Gravitational wave detection with optical lattice atomic clocks

We propose a space-based gravitational wave (GW) detector consisting of two spatially separated, drag-free satellites sharing ultrastable optical laser light over a single baseline. Each satellite contains an optical lattice atomic clock, which serves as a sensitive, narrowband detector of the local frequency of the shared laser light. A synchronized two-clock comparison between the satellites will be sensitive to the effective Doppler shifts induced by incident GWs at a level competitive with other proposed space-based GW detectors, while providing complementary features. The detected signal is a differential frequency shift of the shared laser light due to the relative velocity of the satellites, and the detection window can be tuned through the control sequence applied to the atoms’ internal states. This scheme enables the detection of GWs from continuous, spectrally narrow sources, such as compact binary inspirals, with frequencies ranging from $\sim 3\mathrm{mHz}–10\mathrm{Hz}$ without loss of sensitivity, thereby bridging the detection gap between space-based and terrestrial optical interferometric GW detectors. Our proposed GW detector employs just two satellites, is compatible with integration with an optical interferometric detector, and requires only realistic improvements to existing ground-based clock and laser technologies.

Figure 1

Proposed gravitational wave detector (not to scale). Our detector consists of two identical drag-free satellites, A and B, separated from each other by a distance $d$ along the $x$ axis. Each satellite contains a free-floating reference mass, an ultrastable laser, and a strontium optical lattice clock. A mirror is mounted on the free mass and is used to define the standing wave of light forming the optical lattice and confining the Sr atoms. Some of the laser light from satellite A (orange, dashed line) is sent to satellite B. The light first passes through an acousto-optic modulator driven at frequency $f_A$, which offsets the frequency of the light reaching photodiode $2_B$ in satellite B and enables the phase locking of laser B to laser A through heterodyne detection. Vibrations and thermal drifts of the optics on each satellite can be corrected locally by feeding back on the beat notes at $2f_{A,B}$ on photodiodes $1_{A,B}$. Light from laser B (blue, dotted line) is sent back to satellite A to verify the phase lock, to maintain pointing stability, and to enable operation in the reverse mode, with laser A locked to laser B. A plus-polarized gravitational wave propagating along the $z$ axis induces relative motion between the two free masses (see Appendix pp1), which can be detected using a clock comparison measurement protocol. The satellite configuration and orbit shown here is intended only for illustration of the basic concepts of our detector. A more sophisticated orbital pattern could be employed to increase the rate of rotation and sweep the detector pattern over a larger region. Additional satellites and optical links could also be used for improved sensitivity and localization of GW sources.

Figure 2

Measurement protocols and comparison of gravitational wave sensitivities. (a) (1) A Ramsey pulse sequence performed on the atomic clock transition can be used to detect a GW with $f_{\mathrm{GW}}=3\mathrm{mHz}$, for which a half period matches the atomic linewidth limited interrogation time $T_{\max }=160\mathrm{s}$. From top to bottom, we depict the effect of a GW on the relative position of the two satellites (A and B), the sign of the Doppler shift induced on the transmitted laser light, the accumulated clock signal $s$ [see Eq. (1)], and the pulse sequence (the pink dotted line indicates the atom state readout). (2) The same Ramsey pulse sequence as in panel 1 will measure a reduced signal for a GW of frequency $f_{\mathrm{GW}}=500\mathrm{mHz}$, because the fast Doppler-shift oscillations will average out. (3) A series of shorter Ramsey sequences with $T^{\prime }=1\mathrm{s}$ can be used to detect a $f_{\mathrm{GW}}=500\mathrm{mHz}\mathrm{GW}$, with a reduced sensitivity due to the shorter coherent interrogation time. (4) A DD sequence with 159 periodic $\pi$-pulses separated by $T^{\prime }=1\mathrm{s}$ can instead be employed to detect a GW with $f_{\mathrm{GW}}=500\mathrm{mHz}$, resulting in the same total accumulated signal $s$ as the Ramsey measurement for $f_{\mathrm{GW}}=3\mathrm{mHz}$ as shown in panel 1. (b) Noise-limited strain sensitivity of our detector to a monochromatic GW using a Ramsey sequence with interrogation time $T_{\max }=160\mathrm{s}$ (orange filled region), and a Ramsey sequence with $T^{\prime }=1\mathrm{s}$ (green filled region). The Ramsey sensitivity envelope for optimized Ramsey sequences at each GW frequency is shown (orange dashed line), and the projected strain sensitivity of LISA is plotted for comparison [18]. The clock GW detector consists of one clock per satellite, each with $7\times 10^6$ atoms, with the baseline length optimized for $f_{\mathrm{GW}}\ge 3\mathrm{mHz}$, giving $d=5\times 10^{10}\mathrm{m}$. (c) Noise-limited strain sensitivity of our detector to a monochromatic GW using a DD sequence with 159 periodic $\pi$-pulses with total interrogation time $T_{\max }=160\mathrm{s}$ (purple filled region), and the DD sensitivity envelope for optimized DD sequences at each GW frequency (thick purple line). In both Fig. 2 and 2 the strain sensitivities corresponding to panels 1–4 in Fig. 2 are highlighted. The clock GW detector has narrow regions of reduced sensitivity for each measurement sequence when the time between pulses, ($T_{\max }$, $T^{\prime }$), is an integer multiple of a GW period. The polarization and direction of propagation of a GW can also change the measurable signal for LISA and the clock GW detector. All sensitivity curves and envelopes are averaged over all polarizations and directions of propagation of the GWs (see Appendix pp2).

Figure 3

A GW incident along the $z$ axis periodically changes the light travel distance between $A$ and $B$.

Figure 4

Transfer functions for a detector sensitive to changes in frequency, ${\mathcal{T}}_{\nu }(f)$ (red curve), as compared to a detector sensitive to phase, ${\mathcal{T}}_{\varphi }(f)$ (blue dotted curve). Frequency measurements yield the maximal signal for $f=(n+1/2)c/d$, $n\in {\mathbb{N}}_0$, while the sensitivity is drastically reduced for $f<c/(2d)$. In contrast, phase measurements become significantly less sensitive for frequencies $f\gtrsim c/(2d)$, even without the presence of noise. Here the distance between satellites is $d=5\times 10^{10}\mathrm{m}$, as in the main text.

Figure 5

Transfer function capturing the sensitivity to frequency fluctuations, Eq. (c4). The blue dashed curve shows the case for $n=10$ measurements with no dead time and ${\mathrm{\Omega }}_R=100\mathrm{Hz}$. The orange and red curves show the transfer functions for $n=10$, and $r=0.8$, with ${\mathrm{\Omega }}_R=100\mathrm{Hz}$ (orange curve) and ${\mathrm{\Omega }}_R=10\mathrm{Hz}$ (red dashed curve), and the Ramsey time $T=1\mathrm{s}$. Spikes appear due to the Dick effect at frequencies $f=r/T$. The transfer function attenuates frequencies above ${\mathrm{\Omega }}_R$ and thus acts as an effective low-pass filter.

Figure 6

Fractional frequency instability as a function of averaging time $\tau$ as given in Eq. (c6), for the case of zero dead time ($r=1$). The red curves correspond to an atomic linewidth ${\mathrm{\Delta }}_A=1\mathrm{mHz}$ as in the main text, while the blue curves are for a narrower atomic transition with ${\mathrm{\Delta }}_A=10\mu \mathrm{Hz}$. The thick and dashed lines differ by the received optical power: 95 pW (red thick line), 10 nW (blue thick line) and $1\mu \mathrm{W}$ (blue and red dashed lines). The dotted lines show the atom projection noise limit as given in Eq. (2) of the main text. For short averaging times photon shot noise dominates and the noise scales with $1/\tau$, while at long averaging times atom projection noise dominates and the noise scales as $1/\sqrt{\tau }$.