Utility investigation of artificial time delay in displacement-noise-free interferometers

Laser interferometer gravitational wave detectors are usually limited by displacement noise in their lower frequency band. Recently, theoretical proposals have been put forward to construct schemes of interferometry that are insusceptible to displacement noise as well as classical laser noise. These so-called displacement-noise-free interferometry (DFI) schemes take advantage of the difference between gravitational waves and displacement noise in their effects on light propagation. However, since this difference diminishes in lower frequencies (i.e., $\Omega <c/{\mathcal{L}}_D$, with ${\mathcal{L}}_D$ the size of the detector), shot-noise-limited sensitivity of DFI schemes deteriorates dramatically in these frequencies—exactly the regime in which they are supposed to be superior, thereby limiting their applicability. In this paper, we explore the obvious possibility of increasing the effective size of the detector in the time domain, by introducing artificial time delays (${\mathcal{T}}_D\gg {\mathcal{L}}_D/c$) into the interferometry scheme, with the hope of improving low-frequency sensitivity. We found that sensitivity can only be improved by schemes in which fluctuations in the artificial time delays are not canceled.

Figure 1

2-D space-time diagram comparing conventional Michelson interferometry (which cancels clock noise, but not displacement noise) with DFI in Ref. 6 (cancels displacement noise, but not clock noise).

Figure 2

3-D space-time ($2+1$) diagram illustrating cancellation of both timing and displacement noise.

Figure 3

Double-octahedron geometry, on which all configurations discussed in this paper are based. Here $C_1{D}_1{C}_2{D}_2$ is a square, with $|O{C}_1|=l$, $A_{a}O\perp C_1{D}_1{C}_2{D}_2$, $|A_{a}O|=|B_{a}O|$, $|A_{b}O|=|B_{b}O|$. We also denote ${\alpha }_a\equiv \angle A_a{C}_{1}O$ and ${\alpha }_b\equiv \angle A_b{C}_{1}O$.

Figure 4

(i), (ii): a section of the time-delayed single-octahedron scheme discussed in Sec. 4a. (iii), (iv): a section of the time-delayed double-octahedron schemes discussed in Secs. 4b, 5c.

Figure 5

DFI shot-noise spectrum compared with the total noise spectrum of a conventional GW detector. The arm length of each detector is set to 3 km. The spectrum of a single octahedron without time delay (gray dashed line) follows a $f^{-2}$ power law in the observation band ($\Omega <{\Omega }_{\mathrm{peak}}$). The spectrum of a time-delayed double octahedron with delay-noise cancellation (indicated as diff. shown in a gray solid curve) cannot be better than that of the single octahedron. The spectrum of a time-delayed double octahedron without delay-noise cancellation (indicated as com, shown in black solid curve) follows the three-part power law (2), and thus has a better sensitivity than the single octahedron. However, delay noise would limit the sensitivity at almost all the frequencies. Here the time delay is set to 10 ms.

Figure 6

Time-delay configurations that either fail to cancel displacement noise or fail to improve sensitivity. Left panel: a naive extension of the 3-D octahedron of Ref. 8; motion of $C$ along reflective surface is not canceled, due to time delay. Right panel: 3-D configuration with time-delay-noise cancellation, the entire configuration can be cut in two, along the two shaded triangles, each without time-delay devices, and free from displacement noise and timing noise, and with $\alpha =2$.

Figure 7

Time-delay configurations that do not cancel delay noise. Left panel: building block from a single octahedron. Right panel: full double-octahedron DFI configuration with $\alpha =1$; unlike in the right panel of Fig. 6, cutting this scheme along the shaded triangles does not give two DFI schemes (because timing-noise contributions at $C$ and $C^{\prime }$ are not canceled).