in order to apply the heterodyne detection scheme, the whole magnet rotates around a vertical axis coincident with the optical axis. Recently, we also performed a measurement of the CME in krypton and xenon [9].
M. Bregant et al. / Chemical Physics Letters 410 (2005) 288–292
2. Experimental The apparatus and its layout are discussed in greater detail in [8], thus only a short description will be given here. Fig. 1 shows a schematic drawing of the experimental set-up. A linearly polarised 100 mW laser beam (k = 1064 nm) is frequency locked, using a modified Pound– Drever–Hall technique [10], to a very high finesse Fabry–Perot cavity. The cavity is placed in the warm bore of a cryostat containing a superconducting dipole magnet which provides a magnetic field up to 6 T over a length of 1 m [11,12]. A pair of crossed Glan–Laser polarisers are used to detect small changes in the polarisation state of the light beam after passing through the cavity. To enhance the sensitivity of the system, the magnetic field harmonically changes its direction, driven by a rotation of the cryostat holding the magnet, and a polarisation (ellipticity) modulator is used to effect heterodyne detection. This modulator is an in-house developed device [13] based on the stress birefringence of a BK7 glass window. It gives a controlled ellipticity of the order of 3 · 103, with a working frequency which can be as high as 1 kHz. The magnet rotation frequency is around 0.3 Hz, producing a beat signal with the much higher frequency of the polarization modulator. Polarisers, modulator and cavity are placed inside a high vacuum chamber, normally evacuated down to pressures p < 107 mbar when looking for magnetic vacuum birefringence. The chamber volume can be filled with gases at controlled pressure values from a fraction of a millibar
289
up to atmospheric pressure, measured by a set of capacitive transducers with accuracy <1%. A residual gas analyser (RGA) measures the composition of the residual gas inside the chamber when the pumping system is set off. This RGA has a Channeltron and a Faraday cup as detectors, and can be used for pressures up to 105 mbar. The extincted beam exiting from the second polariser is collected onto a InGaAs photodiode and the resulting photocurrent is then amplified using a low noise current amplifier. Its output is frequency downconverted using a lock-in amplifier which has as a reference oscillator the driving signal of the polarisation modulator. The output low frequency signal is sampled, by using a computer controlled slow ADC, at time intervals corresponding to 11.25 of rotation of the cryostat (32 equally spaced tags on the circumference of the rotating table that supports the cryostat, for a sampling frequency 10 Hz). A fast ADC [14] (sampling frequency 8.2 kHz) is also used to sample the direct signal from the amplifier. Data are saved for offline analysis. The ellipticity w induced on the beam by the Cotton– Mouton effect is related to the anisotropy of the index of refraction Dn = ni n^, where the i and ^ axes are defined with respect to the magnetic field direction. Following [2], we define a birefringence for unit field (1 T) and atmospheric pressure patm as 2 1 patm Dnu ¼ Dn ; ð1Þ B½T p where p is the pressure and B the magnetic field amplitude in Tesla. We have then 0L 1 Z F p w ¼ 2 sin 2h@ B2 ½T dzADnu ; ð2Þ k patm 0
Fig. 1. Schematic drawing of the apparatus: M1, M2, Fabry–Perot cavity mirrors; P1, P2, crossed polarisers; PM, polarisation modulator; RGA, residual gas analyser. The magnet rotates around the cavity axis.
where we have already taken into account the 2F/p amplification factor resulting from a cavity with finesse F (see [2]). The angle h is time-dependent and it is measured between the initial polarisation of the light and the magnetic field B, which lies in the plane perpendicular to the light propagation direction (dipole field). L is the length of the magnet bore. Due to the dependence of the ellipticity on sin 2h, the low frequency ADC signal has a spectral component at twice the magnet rotation frequency Xm, whose amplitude is used to measure w. Since the data sampling frequency is an integer multiple of the magnet rotation frequency, the relevant peak in the spectrum of the sampled signal does not suffer from spectral leakage and is contained in a single frequency bin (see Fig. 2). This frequency has the numeric value of 2 if measured in units of magnet rotation frequency. It is also possible to measure the relative sign of the ellipticity, since acquisition is triggered so that its starting time always corresponds to the same angular position of the rotating magnet, and absolute phase information can also be extracted.
290
M. Bregant et al. / Chemical Physics Letters 410 (2005) 288–292 10
Ellipticity
10
10
10
10
10
-4
-5
-6
-7
-8
-9
0
2
4 6 8 Frequency (units of magnet rotation frequency)
10
Fig. 2. Ellipticity spectrum of the demodulated photodiode signal with 8 mbar of neon and 4.53 T magnetic field. The peak amplitude at 2 units of magnet rotation frequency is 1.12 · 105. The peak at frequency 1 is probably due to the mechanical action of the stray magnetic field on the optical elements. No correlation has been found with the peak at frequency 2. In the spectrum the 1/f noise has been suppressed during the data analysis by using a software filter.
The finesse F in each run is deduced from the measurement of the cavity decay time s, from the relation pcs ; ð3Þ F ¼ d where c is the speed of light and d = 6.4 m is the cavity length. The measurements are repeated for each pressure of neon inside the chamber. Typical s values are around 400 ls, with an average absolute error of 10 ls. The temperature, measured with a probe placed on the outside of the vacuum pipe housing the high finesse cavity, is 289 K. Due to the presence of a localised cold source, given by the cryostat, a temperature gradient exists in the vacuum chamber containing the neon gas. Rough measurements indicate gas temperature values ranging between 287 and 291 K. The Cotton–Mouton anisotropy is dependent on the absolute temperature T: in particular a 1/T dependence is expected for monoatomic gases [2]. Uncertainties in the temperature measurements will not be considered in the global error reported in this Letter, but should be taken into account when comparing our results with other experiments. The magnet is energised at a given field intensity at the beginning of each measurement session and subsequently disconnected from the power supply to allow rotation. Due to the residual ohmic resistance of the high current switch, a current decay is observed. This is monitored using Hall probes and for each data taking run the effective field is considered. The error in the determination of the magnetic field is 0.075 T for field values around 5 T. Measurements have been performed in several runs and with different pressures. The neon bottle we used
was of type 5.0 (purity better than 99.999%). The chamber and the gas line were always evacuated to high vacuum before filling them with the desired gas. The chamber itself has a low static pressure: with no pumping the pressure is below 105 mbar for at least an hour, which is the time necessary to perform the CME measurement. Using the RGA it was also possible to measure that the main components of the residual gas are hydrogen and water vapour. A pressure of neon in the mbar range was normally sufficient to obtain a signal to noise ratio in ellipticity much larger than 100.
3. Results Table 1 lists the different data taking runs performed. Quantities of physical interest were obtained Table 1 The PVLAS data taking runs with neon discussed in this Letter Progressive #
Pressure (mbar)
Cavity decay time s (ls)
Magnetic field (T)
Measured ellipticity (·106)
1 2 3 4 5 6 7 8 9 10
5 5 5 10 15 20 7 10 3 8
375 375 375 375 375 375 420 420 450 440
4.80 4.80 4.80 4.77 4.74 4.73 4.96 4.77 4.77 4.53
7.17 6.88 6.87 13.1 19.6 26.0 11.0 14.6 4.47 11.2
The values of the finesse and of the magnetic field are measured for each run.
M. Bregant et al. / Chemical Physics Letters 410 (2005) 288–292 1.4 10
Reduced birefringence (mbar)
1.2 10
1 10
8 10
6 10
4 10
2 10
291
-14
-14
-14
-15
-15
-15
-15
0 0
5
10
15
20
25
Pressure (mbar) Fig. 3. Reduced birefringence values obtained as a function of pressure. The error bars are obtained by applying standard propagation formula to Eq. (5). The superimposed solid line is a least-squares fit to a linear function (see text).
from a Fourier analysis of the ADC sampled signal (see Fig. 1). For each run i we calculated the ellipticity wi, with its phase ui. To compare the different data sets it is necessary to transform each value into the reduced birefringence Dnri ¼ Dnu pi ;
ð4Þ
which is expressed for a 1 T magnetic field, using Eq. (3) for F: Dnri ¼ wi
k d patm ; 2p si c B2i ½TL
ð5Þ
where k = 1064 nm is the light wavelength, d is the cavity length and c the speed of light. For the field we have considered an effective length L with a mean field B: these values were obtained from a numerical model of the magnet (calculated using the finite elements software package TOSCA [15]) which has been verified by direct measurement of the magnetic field. By plotting each value Dnri against pressure one can fit the data to a linear function, where the angular coefficient is the birefringence Dnu at normal pressure as defined in Eq. (1). The values wi and their associated statistical uncertainties are obtained from a spectral analysis of the recorded fast ADC samples [14]: a typical spectrum of the demodulated photodiode signal, obtained for 8 mbar of pure neon in the chamber, can be seen in Fig. 2. This corresponds to an ellipticity w = 1.12 · 105, i.e., Dn = 9.2 · 1017 with a signal to noise ratio of the order of 103, for an integration time of 360 s. This shows the very high sensitivity of the PVLAS apparatus.
Fig. 3 shows a plot of the reduced birefringence versus pressure for all the runs, together with the best fit of a linear function of the type Dnr = a + Dnup. A leastsquares analysis gives for the angular coefficient the value Dnu = (5.9 ± 0.2) · 1016. The intercept at zero pressure is compatible with zero, being a = (6 ± 9) · 1017. The reduced v2, v2/n has a value of 0.85, corresponding to a probability of 64%. From the resulting phase of the signal (all ui values are equal within ±1), knowing the direction of the polarization of the light with respect to the direction of the magnetic field, we determined the sign of the effect to be positive.
4. Conclusions We summarize in Table 2 the different results available for the CME of Ne.
Table 2 Comparison between experimental data and theoretical calculations for Dnu Experiments This work (k = 1064 nm, T . 290 K) Ref. [3] (k = 514.5 nm, T = 298.15 K) Ref. [4] (k = 790 nm, T = 285 K)
5.9 ± 0.2 2.83 ± 0.15 6.9 ± 2.7
Theory Ref. [5] Ref. [6] Ref. [6] Ref. [7]
6.04 6.869 6.858 6.54 ± 0.23
(k = 514.5 nm, T = 273.15 K) (k = 514.5 nm, T = 273.15 K) (k = 632.8 nm, T = 273.15 K) (k = 1, T = 273.15 K)
All values are multiplied by 1016.
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The theoretical values of Dnu are reported following Rizzo et al. [2]. The values of [5] use the quadratic response method, those of [6] use perturbation theory and finite magnetic field method and those of [7] use highly correlated coupled cluster wave functions.
Acknowledgements We are grateful to all the people at the Laboratori Nazionali di Legnaro who helped us in building the apparatus. In particular we thank Selvino Marigo, of INFN Legnaro, and Aldo Zanetti, of INFN Trieste, for the precious collaboration also during data taking.
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