A nanoelectromechanical weak link composed of a carbon nanotube suspended between two normal electrodes in a gap between two superconducting leads is considered. The nanotube is treated as a movable single-level quantum dot in which the position-dependent superconducting order parameter is induced due to the Cooper pair tunneling. We show that electron tunneling processes significantly affect the state of the mechanical subsystem. We found that at a given direction of the applied voltage between the electrodes, the stationary state of the mechanical subsystem has a Boltzmann form with an effective temperature depended on the parameters of the device. As this takes place, the effective temperature can reach significantly small values (cooling effect). We also demonstrate that nanotube fluctuations strongly affect the dc current through the system. The latter can be used to probe the predicted effects in an experiment.
We demonstrate that a supercurrent can pump energy from a battery that provides a voltage bias into nanomechanical vibrations. Using a device containing a nanowire Josephson weak link as an example we show that a nonlinear coupling between the supercurrent and a static external magnetic field leads to a Lorentz force that excites bending vibrations of the wire at resonance conditions. We also demonstrate the possibility to achieve more than one regime of stationary nonlinear vibrations and how to detect them via the associated dc Josephson currents and we discuss possible applications of such …
We consider a quantum dot with mechanical degrees of freedom which is coupled to superconducting electrodes. A Josephson current is generated by applying a phase difference. In the absence of coupling to vibrations, this setup was previously proposed as a detector of magnetic flux and we wish here to address the effect of the phonon coupling to this detection scheme. We compute the charge on the quantum dot and determine its dependence on the phase difference in the presence of phonon coupling and Coulomb interaction. This allows to identify regions in parameter space with the highest charge t…
We propose and study a spin-orbit interaction based mechanism to actively cool down the torsional vibration of a nanomechanical resonator made by semiconductor materials. We show that the spin-orbit interactions of electrons can induce a coherent coupling between the electron spins and the torsional modes of nanomechanical vibration. This coherent coupling leads to an active cooling for the torsional modes via the dynamical thermalization of the resonator and the spin ensemble.
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…
The electrical response of two-dimensional electron gas to vibrations of a nanomechanical cantilever containing it is studied. Vibrations of perpendicularly oriented cantilevers are experimentally shown to change oppositely the conductivity near their bases. This indicates the piezoelectric nature of electromechanical coupling. A physical model is developed, which quantitatively explains the experiment. It shows that the main origin of the conductivity change is a rapid change in the mechanical stress on the boundary between suspended and non-suspended areas, rather than the stress itself.
Software-defined radio dongles are small and inexpensive receivers well known to amateur radio enthusiasts. When connected to an antenna, they enable monitoring of a wide range of the radio spectrum by conditioning the input signal and transferring a downconverted version of it to a personal computer for software processing. Here, we employ a composite of two such dongles, interfaced with codes written in MATLAB and GNU Radio, as a measuring instrument to study the flexural vibrations of a few-layer graphene nanomechanical resonator. Instead of an antenna, we connect the dongles to the split o…
We discuss two theoretical proposals for controlling the nonequilibrium steady state of nanomechanical resonators using quantum electronic transport. Specifically?, we analyse two approaches to achieve the ground-state cooling of the mechanical vibration coupled to a quantum dot embedded between (i) spin-polarised contacts or (ii) a normal metal and a superconducting contact. Assuming a suitable coupling between the vibrational modes and the charge or spin of the electrons in the quantum dot, we show that ground-state cooling of the mechanical oscillator is within the state of the art for susp…
We study the response of a weakly damped vibrational mode of a nanostring resonator to a moderately strong resonant driving force. Because of the geometry of the experiment, the studied flexural vibrations lack inversion symmetry. As we show, this leads to a nontrivial dependence of the vibration amplitude on the force parameters. For a comparatively weak force, the response has the familiar Duffing form, but for a somewhat stronger force, it becomes significantly different. Concurrently there emerge vibrations at twice the drive frequency, a signature of the broken symmetry. Their amplitude a…
We consider a nanoelectromechanical system composed of a carbon nanotube suspended between two normal leads and coupled to a superconducting scanning tunneling microscope (STM) tip via vacuum tunnel barrier. Treating the nanotube as a single-level quantum dot, it is shown that an applied voltage between the superconducting STM tip and normal leads gives rise to a pumping or a cooling of the mechanical subsystem depending on the direction of the electronic flow. It is also demonstrated that the transition between these two regimes is controlled by the strength of the tunnel coupling between the…
We study a system consisting of a particle adsorbed on a carbon nanotube resonator. The particle is allowed to diffuse along the resonator, in order to enable study of e.g. room temperature mass sensing devices. The system is initialized in a state where only the fundamental vibration mode is excited, and the ring-down of the system is studied by numerically and analytically solving the stochastic equations of motion. We find two mechanisms of dissipation, induced by the diffusing adsorbate. First, short-time correlations between particle and resonator motions means that the net effect of the …
We study the effect of the electron-phonon coupling on vibrational eigenmodes of nano- and micro-mechanical systems made of semiconductors with equivalent energy valleys. We show that the coupling can lead to a strong mode nonlinearity. The mechanism is the lifting of the valley degeneracy by the strain. The redistribution of the electrons between the valleys is controlled by a large ratio of the electron-phonon coupling constant to the electron chemical potential or temperature. We find the quartic in the strain terms in the electron free energy, which determine the amplitude dependence of th…
The detection of mechanical vibrations near the quantum limit is a formidable challenge since the displacement becomes vanishingly small when the number of phonon quanta tends towards zero. An interesting setup for on-chip nanomechanical resonators is that of coupling them to electrical microwave cavities for detection and manipulation. Here we show how to achieve a large cavity coupling energy of up to (2 π) 1 MHz/nm for metallic beam resonators at tens of MHz. We used focused ion beam (FIB) cutting to produce uniform slits down to 10 nm, separating patterned resonators from their gate electr…
We study nonlinear resonance of coupled modes in nano-mechanical systems. To reveal the qualitative features of the dynamics, we consider the limiting cases, where the results can be obtained analytically. For 1:3 resonance, we find the anomalously strong and nonmonotonic dependence of the decay rate of the low frequency mode on its amplitude, if the decay rate of the high-frequency mode is comparatively large. In this case the low-frequency mode driven close to resonance can have several branches of steady-state vibrations with constant amplitude. If the decay rates of the both modes are smal…
We show that the vibrations of a nanomechanical resonator can be cooled to near its quantum ground state by tunnelling injection of electrons from an STM tip. The interplay between two mechanisms for coupling the electronic and mechanical degrees of freedom results in a bias-voltage dependent difference between the probability amplitudes for vibron emission and absorption during tunneling. For a bias voltage just below the Coulomb blockade threshold we find that absorption dominates, which leads to cooling corresponding to an average vibron population of the fundamental bending mode of 0.2.
Nanomechanics has brought mesoscopic physics into the world of vibrations. Because nanomechanical systems are small, fluctuations are significant, the vibrations become nonlinear already for comparatively small amplitudes, and new mechanisms of dissipation come into play. At the same time, the exquisite control of these systems makes them a platform for studying many problems of classical and quantum physics far from thermal equilibrium in a well-characterized setting. This review describes, at a conceptual level, basic theoretical ideas and explicative experiments pertaining to mesoscopic phy…
We present results from a study of the nonlinear intermodal coupling between different flexural vibrational modes of a single high-stress, doubly-clamped silicon nitride nanomechanical beam. The measurements were carried out at 100 mK and the beam was actuated using the magnetomotive technique. We observed the nonlinear behavior of the modes individually and also measured the coupling between them by driving the beam at multiple frequencies. We demonstrate that the different modes of the resonator are coupled to each other by the displacement induced tension in the beam, which also leads to th…
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 study the nonequilibrium steady state of a mechanical resonator in the quantum regime realized by a suspended carbon nanotube quantum dot contacted by two ferromagnets. Because of the spin-orbit interaction and/or an external magnetic field gradient, the spin on the dot couples directly to the flexural eigenmodes. Accordingly, the nanomechanical motion induces inelastic spin flips of the tunneling electrons. A spin-polarized current at finite bias voltage causes either heating or active cooling of the mechanical modes. We show that maximal cooling is achieved at resonant transport when the …
We study the ground-state cooling of a mechanical oscillator linearly coupled to the charge of a quantum dot inserted between a normal metal and a superconducting contact. Such a system can be realized, e.g., by a suspended carbon nanotube quantum dot with a capacitive coupling to a gate contact. Focusing on the subgap transport regime, we analyze the inelastic Andreev reflections which drive the resonator to a nonequilibrium state. For small coupling, we obtain that vibration-assisted reflections can occur through two distinct interference paths. The interference determines the ratio between …
We present a free and a constrained fitting procedure for quantitative signal extraction of nanomechanical systems in the dynamic range and for physical model testing. We demonstrate that applying the free-fitting procedure to the measured frequency response of silicon nitride (SiN) nanomembranes at varying pressure enables us to disentangle the intrinsic membrane vibration properties from the system response, thereby giving quantitative access to the eigenfrequency, quality factor, coupling strength between resonator and drive system, and to system noise. The validity of physical models for q…
Using the transversal vibration resonance of a suspended carbon nanotube as charge detector for its embedded quantum dot, we investigate the case of strong Kondo correlations between a quantum dot and its leads. We demonstrate that even when large Kondo conductance is carried at odd electron number, the charging behaviour remains similar between odd and even quantum dot occupation. While the Kondo conductance is caused by higher order processes, a sequential tunneling only model can describe the time-averaged charge. The gate potentials of maximum current and fastest charge increase display a …
Mechanical displacements of a nanoelectromechanical system (NEMS) shift the electron trajectories and hence perturb phase coherent charge transport through the device. We show theoretically that in the presence of a magnetic feld such quantum-coherent displacements may give rise to an Aharonov-Bohm-type of effect. In particular, we demonstrate that quantum vibrations of a suspended carbon nanotube result in a positive nanotube magnetoresistance, which decreases slowly with the increase of temperature. This effect may enable one to detect quantum displacement fluctuations of a nanomechanical de…
We theoretically investigate the deflection-induced coupling of an electron spin to vibrational motion due to spin-orbit coupling in suspended carbon nanotube quantum dots. Our estimates indicate that, with current capabilities, a quantum dot with an odd number of electrons can serve as a realization of the Jaynes-Cummings model of quantum electrodynamics in the strong-coupling regime. A quantized flexural mode of the suspended tube plays the role of the optical mode and we identify two distinct two-level subspaces, at small and large magnetic field, which can be used as qubits in this setup. …
When a contaminant diffuses on the surface of a nanomechanical resonator, the motions of the two become correlated. Despite being a high-order effect in the resonator-particle coupling, such correlations affect the system dynamics by inducing dissipation of the resonator energy. Here, we consider this diffusion-induced dissipation in the cases of multiple particles adsorbed on carbon nanotube and graphene resonators. By solving the stochastic equations of motion, we simulate the ringdown of the resonator, in order to determine the resonator energy decay rate. We find two different scalings wit…
We propose a general one-dimensional {\em continuous} formulation to analyze the vibrational modes of antenna-like nanomechanical resonators consisting of two symmetric arrays of cantilevers affixed to a central nano-beam. The cantilever arrays can have arbitrary density and length profile along the beam. We obtain the secular equation that allows for the determination of their frequency spectrum and illustrate the results on the particular examples of structures with constant or alternating cantilever length profiles. We show that our analytical results capture the vibration spectrum of such …
We experimentally study stochastic resonance in a nonlinear bistable nanomechanical resonator. The device consists of a PdAu doubly clamped beam serving as a nanomechanical resonator excited capacitively by an adjacent gate electrode and its vibrations are detected optically. The resonator is tuned to its bistability region by an intense pump near a point of equal transition rates between its two metastable states. The pump is amplitude modulated, inducing modulation of the activation barrier between the states. When noise is added to the excitation, the resonator's displacement exhibits noise…
The efficient cooling of the nanomechanical resonators is essential to exploration of quantum properties of the macroscopic or mesoscopic systems. We propose such a laser-cooling scheme for a nanomechanical cantilever, which works even for the low-frequency mechanical mode and under weak cooling lasers. The cantilever is attached by a diamond nitrogen-vacancy center under a strong magnetic field gradient and the cooling is assisted by a dynamical Stark-shift gate. Our scheme can effectively enhance the desired cooling efficiency by avoiding the off-resonant and unexpected carrier transitions, …
We present a theoretical investigation for the full counting statistics of the Andreev tunneling through a quantum dot (QD) embedded between superconducting (SC) and normal leads in the presence of a strong on-site electron-phonon interaction using nonequilibrium Green function method. For this purpose, we generalize the dressed tunneling approximation (DTA) recently developed in dealing with inelastic tunneling in a normal QD system to the Andreev transport issue, which takes account of vibrational effect in evaluation of electronic tunneling self energy in comparison with other simple approa…
We study the quantum transport and the nonequilibrium vibrational states of a quantum dot embedded between a normal and a superconducting lead with the charge on the quantum dot linearly coupled to a harmonic oscillator of frequency $ω$. To the leading order in the charge-vibration interaction, we calculate the current and the nonequilibrium phonon occupation by the Keldsyh Green's function technique. We analyze the inelastic, vibration-assisted tunneling processes in the regime $ω<Δ$, with the superconducting energy gap $Δ$, and for sharp resonant transmission through the dot. When the energy…
We study spin-dependent transport in a suspended carbon nanotube quantum dot in contact with two ferromagnetic leads and with the dot's spin coupled to the flexural mechanical modes. The spin-vibration interaction induces spin-flip processes between the two energy levels of the dot. This interaction arises from the spin-orbit coupling or a magnetic field gradient. The inelastic vibration-assisted spin flips give rise to a mechanical damping and, for an applied bias voltage, to a steady nonequilibrium occupation of the harmonic oscillator. We analyze these effects as function of the energy-leve…
Networks of coupled resonators are an ubiquitous concept in physics, forming the basis of synchronization phenomena, metamaterial formation, nonreciprocal behavior and topological effects. Such systems are typically explored using optical or microwave resonators. In recent years, mechanical resonators have entered the stage as universal building block for resonator networks, both for their well-controlled mechanical properties and for their eigenfrequencies conveniently located in the radio-frequency regime. Vertically oriented nanomechanical pillar resonators are ideally suited for the dense …
The interplay between discrete vibrational and electronic degrees of freedom directly influences the chemical and physical properties of molecular systems. This coupling is typically studied through optical methods such as fluorescence, absorption, and Raman spectroscopy. Molecular electronic devices provide new opportunities for exploring vibration-electronic interactions at the single molecule level. For example, electrons injected from a scanning tunneling microscope tip into a metal can excite vibrational excitations of a molecule in the gap between tip and metal. Here we show how current …
We demonstrate that entanglement of two macroscopic nanoelectromechanical resonators -- coupled to each other via a common detector, a tunnel junction -- can be generated by running a current through the device. We introduce a setup that overcomes generic limitations of proposals suggesting to entangle systems via a shared bath. At the heart of the proposal is an Andreev entangler setup, representing an experimentally feasible way of entangling two nanomechanical oscillators. Instead of relying on the coherence of a (fermionic) bath, in the Andreev entangler setup, a split Cooper-pair that coh…
We show that nitrogen-vacancy (NV) centers in diamond interfaced with a suspended carbon nanotube carrying a dc current can facilitate a spin-nanomechanical hybrid device. We demonstrate that strong magnetomechanical interactions between a single NV spin and the vibrational mode of the suspended nanotube can be engineered and dynamically tuned by external control over the system parameters. This spin-nanomechanical setup with strong, \emph{intrinsic} and \emph{tunable} magnetomechanical couplings allows for the construction of hybrid quantum devices with NV centers and carbon-based nanostructu…
Phonon-assisted electronic tunnelings through a vibrating quantum dot embedded between normal and superconducting leads are studied in the Kondo regime. In such a hybrid device, with the bias applied to the normal lead, we find a series of Kondo sidebands separated by half a phonon energy in the differential conductance, which are distinct from the phonon-assisted sidebands previously observed in the conventional Andreev tunnelings and in systems with only normal leads. These Kondo sidebands originate from the Kondo-Andreev cooperative cotunneling mediated by phonons, which exhibit a novel Kon…
Nanomechanical resonators are used with great success to couple mechanical motion to other degrees of freedom, such as photons, spins, and electrons. Mechanical vibrations can be efficiently cooled and amplified using photons, but not with other degrees of freedom. Here, we demonstrate a simple yet powerful method for cooling, amplification, and self-oscillation using electrons. This is achieved by applying a constant (DC) current of electrons through a suspended nanotube in a dilution fridge. We demonstrate cooling down to $4.6\pm 2.0$ quanta of vibrations. We also observe self-oscillation, w…
We consider a nanoelectromechanical Josephson junction, where a suspended nanowire serves as a superconducting weak link, and show that an applied DC bias voltage an result in suppression of the flexural vibrations of the wire. This cooling effect is achieved through the transfer of vibronic energy quanta first to voltage driven Andreev states and then to extended quasiparticle electronic states. Our analysis, which is performed for a nanowire in the form of a metallic carbon nanotube and in the framework of the density matrix formalism, shows that such self-cooling is possible down to a level…
Fundamental and second-subharmonic Autler-Townes splitting in classical systems
Technical University of Munich（慕尼黑工业大学）；Ain Shams University（艾因夏姆斯大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；TUM Center for Quantum Engineering (ZQE)（慕尼黑工业大学量子工程中心）
Cavity optomechanics allows the characterization of a vibration mode, its cooling and quantum manipulation using electromagnetic fields. Regarding nanomechanical as well as electronic properties, single wall carbon nanotubes are a prototypical experimental system. At cryogenic temperatures, as high quality factor vibrational resonators, they display strong interaction between motion and single-electron tunneling. Here, we demonstrate large optomechanical coupling of a suspended carbon nanotube quantum dot and a microwave cavity, amplified by several orders of magnitude via the nonlinearity of …
Coupled mechanical oscillations were first observed in paired pendulum clocks in the mid-seventeenth century and were extensively studied for their novel sympathetic oscillation dynamics. In this era of nanotechnologies, coupled oscillations have again emerged as subjects of interest when realized in nanomechanical resonators for both practical applications and fundamental studies. However, a key obstacle to the further development of this architecture is the ability to coherently manipulate the coupled oscillations. This limitation arises as a consequence of the usually weak coupling between …
Geometric configuration and energy of a hydrogen molecule centered between two point-shaped tips of equal charge are calculated with the variational quantum Monte-Carlo (QMC) method without the restriction of the Born-Oppenheimer (BO) approximation. Ground state nuclear distribution, stability, and low vibrational excitation are investigated. Ground state results predict significant deviations from the BO treatment that is based on a potential energy surface (PES) obtained with the same QMC accuracy. The quantum mechanical distribution of molecular axis direction and bond length at a sub-nanom…
