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Research in Optical Sciences

Optimal Mass-sensing with a Nano-mechanical Resonator

Daniel Braun

Published Jan 1, 2012• 0 citations
Publisher

Abstract

I report the quantum Cramér-Rao bound on the sensitivity of mass-sensing with a nano-mechanical resonator as function of its quantum state. I then identify the quantum states which allow the largest sensitivity for a given maximum energy. I show that even with a classical state of existing carbon nanotube resonators it should be possible in principle to measure one thousandth of the mass of an electron. Further improvements might enable reaching a regime where one can measure the relativistic change of mass due to absorption of a single photon, or the creation of a chemical bond.

DOI
10.1364/qim.2012.qm3a.4

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I report the quantum Cramér-Rao bound on the sensitivity of mass-sensing with a nano-mechanical resonator as function of its quantum state. I then identify the quantum states which allow the largest sensitivity for a given maximum energy. I show that even with a classical state of existing carbon nanotube resonators it should be possible in principle to measure one thousandth of the mass of an electron. Further improvements might enable reaching a regime where one can measure the relativistic change of mass due to absorption of a single photon, or the creation of a chemical bond.
I establish the fundamental lower bound on the mass that can be measured with a nano-mechanical resonator in a given quantum state based on the fundamental quantum Cramér-Rao bound, and identify the quantum states which will allow the largest sensitivity for a given maximum energy. I show that with existing carbon nanotube resonators it should be possible in principle to measure a thousandth of the mass of an electron, and future improvements might allow to reach a regime where one can measure the relativistic change of mass due to absorption of a single photon, or the creation of a chemical bond.
We determine the quantum Cram\'er-Rao bound for the precision with which the oscillator frequency and damping constant of a damped quantum harmonic oscillator in an arbitrary Gaussian state can be estimated. This goes beyond standard quantum parameter estimation of a single-mode Gaussian state for which typically a mode of fixed frequency is assumed. We present a scheme through which the frequency estimation can nevertheless be based on the known results for single-mode quantum parameter estimation with Gaussian states. Based on these results, we investigate the optimal measurement time. For measuring the oscillator frequency, our results unify previously known partial results and constitute an explicit solution for a general single-mode Gaussian state. Furthermore, we show that with existing carbon nanotube resonators see J. Chaste et al. [Nat. Nanotechnol. 7, 301 (2012)] it should be possible to achieve a mass sensitivity of the order of an electron mass ${\mathrm{Hz}}^{\ensuremath{-}1/2}$.
Due to their small mass and large mechanical stiffness, free-standing carbon nanotubes turn out to be superb high-frequency mechanical resonators that could find applications in mass and force sensing. With resonance frequencies up to the gigahertz regime, one can envision cooling the mechanical motion of such nanotubes to its quantum mechanical ground state at cryogenic temperatures. This opens the possibility to generate highly nonclassical states of the mechanical motion, similar to those created several years ago with electromagnetic radiation in superconducting microwave resonators. Such states are interesting from a fundamental physics point of view, but potentially also for quantum information and sensing applications. While classical activation of a harmonic oscillator will always produce ``quasiclassical'' coherent states, a necessary ingredient for the production of nonclassical states such as Fock states is a nonlinear element, e.g., superconducting qubits in the case of microwaves. This work predicts that for mechanical oscillations, the spin-orbit coupling of the mechanical motion to a single localized electron spin can do the trick by acting as a nonlinear element for phonons.
The quantum Cramér-Rao bound (QCRB) sets a fundamental limit for the measurement of classical signals with detectors operating in the quantum regime. Using linear-response theory and the Heisenberg uncertainty relation, we derive a general condition for achieving such a fundamental limit. When applied to classical displacement measurements with a test mass, this condition leads to an explicit connection between the QCRB and the standard quantum limit that arises from a tradeoff between the measurement imprecision and quantum backaction; the QCRB can be viewed as an outcome of a quantum nondemolition measurement with the backaction evaded. Additionally, we show that the test mass is more a resource for improving measurement sensitivity than a victim of the quantum backaction, which suggests a new approach to enhancing the sensitivity of a broad class of sensors. We illustrate these points with laser interferometric gravitational-wave detectors.
The quantum Cramér-Rao bound (QCRB) stands as a cornerstone of quantum metrology. Yet, akin to its classical counterpart, it provides only local information and overlooks higher-order details. We leverage the theory of higher-order asymptotics to circumvent these issues, providing corrections to the performance of estimators beyond the QCRB. While the QCRB often yields a whole family of optimal states and several optimal measurements, our approach allows us to identify optimal states and measurements within the family that are otherwise equivalent according to the QCRB alone; these are requisite for optimal metrology before reaching the asymptotic limit. These results are especially pertinent when dealing with unitary processes.