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Physical Review Letters

Comment on “Evidence for Quantized Displacement in Macroscopic Nanomechanical Oscillators”

Keith Schwab, M. P. Blencowe, Michael Lee Roukes, A.N. Cleland, S. M. Girvin, Gerard J Milburn, K. L. Ekinci

Published Dec 5, 2005• 19 citations
Publisher

Abstract

In a recent Letter, Gaidarzhy et al. [1] claim to have observed evidence for "quantized displacements" of a high-order mode of a nanomechanical oscillator. We contend that the methods employed by the authors are unsuitable in principle to observe such states for any harmonic mode.
DOI
10.1103/physrevlett.95.248901

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Recent technological advances have enabled the creation of nanomechanical structures with resonance frequencies exceeding 2 GHz. The oscillators are nanomachined single‐crystal structures of silicon, designed to provide two distinct sets of coupled elements with very low and very high frequencies. At higher temperatures — in the so‐called classical regime, the response to an applied driving force is continuous. At millikelvin temperatures these nanomechanical oscillators enter the quantum regime of mechanical motion with a manifestly discrete response to the applied drive. The observed discrete response most likely results from the onset of the quantum regime in these macroscopic nanomechanical oscillators. These new experiments offer exciting prospects for the studies of quantum measurement and quantum computation, involving macroscopic quantum mechanical oscillators.
This is a response to a recently reported comment [F. M. Fernández, J. Math. Phys. 60, 094101 (2019)] on paper [M. Serhan et al., J. Math. Phys. 59, 082105 (2018)] regarding the quantization of the damped harmonic oscillator using a non-Hermitian Hamiltonian with real energy eigenvalues. We assert here that the calculation of Eq. (29) of Serhan et al. [J. Math. Phys. 59, 082105 (2018)] is incorrect, and thus the subsequent steps via the Nikiforov-Uvarov method are affected, and the energy eigenvalues should have been complex. However, we show here that the Hermiticity of the Hamiltonian should be first achieved to make the correct transition from classical Hamiltonian to quantum counterpart, and this can be reached using the symmetrization rule. The canonical quantization on the resulted Hermitian Hamiltonian yields the real energy eigenvalues as exactly given by Eq. (35) of Serhan et al. [J. Math. Phys. 59, 082105 (2018)].