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arXiv:2006.14475 · Physical Review A

Dynamical tunneling of a nanomechanical oscillator

Piyush Jangid, Anil Kumar Chauhan, Sebastian Wüster

Published Oct 14, 2020• 3 citations
arXivHTMLPDF

Abstract

The study of the quantum to classical transition is of fundamental as well as technological importance, and focuses on mesoscopic devices, with a size for which either classical physics or quantum physics can be brought to dominate. A particularly diverse selection of such devices is available in cavity quantum optomechanics. We show that these can be leveraged for the study of dynamical tunneling in a quantum chaotic system. This effect probes the quantum to classical transition deeply, since tunneling rates sensitively depend on the ability of the quantum system to resolve the underlying classical phase space. We show that the effective Planck's constant, which determines this phase space resolution, can be varied over orders of magnitude as a function of tunable parameters in an optomechanical experiment. Specifically, we consider a membrane-in-the-middle configuration of a mechanical oscillator within an optical cavity, where the intracavity field is modulated periodically by the external laser source. We demonstrate that a mixed regular and chaotic phase space can be engineered in one spatial dimension, through a significant quartic optomechanical interaction. For that case, we explore the expected dynamical tunneling rates using Floquet theory and map out values of the effective Planck's constant that should be within practical reach.

DOI
10.1103/physreva.102.043513

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The study of the quantum to classical transition is of fundamental as well as technological importance, and focuses on mesoscopic devices, with a size for which either classical physics or quantum physics can be brought to dominate. A particularly diverse selection of such devices is available in cavity quantum optomechanics. We show that these can be leveraged for the study of dynamical tunneling in a quantum chaotic system. This effect probes the quantum to classical transition deeply, since tunneling rates sensitively depend on the ability of the quantum system to resolve the underlying classical phase space. We show that the effective Planck's constant, which determines this phase space resolution, can be varied over orders of magnitude as a function of tunable parameters in an optomechanical experiment. Specifically, we consider a membrane-in-the-middle configuration of a mechanical oscillator within an optical cavity, where the intracavity field is modulated periodically by the external laser source. We demonstrate that a mixed regular and chaotic phase space can be engineered in one spatial dimension, through a significant quartic optomechanical interaction. For that case, we explore the expected dynamical tunneling rates using Floquet theory and map out values of the effective Planck's constant that should be within practical reach.
The nonlinear dynamics of a mechanical resonator in an optomechanical system with linear, quadratic and cubic photon-vibration interactions (with respect to mechanical displacements) in a modulated driving field under conditions of adiabatic elimination of the optical field is studied. Based on the constructed bifurcation diagrams of the mechanical coordinate and the largest Lyapunov exponent as a function of the modulation amplitude, as well as power spectra, phase portraits and Poincare sections, regions of regular and chaotic dynamics of the optomechanical system are identified. It is also shown that for a certain modulation amplitude in the presence of all three types of interactions, chaotic dynamics of the mechanical resonator (oscillator) is realized, which is replaced by quasi-periodic oscillations in the absence of cubic interaction, and the system returns to chaotic behavior if only linear interaction remains. This non-monotonic dependence of chaotic dynamics on the order of nonlinearity originates from the interplay between parametric driving and effective potential reshaping and manifests that nonlinearity does not always enhance chaos. For an optomechanical system in a membrane-in-the-middle configuration, where only quadratic photon-vibration interaction is present, it is demonstrated that at small modulation amplitudes the mechanical oscillator exhibits quasi-periodic motion in each of the wells of a symmetric two-minimum potential, whereas large modulation amplitudes lead to chaotic motion, involving interwell transitions.
Why are classical theories often sufficient to describe the physics of our world even though everything around us is entirely composed of microscopic quantum systems? The boundary between these two fundamentally dissimilar theories remains an unsolved problem in modern physics. Position measurements of small objects allow us to probe the area where the classical approximation breaks down. In quantum mechanics, Heisenbergâ s uncertainty principle dictates that any measurement of the position must be accompanied by measurement induced back-action---in this case manifested as an uncertainty in the momentum. In recent years, cavity optomechanics has become a powerful tool to perform precise position measurements and investigate their fundamental limitations. The utilization of optical micro-cavities greatly enhances the interaction between light and state-of-the-art nanomechanical oscillators. Therefore, quantum mechanical phenomena have been successfully observed in systems far beyond the microscopic world. In such a cavity optomechanical system, the fluctuations in the position of the oscillator are transduced onto the phase of the light, while fluctuations in the amplitude of the light disturb the momentum of the oscillator during the measurement. As a consequence, correlations are established between the amplitude and phase quadrature of the probe light. However, so far, observation of quantum effects has been limited exclusively to cryogenic experiments, and access to the quantum regime at room temperature has remained an elusive goal because the overwhelming amount of thermal motion masks the weak quantum effects. This thesis describes the engineering of a high-performance cavity optomechanical device and presents experimental results showing, for the first time, the broadband effects of quantum back-action at room temperature. The device strongly couples mechanical and optical modes of exceptionally high quality factors to provide a measurement sensitivity ∼ ⁣104\sim\!10^4∼104 times below the requirement to resolve the zero-point fluctuations of the mechanical oscillator. The quantum back-action is then observed through the correlations created between the probe light and the motion of the nanomechanical oscillator. A so-called â variational measurementâ , which detects the transmitted light in a homodyne detector tuned close to the amplitude quadrature, resolves the quantum noise due to these correlations at the level of 10\% of the thermal noise over more than an octave of Fourier frequencies around mechanical resonance. Moreover, building on this result, an additional experiment demonstrates the ability to achieve quantum enhanced metrology. In this case, the generated quantum correlations are used to cancel quantum noise in the measurement record, which then leads to an improved relative signal-to-noise ratio in measurements of an external force. In conclusion, the successful observation of broadband quantum behavior on a macroscopic object at room temperature is an important milestone in the field of cavity optomechanics. Specifically, this result heralds the rise of optomechanical systems as a platform for quantum physics at room temperature and shows promise for generation of ponderomotive squeezing in room-temperature interferometers.
Hidden structures with well-defined predictability are uncovered in the evolution of a chaotic optomechanical system from the perspective of the εεε-machine. Tuning the frequency of the driving laser can switch off this predictability, and such behaviour corresponds to a phase transition that is deeply related to topological changes in phase space. The transition probabilities between causal states allow us to define an entropy (uncertainty) that serves as an effective order parameter. This phase transition can be readily demonstrated in currently available experiments by monitoring the quadrature of the optical mode. We hope that this work could fundamentally broaden the regimes of cavity micromechanics and nonlinear optics.
Since the early days of quantum mechanics, tunneling has been recognized as one of the hallmarks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has led to various applications in nuclear, atomic, and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape route of a quantum particle from a quasibounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, that is, to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying classical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in periodically modulated optical lattices [2-4]. Moreover, the electromagnetic analog of dynamical tunneling was also obtained with microwaves in billiards [5].
The study of dynamical tunneling in a periodically driven anharmonic potential probes the quantum-classical transition via the experimental control of the effective Planck's constant for the system. In this paper we consider the prospects for observing dynamical tunneling with ultracold atoms in magnetic microtraps on atom chips. We outline the driven anharmonic potentials that are possible using standard magnetic traps and find the Floquet spectrum for one of these as a function of the potential strength, modulation, and effective Planck's constant. We develop an integrable approximation to the nonintegrable Hamiltonian and find that it can explain the behavior of the tunneling rate as a function of the effective Planck's constant in the regular region of parameter space. In the chaotic region we compare our results with the predictions of models that describe chaos-assisted tunneling. Finally, we examine the practicality of performing these experiments in the laboratory with Bose-Einstein condensates.
Cavity optomechanical systems involving an optical parametric amplifier (OPA) can exhibit rich classical and quantum dynamical behaviors. By simply modulating the frequency of the laser pumping the OPA, we find two interesting parameter regimes, with one of them enabling us to study quantum-classical correspondence in system dynamics, while there exist no classical counterparts of the quantum features for the other. For the detuning of the laser pumping around the mechanical frequency, as the parametric gain of OPA increases to a critical value, the classical dynamics of the optical or mechanical modes can experience a transition from the regular periodic oscillation to a period-doubling motion, in which case the light-mechanical entanglement can be well studied by the logarithmic negativity and can manifest the dynamical transition the classical nonlinear dynamics. In addition, the optomechanical entanglement shows a second-order transition characteristic at the critical parametric gain. For the laser detuning being about twice the mechanical frequency, the kind of normal mode splitting comes up in the laser detuning dependence of optomechanical entanglement, which is induced by the squeezing of the optical and mechanical hybrid modes and finds no classical correspondence. The OPA-assisted optomechanical systems therefore offer a simple way to study and exploit quantum manifestations of classical nonlinear dynamics.