Quantum noise in differential-type gravitational-wave interferometer and signal recycling

There exists the standard quantum limit (SQL), derived from Heisenberg’s uncertainty relation, in the sensitivity of laser interferometer gravitational-wave (GW) detectors. However, in the context of a full quantum-mechanical approach, SQL can be overcome using the correlation of shot noise and radiation-pressure noise. So far, signal recycling, which is one of the methods to overcome SQL, is considered only in a recombined-type interferometer such as Advanced LIGO, LCGT, and GEO600. In this paper, we investigated quantum noise and the possibility of signal recycling in a differential-type interferometer. As a result, we found that signal recycling is possible and creates at most three dips in the sensitivity curve of the detector due to two coupled resonators. The additional third dip makes it possible to decrease quantum noise at low frequencies, keeping the moderate sensitivity at high frequencies. Then, taking advantage of the third dip and comparing the sensitivity of a differential-type interferometer with that of a next-generation Japanese GW interferometer, LCGT, we found that signal-to-noise ratio (SNR) of inspiral binary is improved by a factor of $\approx 1.43$ for neutron star binary, $\approx 2.28$ for $50M_{\odot }$ black hole binary, and $\approx 2.94$ for $100M_{\odot }$ black hole binary. We also found that power recycling to increase laser power is possible in our signal-recycling configuration of a detector.

Figure 1

SR configuration of a recombined-type interferometer.

Figure 2

Conventional configuration of a differential-type interferometer, and input and output fields.

Figure 3

SR configuration of a differential-type interferometer and input and output fields.

Figure 4

Characteristic sensitivity curve of a differential-type SR interferometer. Parameters selected are $T=0.14$, $\rho =0.98$, $I_0=I_{\mathrm{SQL}}$, $\varphi =1.4$, $\theta =0.86$. The solid curve is the sensitivity of quadrature mode 1 ($\zeta =\pi /2$) and the dashed curve is that of quadrature mode 2 ($\zeta =0$). The diagonal black line is $h_{\mathrm{SQL}}$.

Figure 5

Number of dips when $n=1$. We numerically solved (48) and showed the number of dips, which has real frequency, with colors. Black, dark gray, and light gray regions have three, two, and one solution, respectively, and white regions have no solution.

Figure 6

Laser power dependence of the positions of resonant frequency for the parameters $\{T=0.14,\varphi =1.4,\theta =0.86\}$. We evaluated (48) and showed the resonant frequencies with a solid line for optical resonance and a dashed line for mechanical resonance.

Figure 7

Number of solutions of GW suppression. We evaluated (49, 50), and showed whether the solution exists or not with colors. Shaded regions have one solution and white regions have no solution. The left panel shows that of the suppression frequency for quadrature mode 1 and the right panel is for quadrature mode 2.

Figure 8

Discrepancy of resonant frequencies with/without the approximation, and the dependence on the reflectivity of the SR mirror $\rho$. Left and right panels show the cases of $\zeta =\pi /2$ (quadrature 1) and $\zeta =0$ (quadrature 2). Other parameters are selected as $T=0.14$, $I_0=I_{\mathrm{SQL}}$, $\varphi =1.4$, $\theta =0.86$. Vertical lines are the dip frequencies predicted by the approximated solutions. Each curve is drawn without the approximation and with $\rho =0.99$, $\rho =0.95$, $\rho =0.90$, $\rho =0.85$, respectively, as indicated in the figure. As $\rho$ is smaller, the dips become shallower and the positions of the dips are slightly shifted. However, the discrepancy is not significant and is within the order of $\mathcal{O}({\tau }^2)$.

Figure 9

Comparison of the sensitivity curves of a differential-type SR interferometer and Advanced LIGO. The solid and dashed curve are the sensitivity curve of the differential type with adjusted parameters listed in Table  and Advanced LIGO. Other straight lines are noises of Advanced LIGO; thermal noise of a suspension, thermal noise of a mirror, and seismic noise as indicated in the Fig. 1.

Figure 10

Comparison of the sensitivity curves of a differential-type SR interferometer and LCGT. Two solid curves are the sensitivity curves of the differential type with adjusted parameters listed in Table . Dotted and dashed curves are the sensitivity curves of LCGT with tuned and detuned configuration, respectively. Other straight lines are LCGT noises; thermal noise of a suspension, thermal noise of a mirror, and seismic noise as indicated in the figure [20]. Diagonal dashed line is $h_{\mathrm{SQL}}$.

Figure 11

Sideband fields in a conventional differential-type interferometer.

Figure 12

Sideband fields in a SR differential-type interferometer.