The final state of the meter is given by Eqs. (3.8) and (3.9). In particular, when we choose θ close to zero, there is tremendous amplification in weak value, i.e., the output value is enhanced by the factor 1/θ which is shown in Eq .(3.10). The amplification is demonstrated in Fig. 3.4.
Figure 3.4: Weak value amplification in modify gravitational-wave detector.
To get a clearer picture of the effect of θ on the sensitivity curve, we assume the gravitational wave signal as a sinusoidal wave h(f ) = 1, and define a quantity σ proportional to the signal-to-noise ratio.
σ = 1
Sh(Ω, θ). (3.18)
The result is shown as in Fig 3.5, we compare σ in different frequency of gravita-tional wave.
The original scheme of weak-value metrology in gravitational-wave detector, if the θ is close to zero,|ψi⟩ and |ψf⟩ is nearly orthogonal that we can enhance the signal at the dark port. In Fig 3.5, we can see the wake-value amplified when θ goes to zero,
Figure 3.5: The brown line presents the σ when detecting the signal in frequency 5Hz, and the blue and yellow line illustrate the σ while detecting the signal in frequency 10Hz and 100Hz, respectively. The amplification of weak value is shown as in red line.
but the σ become smaller. Whether we detect the gravitational signal in any different frequency, the sensitivity is worse after increasing θ. The reason is that in Eq. (3.16), we calculate input-output relation, we derive the noise signal is also be enhanced. We add phase rotators on two arms; thus, we can not eliminate the quadrature term ˆi(Ω) which is the noise from the bright port. The noise signal is also amplified in the use of weak-value metrology in the gravitational-wave detector. Thus this setup can not suppress the standard quantum limit.
Chapter 4 Conclusion
Improving the sensitivity of the detector is crucial in the gravitational wave astronomy in recent years. The higher is the sensitivity, the more events we will detect. Besides, the sensitivity band of the ground-based detector ranges from 10 Hz to 10 kHz. It is also essential to build a broad frequency band detector to detect more sources of gravitational waves. In this thesis, we try to figure out whether it is possible to improve the sensitivity of gravitational-waves detector via weak value metrology.
We are mainly motivated by the goal to overcome the quantum noise, which is a significant part of the detector noises.
We try to follow the calculation of optical processes in conventional gravitational-waves detector and introduce the weak-value metrology. The most important thing is to derive the sensitivity curve of the detector inspired by the weak-value metrology and to compare the performance with the conventional detector.
However, the result is not as good as expected. The reason is that although we am-plify the signal, we also increase the noise, as shown in the Eqs. (3.16) and (3.17).To introduce the weak value metrology in the gravitational-wave interferometer, we add the phase rotator in each arm. But at the same time, we broke the symmetry of the
interferometer. Hence; the noise contributed by ˆi can not be eliminated, it will be detected at the dark port. Overall we amplify not only the signal but also the noise.
For gravitational wave detection, it is vital not only to amplify the signal but more importantly, to improve the sensitivity to enhance the signal-to-noise ratio.
Even if we expand the signal, we can not avoid increasing the noise, which makes no improvement in the sensitivity. Consequently, the use of weak value metrology in the gravitational wave detector did not help us improve the sensitivity in detection.
In future studies, it will still be essential to explore whether there are other meth-ods to improve the sensitivity to detect more gravitational wave signal. Quantum noise, in particular, accounts for a large part of the noise of gravitational wave de-tection, and it is well worth investigating from quantum measurements and quantum optics. As gravitational wave astronomy has flourished, the development of the broad-band and more sensitive detectors is a necessary part of physical development.
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