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 2o)Oy true amplitude squeezing is possible in anharmonic oscillators, either by driving at 2a>o 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. PACS numbers: 06.20.-f, 07.75.+h, 42.50.Dv, 46.10,+zIn recent years, understanding of squeezed light [1,2] has evolved to the point that detection below the shotnoise limit has been demonstrated [3,4], and several applications of these nonclassical states are being considered [5l. Although the emphasis has been in the quantum regime, where the source of noise is the uncertainty principle, there is a classical correspondence [6] which suggests that noise of a thermal or technical origin can be squeezed to minimize its unwanted effects on a particular measurement. Such a reduction of thermal noise in a quadrature component has been observed in a high-0 classical oscillator by parametric excitation and has applications to atomic force microscopy and gravity wave detection [7]. A similar reduction in amplitude uncertainty would be useful for determining the frequency of an anharmonic oscillator in the presence of noise. Since the frequency is amplitude dependent, fluctuations in amplitude will result in fluctuations in the measured frequency. This paper describes three schemes for amplitude squeezing in a classical anharmonic oscillator: by driving parametrically at leoo in the anharmonic and harmonic regimes, and by dephasing in an undriven oscillator. As an illustration, this concept is applied to highprecision mass spectroscopy of a single trapped ion.We begin by considering the motion of a classical anharmonic oscillator parametrically driven at twice the resonant frequency. A simple treatment is presented, with emphasis on the phase diagram, of an undamped resonant oscillator to lowest order in the parametric drive strength and the anharmonicity. (Higher-order expansions, detuning, and damping [8,9] can be neglected for the mass spectroscopy example.) Afterwards, the special cases of no anharmonicity and no parametric drive are considered.The potential for a one-dimensional oscillator with a small (az 2 <£ 1) quartic anharmonic correction whose frequency is modulated at 2G>O by a weak (s<£ 1) parametric drive is U(z,t) -jmo)oz 2 (\ + esmlmot + jaz 2 ).(1)To lowest order in a and e, higher harmonics can be neglected, and one expects oscillation only at cao'>
z(t) =r(/)cos(ft>o* -Bit))= C(f kosttJof + S^sintyo* • ^ In this approximation, C(t) and S(t) are slowly varying (i.e., dC/dt,dS/dt<^r(o Q ). Thus d 2 C/dt 2 and d 2 S/dt 2 can be neglected in the equation of motion, yielding the autonomous system of equations ^f=K(C+yr 2 S), ^-= -...
