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Physical Review Letters

Classical Squeezing of an Oscillator for Subthermal Noise Operation

Vasant Natarajan A, Frank Difilippo, David E. Pritchard

Published Apr 10, 1995• 46 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.74.2855

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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.
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.
Nonlinear forces allow motion of a mechanical oscillator to be squeezed below the zero-point motion. Of existing methods, mechanical parametric amplification is relatively accessible, but previously thought to be limited to 3 dB of squeezing in the steady state. We consider the effect of applying continuous weak measurement and feedback to this system. If the parametric drive is optimally detuned from resonance, correlations between the quadratures of motion allow unlimited steady-state squeezing. Compared to backaction evasion, we demonstrate that the measurement strength, temperature and efficiency requirements for quantum squeezing are significantly relaxed.
Squeezing the quadrature noise of a harmonic oscillator enhances its sensitivity. Stabilization of the antisqueezed quadrature can overcome the 3 dB squeezing limit by parametric modulation. Here, we apply this method to soft-clamped membrane resonators, which hold promise for both classical and quantum sensing applications. We compare piezoelectric and capacitive parametric modulation and observe thermomechanical squeezing by up to 17 and 21 dB, respectively. Finally, we provide a full quantum theory of combining this approach with quantum-limited motion measurement. We conclude that quantum squeezing is feasible at moderate cryogenic temperatures and realistic device parameters.
Improvements in precision beyond the limit set by the vacuum state or zero-point fluctuations of the field are reported for two optical experiments. Both experiments employ the squeezed light from an optical parametric oscillator to reduce the level of fluctuations in one of two quadrature phase amplitudes below the shot-noise level (SNL). The first experiment follows the work of Caves1 on precision interferometry with squeezed states. An increase in signal-to-noise of 2.5 dB relative to the SNL is demonstrated for the detection of phase modulation in a Mach-Zehnder interferometer. In the second experiment amplitude modulation is encoded on a weak signal beam with an acoustooptic modulator. For the case of a squeezed signal beam compared with a beam in a coherent state, an improvement in sensitivity of 2.5 dB is achieved for the detection of amplitude changes of the field. The observed increases in sensitivity in both experiments are currently limited by simple linear losses in propagation and detection and not by the degree of available squeezing.
Squeezed light in the form of a train of pulses, each approximately 100 ps in width, has been generated by parametric down-conversion in KTiOPO4{\mathrm{O}}_{4}O4​ crystals. The measured noise reduction in the quiet quadrature of the field is 0.6 dB below the shot-noise limit. The quadrature noise components are measured with a balanced homodyne detector using a pulse train for the local oscillator. The noise reduction extends over a broad spectrum and has important implications for precision measurement at very short time scales.
We present a single-quadrature feedback scheme able to overcome the conventional 3 dB limit on parametric squeezing. The method is experimentally demonstrated in a micromechanical system based on a cantilever with a magnetic tip. The cantilever is detected at low temperature by a SQUID susceptometer, while parametric pumping is obtained by modulating the magnetic field gradient at twice the cantilever frequency. A maximum squeezing of 11.5 dB and 11.3 dB is observed, respectively, in the response to a sinusoidal test signal and in the thermomechanical noise. So far, the maximum squeezing factor is limited only by the maximum achievable parametric modulation. The proposed technique might be used to squeeze one quadrature of a mechanical resonator below the quantum noise level, even without the need for a quantum limited detector.