Piezoaxionic effect

Axion dark matter (DM) constitutes an oscillating background that violates parity and time-reversal symmtries. Inside piezoelectric crystals, where parity is broken spontaneously, this axion background can result in a stress. We call this new phenomenon “the piezoaxionic effect.” When the frequency of axion DM matches the natural frequency of a bulk acoustic normal mode of the piezoelectric crystal, the piezoaxionic effect is resonantly enhanced and can be read out electrically via the piezoelectric effect. We explore all axion couplings that can give rise to the piezoaxionic effect—the most promising one is the defining coupling of the QCD axion, through the anomaly of the strong sector. We also point our another, subdominant phenomenon present in all dielectrics, namely the “electroaxionic effect.” An axion background can produce an electric displacement field in a crystal which in turn will give rise to a voltage across the crystal. The electroaxionic effect is again largest for the axion coupling to gluons. We find that this model-independent coupling of the QCD axion may be probed through the combination of the piezoaxionic and electroaxionic effects in piezoelectric crystals with aligned nuclear spins, with near-future experimental setups applicable for axion masses between ${10}^{-11}\mathrm{eV}$ and ${10}^{-7}\mathrm{eV}$, a challenging range for most other detection concepts.

Figure 1

The relativistic enhancement factor ${\mathcal{R}}_j$ of Eqs. (38) and (39) as a function of the proton number $Z$, assuming an atomic number of $A\approx 2Z$, for two values of the total electron angular momentum $j=1/2,3/2$. https://github.com/kenvantilburg/piezoaxionic-effect/blob/main/code/supporting_plots_eta20.ipynb.

Figure 2

A simplified illustration of the proposed experimental setup. The axion-induced stress and resulting strain signal in a piezoelectric crystal with polarized nuclear spins is read out capacitively and ultimately measured through a SQUID. The circuit includes variable inductive and capacitive elements to scan around the mechanical resonance frequency, as well as a transformer circuit for impedance matching with the SQUID.

Figure 3

Equivalent electric circuit of the experimental set-up described in Sec. 3a and shown in Fig. 2. The axion signal appears as an in-series voltage, $V_a$, to the piezoelectric crystal with capacitance $C_m$, inductance $L_m$, and resistance $R_m$ around mechanical resonance. Two of the crystal’s faces are connected to electrodes, resulting in a clamped capacitance $C_c$. The electrodes are connected to variable electrical components $L_1$ and $C_1$. The current inside the circuit is read out inductively by a SQUID through a transformer. The SQUID and transformer contribute also a back-impedance $Z_{\mathrm{BI}}$ to the circuit (see text for more details).

Figure 4

Top panel: axion induced voltage $V_a$ from Eq. (61) as function of frequency $f$ for the idealized crystal with parameters shown in Table 3. The piezoaxionic (blue dashed) and electroaxionic (red dashed) contributions are plotted separately, giving the total voltage in black. The piezoaxionic effect changes sign, and adds constructively (destructively) to the axioelectric polarizability effect just above (below) the fundamental mechanical resonance frequency (dashed black vertical line), for our choice of relative sign between the piezoaxionic and electroaxionic coefficients $\xi$ and $\zeta$. Bottom panel: Crystal quality factor $Q$ from Eq. (65) as a function of frequency $f$ for the same idealized crystal. The fundamental and the natural resonance frequencies $f_0$ and $f_{\mathrm{nat}}$ are depicted by the black and dashed gray vertical lines, respectively. (See https://github.com/kenvantilburg/piezoaxionic-effect/blob/main/code/supporting_plots_eta20.ipynb.

Figure 5

Top panel: total impedance as referred to the input circuit of Fig. 3, for fixed $C_1=C_c/3$ and different values of the transformer inductance $L_1$. For small $L_1=0.01\mathrm{mH}$ (black), the resonance frequency $f_{\mathrm{res}}$ below the fundamental mechanical resonance angular frequency $f_0$ (black dashed line) is shifted up above the natural resonance frequency $f_{\mathrm{nat}}$ (gray dashed line) due to the finite value of $C_1$. For increasing $L_1$, the resonance frequency is tuned downwards, as indicated by the blue lines for $L_1=12.0,12.5\mathrm{mH}$. For yet larger inductances $L_1=16.0,16.5\mathrm{mH}$, the resonance frequency associated with the mode above the fundamental mechanical resonance approaches $f_0$ (red lines). Bottom panel: resonance angular frequency ${\omega }_{\mathrm{res}}$ (in units of ${\omega }_0$) as a function of $L_1$ and $C_1$. Blue (red) colors indicate the resonant branch below (above) ${\omega }_0$, like in the top panel. For both panels, the parameters of the idealized crystal used are given in Table 3. See https://github.com/kenvantilburg/piezoaxionic-effect/blob/main/code/supporting_plots_eta20.ipynb.

Figure 6

Amplitude spectral densities of all noise sources considered in the experimental setup with a crystal of dimensions ${\ell }_1=4\mathrm{cm}$ and ${\ell }_{2,3}=40\mathrm{cm}$, variable capacitance $C_1=C_c/10$, and variable inductance $L_1=0.06\mathrm{H}$. All other parameters are the fiducial ones from Table 3. The top panel depicts noise sources expressed as equivalent voltage noise amplitudes $S_V^{1/2}$ of Eqs. (70), (71), and (74), as referred to in the input circuit of Fig. 3. Via Eq. (61), they can be expressed as effective noise amplitudes for ${\zeta }_{11}{\overline{\theta }}_a$ (right axis of top panel) and ${\xi }_{11}{\overline{\theta }}_a$ (middle panel). In the bottom panel, they are shown as effective noise amplitudes for the Schiff moment $\mathsf{S}$ (left axis) and the QCD ${\overline{\theta }}_a$ angle (right axis), via Eqs. (47) and (48). See https://github.com/kenvantilburg/piezoaxionic-effect/blob/main/code/supporting_plots_eta20.ipynb.

Figure 7

Sensitivity to an oscillatory theta angle ${\overline{\theta }}_a$ as a function of frequency for a single shot (red line) and for a scan around the mechanical resonance frequency of the first acoustic thickness expander mode of an idealized crystal with fiducial parameters given by Table 3. The green line shows the prediction for QCD axion DM. See https://github.com/kenvantilburg/piezoaxionic-effect/blob/main/code/sensitivity_eta20.ipynb.

Figure 8

Axion parameter space probed by the setup described in Sec. 3 and parameters given in Table 3 as a function of the axion decay constant $f_a$ (inverse, vertical axis) and the axion mass $m_a$ (the top axis also indicates the corresponding frequency $f$). The blue and red shaded regions show the reach [${\mathrm{SNR}}_{\mathrm{tot}}=1$ with Eq. (82)] by scanning electrically around the first acoustic modes of a series of crystals with 13 different thicknesses. The gold region outlines the ultimate reach assuming mechanical scanning of the frequency for the parameters indicated. The ultimate reach, limited by a combination of SQUID noise and the crystal thermal noise, is shown for different values of ${\eta }_{SQ}$, the parameter indicating how close the SQUID operates to the quantum limit (see Sec. 3c2 and Sec. 3d for more details). Similarly, the sensitivity achieved by the electroaxionic effect alone is outlined by the brown dashed line. The QCD axion relation between the mass and the decay constant [Eq. (3)] is shown in green. The gray shaded regions are disfavored from BH superradiance [90], BBN [91], the structure of the Sun [30], white dwarfs [92], and neutron star binary mergers [93]. The brown dotted line is the projected sensitivity for commercially available PZT crystals with ${\eta }_{SQ}=1$, assuming an $\mathcal{O}(1)$ concentration of ${}_{82}{}^{207}\mathrm{Pb}$. See https://github.com/kenvantilburg/piezoaxionic-effect/blob/main/code/sensitivity_eta20.ipynb.

Figure 9

Sensitivity to the derivative axion-electron coupling $G_{aee}$ of Eq. (83) as a function of mass $m_a$ of the axionlike particle. The setup and parameters are the same as in Fig. 8, with the additional assumption of ${\tilde{\theta }}_1={\tilde{\eta }}_1=1$ in Eqs. (86) and (87). The gray region indicates couplings strongly disfavored by cooling of white dwarf (WD) stars [105]. See https://github.com/kenvantilburg/piezoaxionic-effect/blob/main/code/sensitivity_eta20.ipynb.