The total mass density of the Universe appears to be dominated by dark matter. However, beyond its gravitational interactions at the galactic scale, little is known about its nature1. Several proposals have been advanced in recent years for the detection of dark matter2,
The main components of an optical atomic clock are a sample of cold, trapped atoms that are isolated from the environment and a laser locked to an ultra-stable optical cavity 12 . Optimal for this purpose are atomic species that posses an ultra-narrow optical transition, called a clock transition. The exceptionally small spectral width of this transition combined with a high frequency value for the optical radiation results in highly accurate spectroscopic measurements that have already reached 10−18 (refs 13,14 ). In the ideal case of perfectly isolated atoms, the frequency of the clock transition ω 0 is dictated by the values of fundamental physical constants. In typical applications, clock transitions serve as the most stable frequency references available. In our experiment, however, the different sensitivities of the clock transition and the optical cavity to variations in the fundamental constants 10,11 enable a search for non-gravitational signatures of topological-defect dark matter (TDM).
Here we show that a single optical neutral atomic or ion clock is sensitive to such signatures (Fig. 1). The frequency of the laser is tightly locked to the ultra-stable optical cavity. The beam, after passing through the frequency shifter (FS in Fig. 1), probes the trapped atoms. The shifter correction is actively controlled to keep the frequency of the beam locked to the clock transition. Therefore, changes in the frequency correction correspond to changes in the frequency of the clock transition with respect to the cavity. We define the clock readout, r(t), as the frequency correction at the FS. If there is a non-zero coupling between dark matter (DM) and standard model fields (DM–SM coupling), then when the Earth traverses a DM object, this DM will perturb certain standard model parameter values. In particular, we may expect a transient variation in the electromagnetic fine-structure constant, α, that can be expressed as , where Φ is the DM field and Λ α is the energy scale (which inversely parametrizes the strength of the DM–SM coupling) 5,17,18 . Such a variation will shift the frequency of the electronic clock transition and the frequency of the chosen cavity mode. Different susceptibilities of these two frequencies to variations in α (ω ∝ α 2 for non-relativistic atoms and ω ∝ α for the cavity) make a single optical atomic clock sensitive to hypothetical dark matter objects (see Methods for details) and hence will directly manifest in the readout, r(t). The theory of how atomic clock transitions (with respect to an optical cavity) respond to a variation in α was recently developed 10,11 , and its application in our experiment is discussed in the Methods. The concept that a single optical atomic clock is already sensitive to TDM greatly simplifies establishing a global network of optical atomic clocks aimed at the detection of TDM. With this approach, unlike those described in refs 5,15 , the optical frequencies are compared locally, so that phase-noise-compensated optical fibre links of Earth-sized lengths are no longer needed. This implies that the existing network of optical atomic clocks can serve as a global topological-defect dark-matter observatory without any further developments in the experimental apparatus. It will be complementary to other experimental approaches 19,11 , for example, those originally aimed at gravitational wave detection 20 or cosmic microwave background polarization measurements 21,22 . The capabilities of laser and maser interferometry for dark-matter searches were comprehensively discussed in some recent papers 10,11 .
A single optical atomic clock is sensitive to a transient signal from a hypothetical DM object. Under real experimental conditions, however, the DM signature is expected to be hidden by noise; hence, a potential signal from a DM object cannot be distinguished from other effects. One possible solution is to simultaneously monitor at least two independent channels. If an event occurs that is common to both, but much larger than is possible for a common component estimated from all known physical phenomena, then it may be associated with an as yet unknown interaction. To provide unambiguous evidence for clocks coupling to the DM halo, they should be separated by large distances (on the scale of the Earth). Positive verification of the DM–SM coupling is impossible when the magnitude of other common effects cannot be quantified. Nevertheless, both in distant and co-located arrangements, a measurement of the common component s(t) can still provide a constraint on the transient variation of α and on the magnitude of the DM–SM coupling. In the bottom part of Fig. 2, we show how the readouts of two colocated clocks could respond to a cascade of TDM objects passing through the clocks. This common component can be extracted by cross-correlating the readouts r 1(t) and r 2(t). The constraint on the transient variation of the fine-structure constant can be expressed as:
where A 0 is the amplitude of the cross-correlation peak, K α is a sensitivity coefficient (equal to one in the non-relativistic case 10,11 ), and η T is the ratio of the overall DM signal duration to the length of the cross-correlated readouts, t 2 − t 1 (see Methods for details). The inequality given in equation (1) can be translated into a constraint on the energy scale in the hypothetical DM–SM interaction Lagrangian:
where d is the size of the defect, ρ TDM is the mean DM energy density, is the time between consecutive encounters with DM defects, and v is the relative speed of the topological defects (see Methods for details).
In our measurement (Fig. 2) we used a system of two nearly identical optical-lattice clocks 23,24 placed approximately 10 m apart (see Methods for details). Both clocks shared the same reference optical cavity, which was the dominant source of the common signal. We observed, however, that the majority of this signal was in the low-frequency range. Therefore, we applied a high-pass filter with a frequency cutoff at 0.027 Hz to both readouts (Fig. 3). This frequency cutoff meant that the sensitivity of the measurement was decreased for topological defects with thicknesses exceeding the radius of the Earth; under the assumption of v = 300 km s−1 (ref. 5). The cross-correlation of the two signals is depicted in Fig. 3 and its amplitude A 0 is equal to 0.73 Hz2. The width of the cross-correlation peak reflects the spectral characteristics of the common noise, and the oscillating shape is a result of the applied filter and the fact that the common noise is dominant. The length of each of the two recordings, t 2 − t 1, is equal to 45,700 s. Under the arbitrary assumption that the thickness of a defect d is 10,000 km (comparable to the diameter of the Earth) and that s, we may use equation (1) to obtain an estimate of δα/α < 7 × 10−14 for the constraint on the transient variation in the fine-structure constant. Note that after the application of the high-pass filter, the amplitude of the cross-correlation peak is quite insensitive to the length of the cross-correlated signals for t 2 − t 1 > 1/(0.027 Hz). For instance, if we cross-correlate only a 100-s long portion of the entire readout, then, independent of which portion we choose, the amplitude is A 0 < 1.9 Hz2. However, for this choice of t 2 − t 1 = 100 s, the η T parameter is considerably larger at 0.33, and hence, the constraint inferred from equation (1) is much stronger:
Strong constraints on the time variation of α have been previously reported 7,25 . However, it should be noted that the physical meaning of those constraints are entirely different to that reported here. The limits reported in refs 7,25 are barely sensitive to short transient effects because of the relatively long averaging time needed for each of the measurements. However, every optical neutral atomic and ion clock is sensitive to transient variations in α (as depicted in Fig. 1). Therefore, we suggest using such clock measurements not only to constrain the long-term drifts in α, but also to search for the brief events that could originate from the postulated TDM objects; something which has not yet been done. In this study, the constraint on the coupling between TDM and standard model fields was determined for the first time using optical atomic clocks. We estimate that if clock readouts are analysed via the method we describe in this work, then the constraint obtained from equation (3) can be increased by about two orders of magnitude with a single state-of-the-art optical atomic clock 9,26,27 . In additon to improving future studies in this area, previously acquired data, for example those from refs 7,25 , could be reanalysed using our method.
We can also use our measurement to estimate a constraint on the energy scale Λ α under the assumption of ρ TDM = 0.3 GeV cm−3 (ref. 28). Similar to the considerations regarding α, the results here also depend on the choice of t 2 − t 1. The constraints on Λ α for the entire readout duration, t 2 − t 1 = 45,700 s, and for a small portion thereof, t 2 − t 1 = 100 s, are shown as functions of d as the grey and green lines, respectively, in Fig. 4. The scaling of these constraints with the defect size is proportional to d 3/4. The constraint determined here exceeds the previous limits 16 (see solid black line in Fig. 4) by more than three orders of magnitude (under the conditions considered).
The dashed blue and red lines in Fig. 4 are the idealized constraints presented in a previous proposal 5 for a trans-continental network of Sr optical-lattice clocks and a GPS constellation, respectively. However, the proposal underestimated the sensitivity coefficient K α by a factor of 34 (see Methods). The dotted blue and red lines in Fig. 4 represent the corrected constraints. Thus, it can be seen that our experimental limit already reaches the constraint for the constellation of GPS clocks, despite the fact that ref. 5 assumes more optimistic conditions of yr. It should be noted, however, that the measurements with microwave clocks are very important. They are sensitive not only to variations in α but to a linear combination of fundamental constants. The green dashed line in Fig. 4 depicts the possible constraint achievable using our method given the same ideal conditions considered in ref. 5 ( yr and σ/ω 0 = 10−18, for a 1 s acquisition step). For this purpose, we assume η T = 1/3 for any d. The interpretations of these two approaches are, however, not exactly the same. Our approach provides a direct recipe for the treatment of the experimental readouts, whereas the previous seminal work shows a general concept based on a phase-difference measurement 5 . Furthermore, our constraint is valid for both distant and closely spaced clocks. This implies that even in the case of distant clocks, our method is not limited by the need for a phase-noise-compensated optical fibre link of a length comparable to the size of the Earth 15 , and that, therefore, our limits (Fig. 4) are directly applicable.
The present experiment was performed using two co-located optical clocks. However, the same approach can be applied to distant clocks. For example, if two clocks were to be placed on opposite sides of the Earth and a series of TDM objects were then to traverse the Earth, moving from one clock to other (at v = 300 km s−1), then the cross-correlation peak in Fig. 3 would appear not at Δt = 0 s but at Δt ≈ 42 s. In a real experiment, the problem is more complex: for long measurements, the effects related to the translational and rotational motion of the Earth must be disentangled. To account for these effects, in one of the signals in the cross-correlation function, equation (11), the time t must be replaced with its corrected value: where f is a function of the position r and orientation θ of the Earth with respect to a reference frame. For example, if we assume that the TDM objects are traversing the Earth along the equatorial plane, then the Earth’s rotation can be disentangled from the signal by taking f(t) = (2R ⊕/v) cos(2πt/T day), where R ⊕ is the Earth’s radius and T day is its period of rotation.
In our approach, the readouts of the distant clocks do not need to be correlated in real time. Just as in the standard radio-astronomical technique, very-long-baseline interferometry, they can be locally recorded (with time stamps that are accurate to the level of 1 ms) and cross-correlated later. A concept for archiving data from various types of precise measurements was reported previously 29 . Any number of scenarios involving movement of TDM objects with respect to the Earth, and their associated f functions, can be tested during post-processing of the data. For the correct function, the cross-correlation peak would be expected to occur at Δt = 0. Otherwise, the common DM–SM coupling pulses would be mismatched and would destructively interfere in the cross-correlation response. Such a measurement with distant clocks requires a correct guess regarding the form of the f function; however, it provides richer information about the motion of TDM objects and hence clearer evidence that the clocks are coupling to the dark matter halo. A common weakness of all such searches for hypothetical transient effects is the assumption that some minimal number of expected events will occur during the observation time. Therefore, it will be crucial to establish a long-term, worldwide observation programme to continuously record the signals over years or even decades.
Susceptibility of atoms and cavities to dark matter
The expansion and cooling of the early Universe could have involved phase transitions related to hypothetical DM cosmic fields, which may have resulted in the formation of topological defects stabilized by a self-interaction potential. Such objects, if sufficiently light, would be of macroscopic size. We refer the reader to the review by Vilenkin 6 . Recently, several estimates concerning these postulated objects have been presented; see the Supplementary Information of ref. 5. Extensions of the QED Lagrangian with DM–SM couplings (so-called portals 30 ) may result in variations in standard model parameters. For instance, the relative change in α, for the case of a quadratic scalar portal, can be expressed as where Φ inside is the DM field inside a TDM object and Λ α is the energy scale.
To consider how the electronic transition with respect to the optical cavity responds to the variation of α, it is useful to write down the electronic Schrödinger equation in the Born–Oppenheimer approximation, in SI units, for n electrons and m nuclei: where m e is the electron mass, Z j is the number of protons in the jth nucleus, r ji = |R j – r i | and r ik = |r i – r k | . The coordinates of the jth nucleus and the ith electron are R j and r i , respectively. One may write equation (6) in dimensionless form as follows: where є = E/E h and x i = r i /a 0, x ji = r ji /a 0 and x ik = r ik /a 0, with E h = α 2 m e c 2 and a 0 = ℏ/(m e αc). This simple transformation into the dimensionless form reveals a very important property, namely, that the values of the dimensionless energies ε and their dependence on the dimensionless positions of the nuclei, X j = R j /a 0, do not depend on any fundamental constant. This means that, within this approximation, the energy of the system scales as E ∝ α 2 and its linear dimensions scale in proprtion to α −1 (refs 10,11 ) for any system starting from simple atoms and molecules up to crystals and even amorphous solids. It turns out that this approximation works well for light elements (including those relevant to our experiment, Sr, Si, O and Ti) when relativistic effects can be neglected 31,32 . In particular, the lengths of optical cavity spacers, when made of either single-crystal silicon 33 or completely amorphous ultra-low expansion glass (as in our experiment), follow this simple scaling, L ∝ α −1. Therefore, the frequency of each longitudinal cavity mode (N cav) can be written as
When relativistic effects are also considered, the dependence of the frequency of an electronic transition on the fine-structure constant α can be expressed, in SI units, as: where is a sensitivity factor 31,32 that indicates how sensitive the transition is to α with respect to the non-relativistic case (ω 0 ∝ α 2). For light elements, for which relativistic effects are small, the sensitivity factor is close to zero.
It follows from equations (8) and (9) that the deviation of the frequency of the clock transition with respect to the cavity, , due to the variation of α is: where is the effective sensitivity coefficient, with X representing the element used. For instance, in the case of our clocks, (refs 31,32 ); hence, the effective sensitivity is K α = 1.06. For the much heavier element mercury, K α = 1.8. In the non-relativistic case, K α = 1, which reproduces the result presented in refs 10,11 . In the above considerations based on SI units, we assume that the definition of the second is not perturbed by the presence of DM.
It follows from equation (9) that if we consider the arrangement proposed in ref. 5, that is, a comparison of clock transitions in two distant atomic samples, then the sensitivity coefficient would be , and not , as stated in ref. 5. The constraints on Λ α for optical and microwave atomic clocks that are given in ref. 5, after this correction, are respectively depicted as blue and red dotted lines in Fig. 4.
The readouts from the two atomic clocks shown in Fig. 2 (the frequency corrections from the shifters FS1 and FS2) may be written as r 1(t) = n 1(t) + s(t) and r 2(t) = n 2(t) + s(t), respectively, where s(t) is the common component and n 1(t) and n 2(t) are the independent noise components in the two detectors. The common component can be extracted from the cross-correlation of the two readouts: where t 1 and t 2 are the limits of the time interval over which the signals were recorded. The cross-correlation, which is given by equation (11), has a resonance-like shape centred at Δt = 0 (see Fig. 3). The amplitude of this peak, A 0 = (r 1*r 2)(Δt = 0 s), reflects the magnitude of the common component s(t). As a first approximation, let us assume that s(t) consists of a series of N p square pulses with arbitrary durations T i and the same height s 0, which, in an ideal experiment, would correspond to DM objects of arbitrary dimensions and the same energy density ρ inside (see the bottom of Fig. 2). Then, the amplitude of the cross-correlation peak A 0 can be expressed as: where . If the two detectors do not suffer from any common instrumental noise or drifts (or they are negligible), then s 0 can be identified with the TDM signal δω 0. If the magnitude of the common apparatus effects cannot be quantified, then positive verification of the DM–SM coupling is impossible. Nevertheless, the amplitude of the cross-correlation peak still gives a constraint on its magnitude: where η T = T tot/(t 2−t 1).
It is also instructive to assume that the common apparatus effects are zero and ask how the detection limit of the system presented in Fig. 2 depends on the intrinsic noise and stability of each of the two clocks, which we denote as n 1(t) and n 2(t). Both clocks are the same; hence, both noise components n 1(t) and n 2(t) are characterized by the same standard deviation, σ. It follows from equation (11) that the noise of the cross-correlation of the signals (its standard deviation) is , where N is the number of samples in the readout of each sensor. Under the assumption that either the signal is weak (s 0 ≪ σ) or the mean pulse duration is small (η T ≪ 1), only the leading term must be retained. If the common apparatus effects are negligible and no cross-correlation peak is observed, then a constraint on the strength of the DM–SM coupling can be determined from the noise level (σ) of the clocks’ readouts:
Equation (14) expresses a fundamental limitation of our approach. The important point emerging from this discussion is that the fundamental detection limit of our approach is only determined by the standard deviation of the clock noise at short timescales, not by the system instability (minimum of the Allan variance). Indeed, the goal of the experiment is not an ultra-accurate determination of the frequency of the clock transition in the two sensors but merely the tracking of their common changes as precisely as possible. Therefore, because clock drift is no longer an issue, the readouts can be arbitrarily long, even exceeding the length of the period indicated by the minimum of the Allan variance.
Equations (13) and (14) provide a complete recipe for interpreting the outcome of the experiment depicted in Fig. 2. If a peak is observed in the cross-correlation function (at Δt = 0), then its amplitude determines a constraint on the effect of the DM according to equation (13). Otherwise, if the cross-correlation is flat, the constraint is determined by the standard deviation of a single measurement σ and by the number of samples N, as shown in equation (14). When the influence of the common apparatus effects is estimated to be negligible, a positive verification of the hypothetical DM–SM interaction is possible, and the strength of the TDM signal is . The same approach may be implemented in other types of experiments looking for transient effects of dark matter, such as those based on synchronized optical magnetometers 34,35 .
The constraint on the transient variation in α that is given by equation (1) can be derived directly from equation (10) and inequality (13). The constraint on the energy scale Λ α of the hypothetical DM–SM interaction Lagrangian that is given by equation (2) can be derived by using equation (5) and (following ref. 5) estimating the magnitude of the field inside a TDM object as .
The experimental setup consisted of two optical atomic clocks 12 with neutral 88Sr atoms trapped in optical lattices 36 . Each clock cycle was divided into preparation and interrogation phases. In the preparation phase, using the 1 S 0–1 P 1 transition at 461 nm, hot strontium atoms from the oven were first cooled to 2 mK using a Zeeman slower 37 and a magneto-optical trap 38 . Using the 1 S 0–3 P 1 7.5 kHz intercombination transition, we further cooled the atoms to a few μK. Subsequently, a sample of a few thousand atoms was loaded into a one-dimensional optical lattice operating at 813 nm. In the interrogation phase, the strontium atoms were probed at the 1 S 0–3 P 0 clock transition at 698 nm (429 THz) with the ultra-narrow clock laser stabilized to the optical cavity. This allowed the excitation probability to be measured as a function of the probe light frequency and allowed the feedback correction to this laser frequency to be calculated. This process heats atoms and removes them from the lattice.
In our setup 23,24 , the two optical atomic clocks shared the same optical cavity. The laser that is locked to it is split into two beams, whose frequencies are controlled separately by acousto-optic frequency shifters. The optical cavity consisted of a 100 mm ultra-low expansion glass spacer and fused silica mirrors. The relative instability of the clock laser, measured using the strontium atoms, was lower than 2 × 10−14 on timescales of 2 to 104 s.
The clock laser frequency was digitally stabilized to the atomic transition by feedback corrections applied to the frequency shifters. The frequency corrections were derived from the excitation probabilities recorded on both sides of the clock transition line using a previously described algorithm 39 . One servo-loop cycle (a single readout), required two cycles of the clock.
The presented results were obtained using synchronized 1.3 s cycles of the two clocks, which means that probing of the clock transition (interrogation time, 40 ms) in the two systems was performed in parallel, with corrections being applied simultaneously. The readouts from the two clocks were recorded independently by two different computers with their internal clocks synchronized to coordinated universal time (UTC) by a stratum 1 network-time-protocol server.
The interpretation of the clocks’ response to TDM events is straightforward when their duration is longer than one servo-loop cycle (2.6 s in our case). However, in the limit of one servo-loop cycle, the sensitivity of the clocks to TDM objects could drop by a factor of two for typical servo-loop settings. Nevertheless, if we assume that during the observation time, a sufficiently large number of TDM events occurs (randomly distributed in time), then our approach is also capable of detecting events shorter than one servo-loop cycle. In this case, the probability that such a TDM event will overlap with the interrogation period is smaller than one. On the other hand, the contribution of any single such TDM event to the height of the cross-correlation peak will be overestimated. These two effects cancel out, causing the sensitivity to be the same as that for longer events. For events that partially overlap the interrogation period, we assume that the clocks’ response is proportional to the overlap fraction; however, the signal may vary slightly depending on the type of spectroscopy used (in our case, Rabi spectroscopy).
In the proposal described in ref. 5 the clock transitions in the two distant atomic samples were directly compared via an optical link. Recently it was demonstrated 15 that the requirement for phase-noise-compensated optical fibre links of lengths comparable to the size of the Earth would constitute a limiting factor for such measurements. Our method does not suffer from this type of noise because it does not require such an optical link to operate. In our approach, each channel possesses two optical references, the atomic clock transition and the optical cavity. Therefore, the comparison of these two frequencies can be performed locally and separately for each channel. The readouts from the two channels can then be analysed digitally during data post-processing. In our approach, the clocks’ cycles of operation must be synchronized at the millisecond level, which can be easily achieved using GPS or a standard internet connection. Here, the discussion is restricted to a setup consisting of two optical atomic clocks; however, these remarks also apply to a system composed of a larger number of detectors.
All data supporting the findings of this study are available from the authors upon reasonable request.
How to cite this article: Wcisło, P. et al. Experimental constraint on dark matter detection with optical atomic clocks. Nat. Astron. 1, 0009 (2016).
We are grateful to V. V. Flambaum and Y. V. Stadnik for discussions and crucial remarks concerning the response of an atomic clock transition and an optical cavity to variations in the fine-structure constant, which helped us to properly evaluate our constraints. We also thank W. Ubachs and S. Pustelny for the inspiring discussions. The reported measurements were performed at the National Laboratory FAMO in Toruń, Poland, and were supported by a subsidy from the Polish Ministry of Science and Higher Education. Support has also been received from the project, EMPIR 15SIB03 OC18. This project has received funding from the EMPIR programme co-financed by the Participating States and from the European Union’s Horizon 2020 research and innovation programme. The individual contributors were partially supported by the National Science Centre of Poland through the following projects: 2015/19/D/ST2/02195, DEC-2013/09/N/ST4/00327, 2012/07/B/ST2/00235, DEC-2013/11/D/ST2/02663, 2015/17/B/ST2/02115 and 2014/15/D/ST2/05281. This research was partially supported by the TEAM Programme of the Foundation for Polish Science, which is co-financed by the EU European Regional Development Fund and the COST Action, CM1405 MOLIM. P.W. is supported by the Foundation for Polish Science’s START Programme.