Fundamental quantum interferometry bound for the squeezed-light-enhanced gravitational wave detector GEO 600

The fundamental quantum interferometry bound limits the sensitivity of an interferometer for a given total rate of photons and for a given decoherence rate inside the measurement device. We theoretically show that the recently reported quantum-noise-limited sensitivity of the squeezed-light-enhanced German-British gravitational wave detector GEO 600 is exceedingly close to this bound, given the present amount of optical loss. Furthermore, our result proves that the employed combination of a bright coherent state and a squeezed vacuum state is generally the optimum practical approach for phase estimation with high precision on absolute scales. Based on our analysis we conclude that the application of neither Fock states nor NOON states nor any other sophisticated nonclassical quantum state would have yielded an appreciably higher quantum-noise-limited sensitivity.

Figure 1

A simplified model of the GEO 600 interferometer operating at a dark fringe. $a_0,d_0,e_0$ represent annihilation operators of the corresponding modes at the carrier frequency ${\omega }_0$, while $b_{\pm },c_{\pm },d_{\pm },e_{\pm }$ represent annihilation operators of sideband modes ${\omega }_0\pm \Omega$. Input modes are marked in red. The signal recycling mirror (MSR) with power transmissivity of 1.9% is placed in the signal output port and is responsible for the frequency-dependent amplification of the signal amplitudes. The distances to the two end mirrors oscillate at frequency $\Omega$ with the relative phase shift $\pi$ and amplitude $\chi$.

Figure 2

Model of the GEO 600 interferometer reduced to an equivalent, in terms of sensitivity, two-mode Mach-Zehnder interferometer with relative phase delay $\phi =2\sqrt{2}g\epsilon$. $b_s$ represents a symmetric combination of the sideband modes. The possibility of a more general measurement is illustrated with the dotted line.

Figure 3

GEO 600 noise spectral density normalized to strain ($\Delta h$, black) according to Ref. [5] and predictions of our simplified theoretical model based on given values for the circulating light power and for the given optical loss: no vacuum squeezing (dashed gray), 10 dB vacuum squeezing [dashed red, Eq. (5)] as realized in [5], and 16 dB vacuum squeezing coinciding with the fundamental quantum enhancement bound [solid red, Eq. (12)]. Note that the injection of 16 dB vacuum squeezing is in reach of current technology, since 12.7 dB was already observed, even with imperfect photodetectors [24].

Figure 4

Optimality of the coherent state – squeezed vacuum state (CSV) scheme quantified as the ratio of precisions achievable with the optimal $N$-photon state and the CSV strategy with the same mean number of photons and optimal squeezing. Note that in GEO 600, the effective number of photons used per $1\mathrm{s}$ was approximately $2\times {10}^{22}$. For such a high photon number the CSV strategy would achieve 99.9996% of the optimum quantum strategy's sensitivity for a loss of 38%.