Two-mode thermal-noise squeezing in an electromechanical resonator

Optical coupling control of isolated mechanical resonators

We present a Hamiltonian model describing two pairs of mechanical and optical modes under standard optomechanical interaction. The vibrational modes are mechanically isolated from each other and the optical modes couple evanescently. We recover the ranges for variables of interest, such as mechanical and optical resonant frequencies and naked coupling strengths, using a finite element model for a standard experimental realization. We show that the quantum model, under this parameter range and external optical driving, may be approximated into parametric interaction models for all involved modes. As an example, we study the effect of detuning in the optical resonant frequencies modes and optical driving resolved to mechanical sidebands and show an optical beam splitter with interaction strength dressed by the mechanical excitation number, a mechanical bidirectional coupler, and a two-mode mechanical squeezer where the optical state mediates the interaction strength between the mechanical modes.

Non-Hermitian chiral phononics through optomechanically induced squeezing

Imposing chirality on a physical system engenders unconventional energy flow and responses, such as the Aharonov-Bohm effect¹ and the topological quantum Hall phase for electrons in a symmetry-breaking magnetic field. Recently, great interest has arisen in combining that principle with broken Hermiticity to explore novel topological phases and applications^2-16. Here we report phononic states with unique symmetries and dynamics that are formed when combining the controlled breaking of time-reversal symmetry with non-Hermitian dynamics. Both of these are induced through time-modulated radiation pressure forces in small nano-optomechanical networks. We observe chiral energy flow among mechanical resonators in a synthetic dimension and Aharonov-Bohm tuning of their eigenmodes. Introducing particle-non-conserving squeezing interactions, we observe a non-Hermitian Aharonov-Bohm effect in ring-shaped networks in which mechanical quasiparticles experience parametric gain. The resulting complex mode spectra indicate flux-tuning of squeezing, exceptional points, instabilities and unidirectional phononic amplification. This rich phenomenology points the way to exploring new non-Hermitian topological bosonic phases and applications in sensing and transport that exploit spatiotemporal symmetry breaking.

Unconventional phonon blockade via atom-photon-phonon interaction in hybrid optomechanical systems

Phonon nonlinearities play an important role in hybrid quantum networks and on-chip quantum devices. We investigate the phonon statistics of a mechanical oscillator in hybrid systems composed of an atom and one or two standard optomechanical cavities. An efficiently enhanced atom-phonon interaction can be derived via a tripartite atom-photon-phonon interaction, where the atom-photon coupling depends on the mechanical displacement without practically changing a cavity frequency. This novel mechanism of optomechanical interactions, as predicted recently by Cotrufo et al. [Phys. Rev. Lett.118, 133603 (2017)10.1103/PhysRevLett.118.133603], is fundamentally different from standard ones. In the enhanced atom-phonon coupling, the strong phonon nonlinearity at a single-excitation level is obtained in the originally weak-coupling regime, which leads to the appearance of phonon blockade. Moreover, the optimal parameter regimes are presented both for the cases of one and two cavities. We compared phonon-number correlation functions of different orders for mechanical steady states generated in the one-cavity hybrid system, revealing the occurrence of phonon-induced tunneling and different types of phonon blockade. Our approach offers an alternative method to generate and control a single phonon in the quantum regime and could have potential applications in single-phonon quantum technologies.

Experimental Demonstration of Dual-Band Nano-Electromechanical Valley-Hall Topological Metamaterials

Suppression of undesired backscattering of very-high-frequency elastic signals has been considered as a grand challenge in integrated phononic circuits. Originating from condensed-matter physics, valley-Hall topological insulators provide an intriguing strategy to overcome this challenge. To date, phononic valley-Hall topological insulators have been demonstrated only in bulk acoustic and mechanical systems operating at relatively low frequencies. Here, an integrated nano-electromechanical valley-Hall topological insulator operating in the very-high-frequency regime is experimentally realized. Valley kink states that are backscattering-immune against sharp bends and exhibit the "valley-momentum locking" effect simultaneously in the fundamental (≈60 MHz) and second-order (≈120 MHz) frequency bands are demonstrated. It is further shown that the propagation directions of these dual-band valley kink states are always locked to their valley pseudospins. The results not only enable various applications in very-high-frequency integrated phononic circuits with enhanced robustness and capacity, but also open the door to experimental exploration of mechanical nonlinearities, particularly those involving the fundamental and second-order frequencies, in topologically nontrivial nanostructures.

Stress-Controlled Frequency Tuning and Parametric Amplification of the Vibrations of Coupled Nanomembranes

Noninvasive tuning of the mechanical resonance frequencies of suspended parallel nanomembranes in various monolithic arrays is achieved by piezoelectric control of their tensile stress. Parametric amplification of their thermal fluctuations is shown to be enhanced by the piezoelectric actuation and amplification factors of up to 20 dB in the sub-parametric oscillation threshold regime are observed.

Dynamic modulation of modal coupling in microelectromechanical gyroscopic ring resonators

Understanding and controlling modal coupling in micro/nanomechanical devices is integral to the design of high-accuracy timing references and inertial sensors. However, insight into specific physical mechanisms underlying modal coupling, and the ability to tune such interactions is limited. Here, we demonstrate that tuneable mode coupling can be achieved in capacitive microelectromechanical devices with dynamic electrostatic fields enabling strong coupling between otherwise uncoupled modes. A vacuum-sealed microelectromechanical silicon ring resonator is employed in this work, with relevance to the gyroscopic lateral modes of vibration. It is shown that a parametric pumping scheme can be implemented through capacitive electrodes surrounding the device that allows for the mode coupling strength to be dynamically tuned, as well as allowing greater flexibility in the control of the coupling stiffness. Electrostatic pump based sideband coupling is demonstrated, and compared to conventional strain-mediated sideband operations. Electrostatic coupling is shown to be very efficient, enabling strong, tunable dynamical coupling.

Nondegenerate Parametric Resonance in Large Ensembles of Coupled Micromechanical Cantilevers with Varying Natural Frequencies

We investigate the collective dynamics and nondegenerate parametric resonance (NPR) of coplanar, interdigitated arrays of microcantilevers distinguished by their cantilevers having linearly expanding lengths and thus varying natural frequencies. Within a certain excitation frequency range, the resonators begin oscillating via NPR across the entire array consisting of 200 single-crystal silicon cantilevers. Tunable coupling generated from fringing electrostatic fields provides a mechanism to vary the scope of the NPR. Our experimental results are supported by a reduced-order model that reproduces the leading features of our data including the NPR band. The potential for tailoring the coupled response of suspended mechanical structures using NPR presents new possibilities in mass, force, and energy sensing applications, energy harvesting devices, and optomechanical systems.

Generation of mechanical squeezing and entanglement via mechanical modulations

We discuss the generation of strong stationary mechanical squeezing and entanglement in the modulated two-and three-mode optomechanics. Following the reservoir engineering scheme, the beam-splitter and parametric optomechanical interactions can be simultaneously achieved through appropriately choosing the modulation frequency on mechanical motion, which is essential to strong squeezing and entanglement. In the two-mode modulated optomechanics, squeezing is tunable by the relative ratio of parametric and beam-splitter couplings, and also robust to thermal noise due to the simultaneously optically induced cooling process. In the three-mode modulated optomechanics, strong EPR-type entanglement is also attainable, which can surpass the 3dB limit of nondegenerate parametric interaction. However, the ideal entanglement is impossible since only one of mechanical Bogoliubov modes is cooled by the cavity mode, which also makes the entanglement fragile to the mechanical noise.

Electrical tuning of elastic wave propagation in nanomechanical lattices at MHz frequencies

Nanoelectromechanical systems (NEMS) that operate in the megahertz (MHz) regime allow energy transducibility between different physical domains. For example, they convert optical or electrical signals into mechanical motions and vice versa¹. This coupling of different physical quantities leads to frequency-tunable NEMS resonators via electromechanical non-linearities^2-4. NEMS platforms with single- or low-degrees of freedom have been employed to demonstrate quantum-like effects, such as mode cooling⁵, mechanically induced transparency⁵, Rabi oscillation^6,7, two-mode squeezing⁸ and phonon lasing⁹. Periodic arrays of NEMS resonators with architected unit cells enable fundamental studies of lattice-based solid-state phenomena, such as bandgaps^10,11, energy transport^10-12, non-linear dynamics and localization^13,14, and topological properties¹⁵, directly transferrable to on-chip devices. Here we describe one-dimensional, non-linear, nanoelectromechanical lattices (NEML) with active control of the frequency band dispersion in the radio-frequency domain (10-30 MHz). The design of our systems is inspired by NEMS-based phonon waveguides^10,11 and includes the voltage-induced frequency tuning of the individual resonators^2-4. Our NEMLs consist of a periodic arrangement of mechanically coupled, free-standing nanomembranes with circular clamped boundaries. This design forms a flexural phononic crystal with a well-defined bandgap, 1.8 MHz wide. The application of a d.c. gate voltage creates voltage-dependent on-site potentials, which can significantly shift the frequency bands of the device. Additionally, a dynamic modulation of the voltage triggers non-linear effects, which induce the formation of a phononic bandgap in the acoustic branch, analogous to Peierls transition in condensed matter¹⁶. The gating approach employed here makes the devices more compact than recently proposed systems, whose tunability mostly relies on materials' compliance^17,18 and mechanical non-linearities^19-22.

Quantum Simulation with a Trilinear Hamiltonian

Interaction among harmonic oscillators described by a trilinear Hamiltonian ℏξ(a^{†}bc+ab^{†}c^{†}) is one of the most fundamental models in quantum optics. By employing the anharmonicity of the Coulomb potential in a linear trapped three-ion crystal, we experimentally implement it among three normal modes of motion in the strong-coupling regime, where the coupling strength is much larger than the decoherence rate of the ion motion. We use it to simulate the interaction of an atom and light as described by the Tavis-Cummings model and the process of nondegenerate parametric down-conversion in the regime of a depleted pump.

Nonlinear couplings and energy transfers in micro- and nano-mechanical resonators: intermodal coupling, internal resonance and synchronization

Extensive development of micro/nano-electromechanical systems (MEMS/NEMS) has resulted in technologies that exhibit excellent performance over a wide range of applications in both applied (e.g. sensing, imaging, timing and signal processing) and fundamental sciences (e.g. quantum-level problems). Many of these outstanding applications benefit from resonance phenomena by employing micro/nanoscale mechanical resonators often fabricated into a beam-, membrane- or plate-type structure. During the early development stage, one of the vibrational modes (typically the fundamental mode) of a resonator is considered in the design and application. In the past decade, however, there has been a growing interest in using more than one vibrational mode for the enhanced functionality of MEMS/NEMS. In this paper, we review recent research efforts to investigate the nonlinear coupling and energy transfers between multiple modes in micro/nano-mechanical resonators, focusing especially on intermodal coupling, internal resonance and synchronization.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Sensitivity of the mixing-current technique for the detection of mechanical motion in the coherent tunnelling regime

In a recent publication we have studied theoretically the sensitivity of the mixing-current technique to detect nanomechanical motion by coupling the oscillator to a single-electron transistor in the incoherent tunnelling regime: [Formula: see text], where Γ is the tunnelling rate, T is the electronic temperature, [Formula: see text] and k _B are the Planck and the Boltzmann constant, respectively. In this work we consider the same problem when the detection device is a quantum dot in the coherent tunnelling regime ([Formula: see text]). In order to reach the best sensitivity we find that one should enter the strong coupling regime, as described in the recent publication (Micchi et al 2015 Phys. Rev. Lett. 115 206802) where a mechanical bistability is described. In this regime the electronic detection device strongly modifies the effective potential of the oscillator and the non-linearities determine the form of the displacement fluctuation spectrum. We find theoretical upper bounds to the sensitivity for the detection of the oscillation amplitude of the oscillator. It turns out that it is convenient to work as close as possible to the bistability.

Parametric Oscillation, Frequency Mixing, and Injection Locking of Strongly Coupled Nanomechanical Resonator Modes

We study locking phenomena of two strongly coupled, high quality factor nanomechanical resonator modes to a common parametric drive at a single drive frequency in different parametric driving regimes. By controlled dielectric gradient forces we tune the resonance frequencies of the flexural in-plane and out-of-plane oscillation of the high stress silicon nitride string through their mutual avoided crossing. For the case of the strong common parametric drive signal-idler generation via nondegenerate parametric two-mode oscillation is observed. Broadband frequency tuning of the very narrow linewidth signal and idler resonances is demonstrated. When the resonance frequencies of the signal and idler get closer to each other, partial injection locking, injection pulling, and complete injection locking to half of the drive frequency occurs depending on the pump strength. Furthermore, satellite resonances, symmetrically offset from the signal and idler by their beat note, are observed, which can be attributed to degenerate four-wave mixing in the highly nonlinear mechanical oscillations.

Optically levitated nanoparticle as a model system for stochastic bistable dynamics

Nano-mechanical resonators have gained an increasing importance in nanotechnology owing to their contributions to both fundamental and applied science. Yet, their small dimensions and mass raises some challenges as their dynamics gets dominated by nonlinearities that degrade their performance, for instance in sensing applications. Here, we report on the precise control of the nonlinear and stochastic bistable dynamics of a levitated nanoparticle in high vacuum. We demonstrate how it can lead to efficient signal amplification schemes, including stochastic resonance. This work contributes to showing the use of levitated nanoparticles as a model system for stochastic bistable dynamics, with applications to a wide variety of fields.

Stable Optical Trap from a Single Optical Field Utilizing Birefringence

We report a stable double optical spring effect in an optical cavity pumped with a single optical field that arises as a result of birefringence. One end of the cavity is formed by a multilayer Al_{0.92}Ga_{0.08}As/GaAs stack supported by a microfabricated cantilever with a natural mode frequency of 274 Hz. The optical spring shifts the resonance to 21 kHz, corresponding to a suppression of low frequency vibrations by a factor of about 5 000. The stable nature of the optical trap allows the cavity to be operated without any external feedback and with only a single optical field incident.

Quantum Backaction Evading Measurement of Collective Mechanical Modes

The standard quantum limit constrains the precision of an oscillator position measurement. It arises from a balance between the imprecision and the quantum backaction of the measurement. However, a measurement of only a single quadrature of the oscillator can evade the backaction and be made with arbitrary precision. Here we demonstrate quantum backaction evading measurements of a collective quadrature of two mechanical oscillators, both coupled to a common microwave cavity. The work allows for quantum state tomography of two mechanical oscillators, and provides a foundation for macroscopic mechanical entanglement and force sensing beyond conventional quantum limits.

Dynamical strong coupling and parametric amplification of mechanical modes of graphene drums

Mechanical resonators are ubiquitous in modern information technology. With the possibility of coupling them to electromagnetic and plasmonic modes, they hold promise as the key building blocks in future quantum information technology. Graphene-based resonators are of interest for technological applications due to their high resonant frequencies, multiple mechanical modes and low mass. The tension-mediated nonlinear coupling between various modes of the resonator can be excited in a controllable manner. Here we engineer a graphene resonator with large frequency tunability at low temperatures, resulting in a large intermodal coupling strength. We observe the emergence of new eigenmodes and amplification of the coupled modes using red and blue parametric excitation, respectively. We demonstrate that the dynamical intermodal coupling is tunable. A cooperativity of 60 between two resonant modes of ∼100 MHz is achieved in the strong coupling regime. The ability to dynamically control the coupling between the high-frequency eigenmodes of a mechanical system opens up the possibility of quantum mechanical experiments at low temperatures.

Correlated anomalous phase diffusion of coupled phononic modes in a sideband-driven resonator

The dynamical backaction from a periodically driven optical cavity can reduce the damping of a mechanical resonator, leading to parametric instability accompanied by self-sustained oscillations. Here we study experimentally and theoretically new aspects of the backaction and the discrete time-translation symmetry of a driven system using a micromechanical resonator with two nonlinearly coupled vibrational modes with strongly differing frequencies and decay rates. We find self-sustained oscillations in both the low- and high-frequency modes. Their frequencies and amplitudes are determined by the nonlinearity, which also leads to bistability and hysteresis. The phase fluctuations of the two modes show near-perfect anti-correlation, a consequence of the discrete time-translation symmetry. Concurrently, the phase of each mode undergoes anomalous diffusion. The phase variance follows a power law time dependence, with an exponent determined by the 1/f-type resonator frequency noise. Our findings enable compensating for the fluctuations using a feedback scheme to achieve stable frequency downconversion.

Dynamical backaction from a periodically driven cavity can reduce the damping of a mechanical resonator causing parametric instability. Here, the authors observe simultaneous self-sustained oscillations in both the mechanical and cavity modes and their correlated phase diffusion.

An electromechanical Ising Hamiltonian

Solving intractable mathematical problems in simulators composed of atoms, ions, photons, or electrons has recently emerged as a subject of intense interest. We extend this concept to phonons that are localized in spectrally pure resonances in an electromechanical system that enables their interactions to be exquisitely fashioned via electrical means. We harness this platform to emulate the Ising Hamiltonian whose spin 1/2 particles are replicated by the phase bistable vibrations from the parametric resonances of multiple modes. The coupling between the mechanical spins is created by generating two-mode squeezed states, which impart correlations between modes that can imitate a random, ferromagnetic state or an antiferromagnetic state on demand. These results suggest that an electromechanical simulator could be built for the Ising Hamiltonian in a nontrivial configuration, namely, for a large number of spins with multiple degrees of coupling.

Dynamical Two-Mode Squeezing of Thermal Fluctuations in a Cavity Optomechanical System

We report the experimental observation of two-mode squeezing in the oscillation quadratures of a thermal micro-oscillator. This effect is obtained by parametric modulation of the optical spring in a cavity optomechanical system. In addition to stationary variance measurements, we describe the dynamic behavior in the regime of pulsed parametric excitation, showing an enhanced squeezing effect surpassing the stationary 3 dB limit. While the present experiment is in the classical regime, our technique can be exploited to produce entangled, macroscopic quantum optomechanical modes.

Squeezing of Quantum Noise of Motion in a Micromechanical Resonator

A pair of conjugate observables, such as the quadrature amplitudes of harmonic motion, have fundamental fluctuations that are bound by the Heisenberg uncertainty relation. However, in a squeezed quantum state, fluctuations of a quantity can be reduced below the standard quantum limit, at the cost of increased fluctuations of the conjugate variable. Here we prepare a nearly macroscopic moving body, realized as a micromechanical resonator, in a squeezed quantum state. We obtain squeezing of one quadrature amplitude 1.1±0.4  dB below the standard quantum limit, thus achieving a long-standing goal of obtaining motional squeezing in a macroscopic object.

Phonon Cooling by an Optomechanical Heat Pump

We propose and analyze theoretically a cavity optomechanical analog of a heat pump that uses a polariton fluid to cool mechanical modes coupled to a single precooled phonon mode via external modulation of the substrate of the mechanical resonator. This approach permits us to cool phonon modes of arbitrary frequencies not limited by the cavity-optical field detuning deep into the quantum regime from room temperature.

Thermomechanical Two-Mode Squeezing in an Ultrahigh-Q Membrane Resonator

We realize a quantum-compatible multimode interaction in an ultrahigh Q mechanical resonator via a reservoir-mediated parametric coupling. We use this interaction to demonstrate nondegenerate parametric amplification and thermomechanical noise squeezing, finding excellent agreement with a theoretical model of this interaction over a large dynamic range. This realization of strong multimode nonlinearities in a mechanical platform compatible with quantum-limited optical detection and cooling makes this a powerful system for nonlinear approaches to quantum metrology, transduction between optical and phononic fields, and the quantum manipulation of phononic degrees of freedom.