Acoustic tests of Lorentz symmetry using quartz oscillators

We propose and demonstrate a test of Lorentz symmetry based on new, compact, and reliable quartz oscillator technology. Violations of Lorentz invariance in the matter and photon-sector of the standard model extension (SME) generate anisotropies in particles' inertial masses and the elastic constants, giving rise to measurable anisotopies in the resonance frequencies of acoustic modes in solids. A first realization of such a"phonon-sector"test of Lorentz symmetry using room-temperature SC-cut crystals provides a limit of $\tilde c_Q^{\rm n}=(-1.8 \pm 2.2)\times 10^{-14}$\,GeV on the most weakly constrained neutron-sector $c-$coefficient of the SME. Future experiments with cryogenic oscillators promise significant improvements in accuracy, opening up the potential for improved limits on Lorentz violation in the neutron, proton, electron and photon sector.

olations (as described by the Standard Model Extension, SME) to (−1.8 ± 2.2) × 10 −14 GeV. This improves upon previous laboratory bounds by almost six orders of magnitude when compared to the best "clean" limit of 10 −8 GeV [28], and still by almost three orders of magnitude compared to a limit of 10 −11 GeV that can be derived from previous experiments by assuming vanishing gravitational-sector coefficients. Our result further compares favorably to a previous astrophysical bound of 2 × 10 −13 GeV [29]. This result closes all possibilities for Lorentz-violating anisotropies in the inertial mass of neutrons, protons and electron out at the ∼ 10 −14 GeV level or above. Future experiments with cryogenic oscillators could be used to perform more sensitive tests of Lorentz symmetry in the proton, neutron, electron, and photon sectors.
Our method is the first to compare bulk elastic wave oscillations in different directions to constrain Lorentz symmetry. It is based on compact, reliable technology readily suited for operation on a turntable or being carried on small air and space vehicles. This allows further experiments where changing gravitational potentials are desirable, such as studies of the equivalence principle. Our oscillators are relatively insensitive to external influences such as temperature changes and vibrations. They are sensitive to many sectors of Lorentz symmetry violation (e.g. photon, neutron, proton and electron) because the properties of all these affect the speed of sound in solids.
Hundreds of limits on Lorentz invariance violations do exist [30], but there are as many gaps where large signals may lie undetected. The gaps are typically left behind by other technologies because filling them would take rotating setups, long-term data taking, or operation in difficult environments. Our method is well suited for this task. We do note, however, that nonstandard phonon-sector signals may exist for reasons other than encoded in the SME, and this may lead to genuinely new tests of fundamental laws of physics. Examples may include searches for topological dark matter by a worldwide clock network [57].

A. Standard Model Extension
We use a phenomenological framework known as the Standard Model Extension (SME) [1][2][3] to describe the effects of Lorentz violation. It augments the Standard Model with new combinations of known particles and fields that lead to Lorentz violation, subject to the requirement that the theory must respect conservation of energy and momentum, renormalizability, gaugeinvariance, and observer Lorentz covariance (i.e., an experimental outcome may depend on its components motion, but not on the coordinates used to describe it). The new terms in the SME are parameterized by tensors, whose component values are collectively known as Lorentz-violation coefficients. If all such coefficients are zero, Lorentz symmetry is exact.
The value of these Lorentz-violating coefficients are by definition frame-dependent, but can be taken as approximately constant in any frame that is inertial on all timescales relevant to the experiment. It is conventional to use a sun-centered celestial equatorial reference frame. Quantities in this frame will be denoted by capital indices T, X, Y, Z. The time coordinate T has its origin at the 2000 vernal equinox. The Z axis is directed north and parallel to the rotational axis of the Earth at T = 0. The X axis points from the Sun towards the vernal equinox, while the Y axis completes a right-handed system [30].
The coefficients c µν of interest here enter the Lagrangian of a free Dirac fermion w by substituting the Dirac matrix γ → γ ν + c µν γ µ , where ν = 0, 1, 2, 3 are the space-time coordinates [7]. The term enters the nonrelativistic Schrödinger hamiltonian of a particle by the substitution where p j are the components of momentum and velocity, j, k = 1, 2, 3, and m is a particle mass. Thus, the c coefficients describe anisotropies of the inertial mass of the particle that depend on the direction of its motion. Though protons and neutrons are composite particles, they are approximated as Dirac fermions for the purpose of parameterizing Lorentz violation at low energies. The electron, proton, and neutron tensors c e µν , c p µν , and c n µν , are independent of one another, symmetric, and traceless with 9 independent degrees of freedom each.
For a composite object T that consists of n w particles of species w, the effects of Lorentz violation in Eq. (1) are given by effective coefficients where M is the total mass. Not all of the basic c µν coeffcients are physical. The physical combinations are conventionally expressed by the combinations [? ] We will use these combinations throughout when stating experimental results. We note that by these definitions, c T J = mc T J .

B. Relative significance of the terms
The components ofc encode independent degrees of freedom for Lorentz violation in the neutron sector. Knowledge of one or many of them does not imply anything about the remaining ones. We may illuminate this by an analogy to the well-known experiments constraining Lorentz symmetry in the photon sector, restricting ourselves to the "minimal" effects that do not depend on photon energy.
In the photon sector, several different modes of Lorentz violation exist, that are characterized by how they transform under Lorentz boosts and rotations. Some of them, theκ e+ andκ o− have been bounded to an accuracy of 10 −34 [14]. Despite this, other coefficients (denotedκ e− andκ o+ ) remain unexplored by these type of observations. They are best studied by laboratory experiments, which have been continuously improved from the original ones by Michelson-Morley, Kennedy-Thorndike and Ives-Stilwell [13] to modern ones that reach down to sensitivities of 10 −18 [4-6, 8, 10]. But even those tests leave behind a last remaining coefficient. This one,κ tr , is arguably the hardest to measure, as dedicated experiments were set up to measure this remaining coefficient, and now Lorentz violation in the minimal photon sector is completely ruled out [6,11,12].
By comparison, the neutron sectors is less well studied. Here also, some components of the coefficients for Lorentz violation have been ruled out with high precision, but others remain tested at low precision. We will review these bounds in the next section. Improving the bound on the least well-studied coefficientc Q , closing the largest remaining gap in the verification of neutron-sector Lorentz symmetry.

C. Existing bounds
The most sensitive experiments to determine limits on thec n for neutrons are based on magnetometry [7,22,25,[31][32][33]. In particular, a Neon/Rubidium/Potassium co-magnetometer has been used, which simultaneously sense the influence of background magnetic fields and the signal for Lorentz violation. This bounds the four spatial componentsc n J ,c to the very low level of 10 −29 [25]. Limits on the fifth, c n Q , are not available from this experiment, but are available from tests of the weak equivalence principle [28] and astrophysics [29]. The best "clean" laboratory limit is |c n Q | < 10 −8 GeV [28]. Assuming that the αa eff -coefficients vanish, an improved limit of 10 −11 GeV is available [28]. Astrophysics studies of the stability of cosmic ray protons yields |c Q | < 2 × 10 −13 GeV, |c T J | < 5 × 10 −14 GeV [29]. The temporalc n T T have been measured in atom interferometry [34].

II. EXPERIMENT
The principle of our search for anisotropic inertial masses is simple: We use a quartz oscillator performing nominally 10-MHz oscillations on a turntable. If the inertial mass of the quartz material in on direction is fractionally higher by δm/m than in an orthogonal direction, then rotating the crystal leads to a modulation of the oscillation frequency by δν ν = − 1 2 δm/m. We use shear oscillations in an sc-cut quartz crystal. This modulation can be measured, either by comparison to a stationary reference or by comparison to a second oscillator on the turntable, rotated by 90 • relative to the first.

A. Influence of Lorentz violation on mechanical resonance
Finding the sensitivity of mechanical resonators to Lorentz violation is possible by perturbation theory for each eigenmode. Our experiment uses a Stress Compensated (SC) cut [35] crystal Bulk Acoustic Wave (BAW) piezoelectric plate resonator working at the third overtone (OT) of the thickness shear mode. This resonator is housed in an oven at the temperature of around 85 • C where the vibrational mode exhibits zero temperature coefficient of its oscillation frequency. The SC cut is doubly rotated relative to the crystal axis by a first angle of θ = 34.11 • and a second angle φ = 21.93 • . This also results in zero stress dependence of the frequency, which reduces dependence of the frequency on the mounting of the crystal, amplitude variations of the oscillation, and ageing [36].

B. Unperturbed modes
The eigenmodes of doubly rotated piezoelectric plate resonators have been studied in detail [37,38]. Due to high Q-factors (typically slightly above 10 6 at room temperature) the eigenmodes may be considered isolated mechanical oscillators. We introduce a plate coordinate system x [i] (i = 1, 2, 3) in which x [2] is normal to the major surfaces of the doubly rotated quartz blank, x [1] is directed along the axis of the second rotation, and x [3] is completing a right-handed system (Tab. I). We denote u [i] (t, x) the components of the displacement of a volume element at x [i] as function of time t. We start by finding the modes that depend only on the x [2] coordinate ("thickness modes"), where Thec [2nr2] are the piezoelectrically stiffened elastic constants rotated into the blank coordinate system. Solving the last equation yields three eigenvectors ( These eigenvectors are used as the basis of new "thickness" coordinates x (1−3) , organized such that x (i) has its largest component along x [i] . Analysis in the thickness mode coordinates then yields three mode families, known as quasi-longitudinal (A-) mode, fast shear (B-) mode and slow shear (C-) mode. For each family, the amplitude of one of the displacement components in x (i) direction is large while the others are small. Due to this smallness, the modes nearly decouple, which makes it possible to find accurate closed-form expressions for the eigenmodes.
The modes of interest here have the largest displacement component along x (1) which is approximately along the x [1] axis. It can be written as [38] and H m is the Hermite polynomial of order m. For the mode used in a third-overtone sc cut crystal at 10 MHz, we have n = 3, m = p = 0, M ′ 1n = 5.3273, P ′ 1n = 6.3858, c (1) = 3.4379, R is the radius of the blank and h 0 is the thickness, which is 0.54094 mm to make the resonance frequency 10 MHz [38].

C. Perturbation due to Lorentz violation
Since the motion of the volume elements is primarily in x [1] direction, Eq. (1) predicts that x [1] Axis of second crystal rotation; approximately direction of shear x [2] Normal to major blank surface x [3] Completes right-handed system Thickness x (i) Parallel to thickness modes Lab x 1 = x Horizontally pointing South x 2 = y Horizontally pointing East Parallel to Earth's axis pointing North which is equivalent to a re-scaling of the inertial mass by 1 + 2c [xx] + c 00 . This leads to a relative change in the resonance frequency of To study a simple case first, we may assume that all coefficients of Eq. (3) were zero except forc Q , that the experiment with two rotating quartz oscillators was located at the equator with the [x]-axis horizontal and rotated around a vertical axis at an angular velocity of ω t . This would lead to a modulation amplitude of δν/ν = c Q Q /4, where the superscript Q indicates we are using the effective combination of coefficients for quartz and M = m n + m p For the general case, we calculate the components c [xx] in the crystal frame, rotating on the turntable, from the c µν in the sun-centered frame. This involves Lorentz boosts and rotations [7]. We denote ω t the angular velocity of the turntable measured in the lab frame, ω ⊕ ≈ 2π/(23h56min), and Ω ⊕ = 2π/(1 year) the sidereal angular velocities of Earth's rotation and orbit, respectively; χ is the co-latitude of the lab in which the experiment is performed (χ ≈ 52.13 • for Berkeley, California, and 148.05 • for Perth, Australia); and η ≈ 23.4 • the angle between the ecliptic and Earth's equatorial plane. The signal includes contributions of order 1, or suppressed by either the Earth's orbital velocity β ⊕ ≈ 10 −4 . We neglect contributions from signals suppressed by higher powers of β ⊕ or by the velocity of the laboratory due to Earth's rotation β L ≈ 10 −6 . We express the measured frequency variation as a Fourier series where the factor of 1/8 is to simplify the Fourier coefficients C lmn , S lmn and The Fourier coefficients are listed in Tables II and III. For the purpose of these tables, we use the definitions similar to Eq. (3) but without the factor of particle mass. In particular, since m n ≈ m p and since naturally abundant quartz contains to very good approximation the same number of neutrons and protons, so D. Preliminary experiment with room-temperature oscillators Our experiment (Fig. 1) uses active rotation at a frequency of ω t = 2π × 0.36 Hz on a precision airbearing turntable. Relative to experiments based solely on Earth's rotation, this increase the signal frequencies and thus allows us to suppress the drift of the oscillators due, e.g., to ageing or temperature instability. The turntable (Professional Instruments, model 10R-606) is specified to 0.1µrad tilt of the rotation axis and < 25 nm radial and axial wobble, and has a specified stiffness of 10 Nm/µradian. We use two oscillators that are rotating on the turntable and that are directly compared on the turntable. This avoids the need to bring the signals in or out of the turntable. (The target accuracy of 10 −13 out of 10 MHz requires us to detect phase modulations of microradian-size; any modulations introduced when transmitting the signal from the turntable to the stationary laboratory frame would be synchronized with the putative signal, and none of the available methods can be trusted to not introduce tiny phase or amplitude modulations). The oscillators (Stanford Research Systems SC-10) are signal generators classified as ovenized voltage-controlled crystal oscillators (OCXO) based on quartz SC-cut BAW resonators. The oscillators are specified to an Alan variance of 2 × 10 −12 at 1 second averaging time. All components are highly reliable and covered with a mu-metal shield, allowing the experiment to take uninterrupted data over long stretches of time.
Directly comparing the frequency of the oscillators via an available frequency counter is limited to a resolution of about 10 −11 in one second by the ∼ 100−ps timing resolution of the device. Much higher resolution can be achieved by using a double-balanced mixer (Mini-Circuits RPD-1) to measure the phase difference between the oscillators. We use the mixer's internal signal transformers to provide galvanic isolation between the quartz oscillators and the direct-current (dc) circuits (the RPD-1 allows the three ports to have separate grounds) to avoid dc signal errors through ground loops, given the large supply current of the quartz ovens. The output II: Signal components for one rotating crystal oscillator compared against a stationary reference that is not affected by the c−coefficients. Signal components suppressed by β 2 ⊕ and higher powers have been omitted. These coefficients are to be inserted in Eq. (9) and are multiplied by 1/8 to give the frequency change.
l, m, n cos sin signal of the mixer is pre-amplified 1000 times and the resulting voltage U is digitized on the turntable. The digital signal is brought out of the turntable via a universal serial bus (USB) connection through slip-ring contacts. Power at 15 V is also supplied via sliprings.
On timescales much longer than the rotation period of our turntable, we phase-lock the oscillators together so that the mixer may always operate close to 90 • phase difference, i.e., near-zero output signal. The effective frequency-to-voltage conversion factor measured at the mixer output is thus zero at extremely low frequencies, where any voltages are removed by the feedback loop; at high frequencies, where the feedback is ineffective, the factor is given by purely by the mixer itself. We measure the conversion efficiency of the mixer as a frequency discriminator by replacing one of the quartz oscillators with a digital synthesizer which provides a known frequency modulation. Fig. 2 shows the measured response function. At our signal frequency of 2ω t ∼ 2π × 0.76 Hz, we obtain δν = ∆U/(1.3 V/Hz).
The turntable is driven by an unregulated dc motor. Even small changes of the rotation rate accumulate to a large angle offset over time. We therefore use a lightgate as a rotation encoder that delivers one pulse per turn to the computer, re-setting the angle scale of the turntable rotation. The computer then interpolates linearly assuming a constant rotation rate during one turn.
The system proved to be extremely reliable and capable of unattended operation. Fig. 3 shows the amplitude Fourier transform of 120.0 hours of data (about 164,000 turntable rotations). Zooming into the region close to the expected signals around 2ω t reveals sine and cosine amplitudes that are normally distributed with a standard deviation of σ 2 = A 2 c = 32 µV after amplification. The measured signal at 2ω t is −26 µV. This   The signal for Lorentz violation (Tab. III) has components at various frequencies around 2ω t . At our present accuracy, we restrict our analysis to the effect of c Q Q , which causes a signal proportional to cos(2ω t T ). For its amplitude, we find δν/ν = 1 2 sin 2 χc Q Q ≃ 0.31c Q Q . In the experiment, however, the axes of the oscillators were oriented 45 • relative to the rotation axis, which we take into account by a factor of sin 45 • . We thus find c Q Q = (−0.9 ± 1.1) × 10 −14 on the effective coefficient for naturally abundant quartz, which translates into a limit ofc p Q = (−1.8 ± 2.2) × 10 −14 GeV on the neutron-sector coefficient.
Systematic effects of quartz oscillators such as aging, temperature fluctuations and thermal hysteresis, acceleration, magnetic fields, power supply voltage, load impedance, electric fields, ionizing radiation, and ground  loops, are well-understood. At our current resolution, they are negligible so we just discuss the largest two: We measured the acceleration sensitivity of our quartz oscillators by inverting them relative to Earth's gravitational acceleration g. For the most sensitive axis, we find δν ∼ 20 mHz/2g. The turntable wobble is specified to be less than 25 nm radially and axially. If we conservatively assume that this wobble contributes a 2ω t frequency component (in reality, the energy of the wobble is likely spread out over many Fourier components), the corresponding acceleration is 25 nm×4ω 2 t ∼ 0.14×10 −6 g, which produces frequency changes of 2.8 nHz. Changing magnetic fields induce voltages into our wiring. Assuming 1 Gauss and an enclosed area of 1 cm 2 at the turntable frequency 2ω t (conservatively assuming that all the magnetic field will contribute to the second harmonic of the turn table rate), we obtain an induced voltage of ∼ 40 nV, comparable to our signal size. For this reason, we enclose the entire setup up to and including the amplifier in a two-layer mu-metal shield, which should reduce this influence at least ∼ 100− fold.

E. Cryogenic experiment
The quartz bulk acoustic wave (BAW) technology provides the most stable oscillators in the medium and high frequency range (1 − 50 MHz) between 1 and 30 seconds of averaging time. Such oscillators are also the most stable macroscopic mechanical harmonic oscillators, with fractional frequency stabilities as low as 2.5 × 10 −14 [39] for room temperature devices. Although, over the last decade there has been no major improvement, mainly due to the quartz resonator self-noise, it is generally believed that no further progress can be made at room temperature. For this reason, the electrodeless (or BVA) [40] BAW quartz resonators (see Fig. 5  tigated for cryogenic operation. These investigations reveal extremely high values of the quality factors exceeding 10 9 [41,42] as well as an ability to operate at high overtones (OTs) [43] providing a new platform for many physical experiments [44,45], for example detection of high frequency gravitational waves [46] and cooling a macroscopic object to its ground state for tests of fundamental physics [43,47]. Table IV gives values of quality factor for some OTs measured in a 4K environment. Such a significant increase of the quality factor may result in reduction of the oscillator fractional frequency stability. Assuming that the dominant flicker noise of the resonator at 1 Hz from the carrier (S φ (1Hz) ∼ −130 dBrd 2 Hz ) does not change between cryogenic and room temperatures, the Allan deviation of a cryogenic source may be estimated to achieve a level of

) have been inves-
BAW resonator ageing is a systematic drift of its res-onant frequency that can be typically observed at long averaging times. This process usually gives a slope of τ 1 (where τ is the integration time) in the Alan Deviation curve dominating over τ 1/2 law resulting from thermal fluctuations for averaging times over 10 3 seconds. The ageing process can be caused by a number of effects primarily related to manufacturing. This process is the most prominent during the first months of the oscillator continuous operation, which gradually decreases during that time. For the current experiment, the ageing is about 2 × 10 −10 pp/day. For ultra-stable oscillators this can be reduced to 3 × 10 −12 pp/day or 1 × 10 −9 pp/year after at least 90 days of continuous operation. Another source of stability improvement is associated with the relation between the flicker and white noise in typical BAW oscillators. While the white noise is connected to the signal-to-noise ratio and could be thus reduced by increasing the oscillation power, the flicker noise drops with decreasing power. This situation results in a compromise between the middle term stability (flicker noise region) and short term stability (white noise region). At cryogenic temperatures the white noise is naturally reduced according to the Nyquist relation, thus giving more room for flicker noise improvement by oscillator power reduction. The Nyquist noise limit for BAW resonators has been recently demonstrated at liquid helium temperatures [48] and unequivocally demonstrates the drop in this limit.
Nevertheless, practical realization of such a cryogenic BAW clock is associated with technical difficulties [49,50]. So far only moderate long temperature stability improvement has been demonstrated [50,51]. The main problem is the absence of a frequency-temperature turn-over point giving rise to significant fluctuations. Additionally, the absence of reliable low temperature components at the medium and high frequency range makes the oscillator design a challenging problem. Whereas, the first problem may be overcome by a design of a special cut for cryogenic temperatures, the second is solvable by shifting from semiconductor to superconductor technology. Furthermore, for realizing Lorentz violation experiments, which utilize two oscillators, one just needs to match temperature coefficients of two orthogonally orientated resonators, so as to read out a stable beat frequency, in a similar way to the cryogenic sapphire oscillator tests in the photon sector [6]. This may relax the requirements for a turnover point for these types of measurements.

III. SUMMARY AND OUTLOOK
We have presented a new method for testing Lorentz symmetry, frequency comparisons between quartz crystal oscillators. Such oscillators are commercially available and highly reliable. While their stability today is surpassed by modern atomic clocks (especially optical clocks), many tests of Lorentz symmetry are not limited by signal-to-noise, but often by systematic effects from wobble and tilt of the turntable, and the ability to take data over long stretches of time. Quartz oscillators are compact, simple to apply and to shield from environmental influences. Their low acceleration sensitivity makes them relatively immune to wobble and tilt. Maintenancefree operation allows for long-term data taking which helps to make up for the reduced stability. As a demonstration, we have improved the laboratory limit on the neutron-c Q coefficient by six orders of magnitude, surpassing even current astrophysics bounds. Currently cryogenic oscillators are under development at University of Western Australia and FEMTO-ST, and promise strong improvements in stability and the sensitivity to Lorentz violating coefficients.
By analogy to photon sector experiments, we believe our method can be strongly improved. Photon sector experiments have gained four orders of magnitude in sensitivity over the last twelve years, through higher quality factors resonance and cryogenic operation. A cryogenic version of our experiment may increase the quality factor of the resonance about 10,000 fold and may strongly reduce the temperature coefficient of the oscillators. We thus estimate that three to four orders of magnitude improvement are realistic. This new technology may also lead to milligram-scale mechanical oscillators at the quantum limit and may see a new brand of ultra-stable oscillators.
Future versions of the experiment will be able to bound a large class of Fermion-and photon sector coefficients. This will close many loopholes in the verification of Lorentz violation. Taking data over a year would allow us to independently measure all coefficients for Lorentz violation in Tab. III and II, as in [17,[54][55][56]. The experiment is in principle sensitive to any Lorentz violation that changes the inertial masses of protons and neutrons. The influence of anisotropic inertial masses of electrons is suppressed by their lower mass. Unsuppressed sensitivity to electron-and photon-sector terms, however, arises from the fact that bindings in crystals are determined by the properties of electrons and electromagnetism [20,52,53]. These parameters have been measured in optical Michelson-Morley experiments [4][5][6], Dysprosium microwave spectroscopy [19] and ion traps [27]. If the cryogenic experiment reaches a sensitivity to frequency changes on the 10 −18 , testing for the photoñ κ e− ,κ o+ terms becomes interesting. Other applications of crystal oscillators might include the search for topological dark matter [57].
under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Con-tract DE-AC52-07NA27344.