Observation of a kilogram-scale oscillator near its quantum ground state

We introduce a novel cooling technique capable of approaching the quantum ground state of a kilogram-scale system—an interferometric gravitational wave detector. The detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO) operate within a factor of 10 of the standard quantum limit (SQL), providing a displacement sensitivity of 10−18 m in a 100 Hz band centered on 150 Hz. With a new feedback strategy, we dynamically shift the resonant frequency of a 2.7 kg pendulum mode to lie within this optimal band, where its effective temperature falls as low as 1.4 μK, and its occupation number reaches about 200 quanta. This work shows how the exquisite sensitivity necessary to detect gravitational waves can be made available to probe the validity of quantum mechanics on an enormous mass scale.


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Abstract. We introduce a novel cooling technique capable of approaching the quantum ground state of a kilogram-scale system-an interferometric gravitational wave detector. The detectors of the Laser Interferometer Gravitational-wave Observatory (LIGO) operate within a factor of 10 of the standard quantum limit (SQL), providing a displacement sensitivity of 10 −18 m in a 100 Hz band centered on 150 Hz. With a new feedback strategy, we dynamically shift the resonant frequency of a 2.7 kg pendulum mode to lie within this optimal band, where its effective temperature falls as low as 1.4 µK, and its occupation number reaches about 200 quanta. This work shows how the exquisite sensitivity necessary to detect gravitational waves can be made available to probe the validity of quantum mechanics on an enormous mass scale.
Observation of quantum effects such as ground state cooling [1]- [15], quantum jumps [16], optical squeezing [17], mechanical squeezing [18]- [20] and entanglement [21]- [26] that involve macroscopic mechanical systems are the subject of intense experimental effort [27]. The first step toward engineering a non-classical state of a mechanical oscillator is to cool it, minimizing the thermal occupation number of the mode. Any mechanical coupling to the environment admits thermal noise that randomly drives the system's motion, as dictated by the fluctuation-dissipation theorem [28], but 'cold' frictionless forces, such as optical or electronic feedback, can suppress this motion, hence cooling the oscillator.
Two types of forces have recently proven valuable for cooling. The first is a frictionless damping force, originating either from an electronic servo system ('cold damping') [4,29,30] or from photothermal or radiation pressure forces in a detuned cavity ('cavity cooling') [1]- [3], [5,6,8]; this force reduces the motion of the oscillator while also diminishing its quality factor. The second is an optical restoring force, which increases the resonant frequency of the oscillator without additional friction, effectively increasing its quality factor [7,10]. To reach the quantum regime in experiments exploiting these techniques, a low noise oscillator's position must be monitored by a highly sensitive readout device. By providing both of these features, the Laser Interferometer Gravitational-wave Observatory (LIGO) interferometers present a unique opportunity to cool kilogram-scale mirrors to enticingly low temperatures. Although the LIGO interferometers do not have sufficiently large optical restoring forces for the second effect to be significant, their active control systems may instead be used to reproduce the effect.

The LIGO interferometers
LIGO operates three kilometer-scale interferometric detectors with the goal of directly detecting gravitational waves of astrophysical origin [31,32]. The measurements reported here were performed at LIGO's Hanford Observatory. The detector shown in figure 1 comprises a Michelson interferometer with a 4 km long Fabry-Perot cavity of finesse 220 placed in each arm to increase the sensitivity of the detector. Each mirror of the interferometer has mass M = 10.8 kg, and is suspended from a vibration-isolated platform on a fine wire to form a pendulum with frequency p = 0.74 Hz, to shield it from external forces and to enable it to respond to a gravitational wave as a mechanically free mass above the natural resonant frequency. To minimize the effects of laser shot noise, the interferometer operates with high power levels; approximately 400 W of laser power of wavelength 1064 nm is incident on the beam splitter, resulting in over 15 kW of laser power circulating in each arm cavity. The present detectors are sensitive to changes in relative mirror displacements of about 10 −18 m in a 100 Hz band centered around 150 Hz (figure 2). This low noise level allows for the preparation of lowenergy states for the oscillator mode considered next.
The four mirrors of the LIGO interferometer (figure 1) are each an extended object with a displacement x i (i = 1, . . . , 4) defined along the optical beam axis. The servo control system that keeps the interferometer mirrors at the resonant operating point is an essential component of this study. While all longitudinal and angular degrees of freedom of the mirrors are actively controlled, our discussion is limited to the differential arm cavity motion, which is the degree of freedom excited by a passing gravitational wave, and hence also the most sensitive to mirror displacements. This mode corresponds to the differential motion of the centers of mass of the four mirrors, 4 ), and has a reduced mass of M r = 2.7 kg. A signal proportional to differential length changes is measured at the antisymmetric output of the beam splitter, as shown in figure 1. This signal is filtered by a servo compensation network before being applied as a force on the differential degree of freedom by voice coils that actuate magnets affixed to the mirrors.

The cooling mechanism
The degree of freedom that is of interest as a quantum particle is the differential mirror motion x c . However, optical measurements probe the location of the mirror surface (averaged over the optical beam), which differs from center-of-mass location due to the mirror's internal thermal noise, and include a sensing noise due to the laser shot noise. Combining these noises into a total displacement noise X N , the output signal is written as The center-of-mass motion is also subject to a noise force F N (including, for example, the thermally driven motion of the mirror suspensions and the seismic motion of the ground that couples through the suspensions) and a feedback force that is proportional to x s . The resulting equation of motion in the frequency domain is given by: Here K ( ) is the frequency-domain feedback filter kernel, and the φ( ) term accounts for mechanical damping. For a viscously damped pendulum with quality factor Q p = p / p . To control the differential degree of freedom, an optical signal proportional to mirror displacement is measured on the photodetector (PD), and fed back as a differential force on the mirrors, after appropriate filtering to form restoring and damping forces. (b) The spectral displacement noise density of the differential mode of motion of the LIGO 4 km interferometer at the Hanford Observatory is shown. Also shown is the target sensitivity and the quantum noise contribution, which consists of shot noise above 30 Hz and radiation pressure noise below. The standard quantum limit (SQL) is also shown, and the closest approach to the measured sensitivity is about a factor of 10 near 150 Hz. ( p and p correspond to the real part and twice the imaginary part of the complex eigenfrequency of the pendulum), φ( ) = 1/Q p . If the damping is not viscous, but instead caused by internal friction, φ( ) takes on a more complex form [28]. Combining equations (1) and (2), the equation of motion for the center-of-mass is obtained: In this experiment, the control kernel is adjusted so that with eff and eff much larger than p and p , respectively, such that the modified dynamics of x c are given by a damped oscillator driven by random forces: An electro-optical potential well in which the mirrors oscillate is thus created. The output of our experiment measures x s , and in order to deduce true mirror motion x c , the limiting sources of noise must be considered. If noise predominantly drives the centerof-mass motion, i.e. F N K ( )X N , then x s ≈ x c (see equation (1)) and the measured signal corresponds to the center-of-mass motion. However, in the case that surface or sensing noise dominates, i.e. K ( )X N F N , then a correction factor must be applied to the measured signal to deduce the center-of-mass motion. Taking equations (1) and (5), in the limit that F N = 0, we obtain If the levels of each noise X N and F N are not precisely known, then one can make a conservative correction by applying a factor max(1, |K ( )/M r 2 |) to determine the worst possible centerof mass motion, thereby accounting for the fact that the servo can inject noise back onto the oscillator. The effective temperature of the mode may then be obtained: where S x s is the single-sided power spectral density of the measured motion x s and S x d includes the correction factor. At large feedback gains, the measured noise S x s may be arbitrarily suppressed, however, the mirror motion will reach a finite level as limited by the detection noise X N . This 'squashing' effect has been explored previously [9,33], and the calculation of S x d avoids underestimates of the mirror motion. It is impossible to reliably measure the mirror motion at arbitrarily high frequencies, and the integral in equation (8) will diverge in any real system. The integration must therefore be limited in its frequency band, as is later discussed. Finally, the corresponding occupation number may be determined by K ( ) of equation (4) is formed by convolving the position-dependent output signal with filter functions corresponding to the real and imaginary parts of the feedback kernel K ( ).
In the LIGO feedback system, there are additional filters and propagation delays that cause deviations from the ideal cold, damped spring, at high and low frequencies. Below 100 Hz, K ( ) increases sharply to suppress seismically driven motion; at high frequencies (above a few kilohertz), K ( ) decreases precipitously to prevent the control system from feeding shot noise back onto the mirrors. However, in the frequency band important for this measurement (near the electro-optical resonance), the feedback is well approximated by a spring and damping force, as shown in figure 2.

Measurement results and discussion
The servo control loops of the LIGO interferometers are optimized to minimize noise coupling to measurement of the differential mode motion of the mirrors. The modifications to the servo loops to create a nearly ideal cold spring at eff = 140 Hz do not significantly affect the noise limits, shown in figure 1. Figure 3 shows the amplitude spectral density of mirror displacement for varying levels of cold damping. To infer the effective temperature of the mode, its effective frequency eff and an estimation of the root-mean-square displacement fluctuation δx rms must be determined. First the differential mirror motion is driven and the response is measured, as shown in figure 2. These response functions are fit to a damped oscillator model; eff and Q eff are products of the fit. Then δx rms is computed by integrating the spectrum in the band from 100 to 170 Hz, as described in equation (8). The sensitivity in this frequency band is limited by laser shot noise that enters into X N . To correct for the finite integration band, the result is scaled by setting our measured spectrum equal to the integral over the same frequency band of a thermally driven oscillator spectrum, In this way, a minimum effective temperature T eff = 1.4 ± 0.2 µK is measured, corresponding to thermal occupation number N eff = 234 ± 35. Systematic error of 15% in the calibration dominates statistical error in these uncertainty estimates. The limits to integration were chosen as a compromise between having a wide limit, and choosing frequencies at which mirror motion is sensed. In the limit that the width of the integration band approaches 0, the lowest temperature achieved approaches 0.9 µK. For larger integration limits, the temperature diverges because of the increased uncertainty at high frequency caused by shot noise (as occurs in all experiments). The spectra in figure 3 are predominantly limited by shot noise in the measurement band. It may at first appear unusual to associate a temperature with a device limited by shot noise, rather than thermal noise. However, the above calculations are justified, since the ultimate limit to experiments such as this is known to arise from optical noise [34].

Cooling to the quantum limit
An interesting question arises as to whether this technique can lead to ground state cooling of the electromechanical oscillator. To mitigate the shot noise limit, which arises due to the fluctuating number of photons detected, the laser power could be increased. However, radiation pressure noise (a fluctuating force exerted on the mirrors due to the shot noise of the laser) increases with laser power and will ultimately limit the sensitivity. The SQL is obtained when shot noise and radiation pressure noise contribute equally to the total quantum noise [35]. Hence, the continuous displacement measurement required for servo feedback does introduce an additional term to the uncertainty relation for the oscillator position and momentum fluctuations due to measurement-induced steady state decoherence. If, however, the classical noises (such as thermal) are reduced significantly below the SQL, active feedback, with the appropriate control kernel, is capable of cooling the electro-optic oscillator to its motional ground state [36].

Future prospects with LIGO
In the coming years, two upgrades of the LIGO detectors are planned. The first, Enhanced LIGO, is presently underway with an expected completion date in 2009, and seeks to improve the sensitivity of the instruments above 40 Hz. The improvement in displacement sensitivity in the frequency band around 150 Hz, where the cold spring measurements were performed, is expected to be about a factor of 2. Subsequently, a major upgrade, Advanced LIGO, expected to be completed in 2014, should give a factor of 10-15 improvement in displacement sensitivity relative to that of the detector used for this work (with a concomitant factor of 4 increase in mass). In Advanced LIGO, the laser power circulating in the Fabry-Perot cavities should exceed 800 kW, permitting strong restoring forces to be generated optically. Enhanced LIGO is expected to reach ∼6 times lower occupation number, approaching 40 quanta, and with Advanced LIGO, the detectors will be operating at the SQL, allowing the ground state to be approached.
As they approach the SQL, these devices should enable novel experimental demonstrations of quantum theory that involve kilogram-scale test masses [25,37,38]. The present work, reaching microkelvin temperatures, provides evidence that interferometric gravitational wave detectors, designed as sensitive probes of general relativity and astrophysical phenomena, can also become sensitive probes of macroscopic quantum mechanics.