Gravitational wave detection beyond the standard quantum limit using a negative-mass spin system and virtual rigidity

Gravitational wave detectors (GWDs), which have brought about a new era in astronomy, have reached such a level of maturity that further improvement necessitates quantum-noise-evading techniques. Numerous proposals to this end have been discussed in the literature, e.g., invoking frequency-dependent squeezing or replacing the current Michelson interferometer topology by that of the quantum speedmeter. Recently, a proposal based on the linking of a standard interferometer to a negative-mass spin system via entangled light has offered an unintrusive and small-scale new approach to quantum noise evasion in GWDs [Phys. Rev. Lett. 121, 031101 (2018)]. The solution proposed therein does not require modifications to the highly refined core optics of the present GWD design and, when compared to previous proposals, is less prone to losses and imperfections of the interferometer. In the present article, we refine this scheme to an extent that the requirements on the auxiliary spin system are feasible with state-of-the-art implementations. This is accomplished by matching the effective (rather than intrinsic) susceptibilities of the interferometer and spin system using the virtual rigidity concept, which, in terms of implementation, requires only suitable choices of the various homodyne, probe, and squeezing phases.

Figure 1

The GW interferometer ($I$) and the spin oscillator ($S$) are probed and detected in parallel by laser beams (solid arrows) with different wavelengths but entangled fluctuations (dashed arrows). These two-mode-squeezing correlations are achieved by means of a sum frequency generator in combination with an optical parametric oscillator. The collective spin of the atomic ensemble precesses around the magnetic field $\overrightarrow{B}$, forming a spin oscillator. With respect to the squeeze angle of this process, we reference the probe ${\varphi }_{I,S}$ and detection ${\zeta }_{I,S}$ phases of the respective systems. The output fields impinge on detectors $D_{I,S}$, and the resulting measurements are suitably combined to cancel the joint meter noise. A possible practical implementation of the present conceptual schematic is presented in Ref. [23].

Figure 2

Geometrical representation of the virtual rigidity effect. The backaction and imprecision noise quadratures for $I$ and $S$ are depicted with respect to the quadrature bases $({\widehat{a}}_{I,S}^c,{\widehat{a}}_{I,S}^s)$, respectively. Homodyne measurement quadratures are chosen to be ${\widehat{b}}_I^s$ and ${\widehat{b}}_S^s$ (i.e., ${\zeta }_I={\zeta }_S=\pi /2$) so that the corresponding imprecision noise quadratures (vertical bold arrow) coincide with the vertical axes, ${\widehat{a}}_I^s$ and ${\widehat{a}}_S^s$. For $I$, we make the “standard” choice of probe phase ${\varphi }_I=0$ such that the orthogonal quadrature ${\widehat{a}}_I^c$ acts as backaction. However, for $S$, a probe phase $\varphi ={\varphi }_S$ relative to ${\widehat{a}}_S^c$ is introduced.

Figure 3

Optimal squeezing as a function of the Fourier frequency for different values of the spin cooperativity ${\mathcal{C}}_S$. The bare spin resonance ${\mathrm{\Omega }}_S/(2\pi )$ is indicated by the vertical line and coincides to a good approximation with the minimum of $r_{\mathrm{opt}}(\mathrm{\Omega })$.

Figure 4

Normalized position sensitivity $\sqrt{|{\chi }_I(\mathrm{\Omega }){|}^{2}S}$ for the spin-GWD scheme benchmarked against a standard interferometer. Also shown is the SQL.

Figure 5

Sensitivity gain relative to standard interferometer $\sqrt{G}\equiv \sqrt{S_{\mathrm{std}}/S}$. The vertical line indicates ${\mathrm{\Omega }}_S/(2\pi )$. Gain values predicted by simplified expressions for low (61) and high (62) frequencies are indicated by horizontal line segments. We also indicate the gain at $\mathrm{\Omega }={\mathrm{\Omega }}_S$, near the minimum, according to the approximate formula Eq. (57).