6.2 Quantum speed meter

6.2.1 Quantum speed-meter topologies

A quantum speed meter epitomizes the approach to the broadband SQL beating, in some sense, opposite to the one based on the quantum noises cross-correlation tailoring with filter cavities, considered above. Here, instead of fitting the quantum noise spectral dependence to the Fabry–Pérot–Michelson interferometer optomechanical coupling factor (389View Equation), the interferometer topology is modified in such a way as to mold the new optomechanical coupling factor 𝒦SM (Ω ) so that it turns out frequency-independent in the low- and medium-frequency range, thus making the frequency-dependent cross-correlation not necessary.

The general approach to speed measurement is to use pairs of position measurements separated by a time delay τ ≲ 1 ∕Ω, where Ω is the characteristic signal frequency (cf. the simplified consideration in Section 4.5). Ideally, the successive measurements should be coherent, i.e., they should be performed by the same photons. In effect, the velocity v of the test mass is measured in this way, which gives the necessary frequency dependence of the 𝒦SM (Ω).

In Section 4.5, we have considered the simplest toy scheme that implements this principle and which was first proposed by Braginsky and Khalili in [21Jump To The Next Citation Point]. Also in this paper, a modified version of this scheme, called the sloshing-cavity speed meter, was proposed. This version uses two coupled resonators (e.g., microwave ones), as shown in Figure 40View Image (left), one of which (2), the sloshing cavity, is pumped on resonance through the input waveguide, so that another one (1) becomes excited at its eigenfrequency ωe. The eigenfrequency of resonator 1 is modulated by the position x of the test mass and puts a voltage signal proportional to position x into resonator 2, and a voltage signal proportional to velocity dx ∕dt into resonator 1. The velocity signal flows from resonator 1 into an output waveguide, from which it is monitored. One can understand the production of this velocity signal as follows. The coupling between the resonators causes voltage signals to slosh periodically from one resonator to the other at frequency Ω. After each cycle of sloshing, the sign of the signal is reversed, so the net signal in resonator 1 is proportional to the difference of the position at times t and t + 2π ∕Ω, thus implementing the same principle of the double position measurement.

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Figure 40: Left: schematic diagram of the microwave speed meter on coupled cavities as given in [21Jump To The Next Citation Point]. Right: optical version of coupled-cavities speed meter proposed in [127Jump To The Next Citation Point].

Later, the optical version of the sloshing-cavity speed-meter scheme suitable for large-scale laser GW detectors was developed [20, 126, 127Jump To The Next Citation Point]. The most elaborated variant proposed in [127Jump To The Next Citation Point] is shown in Figure 40View Image (right). Here, the differential mode of a Michelson interferometer serves as the resonator 1 of the initial scheme of [21], and an additional kilometer-scale Fabry–Pérot cavity – as the resonator 2, thus making a practical interferometer configuration.

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Figure 41: Two possible optical realizations of zero area Sagnac speed meter. Left panel: The ring cavities can be used to spatially separate the ingoing fields from the outgoing ones, in order to redirect output light from one arm to another [42Jump To The Next Citation Point]. Right panel: The same goal can be achieved using an optical circulator consisting of the polarization beamsplitter (PBS) and two λ ∕4-plates [85Jump To The Next Citation Point, 50Jump To The Next Citation Point].

In parallel, it was realized by Chen and Khalili [42Jump To The Next Citation Point, 85Jump To The Next Citation Point] that the zero area Sagnac interferometer [140, 11, 145] actually implements the initial double-measurement variant of the quantum speed meter, shown in Figure 26View Image. Further analysis with account for optical losses was performed in [50Jump To The Next Citation Point] and with detuned signal-recycling in [113Jump To The Next Citation Point]. Suggested configurations are pictured in Figure 41View Image. The core idea is that light from the laser gets split by the beamsplitter (BS) and directed to Fabry–Pérot cavities in the arms, exactly as in conventional Fabry–Pérot–Michelson interferometers. However, after it leaves the cavity, it does not go back to the beamsplitter, but rather enters the cavity in the other arm, and only afterwards returns to the beamsplitter, and finally to the photo detector at the dark port. The scheme of [42Jump To The Next Citation Point] uses ring Fabry–Pérot cavities in the arms to spatially separate ingoing and outgoing light beams to redirect the light leaving the first arm to the second one evading the output beamsplitter. The variant analyzed in [85, 50Jump To The Next Citation Point] uses polarized optics for the same purposes: light beams after ordinary beamsplitter, having linear (e.g., vertical) polarization, pass through the polarized beamsplitter (PBS), then meet the λ∕4 plates that transform their linear polarization into a circular one, and then enter the Fabry–Pérot cavity. After reflection from the Fabry–Pérot cavity, light passes through a λ∕4-plate again, changing its polarization again to linear, but orthogonal to the initial one. As a result, the PBS reflects it and redirects to another arm of the interferometer where it passes through the same stages, restoring finally the initial polarization and comes out of the interferometer. With the exception of the implementation method for this round-robin pass of the light through the interferometer, both schemes have the same performance, and the same appellation Sagnac speed meter will be used for them below.

Visiting both arms, counter propagating light beams acquire phase shifts proportional to a sum of end mirrors displacements of both cavities taken with time delay equal to average single cavity storage time τarm:

δ ϕR ∝ xN (t) + xE (t + τarm), δϕL ∝ xE (t) + xN (t + τarm). (457 )
After recombining at the beamsplitter and photo detection the output signal will be proportional to the phase difference of clockwise (R) and counter clockwise (L) propagating light beams:
δϕR − δϕL ∝ [xN (t) − xN(t + τarm )] − [xE(t) − xE(t + τarm)] ∝ ˙xN (t) − x˙E (t) + O (τarm ) (458 )
that, for frequencies −1 ≪ τarm, are proportional to the relative velocity of the interferometer end test masses.

Both versions of the optical speed meter, the sloshing cavity and the Sagnac ones, promise about the same sensitivity, and the choice between them depends mostly on the relative implementation cost of these schemes. Below we consider in more detail the Sagnac speed meter, which does not require the additional long sloshing cavity.

We will not present here the full analysis of the Sagnac topology similar to the one we have provided for the Fabry–Pérot–Michelson one. The reader can find it in [42Jump To The Next Citation Point, 50Jump To The Next Citation Point]. We limit ourselves by the particular case of the resonance tuned interferometer (that is, no signal recycling and resonance tuned arm cavities). It seems that the detuned Sagnac interferometer can provide a quite interesting regime, in particular, the negative inertia one [113]. However, for now (2011) the exhaustive analysis of these regimes is yet to be done. We assume that the squeezed light can be injected into the interferometer dark port, but consider only the particular case of the classical optimization, 𝜃 = 0, which gives the best broadband sensitivity for a given optical power.

6.2.2 Speed-meter sensitivity, no optical losses

In order to reveal the main properties of the quantum speed meter, start with the simplified case of the lossless interferometer and the ideal photodetector. In this case, the sum quantum noise power (double-sided) spectral density of the speed meter can be written in a form similar to the one for the Fabry–Pérot–Michelson interferometer [see, e.g., Eqs. (386View Equation), (387View Equation) and (388View Equation)]:

ShSQL(Ω )[ e−2r + e2r cot2ϕLO ] Sh(Ω ) = -------- ------------------− 2e2r cotϕLO + 𝒦SM (Ω)e2r , (459 ) 2 𝒦SM (Ω )
but with a different form of the optomechanical coupling factor, see [42]:
---4J-γ---- 𝒦SM (Ω ) = (γ2 + Ω2)2. (460 )
The factor J here is still defined by Eq. (369View Equation), but the circulating power is now twice as high as that of the position meter, for the given input power, because after leaving the beamsplitter, here each of the “north” and “east” beams visit both arms sequentially.

The key advantage of speed meters over position meters is that at low frequencies, Ω < γ, 𝒦SM is approximately constant and reaches the maximum there:

4J 𝒦SM (Ω ≪ γ ) ≈ arccot𝒦SM (0) = arccot --. (461 ) γ3
As a consequence, a frequency-independent readout quadrature optimized for low frequencies can be used:
4J ϕLO = arccot𝒦SM (0 ) = arccot-3-, (462 ) γ
which gives the following power (double-sided) spectral density
h [ −2r 4 2 2 2 ] h SSQL-(Ω-) --e----- Ω-(2γ--+-Ω--)- 2r S (Ω ) = 2 𝒦SM (Ω ) + γ8 𝒦SM (Ω)e . (463 )
Here, the radiation-pressure noise (the second term in brackets) is significantly suppressed in low frequencies (Ω <∼ γ), and ShSMLF can beat the SQL in a broad frequency band.

This spectral density is plotted in Figure 42View Image (left). For comparison, spectral densities for the lossless ordinary Fabry–Pérot–Michelson interferometer without and with squeezing, as well as for the ideal post-filtering configuration [see Eq. (408View Equation)] are also given. One might conclude from these plots that the Fabry–Pérot–Michelson interferometer with the additional filter cavities is clearly better than the speed meter. However, below we demonstrate that optical losses change this picture significantly.

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Figure 42: Examples of the sum quantum noise power (double-sided) spectral densities of the Sagnac speed-meter interferometer (thick solid line) in comparison with the Fabry–Pérot–Michelson based topologies considerd above (dashed lines). Left: no optical losses, right: with optical losses, ηd = 0.95, the losses part of the bandwidth − 1 γ2 = 1.875 s (which corresponds to the losses −4 Aarm = 10 per bounce in the 4 km length arms). “Ordinary”: no squeezing, ϕLO = π ∕2. “Squeezed”: 10 dB squeezing, 𝜃 = 0, ϕLO = π ∕2. “Post-filtering”: 10 dB squeezing, 𝜃 = 0, ideal frequency-dependent homodyne angle [see Eq. (408View Equation)]. For the Fabry–Pérot–Michelson-based topologies, J = J aLIGO and γ = 2π × 500 s− 1. In the speed-meter case, J = 2J aLIGO and the bandwidth is set to provide the same high-frequency noise as in the other plots (− 1 γ = 2π × 385 s in the lossless case and −1 γ = 2π × 360 s in the lossy one).

6.2.3 Optical losses in speed meters

In speed meters, optical losses in the arm cavities could noticeably affect the sum noise at low frequencies, even if

2 γ2- 2 𝜖arm ≡ γ ≪ 𝜖d, (464 ) 1
because the radiation pressure noise component created by the arm cavity losses has a frequency dependence similar to the one for position meters (remember that 𝒦 ∕𝒦SM → ∞ if Ω → 0; see Eqs. (389View Equation), (460View Equation)). In this paper, we will use the following expression for the lossy speed-meter sum noise, which takes these losses into account (more detailed treatment of the lossy speed meter can be found in papers [127, 50]):
h { [ ] h SSQL(Ω-) ---1---- − 2r 2r 2 ---𝜖2d--- S (Ω ) = 2 𝒦SM (Ω ) e + e cot ϕLO + sin2 ϕLO } − 2e2r cot ϕLO + 𝒦SM (Ω )e2r + 𝜖2 𝒦 (Ω) . (465 ) arm
The low-frequency optimized detection angle, in presence of loss, is
𝒦SM (0) 4J∕γ3 ϕLO = arccot-----2-−2r = arccot -----2-−2r, (466 ) 1 + 𝜖de 1 + 𝜖de
which gives
{ [ ] } h ShSQL(Ω-) e−2r +-𝜖2d -𝒦SM-(Ω-)- Ω4-(2γ2-+-Ω2)2 2r 2 2 S (Ω ) = 2 𝒦 (Ω) + 1 + 𝜖2e −2r γ8 e + 𝜖d + 𝜖arm 𝒦(Ω ) (467 ) SM d
[compare with Eq. (463View Equation) and note the additional residual back-action term similar to one in Eq. (408View Equation)].

This spectral density is plotted in Figure 42View Image (right), together with the lossy variants of the same configurations as in Figure 42View Image (left), for the same moderately optimistic value of ηd = 0.95, the losses part of the bandwidth and for − 1 γ2 = 1.875 s [which corresponds to the losses −4 Aarm = 10 per bounce in the 4 km length arms, see Eq. (322View Equation)]. These plots demonstrate that the speed meter in more robust with respect to optical losses than the filter cavities based configuration and is able to provide better sensitivity at very low frequencies.

It should also be noted that we have not taken into account here optical losses in the filter cavity. Comparison of Figure 42View Image with Figure 39View Image, where the noise spectral density for the more realistic lossy–filter-cavity cases are plotted, shows that the speed meter has advantage over, at least, the short and medium length (tens or hundred of meters) filter cavities. In the choice between very long (and hence expensive) kilometer scale filter cavities and the speed meter, the decision depends, probably, on the implementation costs of both configurations.

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