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arXiv:1608.03068 · Physical review. B./Physical review. B

Creating arbitrary quantum vibrational states in a carbon nanotube

Heng Wang, Guido Burkard

Published Nov 10, 2016• 3 citations
arXivHTMLPDF

Abstract

Due to their small mass and large mechanical stiffness, free-standing carbon nanotubes turn out to be superb high-frequency mechanical resonators that could find applications in mass and force sensing. With resonance frequencies up to the gigahertz regime, one can envision cooling the mechanical motion of such nanotubes to its quantum mechanical ground state at cryogenic temperatures. This opens the possibility to generate highly nonclassical states of the mechanical motion, similar to those created several years ago with electromagnetic radiation in superconducting microwave resonators. Such states are interesting from a fundamental physics point of view, but potentially also for quantum information and sensing applications. While classical activation of a harmonic oscillator will always produce ``quasiclassical'' coherent states, a necessary ingredient for the production of nonclassical states such as Fock states is a nonlinear element, e.g., superconducting qubits in the case of microwaves. This work predicts that for mechanical oscillations, the spin-orbit coupling of the mechanical motion to a single localized electron spin can do the trick by acting as a nonlinear element for phonons.

DOI
10.1103/physrevb.94.205413

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Due to their small mass and large mechanical stiffness, free-standing carbon nanotubes turn out to be superb high-frequency mechanical resonators that could find applications in mass and force sensing. With resonance frequencies up to the gigahertz regime, one can envision cooling the mechanical motion of such nanotubes to its quantum mechanical ground state at cryogenic temperatures. This opens the possibility to generate highly nonclassical states of the mechanical motion, similar to those created several years ago with electromagnetic radiation in superconducting microwave resonators. Such states are interesting from a fundamental physics point of view, but potentially also for quantum information and sensing applications. While classical activation of a harmonic oscillator will always produce ``quasiclassical'' coherent states, a necessary ingredient for the production of nonclassical states such as Fock states is a nonlinear element, e.g., superconducting qubits in the case of microwaves. This work predicts that for mechanical oscillations, the spin-orbit coupling of the mechanical motion to a single localized electron spin can do the trick by acting as a nonlinear element for phonons.
The quantum behaviour of mechanical resonators is a new and emerging field driven by recent experiments reaching the quantum ground state. The high frequency, small mass, and large quality-factor of carbon nanotube resonators make them attractive for quantum nanomechanical applications. A common element in experiments achieving the resonator ground state is a second quantum system, such as coherent photons or a superconducting device, coupled to the resonators motion. For nanotubes, however, this is a challenge due to their small size. Here, we couple a carbon nanoelectromechanical (NEMS) device to a superconducting circuit. Suspended carbon nanotubes act as both superconducting junctions and moving elements in a Superconducting Quantum Interference Device (SQUID). We observe a strong modulation of the flux through the SQUID from displacements of the nanotube. Incorporating this SQUID into superconducting resonators and qubits should enable the detection and manipulation of nanotube mechanical quantum states at the single-phonon level.
Abstract It is an open question whether mechanical resonators can be made nonlinear with vibrations approaching the quantum ground state. This requires the engineering of a mechanical nonlinearity far beyond what has been realized so far. Here we discover a mechanism to boost the Duffing nonlinearity by coupling the vibrations of a nanotube resonator to single-electron tunnelling and by operating the system in the ultrastrong-coupling regime. We find that thermal vibrations become highly nonlinear when lowering the temperature. The average vibration amplitude at the lowest temperature is 13 times the zero-point motion, with approximately 42% of the thermal energy stored in the anharmonic part of the potential. Our work may enable the realization of mechanical Schrödinger cat states, mechanical qubits and quantum simulators emulating the electron–phonon coupling.
This thesis deals with the implementation and demonstration of an optomechanical system that is composed of a superconducting microwave resonator and a suspended carbon nanotube (CNT). This is of particular interest, as CNTS - being macromolecules - have outstanding electronic and mechanical properties. Single electron effects combined with large mechanical resonance frequencies, quality factors, and zero-point motion amplitudes make possible promising applications in the field of quantum information processing or fundamental research, e.g. the synthesis of arbitrary quantum states. Besides the assembly of a high frequency measurement setup for cryogenic temperatures in the millikelvin range, introductory chapters discuss electronic and mechanical characteristics of carbon nanotubes. Here, two different fabrication techniques are described, the overgrowth and the fork transfer process. The physics of superconducting microwave resonators in coplanar waveguide geometry is introduced. Their high frequency behaviour at low temperatures is described by a transition from Mattis-Bardeen to two-level system theory. This serves as a means to investigate the high frequency material properties of a molybdenum/rhenium alloy, which is particularly suitable for the overgrowth process. Device design adjustments and considerations are described that were necessary for the integration of CNTs into hybrid geometries using the fork transfer process. Through this, the (previously demonstrated) electronic coupling of a carbon nanotube to a microwave resonator could be successfully experimentally demonstrated. There, the transmission of the resonator is sensitive to single electron tunneling processes in the CNT. The central point of this thesis is the introduction to the field of dispersive optomechanics. Here, the interaction of a vibrating carbon nanotube and a microwave field is investigated. A mechanism based on the quantum capacitance enhances this otherwise negligible coupling by orders of magnitude. Through an optomechanically induced transparency experiment, this coupling was experimentally quantified. A comparison to the beforehand deduced theory shows a very good agreement. In conclusion, subsequent prospective experiments towards optomechanical cooling and motion manipulation are explained and preliminary experimental data are discussed.
In the late 1960s and early 1970s, the development of mechanical resonators was pushed forward in efforts to record gravitational waves. In order to push down the limit of the smallest detectable force, the radiation circulating in a Fabry-Perot cavity was demonstrated to modify the damping of the cavity end-mirror suspended to act as a mechanical resonator. A few years later, motivated by the study of the internal properties of atoms and ions, the radiation pressure of light was realized as a means to cool atomic motion. In the following years, the ground-state cooling of ion motion was achieved in landmark studies. For mechanical resonators parametrically coupled to Fabry-Perot cavities, it took much longer to reach quantum ground-state cooling, both in the microwave domain, where the original experiments took place, and in the optical domain. Ground-state cooling in both regimes was reached by three groups during the preparation of this thesis and was largely due to advancements in the fabrication methods. \n \nGround-state cooling allows, in principle, the quantum nature of mechanical resonators to be probed and harnessed. Microwave domain devices, which are focused on in this thesis, have the advantage that nonlinear elements (superconducting quantum bits, qubits) are readily available. These elements enable nonclassical state preparation of the qubit and the subsequent transfer to the microwave cavity. The goal is to transfer these quantum states to mechanical resonators. This would allow mechanics to be used as a quantum mechanical memory in quantum information processing thanks to their long lifetimes. \n \nIn this thesis, various aspects of microwave domain circuit optomechanics are studied. The main goal of this thesis is to study how to enhance the coupling between a quantum regime microwave device and a mechanical resonator by using Josephson junction –based qubits. Coherent interaction between a superconducting qubit and a classical driven mechanical field is demonstrated. The analysis presented in this thesis predicts that ground-state cooling and coherent state transfer is possible in the single-phonon regime with transmon-regime qubits. Also, traditional linear optomechanical systems are considered in which the mechanical resonator is coupled to an electrical resonator. Such systems exhibit a rich spectrum of phenomena from which the quantum-limited amplification and multimode effects are considered in this thesis. It is shown that the system operates near quantum-limited amplification, which could be reached if the mechanical resonator was in the ground state.
The ability to access a broad range of quantum states with mechanical oscillators has many applications and is an enduring ambition in the fields of opto- and electromechanics. However, as mechanical oscillators are linear at the quantum scale, arbitrary quantum control over them requires an extrinsic nonlinearity. In this thesis, I aim to provide this extrinsic nonlinearity and establish quantum control over the motion of a suspended aluminum disk by coupling it strongly to a superconducting qubit. To this end, I design and fabricate a device, when operated at its maximum coupling strength, could enable phonon-number-resolved measurements and arbitrary quantum control over the mechanical motion. However, limited by unclear reasons that break the qubit readout, I operate at approximately a quarter of the maximum coupling strength. Nevertheless, at this smaller coupling strength, I demonstrate the preparation of a non-Gaussian nonclassical state of motion.Because of the large coupling achieved in this work, the sideband transitions are phonon-number-sensitive, selectively altering the phonon populations in only a few Fock states. Using these phonon-number-sensitive sideband transitions, I dissipatively stabilize the mechanical oscillator into a highly-energized sub-Poissonian state, where its energy fluctuations are below the classical limit. This result represents a major step toward the long-time ambition of accessing a broad range of nonclassical states with macroscopic mechanical oscillators. Moreover, requiring neither number-resolution nor coherent control of the qubit, the dissipative stabilization technique demonstrated in this work also provides an accessible path toward preparing nonclassical states in other harmonic oscillators coupled to qubits.