PHYSICS LETTERS A

ELSEV1ER

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A very high quality facior sapphire dielectric resonator transducer is shown to be an extremely sensitive readout device for a resonant bar gravitational wave antenna. The transducer has the possibility of allowing operation below the standard quantum limit. Tests on <i prototype transducer with quality factor up to Q s" 4 x lO" are described.

I. Introduction

Al present ihc sensitivity reached by cryogenic gravitational wave antennas developed by several groups around the world is enough 10 register a perturbation of inc metric icnsor h” I x 10"'* [1]. However, the estimates of the rate of supernova explosions in our galaxy, which is the only known source of such a perturbation, give a value of about 1 per 10-30 years [2]. In order lo detect a few events every year, il is necessary lo register gravity wave bursts from the nearest cluster of galaxies with a sensitivity of h of order lO"^. The sensitivity of low temperature resonant bar antennas is practically limited by the readout devices which consist either of SQUID based transducers or microwave parametric transducers. In this work we describe the pcrfor-

mance of a transducer based on a cryogenic sapphire dielectric resonator operating in whispering gallery modes [3.4]. The extremely high quality facior of a sapphire transducer, operating at liquid helium lent-peratures. allows the development of new transducers with improved perfoniiance.

The ultimate sensitivity of the parametric displacement transducer is described by the simple expression [5]

where /n is the transducer resonance frequency. ct//a^. is the tuning coefficient, qq is the qualiiv facior. k is Bolizmann's constant, W is the applied pump power, T is the noise temperature of the pump source and t is the averaging lime. It is evident that the facior

Now ai the Phy&lct Dcpanmcni. mmcow Suie University. Mrscow 119899. Russian Federation. ^: E-mail: mikc^earwax.ptl.uwt.cdu.au-

Eli-cvicr Science B^. SSDI 0375-9601(95)00959.0

\'> useful 10 characlcrisc (he quality of ihc transducer itself. For (he re-entrant cavity transducer [6] Z - -x 10⁷ MHz//.im. Such a transducer is used as a displacement sensor for (he University of Western Australia gravitational wave detector. The sapphire transducer we report now has Z = 3 X 10⁸ MHz/^m. which is one order of magnitude greater than the re-cntrani cavity.

As shown in Ref. |7]. there is a fundamental limit for accuracy of a standard (amplilude-and-phase) measurement due to the back action effect. However. if a special form of pump source signal is used [8]. it is possible, in principle, to monitor one quadrature component of the mechanical oscillator and redirect back .iction perturbations to the other. In this case the excitation voltage applied to the transducer should consist of two components with frequencies /o -/„, and /o+/n, (double frequency pump); here /,„ I^{s a} resonant frequency of a mechanical oscillations. The necessary condition for the back action evading (BAG) operation is that the transducer bandwidth fo/Qo ^has ^ be small compared to /„,. For the re-entrant cavity /o° 10¹⁰ Hz. qq” 1 X 10^s and this condition is not satisfied for /„, ” 10³ Hz. However the quality factor of the sapphire transducer was measured to be as large as Qo - 3.9 X lO" at /0¹⁰ Hz making BAE possible.

Thus it is possible to improve the sensitivity of the gravitational wave detector and under certain conditions to overcome the standard quantum limit using the sapphire transducer.

2. Measurements of low temperature quality fac tor and tuning coefficient of a sapphire transducer

Whispering gallery mode resonant transducers with high quality factors operating at room temperature have been developed previously [4]. In this work we have tested a similar transducer as a prototype registering device for a cryogenic gravitational wave antenna. The cryogenic transducer design is shown schematically in Fig. 1. The transducer consists of two coaxial sapphire mushrooms shielded with the niobium cavity. The lop of the cavity is a membrane with a resonant frequency 1.2 kHz when loaded by a sapphire mushroom. A piston driven by a stepper

Fig. 1. Schematic diagram of the transducer anil stepper motors for coupling adjustment and spuing iNlfusUncnt.

motor is used 10 provide a pressure on the top of (he membrane for calibration. Spacing variations between 0-KX) Jxm could be achieved. The high frequency of the membrane and overall rigid design is intended to reduce the effect of seismic noise. The sapphire di^k-. .ire 14.9 mm thick and 30 mm in diameter. The axial shift and skew of the disks does not exceed 50 ^.m. The microwave pump source is coupled to the transducer by an electric probe. The probe position can be adjusted by a stepper motor. The entire system is placed in a vacuum can and cooled with liquid helium in a conventional cryostaL

To measure the quality factor a fast frequency sweep method in reflection was used [9]. As the sweep rate across a cavity is incrc.ised. at a certain rate the response starts to ring. The decaying cavity field mixes with the swept signal producing a decaying signal of varying beat frequency. The time constant of the decay is equal to the resonant field amplitude decay time, and by measuring the time for half the amplitude 10 decay (/,/:>) the resonators Q can be calculated by

/. f^^.^ ^Q=^- ⁽¹⁾

Fig. 2- The dependence of the cigcnficqucncy of ihc E^,, mode an inc gap spacing between disk”.

We have tested the behaviour of B-type whispering-gallery modes with azimuth field variation numbers from 7 to 9. The low temperature frequency and unloaded Q dependences for Ihe transducer modes EtJ., and £7,1 are shown as functions of gap spacing in Figs. 2 and 3 respectively. The maximum value of the tuning coefficient was 8//3L - 1.7 ± 0.2 MHz/^im, which is comparable to the room temperature result of greater than 2 MHz/<im [4]. The difference between this result and the room tempera-tun: one can be explained by the slight non-parallel alignment of the disks. The maximum observed quality factor was fig = (3.9 ± 0.2) x 10'. This value varied in the range of 5 X 10'-3.9 X 10'. and was severely degraded if a low Q resonance tuned close to ihe frequency of Ihe selected mode due to reactive coupling between modes [10.11].

The temperature dependence of Qy is shown in Fig. 4. The figure shows a strong step in qq at Ihe

niobium superconducting transition temperature. This indicates that a significant pan of the microwave field energy is outside the sapphire disks.

The above results demonstrate the very high achievable quality factor in the sapphire transducer. The results are comparable to (he quality factor of fixed frequency sapphire resonators. It seems not unrealistic to expect a quality factor above 10⁹ in larger diameter disks of higher purity [12]. However, the experiments show that the quality factor decreases for the gap length value </>40 /im. This effect could limit Ihe dynamic range of the future transducers.

The properties of a sapphire dielectric microwave transducer for the gravitational wave antennas have been described. The obtained Qo - 3.9 x 10" is the best value ever measured for a microwave transducer. The application of this transducer can improve total sensitivity of the antennas. In future work we intend to apply the methods discussed here to a vibration isolated resonator, with a view to creating a practical prototype BAE transducer.

We are grateful to Professor V.B. Braginsky for his interest in this work, to Dr. A.G. Mann. PJ.

Turner and A.N. Luiien for their assistance with experiments. This work was supported by the Australian Research Council and a Department of Indus-ny. Technology and Regional Development International Science and Technology cooperation grant.