Mechanical Thermal Noise Research Articles

Noise Limits in Thin-Film Magnetoelectric Sensors With Magnetic Frequency Conversion

To enable the measurement of low-frequency magnetic signals with cantilever type thin-film magnetoelectric sensors, magnetic frequency conversion transfers the frequency of the desired signal into the mechanical resonance of the cantilever. The system electronics for the realization of this approach and the approach itself introduce additional noise sources as compared with direct detection, which lowers the limit of detection. In this paper, the magnetic frequency conversion noise sources are reviewed, discussed, and evaluated for our setup. The model for the nonlinear transfer process is implemented in the time domain. This enables the consideration of the pump noise in a noise equivalent circuit. For the sensor type under investigation, the dominant noise near its optimal working point originates from the pump source. If the noise of the pump can be decreased and magnetic excess noise is not dominant, the noise limit is the thermal-mechanical noise of the sensor. The implementation of a filter after the excitation source decreases the limit of detection to 60 pT/ $\sqrt {\text {Hz}}$ at 10 Hz.

A micromachined microphone based on the electret membrane and field-effect-transistor mechano-electrical transduction

Most commercially available MEMS (Micro-Electro-Mechanical-System) microphones use a capacitive method, and their structure and performance are saturated to some extent. However, due to the limitation of the capacitive transduction method, the roll-off at the low frequency is inevitable, and there is a limit in reducing the mechanical thermal noise caused by the squeeze film damping that occurs between membrane and backplate structure. Proposed electret membrane and FET (Field Effect Transistor) based MEMS microphone detects a change in the source-drain current according to the gate voltage change of the FET induced by vibrating the membrane with the fixed charges. In the case of a MEMS microphone using the proposed FET, low frequency roll-off according to the energy conversion does not occur, and the size of the backplate can be drastically reduced as compared with the conventional MEMS microphone, thereby further reducing the mechanical thermal noise, leading to the possibility of achieving the higher SNR (Signal to Noise Ratio) than 65 dBA. Conventional metal-oxide-semiconductor fabrication process and micromachining process were used. In this study, design, fabrication and performance test of the proposed FET based MEMS microphone are conducted. [Work supported by CMTC, UM15304RD3.]

Quantum-enhanced accelerometry with a nonlinear electromechanical circuit

It is known that placing a mechanical oscillator in a superposition of coherent states allows, in theory, a measurement of a linear force whose sensitivity increases with the amplitude of the mechanical oscillations, a uniquely quantum effect. Further, entangled versions of these states across a network of $n$ mechanical oscillators enables a measurement whose sensitivity increases linearly with $n$, thus improving over the classical scaling by $\sqrt{n}$. One of the key challenges in exploiting this effect is processing the signal so that it can be readily measured; linear processing is insufficient. Here we show that a Kerr oscillator will not only create the necessary states, but also perform the required processing, transforming the quantum phase imprinted by the force signal into a shift in amplitude measurable with homodyne detection. This allows us to design a relatively simple quantum electro-mechanical circuit that can demonstrate the core quantum effect at the heart of this scheme, amplitude-dependent force sensitivity. We derive analytic expressions for the performance of the circuit, including thermal mechanical noise and photon loss. We discuss the experimental challenges in implementing the scheme with near-term technology.

Generating EPR-type entanglement of degenerate optomechanical parametric oscillators

Einstein–Podolski–Rosen (EPR) entanglement states are achievable by combining two single-mode position and momentum squeezed states at a 50:50 beam splitter (BS). To generate the EPR mechanical entanglement, we consider the system consisted of two parametric optomechanical resonators, where two mechanical oscillators are linearly coupled. The linear coupling forms the symmetric and antisymmetric combinations of two mechanical modes, parallel to a 50:50 BS mixing. In the weak optomechanical coupling regime and via applying the opposite phases of parametric interactions, the symmetric and antisymmetric mechanical modes can be position and momentum squeezed, respectively. Therefore, two original mechanical modes are EPR entangled. Moreover, the mechanical thermal noise can decrease the entanglement. But with the parametric interaction enhanced optomechanical cooling, the influence of thermal noise on entanglement can be significantly suppressed, and the mechanical entanglement can be generated under a relatively high temperature. We also discuss the critical thermal occupation where the entanglement disappears, which is proportional to the optomechanical cooperativity parameter.

Thermal-Mechanical Noise in Resonant Thin-Film Magnetoelectric Sensors

Thin-film magnetoelectric sensors, i.e., composites of magnetostrictive and piezoelectric materials, are able to measure very low magnetic flux densities in the picotesla range. In order to further improve the limit of detection it is of high importance to understand and quantify the relevant noise sources. In this paper, a common model for the deflection noise in vibrational structures is applied to the cantilever structure of resonant magnetoelectric sensors. By means of deflection and noise measurements the existence of thermal-mechanical noise even in sensor structures with a size in the centimeter range is proven. Based on these findings a noise equivalent circuit is suggested which allows not only the distinction between the impact of different sensor-intrinsic noise sources and also the involvement of the preamplifier noise. We found that the thermal-mechanical noise is the dominant noise source if direct signal detection is performed at the first bending resonance frequency of the sensor. However, this kind of noise is not the limiting influence when applying magnetic frequency-conversion techniques.

Generating spin squeezing states and Greenberger-Horne-Zeilinger entanglement using a hybrid phonon-spin ensemble in diamond

Quantum squeezing and entanglement of spins can be used to improve the sensitivity in quantum metrology. Here we propose a scheme to create collective coupling of an ensemble of spins to mechanical vibrational mode actuated by an external magnetic field. We find an evolution time where the mechanical motion decouples from the spins, and the accumulated geometric phase yields a squeezing of $5.9~\text{dB}$ for $20$ spins. We also show the creation of a Greenberger-Horne-Zeilinger spin state for $20$ spins with a fidelity of $\sim 0.62$ at cryogenic temperature. The numerical simulations show that the geometric-phase based scheme is mostly immune to thermal mechanical noise.

Single-photon nonlinearities in a strongly driven optomechanical system with quadratic coupling

Optical nonlinearities at the single-photon level are explored in a quadratically coupled optomechanical system, where the cavity frequency is coupled to the square of the mechanical displacement. The effective nonlinear interaction between photons and phonons is enhanced by a strong driving field, which allows one to implement the single-photon nonlinearities even if the single-photon coupling strength ${g}_{0}$ is much lower than the cavity decay rate $\ensuremath{\kappa}$. The photon statistical properties are discussed by calculating the second-order correlation function both analytically and numerically. The results show that the single-photon nonlinearities are robust against mechanical thermal noise in the strong-coupling and sideband-resolved regime, and photon blockade and photon-induced tunneling can be realized with experimentally accessible parameters.

A mechanical–thermal noise analysis of a nonlinear microgyroscope

The mechanical–thermal noise (MTN) equivalent rotation rate (Ωn) is computed by using the linear approximation of the system response and the nonlinear “slow” system. The slow system, which is obtained using the method of multiple scales, is used to identify the linear single-valued response of the system. The linear estimate of the noise equivalent rate fails as the drive direction stroke increases. It becomes imperative in these conditions to use a more complex nonlinear estimate of the noise equivalent rate developed here for the first time in literature. The proposed design achieves a high performance regarding noise equivalent rotation rate.

A Noise Reduction Method for Dual-Mass Micro-Electromechanical Gyroscopes Based on Sample Entropy Empirical Mode Decomposition and Time-Frequency Peak Filtering

The different noise components in a dual-mass micro-electromechanical system (MEMS) gyroscope structure is analyzed in this paper, including mechanical-thermal noise (MTN), electronic-thermal noise (ETN), flicker noise (FN) and Coriolis signal in-phase noise (IPN). The structure equivalent electronic model is established, and an improved white Gaussian noise reduction method for dual-mass MEMS gyroscopes is proposed which is based on sample entropy empirical mode decomposition (SEEMD) and time-frequency peak filtering (TFPF). There is a contradiction in TFPS, i.e., selecting a short window length may lead to good preservation of signal amplitude but bad random noise reduction, whereas selecting a long window length may lead to serious attenuation of the signal amplitude but effective random noise reduction. In order to achieve a good tradeoff between valid signal amplitude preservation and random noise reduction, SEEMD is adopted to improve TFPF. Firstly, the original signal is decomposed into intrinsic mode functions (IMFs) by EMD, and the SE of each IMF is calculated in order to classify the numerous IMFs into three different components; then short window TFPF is employed for low frequency component of IMFs, and long window TFPF is employed for high frequency component of IMFs, and the noise component of IMFs is wiped off directly; at last the final signal is obtained after reconstruction. Rotation experimental and temperature experimental are carried out to verify the proposed SEEMD-TFPF algorithm, the verification and comparison results show that the de-noising performance of SEEMD-TFPF is better than that achievable with the traditional wavelet, Kalman filter and fixed window length TFPF methods.

Observation of strong radiation pressure forces from squeezed light on a mechanical oscillator

Quantum enhanced sensing is a powerful technique in which nonclassical states are used to improve the sensitivity of a measurement. For enhanced mechanical displacement sensing, squeezed states of light have been shown to reduce the photon counting noise that limits the measurement noise floor. It has long been predicted, however, that suppressing the noise floor with squeezed light should produce an unavoidable increase in radiation pressure noise that drives the mechanical system. Such nonclassical radiation pressure forces have thus far been hidden by insufficient measurement strengths and residual thermal mechanical motion. Since the ultimate measurement sensitivity relies on the delicate balance between these two noise sources, the limits of the quantum enhancement have not been observed. Using a microwave cavity optomechanical system, we observe the nonclassical radiation pressure noise that necessarily accompanies any quantum enhancement of the measurement precision. By varying both the magnitude and phase of the squeezing, we optimize the fundamental trade-off between mechanical imprecision and backaction noise in accordance with the Heisenberg uncertainty principle. As the strength of the measurement is further increased, radiation pressure forces eventually dominate the mechanical motion. In this regime, the optomechanical interaction can be exploited as an efficient quantum nondemolition (QND) measurement of the amplitude fluctuations of the light field. By overwhelming mechanical thermal noise with radiation pressure by two orders of magnitude, we demonstrate a mechanically-mediated measurement of the squeezing with an effective homodyne efficiency of 94%. Thus, with strong radiation pressures forces, mechanical motion enhances the measurement of nonclassical light, just as nonclassical light enhances the measurement of the motion.

A Torsion MEMS Magnetic Sensor with Permanent Magnet and Fiber-optic Detection

A novel microelectromechanical system (MEMS) magnetic sensor based on fiber-optic detection is presented in this paper. The magnetic field is detected by measuring the tilting motion of a mechanical suspension with permanent magnet attached. When the magnet is magnetized in different directions, the magnetic sensor can detect the in-plane or out-of-plane magnetic field. The MEMS structure was designed analytically, fabricated by using a bulk micromachining process, and assembled with the permanent magnet manually. Finally, the magnetic-sensing capability of an as-prepared sensor was tested by assembling it into the fiber-optic detection system. The testing result shows a sensitivity of 2.86 mV/ $\mu \text{T}$ for the in-plane magnetic field and 6.57 mV/ $\mu \text{T}$ for the out-of-plane. According to the mechanical–thermal noise theory, estimated resolutions are, respectively, close to 142 and 62 nT for the in-plane and out-of-plane magnetic field, which suggest the adequacy for measurement in earth magnetic field range. The reported magnetic sensor shows the capability to fulfill the high-resolution detection of low-frequency signals, benefitted from the utilization of permanent magnet and fiber-optic detection. Moreover, the sensitive direction of reported magnetic sensor can be easily switched by varying its magnetized direction, which, therefore, paves the way to the integrated tri-axis magnetic sensor.

Hybrid Einstein-Podolsky-Rosen steering in an atom-optomechanical system

Einstein-Podolsky-Rosen steering manifests a type of quantum correlations intermediate between entanglement and Bell nonlocality. In this paper we propose a scheme for realizing hybrid atom-mechanical quantum steering in the steady-state regime. In our scheme, an optomechanical two-mode cavity is coupled to a distant ensemble of double-$\ensuremath{\lambda}$ atoms in a cascade way, which induces the dissipative interaction between the mechanics and the internal atomic states. We show that strong cavity dissipation can lead to two-way atom-mechanical steering and the optimal steering exhibited in an approximate two-mode squeezed-vacuum atom-mechanical state. Moreover, we find that for the present cascade coupling scheme, the one-way steering from the mechanical oscillator to the atoms is achievable under certain conditions, while the reverse one-way steering is impossible to obtain. The reason for achieving the asymmetric steering is analyzed and the effect of mechanical thermal noise on the steering is also studied.

Acoustical-Thermal Noise in a Capacitive MEMS Microphone

Since the size of silicon capacitive micro-electro-mechanical system (MEMS) microphones is getting smaller, the mechanical-thermal noise especially acoustical-thermal noise from the MEMS transducer becomes important taking a significant portion of the total noise. Acoustical-thermal noise caused by the flow resistance of microphone components, including the sound port on the package, perforation holes on the backplate, and the vent holes on the diaphragm, was investigated with lumped element models in this paper. The size of the ventilation holes on the backplate and the air-gap distance between the diaphragm and the backplate were designed to minimize the flow resistance. To estimate the acoustical-thermal noise sources, Johnson-Nyquist's relation was used for the flow resistance, and then the signal-to-noise ratio (SNR) was predicted only with acoustical-thermal noise. A capacitive MEMS microphone which has dual backplates to take advantage of the dual transductions has been compared with the single backplate microphone in terms of SNR. Although the advantage of dual transductions leads to SNR increases by $\sim 3$ dB, the prediction shows that the SNR of the double backplate microphones turns out to be merely 7% higher than the single backplate due to the viscous flow resistance and squeeze film damping caused by the additional backplate.

Real-Time Operation and Characterization of a High-Performance Time-Based Accelerometer

An accelerometer based on the electrostatic pull-in time of a microstructure is presented in this paper. The device uses a parallel-plate overdamped microstructure, and real-time control operation is performed using a field programmable gate array and high precision digital-to-analog converters. Both open-loop and closed-loop measurements are presented. The low noise is a key feature of this approach, which is limited only by the mechanical–thermal noise of the microstructure used, 2 $\mu \text{g}/\surd $ Hz as shown in the open-loop results (3 $\mu \text{g}/\surd $ Hz in closed-loop operation). The time readout method has extremely high-resolution capabilities. The pull-in time sensitivity can be adjusted up to 1.6 $\mu \text{s}/\mu \text{g}$ , and the electrostatic feedback voltage sensitivity is 61.3 $\text{V}^{2}$ /g. The closed-loop control allows operation of the accelerometer in a much larger range than in open-loop (±500 mg have been achieved) and the linearity is greatly improved ( $\mu \text{g}$ has been measured over 45 h in open loop and 250 $\mu \text{g}$ in closed loop. [2015-0074]

Detection of genuine tripartite entanglement and steering in hybrid optomechanics.

Multipartite quantum entanglement is a key resource for ensuring security in quantum network. We show that by using a unified parameter in terms of reduced noise variances one can determine different types of tripartite entanglement of a given state generated in a hybrid optomechanical system, where an atomic ensemble is located inside a single-mode cavity with a movable mirror, with different thresholds for each type. In particular, the special quantum states which allow both entanglement and steering genuinely shared among atom-light-mirror modes can be observed, even though there is no direct interaction between the mirror and the atomic ensemble. We further show the robustness against mechanical thermal noise and damping, the relaxation time of atomic ensemble, as well as the effect of gain factors involved in the criteria. Our analysis provides an experimentally achievable method to determine the type of tripartite quantum correlation in a way.

An improved interface and noise analysis of a turning fork microgyroscope structure

This paper analyzes different noise components in MEMS gyroscope silicon structure, including mechanical–thermal noise (MTN), electronic-thermal noise (ETN), flicker noise (FN) and Coriolis signal in-phase noise (IPN). The structure equivalent electronic model is established, and the improved differential interface is proposed based on weak signal detection technology, after that, the noise components in silicon structure are introduced and analyzed in sense open loop. The quadrature error (QE) signal automatically cancellation loop is proposed, and the results of the experiment indicate that the equivalent angular rates of QE and IPN are 46°/s and 4.55°/s respectively. The interfaces contrast experiments show that the DC noise and the useful signal amplitudes of differential and single-side detection interfaces are −49.8dBmV, −16.8dBmV and −39.8dBmV (−42.1dBmV), −22.1dBmV (−22.2dBmV), which confirms the differential interface has better SNR. The carrier experiments also illustrate that higher carrier frequency (from 500kHz to 10MHz) can restrain DC noise (from −19.8dBmV to −54.2dBmV) better, which demonstrate the FN is the dominant noise component of the silicon structure under normal temperature. The temperature experiments show the DC noise enhances from −48.5dBmV to −14.6dBmV over the range 20°C to 60°C while the useful signal amplitude remains around -16.6dBmV, and this phenomenon indicates the MTN and ETN become the dominant structure noise components gradually with temperature rising.

Quantum state transfer between two distant membranes

We investigate the quantum state transfer between two membranes in coupled cavities for the case before the cavity fields arrive at the steady state. Three different schemes are presented. The first one is based on the double swapping, i.e., one of the two collective cavity modes interacts with one membrane first and later with the other membrane. The second scheme is to use the mechanical dark mode, which decouples from the cavity fields and evolves adiabatically from one mechanical mode to the other. The third scheme depends on the interactions of the two oscillators with the same continuous, collective cavity mode. In comparison, the first scheme is more robust to the mechanical thermal noise, the second one depresses the cavity dissipation, and the third one merges the merits of the two former schemes.

Efficient Scheme for Perfect Collective Einstein-Podolsky-Rosen Steering.

A practical scheme for the demonstration of perfect one-sided device-independent quantum secret sharing is proposed. The scheme involves a three-mode optomechanical system in which a pair of independent cavity modes is driven by short laser pulses and interact with a movable mirror. We demonstrate that by tuning the laser frequency to the blue (anti-Stokes) sideband of the average frequency of the cavity modes, the modes become mutually coherent and then may collectively steer the mirror mode to a perfect Einstein-Podolsky-Rosen state. The scheme is shown to be experimentally feasible, it is robust against the frequency difference between the modes, mechanical thermal noise and damping, and coupling strengths of the cavity modes to the mirror.

Offset Suppression in a Micromachined Lorentz Force Magnetic Sensor by Current Chopping

We present a method to reduce offset in micromachined Lorentz force magnetic sensors by chopping the Lorentz force bias current. By switching the polarity of this current, the sensitivity of the magnetic sensor alternates its sign, whereas the offset remains the same. A residual offset of 31 μT is obtained from the initial offset of 25 mT. The proposed method significantly reduces the long-term drift of the magnetic sensor at the same time. A 9-h measurement shows the maximum drift error is reduced from ±500 μT to ±1 μT. Allan deviation measurements demonstrate that the longterm drift is reduced by a factor of 120. With 0.9 mA <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rms</sub> bias current the white noise is 400 nT/rt-Hz, limited by the thermal-mechanical noise.

Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit

Optomechanics allows the transduction of weak forces to optical fields, with many efforts approaching the standard quantum limit. We consider force-sensing using a mirror-in-the-middle setup and use two coupled cavity modes originated from normal mode splitting for separating pump and probe fields. We find that this two-mode model can be reduced to an effective single-mode model, if we drive the pump mode strongly and detect the signal from the weak probe mode. The optimal force detection sensitivity at zero frequency (DC) is calculated and we show that one can beat the standard quantum limit by driving the cavity close to instability. The best sensitivity achievable is limited by mechanical thermal noise and by optical losses. We also find that the bandwidth where optimal sensitivity is maintained is proportional to the cavity damping in the resolved sideband regime. Finally, the squeezing spectrum of the output signal is calculated, and it shows almost perfect squeezing at DC is possible by using a high quality factor and low thermal phonon-number mechanical oscillator.