Sensitivity of a speed meter

Since the early 1990s it has been known that the measurement of momentum, known to be a quantum non-demolition (QND) observable, offers the ability to surpass the SQL in interferometric measurement [65]. Ideally, the back-action applied to test masses by a mea- surement of momentum—a consequence of the Heisenberg Uncertainty Principle—does not affect its future value and so momentum can in principle be measured to arbitrary preci- sion. Velocity is an appropriate observable to measure momentum and also approximates a QND scheme due to its relation to momentum. Interferometers that measure velocity are called speed meters, and their principle of operation is as follows. Light from a laser enters the interferometer as it would for a position-meter, and accumulates a phase shift propor- tional to the propagation and the signal from any gravitational waves or disturbances in the positions of the test masses. As the light reflects from the test masses it imparts radiation pressure arising from its classical amplitude and the amplitude quantum vacuum fluctua- tions as discussed in section 4.1.1 for a Fabry-Perot Michelson interferometer. Within the interferometer there must be a mechanism to impart a phase shift equivalent to 180° to one light field to create a second light field that samples the same mode. Propagating through the interferometer, the radiation pressure imparted to the mirrors by one field is superim- posed upon the radiation pressure imparted from the other, and as these effects are out of phase within the light travel time the radiation pressure force can be suppressed.

Input ETMY ITMX ETMX ITMY BS Sloshing cavity Output SR

Figure 4.4: Layout of a Fabry-Perot Michelson interferometer with a sloshing cavity as presented in ref. [86]. The light leaving the Fabry-Perot Michelson interferometer is coupled into a sloshing cavity via a beam splitter where it receives a phase shift, and it re-enters the interferometer via the recycling mirror to the left of the sloshing beam splitter. The light incident upon the beam splitter then contains light that has sampled the mirrors at two points in time, leading to a speed meter effect.

meters are also being considered as alternatives to the proposed Michelson interferome- ters in the Einstein Telescope [132, 133]. We consider here two speed meter topologies to highlight the significantly different forms in which a speed meter interferometer can take.

4.1.2.1 The Michelson-type speed meter

Initial suggestions for the application of speed meter type interferometers in the field of gravitational wave detection were focused on a Fabry-Perot Michelson interferometer topol- ogy with the addition of a sloshing cavity at the output port [85, 86] as shown in figure 4.4. Here the 180° phase shift is imparted to the light by the addition of a beam splitter and slosh- ing cavity at the output of the interferometer. The light returning from the sloshing cavity is either re-injected into the interferometer or transmits through the beam splitter where it reflects from a signal recycling mirror (SR). The light at the output of this interferometer then contains reduced quantum radiation pressure noise.

4.1.2.2 The Sagnac-type speed meter

It was realised by Chen that the zero-area Sagnac interferometer topology is a speed me- ter [87]. This interferometer is arranged such that incident photons enter into two counter- propagating modes which sample the position of the test masses at different intervals. The Sagnac interferometer is sensitive to the rotation of the Earth via the area enclosed by its arms, and so to avoid this the propagation of the light is arranged in a zero-area configu- ration to cancel the rotation-induced phase accumulation from each arm. The remaining signal at the output contains information of the difference in round-trip phase of the two counter-propagating modes due to test mass motion. Given two test mass positions 𝑥𝐴 and 𝑥𝐵 in arms 𝐴 and 𝐵, respectively, over a time interval of Δ𝑡 each counter-propagating mode will measure phase changes 𝛿𝜙𝐴and 𝛿𝜙𝐵 arising from motion of the arms less than the light propagation time [87]:

𝛿𝜙_𝐴∝ Δ𝑥_𝐴(𝑡) + Δ𝑥𝐵(𝑡 + Δ𝑡) (4.16)

𝛿𝜙_𝐵 ∝ Δ𝑥_𝐵(𝑡) + Δ𝑥_𝐴(𝑡 + Δ𝑡) . (4.17)

At the output port, the combined signal will then be the difference of phase, 𝛿𝜙_𝐴− 𝛿𝜙_𝐵 ∝(Δ𝑥_𝐴(𝑡) − Δ𝑥_𝐴(𝑡 + Δ𝑡))−(Δ𝑥_𝐵(𝑡) − Δ𝑥_𝐵(𝑡 + Δ𝑡))

∝ Δ ̇𝑥_𝐴(𝑡) − Δ ̇𝑥_𝐵(𝑡) , (4.18)

which shows that the signal is proportional to the relative velocity of the test masses. The output port is automatically at the dark fringe for the carrier light as long as the motion of the test masses is slower than the light propagation time. The output is not dark for the signal sidebands, and as they contain components from the test masses sampled at different times the signal is proportional to test mass speed.

The layout of a Sagnac speed meter interferometer can be arranged in different forms [91], and we show one based on a zero-area Sagnac enhanced with ring cavities as arms in fig- ure 4.5.

4.1.2.3 Input-output relations

The same approach to that for a Fabry-Perot Michelson interferometer in section 4.1.1 can be taken to calculate the response and noise of a speed meter, but with a value of 𝜅 modified for a speed-meter [87],

Input

Cavity X

BS Output

Cavity Y

Figure 4.5: Layout of a zero-area Sagnac speed meter with ring cavities. The input light is split at the beam splitter where it forms two counter-propagating modes within the inner Sagnac mirrors, denoted by black arrows. At each ITM, the light is partially transmitted into the arm cavities, and upon exiting the cavities this light is either sent back to the beam splitter or to the next cavity. The recombined light at the beam splitter contains fields that have interacted with all of the mirrors and the difference in phase between the counter-propagating modes provides a signal proportional to relative test mass velocity.

and as the round-trip phase includes both arms and an extra reflection from or transmission through the beam splitter, it is also modified:

𝛽_SM = 2𝛽_FP+ 𝜋

2. (4.20)

The response of a Sagnac speed meter to differential arm cavity motion is shown in fig- ure 4.6 for parameters identical to that of figure 4.1. Notice that below the cavity pole, the response vanishes towards dc, consistent with a speed measurement. The higher response above the cavity pole is a consequence of the fact that the light samples the interferometer in both directions. For fair comparisons to the Michelson interferometer the choice may be made to alter the input power and readout angle of one with respect to the other.

The corresponding quantum noise at the output port is shown in figure 4.7. Note that the noise is unity at high frequencies as with the Fabry-Perot Michelson interferometer, but is suppressed at low frequencies due to the cancellation of back-action due to radiation pressure from quantum vacuum fluctuations. The cancellation is not perfect due to the time delay between the two consecutive visits of the arm cavities by the counter-propagating

1020 1021 1022 1023 1024 Resp onse (√ Hz ) Sagnac speed-meter Michelson position-meter 100 101 102 103 104 105 Frequency (Hz) −180 −135 −90 −45 0 45 90 135 180 Phase (° )

Figure 4.6: Response of a Sagnac speed meter to differential arm cavity motion. In contrast to the Michelson interferometer, the Sagnac speed meter has response proportional to frequency below the cavity pole.

modes.

While the response in a Sagnac speed meter is reduced at low frequencies, the quantum noise is further reduced and so the overall quantum noise limited sensitivity is improved over an equivalent Fabry-Perot Michelson interferometer in the absence of loss, as shown in figure 4.8. For lossy speed meters the sensitivity is degraded. In the next subsection we consider loss in the case of a Sagnac speed meter but loss in any QND interferometers significantly affects sensitivity.

4.1.2.4 Loss in Sagnac speed meters

The QND behaviour of the interferometer arises from the fact that the output port contains only commutative time-dependent momentum information. Time-independent position in- formation can, however, enter the output port of the interferometer in the presence of cer- tain types of loss [88]. For symmetric loss, such as from balanced but imperfect reflectivity of the ETMs or substrate absorption in the ITMs, incoherent vacuum fluctuations can en- ter the interferometer at a point after the light has been split into the counter-propagating modes and this affects sensitivity. It has also been shown that asymmetric loss results in a

100 101 102 103 104 105 Frequency (Hz) 100 102 104 106 108 1010 Normalise d noise ( 1 Hz ) _{Sagnac speed-meter} Michelson position-meter Quantum shot noise

Figure 4.7: Quantum noise of a Sagnac speed meter at the output port, normalised to quantum shot noise. Like the Michelson interferometer, the high frequency noise contribution arises from quantum shot noise from incoherent vacuum fluctuations entering the interferometer. In contrast to the Michelson interferometer, the Sagnac speed meter has flat noise at low frequencies below a transition region, as the test mass noise fluctuations are cancelled by the counter-propagating modes in the instance where the quantum radiation pressure forces are balanced.

100 101 102 103 104 105 Frequency (Hz) 10−24 10−23 10−22 10−21 10−20 10−19 10−18 Sensitivity ( 1 √ Hz

) Sagnac speed-meter_{Michelson position-meter}

SQL

Figure 4.8: Quantum noise limited sensitivity of a Sagnac speed meter at the output port to dif- ferential arm cavity motion. In contrast to the Michelson interferometer, the Sagnac speed meter sensitivity at low frequencies follows the gradient of the SQL due to its reduced quantum noise. This improved sensitivity is in practice difficult to achieve as the presence of loss in the interferometer introduces a Michelson-like sensitivity slope at a frequency proportional to the level of loss.

greater decrease in sensitivity [89], considering effects from imperfect beam splitting and imbalanced ITM reflectivity in a Sagnac speed meter.

The optic in a Sagnac speed meter most susceptible to asymmetries is typically the beam splitter, as coatings typically cannot be manufactured with better than around 0.1 % tol- erance in the amplitude reflectivity at the standard wavelength for detectors1_{. Imperfect} splitting leads to different power in the counter-propagating modes which leads to asym- metries. In the Sagnac speed meter some of the light that would otherwise have exited at the input port of the interferometer (towards the input laser) instead exits at the output port due to the imbalanced beam splitter, carrying time-independent signal and therefore dam- aging the sensitivity. This appears on displacement sensitivity curves as an additional 1

𝑓 slope at low frequencies such that it resembles the 1

𝑓2 displacement sensitivity of a Michel-

son interferometer. The minimisation of loss is therefore critical in the design of a Sagnac speed meter experiment.