We have observed nonlinear transduction of the thermomechanical motion of a nanomechanical resonator when detected as laser transmission through a sideband unresolved optomechanical cavity. Nonlinear detection mechanisms are of considerable interest as special cases allow for quantum nondemolition measurements of the mechanical resonator's energy. We investigate the origin of the nonlinearity in the optomechanical detection apparatus and derive a theoretical framework for the nonlinear signal transduction, and the optical spring effect, from both nonlinearities in the optical transfer function and second order optomechanical coupling. By measuring the dependence of the linear and nonlinear signal transduction -as well as the mechanical frequency shift -on laser detuning from optical resonance, we provide estimates of the contributions from the linear and quadratic optomechanical couplings.Cavity optomechanics has resulted in new levels of extremely precise displacement transduction [1, 2] of ultrahigh frequency resonators [3]. This has created much interest in pursuing quantum measurements [4] of nanomechancial devices [5][6][7], as well as dynamical back action cooling [8][9][10][11].One of the most fundamental, and as of yet unattained, quantum measurements that could be performed is that of the quantized energy eigenstates of a nanomechanical resonator (as has been demonstrated with an electron in a cyclotron orbit [12]). To achieve this, one cannot measure the displacement of the resonator [13], but instead must measure the energy directly -preferably without destroying the quantum state, a so-called quantum non-demolition (QND) measurement. Whereas the accuracy in continuously measuring two conjugate quantities is limited by the Heisenberg uncertainty principle to the standard quantum limit (SQL) [13], QND measurements allow for continuous measurements of an observable to be taken to arbitrary precision [14][15][16][17]. Here our interest lies in a QND measurement of the energy, and thereby the number of phonons [18]. In an optomechanical system, this is expected to be possible by having strong second order optomechanical coupling [19][20][21][22]. This has been demonstrated in membrane-in-the-middle Fabry-Pérot cavities [23,24], however it has been pointed out there remains first order coupling between the two optical modes, possibly obscuring QND measurements [25].Signal from second order optomechanical coupling, hence measurement of x 2 , will display mechanical peaks at twice the fundamental frequency. However, we would also expect that nonlinear transduction of the displacement, x, of a mechanical resonator from a nonlinear optical transfer function would also appear at harmonics of the mechanical resonance frequency, as has been observed [26][27][28].In this Letter we report observation of peaks in the mechanical power spectra at exactly twice the fundamental mechanical frequency, as shown in Fig. 1. We derive a model for the origin of the harmonic signal, as well as the optical spring effect, from bo...
The structure of unit weighing matrices of order n and weights 2, 3 and 4 are studied. We show that the number of inequivalent unit weighing matrices UW (n, 4) depends on the number of decomposition of n into sums of non-negative multiples of some specific positive integers. Two interesting sporadic cases are presented in order to demonstrate the complexities involved in the classification of weights larger than 4.
Inspired by the many applications of mutually unbiased Hadamard matrices, we study mutually unbiased weighing matrices. These matrices are studied for small orders and weights in both the real and complex setting. Our results make use of and examine the sharpness of a very important existing upper bound for the number of mutually unbiased weighing matrices.We will start off with a very well-known result (see [2]). Lemma 1. Let H and K be real unbiased weighing matrices with order n and weight w, then w must be a perfect square.Proof. Since both H and K are integer matrices, HK T = √ wL must be an integer matrix as well.Lemma 2. Let W = {W 1 , · · · ,W k } be a set of mutually unbiased weighing matrices of order m with weight w and X = {X 1 , · · · , X l } be a set of mutually unbiased weighing matrices of order n with weight w. Then there exist p = min(k, l) mutually unbiased weighing matrices of order m + n and weight w.Proof. The set W 1 ⊕ X 1 ,W 2 ⊕ X 2 , · · · ,W p ⊕ X p gives the desired result, where ⊕ denotes the standard direct sum of matrices (i.e., A ⊕ B = diag(A, B)).
Mechanical modes are a potentially useful resource for quantum information applications, such as quantum-level wavelength transducers, due to their ability to interact with electromagnetic radiation across the spectrum. A significant challenge for wavelength transducers is thermomechanical noise in the mechanical mode, which pollutes the transduced signal with thermal states. In this paper, we eliminate thermomechanical noise in the GHz-frequency mechanical breathing mode of a piezoelectric optomechanical crystal using cryogenic cooling in a dilution refrigerator. We optically measure an average thermal occupancy of the mechanical mode of only 0.7 ± 0.4 phonons, providing a path towards low-noise microwave-to-optical conversion in the quantum regime.Quantum information science may have begun with atoms and optical photons, but it has expanded to include numerous experimental platforms that have demonstrated quantum behavior. Examples of quantum devices now include a wide array of well-controlled natural systems including neutral atoms [1], ions [2, 3], electronic [4,5] and nuclear spins [6,7], as well as fabricated systems such as quantum dots [8], superconducting circuits [9][10][11], and mechanical devices [12][13][14][15]. Each system has a unique set of properties that make them advantageous for specific applications: for example, long coherence times make atomic systems a natural candidate for quantum memories [16][17][18][19], while the flexible nature of fabrication makes superconducting circuitry ideal for creating quantum processing gates [20]. This has culminated in the vision of hybrid quantum systems that can link multiple sub-systems into complex quantum machines [21,22].One major challenge in creating hybrid quantum systems arises from transferring quantum information between the different sub-systems. Photons are the obvious medium for transferring of information as most quantum systems can interact with light, however the relevant wavelength varies widely. Furthermore, quantum information has been effectively transferred over long distances using optical photons [23][24][25]. This has spurred interest in the development of mechanical wavelength transducers -devices that use photon-phonon interactions to coherently convert photons between different wavelengths while preserving quantum information. In particular, interest lies in converting between the microwave frequencies used in superconducting quantum processors and telecom-wavelength optical photons used in fiber networks for the purposes of networking quantum processors [21]. At its core, a mechanical wavelength transducer consists of a mechanical element coupled to two electromagnetic resonances at the desired input/output wavelengths. State-of-the-art transducers have demonstrated wavelength conversion for classical signals within the optical wavelength range [26][27][28], within the microwave regime [29,30], the up-conversion of microwave tones to optical wavelengths [31][32][33], and the bidirectional conversion of microwave and optical si...
On-chip cavity optomechanics, in which strong co-localization of light and mechanical motion is engineered, relies on efficient coupling of light both into and out of the on-chip optical resonator. Here we detail our particular style of tapered and dimpled optical fibers, pioneered by the Painter group at Caltech, which are a versatile and reliable solution to efficient on-chip coupling. First, a brief overview of tapered, single mode fibers is presented, in which the single mode cutoff diameter is highlighted. The apparatus used to create a dimpled tapered fiber is then described, followed by a comprehensive account of the procedure by which a dimpled tapered fiber is produced and mounted in our system. The custom-built optical access vacuum chambers in which our on-chip optomechanical measurements are performed are then discussed.Finally, the process by which our optomechanical devices are fabricated and the method by which we explore their optical and mechanical properties is explained. It is our expectation that this manuscript will enable the novice to develop advanced optomechanical experiments.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.