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arXiv:1507.04209 · Physical Review Letters

Squeezing of Quantum Noise of Motion in a Micromechanical Resonator

J.-M. Pirkkalainen, Erno Damskägg, M. Brandt, Francesco Massel, Mika A. Sillanpää

Published Dec 7, 2015• 400 citations
arXivHTMLPDF

Abstract

A pair of conjugate observables, such as the quadrature amplitudes of harmonic motion, have fundamental fluctuations that are bound by the Heisenberg uncertainty relation. However, in a squeezed quantum state, fluctuations of a quantity can be reduced below the standard quantum limit, at the cost of increased fluctuations of the conjugate variable. Here we prepare a nearly macroscopic moving body, realized as a micromechanical resonator, in a squeezed quantum state. We obtain squeezing of one quadrature amplitude 1.1±0.4 dB below the standard quantum limit, thus achieving a long-standing goal of obtaining motional squeezing in a macroscopic object.

DOI
10.1103/physrevlett.115.243601

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A pair of conjugate observables, such as the quadrature amplitudes of harmonic motion, have fundamental fluctuations that are bound by the Heisenberg uncertainty relation. However, in a squeezed quantum state, fluctuations of a quantity can be reduced below the standard quantum limit, at the cost of increased fluctuations of the conjugate variable. Here we prepare a nearly macroscopic moving body, realized as a micromechanical resonator, in a squeezed quantum state. We obtain squeezing of one quadrature amplitude 1.1±0.4 dB below the standard quantum limit, thus achieving a long-standing goal of obtaining motional squeezing in a macroscopic object.
A pair of conjugate observables, such as the quadrature amplitudes of harmonic motion, have fundamental fluctuations that are bound by the Heisenberg uncertainty relation.However, in a squeezed quantum state, fluctuations of a quantity can be reduced below the standard quantum limit, at the cost of increased fluctuations of the conjugate variable.Here we prepare a nearly macroscopic moving body, realized as a micromechanical resonator, in a squeezed quantum state.We obtain squeezing of one quadrature amplitude 1.1 Æ 0.4 dB below the standard quantum limit, thus achieving a long-standing goal of obtaining motional squeezing in a macroscopic object.
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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.
We present a comprehensive quantum framework to analyze the dynamics of a nanomechanical cantilever subjected to combined additive and parametric noise. Using the Wigner phase-space representation, we derive and numerically solve c-number stochastic differential equations that accurately capture the evolution of the quantum state under realistic noise conditions. This operator-to-c-number mapping makes the role of quadrature noise sources explicit and provides a transparent route to evaluating squeezing. Our simulations reveal distinct dynamical regimes, demonstrating a critical trade-off determined by the parametric drive strength g0. In the stable regime, parametric modulation suppresses quantum fluctuations, achieving squeezing of about −1.7 dB relative to the vacuum noise limit. For stronger parametric driving, transient squeezing levels approaching −5 dB are observed, but these gains are accompanied by rapid growth of the conjugate quadrature and eventual instability. We further connect these dynamics to effective quality factor enhancement, clarifying the implications for nano-cantilever sensor performance. An explicit check of the Heisenberg uncertainty relation confirms that our simulations remain physically consistent. Together, these results establish a unified framework that links fundamental quantum noise dynamics with practical benchmarks for quantum-enhanced sensing.
Microscopic systems exhibit many intriguing quantum mechanical phenomena which cannot be explained by classical physics. Quantum mechanics has succesfully explained the behaviour of subatomic particles, atoms, and electromagnetic waves where entanglement, wave-particle duality and superposition of particles has been observed. In macroscopic scales, quantum mechanics explains the microscopic origin of electronic properties of solids, and quantum coherence effects are routinely observed in superfluids and superconductors. However, most macroscopic objects behave according to the laws of classical physics due to decoherence effects caused by interaction with the environment. Mechanical systems are ideal for studying quantum behavior of macroscopic objects as they can be measured with high accuracy, and their interaction with the environment can be controlled. Cavity optomechanics studies the interaction between light and mechanical objects through radiation pressure force. Recent developments in the field have made it possible to study quantum behaviour in macroscopic mechanical objects. For example, mechanical oscillators coupled to either microwave or optical light have been cooled to their quantum ground state. This thesis studies a cavity optomechanical system in which micromechanical aluminium membrane resonator is coupled to a superconducting microwave circuit resonator. This work focuses on the phenomena relating to the Heisenberg's uncertainty principle which sets a fundamental limit to the precision in which certain pairs of properties of a system can be measured. Due to uncertainty principle, even if a harmonic oscillator is cooled to the ground state, it still has random motion called the zero-point motion. This sets a fundamental limit on how accurately one can measure the motion of the resonator. In a squeezed quantum state, fluctuations of a quantity can be reduced below this quantum limit, at the cost of increased fluctuations in the pair variable. In this work, motion of a nearly macroscopic object is measured in a squeezed state. Due to uncertainty principle, a measurement amplifying an oscillatory signal always adds noise. However, when information in one part of the signal is lost, one can amplify and measure the other part perfectly. This kind of nearly noiseless phase-sensitive measurement, and squeezing of microwave fields is demonstrated in this work. An optomechanical device that simultaneously performs high-gain amplification and frequency conversion is created. Collective dynamics of a multimode optomechanical system is studied. When measuring the oscillator position continuously, the system is perturbed by the measuring. This can be avoided in quantum back-action evading measurement, which is demonstrated for collective motion of two mechanical resonators. Finally, an entanglement of centre-of-mass motion of two nearly macroscopic objects is created and stabilised.