Preferred-frame and $CP$-violation tests with polarized electrons

We used a torsion pendulum containing $\approx {10}^{23}$ polarized electrons to search new interactions that couple to electron spin. We limit $CP$-violating interactions between the pendulum’s electrons and unpolarized matter in the Earth or the Sun, test for rotation and boost-dependent preferred-frame effects using the Earth’s rotation and velocity with respect to the entire cosmos, and search for exotic velocity-dependent potentials between polarized electrons and unpolarized matter in the Sun and Moon. We find $CP$-violating parameters $|g_{\mathrm{P}}^e{g}_{\mathrm{S}}^N|/(\hslash c)<9.4\times {10}^{-37}$ and $|g_{\mathrm{A}}^e{g}_{\mathrm{V}}^N|/(\hslash c)<1.2\times {10}^{-56}$ for $\lambda >1\mathrm{AU}$. We test for preferred-frame interactions of the form $V=-{\mathit{\sigma }}^e·\mathit{A}$, $V=-B{\mathit{\sigma }}^e·\mathit{v}/c$, or $V=-\sum _{}^{}{\sigma }_i^e{C}_{ij}{v}_j/c$, where $\mathit{v}$ is the velocity of the Earth with respect to the cosmic microwave background restframe and $i$, $j$ represent the equatorial inertial coordinates $X$, $Y$, and $Z$. We constrain all 3 components of $A$, obtaining $1\sigma$ upper limits $|{\mathit{A}}_{X,Y}|\le 1.5\times {10}^{-22}\mathrm{eV}$ and $|{\mathit{A}}_Z|\le 4.4\times {10}^{-21}\mathrm{eV}$ that may be compared to the benchmark value $m_e^2/M_{\mathrm{Planck}}=2\times {10}^{-17}\mathrm{eV}$. Interpreting our constraint on $\mathit{A}$ in terms of noncommutative geometry, we obtain an upper bound of $(355l_{\mathrm{GUT}}{)}^2$ on the minimum observable area, where $l_{\mathrm{GUT}}=\hslash c/({10}^{16}\mathrm{GeV})$ is the grand unification length. We find that $|B|\le 1.2\times {10}^{-19}\mathrm{eV}$. All 9 components of $\mathit{C}$ are constrained at the ${10}^{-17}$ to ${10}^{-18}\mathrm{eV}$ level. We determine 9 linear combinations of parameters of the standard model extension; rotational-noninvariant and boost-noninvariant terms are limited at roughly the ${10}^{-31}\mathrm{GeV}$ and ${10}^{-27}\mathrm{GeV}$ levels, respectively. Finally, we find that the gravitational mass of an electron spinning toward the galactic center differs by less than about 1 part in ${10}^{21}$ from an electron spinning in the opposite direction. As a byproduct of this work, the density of polarized electrons in $\mathrm{Sm}{\mathrm{Co}}_5$ was measured to be $(4.19\pm 0.19)\times {10}^{22}{\mathrm{cm}}^{-3}$ at a field of 9.6 kG.

Figure 1

Scale cross section of the Eöt-Wash rotating torsion balance; 1: torsion fiber, 2: autocollimator, 3: $Q_{21}$ gravity-gradient compensator, 4: nested magnetic and electrostatic shields, 5: turntable, 6: $Q_{31}$ gravity-gradient compensator, 7: vacuum vessel, 8: spin pendulum, 9: thermal shield, 10: 3-axis Helmholtz coil system, 11: thermally controlled feet, 12: mode spoiler, 13: corotating slip-ring assembly, 14: upper fiber attachment mechanism. Two additional layers of magnetic shielding (not shown for clarity) are immediately inside and outside of the vacuum vessel.

Figure 2

Spectral power density in the twist signal. The horizontal scale shows the frequency in units of the turntable rotation frequency. The smooth curve shows the expected thermal noise density for this pendulum which had $Q\approx 2000$. At our signal frequency (shown by the arrow) the noise is predominantly thermal. Reproducible irregularities in the turntable rotation rate produce peaks at integer values of $f/f_{\mathrm{tt}}$. Spurious peaks at odd integer values were suppressed by combining data taken with 2 opposite orientations of the pendulum in the rotating apparatus.

Figure 3

Scale drawing of the spin pendulum. The light green and darker blue volumes are Alnico and $\mathrm{Sm}{\mathrm{Co}}_5$, respectively. Upper left: top view of a single “puck”; the spin moment points to the right. Lower right: the assembled pendulum with the magnetic shield shown cut away to reveal the 4 pucks inside. Two of the 4 mirrors (light gold) used to monitor the pendulum twist are prominent. Arrows with filled heads show the relative densities and directions of the electron spins, open-headed arrows show the directions of $\mathit{B}$. The 8 tabs on the shield held small screws that we used to tune out the pendulum’s residual $q_{21}$ gravitational moment.

Figure 4

Dynamical calibration of the torque scale, showing the pendulum twist as a function of time. At $t=377.7\mathrm{s}$ the turntable rotation frequency was abruptly changed from 0.555 56 mHz to 0.556 11 mHz. The resulting change in the pendulum’s oscillation amplitude and phase calibrated the angular deflection scale and determined the pendulum’s free-oscillation frequency $f_0$. The smooth curve is the best fit.

Figure 5

Daily signals produced by the Lorentz-boost-violating term $C_{ij}$. The daily signals can be essentially constant (i.e. with a weak annual modulation), modulated at the sidereal frequency, or modulated at both the sidereal and annual frequencies. Solid blue line: $C_{XX}={10}^{-18}\mathrm{eV}$; dashed red line $C_{YY}={10}^{-18}\mathrm{eV}$. The points show the actual times at which data were acquired.

Figure 6

Expected ${\beta }_{\mathrm{E}}$ signal corresponding to $C_{XX}={10}^{-18}\mathrm{eV}$. Each point is placed at the time of one of the cuts in our complete data set. Almost 4000 additional data points extending out to 2825 on the horizontal axis are omitted to keep the figure readable. The annual modulation arises from ${\mathit{v}}_{\odot }$, the steady component comes from ${\mathit{v}}_{\mathrm{CMB}}$.

Figure 7

Earth source integrals defined in Eq. (24). The vertical axis shows $I(\lambda )$ divided by $\lambda$. To obtain constraints on $g_{\mathrm{P}}^e{g}_{\mathrm{P}}^e$ or $g_{\mathrm{P}}^e{g}_{\mathrm{P}}^p$, rather than on $g_{\mathrm{P}}^e{g}_{\mathrm{P}}^N$, simply multiply $I(\lambda )$ by $[Z/\mu {]}_{\oplus }\approx 0.48$.

Figure 8

Scale drawing of the “zero-moment” pendulum. The pendulum body was made from 7075 aluminum. The 4 vertical screws (shaded) were adjusted to trim away the residual $q_{21}$ moments of the pendulum. The “ball-cone” device on top of the pendulum allowed us to rotate the pendulum with respect to the torsion fiber. The entire pendulum, including the mirrors, was coated with gold.

Figure 9

Spin-pendulum data from a set of runs at ${\varphi }_{\mathrm{d}}=112.5\mathrm{deg}$ that spanned a duration of 113 h. The sidereal average of the signals has been set to zero. The dashed curves show the signal from a hypothetical $\mathit{A}=(2.5\times {10}^{-20}\mathrm{eV})\widehat{\mathit{x}}$ which gives out-of-phase sine waves in ${\beta }_N$ and ${\beta }_E$. The solid curves show the best sidereal fit, which yields $A_x=(-0.20\pm 0.76)\times {10}^{-21}\mathrm{eV}$ and $A_y=(-0.23\pm 0.76)\times {10}^{-21}\mathrm{eV}$. (Note that the hypothetical signal in Fig. 2 of Ref. 1 was actually $\mathit{A}=(2.5\times {10}^{-20}\mathrm{eV})\widehat{\mathit{x}}$.)

Figure 10

Systematic error study using the zero-moment pendulum. Data were taken for 4 orthogonal directions of the pendulum with respect to the turntable and for two opposite orientations of the pendulum with respect to its suspension system and fitted with Eq. (37). The ellipses show the values and $1\sigma$ uncertainties in the extracted signals associated with the pendulum, the turntable, and the suspension system, respectively. Systematic effects associated with the pendulum itself were negligible.

Figure 11

Extraction of lab-fixed signals from the October 2006 data, showing the N and E components of the twist signals, obtained using Eq. (37). Ellipses indicate $1\sigma$ statistical errors. The turntable systematic changed each time we opened up the vacuum system to change ${\varphi }_{\mathrm{ds}}$, while the suspension systematic remained constant at a value very close to that seen in the zero-moment data plotted in Fig. 10. The red vertical lines show the expected gyro-compass signal; the length of the line reflects the uncertainty in the $R$ value.

Figure 12

Daily variation of the ambient magnetic field measured with a flux-gate magnetometer.

Figure 13

Upper limits on the ghost condensate parameter $M/F$ as a function of $M$. The shaded area is excluded at 95% confidence. The vertical dashed lines show values of $M$ having particular cosmological significance.

Figure 14

Upper limits on $|g_{\mathrm{P}}^e{g}_{\mathrm{S}}^N|/(\hslash c)$ as a function of interaction range $\lambda$; the shaded region is excluded at 95% confidence. Our results and previous work by Youdin et al. [31], Ni et al. [32], and Wineland et al. [33] are indicated by solid, dash-dotted, dotted, and dashed lines, respectively. Our work does not provide constraints for $10\mathrm{km}<\lambda <{10}^3\mathrm{km}$ because integration over the terrestrial surrounding is not reliable in this regime (see Ref. 20).

Figure 15

Constraints on the velocity-dependent couplings $f_v$ and $f_{\perp }$ inferred from the infinite-range results in Table . The shaded area is excluded at 95% confidence. In making this plot from the $\lambda =\infty$ results in Table  we neglected the eccentricities of the earth’s and moon’s orbits.

Figure 16

Performance of the feetback system showing the tilt measured by corotating level sensors. Upper panel: “feetback” off. The rapid fluctuations arise from imperfections in the turntable bearing, slower variations arise from the varying tilt of the laboratory floor. Lower panel: “feetback” on.

Figure 17

Cross section of a foot of the Eöt-Wash rotating torsion balance.

Figure 18

Thermal model and flow chart of the simplified feetback loop implemented at the Eöt-Wash II rotating torsion balance.