Sensitivity below the standard quantum limit in gravitational wave detectors with Michelson-Fabry-Perot readout

We calculate the quantum noise limited displacement sensitivity of a Michelson-Fabry-Perot (MFP) interferometer with detuned cavities, followed by phase-sensitive homodyne detection. We show that the standard quantum limit can be surpassed even with resonant cavities and without any signal-recycling mirror nor additional cavities. Indeed, thanks to the homodyne detection, the output field quadrature can be chosen in such a way to cancel the effect of input amplitude fluctuations, i.e., eliminating the force noise. With detuned cavities, the modified opto-mechanical susceptivity allows to reach unlimited sensitivity for large enough (yet finite) optical power. Our expressions include mirror losses and cavity delay effect, for a realistic comparison with experiments. Our study is particularly devoted to gravitational wave detectors and we consider both an interferometer with free-falling mirrors, and a MFP interferometer as readout for a massive detector. In the latter case, the sensitivity curve of the recently conceived “DUAL” detector, based on two acoustic modes, is obtained.

Figure 1

Left: the optical configuration considered is a Michelson-Fabry-Perot interferometer with the additional free choice of the detected field quadrature. Right: mechanical scheme, with self- and cross- susceptivities.

Figure 2

Sensitivity $S_L$ as a function of input power $p$, for ${\Omega }_{\mathrm{cav}}\to \infty$, ${\Gamma }_m=1$, and constant, real $\chi$. Solid line: “normal case” ($\Psi =0$, $w=\pi /2$); dashed line: general resonant case ($\Psi =0$, $w=\arctan (-1/p\chi )$); dotted lines: $\Psi =-0.4$ and $w=\pi /2$ (a), $w=2.0$ (b), $w=2.3$ (c).

Figure 3

Sensitivity $S_L$ in the case of free-falling mirrors, as a function of the frequency $\Omega$, for ${\Omega }_{\mathrm{cav}}\to \infty$, ${\Gamma }_m=1$, $p=1$. Solid line: “normal case” ($\Psi =0$, $w=\pi /2$); dashed line: resonant case limit $S_L^0$; dash-dotted line: SQL; dotted lines: $\Psi =0$ and $w=1.2$ (a), $w=0.8$ (b), $w=0.4$ (c), $w=0.2$ (d).

Figure 4

Sensitivity $S_L$ in the case of free-falling mirrors, as a function of the frequency $\Omega$, for ${\Omega }_{\mathrm{cav}}\to \infty$, ${\Gamma }_m=1$, $p=1$. Solid line: “normal case” ($\Psi =0$, $w=\pi /2$); dashed line: resonant case limit $S_L^0$; dash-dotted line: SQL; dotted lines: $w=\pi /2$ and $\Psi =0.4$ (a), $\Psi =0.6$ (b), $\Psi =0.8$ (c), $\Psi =0.9$ (d), $\Psi =1.1$ (e), $\Psi =1.2$ (f), $\Psi =1.4$ (g).

Figure 5

Sensitivity $S_L$ in the case of free-falling mirrors, as a function of the frequency $\Omega$, for ${\Omega }_{\mathrm{cav}}\to \infty$, ${\Gamma }_m=1$, $p=1$. Black solid line: “normal case” ($\Psi =0$, $w=\pi /2$); long-dashed straight line: resonant case limit $S_L^0$; dash-dotted line: SQL; dashed lines: the phase $w$ is optimized for each frequency, $\Psi =0.06$ (a), $\Psi =0.1$ (b), $\Psi =0.2$ (c), $\Psi =0.3$ (d), $\Psi =0.4$ (e); gray solid line: both $w$ and $\Psi$ are optimized for each frequency. In the inset: phase $w$ optimized for each frequency, for the values of $\Psi$ used in the main plot.

Figure 6

Sensitivity $S_L$ in the case of free-falling mirrors, as a function of the frequency $\Omega$, for ${\Omega }_{\mathrm{cav}}\to \infty$, $p=1$. Solid lines: “normal case” ($\Psi =0$, $w=\pi /2$, from lower to upper curve ${\Gamma }_m=1$, 0.8, 0.5); dashed line: resonant case limit $S_L^0$ for ${\Gamma }_m=0$; dash-dotted line: SQL for ${\Gamma }_m=0$; dotted lines: $w=\pi /2$, $\Psi =0.6$, and from lower to upper curve ${\Gamma }_m=1$, 0.8, 0.5.

Figure 7

Sensitivity $S_L$ in the case of free-falling mirrors, as a function of the frequency $\Omega$, for ${\Gamma }_m=1$, $p=1$. Thin, dark lines (black): “normal case” ($\Psi =0$, $w=\pi /2$); thick, light lines (red): $\Psi =0.6$, $w=1.3$. Solid lines: ${\Omega }_{\mathrm{cav}}\to \infty$; dotted lines: ${\Omega }_{\mathrm{cav}}=3{\Omega }_{\mathrm{SQL}}$; dashed lines: ${\Omega }_{\mathrm{cav}}=0.3{\Omega }_{\mathrm{SQL}}$. Dash-dotted line: SQL.

Figure 8

Sensitivity to a gravitational wave $S_{\mathrm{GW}}$ of a DUAL detector, as a function of the frequency $\Omega$, in arbitrary units. Dark, solid line (black): ${\mathrm{SQL}}_{\mathrm{GW}}$. Light solid line (red): “normal case” (resonant cavities, $w=\pi /2$) with $p=0.4$. Dotted lines: “normal case” with $p=2$ (a), $p=0.1$ (b).

Figure 9

Sensitivity to a gravitational wave $S_{\mathrm{GW}}$ of a DUAL detector, as a function of the frequency $\Omega$, in arbitrary units. Dark, solid line (black): ${\mathrm{SQL}}_{\mathrm{GW}}$, the other curves are for $p=0.4$; dashed line: resonant case limit; light solid line (red): “normal case”; dotted lines: $w=\pi /2$ and $\Psi =-0.5$ (a), $\Psi =0.5$ (b).