A search for light scalar dark matter in the radio-frequency band with atomic spectroscopy

Introduction - The fundamental constants (FC) of nature are invariant in time within the Standard Model (SM) of particle physics, but become dynamical in a number of theories beyond the SM. This possibility has motivated diverse studies that constrain both present-day FC drifts, and changes of FC between the present time and an earlier time in the universe (see, for example, reviews Safronova et al. (2018a); Uzan (2015) and references therein).

The FC are expected to oscillate in cases where the SM fields couple to an ultra-light scalar field, coherently oscillating to account for the observed dark matter (DM) density. Within this class of models, FC oscillations are expected to occur at the Compton frequency of the DM particle, ${\omega }_C=m_{\varphi }$ ¹¹1We use natural units, where $\hslash =c=1$., where $m_{\varphi }$ is the particle’s mass. Such a scenario is particularly motivated in two main cases: (i) where the DM candidate is a dilaton, and its coupling to the SM particles is dictated by scale invariance Arvanitaki et al. (2015); Graham et al. (2015a); and (ii) the DM is the relaxion field Graham et al. (2015b), dynamically misaligned from its local minima Banerjee et al. (2018), and its coupling to the SM fields arises due to its mixing with the Higgs Flacke et al. (2017). Relaxions may form gravitationally bound objects Banerjee et al. (2019), thereby increasing the local DM density and enhancing the observability of the scenario.

There are several proposed schemes to probe light DM that has scalar couplings to SM matter. These include suggestions to look for variations of the fine-structure constant $\alpha$ using atomic clocks Stadnik and Flambaum (2015a, 2016a); Arvanitaki et al. (2015); Safronova et al. (2018b); Savalle et al. (2019), as well as variations in the length of solid objects Stadnik and Flambaum (2015a, 2016a); Arvanitaki et al. (2016); Geraci et al. (2018) which would arise from oscillations in $\alpha$ or the electron mass $m_e$. Direct detection of light scalar DM by probing a DM-induced equivalence-principle (EP)-violating force has been also suggested Graham et al. (2016); Hees et al. (2018). Existing limits on scalar DM-SM matter interactions come from astrophysical considerations Stadnik and Flambaum (2015b) as well as table-top experiments including radio-frequency (rf) spectroscopy in atomic Dy Van Tilburg et al. (2015); Stadnik and Flambaum (2016b); Leefer et al. (2016), long-term comparison of Cs and Rb clocks Hees et al. (2016); Stadnik and Flambaum (2016b), a network of atomic clocks Wcisło et al. (2018), EP and fifth-force (FF) experiments Touboul et al. (2017); Bergé et al. (2018); Adelberger et al. (2009); Wagner et al. (2012). In an ongoing experiment Kennedy et al. (2018), a comparison of a Sr atomic clock with a Si cavity Stadnik and Flambaum (2016a) is employed to probe scalar DM couplings at frequencies up to 10 Hz, while in Aharony et al. (2019), a similar scheme involving spectroscopy with a single Sr${}^{+}$ ion was used to probe the 1 Hz-1 MHz region (mass range $4\cdot {10}^{-15}-4\cdot {10}^{-9}$ eV).

The most stringent bounds to-date on scalar DM-SM matter couplings are the result of searches with atomic probes in the regime below the Hz-level, or tests of gravity in EP and FF apparatus. The latter experiments provide constraints up to a frequency of $\sim$ ${10}^9$ Hz. Here we present an atomic spectroscopy method for detection of rapid variations of $\alpha$ and $m_e$ that extends the frequency range probed well into the rf-band, up to 100 MHz. The rf-band is already accessible by EP/FF searches indirectly. Direct probing of fast FC variations with atoms, however, offers a conceptually different approach to studying scalar DM in the particular regime. As we will see, the rf-frequency regime is especially interesting for searches for FC oscillations associated with the presence of astronomical-size DM objects.

Our method involves a search for oscillations in the energy spacing $\Delta E$ between two electronic levels in atomic ${}^{133}$Cs, the ground state 6s${}^2$S${}_{1/2}$ and excited state 6p${}^2$P${}_{3/2}$, while the corresponding transition is resonantly excited with continuous-wave laser light of frequency $f_L\approx f_{at}$, where $f_{at}\equiv \Delta E/2\pi$. As this level spacing is approximately proportional to the Rydberg constant $\propto m_e{\alpha }^2$, our scheme is sensitive to oscillations of $m_e$ and $\alpha$.

The assumption of stable frequency $f_L$ in its comparison with $f_{at}$ requires some discussion. If $\alpha$ and $m_e$ oscillate, so does the length $L_r$ of the laser resonator, since it is a multiple of the Bohr radius $1/\alpha m_e$ Stadnik and Flambaum (2015a); Kozlov and Budker (2018), leading to a modulation in $f_L$. This modulation depends on a different combination of $\delta \alpha /{\alpha }_0$ and $\delta m_e/m_{e,0}$, compared to that of (4), and can occur at frequencies as large as $f_{c1}=v_s/L_r\approx$ 50 kHz, where $L_r\approx$ 0.12 m is the linear dimension of the resonator and $v_s\approx$ 6000 m/s is the speed of sound in the stainless-steel-made resonator structure. At frequencies below $f_{c1}$, the induced fractional oscillation in $f_L$ has amplitude Kozlov and Budker (2018):

Experiment - To search for fast variations in the Cs 6S${}_{1/2}\to$ 6P${}_{3/2}$ transition frequency, we employ polarization spectroscopy in a vapor cell Demtröder (2015) (see fig. 0(a)). The 7 cm long cell is placed inside a four-layer magnetic shield and maintained at room temperature. Two counter-propagating laser beams, termed pump and probe, are overlapped inside the cell. The circularly polarized pump induces birefringence in the Cs vapor. Analysis of the polarization of the linearly polarized probe with a balanced polarimeter yields a dispersive-shape feature against laser frequency, for each of the hyperfine components of the transition. These features have narrow widths, nearly limited by the $\approx$ 5.2 MHz natural linewidth of the transition, and serve as calibrated frequency discriminators. A typical polarization-spectroscopy signal is shown in fig. 0(b). Fast changes in $f_{at}$ will appear as amplitude oscillation in the polarimeter output. The quality factor of this oscillation is related to the coherence of the field $\varphi$ of eq. (1) and is given by $\omega /\Delta \omega \approx 2\pi /v_{DM}^2\approx 6\cdot {10}^6$, where $v_{DM}={10}^{-3}$ is the virial velocity of the DM field Krauss et al. (1985) (In the case of a relaxion halo, a longer coherence time is expected, resulting in a larger quality factor Banerjee et al. (2019).) Within the 20 kHz-100 MHz band probed in the experiment, the expected spectral width $\Delta \omega /2\pi$ of the oscillation is in the range 3 mHz - 17 Hz.

To account for the decrease in the atomic response at frequencies above the transition linewidth, and other response non-uniformities in the apparatus, a frequency calibration is required. This is done by imposing frequency modulation on the laser light with the use of an electro-optic modulator (EOM), and comparing the amplitudes of this modulation, as measured with the atoms and with a Fabry-Perot cavity of known characteristics that serves as a calibration reference.

During an experiment, the laser frequency is tuned to excite atoms from the $F=3$ hyperfine level of the ground state to the $F^{\prime }=2$ level of the excited state. The output of the balanced polarimeter is measured with a spectrum analyzer (Keysight N9320B). To produce a high resolution noise power spectrum in the 20 kHz-100 MHz range, measurements in $\approx 22,000$ frequency windows are required, each of which consists of 461 bins; a bin is 10 Hz wide and corresponds to integration time of 5 ms. Approximately 22 hr is required to acquire such a spectrum. To ensure long-term frequency stability of the laser, its frequency is stabilized to the atomic resonance. This is achieved by modulating $f_L$ at 167 Hz with an amplitude of 200 kHz, and demodulation of the measured probe beam power with a lock-in amplifier provides an error signal, to which the laser frequency is stabilized with a bandwidth of 2 Hz.

Data analysis - A set of three high-resolution noise-power spectra in the 20 kHz-100 MHz range were acquired and analyzed to probe fast oscillations in $f_{at}$. The mean and variance of each spectrum were computed in several selected frequency regions and were found to be consistent among the three spectra to within 2%. The slope of the $F=3\to F^{\prime }=2$ feature in the polarization spectrum of fig. 0(b), relevant to the sensitivity in detecting oscillations in $f_{at}$, was stable to within 6% during the entire 66 hr long acquisition run. An averaged spectrum was computed from the three high-resolution power spectra. The sensitivity in detection of FC oscillations at a given frequency is related to the fluctuations of the noise power level in that spectrum, within the particular frequency range. For each frequency bin within this range, the noise fluctuations define a global power threshold (i.e. accounting for the look elsewhere effect) at the 95% confidence level Scargle (1982). Any peak above this threshold must be then investigated for possible detection.

A number of peaks were present in the averaged spectrum whose power exceeded this threshold. The origin of these peaks was investigated by comparing their power with the laser frequency tuned on- and off- the $F=3\to F^{\prime }=2$ resonance, or with use of a second Ti:Sapphire laser and external-cavity diode laser to acquire data. These lasers have different technical noise spectra compared to that of the primary laser system. All features exceeding the 5% threshold were traced to either laser technical noise or rf-pickup in the apparatus, and in the majority of cases their power was measured accurately and subtracted out, such that the residual power was bellow the detection threshold.

No signal of unknown origin with power above the threshold was detected. In its absence, an upper limit is placed on possible oscillations of the frequency $f_{at}$, which is presented in fig. 2 at the 95% (CL).

Constraints on scalar DM couplings - We use the obtained bounds on $\delta f_{at}/f_{at}$ to constrain the parameters $g_{\gamma }$ and $g_e$ of Eqs (2) and (3). With the assumption that DM-induced oscillations in $f_a$ arise solely due to either the coupling to the photon or to the electron, we set bounds on the corresponding coupling constants, and present these in fig. 3a and fig. 3b. In the same plots, corresponding limits derived from analysis Hees et al. (2018) of results of EP experiments, as well as limits derived from naturalness are also shown. In the case of a scalar field $\varphi$, naturalness requires that radiative corrections to the mass $m_{\varphi }$, arising due to its interactions, be smaller than the mass itself Arvanitaki et al. (2016); Graham et al. (2016). In the present work, where a DM field that has scalar couplings to SM matter is considered, this requirement leads to the constraints: $|g_e|<4\pi m_{\varphi }/\Lambda$, $|g_{\gamma }|<16\pi m_{\varphi }/{\Lambda }^2$, where $\Lambda$ is the cut-off scale for the Higgs mass Banerjee et al. (2018).

To obtain the bounds of fig. 3a and 3b, $g_{\gamma }$ and $g_e$ were treated independently. Within the relaxion DM model Banerjee et al. (2018), however, this assumption is not valid. These couplings are related; both acquire values dependent on the relaxion-Higgs mixing, which is parametrized in terms of a mixing angle $\theta$. For the range of mass $m_{\varphi }$ probed in this work (${10}^{-10}$-${10}^{-6}$ eV), the contribution of the relaxion coupling to the electron is expected to dominate the oscillations in $f_a$ Banerjee et al. (2018). One can therefore assume that $\delta f_{at}/f_{at}\approx \delta m_e/m_e$ in Eq. (4) and employ the defining relation between $g_e$ and the mixing angle $\theta$, to constrain $\theta$ within the investigated $m_{\varphi }$ region. The parameter $g_e$ is given by Banerjee et al. (2018) :

An enhancement in the amplitude of FC oscillations is expected in the presence of an astronomical-scale DM object around the earth or in its vicinity. Such an enhancement would occur due to an increase in the local DM density ${\rho }_{DM}$ [see eqns (1), (2), (3)]. Searches for transient variations of $\alpha$ using a network of GPS satellites Roberts et al. (2017) and a terrestrial network of remotely located atomic clocks Wcisło et al. (2018), have provided constraints on topological DM. Here we consider the scenario of an earth-bound relaxion halo, examined in Banerjee et al. (2019). We make use of the computed DM density ${\rho }_{DM}$ Banerjee et al. (2019) at the surface of the Earth, to provide more stringent constraints on the couplings $g_{\gamma }$ and $g_e$ than these shown in fig. 3 (conditional on the existence of the relaxion halo). These tighter bounds are presented in fig. 4. In the presence of the halo, the enhancement in ${\rho }_{DM}$ is expected to be mostly pronounced in the mass range ${10}^{-12}$-${10}^{-8}$ eV. We note that the corresponding frequency regime ${10}^4$-${10}^8$ Hz has been out of reach for most of the laboratory searches for variations of FC, which with the exception of the recent work Aharony et al. (2019), have been mostly sensitive to frequencies below 1 Hz.

Discussion and outlook - The obtained constraints on light DM scalar interactions with SM matter extend the frequency range investigated with atomic probes to the ${10}^8$ Hz, a regime not previously searched directly. More stringent bounds on the scalar coupling to the photon and the electron exist within the 20 kHz-100 MHz range of this work. Such constraints, however, are derived from EP and FF experiments, which are not directly sensitive to rapid oscillations of FC, as is the method demonstrated here.

The sensitivity in detection of DM-induced FC variations for given bound on $\delta f_{at}/f_{at}$ is inversely proportional to the frequency/mass probed [see Eq. (4)]. Use of atomic clocks is therefore advantageous, in that these probes typically search the sub-Hz regime, with resulting DM constraints competing against those provided by EP and FF studies. Without assuming any enhancement in the local DM density due to the Earth, our polarization spectroscopy scheme is unlikely to approach a sensitivity level comparable to that offered by EP/FF searches because of the higher frequencies being probed. However, searching for rapid FC oscillations in the rf-band allows to access a frequency range, which as discussed, might provide enhanced sensitivity in detection of FC variations within scenarios of an earth-bound DM-halo. As seen in fig. 4, the bounds from polarization spectroscopy are at regions better than those set by EP results. Improvements in detection of modulation in the Cs energy levels involved in the experiment, would enable a search deeper into the parameter space of $g_{\gamma }$ and $g_e$.

Several apparatus upgrades are under way. The primary improvement involves a change in the scheme employed to obtain the frequency spectrum of the polarization-spectroscopy signal. The spectrum analyzer currently used will be replaced with fast data-acquisition electronics with the ability to perform real-time high-resolution spectral analysis, thereby increasing the effective integration time drastically. An optimization of the signal-to-noise ratio in the polarization spectroscopy setup is also being explored. These improvements, combined with a longer integration time of up to $\sim {10}^4$ hr, should result in a sensitivity enhancement in excess of ${10}^3$. This level of improvement is sufficient to explore scalar interactions between DM and SM matter with a sensitivity better than that of other methods in a significant fraction of the parameter space accessible by polarization spectroscopy.