A force-measuring system composed of a macroscopic harmonic oscillator, mechanical-electrical transducer, and linear amplifier is described and the sensitivity is calculated for a "back-action evasion" mode of operation. It is shown that it is possible to surpass the standard amplifier limit with this device, and the specific requirements to do so are given. These results may also apply to a quantum-limited amplifier and thus give the requirements to perform quantum-nondemolition measurements.
Several high-precision physics experiments are approaching a level of sensitivity at which the intrinsic quantum nature of the experimental apparatus is the dominant source of fluctuations limiting the sensitivity of the measurements. This quantum limit is embodied by the Heisenberg uncertainty principle, which prohibits arbitrarily precise simultaneous measurements of two conjugate observables of a system but allows one-time measurements of a single observable with any precision. The dynamical evolution of a system immediately following a measurement limits the class of observables that may be measured repeatedly with arbitrary precision, with the influence of the measurement apparatus on the system being confined strictly to the conjugate observables. Observables having this feature, and the corresponding measurements performed on them have been named quantum nondemolition or back-action evasion observables. In a previous review (Caves et al., 1980, Rev. Mod. Phys. 52, 341) a quantum-mechanical analysis of quantum nondemolition measurements of a harmonic oscillator was presented. The present review summarizes the experimental progress on quantum nondemolition measurements and the classical models developed to describe and guide the development of practical implementations of quantum nondemolition measurements. The relationship between the classical and quantum theoretical models is also reviewed. The concept of quantum nondemolition and back-action evasion measurements originated in the context of measurements on a macroscopic mechanical harmonic oscillator, though these techniques may be useful in other experimental contexts as well, as is discussed in the last part of this review. [S0034-6861(96)00103-1]
The monitoring of a quantum-mechanical harmonic oscillator on which a classical force acts is important in a variety of high-precision experiments, such as the attempt to detect gravitational radiation. This paper reviews the standard techniques for monitoring the oscillator, and introduces a new technique which, in principle, can determine the details of the force with arbitrary accuracy, despite the quantum properties of the oscillator. The standard method for monitoring the oscillator is the "amplitude-and-phase" method (position or momentum transducer with output fed through a narrow-band amplifier). The accuracy obtainable by this method is limited by the uncertainty principle ("standard quantum limit"). To do better requires a measurement of the type which Braginsky has called "quantum nondemolition." A well known quantum nondemolition technique is "quantum counting," which can detect an arbitrarily weak classical force, but which cannot provide good accuracy in determining its precise time dependence. This paper considers extensively a new type of quantum nondemolition measurement---a "back-action-evading" measurement of the real part
X1 (or the imaginary part
X2) of the oscillator's complex amplitude. In principle
X1 can be measured "arbitrarily quickly and arbitrarily accurately," and a sequence of such measurements can lead to an arbitrarily accurate monitoring of the classical force. The authors describe explicit Gedanken experiments which demonstrate that
X1 can be measured arbitrarily quickly and arbitrarily accurately. In these experiments the measuring apparatus must be coupled to both the position (position transducer) and the momentum (momentum transducer) of the oscillator, and both couplings must be modulated sinusoidally. For a given measurement time the strength of the coupling determines the accuracy of the measurement; for arbitrarily strong coupling the measurement can be arbitrarily accurate. The "momentum transducer" is constructed by combining a "velocity transducer" with a "negative capacitor" or "negative spring." The modulated couplings are provided by an external, classical generator, which can be realized as a harmonic oscillator excited in an arbitrarily energetic, coherent state. One can avoid the use of two transducers by making "stroboscopic measurements" of
X1, in which one measures position (or momentum) at half-cycle intervals. Alternatively, one can make "continuous single-transducer" measurements of
X1 by modulating appropriately the output of a single transducer (position or momentum), and then filtering the output to pick out the information about
X1 and reject information about
X2. Continuous single-transducer measurements are useful in the case of weak coupling. In this case long measurement times are required to achieve good accuracy, and continuous single-transducer measurements are almost as good as perfectly coupled two-transducer measurements. Finally, the authors develop a theory of quantum nondemolition measurement for arbitrary systems. This paper (Paper I) concentrates on issues of principle; a sequel (Paper II) will consider issues of practice.
Limitations on the sensitivity for detecting a weak classical force acting on a harmonic oscillator are imposed by the quantum mechanical properties associated with a "standard" amplitude-and-phase measurement and, classically, by the back reaction noise associated with the finite noise temperature of the amplifier used to process the measurement. These are known as the "standard quantum limit" and the "standard amplifier limit", respectively. We present the theoretical motivation behind the quest to circumvent these limits, and examine a single-transducer back-action evading measurement scheme designed to perform a phase-sensitive coupling to the oscillator, with the concomitant circumvention of the standard amplifier limit via the "squeezing" of amplifier back reaction noise. The applicability of squeezing in the detection of gravitational radiation is explored via the dependence of detection sensitivity on the physical temperature and quality factor of the resonant-bar gravitational radiation antenna, on the noise temperature of the amplifier, and on the squeezing factor of the back-action evading measurement; the success of back-action evasion in improving upon the optimum amplitude-and-phase detection sensitivities depends critically upon these parameters. Using the LSU superconducting dual-cavity accelerometer as a test platform, we present direct evidence for the establishment of a phase-sensitive coupling to an oscillator, along with a variety of indirect corroborating evidence. This data indicates that our phase-sensitive measurement scheme is indeed more sensitive to one component of the oscillator than to the other component. We also present the first evidence for the existence of back-action evasion of amplifier back reaction noise. We show theoretical expectations and experimental results for the dependence of squeezing on: input signal phases, amplitudes, and frequencies; amplifier back reaction levels; background noise; mechanical oscillator frequency; and coherent carrier contribution at the cavity resonance frequency. Squeezing factors of up to fifteen were achieved by our back-action evading measurement scheme.
We discuss limitations in precision measurements of a weak classical force coupled to quantum mechanical systems, the so-called standard quantum limit (SQL). Among the several contexts exploiting the measurement of classical signals, gravitational wave (GW) detection is of paramount importance. In this framework, we analyze the quantum limited sensitivity of a free test mass, a quantum mechanical harmonic oscillator, two harmonic oscillators with equal masses and different resonance frequencies, and finally two mechanical oscillators with different masses and resonating at the same frequency. The sensitivity analysis of the latter two cases illustrates the potentialities of back-action reduction and classical impedance matching schemes, respectively. By examining coupled quantum oscillators as detectors of classical signals, we found a viable path to approach the SQL for planned or operating GW detectors, such as DUAL and AURIGA.
Aspects of quantum limits for the detection of a classical force The development of the theory of QND measurements was triggered by the problem of detecting a small classical force that acts on a quantum probe―e.g. on a free mass or an oscillator, see the review article and the monograph. This problem arose most especially in gravitational-wave experiments. Astrophysical estimates indicate that the gravitational waves from cosmic sources should be so strong that they behave almost perfectly classically. During the measuring time the number of gravitons that pass within one wavelength of the measuring device is enormously larger than unity; and correspondingly, as would be the case if the waves were electromagnetic, quantum effects in the radiation are utterly negligible. A central feature of gravitational waves is their extremely weak interaction with matter. Because of this weakness, the response of the measuring device's probe object to the waves is so small that it may be comparable to the object's quantum mechanical uncertainties. At the same time, the weakness of the interaction guarantees that there is almost no back action of the probe on the gravitational-wave field. As a result, the gravitational wave acts on the probe as through it were a precisely classical force (i.e. a force that is independent of the probe's quantum state).