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arXiv:1104.5665 · New Journal of Physics

Steady-state negative Wigner functions of nonlinear nanomechanical oscillators

Simon Rips, Martin Kiffner, Ignacio Wilson-Rae, Michael J. Hartmann

Published Feb 20, 2012• 95 citations
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Abstract

We propose a scheme for preparing 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 selectively addressed. We show that this enables the preparation of the nanomechanical oscillator in a single-phonon Fock state. Our scheme can, for example, be implemented with a carbon nanotube dispersively coupled to the evanescent field of a state of the art whispering gallery mode microcavity.

DOI
10.1088/1367-2630/14/2/023042

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We propose a scheme for preparing 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 selectively addressed. We show that this enables the preparation of the nanomechanical oscillator in a single-phonon Fock state. Our scheme can, for example, be implemented with a carbon nanotube dispersively coupled to the evanescent field of a state of the art whispering gallery mode microcavity.
In this paper we propose a scheme for the generation of a nonclassical non-Gaussian motional state of a mechanical resonator (MR) in the three-mode optomechanical system in which two linearly coupled single-mode cavities interact dispersively with the mechanical oscillator simultaneously. With one cavity driven by a weak laser field and by properly tuning the driving frequency, a desirable phononic Liouvillian superoperator can be obtained by engineering the selective interaction Hamiltonian confined to Fock subspaces. It is shown that the MR can be driven dissipatively into a steady non-Gaussian nonclassical state, which possesses sub-Poisson statistics, although its Wigner function is positive. The present scheme can be useful for obtaining single phonons.
Non-Gaussian quantum states of mechanical motion exhibiting Wigner negativity offer promising capabilities for quantum technologies and tests of fundamental physics. Within the field of cavity quantum optomechanics, it is commonly held that deterministic preparation of mechanical Wigner negativity in the unresolved-sideband regime is not possible, as the intracavity interaction Hamiltonian is linear in mechanical position. Here, we show that, despite this, by accounting for the nonlinearity of the cavity response with mechanical position, mechanical Wigner negativity can be prepared deterministically in the unresolved-sideband regime, without additional nonlinearities, nonclassical drives, or conditional measurements. In particular, we find that Wigner negativity can be prepared with an optical pulse, even without single-photon strong coupling, and the negativity persists in the steady state of a continuously driven system. Our results deepen our understanding of cavity-enhanced radiation pressure and establish a pathway for deterministic preparation of nonclassical mechanical states in the unresolved sideband regime.
The ability to prepare a macroscopic mechanical resonator into a quantum superposition state is an outstanding goal of cavity optomechanics. Here, we propose a technique to generate cat states of motion using the intrinsic nonlinearity of a dispersive optomechanical interaction. By applying a bichromatic drive to an optomechanical cavity, our protocol enhances the inherent second-order processes of the system, inducing the requisite two-phonon dissipation. We show that this nonlinear sideband cooling technique can dissipatively engineer a mechanical resonator into a cat state, which we verify using the full Hamiltonian and an adiabatically reduced model. While the fidelity of the cat state is maximized in the single-photon, strong-coupling regime, we demonstrate that Wigner negativity persists even for weak coupling. Finally, we show that our cat state generation protocol is robust to significant thermal decoherence of the mechanical mode, indicating that such a procedure may be feasible for near-term experimental systems.
We address quantitatively the relationship between the nonlinearity of a mechanical resonator and the nonclassicality of its ground state. In particular, we analyze the nonclassical properties of the nonlinear Duffing oscillator (being driven or not) as a paradigmatic example of a nonlinear nanomechanical resonator. We first discuss how to quantify the nonlinearity of this system and then show that the nonclassicality of the ground state, as measured by the volume occupied by the negative part of the Wigner function, monotonically increases with the nonlinearity in all the working regimes addressed in our study. Our results show quantitatively that nonlinearity is a resource to create nonclassical states in mechanical systems.
We propose a scheme for generation of a nonlinear coherent state (NCS) of a mechanical resonator (MR) in an optomechanical micro-cavity, in which a two-level quantum dot (QD) and the microcavity are respectively driven by a strong laser and a weak laser. This microcavity can be engineered within a photonic band-gap (PBG) material. By properly tuning the frequency of the weak driving field, two-photon blockade phenomenon occurs. The QD-cavity subsystem can evolve into a dark state due to the damping of the microcavity and the elimination of the decay rate of the QD at selected frequencies in the PBG material. In this situation, the phonon mode of the MR can be prepared into a NCS, which is a non-classical state and possesses the sub-Poisson statistics. We also demonstrate the Wigner function of the NCS, which negativity implies its non-classicality.
We address quantitatively the relationship between the nonlinearity of a mechanical resonator and the nonclassicality of its ground state. In particular, we analyze the nonclassical properties of the nonlinear Duffing oscillator (being driven or not) as a paradigmatic example of a nonlinear nanomechanical resonator. We first discuss how to quantify the nonlinearity of this system and then show that the nonclassicality of the ground state, as measured by the volume occupied by the negative part of the Wigner function, monotonically increases with the nonlinearity in all the working regimes addressed in our study. Our results show quantitatively that nonlinearity is a resource to create nonclassical states in mechanical systems.