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# QND techniques

## Ponderomotive squeezing

Let us first explain the source of quantum noise. See the left two panels of the figure below. The number of photons in the laser light fluctuates both in the same phase as the non-fluctuating light (classical light) and in the opposite phase; or we can use the term "quadrature". The former fluctuation causes the amplitude modulation and the latter causes the phase modulation to the light. The fluctuation can appear in any quadrature randomly. What we can tell is the expected variance of the fluctuation, which is shown in the shaded area. This quantum fluctuation exists even without the classical light component; in this case we call it a vacuum fluctuation.

In a gravitational-wave detector, the interferometer is operated in the dark fringe, that is, all the classical light goes back toward the laser and the differential signal comes to the dark port. The fluctuation entering from the bright port with the laser light does not come through the dark port. Instead the vacuum fluctuation entering from the dark port comes back to the dark port, and this is the source of quantum noise.

Let us define the amplitude quadrature component and the phase quadrature component of the vacuum fluctuation entering from the dark port a1 and a2, respectively. The output field (b1,b2) is described by

where K means an opto-mechanical coupling coefficient, beta is a phase shift term, and hSQL is the SQL

in strain. If we obtain the signal component (h) and the noise component (a1 and a2) in the b2 quadrature, the signal-to-noise ratio will read

We used the definition that the power spectra of a1 and a2 are both unity.

The signal appears in the phase quadrature, so the measurement in the phase quadrature means that we take the maximum signal. On the other hand, we can measure in a quadrature with the best signal-to-noise ratio. The output in the quadrature zeta is given by b1*cos(zeta)+b2*sin(zeta). The signal-to-noise ratio will read

which can be smaller than the hSQL by choosing zeta to be arccot(K). See the right panel of the figure above. What we just did is to take the signal and noise in the quadrature where the noise ellipse is the thinnest. The radiation-pressure effect ponderomotively squeezes the input coherent vacuum. In the optimal quadrature, the signal appears smaller than in the phase quadrature but the signal-to-noise ratio is the best. It can be also explained in such a way that we use the information of a1 by changing the readout quadrature and cancel out the a1 term appearing as radiation pressure noise. Note that this K is not a constant in frequency, so the cancellation of radiation pressure noise can be realized only at a certain frequency unless we use a so-called filter cavity, which will be explained in the next paragraph.

## Frequency-dependent squeezing and variational readout

A number of QND techniques are described in one figure (the left panel of the figure above). One way to realize the cancellation of radiation pressure noise in broadband is to use a so-called filter cavity. While the readout quadrature must be fixed, the output field can be rotated in the quadrature plane so that the thinnest part of the noise ellipse always comes on to the readout quadrature. This technique is called variational readout. Note that we actually need two cavities to realize the idea, although only one cavity is drawn in the figure. The right panel (top) shows the spectrum. Without the optical losses, the sensitivity could be limited only by shot noise, which could be reduced by increasing the laser power. The optical losses taken into account, however, the sensitivity exceeds the SQL only at around a certain frequency.

Injection of a squeezed vacuum to a gravitational-wave detector has been considered for many years. Instead of the coherent vacuum, a squeezed vacuum generated by a non-linear crystal is injected to the interferometer. The filter cavity can be used with the input squeezing. The right panel (bottom) shows that the frequency-dependent input squeezing is not as weak against the optical losses as the variational readout. This is because the input squeezing does not require reducing the amount of the signal like the variational readout.

## Optical spring

We also described recycling mirrors in the figure above. The mirror placed at the dark port is called signal recycling mirror, and changing the resonant condition of the signal-recycling cavity let us overcome the SQL in a different way. If this cavity is locked somewhere in between the resonance and the anti-resonance for the carrier light, the signal coming from the interferometer, which is in the phase quadrature, is reinjected to the interferometer in a quadrature somewhere between the phase and the amplitude quadratures. Then a part of the signal field couples with the carrier light in the amplitude quadrature, driving the mirrors by the radiation pressure. The motion of the mirror generates a signal field in the phase quadrature and it comes to the dark port. This loop results in new dynamics with an optical spring that increases the signal at around a certain frequency. As is shown in the figure below, the sensitivity can overcome the SQL at around the spring frequency.

## Other QND techniques

Another QND technique is a so-called speedmeter. The variational readout has weakness against optical losses and the optical spring realizes the QND in a limited frequency band. The speedmeter realizes the QND in broadband and has no weakness against the optical loss at the photo-detection, which is currently the largest of all optical losses. The basic idea is to inject the vacuum to the interferometer more than twice so that radiation pressure can cancel out. Unlike the optical spring, the reinjection is not done immediately but after a significant phase shift.

A so-called displacement-noise-free interferometer could be regarded a QND device in the sense that it obtain the information of the gravitational waves without radiation pressure noise, although it actually does not sense any motion of the test masses.

## References

* H.J.Kimble et al, Phys. Rev. D

, 65, 022002 (2002)

The filter cavity is introduced in this paper.

* A.Buonanno and Y.Chen, Phys. Rev. D

, 64, 042006 (2001)

The optical spring with a signal-recycled interferometer is introduced in this paper.

* O.Miyakawa et al, Phys. Rev. D

, 74, 022001 (2006)

The experimental demonstration of the optical spring is shown in this paper.

* P.Purdue and Y.Chen, Phys. Rev. D

, 66, 122004 (2002)

The Sagnac-type speedmeter is introduced in this paper.

* Y.Chen et al, Phys. Rev. Lett

, 97, 151103 (2006)

The 3-dimensional displacement-noise-free interferometer is introduced in this paper.