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Physical Review Letters

Classical amplitude squeezing for precision measurements

Frank Difilippo, Vasant Natarajan A, Kevin Robert Boyce, David E. Pritchard

Published May 11, 1992• 38 citations
Publisher

Abstract

In analogy to squeezing of light, noise in a classical oscillator can be squeezed to reduce amplitude uncertainty. While this can be achieved to some extent in a harmonic oscillator parametrically driven at 2${\mathrm{\ensuremath{\omega}}}_{0}$, true amplitude squeezing is possible in anharmonic oscillators, either by driving at 2${\mathrm{\ensuremath{\omega}}}_{0}$ or allowing amplitude-dependent dephasing. These techniques can reduce the uncertainty in measurements of the frequency of an oscillator; for example, the thermal uncertainty in the relativistic frequency shift in single ion mass spectroscopy can be reduced by more than a factor of 5.

DOI
10.1103/physrevlett.68.2859

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In analogy to squeezing of light, noise in a classical oscillator can be squeezed to reduce amplitude uncertainty. While this can be achieved to some extent in a harmonic oscillator parametrically driven at 2\ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​, true amplitude squeezing is possible in anharmonic oscillators, either by driving at 2\ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​ or allowing amplitude-dependent dephasing. These techniques can reduce the uncertainty in measurements of the frequency of an oscillator; for example, the thermal uncertainty in the relativistic frequency shift in single ion mass spectroscopy can be reduced by more than a factor of 5.
We have demonstrated that, analogous to the use of quantum mechanical squeezing to beat the quantum shot noise limit, classical squeezing can be used to reduce the effect of thermal amplitude noise in precision measurements. The classical oscillator under study is a single ion in a Penning trap whose motion is cooled by coupling to a thermal bath. The thermal noise distribution is quadrature squeezed by parametric amplification. We observe 6 dB of noise reduction below the equilibrium value when the squeezing axis is 90\ifmmode^\circ\else\textdegree\fi{} out of phase with the coherent signal drive.
Some of the most enduring questions in physics--including the quantum measurement problem and the quantization of gravity--involve the interaction of a quantum system with a classical environment. Two linearly coupled harmonic oscillators provide a simple, exactly soluble model for exploring such interaction. Even the ground state of a pair of identical oscillators exhibits effects on the quantum nature of one oscillator, e.g., a diminution of position uncertainty, and an increase in momentum uncertainty and uncertainty product, from their unperturbed values. Interaction between quantum and classical oscillators is simulated by constructing a quantum state with one oscillator initially in its ground state, the other in a coherent or Glauber state. The subsequent wave function for this state is calculated exactly, both for identical and distinct oscillators. The reduced probability distribution for the quantum oscillator, and its position and momentum expectation values and uncertainties, are obtained from this wave function. The oscillator acquires an oscillation amplitude corresponding to a beating between the normal modes of the system; the behavior of the position and momentum uncertainties can become quite complicated. For oscillators with equal unperturbed frequencies, i.e., at resonance, the uncertainties exhibit a time-dependent quantum squeezing which can be extreme.
While it is true that quantum mechanics imposes strict limits on the precision of measurements, we can sometimes improve measurement precision without violating the uncertainty principle. The rub is that this improvement may be formidable to achieve in practice—and always comes at the expense of a loss in the precision of a complementary measurement. Last October a team at AT&T Bell Labs succeeded in reducing the noise observed from an optical cavity below the “shot noise”—the level of fluctuations associated with variations in the vacuum fields, which is the ultimate limit to accuracy that would be predicted by semiclassical models. Although the Bell Labs experiment reduced the noise below this level by only 7 to 10%, it bolstered hopes that continued progress might eventually yield reductions in noise by factors of ten. Manipulation of the quantum noise limitations might be absolutely critical to such precision measurements as detection of gravity waves, whose faint signals are likely to be near the measurement limit of laser interferometers or mechanical oscillators. The phenomenon might possibly find applications in optical communication or optical memory systems.
The complex amplitude ${X}_{1}+i{X}_{2}\ensuremath{\equiv}(x+\frac{\mathrm{ip}}{m\ensuremath{\omega}}){e}^{i\ensuremath{\omega}t}$ of a harmonic oscillator is constant in the absence of driving forces. Although the uncertainty principle forbids precise measurements of X1{X}_{1}X1​ and X2{X}_{2}X2​ simultaneously ($\ensuremath{\Delta}{X}_{1}\ensuremath{\Delta}{X}_{2}>~\frac{\ensuremath{\hbar}}{2m\ensuremath{\omega}}$), X1{X}_{1}X1​ alone can be measured precisely and continuously ("quantum nondemolition measurement"). Examples are given of measuring systems that do this job. Such systems might play a crucial role in gravitational-wave detection and elsewhere.
We showed in Section 5.2 of the present chapter that according to quantum theory the value of the electric field of an electromagnetic wave could not be predicted to arbitrarily high precision, this being a consequence of the uncertainty relation satisfied by the two quadrature components of the field (Equation 5.32), which imposed a finite limit on the product of their variances. This limit is more than an abstract theoretical limit; it is often the overriding factor determining the resolution of high-precision optical measurements. In such situations measurements by photodetectors on the field exhibit uncontrollable fluctuations of quantum origin known as quantum noise . Fortunately, as we shall show in this complement, it is possible to overcome this by using non-classical states of the radiation field, namely the squeezed states , provided measurements are made of a single one of a pair of conjugate field variables. A field mode prepared in such a state exhibits reduced quantum noise in one of the variables at the expense of increased noise in the other, so that the uncertainty relation involving their variances is still satisfied. The potential of such states for increasing the obtainable precision of optical measurements is self-evident. However, the practical realization of the potential benefits requires delicate experimental techniques that we describe only briefly here.
A squeezed state of the electromagnetic field has less noise in one of its two amplitude components than that of a coherent state. This property makes squeezed states potentially useful in low noise and precision measurement. Squeezed states may be generated by, among other things, degenerate parametric amplifiers and four-wave mixers. The reduced noise in one of the amplitude components is most easily observed via a homodyne detector. A number of experimental efforts to generate and detect squeezed states are under way. Here, in describing experimental techniques, the experiments' carried out by Slusher et al. are emphasized. A number of techniques were employed to successfully generate squeezed light. An atomic beam of sodium provided the optical nonlinearity in a cavity-backward four-wave mixer. By employing pairs of Fabry-Perot modes spaced symmetrically about the pump frequency, squeezed light was generated in nondegenerate conditions where Rayleigh scattering of pump light could be avoided. To observe squeezing, a number of nonidealities such as phase jitter and detector inefficiencies also had to be overcome.