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Suppression of Brownian motion by gravity field
Volume 92A, number 3
PHYSICS LETTERS
1 November 1982
SUPPRESSION OF BROWNIAN MOTION BY GRAVITY FIELD K. KURODA, K. TSUBONO and H. HIRAKAWA Department of Physics, The University of Tokyo, Bunkyo, Tokyo 113, Japan Received 13 April 1982
It is demonstrated that the Brownian motion of a vibrational mode of a torsional oscillator at 41 Hz was suppressed by a feedback loop which incorporates the newtonian gravity field ofa rotating bar. An effective temperature Te 35 K was obtained.
In thermal equilibrium at a temperature T, a vibrational mode of a macroscopic oscillator is excited with mean energy kT. There are several methods to decrease the amplitude of this so-called Brownian motion [1]. The first is to damp the oscillator with a cold load, aload of low temperature [2]. The second method, called feedback damping [3], is to use a negative feedback mechanism suppressing the output noise amplitude. Considering an artificial cold resistor [41for a load of the oscillator, one understands that the above two methods have much in common. In this letter we show a feedback damping method which uses the grayity field of a rotating body as a part of its feedback loop. This method of feedback using gravity field will be particularly useful in a low-temperature low-noise experiment including detection of gravitational radiation. The deep penetrability of the gravitational signal makes it easy to construct a feedback loop across the walls of the cryostat. A torsional oscillator with mass quadrupole moment interacts with the quadrupole part of the dynamic gravitational potential [5] produced around a bar of quadrupole moment q rotating with angular frequency w0 (fig. 1). For reasons of symmetry the force on the oscifiator of resonant frequency 2w0 disappears when the oscillator is located on the rotation plane of the bar, i~’= 0. For a small value of the rotor inclination angle ~,L’, the gravitational force acting on the oscifiator located at distance R is shown to be
R
4
,7xY
~
---i
I Fig. 1. A torsional oscillator interacts with the gravity field produced around a rotating bar. The rotation axis makes an angle q from the neutral position around a vertical axis.
referred to the generalized coordinate where ~xy ~S the dynamic part of the oscifiator quadrupole moment. When the phase ~r of the rotor is locked in quadrature to the oscifiator phase Ø~and the inclination angle ~i is set proportional to the force fed back to the osdilator is proportional to thus establishing a necessary feedback loop for damping. We constructed a system consisting of a 46 kg steel rotor driven at 20 Hz and a 12 kg high-Q torsional os~,
~,
—~,
cifiator of frequency 41 Hz and reduced mass ~.z= 14 kg, both housed in vacuum tanks separately, and computer aided phase locking loop (PLL) for ~r and control for i~i.The effective quality factor Qe of this feedback system is given by —
l/Qe 1/Q0 + l/Q~, (2) where Q0 is the quality factor of the system including —
F = 9Gqq~~(~P/R 5) cos(2w0t + ~r), 0 031-9163/82/0000-—0000/$02.75 © 1982 North-Holland
121
Volume 92A, number 3
PHYSICS LETTERS
/~/ ~200
100
I
1
where a = w~T/Qe represents the effect of the required signal integration time r and assumed as a 1. Since the output signal from the transducer has its equivalent displacement noise spectrum (~n2),to perform normal PLL action the signal must be passed through a filter with the time constant r, which is required to satisfy r~~iw~(~fl2>/kTe. In our case we had r lOs. The result of the feedback damping at T 0 = 300 K and R = 1.4 mis shown in fig. 2. It agrees well with eq. (4). The lowest temperature of the system obtained was 35 K, a value limited by the insufficient lock range of PLL. We thank K. Oide for his earlier participation with us in this experiment. This work was supported by a <~
300
0
1 November 1982
25
X10 Fig. 2. Effective temperature TeQeobtained by feedback damping at distanceR = 1.4 m. Q~was calculated from eq. (2). The line shows the theory [eq. (4)] with r 10 s. The lowest temperature attained was 35 K.
grant
from the Yamada Science Foundation.
References the transducer electric circuit and was equal to 1.7 X 10g. l/Q~’is the feedback ioop gain which could be adjusted by transducer and amplifier gains. The effective temperature Te of the fluctuation and system temperature I’0 satisfies TeIQe = T 0/Q0 (3) if an ideal noiseless transducer is available. In practice we have TeIQe = T0/Q0 +aT0/Q0,
122
(4)
[11 C. Kittel, Elementary statistical
physics (Wiley, New
York, 1958) Sec. 30. [2] H. Hirakawa, S. Hiramatsu and Y. Ogawa, Phys. Lett. 63A (1977) [3] J.M.W. Milatz199. and J.J. van Zolingen, Physica 19 (1953) 181. [4] K. Oide, Y. Ogawa and H. Hirakawa, Japan J. Appl. Phys. 17 (1978) 429.
[5] K. Oide, K. Tsubono and H. Hirakawa, Japan J. Appl. Phys. 19 (1980) L123.
PHYSICS LETTERS
1 November 1982
SUPPRESSION OF BROWNIAN MOTION BY GRAVITY FIELD K. KURODA, K. TSUBONO and H. HIRAKAWA Department of Physics, The University of Tokyo, Bunkyo, Tokyo 113, Japan Received 13 April 1982
It is demonstrated that the Brownian motion of a vibrational mode of a torsional oscillator at 41 Hz was suppressed by a feedback loop which incorporates the newtonian gravity field ofa rotating bar. An effective temperature Te 35 K was obtained.
In thermal equilibrium at a temperature T, a vibrational mode of a macroscopic oscillator is excited with mean energy kT. There are several methods to decrease the amplitude of this so-called Brownian motion [1]. The first is to damp the oscillator with a cold load, aload of low temperature [2]. The second method, called feedback damping [3], is to use a negative feedback mechanism suppressing the output noise amplitude. Considering an artificial cold resistor [41for a load of the oscillator, one understands that the above two methods have much in common. In this letter we show a feedback damping method which uses the grayity field of a rotating body as a part of its feedback loop. This method of feedback using gravity field will be particularly useful in a low-temperature low-noise experiment including detection of gravitational radiation. The deep penetrability of the gravitational signal makes it easy to construct a feedback loop across the walls of the cryostat. A torsional oscillator with mass quadrupole moment interacts with the quadrupole part of the dynamic gravitational potential [5] produced around a bar of quadrupole moment q rotating with angular frequency w0 (fig. 1). For reasons of symmetry the force on the oscifiator of resonant frequency 2w0 disappears when the oscillator is located on the rotation plane of the bar, i~’= 0. For a small value of the rotor inclination angle ~,L’, the gravitational force acting on the oscifiator located at distance R is shown to be
R
4
,7xY
~
---i
I Fig. 1. A torsional oscillator interacts with the gravity field produced around a rotating bar. The rotation axis makes an angle q from the neutral position around a vertical axis.
referred to the generalized coordinate where ~xy ~S the dynamic part of the oscifiator quadrupole moment. When the phase ~r of the rotor is locked in quadrature to the oscifiator phase Ø~and the inclination angle ~i is set proportional to the force fed back to the osdilator is proportional to thus establishing a necessary feedback loop for damping. We constructed a system consisting of a 46 kg steel rotor driven at 20 Hz and a 12 kg high-Q torsional os~,
~,
—~,
cifiator of frequency 41 Hz and reduced mass ~.z= 14 kg, both housed in vacuum tanks separately, and computer aided phase locking loop (PLL) for ~r and control for i~i.The effective quality factor Qe of this feedback system is given by —
l/Qe 1/Q0 + l/Q~, (2) where Q0 is the quality factor of the system including —
F = 9Gqq~~(~P/R 5) cos(2w0t + ~r), 0 031-9163/82/0000-—0000/$02.75 © 1982 North-Holland
121
Volume 92A, number 3
PHYSICS LETTERS
/~/ ~200
100
I
1
where a = w~T/Qe represents the effect of the required signal integration time r and assumed as a 1. Since the output signal from the transducer has its equivalent displacement noise spectrum (~n2),to perform normal PLL action the signal must be passed through a filter with the time constant r, which is required to satisfy r~~iw~(~fl2>/kTe. In our case we had r lOs. The result of the feedback damping at T 0 = 300 K and R = 1.4 mis shown in fig. 2. It agrees well with eq. (4). The lowest temperature of the system obtained was 35 K, a value limited by the insufficient lock range of PLL. We thank K. Oide for his earlier participation with us in this experiment. This work was supported by a <~
300
0
1 November 1982
25
X10 Fig. 2. Effective temperature TeQeobtained by feedback damping at distanceR = 1.4 m. Q~was calculated from eq. (2). The line shows the theory [eq. (4)] with r 10 s. The lowest temperature attained was 35 K.
grant
from the Yamada Science Foundation.
References the transducer electric circuit and was equal to 1.7 X 10g. l/Q~’is the feedback ioop gain which could be adjusted by transducer and amplifier gains. The effective temperature Te of the fluctuation and system temperature I’0 satisfies TeIQe = T 0/Q0 (3) if an ideal noiseless transducer is available. In practice we have TeIQe = T0/Q0 +aT0/Q0,
122
(4)
[11 C. Kittel, Elementary statistical
physics (Wiley, New
York, 1958) Sec. 30. [2] H. Hirakawa, S. Hiramatsu and Y. Ogawa, Phys. Lett. 63A (1977) [3] J.M.W. Milatz199. and J.J. van Zolingen, Physica 19 (1953) 181. [4] K. Oide, Y. Ogawa and H. Hirakawa, Japan J. Appl. Phys. 17 (1978) 429.
[5] K. Oide, K. Tsubono and H. Hirakawa, Japan J. Appl. Phys. 19 (1980) L123.