1 Introduction↩︎

VMB is an effect predicted by QED: Vacuum in the presence of a magnetic field will exhibit slightly different indices of refraction for light polarized parallel or perpendicular to the magnetic field [1], [2]. While a host of experiments have attempted to observe VMB over the last 40 years, a measurement confirming its existence has proved elusive.¹ This is mainly due to the interaction strength being so small as the standard QED prediction gives a change in the differential index of refraction as a function of the magnetic field strength of \[n_{\parallel}-n_{\perp}=3A_eB^2.\] With \(A_e=1.32\times10^{-24}\rm\,T^{-2}\) this leads to an effect of only \(3.96\times10^{-24}\rm\,T^{-2}\) .

A typical strategy to measure VMB is to pass a laser through a magnetic field and observe the change in the polarization state of the laser at the output with a polarimetric sensing scheme. The first measurements of this kind were performed by the BFRT (Brookhaven | Fermilab | Rochester | Trieste) collaboration using an optical delay line to increase the interaction length between the lasers and the magnetic field [4]. Modern experiments now use optical cavities to further extend the optical path length of the laser through the magnetic field. It is also essential to modulate the magnetic field to produce a measurable signal, and higher modulation frequencies typically correspond to better sensitivities [5]. BFRT performed the modulation of the magnetic field by ramping the current of a superconducting dipole magnet achieving a modulation frequency of 32 mHz. However the modulation can be done in a number of ways. PVLAS, for example, opted to rotate permanent magnets [5] to increase the signal frequency to 16 Hz, while other experiments such as BMV and OVAL utilized pulsed magnets to achieve even higher characteristic frequencies above 100 Hz [6], [7]. The most sensitive measurement to date was from PVLAS with a sensitivity in the differential index of refraction of \(3\times10^{-23}\rm\,T^{-2}\), within a factor of 7 of the prediction from QED.

An alternative to the polarimetric approach was proposed in Hall et al. (2000) [8]. Three fields are stabilized to three resonances of an optical cavity with the center resonance polarized orthogonally to the magnetic field and the other two resonances polarized parallel to it. In this case, the finesse of the optical cavity is used to improve precision with which the lasers can be stabilized to the cavity resonances. This is unlike the polarimetric based technique in which the finesse of the cavity amplifies the signal. The VMB effect can then be measured by observing the relative changes between the two outer resonances and the center resonance when the magnetic field is modulated. Here, it is necessary to use three resonances to decouple the measurement from the length noise of the cavity.

This paper provides a demonstration of the sensing scheme for this technique using a 19 m optical cavity without a magnetic field. A heterodyne detection system is used to read out the frequency changes between the cavity resonances. The ultimate goal of this prototype setup is to demonstrate that this technique is capable of achieving a sensitivity sufficient to measure VMB using the magnet string assembled for the Any Light Particle Search II (ALPS II) at DESY in Hamburg, Germany. In ALPS II, 24 superconducting dipole magnets formerly used by the Hadron Electron Ring Accelerator (HERA) accelerator, each generating a 5.3 T dipole field over 8.8 m of length have been straightened and aligned to give a total \(B^2 L\) of 5990 \(\rm T^2\cdot m\) [9]. This would correspond to a predicted optical pathlength difference of \(2.37\times10^{-20}\) m between fields polarized parallel and perpendicular to the direction of the magnetic field. This magnet string would therefore be able to produce a VMB signal hundreds of times larger than the current generation of experiments discussed earlier. Furthermore, ALPS II has shown it is possible to operate a high-finesse optical cavity whose eigenmode propagates through the bore of the magnet string. A characteristic modulation frequency of 0.3 mHz was demonstrated in the ALPS II magnet string by ramping and discharging the current to generate 4 pulses over the course of 4 hours as is shown in Figure 1 (a). It may also be possible to increase this frequency up to 5 mHz with upgrades to the system. Such low modulation frequencies present a challenge, as environmental noise tends to rapidly increase at lower frequencies.

The intrinsic birefringence noise of the cavity itself may also limit the sensitivity of the experiment at these frequencies as was the case in several of the previous generation of VMB experiments. Current literature on the intrinsic birefringence noise of optical cavities appears to show that it has an ASD that is inversely proportional to both the square root of the frequency and the \(1/e^2\) beam radius of the cavity eigenmode on the mirror surface \(w_0\) [5], [10], [11]. This noise level can be roughly predicted with the following equation. \[S \sim \left(\frac{\rm 1 mm}{w_0}\right) \cdot\sqrt{\left(\frac{\rm 1\,Hz}{f}\right)} \cdot(1\,-\,2)\cdot10^{-18}\rm\,m/\sqrt{Hz}\] Here the term \((1\,-\,2)\) is a factor between one and two that accounts for the uncertainty due to the range of values across the measurements in these papers. For a signal frequency is at 0.3 mHz and a beam radius of 9 mm on the cavity mirrors (as is the case for a 245 m cavity whose eigenmode propagates through the ALPS II magnet string) this equation predicts a birefringence noise between \(0.6\times10^{-17}\rm\,m/\sqrt{Hz}\) and \(1.2\times10^{-17}\rm\,m/\sqrt{Hz}\). This would mean that even if the measurement is limit by the cavity birefringence noise, a measurement of VMB at the QED prediction using the ALPS II magnet string could achieve a signal-to-noise ratio of one after only one to four days of integration time. The outlook therefore appears to be quite encouraging.

This paper is organized as follows. In Section 2 the design of the optical system is discussed and in Section 3 the experimental setup implemented on a 19 m long cavity is presented. The results of measurements using this system are given in Section 4. This includes a measurement of the noise of the setup as well as the static birefringence of the optical cavity. In Section 5 conclusions from the experiment are discussed along with the prospects of measuring VMB with a full-scale experiment using a 245 m cavity and the ALPS II magnet string.

2 Optical System↩︎

The measurement works by encoding the birefringence signal on the interference beatnotes between three different laser fields. To achieve this, three separate linearly polarized laser fields are stabilized to different resonances of the cavity using the PDH laser frequency stabilization technique [12]–[14]. A laser on resonance with an optical cavity must propagate for some integer number of cycles over a round trip through the cavity. This leads to the following condition. \[2nkL+\theta_{r1}+\theta_{r2} = 2\pi N\] In this equation \(n\) is the index of refraction of the medium between the mirrors, \(k\) is the wave vector of the laser, \(L\) is the single pass distance between the mirrors, and the \(\theta\) terms are the phase of the reflection coefficients of each of the cavity mirrors. \(N\) is the integer number of the cycles experienced by the laser after a single round trip through the cavity.

Assuming no dispersion effects in the coatings, the frequency difference between two lasers with orthogonal polarization states that are stabilized to different cavity resonances can be expressed, to first order, by the following equation. \[\Delta\nu\simeq f_0(\Delta N-\Delta\theta/2\pi + N\Delta n) \label{Eq:Del95nu}\tag{1}\] In the above equation \(f_0=c/2nL\) is the FSR of the cavity. \(\Delta\theta\) gives the difference in the total reflected phase from the mirror coatings \(\theta_{\parallel}-\theta_{\perp}\), for the orthogonal polarization states. This is due to the birefringence of their dielectric coatings and will lead to an offset in the difference frequency between the lasers from some integer multiple of FSRs by \(\Delta \nu_\theta=f_0\Delta\theta/2\pi\). Fluctuations in the differential reflected phase from the mirror coatings will lead to the birefringence noise of the cavity and will be referred to as \(\delta\nu_\theta\).

Perhaps counterintuitively, the finesse of the cavity does not appear in Equation 1 . The reason for this is once again that a higher finesse allows the resonances to act as better discriminator of the signal, but this does not amplify the signal itself.

If there is some birefringent medium between the mirrors, orthogonal polarization states will experience a small difference in the refractive index of this medium (\(\Delta n\) in the previous equation) which also appears as an offset in the difference frequency of the lasers \(\Delta\nu_n = \nu\Delta n\) (as the laser frequency \(\nu\) is equal to \(Nf_0\)). Fluctuations in the differential refractive index² experienced by the two fields, denoted by \(\delta n\), will cause the frequency difference between the resonances to fluctuate by some amount \(\delta \nu_n=\nu\,\delta n\). The fluctuation in the birefringence of the cavity generated by VMB at the magnet modulation frequency will be referred to as \(\delta n_{_{\rm VMB}}\) and will introduce a fluctuation in the frequency difference between the resonances of the cavity for the orthogonally polarized fields of \(\delta \nu_{_{\rm VMB}} = \nu \,\delta n_{_{\rm VMB}}L_{_{\rm B}}/L\) (here the length of the magnetic field \(L_{_{\rm B}}\) must be distinguished from the length of the cavity \(L\)). This can also be expressed as a differential change in the cavity length of \(\delta L_{_{\rm VMB}}= L_{_{\rm B}}\delta n_{_{\rm VMB}}\) experienced by the orthogonal polarization states. An example of a projection of the waveform of \(\delta L_{_{\rm VMB}}= L_{_{\rm B}}\delta n_{_{\rm VMB}}\) is shown for the ALPS II magnet modulation data in Figure 1 (b). From this plot it is clear that the waveforms may not be sinusoidal. It should also be noted that in this manuscript lengths such as \(L\) and \(\delta L\) will refer to the single pass lengths (the physical distance between the mirrors) and their changes respectively.

Equation 1 also shows why only using two fields stabilized to the cavity resonances may be insufficient to reach the sensitivity needed to measure the VMB effect. This comes from the fact that the cavity FSR is not static and itself fluctuates due to the length noise of the cavity, ultimately leading to a noise term in the equation \(\delta \nu_L=\Delta N \, f_0 \, \delta L/L\). Therefore, if the frequency of the interference beatnote between the lasers (\(\Delta N \, f_0\)) is 50 MHz for a VMB measurement using the ALPS II magnet string with a 245 m cavity, length noise of 1 μm/\(\sqrt{\rm Hz}\)³ would produce noise in the differential cavity length measurement of \(1.8\times 10^{-13}\) m/\(\sqrt{\rm Hz}\), making an observation of the VMB effect practically impossible.

An additional field sensing the frequency changes of a third resonance can be used to cancel the coupling of the cavity length noise to the measurement. First, a field \(E_{s_0}\), polarized perpendicular to the direction of the magnetic field, is stabilized to a resonance of the cavity. Two other fields \(E_{p_-}\) and \(E_{p_+}\), with polarization states aligned to the magnetic field, are then stabilized to cavity resonances some integer number \(\Delta N\) of FSRs above and below the resonance occupied by \(E_{s_0}\). Figure 2 (a) shows a diagram of the resulting frequency spectrum for a symmetric spacing between the resonances. Ignoring \(E_{\rm H}\), this spectrum shows that changes in the FSR, again denoted by \(f_0\), will lead to changes in the spacing between the three resonances. It is also apparent that this coupling can be canceled by subtracting the time series of the frequency of the interference beatnote between \(E_{p_-}\) and \(E_{s_0}\), given by \(\Delta\nu_-(t)\), from the time series of the frequency of the interference beatnote between \(E_{p_+}\) and \(E_{s_0}\), given by \(\Delta\nu_+(t)\). This can be seen explicitly in the following equation. \[\frac{\Delta\nu_+ - \Delta\nu_-}{2} \simeq -f_0\Delta\theta/2\pi + \nu\Delta n + \delta\nu_{_{\theta}} + \delta\nu_{_{\rm VMB}}\] Here, the coupling to changes in \(f_0\) is drastically reduced as typical values of \(\Delta\theta\) are on the order of μrad. Again, \(\delta\nu_{_{\theta}}\) is due to the intrinsic birefringence noise of the cavity and \(\delta\nu_{_{\rm VMB}}\) is the signal induced by VMB.

The differential optical path length changes for a field with a polarization state aligned to the magnetic field versus a field with a polarization state aligned perpendicular to the magnetic field can be calibrated using the following equation. \[\Delta L_n\simeq \frac{L}{2}\frac{\Delta\nu_+ - \Delta\nu_-}{\nu} \label{Eq:f95to95dL}\tag{2}\] While using a symmetric configuration of the cavity resonances will simplify the calibration of the signal strength and also slightly reduce the impact of uncorrelated noise in each of the laser frequency stabilization systems, it is not explicitly necessary. One complication can arise when using this configuration in that the frequencies of the laser beatnotes \(\Delta\nu_-(t)\) and \(\Delta\nu_+(t)\) will be nearly the same, posing a challenge for the readout of their frequency noise.

This can be addressed by using a heterodyne detection system in transmission of the cavity to sense the frequencies changes of the resonances. An additional laser provides a fourth field \(E_h\) that is phase locked to \(E_{s_0}\) at a frequency \(f_{\rm PLL}\) off resonance of the cavity. This laser is used as the local oscillator to form interference beatnotes with each of the three fields at frequencies \(f_{\rm PLL}\), \(f_- = f_{\rm PLL}-\Delta\nu_-\), and \(f_+ = f_{\rm PLL}+\Delta\nu_+\).

The formation of the interference beatnotes between the fields stabilized to the cavity and the local oscillator laser will require some manipulation of the polarization states as \(E_{p_-}\) and \(E_{p_+}\) are orthogonally polarized to \(E_{s_0}\). After combining the four fields with a power beamsplitter, this can be achieved by first passing them through a QWP oriented with its fast axis at \(45^\circ\) with respect to the vertical and then through a polarizing beamsplitter or a polarizer. If the polarization state of \(E_{h}\) is set to be parallel with \(E_{s_0}\), after passing through the QWP all four fields will be circularly polarized albeit \(E_{p_-}\) and \(E_{p_+}\) will still be orthogonal to \(E_{s_0}\) and \(E_{h}\). This will not be the case after the polarizer and the interference beatnotes can then be measured by a photodetector.

The differential length changes induced by the VMB effect can then be found with the following superposition. \[\Delta L_n \simeq \frac{L}{2\nu}(f_+ + f_- -2f_{\rm PLL})\] When using this system a symmetric configuration of the three resonances is advantageous as any residual phase noise in the phase lock loop will be common mode to all three beatnotes. To eliminate the length noise coupling, \(\delta\nu_L\), asymmetric configurations will require a superposition of the frequency data from the interference beatnotes that does not cancel the residual phase noise of the phase lock loop.

If the fluctuations in the time series of the beatnote frequencies about some mean value is defined as \(\delta f =f-\bar f\), the following equation can be used to calibrate the birefringence induced by the VMB effect in terms of these fluctations. \[\delta n_{_{\rm VMB}} \simeq \frac{\delta f_+ + \delta f_- -2\delta f_{\rm PLL}}{2\nu} \label{Eq:del95f}\tag{3}\] In this work, the laser frequency is \(\nu=282\rm\,THz\). If sufficient precision in the laser frequency stabilization loops can be achieved, the measurement will likely be limited by the intrinsic birefringence noise of the cavity \(\delta\theta\). While the amplitude of the cavity birefringence noise is expected to be much larger than the amplitude of \(\delta n_{_{\rm VMB}}\), the measured data can be correlated with the waveform of the magnetic field squared and integrated over long periods of time to increase the signal-to-noise ratio.

3 Prototype Setup↩︎

One of the primary challenges in using the three resonance technique is the very precise stabilization of the laser fields to the cavity resonances. If a precision of \(10^{-18}\) m in \(\delta L_n\) in the control system is targeted for a 245 m cavity with a finesse of 100,000 using the ALPS II magnet string, this would require stabilizing the laser frequencies to 0.2 ppm of the cavity FWHM.

A setup was built to test the three-resonance technique on a 19 m cavity. A diagram of the experiment can be seen in Figure 3. The three laser fields were generated with two lasers operating with a wavelength of 1064 nm, one of which, L1 in the diagram, is split into two paths. Each path is then frequency shifted by its own AOM in a double pass configuration (not shown in Figure 3) to reduce the coupling between the frequency shift induced by the AOM and the alignment of the laser. An EOM in each path also generates phase modulation sidebands such that each field can be stabilized to a cavity resonance using the PDH technique. A single photodectector in reflection of the cavity (labeled \(\rm PD_r\) in Figure 3) is used as the sensor in both loops. The feedback on one of the paths is sent directly to the frequency of L1 (the field emerging from this path will be referred to as \(E_{p_+}\)), while in the other path the AOM is used as a frequency actuator (this field will be referred to as \(E_{p_-}\)). ⁴ The two paths each have a HWP (labeled \(\lambda/2\) in the Figure 3) and are combined at a power beam splitter to allow any configuration of linear polarization states to be used. Faraday isolators, not shown in Figure 3 for simplicity, are used to prevent light from being reflected back to the lasers.

While one port of the power beam splitter directs the fields to the cavity, at the other port they are incident on the photodetector \(\rm PD_i\). This photodetector can be used to directly sense the interference beatnote between \(E_{p_+}\) and \(E_{p_-}\) without the additional complexity of both fields also interacting with the cavity. The frequency and amplitude of the beatnote is read out with a digital phasemeter. It should be noted here that when the fields are configured to have orthogonal polarization states the beatnote amplitude is significantly reduced, however it was still possible to measure it with the phasemeter due to these states not being purely orthogonal. It is planned for the implementation of a wave-plate and polarizing beam splitter before this photodetector in the future to increase the amplitude of the interference beatnote (or allow for the beatnote to be observed at all if the purity of the polarization states are improved).

A second laser on the opposite side of the cavity, L2 in Figure 3, provides the third field which will be referred to as \(E_{s_0}\). Although this setup differs from the technique described in the previous section in which all three fields are injected from the same side of the cavity, it is possible to inject the fields from either side. The only complication in this case is the loss of power in reflection from a cavity that is close to impedance matched.⁵ This laser (L2) also has an EOM in its beam path such that it can be PDH stabilized to a cavity resonance using \(\rm PD_t\) as a sensor. \(\rm PD_t\) is also used to sense the interference beatnotes between the three fields and the local oscillator fields and is the primary sensor for the measurement. The frequencies of the interference beatnotes are measured using a digital phasemeter as shown in Figure 3.

Since a power beam splitter is used to combine \(E_{s_0}\) with the fields from L1 in transmission of the cavity, the setup does not restrict what polarization states the fields can be injected to the cavity. This allows for null measurements with all fields in the same polarization state to test the sensitivity of the setup independent of polarization effects. However, when the setup is configured to measure the dynamic birefringence of the cavity, \(E_{s_0}\) is aligned to an orthogonal polarization axes of the cavity from \(E_{p_+}\) and \(E_{p_-}\).

3.1 Heterodyne sensing↩︎

Instead of using an additional laser, the phase modulation sidebands used for the L2 PDH stabilization system are used as the local oscillator fields. Also, as Figure 4 shows, the configuration of frequencies used to measure the cavity birefringence was different from what was described in Section 2. This was done as only two lasers and two AOM were available for the setup, so implementing an local oscillator laser for the readout was not possible. Here, \(E_{p_-}\) was stabilized to the lowest frequency resonance of the three, \(E_{p_+}\) was stabilized to the middle resonance and \(E_{s_0}\) was stabilized to the highest frequency resonance.⁶ Additionally, while the beatnote \(f_+\) was measured, the frequency difference between the upper and lower resonance \(f_2\) was measured. Therefore, the superposition of the measurements of the beatnote frequencies that revealed the cavities birefringence was \(2\,\delta f_+-\delta f_2\).⁷

Using the phase modulation sidebands for the L2 PDH loop, labeled as \(E_{s_l}\) and \(E_{s_u}\) in Figure 4, had several benefits as that it did not require an additional field or phase lock loop. Instead, the sidebands should perfectly track the frequency changes of L2. With the phase modulation sidebands of \(E_{s_0}\) at 2.5 MHz, this meant that the actual beatnote frequency used to monitor \(\delta f_+\) was at \(f_+=f_0+\Delta\nu_\theta+\rm2.5\,MHz\) or \(10.39171\pm0.00003\) MHz, since the upper sideband was used in this case. To monitor changes in \(\delta f_2\) the lower sideband was used meaning the frequency of the beatnote was \(f_2=2f_0+\Delta\nu_\theta-\rm2.5\,MHz\), or \(13.28343\pm0.00006\) MHz. In both cases, the uncertainty in the beatnote frequencies was related to the length drift of the cavity changing the FSR, with \(f_2\) having a larger uncertainty due to its coupling to this effect being twice as strong.

The phasemeters used to readout the frequencies of the beatnotes had synchronized clocks and operated with a bandwidth of 10 kHz. This initial sampling rate of the analog-to-digital converters was \(1\times10^9\) samples per second with the phasemeter data then being down sampled to 596 Hz after the data was filtered by a finite-impulse-response filter. The data stream from the phasemeter consisted of a continuous data set of the frequency, phase, and amplitude of the beatnote it was measuring.

Future upgrades are planned for the setup that will allow for the configuration discussed in Section 2 to be used, but the results discussed in Section 4 nevertheless demonstrate that the concept is feasible while also providing information critical to its optimization.

3.2 Optical cavity↩︎

A detailed characterization of the optical cavity used in this study is available in Spector and Kozlowski (2024) [15]. The cavity is formed by two 2" diameter mirrors with a single pass length of 18.99 m leading to an FSR of 7.892 MHz. The nominal storage time is 1.30 ms, corresponding to a finesse of 32,220. The input mirror of the cavity has a transmissivity of \(90.0\pm0.5\) ppm while the output mirror transmissivity is \(89.8\pm0.4\) ppm. The excess optical losses in addition to the mirror transmissivities was found to be 15 ppm. The power buildup of the cavity is 9470 and with 20 mW typically injected for a measurement the cavity circulating power reaches nearly 200 W. Typically values for the field overlap between the injected lasers and the cavity eigenmode were measured to be between 92% and 97%. The geometry of the cavity is nearly confocal with an average radius of curvature of the mirrors measured via the higher order mode spacing to be 19.95 m. This helped reduce the size of the beam spots on the cavity mirrors while still avoiding degeneracies of higher order mode.

Section 4.2 describes how the static birefringence of the cavity was measured by observing the change in the beatnote frequency of the fields stablized to the cavity resonances while rotating the polarization state of one of the fields. By doing this a \(8.5\pm0.04\) Hz peak to peak change in the beatnote frequency was observed. This indicates a static birefringence in the cavity of \((0.538\pm 0.003)\times10^{-6}\) leading to \(\Delta \theta=(3.38\pm 0.02)\times10^{-6}\) rad. The major and minor polarization axes of the cavity are aligned at \(40^\circ\) with respect to the vertical and horizontal directions and the polarization states of the input fields are aligned to them. For an actual measurement of VMB the mirrors would need to be rotated to ensure the birefringence of the cavity is aligned to the direction of the magnetic field. This has not yet been attempted in this setup.

4 Results↩︎

A time series of a 20 hour measurement of the frequency changes between three resonances of the 19 m cavity is shown in Figure 5 (a). To help show data at relevant frequencies, a moving average filter is applied to the data for this plot. The frequency drift of \(\delta f_2\) is shown as the green trace, \(\delta f_+\) is shown as the blue trace, and \(2\,\delta f_+-\delta f_2\) is shown as the orange-red trace. The purple trace gives the temperature control signal sent to the laser calibrated in terms of the frequency changes of \(f_0\). While the frequency drift of the laser may play a small role in this data, the frequency changes are primarily driven by the length changes of the cavity. From this plot it is apparent that the length drift of the cavity is common to both \(\delta f_+\) and \(\delta f_2\) (with \(\delta f_2\) showing twice the effect as it is spaced at two FSRs), but this can be canceled using the superposition \(2\,\delta f_+-\delta f_2\). It is clear the subtraction helps reduce the noise of the measurement as the standard deviation of \(2\,\delta f_+-\delta f_2\) from its mean value shown in this figure is only 0.48 Hz compared to 7.07 Hz for \(\delta f_+\) and 13.86 Hz for \(\delta f_2\). The standard deviation of the calibrated control signal from its mean value over this time was 7.35 Hz.

The Allan deviation of the frequency drifts from this measurement is shown in Figure 5 (b) with error bars that give the \(1\,\sigma\) statistical uncertainty in the data.⁸ Here, an additional null measurement is shown in yellow, in which all three fields stabilized to the laser resonances were in the same polarization state. This measurement was performed to assess whether the measurements were limited by the control system, specifically the precision of the PDH loops, versus polarization effects in the setup. The Allan deviation shows that both the normal and null measurement of \(2\,\delta f_+-\delta f_2\) demonstrate significantly better stability for averaging times longer than 100 s, compared to the measurement of the frequency drift of \(\delta f_+\). It is also clear that the stability of \(\delta f_+\) for these times is limited by the cavity length noise due to its agreement with the control signal. In addition to this, the plot shows good agreement between the null measurement of \(2\delta f_+-\delta f_2\), shown in yellow, and the baseline setup that could sense the cavity birefringence, shown in orange-red, out to averaging times of 1000 s. The differences in the data for longer averaging times is believed to be due to the measurements experiencing slightly different environmental noise during their respective runs,⁹ but the results would be expected to converge to similar values at these times if more data were to be taken. The ASD of the noise achieved in each of the measurements is shown in Figure 5 (c). Here, the \(x\)-axis gives the Fourier frequency of the noise, while the \(y\)-axis to the left gives the sensitivity of the measurement in terms of the differential single-pass optical path-length inside the cavity and the \(y\)-axis on the right shows the sensitivity relative to the FWHM of the cavity. In this plot, \(\delta f_+\) is again shown as the blue trace, with the normal measurement of \(2\,\delta f_+-\delta f_2\) shown in orange red, and the null measurement is shown in yellow. The data from the temperature control signal calibrated in terms of noise in \(f_0\) is shown in purple. The noise in the normal and null measurements of \(2\,\delta f_+-\delta f_2\) show relatively good agreement across the full range of frequencies. As expected, the cavity length noise of the cavity leads to an increase in the noise of \(\delta f_+\) below 1 mHz. The fact that the amplitude spectral density of the noise in the measurements of \(2\,\delta f_+-\delta f_2\) is below the noise measured in \(\delta f_+\) at such frequencies is a critical result as it demonstrates that the prototype setup is capable of a sensitivities below the noise floor established by the cavity length noise (shown in purple).

In the measurements of \(2\,\delta f_+-\delta f_2\) in the frequency region between 0.4 mHz and 4 mHz an amplitude spectral density of between 0.3 pm/\(\sqrt{\rm Hz}\) and 0.1 pm/\(\sqrt{\rm Hz}\) or between 2%/\(\sqrt{\rm Hz}\) and 0.6%/\(\sqrt{\rm Hz}\) of the FWHM of the cavity was achieved. It is also apparent that the current noise floor of \(\delta f_+\) is roughly a factor of 2 better than the measurements of \(2\,\delta f_+-\delta f_2\) from 4 mHz to roughly 100 mHz.

4.1 Out-of-loop Noise↩︎

In principle, the amplitude modulation on the laser at the EOM driving frequency should only be present in reflection of an optical cavity when the frequency of the laser is not exactly on resonance. This is, however, not always the case and there are a number of effects that can cause RAM at this frequency, such as etalons forming in the optics and polarization modulation introduced by the EOM. RAM can cause problems in the PDH loops as it can lead to offsets at the error point. If the RAM changes at all this will cause error point noise that the gain of the loop does not suppress. An example of this is also shown in Figure 5 (c) as the light blue trace. Here, the noise at the PDH error point of the \(E_{p_+}\) PDH control loop was measured with the cavity blocked and then calibrated in terms of the error in the measurement of the cavity length \(\Delta L_{_{V}}\), using the following equation (a derivation of this equation and a discussion of the assumptions it relies on are given in Appendix 6.4). \[\Delta L_{_{V}}(t) \simeq \frac{\lambda}{2}\frac{f_0}{m_{_{\rm ES}}}\left(1-\eta^2\frac{2T_i}{\rho}\right)V(t) \label{Eq:OOL}\tag{4}\] Here, \(V(t)\) is the time series of the voltage noise measured at the error point of the PDH lock with the cavity blocked. \(\lambda\) is the laser wavelength and \(f_0\) is the cavity FSR. \(m_{_{\rm ES}}\) is the slope of the PDH error signal in V/Hz (details on this measurement can be found in Appendix 7). \(\eta\) is the field overlap between the cavity and the laser, \(T_i\) is the power transmissivity of the input coupling mirror, and \(\rho\) is the total optical losses that the circulating field experiences during one round trip through the cavity including the mirror transmissivities, as well as scattering and absorption in the reflective coatings of the mirrors. In this case the term \(\frac{2T_i}{\rho}\) gives the field reflection coefficient of the cavity on resonance. In Figure 5 (c) it appears as though the RAM induced out-of-loop noise measured in the loop controlling the frequency of \(E_{p_+}\) may explain the excess noise seen in the measurement of \(2\,\delta f_+-\delta f_2\) (the orange and yellow traces) at frequencies below 2 mHz. Therefore, mitigating this noise may significantly improve the sensitivity of the setup at low frequencies. The error signal noise with the cavity blocked was also measured at the error points of the loops controlling the frequencies of \(E_{s_0}\) and \(E_{p_-}\) and was significantly lower than the noise measured at the \(E_{p_+}\) at \(\rm PD_{r}\). In addition to this, the noise of \(E_{p_+}\) was also measured at \(\rm PD_{i}\), but was also lower after calibration than the out-of-loop noise in the \(E_{p_+}\) frequency control loop measured at \(\rm PD_{r}\). It therefore appears that the measurement of the cavity birefringence is currently limited by the RAM of \(E_{p_+}\) at the EOM frequency, that is generated at some location in the setup after the beam splitter that combines \(E_{p_+}\) and \(E_{p_-}\). The source of this noise is not well understood, however it could be related to an etalon forming due to a spurious reflection from an optic between this beam splitter and the input coupling mirror of the cavity. This is currently under investigation, but if it is indeed the cause of the out-of-loop noise the input optics will need to be changed to eliminate the etalon.

The noise at frequencies above 1 mHz in the measurement of \(2\,\delta f_+-\delta f_2\) also merits further study as the measurement appears to limited by some unknown source of noise. This could be related to stray light, cross-couplings to the alignment noise of the input optics, or other effects induce by the RAM of the lasers at the EOM frequencies that were not observable in the measurements described here.

4.2 Cavity Birefringence Measurement↩︎

The static birefringence of the cavity was measured by a similar method as was performed in Hall et al. (2000) [8]. Here \(E_{p_+}\) and \(E_{p_-}\) were stabilized to resonances of the cavity, then the polarization states of both fields was change by rotating the HWP directly before the cavity. The cavity birefringence can be found from the amplitude of the changes induced in \(\delta f_-\), the fluctuations in the frequency difference between these fields. As the fields are aligned to orthogonal polarization states, when \(E_{p_+}\) is aligned to the slow axis of the cavity and \(E_{p_-}\) to the fast axis, \(f_-\) will be shifted by \(\Delta\nu_{_\theta}\). On the other hand, when \(E_{p_+}\) is aligned to the fast axis and \(E_{p_-}\) to the slow axis, after the HWP has rotated by \(45^\circ\), \(f_-\) will be shifted by \(-\Delta\nu_{_\theta}\). Therefore rotating the HWP in the \(E_{p_+}\) path should produce an oscillation in \(f_-\) with an amplitude of \(2\Delta\nu_{_\theta}\) at a frequency four times the rotating frequency of the HWP.

The dark red trace in Figure 6 (a) shows a 300 second section of nearly 15 hour measurement of \(\delta f_-\) while rotating the HWP and sensing the beatnote with \(\rm PD_i\).To find the amplitude of the oscillations in the beatnote frequency, the following model, a sine wave with a linear component, was fit to the sections of the data spanning 50 seconds.¹⁰ \[a_1\sin{\left(2\pi a_2 (t'-t_0) +a_3 \right) + a_4 +a_5 (t'-t_0)}\] Here the parameters \(a_n\) were the various free parameters in the fit with \(a_1\) representing the amplitude data, \(a_2\) representing the frequency, \(a_3\) the phase, \(a_4\) the static offset, and \(a_5\) the slope of the linear component of the data. The section of data being modeled was centered on \(t_0\) with \(t'\) being the series of times 25 seconds before and after \(t_0\). Thus for every time \(t\) in the data a set of model parameter \(a_n(t)\) could be produced. The pink curve in Figure 6 (a) shows the result of plotting \(a_1(t)\sin(a_3(t))+a_4\). Here the dotted line gives \(a_4(t)\) while the dashed lines show the maximum and minimum of the oscillations or \(a_4(t)+a_1(t)\) and \(a_4(t)-a_1(t)\).

The amplitude data (\(a_1(t)\)) for the full measurement is plotted as the darkest line in Figure 6 (b). A time series of the median of \(a_1(t)\) calculated over sections of data 10 minutes long is also shown in this plot as the red line, while the gray x’s show the median amplitude calculated over three hour sections of the data. The median peak-to-peak amplitude over the entire run was \(8.5\pm0.04\,\rm Hz\). The uncertainty in this value is the standard deviation of the median calculated over the three hour sections of data. Since the polarization states of both fields were rotated, the peak-to-peak amplitude of these oscillations gives twice the frequency splitting of the resonances, which leads to \(\Delta\nu_{_\theta} = 4.25\pm0.02\,\rm Hz\). This corresponds to a cavity birefringence of \((0.538\pm 0.003)\times10^{-6}\).

Figure 6 (c) shows the result of taking the ASD of \(2a_1(t)\). This gave a level between \(5\times10^{-14}\rm\,m/{\sqrt{Hz}}\) and \(2\times10^{-13}\rm\,m/{\sqrt{Hz}}\) for frequencies below 4 mHz. It is worthwhile to note that at frequencies close to 100 μHz this measurement had a lower noise level than was achieved in the measurements of the frequency drift of \(\delta f_+\) and \(2\,\delta f_+-\delta f_2\) without rotating the HWP. This shows that rotating the polarization state of one of the laser and observing the oscillations of the beatnote frequencies can lead to a better noise performance at low frequencies than the baseline design. This happens as the sensitivity at the signal frequency (four times to HWP rotation frequency) is relevant to the measurement noise induced by the PDH. Since this corresponds to higher frequencies (20 mHz to 40 mHz in this case) than the magnet modulation frequency (0.4 mHz to 4 mHz), the noise levels are typically also lower.

The prospects for this may be limited though, as it is clear that the rotation of the HWP is already introducing excess noise as the ASD of the amplitude oscillations shown in Figure 6 (c) is well above the level seen in the measurement of \(\delta f_-\) HWP was not rotated (not shown in Figure 5), at the rotation frequency between 20 mHz and 40 mHz. If the measurements of the amplitude oscillations while rotating the HWP were limited by the same noise that was present when the HWP was not rotated, then Figure 6 (c) should have shown a noise level at roughly \(4\times10^{-14}\rm\,m/{\sqrt{Hz}}\). The reason for this is still under investigation. Furthermore, this will only improve the sensitivity of the setup if the limiting noise will not be upconverted to the frequency of oscillations in the beatnote. If the setup is limited by effects inside the cavity, for example, its intrinsic birefringence noise, this noise will still couple to the measurement.

5 Conclusions↩︎

This work represents the first demonstration of the measurement of the birefringence of an optical cavity by measuring the changes in three different resonances as was first described in Hall et al. (2000) [8]. This experiment was performed on a 19 m optical cavity, meaning that the setup is already large enough to potentially incorporate a super conducting dipole magnet like those used in particle accelerators.

The ultimate goal of the experiment is to demonstrate a sensitivity sufficient to test the QED prediction of the VMB effect using the string of 24 superconducting dipole magnets assembled for the ALPS II experiment. While a measurement of VMB at the predicted amplitude would be a first macroscopic confirmation of QED, a deviation in amplitude or absence of the effect would serve as evidence for new theories of non-linear electrodynamics or new physics. Through a similar mechanism of birefringence induced by VMB, other hypothetical particles beyond the standard model such as minicharged or millicharged particles [16] can induce polarization effects, such that sensitive measurements of the quantum vacuum also serve as probes of these theories [17].

According to the QED prediction, the ALPS II magnet string is capable of producing a VMB signal with an amplitude of \(2.37\times10^{-20}\) m in terms of the differential single pass length for orthogonal polarization states. The sensing scheme would therefore require a differential length sensitivity at the characteristic frequency of the modulation of the magnetic field on the order of \(10^{-17}\rm\,m/\sqrt{Hz}\), or a fractional sensitivity in terms of the FWHM of the cavity of \(2\,\rm ppm\,/\sqrt{Hz}\) assuming a cavity finesse of 100,000. To reach this goal, the sensitivity of the prototype must improve three to four orders of magnitude from its current level in the relevant frequency range of between \(2\times10^{-2}\rm\,/\sqrt{Hz}\) and \(3\times10^{-3}\rm\,/\sqrt{Hz}\) in terms of the cavity linewidth.

There is evidence that below 1 mHz the setup is limited by the out-of-loop noise induced by RAM on one of the lasers. This is currently being addressed with a redesign of the optics on the injection path to the cavity. Future plans also include the implementation of a number of additional systems including a feedback control loop to actively suppress the RAM introduced directly at the EOMs used in the PDH frequency stabilization systems. The aim of this effort will be to improve the sensitivity of the prototype to a level capable of measuring the intrinsic birefringence noise of the cavity.

The prospects for achieving this appear to be promising as a frequency stabilization system was demonstrated in Kedar et al. (2024) [18] with a fractional stability down to \(0.55\rm\,ppm/\sqrt{\rm Hz}\) with respect to a cavity linewidth at 100 mHz and \(1.2\rm\,ppm/\sqrt{\rm Hz}\) at 10 mHz (data at 1 mHz is not shown). This is comparable to what a VMB measurement using the ALPS II magnet string would require.

One other factor that has limited the previous generation of VMB was the intrinsic birefringence noise of the optical cavity. Projections based on the results from measurements of other high-finesse optical cavities [5], [10], [11] suggest that this noise should be on the level of \(1\times10^{-17}\rm\,m/\sqrt{Hz}\) at 1 mHz for a 245 m cavity with a 9 mm beam radius on the mirrors. This would allow the system to measure a frequency change between the resonances induced by the VMB effect at the strength predicted by QED, with a signal-to-noise ratio of 3 after 1.6 million seconds of integration time, even if it is limited by this noise.

Appendix↩︎

6 Noise Projections↩︎

In the following appendices 6.1, 6.2, and 6.3, projections of the measurement noise due to laser shot noise, polarization noise, and dispersion in the mirror coatings are calculated, respectively.

6.1 Laser Shot Noise↩︎

With 1 mW of power incident on the cavity from each of the lasers, the projected shot noise limit of the PDH sensing is projected to be \(6\times10^{-19}\rm\,m/\sqrt{Hz}\) [14]. Therefore the expected impact on the sensitivity of the measurement is expected to be less than \(1\times10^{-18}\rm\,m/\sqrt{Hz}\), well below the noise measured in the setup. Here the increase from \(6\times10^{-19}\rm\,m/\sqrt{Hz}\) to \(1\times10^{-18}\rm\,m/\sqrt{Hz}\) is found by calculating the quadratic sum of the shot noise for the superposition of the beatnotes.

6.2 Laser polarization noise↩︎

The stability of the polarization states of the lasers at the input of the cavity was also considered. In Section 3.2 it was noted that the major and minor polarization axes of the cavity show a static length difference of 0.5 pm with these axes oriented \(40^\circ\) with respect to the horizontal and vertical directions. The cause of the cavity birefringence is not known, but is believed to be related to stress induced by the mirror mounts. Long term measurements of these fields using a diagnostic polarimeter were performed at a position just before the cavity mirror as well as in transmission of the cavity. These suggested no evidence of polarization noise in the lasers fields to within the 4 mrad sensitivity of the device. Therefore, differential length noise induced by changes in the polarization states of the lasers at the cavity input can be excluded above \(4\times 10^{-18}\rm\, m/\sqrt{Hz}\).

6.3 Dispersion in the mirror coatings↩︎

The effects of dispersion in the dielectric coatings of the cavity mirrors were also calculated. The length noise of the cavity could couple to the three-resonance measurement if the optical coatings of the cavity mirrors show a strong dispersion as each of the three fields would experience a different optical path-length through the cavity and thus a different free spectral range. With a group delay dispersion in the reflectance of the optical coatings expected to be on the order of -8 \(\rm fs^2\) (data from manufacturer) and a frequency spacing between the beatnotes of 8 MHz, this would correspond to a differential FSR between the upper and lower sidebands of only 0.3 μHz, less than the changes induced by the VMB signal if a 245 m cavity is used with the ALPS II magnet string. The changes in the frequencies of the beatnotes would then be 0.3 μHz times the strain noise of the cavity which is at most \(10^{-6}/\sqrt{\rm Hz}\) at the relevant frequencies.

6.4 Coupling of RAM to PDH loops↩︎

RAM at the EOM sideband frequency can lead to offsets forming in the PDH frequency stabilization loops. If the amplitude itself of the RAM changes, this will cause the offsets to change and the loops to exhibit out-of-loop noise. To calculate the impact of the RAM on the error signal offset, first, a field with both phase and amplitude modulation should be considered. \[E = E_0 e^{i(\omega t + a_\phi \sin\Omega t) + a_0 \sin\Omega t}\] Here \(\omega\) is the angular frequency of the carrier field, while \(\Omega\) is the angular frequency driving the EOM. The modulation depth of the phase modulation is given by \(a_\phi\) while the modulation depth of the amplitude modulation is given by \(a_0\). In both case we assume \(a\ll1\). The power in this field measured by a photodetector can then be expressed as the following using the small angle approximation and ignoring higher order terms in \(a\). \[P \simeq P_0 + 2P_0a_0 \sin\Omega t \label{APP95EQ:Pow}\tag{5}\] In this case the power in the carrier is given by \(P_0\). Therefore the photodetector will measure a power modulation with a relative peak-to-peak amplitude of \(4a_0\).

When considering how the field interacts with the cavity it is convenient to express the field as a carrier with real and imaginary sidebands due respectively to the amplitude and phase modulation. \[\begin{align} E \simeq E_0 e^{i\omega t} \bigg[ 1 &+ i\frac{a_\phi}{2} \left(e^{i\Omega t} -e^{-i\Omega t} \right) \\ & + \frac{a_0}{2} \left(e^{i\Omega t} -e^{-i\Omega t} \right) \bigg] \addtocounter{equation}{1}\end{align}\] If this field is then incident on a cavity and the carrier is resonant, the first term in the brackets will experience the cavity reflectivity, while the sidebands which are assumed to be well off the cavity resonance will not. Therefore, the field in reflection of the cavity can be expressed as the following assuming perfect spatial overlap between this field and the cavity eigenmode. \[\begin{align} E \simeq E_0 e^{i\omega t} \bigg[ \mathcal{R}(\omega) &+ i\frac{a_\phi}{2} \left(e^{i\Omega t} -e^{-i\Omega t} \right) \\ & + \frac{a_0}{2} \left(e^{i\Omega t} -e^{-i\Omega t} \right) \bigg] \addtocounter{equation}{1}\end{align}\] The cavity reflectivity can be represented by the following approximation[15]. \[\mathcal{R}(\omega)\simeq 1-\frac{T_i}{\frac{\rho}{2}-i\frac{\omega-\omega_0}{f_0}}\] Here \(\omega_0\) refers to the angular frequency of the nearest cavity resonance, while \(T_i\) is the power transmissivity of the input mirror, \(f_0\) is the cavity FSR, and \(\rho\) is the total roundtrip optical losses experienced by a field propagating through the length of the cavity including the transmissivities of the mirrors and the scattering and absorption losses upon reflection from their reflective coatings.

Assuming a small frequency offset between the field and the resonance of the cavity such that \(\Delta\omega=\omega-\omega_0\ll\frac{f_0\rho}{2}\) the previous expression for \(\mathcal{R}(\omega)\) can be simplified to the following. \[\mathcal{R}(\omega)\simeq 1-\frac{2T_i}{\rho} + i\frac{4T_i}{\rho^2 f_0}\Delta\omega\] Using the expression above, the power in reflection of the cavity can be calculated with the following equation. \[\begin{align} P \simeq & P_0 \bigg[ \left( 1-\frac{2T_i}{\rho} \right)^2 + \\&+ \frac{8T_i a_\phi}{\rho^2 f_0}\Delta\omega \sin \Omega t\\ & +2a_0\left( 1-\frac{2T_i}{\rho} \right) \sin \Omega t\bigg] \addtocounter{equation}{1}\end{align}\] To generate the error signal the power measured at the photodetector is then demodulated at the angular frequency \(\Omega\). The amplitude of the error signal in voltage is then proportional to the following expression. \[V_{\rm ES} \propto \frac{m_{\rm ES}}{2\pi}\Delta\omega +2a_0 \left( 1-\frac{2T_i}{\rho} \right) \label{APP95EQ:ES95prop}\tag{6}\] By only considering the relationship of the proportionality of the error signal, we do not need to consider the gain of the photodetector or the electronics used to perform the demodulation. In this equation \(m_{\rm ES}\) refers the slope of the error signal in V/Hz due to the phase modulation and was substituted for the term \(\frac{8T_i a_\phi}{\rho^2 f_0}\). The second term on the right side of this equation is proportional to the error signal offset induced by the RAM and will be refered to as \(\Delta V_{\rm ES}\). Its apparent in this equation, that the coupling of the offset \(\Delta V_{\rm ES}\) from the RAM to the error signal is proportional to the field reflectivity of the cavity \(\left( 1-\frac{2T_i}{\rho} \right)\). In the case of perfectly impedance matched cavity (\(\rho=2T_i\)) with a perfect mode matched field, this coupling would go to zero.

Returning to Equation 5 , if the cavity is blocked and the same assumptions of 6 are used to calculate the measured error signal offset it would generate the following relation. \[\Delta V_{\rm ES}' \propto 2a_0\] Therefore, to scale error signal offset measured with the cavity blocked \(\Delta V_{\rm ES}'\), in terms of the actual error signal offset induced in the PDH lock, the following relation can be used. \[\Delta V_{\rm ES} \propto \left( 1-\frac{2T_i}{\rho} \right)\Delta V_{\rm ES}'\] To express this in terms of an error in the measurement of single pass cavity length \(\Delta L_V\), \(\Delta V_{\rm ES}\) can be divided by the error signal slope \(m_{\rm ES}\), and then multiplied by \(\frac{\lambda f_0}{2}\), where \(\lambda\) is the wavelength of the laser. The factor of 2 is due the fact that the single pass cavity length (19 m in the case of this manuscript) is being considered and not the full round trip length.

This results in the following expression that was used to project the measured out-of-loop noise as the light blue trace in Figure 5 (c). \[\Delta L_{V} \simeq \frac{\lambda f_0}{2m_{\rm ES}}\left( 1-\frac{2T_i}{\rho} \right)\Delta V_{\rm ES}'\] This is still not quite the same as the Equation 4 in the text as up to this point, perfect spatial overlap between the field and the cavity eigenmode was assumed. If instead, some component of the field is not in the cavity eigenmode, its contribution to the error signal will not be reduced by \(\left( 1-\frac{2T_i}{\rho} \right)\).

Its offset must therefore be expressed in terms of the amount of power in the spatial mode of the cavity eigenmode \(\eta^2P_0\), and the amount of power that is not \((1-\eta)^2P_0\). In this case \(\eta\) will refer to the field overlap between the field and the cavity eigenmode. With this in mind, the scaling of the error signal voltage measured with the cavity blocked can be expressed with the following \[\Delta V_{\rm ES} \propto \left( 1-\frac{2T_i}{\rho} \right)\eta^2\Delta V_{\rm ES}' +\left(1-\eta^2\right)\Delta V_{\rm ES}'\] In this case the first term is the in mode contribution of the RAM to the PDH error signal offset while the second term is the out of mode contribution.

Simplifying this previous equation and expressing it in terms of \(\Delta L_{V}\) gives the form of Equation 4 discussed in the text. \[\Delta L_{V} \simeq \frac{\lambda f_0}{2m_{\rm ES}}\left( 1-\eta^2\frac{2T_i}{\rho} \right)\Delta V_{\rm ES}'\] It is interesting to note here that the first order coupling could be eliminated in the case of a overcoupled cavity (\(2T_i>\rho\)) if the spatial overlap could be maintained such that \(\eta^2=\frac{\rho}{2T_i}\). Practically, that would be very difficult to achieve, but it speaks to the importance of optimizing the spatial overlap and impedance matching to reduce the coupling of the RAM. Furthermore, this derivation relies on the assumption that the field producing the amplitude modulation shares the same spatial eigenmode as the carrier field. This is not necessarily the case and a more thorough exploration of this effect would need to consider the spatial overlap of this field and the carrier, separate from the spatial overlap between the carrier and the cavity eigenmode. Nevertheless, the treatment above is believed to be sufficient to assess the impact of the RAM of the laser on the out-of-loop noise in the PDH frequency stabilization loops.

7 Error Signal Measurement↩︎

The error signals of the PDH loops controlling the frequencies of the fields \(E_{p_-}\) and \(E_{p_+}\) were directly measured with respect to the difference frequency between the field and the cavity resonance. This was done by locking the field not being measured to a cavity resonance using its PDH loop and then feeding back to the frequency of the L1 laser via its piezo actuator. The field of the error signal being measured was then phase locked to the field stabilized to the cavity with \(\rm PD_i\) in Figure 3 used to sense the beatnote. The offset frequency of the phase lock loop was then centered on the FSR of the cavity with an additional frequency modulation with apeak-to-peak amplitude of 1 kHz or roughly 4 cavity line-widths, and a modulation frequency of 1 Hz. Phase modulation sidebands were still generated on the field being tested using its EOM and the signal from \(\rm PD_r\) was still demodulated at the EOM driving frequency to generate the error signal that was then measured. With this, over the course of a 10 minute measurement, 1200 error signals were generated and measured with a parallel data stream measuring the difference frequency \(\delta f_-\).

The results of overlaying 600 error signals, those when the error signal was scanned with an increasing frequency, are shown in Figure 7 as the transparent blue lines. It is difficult to distinguish the individual error signals here as there was little difference between the scans.¹¹ Here the horizontal axis represent the frequency offset in Hz from the error point measured in \(\delta f_-\) at \(\rm PD_i\) and the vertical axis is the error signal in V. The orange line shows a representation of the median slope obtained at the error point (\(m_{\rm ES}\)) of \((3.69\pm0.07)\times10^{-4}\rm\,V/Hz\). This was the value of \(m_{\rm ES}\) used to calibrate the measurement of the out-of-loop noise for the PDH stabilization loop of \(E_{p_-}\) with Equation 4 .

Acknowledgments↩︎

The work is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe" – 390833306 and - grant number WI 1643/2-1, the Partnership for Innovation, Education and Research (PIER) of DESY and Universität Hamburg under PIER Seed Project - PIF-2022-18, the German Volkswagen Stiftung, the National Science Foundation - grant numbers PHY-2110705 365 and PHY-1802006, the Heising Simons Foundation - grant numbers 2015-154 and 2020-1841, the UK Science and Technologies Facilities Council - grant number ST/T006331/1.

References↩︎

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1. Although measurements of neutron stars may suggest evidence for the existence of VMB at gamma ray frequencies [3]↩︎

2. Throughout the manuscript the term birefringence will be used interchangeably with the phrase differential index of refraction for orthogonally polarized fields.↩︎

3. This is a typical value of the length noise for this cavity in the frequency range of interest.↩︎

4. It should be noted that it is important that an AOM is still used to frequency shift \(E_{p_+}\) as it was not possible to completely eliminate the stray light from the primary that is not frequency shift by the AOM. If \(E_{p_+}\) was not frequency shifted by the AOM, the stray light from the unshifted primary beam of \(E_{p_-}\) will interact with \(E_{p_+}\) at \(\rm PD_r\) and introduce additional noise to its frequency stabilization loop.↩︎

5. Impedance matched refers to the condition where the transmissivity of the mirror used for the input coupling is equal to all other optical losses in a round trip through the cavity, including the transmissivity of the output coupling mirror and the scattering and absorption upon reflection from each mirror surface.↩︎

6. This was necessary to be able to filter the beatnote frequencies from the sideband frequencies used by resonant EOMs in the laser frequency stabilization control loops.↩︎

7. Assuming that all of the frequency control loops have a similar noise floor with no common mode noise, the symmetric configuration of \(E_{p_+}\) and \(E_{p_-}\) stabilized to resonances above and below \(E_{s_0}\) and superposition beatnote frequencies that allows for the sensing of \(\delta \nu_+-\delta \nu_+\) should achieve a factor of 2 better sensitivity over the current configuration.↩︎

8. The Allan deviation of \(\delta f_2\) is not shown, although it shows roughly the same features as \(\delta f_+\) with, but with twice the coupling of the length noise.↩︎

9. At the longest averaging time of 28,000 s the statistical error shows some overlap between the two measurements.↩︎

10. The rotation frequency of the HWP varied considerably over the course of the run, starting at 10 mHz, but by the end of run experiences periods where the frequencies was as low as 5 mHz. 50 second sections of data were then chosen to make sure that even the lowest frequency oscillations at 20 mHz still experienced a full cycle.↩︎

11. It should be noted that the error signal and frequency data in this plot were processed with a filter that out put the moving average of 3 ms of data. Furthermore, the FSR of the cavity was tracked during the measurement and a shift was applied to the frequency data in post-processing to ensure that the error point of the error signal was fixed to a 0 Hz frequency offset.↩︎