1.1 Analysis and design of advanced gravitational-wave in- terferometers: beating the Standard Quantum Limit
1.1.4 Speed-meter interferometers with Michelson topology
As was mentioned in Sec. 1.1.1, measuring a QND observable explicitly can make a gravitational- wave interferometer immune to the Standard Quantum Limit. For a free test mass, the momentum is such an observable, by virtue of having commuting Heisenberg operators at different times. Mo- tivated by this fact, Braginsky and Khalili proposed measuring the speed of a free test mass, which is closely related to its momentum [27]. Two gedanken designs were studied in Ref. [27], with the second one deemed easier to realize in gravitational-wave detectors. 6 This second gedanken design requires two weakly coupled resonators with equal eigenfrequency. Resonator 1 is pumped on resonance, while resonator 2 is left empty. Any change of the position of one end of resonator 1 causes a length change that phase modulates the carrier field, generating signal sidebands; no signals are generated inside resonator 2, since it is empty. As a property of weakly coupled res- onators, the signal sidebands generated in resonator 1 “slosh” [move back and forth] between these two resonators,flipping signeach time they return into resonator 1, thereby canceling any sensitivity to time-independent position. For motions with frequencies below thesloshing frequency, speed is recorded in the sideband fields extracted from resonator 1; at higher frequencies the output signal is a combination of speed and higher time derivatives of position.
Braginsky, Gorodetsky, Khalili and Thorne (BGKT) analyzed a microwave version of this orig- inal “speed-meter design,” and proposed an optical version modeled straightforwardly from the microwave system, withfourkilometer-scale cavities. Purdue [29] analyzed the proposal of BGKT in detail, showing that a broadband QND performance can indeed be achieved with ordinary ho- modyne detection. The QND performance of the speed meter is shown to be characterized by a spectrum that beats the Standard Quantum Limit by a relatively constant factor below the “sloshing frequency.” A plausible amount of circulating power (megawatt scale, similar to the requirement of the QND position meter proposed by KLMTV [15]) is required for the speed meter to beat the SQL by a significant amount. However, as Purdue found, an exorbitant amount of pumping power (nearly gigawatt scale) is needed to achieve the required circulating power. In addition, a large amount of light (nearly megawatt level) comes out of the interferometer together with the signal light, complicating the photodetection process.
Based on the work of Purdue, I invented a mathematically equivalent configuration that can fit more easily into the facility of LIGO, and can solve the problem of high pumping power and high
6The first gedanken design described in Ref. [27] was then regarded as harder to realize — but that is no longer true, see Sec. 1.1.5.
Laser
Squeezed
Vacuum Circulator
Homodyne detector
Filter Cavity I
Filter Cavity II
Arm Cavity
Arm Cavity
Sloshing Cavity RSE Mirror
Power-Recycling Mirror
Figure 1.3: Optical topology of the Purdue-Chen speed meter. A kilometer-scale sloshing cavity is added at the dark port of a Michelson interferometer, folded back to share the vacuum tube with one of the arms. The folding mirror is left somewhat transmissive to allow the extraction of signal.
Cavities can be used in the arms to enhance the circulating power, but an RSE mirror must added to compensate the effect of the arm-cavity on the signal sidebands. Input squeezing and variational readout can be implemented using the proposal of KLMTV, with a circulator and two kilometer-scale optical filter cavities.
5 10 50 100 500 1000 0.5
1 5 10 50
[Sh (f)/SSQL (100 Hz)]1/2
f (Hz) SVPM CPM
SVSM
SISM
SVSM SQL
Figure 1.4: Quantum noise spectra of Purdue-Chen speed meters [one squeezed-input speed meter (SISM) and two squeezed variational speed meters (SVSM) with different parameters] and com- parison with conventional interferometer (conventional position meter, CPM) and KLMTV inter- ferometer (squeezed variational position meter, SVPM). The circulating power of all configurations is 820 kW, with input squeeze factor e2R = 10 (CPM does not have squeezing). Optical-loss level thought to be practical in the next decade is used.
static output power. (See Fig. 1.3 for its optical topology.) This design is analyzed by Purdue and me in Chapter 5. In this Purdue-Chen speed meter, resonator 1 is the antisymmetric mode of a Michelson interferometer. The symmetric mode can be pumped in the usual way from the bright port, which then couples the antisymmetric motions of the mirrors to the antisymmetric optical mode in a manner of a conventional interferometer such as LIGO-I. Power recycling techniques can also be used to enhance the circulating power, as in LIGO-I. An additional cavity with equal length is placed in the dark port, forming resonator 2, which is empty. The (highly reflective) common input mirror of the two optical resonators is called the sloshing mirror. In practice, the sloshing cavity will have to be folded back into one of the interferometer arms; the folding mirror can be made partially transmissive, forming an output coupler which allows signal sidebands to be extracted from resonator 1. [In our analysis, one of the two ports opened by the output coupler was closed for ease of treatment.] In principle, there would be no static output light, since the detection is made at a dark port. Arm cavities can also be used to further enhance the circulating power and decrease the power going through the beamsplitter; their effect on the signal sideband can be removed by putting an RSE mirror at the dark port, making an impedance-matched cavity with the arm-cavity input mirror.
The QND performance of this Purdue-Chen speed meter can be further enhanced by the use of input squeezing and variational output techniques on the speed meter. As we show in Sec. 6.4.1, input squeezing with frequency-independent squeeze angle can increase theeffective optical power
by the squeeze factor (as in Caves’ original proposal for conventional interferometers [11]; see also KLMTV [15]), thereby enhancing the sensitivity by the same factor (in power) for frequencies below the sloshing frequency. We also show that, using two detuned FP cavities as optical filters in the output can enhance high-frequency performance greatly. As a by-product of our research, in Sec. 6.7, we work out the most general frequency-dependent rotation angle in quadrature fields achievable by detuned (high finesse) FP cavities, and give a prescription for solving for the corresponding filter parameters needed — an issue left untackled by KLMTV [15].
Finally, Purdue and I study the influence of optical losses in speed-meter interferometers. The mirror quality thought achievable by the next decade (10 ppm loss per bounce) dictates that the sloshing cavity, as well as the (optional) output filters have lengths of kilometers, in order to achieve a sensitivity a factor 5 (in amplitude) below the Standard Quantum Limit. This reduces the practical- ity of adding these cavities. On the other hand, these speed meters were able to achieve a broadband QND performance with one such additional cavity (one less than the KLMTV interferometers); and were found to be significantly less susceptible to losses than the KLMTV interferometers, due to the shape of their transfer functions. The full noise spectra of speed meter designs, with optical losses included are summarized and compared with KLMTV interferometers (QND position meters) in Sec. 6.5.3. (See Fig. 1.4).