Resonant mode for gravitational wave detectors based on atom interferometry

We describe an atom interferometric gravitational wave detector design that can operate in a resonant mode for increased sensitivity. By oscillating the positions of the atomic wave packets, this resonant detection mode allows for coherently enhanced, narrow-band sensitivity at target frequencies. The proposed detector is flexible and can be rapidly switched between broadband and narrow-band detection modes. For instance, a binary discovered in broadband mode can subsequently be studied further as the inspiral evolves by using a tailored narrow-band detector response. In addition to functioning like a lock-in amplifier for astrophysical events, the enhanced sensitivity of the resonant approach also opens up the possibility of searching for important cosmological signals, including the stochastic gravitational wave background produced by inflation. We give an example of detector parameters which would allow detection of inflationary gravitational waves down to ${\mathrm{\Omega }}_{\mathrm{GW}}\sim {10}^{-14}$ for a two-satellite space-based detector.

Figure 1

A space-time diagram of an example resonant atom interferometer detector sequence. The detector consists of two atom interferometers, one at position $x_1$ and the other at position $x_2$, where $x_2=x_1+L$ and $L$ is the baseline. The spacetime trajectories of the atoms are shown in blue for the ground state and red for the excited state. Pulses of light (thin black lines) are sent back and forth from each end of the baseline and interact with the atoms (interactions shown as black dots), transferring momentum to the atoms and changing their internal state. Whether or not an interaction occurs is controlled by matching the frequency of the light pulses to the Doppler shift of the atoms. The sequence shown consists of two single-photon transitions for each atom optic ($n=2$, $2\hslash k$ momentum transferred) and a resonant enhancement of $Q=3$ (three diamonds). The interrogation time $T$ depicted here is near the limit for this baseline $T\approx 2(n-1)L/c$, so the gaps between the sequential laser pulses are small.

Figure 2

A space-time diagram of an LMT enhanced resonant atom interferometer sequence. This pulse sequence uses an LMT enhancement of $n=6$ ($6\hslash k$ momentum transfer per LMT atom optic) and a resonant enhancement of $Q=3$ (three diamonds). The dashed curve is the atom trajectory of the upper arm of the interferometer and the solid curve is the lower arm. Pulses of light are shown as gray lines traveling in alternating directions from each end of the baseline. The midpoint of each LMT $\pi$-pulse sequence is indicated with a dark gray photon path, while the ($2n-1$) axillary $\pi$ pulses are shown in light gray. The first and last pulse sequences are the LMT $\frac{\pi }{2}$ pulses and consist of ($n-1$) axillary $\pi$ pulses (light gray) and one $\frac{\pi }{2}$ pulse (dark gray). As in Fig. 1, the atom ground state is shown in blue and the excited state is red, and black dots indicate atom-laser interaction points. For simplicity, only one LMT resonant interferometer is shown here (at position $x_0$), but in the detector configuration there would be two interferometers at different positions, as in Fig. 1. For the parameters chosen here, the light travel time of the $n$ pulses in each LMT atom optic is comparable to the interrogation time $T$ of the interferometer, so the atom trajectories exhibit substantial curvature. Also, the recoil velocity $v_r$ has been greatly exaggerated to show detail, but in any real detector we always have $v_{r}T\ll L$.

Figure 3

Strain sensitivity for a detector using resonantly enhanced LMT atom interferometry. The design is based on single-photon transitions in Sr and uses a heterodyne laser link over a single baseline of length $L=4.4\times 10^7\mathrm{m}$. The atom interferometer phase noise is taken to be $\delta {\varphi }_a={10}^{-5}\mathrm{rad}/\sqrt{\mathrm{Hz}}$, and the design assumes a telescope diameter of $d=50\mathrm{cm}$ which is large enough to keep the contribution of photon shot noise less than $\delta {\varphi }_a$ for all frequencies in the detection band. The sensitivity curves for several possible pulse sequences are shown (red, green, blue, orange) with different choices for the resonant enhancement $Q$ and LMT $n$. The black curve shows the peak on-resonance response at each frequency that can be reached with an appropriate choice of $n$ and $Q$. In all cases, the parameters of the sequence are constrained so that the total number of pulses does not exceed $n_{\max }=10^3$ and the total interferometer duration is less than $T_{\max }=300\mathrm{s}$. The shaded regions below the black curve indicate the value of $n$ used in each range. The repetition rate of the detector is $f_{\text{rep}}=\frac{\chi }{2T}$, where we assume sufficient multiplexing to reach $\chi =10$ samples per gravitational wave period.

Figure 4

Stochastic gravitational wave sensitivity. The black curve shows the sensitivity to ${\mathrm{\Omega }}_{\mathrm{GM}}$ using the same detector design as Fig. 3, with baseline length $L=4.4\times 10^7\mathrm{m}$, phase readout noise $\delta {\varphi }_a={10}^{-5}\mathrm{rad}/\sqrt{\mathrm{Hz}}$, and telescope size $d=50\mathrm{cm}$. This curve is constructed from Eq. (11) using the peak on-resonance response for $h_n(f)$ (the black curve from Fig. 3), a bandwidth of $\mathrm{\Delta }f=f/Q$ (using the appropriate $Q$ at each $f$), and a ${\tau }_{\mathrm{int}}=1\mathrm{year}$ integration time. As before, this is not the broadband sensitivity of the detector, but rather it shows the sensitivity that can be reached at each frequency after operating a narrow-band resonant detector there for ${\tau }_{\mathrm{int}}=1\mathrm{year}$. To help visualize this, the red curve shows the ${\mathrm{\Omega }}_{\mathrm{GM}}$ sensitivity for a specific sequence [$h_n$ given by Eq. (9)] with resonance frequency $f_r=0.15\mathrm{Hz}$, $Q=44$, and $6\hslash k$ LMT atom optics. This frequency is in a range that is interesting for cosmological signals and may avoid white dwarf binary confusion noise, shown in the gray band (whose width shows an estimated uncertainty) [33]. The purple line shows an estimate for the power spectrum that could arise from GWs produced during slow roll inflation with $r\sim 0.1$ [35], the orange line depicts GWs from models such as axion inflation [37] that lead to enhanced GW production at frequencies $\sim 1\mathrm{Hz}$ while remaining consistent with cosmic microwave background (CMB) bounds on $r$ (measured at ${10}^{-18}\mathrm{Hz}$), the blue line shows the expected GW strains produced by a network of cosmic strings with tension $G\mu \sim {10}^{-16}$ [38] and the green curve (RS1) is an example of the spectrum that might be produced by a phase transition in the early Universe [39].