We investigate nonequilibrium entanglement generation in a chain of harmonic oscillators with time-dependent linear coupling. We use optimal control theory to determine the coupling modulation that leads to maximum logarithmic negativity for a pair of opposite oscillators and show that it corresponds to a synchronization of the eigenmodes of the chain. We further analytically relate the maximum attainable entanglement to the irreversible work done to produce it, thus bridging nonequilibrium entanglement production and nonequilibrium thermodynamics.
We investigate the coupled-system dynamics of two-level quantum dots placed on a vibrating nanomechanical resonator. The ensemble of quantum dots exhibits superradiance features which are transferred to the mechanical degrees of freedom representing fast quantum dynamics and enhanced phonon emission in a nanomechanical setup, resembling of the superradiance effect.
A scheme is presented for entangling two separated nanomechanical oscillators by injecting broad band squeezed vacuum light and laser light into the ring cavity. We work in the resolved sideband regime. We find that in order to obtain the maximum entanglement of the two oscillators, the squeezing parameter of the input light should be about 1. We report significant entanglement over a very wide range of power levels of the pump and temperatures of the environment.
We show how to prepare and directly measure the squeezed states of nanomechanical oscillators. An intense pulse interacts with a dielectric mirror in a cavity. The quadratic coupling between the optical pulse and the oscillator results in the reduction of a quadrature variance of the massive oscillator. Differently from others, the proposed scheme here requires neither any post-action on the optical field nor a preparation of a nonclassical optical field. Depending on the initial temperature of the system, the squeezing of the variance under the vacuum limit can be achieved only by a few repet…
We propose a scheme to measure the quantum state of a nanomechanical oscillator cooled near its ground state of vibrational motion. This is an extension of the nonlinear atomic homodyning technique scheme first developed to measure the intracavity field in a micromaser. It involves the use of a detector-atom that is simultaneously coupled to the cantilever via a magnetic interaction and to (classical) optical fields via a Raman transition. We show that the probability for the atom to be found in the excited state is a direct measure of the Wigner characteristic function of the nanomechanical o…
A patterned Si nanobeam is formed which supports co-localized acoustic and optical resonances that are coupled via radiation pressure. Starting from a bath temperature of T=20K, the 3.68GHz nanomechanical mode is cooled into its quantum mechanical ground state utilizing optical radiation pressure. The mechanical mode displacement fluctuations, imprinted on the transmitted cooling laser beam, indicate that a final phonon mode occupancy of 0.85 +-0.04 is obtained.
We propose a scheme via three-level cascade atoms to entangle two optomechanical oscillators as well as two-mode fields. We show that two movable mirrors and two-mode fields can be entangled even for bad cavity limit. We also study entanglement of the output two-mode fields in frequency domain. The results show that the frequency of the mirror oscillation and the injected atomic coherence affect the output entanglement of the two-mode fields.
We show that squeezing of a nanomechanical mirror can be generated by injecting broad band squeezed vacuum light and laser light into the cavity. We work in the resolved sideband regime. We find that in order to obtain the maximum momentum squeezing of the movable mirror, the squeezing parameter of the input light should be about 1. We can obtain more than 70% squeezing. Besides, for a fixed squeezing parameter, decreasing the temperature of the environment or increasing the laser power increases the momentum squeezing. We find very large squeezing with respect to thermal fluctuations, for ins…
We study how an optical parametric amplifier inside the cavity can affect the normal mode splitting behavior of the coupled movable mirror and the cavity field. We work in the resolved sideband regime. The spectra exhibit a double-peak structure as the parametric gain is increased. Moreover, for a fixed parametric gain, the double-peak structure of the spectrum is more pronounced with increasing the input laser power. We give results for mode splitting. The widths of the split lines are sensitive to parametric gain.
In a recent letter, Gaidarzhy et al. claim to have observed evidence for quantized displacements of a nanomechanical oscillator. We contend that the evidence, analysis, claims, and conclusions presented are contrary to expectations from fundamentals of quantum mechanics and elasticity theory, and that the method used by the authors is unsuitable in principle to observe the quantized energy states of a nanomechanical structure.
Typical of modern quantum technologies employing nanomechanical oscillators is to demand few mechanical quantum excitations, for instance, to prolong coherence times of a particular task or, to engineer a specific non-classical state. For this reason, we devoted the present work to exhibit how to bring an initial thermalized nanomechanical oscillator near to its ground state. Particularly, we focus on extending the novel results of D. D. B. Rao \textit{et al.}, Phys. Rev. Lett. \textbf{117}, 077203 (2016), where a mechanical object can be heated up, squeezed, or cooled down near to its ground …
Cavity-enhanced radiation pressure coupling between optical and mechanical degrees of freedom allows quantum-limited position measurements and gives rise to dynamical backaction enabling amplification and cooling of mechanical motion. Here we demonstrate purely dispersive coupling of high Q nanomechanical oscillators to an ultra-high finesse optical microresonator via its evanescent field, extending cavity optomechanics to nanomechanical oscillators. Dynamical backaction mediated by the optical dipole force is observed, leading to laser-like coherent nanomechanical oscillations solely due to r…
We propose a scheme to prepare nanomechanical oscillators in nonclassical steady states, characterized by a pronounced negative Wigner function. In our optomechanical approach, the mechanical oscillator couples to multiple laser driven resonances of an optical cavity. By lowering the resonance frequency of the oscillator via an inhomogeneous electrostatic field, we significantly enhance its intrinsic geometric nonlinearity per phonon. This causes the motional sidebands to split into separate spectral lines for each phonon number and transitions between individual phonon Fock states can be sele…
The coherent states that describe the classical motion of a mechanical oscillator do not have well-defined energy, but are rather quantum superpositions of equally-spaced energy eigenstates. Revealing this quantized structure is only possible with an apparatus that measures the mechanical energy with a precision greater than the energy of a single phonon, $\hbarω_\text{m}$. One way to achieve this sensitivity is by engineering a strong but nonresonant interaction between the oscillator and an atom. In a system with sufficient quantum coherence, this interaction allows one to distinguish differ…
We study the quantum properties of a nanomechanical oscillator via the squeezing of the oscillator amplitude. The static longitudinal compressive force $F_0$ close to a critical value at the Euler buckling instability leads to an anharmonic term in the Hamiltonian and thus the squeezing properties of the nanomechanical oscillator are to be obtained from the Hamiltonian of the form $H= a^†a+β(a^†+a)^4/4$. This Hamiltonian has no exact solution unlike the other known models of nonlinear interactions of the forms $a^{†2}a^2$, $(a^†a)^2$ and $a^{\dag4}+a^4-(a^{\dag2}a^2+a^2a^{\dag2})$ previously e…
A one-dimensional topological superconductor features a single fermionic zero mode that is delocalized over two Majorana bound states located at the ends of the system. We study a pair of spatially separated nanomechanical oscillators tunnel-coupled to these Majorana modes. Most interestingly, we demonstrate that the combination of electron-phonon coupling and a finite charging energy on the mesoscopic topological superconductor can lead to an effective superexchange between the oscillators via the non-local fermionic zero mode. We further show that this teleportation mechanism leads to entang…
We measure the response and thermal motion of a high-Q nanomechanical oscillator coupled to a superconducting microwave cavity in the resolved-sideband regime where the oscillator's resonance frequency exceeds the cavity's linewidth. The coupling between the microwave field and mechanical motion is strong enough for radiation pressure to overwhelm the intrinsic mechanical damping. This radiation-pressure damping cools the fundamental mechanical mode by a factor of 5 below the thermal equilibrium temperature in a dilution refrigerator to a phonon occupancy of 140 quanta.
A single Nitrogen Vacancy (NV) center hosted in a diamond nanocrystal is positioned at the extremity of a SiC nanowire. This novel hybrid system couples the degrees of freedom of two radically different systems, i.e. a nanomechanical oscillator and a single quantum object. The dynamics of the nano-resonator is probed through time resolved nanocrystal fluorescence and photon correlation measurements, conveying the influence of a mechanical degree of freedom given to a non-classical photon emitter. Moreover, by immersing the system in a strong magnetic field gradient, we induce a magnetic coupli…
We report the observation of discrete displacement of nanomechanical oscillators with gigahertz-range resonance frequencies at millikelvin temperatures. The oscillators are nanomachined single-crystal structures of silicon, designed to provide two distinct sets of coupled elements with very low and very high frequencies. With this novel design, femtometer-level displacement of the frequency-determining element is amplified into collective motion of the entire micron-sized structure. The observed discrete response possibly results from energy quantization at the onset of the quantum regime in t…
The question of how swiftly entanglement spreads over a system has attracted vital interest. In this regard, the out-of-time ordered correlator (OTOC) is a quantitative measure of the entanglement spreading process. Particular interest concerns the propagation of quantum correlations in the lattice systems, {\it e.g.}, spin chains. In a seminal paper D. A. Roberts, D. Stanford and L. Susskind, J. High Energy Phys. 03, 051, (2015) the concept of the OTOC's radius was introduced. The radius of the OTOC defines the front line reached by the spread of entanglement. Beyond this radius operators com…
The observation of quantized nanomechanical oscillations by detecting femtometer-scale displacements is a significant challenge for experimentalists. We propose that phonon blockade can serve as a signature of quantum behavior in nanomechanical resonators. In analogy to photon blockade and Coulomb blockade for electrons, the main idea for phonon blockade is that the second phonon cannot be excited when there is one phonon in the nonlinear oscillator. To realize phonon blockade, a superconducting quantum two-level system is coupled to the nanomechanical resonator and is used to induce the phono…
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 study the consequence of the frequency errors of individual oscillators on the scalability of quantum computing based on nanomechanical resonators. We show the fidelity change of the quantum operation due to the frequency shifts numerically. We present a method to perfectly compensate for these negative effects. Our method is robust to whatever large frequency errors.
We present a theory describing the semiclassical dynamics of a superconducting flux qubit inductively coupled to a nanomechanical oscillator. Focusing on the influence of the qubit on the mechanical element, and on the nonlinear phenomena displayed by this device, we show that it exhibits retardation effects and self-excited oscillations. These can be harnessed for the generation of non-classical states of the mechanical oscillator. In addition, we find that this system shares several fundamental properties with cavity optomechanical systems, and elucidate the analogy between these two classes…
Optomechanical cavities in the well-resolved-sideband regime are ideally suited for the study of a myriad of quantum phenomena with mechanical systems, including backaction-evading measurements, mechanical squeezing, and generation of non-classical states. For these experiments, the mechanical oscillator should be prepared in its ground state; residual motion beyond the zero-point motion must be negligible. The requisite cooling of the mechanical motion can be achieved using the radiation pressure of light in the cavity by selectively driving the anti-Stokes optomechanical transition. To date,…
We investigate the coupling of a nanomechanical oscillator in the quantum regime with molecular (electric) dipoles. We find theoretically that the cantilever can produce single-mode squeezing of the center-of-mass motion of an isolated trapped molecule and two-mode squeezing of the phonons of an array of molecules. This work opens up the possibility of manipulating dipolar crystals, which have been recently proposed as quantum memory, and more generally, is indicative of the promise of nanoscale cantilevers for the quantum detection and control of atomic and molecular systems.
We present a ground state cooling scheme for a nanomechanical resonator linearly coupled with a LC oscillator. The linear coupling, when periodically modulated at red detuning, up-converts the low-frequency nanomechanical mode to the high-frequency LC oscillator mode and generates backaction force that can cool the resonator to its ground state in the resolved-sideband regime. We also study the effect of the quantum backaction noise on the cooling due to the counter rotating term in the linear coupling. The scheme can be compared with laser cooling for the atomic systems and can be realized in…
We investigate theoretically the conditions under which a multi-mode nanomechanical resonator, operated as a purely mechanical parametric oscillator, can be driven into highly nonclassical states. We find that when the device can be cooled to near its ground state, and certain mode matching conditions are satisfied, it is possible to prepare distinct resonator modes in quantum entangled states that violate Bell inequalities with homodyne quadrature measurements. We analyze the parameter regimes for such Bell inequality violations, and while experimentally challenging, we believe that the reali…
We consider the thermodynamic properties of the squeezed vacuum state of a frequency--modulated quantum harmonic oscillator. We analytically relate the squeezing parameter to the irreversible work and the degree of nonadiabaticity of the frequency transformation. We furthermore determine the optimal modulation that leads to maximal squeezing, and discuss its implementation as well as the detection of squeezing in single cold ion traps.
Nano- and micromechanical oscillators with high quality (Q) factors have gained much attention for their potential application as ultrasensitive detectors. In contrast to micro-fabricated devices, optically trapped nanoparticles in vacuum do not suffer from clamping losses, hence leading to much larger Q-factors. We find that for a levitated nanoparticle the thermal energy suffices to drive the motion of the nanoparticle into the nonlinear regime. First, we experimentally measure and fully characterize the frequency fluctuations originating from thermal motion and nonlinearities. Second, we de…
An experimental demonstration of a non-classical state of a nanomechanical resonator is still an outstanding task. In this paper we show how the resonator can be cooled and driven into a squeezed state by a bichromatic microwave coupling to a charge qubit. The stationary oscillator state exhibits a reduced noise in one of the quadrature components by a factor of 0.5 - 0.2. These values are obtained for a 100 MHz resonator with a Q-value of 10$^4$ to 10$^5$ and for support temperatures of T $\approx$ 25 mK. We show that the coupling to the charge qubit can also be used to detect the squeezed st…
We use a Born-Markov approximated master equation approach to study the symmetrized-in-frequency current noise spectrum and the oscillator steady state of a nanoelectromechanical system where a nanoscale resonator is coupled linearly via its momentum to a quantum point contact (QPC). Our current noise spectra exhibit clear signatures of the quantum correlations between the QPC current and the back-action force on the oscillator at a value of the relative tunneling phase (η= -π/2) where such correlations are expected to be maximized. We also show that the steady state of the oscillator obeys a …
We show how the dynamics of the Dicke model after a quench from the ground-state configuration of the normal phase into the superradiant phase can be described for a limited time by a simple inverted harmonic oscillator model and that this limited time approaches infinity in the thermodynamic limit. Although we specifically discuss the Dicke model, the presented mechanism can be also used to describe dynamical quantum phase transitions in other systems and opens a new avenue in simulations of physical phenomena associated with an inverted harmonic oscillator.
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…
We present optical sideband spectroscopy measurements of a mesoscopic mechanical oscillator cooled near its quantum ground state. The mechanical oscillator, corresponding to a 3.99GHz acoustic mode of a patterned silicon nanobeam, is coupled via radiation pressure to a pair of co-localized 200THz optical modes. The mechanical mode is cooled close to its quantum ground state from a bath temperature of 18K using radiation pressure back-action stemming from the optical pumping of one of the optical cavity resonances. An optical probe beam, resonant with the second optical cavity resonance, is use…
This study is focused on the quantum dynamics of a nitrogen-vacancy (NV) center coupled to a nonlinear, periodically driven mechanical oscillator. For a continuous periodic driving that depends on the position of the oscillator, the mechanical motion is described by Mathieu elliptic functions. This solution is employed to study the dynamics of the quantum spin system including environmental effects and to evaluate the purity and the von Neumann entropy of the NV-spin. The unitary generation of coherence is addressed. We observe that the production of coherence through a unitary transformation …
We considered a nanoelectromechanical system consisting of a movable Cooper-pair box qubit which is subject to an electrostatic field, and coupled to the two bulk superconductors via tunneling processes. We suggest that qubit dynamics is related to the one of a quantum oscillator and demonstrate that a bias voltage applied between superconductors generates states represented by the entanglement of qubit states and coherent states of the oscillator if certain resonant conditions are fulfilled. It is shown that a structure of this entanglement may be controlled by the bias voltage in a way that …
Mechanical oscillators have been demonstrated with very high quality factors over a wide range of frequencies. These also couple to a wide variety of fields and forces, making them ideal as sensors. The realization of a mechanically-based quantum bit could therefore provide an important new platform for quantum computation and sensing. Here we show that by coupling one of the flexural modes of a suspended carbon nanotube to the charge states of a double quantum dot defined in the nanotube, it is possible to induce sufficient anharmonicity in the mechanical oscillator so that the coupled system…
In real-time quantum feedback protocols, the record of a continuous measurement is used to stabilize a desired quantum state. Recent years have seen highly successful applications in a variety of well-isolated micro-systems, including microwave photons and superconducting qubits. By contrast, the ability to stabilize the quantum state of a tangibly massive object, such as a nanomechanical oscillator, remains a difficult challenge: The main obstacle is environmental decoherence, which places stringent requirements on the timescale in which the state must be measured. Here we describe a position…
In the context of de Broglie-Bohm pilot-wave theory, violations of the Born rule are allowed and can be considered as describing nonequilibrium distributions. We study the effects of interactions on quantum relaxation towards equilibrium for a system of one-dimensional coupled harmonic oscillators. We show by numerical simulations that interactions can delay or even prevent complete relaxation for some initial states. We also discuss how this effect might be relevant for cosmological scenarios and how nonequilibrium could be detected in some models.
We formulate exact generalized nonequilibrium fluctuation relations for the quantum mechanical harmonic oscillator coupled to multiple harmonic baths. Each of the different baths is prepared in its own individual (in general nonthermal) state. Starting from the exact solution for the oscillator dynamics we study fluctuations of the oscillator position as well as of the energy current through the oscillator under general nonequilibrium conditions. In particular, we formulate a fluctuation-dissipation relation for the oscillator position autocorrelation function that generalizes the standard res…
We show that a parametrically coupled qubit can be used to fully reconstruct the quantum state of a harmonic oscillator, even when both systems are subject to decoherence. By controlling the coupling strength of the qubit over time, the characteristic function of the oscillator at any phase space point can be directly measured by combining the expectation values of two Pauli operators. The effect of decoherence can be filtered out from the measured data, provided a sufficient number of experimental runs is performed. In situations where full state reconstruction is not practical or not necessa…
We propose a method to achieve high degree control of nanomechanical oscillators by coupling their mechanical motion to single spins. By manipulating the spin alone and measuring its quantum state heralds the cooling or squeezing of the oscillator even for weak spin-oscillator couplings. We analytically show that the asymptotic behavior of the oscillator is determined by a spin-induced thermal filter function whose overlap with the initial thermal distribution of the oscillator determines its cooling, heating or squeezing. Counterintuitively, the rate of cooling dependence on the instantaneous…
We introduce a functional perturbative method for treating weakly nonlinear systems coupled with a quantum field bath. We demonstrate using this method to obtain the covariance matrix elements and the correlation functions of a quantum anharmonic oscillator interacting with a heat bath. We identify a fluctuation-dissipation relation based on the nonequilibrium dynamics of this nonlinear open quantum system. To establish its connection with dynamical equilibration, we further examine the energy flows between the anharmonic oscillator and the bath field. The vanishing of the net flow {is an indi…
We consider a quantum harmonic oscillator coupled to a general nonequilibrium environment. We show that the decoherence factor can be expressed in terms of a measurable effective temperature, defined via a generalized fluctuation-dissipation relation. We further propose a simple experimental scheme to determine the time-dependent effective temperature in a linear Paul trap with engineered reservoirs. Our formalism allows quantitative description of nonequilibrium decoherence in the presence of an arbitrary number of non-Markovian noise sources in a unified manner.
We extend the quantum theory of dissipation in the context of system-reservoir model, where the reservoir in question is kept in a nonequilibrium condition. Based on a systematic separation of time scales involved in the dynamics, appropriate generalizations of the fluctuation-dissipation and Einstein's relations have been pointed out. We show that the Wigner-Weisskopf decay of the system mode results in a rate constant which depending on the relaxation of nonequilibrium bath is dynamically modified. We also calculate the time-dependent spectra of a cavity mode with a suitable gain when the ca…
Continuing our inquiry into the conditions when fluctuation-dissipation relations (FDR) may appear in the context of nonequilibrium dynamics of open quantum systems (over and beyond the conventional FDR from linear response theory) we turn to nonGaussian systems and consider this issue for an anharmonic oscillator interacting with a scalar quantum field bath. We present the general {nonperturbative} expressions for the rate of energy (power) exchange between the anharmonic oscillator and the thermal bath. For the cases that a stable final equilibrium state exists, and the nonstationary compone…
We propose and analyze a setup to achieve strong coupling between a single trapped atom and a mechanical oscillator. The interaction between the motion of the atom and the mechanical oscillator is mediated by a quantized light field in a laser driven high-finesse cavity. In particular, we show that high fidelity transfer of quantum states between the atom and the mechanical oscillator is in reach for existing or near future experimental parameters. Our setup provides the basic toolbox for coherent manipulation, preparation and measurement of micro- and nanomechanical oscillators via the tools …
Recent experiments aim at cooling nanomechanical resonators to the ground state by coupling them to non-equilibrium environments in order to observe quantum effects such as entanglement. This raises the general question of how such environments affect entanglement. Here we show that there is an optimal dissipation strength for which the entanglement between two coupled oscillators is maximized. Our results are established with the help of a general framework of exact quantum Langevin equations valid for arbitrary bath spectra, in and out of equilibrium. We point out why the commonly employed L…
In this article we discuss and compare different ways to engineer an interface between ultracold atoms and micro- and nanomechanical oscillators. We start by analyzing a direct mechanical coupling of a single atom or ion to a mechanical oscillator and show that the very different masses of the two systems place a limit on the achievable coupling constant in this scheme. We then discuss several promising strategies for enhancing the coupling: collective enhancement by using a large number of atoms in an optical lattice in free space, coupling schemes based on high-finesse optical cavities, and …
This is the second of a series of three papers examining how viable it is for entanglement to be sustained at high temperatures for quantum systems in thermal equilibrium (Case A), in nonequilibrium (Case B) and in nonequilibrium steady state conditions (Case C). The system we analyze here consists of two coupled quantum harmonic oscillators each interacting with its own bath described by a scalar field, set at temperatures $T_1 > T_2$. For \textit{constant bilinear inter-oscillator coupling} studied here (Case C1) owing to the Gaussian nature, the problem can be solved exactly at arbitrary te…
For suitable parameters, the classical Duffing oscillator has a known bistability in its stationary states, with low- and high-amplitude branches. As expected from the analogy with a particle in a double-well potential, transitions between these states become possible either at finite temperature, or in the quantum regime due to tunneling. In this analogy, besides local stability, one can also discuss global stability by comparing the two potential minima. For the Duffing oscillator, the stationary states emerge dynamically so that a priori, a potential-minimum criterion for them does not exis…
The possibility of maintaining entanglement in a quantum system at finite, even high, temperatures -- the so-called `hot entanglement' -- has obvious practical interest, but also requires closer theoretical scrutiny. Since quantum entanglement in a system evolves in time and is continuously subjected to environmental degradation, a nonequilibrium description by way of open quantum systems is called for. To identify the key issues and the contributing factors that may permit `hot entanglement' to exist, or the lack thereof, we carry out a model study of two spatially-separated, coupled oscillat…
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…
