We analyze the response of a nanomechanical resonator to an external drive when it is also coupled to a single-electron transistor (SET). The interaction between the SET electrons and the mechanical resonator depends on the amplitude of the mechanical motion leading to a strongly non-linear response to the drive which is similar to that of a Duffing oscillator. We show that the average dynamics of the resonator is well-described by a simple effective model which incorporates damping and frequency renormalization terms which are amplitude dependent. We also find that for a certain range of parameters the system displays interesting bistable dynamics in which noise arising from charge fluctuations causes the resonator to switch slowly between different dynamical states.
We analyze the noise properties of two single electron transistors (SETs) coupled via a shared voltage gate consisting of a nanomechanical resonator. Working in the regime where the resonator can be treated as a classical system, we find that the SETs act on the resonator like two independent heat baths. The coupling to the resonator generates positive correlations in the currents flowing through each of the SETs as well as between the two currents. In the regime where the dynamics of the resonator is dominated by the back-action of the SETs, these positive correlations can lead to parametrica…
We study a general model describing a self-detecting single electron transistor realized by a suspended carbon nanotube actuated by a nearby antenna. The main features of the device, recently observed in a number of experiments, are accurately reproduced. When the device is in a low current-carrying state, a peak in the current signals a mechanical resonance. On the contrary, a dip in the current is found in high current-carrying states. In the nonlinear vibration regime of the resonator, we are able to reproduce quantitatively the characteristic asymmetric shape of the current-frequency curve…
A single-electron transistor incorporated as part of a nanomechanical resonator represents an extreme limit of electron-phonon coupling. While it allows for fast and sensitive electromechanical measurements, it also introduces backaction forces from electron tunnelling which randomly perturb the mechanical state. Despite the stochastic nature of this backaction, under conditions of strong coupling it is predicted to create self-sustaining coherent mechanical oscillations. Here, we verify this prediction using time-resolved measurements of a vibrating carbon nanotube transistor. This electromec…
Graphene-based electromechanical resonators have attracted much interest recently because of the outstanding mechanical and electrical properties of graphene and their various applications. However, the coupling between mechanical motion and charge transport has not been explored in graphene. Here, we studied the mechanical properties of a suspended 50-nm-wide graphene nanoribbon, which also acts as a single-electron transistor (SET) at low temperature. Using the SET as a sensitive detector, we found that the resonance frequency could be tuned from 82 MHz to 100 MHz and the quality factor exce…
We investigate a superconducting single-electron transistor capacitively coupled to a nanomechanical oscillator and focus on the double Josephson quasiparticle resonance. The existence of two coherent Cooper pair tunneling events is shown to lead to pronounced backaction effects. Measuring the current and the shot noise provides a direct way of gaining information on the state of the oscillator. In addition to an analytical discussion of the linear-response regime, we discuss and compare results of higher-order approximation schemes and a fully numerical solution. We find that cooling of the m…
We present an analysis of the dynamics of a nanomechanical resonator coupled to a superconducting single electron transistor (SSET) in the vicinity of the Josephson quasiparticle (JQP) and double Josephson quasiparticle (DJQP) resonances. For weak coupling and wide separation of dynamical timescales, we find that for either superconducting resonance the dynamics of the resonator is given by a Fokker-Planck equation, i.e., the SSET behaves effectively as an equilibrium heat bath, characterised by an effective temperature, which also damps the resonator and renormalizes its frequency. Depending …
We analyse the quantum dynamics of a nanomechanical resonator coupled to a normal-state single-electron transistor (SET). Starting from a microscopic description of the system, we derive a master equation for the SET island charge and resonator which is valid in the limit of weak electro-mechanical coupling. Using this master equation we show that, apart from brief transients, the resonator always behaves like a damped harmonic oscillator with a shifted frequency and relaxes into a thermal-like steady state. Although the behaviour remains qualitatively the same, we find that the magnitude of t…
We present a scheme for photonic transistors based on photons and phonons in a cavity electromechanical system, which is consisted of a superconducting microwave cavity coupled to a nanomechanical resonator. Control of the propagation of photons is achieved through the interaction of microwave field (photons) and nanomechanical vibrations (phonons). By calculating the transmission spectrum of the signal field, we show that the signal field can be efficiently attenuated or amplified, depending on the power of a second `gating'(pump) field. This scheme may be a promising candidate for single-pho…
We propose a new mechanism of friction in resonantly driven vibrational systems. The form of the friction force follows from the time- and spatial-symmetry arguments. We consider a microscopic mechanism of this resonant force in nanomechanical systems. The friction can be negative, leading to an instability of forced vibrations of a nanoresonator and the onset of self-sustained oscillations in the rotating frame.
We study nanomechanical resonators with frequency fluctuations due to diffusion of absorbed particles. The diffusion depends on the vibration amplitude through inertial effect. We find that, if the diffusion coefficient is sufficiently large, the resonator response to periodic driving displays bistability. The lifetime of the coexisting vibrational states scales exponentially with the diffusion coefficient. It also displays a characteristic scaling dependence on the distance to bifurcation points.
We investigate the nonlinear response of a vibrating suspended nanomechanical beam on external periodic driving. The amplitude of the fundamental transverse mode behaves thereby like a weakly damped quantum particle in a driven anharmonic potential. Upon using a Born-Markovian master equation, we calculate the fundamental mode amplitude for varying driving frequencies. In the nonlinear regime, we observe resonances which are absent in the corresponding classical model. They are shown to be associated with resonant multi-phonon excitations. Furthermore, we identify resonant tunneling in a dynam…
A "spin-flip transistor" is a lateral spin valve consisting of ferromagnetic source drain contacts to a thin-film normal-metal island with an electrically floating ferromagnetic base contact on top. We analyze the \emph{dc}-current-driven magnetization dynamics of spin-flip transistors in which the source-drain contacts are magnetized perpendicularly to the device plane by magnetoelectronic circuit theory and the macrospin Landau-Lifshitz-Gilbert equation. Spin flip scattering and spin pumping effects are taken into account. We find a steady-state rotation of the base magnetization at GHz freq…
Quantum mechanics demands that the act of measurement must affect the measured object. When a linear amplifier is used to continuously monitor the position of an object, the Heisenberg uncertainty relationship requires that the object be driven by force impulses, called back-action. Here we measure the back-action of a superconducting single-electron transistor (SSET) on a radiofrequency nanomechanical resonator. The conductance of the SSET, which is capacitively coupled to the resonator, provides a sensitive probe of the latter's position;back-action effects manifest themselves as an effectiv…
We review a recent theoretical development based on non-equilibrium Green's function formalism to study heat transport in nanomechanical devices modeled by phononic systems of coupled quantum oscillators driven by ac forces and connected to phononic reservoirs. We present the relevant equations to calculate the heat currents flowing along different regions of the setup, as well as the power developed by the time-dependent forces. We also present different strategies to evaluate the Green's functions exactly or approximately within the weak driving regime. We finally discuss the different mecha…
We show that the nanoresonator position can be squeezed significantly below the ground state level by measuring the nanoresonator with a quantum point contact or a single-electron transistor and applying a periodic voltage across the detector. The mechanism of squeezing is basically a generalization of quantum nondemolition measurement of an oscillator to the case of continuous measurement by a weakly coupled detector. The quantum feedback is necessary to prevent the ``heating'' due to measurement back-action. We also discuss a procedure of experimental verification of the squeezed state.
The response of a coupled array of nonlinear oscillators to parametric excitation is calculated in the weak nonlinear limit using secular perturbation theory. Exact results for small arrays of oscillators are used to guide the analysis of the numerical integration of the model equations of motion for large arrays. The results provide a qualitative explanation for a recent experiment [Buks and Roukes, cond-mat/0008211, to appear in J. MEMS (2002)] involving a parametrically-excited micromechanical resonator array. Future experiments are suggested that could provide quantitative tests of the the…
Because of the broken time-translation symmetry, in periodically driven vibrational systems fluctuations of different vibration components have different intensities. Fluctuations of one of the components are often squeezed, whereas fluctuations of the other component, which is shifted in phase by π/2, are increased. Squeezing is a multifaceted phenomenon; it attracts much attention from the perspective of high-precision measurements. Here we demonstrate a new and hitherto unappreciated side of squeezing: its direct manifestation in the spectra of driven vibrational systems. With a weakly damp…
Nanomechanical resonators, machined out of Silicon-on-Insulator wafers, are operated in the nonlinear regime to investigate higher-order mechanical mixing at radio frequencies, relevant to signal processing and nonlinear dynamics on nanometer scales. Driven by two neighboring frequencies the resonators generate rich power spectra exhibiting a multitude of satellite peaks. This nonlinear response is studied and compared to $n^{th}$-order perturbation theory and nonperturbative numerical calculations.
We study a resonator coupled to a generic detector and calculate the noise spectra of the two sub-systems. We describe the coupled system by a closed, linear, set of Langevin equations and derive a general form for the finite frequency noise of both the resonator and the detector. The resonator spectrum is the well-known thermal form with an effective damping, frequency shift and diffusion term. In contrast, the detector noise shows a rather striking Fano-like resonance, i.e. there is a resonance at the renormalized frequency, and an anti-resonance at the bare resonator frequency. As examples …
We analyze squeezing of the nanoresonator state produced by periodic measurement of position by a quantum point contact or a single-electron transistor. The mechanism of squeezing is the stroboscopic quantum nondemolition measurement generalized to the case of continuous measurement by a weakly coupled detector. The magnitude of squeezing is calculated for the harmonic and stroboscopic modulations of measurement, taking into account detector efficiency and nanoresonator quality factor. We also analyze the operation of the quantum feedback, which prevents fluctuations of the wavepacket center d…
We demonstrate the formation of confinement potentials in suspended nanostructures induced by the geometry of the devices. We then propose a setup for measuring the resulting geometric phase change of electronic wave functions in such a mechanical nanostructure. The device consists of a suspended loop through which a phase coherent current is driven. Combination of two and more geometrically induced potentials can be applied for creating mechanical quantum bit states.
We present measurements on nanomechanical resonators machined from Silicon-on-Insulator substrates. The resonators are designed as freely suspended Au/Si beams of lengths on the order of 1 - 4 um and a thickness of 200 nm. The beams are driven into nonlinear response by an applied modulation at radio frequencies and a magnetic field in plane. The strong hysteresis of the magnetomotive response allows sensitive charge detection by varying the electrostatic potential of a gate electrode.
We consider a new type of cooling mechanism for a suspended nanowire acting as a weak link between two superconductive electrodes. By applying a bias voltage over the system, we show that the system can be viewed as a refrigerator for the nanomechanical vibrations, where energy is continuously transferred from the vibrational degrees of freedom to the extended quasiparticle states in the leads through the periodic modulation of the inter-Andreev level separation. The necessary coupling between the electronic and mechanical degrees of freedom responsible for this energy-transfer can be achieved…
We investigate electrical transport through a single-electron transistor coupled to a nanomechanical oscillator. Using a combination of a master-equation approach and a numerical Monte Carlo method, we calculate the average current and the current noise in the strong-coupling regime, studying deviations from previously derived analytic results valid in the limit of weak-coupling. After generalizing the weak-coupling theory to enable the calculation of higher cumulants of the current, we use our numerical approach to study how the third cumulant is affected in the strong-coupling regime. In thi…
We consider the charge shuttle proposed by Gorelik {\em et al.} driven by a time-dependent voltage bias. In the case of asymmetric setup, the system behaves as a rachet. For pure AC drive, the rectified current shows a complex frequency dependent response characterized by frequency locking at fracional values of the external frequency. Due to the non-linear dynamics of the shuttle, the rachet effect is present also for very low frequencies.
We observe direct mechanical mixing in nanoelectromechanical transistors fabricated in semiconductor materials operating in the radio frequency band of 10-1000 MHz. The device is made of a mechanically flexible pillar with a length of 240 nm and a diameter of 50 nm, placed between two electrodes in an impedance matched coplanar wave guide. We find a non-linear IV-characteristic, which enables radio frequency mixing of two electromagnetic signals via the nanomechanical transistor. Potential applications for this mixer are ultrasensitive displacement detection or signal processing in communicati…
We demonstrate robust and reliable signatures for the transition from quantum to classical behavior in the position probability distribution of a damped double-well system using the Qunatum State Diffusion approach to open quantum systems. We argue that these signatures are within experimental reach, for example in a doubly-clamped nanomechanical beam.
We describe the measurement and modeling of amplitude noise and phase noise in ultra-high Q nanomechanical resonators made from stoichiometric silicon nitride. With quality factors exceeding 2 million, the resonators' noise performance is studied with high precision. We find that the amplitude noise can be well described by the thermomechanical model, however, the resonators exhibit sizable extra phase noise due to their intrinsic frequency fluctuations. We develop a method to extract the resonator frequency fluctuation of a driven resonator and obtain a noise spectrum with dependence, which c…
We seek the first indications that a nanoelectromechanical system (NEMS) is entering the quantum domain as its mass and temperature are decreased. We find them by studying the transition from classical to quantum behavior of a driven nonlinear Duffing resonator. Numerical solutions of the equations of motion, operating in the bistable regime of the resonator, demonstrate that the quantum Wigner function gradually deviates from the corresponding classical phase-space probability density. These clear differences that develop due to nonlinearity can serve as experimental evidence, in the near fut…
The dynamics of nanomechanical resonators driven by both low- and high-frequency signals is studied. Considering, as an example, resonators made of a doubly-clamped beam with magnetomotive driving, it is shown that three-frequency resonances arise due to the interaction of the above frequencies. Properties and characteristics of these resonances are determined analytically for linear and nonlinear modes of the resonator excitation. It is shown that in opposite to the conventional two-frequency resonance, the central frequency of these resonances, their linear dynamic range, and the critical va…
Self excitation is a mechanism which is ubiquitous for electromechanical power devices such as electrical generators. This is conventionally achieved by making use of the magnetic field component in electrical generators [1], where a good example are the overall visible wind farm turbines [2]. In other words, a static force, like wind acting on the rotor blades, can generate a resonant excitation at a certain mechanical frequency. For nanomechanical systems [3,4,5] such a self excitation (SE) mechanism is highly desirable as well, since it can generate mechanical oscillations at radio frequenc…
We propose a scheme able to entangle at the steady state a nanomechanical resonator with a microwave cavity mode of a driven superconducting coplanar waveguide. The nanomechanical resonator is capacitively coupled with the central conductor of the waveguide and stationary entanglement is achievable up to temperatures of tens of milliKelvin.
Because of their nonlinearity, vibrational modes of resonantly driven nanomechanical systems have coexisting stable states of forced vibrations in a certain range of the amplitude of the driving force. Depending on its phase, which encodes binary information, a signal at the same frequency increases or decreases the force amplitude. The resulting force amplitude can be outside the range of bistability. The values of the mode amplitude differ significantly on the opposite sides of the bistability region. Therefore the mode amplitude is very sensitive to the signal phase. This suggests using a d…
Dynamic interactions between two oscillating micromechanical cantilevers are studied. In the experiment, the tip of a high-frequency cantilever is positioned near the surface of a second low-frequency cantilever. Due to the highly nonlinear interaction forces between the two surfaces, thermal oscillations of the low-frequency cantilever modulate the driven oscillations of the high-frequency cantilever. The dissipations and the frequencies of the two cantilevers are shown to be coupled, and a simple model for the interactions is presented. The interactions studied here may be useful for the des…
Two elastically coupled nanomechanical resonators driven independently near their resonance frequencies show intricate nonlinear dynamics. The dynamics provide a scheme for realizing a nanomechanical system with tunable frequency and nonlinear properties. For large vibration amplitudes the system develops spontaneous oscillations of amplitude modulation that also show period doubling transitions and chaos. The complex nonlinear dynamics are quantitatively predicted by a simple theoretical model.
We report quantitative experimental measurements of the nonlinear response of a radiofrequency mechanical resonator, with very high quality factor, driven by a large swept-frequency force. We directly measure the noise-free transition dynamics between the two basins of attraction that appear in the nonlinear regime, and find good agreement with those predicted by the one-dimensional Duffing equation of motion. We then measure the response of the transition rates to controlled levels of white noise, and extract the activation energy from each basin. The measurements of the noise-induced transit…
Statistical Properties of a Fluctuation-Driven Nanomechanical Duffing Resonator
Boston University（波士顿大学）；Virginia Tech（弗吉尼亚理工大学）；James Madison University（詹姆斯麦迪逊大学）；Sabanci University（萨班哲大学）；Bilkent University（比尔肯特大学）；Gordon College（戈登学院）
We report a theoretical study of time-dependent transport in a ballistic graphene field effect transistor. We develop a model based on Floquet theory describing Dirac electron transmission through a harmonically driven potential barrier. Photon-assisted tunneling results in excitation of quasibound states at the barrier. Under resonance condition, the excitation of the quasibound states leads to promotion of higher-order sidebands and enhanced higher harmonics of the source-drain conductance. The resonances in the main transmission channel are of the Fano form, while they are of the Breit-Wign…
Phononic frequency combs have been attracting an increasing attention both as a qualitatively new type of nonlinear phenomena in vibrational systems and from the point of view of applications. It is commonly believed that at least two modes must be involved in generating a comb. In this paper we demonstrate that a comb can be generated by a single nanomechanical mode driven by a resonant monochromatic drive. The comb emerges where the drive is still weak, so that the anharmonic part of the mode potential energy remains small. We relate the effect to a negative nonlinear friction induced by the…
Quasicrystal Architected Nanomechanical Resonators via Data-Driven Design
Department of Precision and Microsystems Engineering, Delft University of Technology（代尔夫特理工大学精密与微系统工程系）；Department of Mechanical Engineering, Pohang University of Science and Technology（浦项科技大学机械工程系）
Nanomechanical devices have attracted the interest of a growing interdisciplinary research community, since they can be used as highly sensitive transducers for various physical quantities. Exquisite control over these systems facilitates experiments on the foundations of physics. Here, we demonstrate that an optically trapped silicon nanorod, set into rotation at MHz frequencies, can be locked to an external clock, transducing the properties of the time standard to the rod's motion with the remarkable frequency stability $f_{\rm r}/Δf_{\rm r}$ of $7.7 \times 10^{11}$. While the dynamics of th…
Magnetic skyrmion is a topologically protected domain-wall structure at nanoscale, which could serve as a basic building block for advanced spintronic devices. Here, we propose a microwave field-driven skyrmionic device with the transistor-like function, where the motion of a skyrmion in a voltage-gated ferromagnetic nanotrack is studied by micromagnetic simulations. It is demonstrated that the microwave field can drive the motion of a skyrmion by exciting propagating spin waves, and the skyrmion motion can be governed by a gate voltage. We also investigate the microwave current-assisted creat…
The ultimate sensitivities achieved in force or mass sensing are limited by the employed nanomechanical probes thermal noise. Its proper understanding is critical for ultimate operation and any deviation from the underlying fluctuation dissipation theorem should be carefully inspected. Here we investigate an ultrasensitive vectorial force-field sensor, a singly clamped nanowire oscillating along two quasi frequency degenerated transverse directions. Immersing the nanowire in a non-conservative optical force field causes dramatic modifications of its thermal noise and driven dynamics. In region…
We measure the energy relaxation rate of single- and few-layer molybdenum disulphide ($\mathrm{MoS_2}$) nanomechanical resonators by detecting the resonator ring-down. Recent experiments on these devices show a remarkably low quality (Q)-factor when taking spectrum measurements at room temperature. The origin of the low spectral Q-factor is an open question, and it has been proposed that besides dissipative processes, frequency fluctuations contribute significantly to the resonance line-width. The spectral measurements performed thus far however, do not allow one to distinguish these two proce…
We propose a scheme for generating the Schrödinger cat state based on geometric operations by a nanomechanical resonator coupled to a superconducting charge qubit. The charge qubit, driven by two strong classical fields, interacts with a high-frequency phonon mode of the nanomechanical resonator. During the operation, the charge qubit undergoes no real transitions, while the phonon mode of the nanomechanical resonator is displaced along different paths in the phase space, dependent on the states of the charge qubit, which yields the Schrödinger cat state. The robustness of the scheme is justif…
We consider a theoretical model for a nonlinear nanomechanical resonator coupled to a superconducting microwave resonator. The nanomechanical resonator is driven parametrically at twice its resonance frequency, while the superconducting microwave resonator is driven with two tones that differ in frequency by an amount equal to the parametric driving frequency. We show that the semi-classical approximation of this system has an interesting fixed point bifurcation structure. In the semi-classical dynamics a transition from stable fixed points to limit cycles is observed as one moves from positiv…
In strained mechanical resonators, the concurrence of tensile stress and geometric nonlinearity dramatically reduces dissipation. This phenomenon, dissipation dilution, is employed in mirror suspensions of gravitational wave interferometers and at the nanoscale, where soft-clamping and strain engineering have allowed extremely high quality factors. However, these techniques have so far only been applied in amorphous materials, specifically silicon nitride. Crystalline materials exhibit significantly lower intrinsic damping at cryogenic temperatures, due to the absence of two level systems in t…
Much experimental effort is invested these days in fabricating nanoelectromechanical systems (NEMS) that are sufficiently small, cold, and clean, so as to approach quantum mechanical behavior as their typical quantum energy scale $\hbarΩ$ becomes comparable to that of the ambient thermal energy $k_{B}T$. Such systems will hopefully enable one to observe the quantum behavior of human-made objects, and test some of the basic principles of quantum mechanics. Here we expand and elaborate on our recent suggestion [PRL 99 (2007) 040404] to exploit the nonlinear nature of a nanoresonator in order to …
We analyze the spectral properties of a resonator coupled to a superconducting single electron transistor (SSET) close to the Josephson quasiparticle resonance. Focussing on the regime where the resonator is driven into a limit-cycle state by the SSET, we investigate the behavior of the resonator linewidth and the energy relaxation rate which control the widths of the main features in the resonator spectra. We find that the linewidth becomes very narrow in the limit-cycle regime, where it is dominated by a slow phase diffusion process, as in a laser. The overall phase diffusion rate is determi…
Elastic strain engineering utilizes stress to realize unusual material properties. For instance, strain can be used to enhance the electron mobility of a semiconductor, enabling more efficient solar cells and smaller, faster transistors. In the context of nanomechanics, the pursuit of resonators with ultra-high coherence has led to intense study of a complementary strain engineering technique, "dissipation dilution", whereby the stiffness of a material is effectively increased without added loss. Dissipation dilution is known to underlie the anomalously high Q factor of Si$_3$N$_4$ nanomechani…
Phase relaxation of electrons transferring through an electromechanical transistor is studied using the Aharonov-Bohm interferometer. With the approach of quantum master equation, the phase properties of an electron are numerically analyzed based on the interference fringes. Coherence of electron is partially destroyed by its scattering on excited levels of the local nanomechanical oscillator. Transmission amplitudes with respect to two adjacent mechanical vibrational levels have a phase difference of $π$. The character of phase shift by $π$ depends on the oscillator frequency only and is robu…
We study time-dependent electron transport and quantum noise in a ballistic graphene field effect transistor driven by an ac gate potential. The non-linear response to the ac signal is computed through Floquet theory for scattering states and Landauer-Büttiker theory for charge current and its fluctuations. Photon-assisted excitation of a quasibound state in the top-gate barrier leads to resonances in transmission that strongly influence the noise properties. For strong doping of graphene under source and drain contacts, when electrons are transmitted through the channel via evanescent waves, …
We investigate electric current in a single-electron tunnelling device weakly coupled to an ac-driven underdamped harmonic nanomechanical oscillator. In the linear regime, the current can respond to the external frequency in a resonant as well as in an anti-resonant fashion. The main resonance is accompanied by an additional resonance at a half of the external frequency.
