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arXiv:1211.4315

Quantum-state steering in optomechanical devices

Mueller-Ebhardt, Helge, Haixing Miao, S. L. Danilishin, Yanbei Chen

Published Nov 19, 2012• 2 citations
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Abstract

We show that optomechanical systems in the quantum regime can be used to demonstrate EPR-type quantum entanglement between the optical field and the mechanical oscillator, via quantum-state steering. Namely, the conditional quantum state of the mechanical oscillator can be steered into different quantum states depending the choice made on which quadrature of the out-going field is to be measured via homodyne detection. More specifically, if quantum radiation pressure force dominates over thermal force, the oscillator's quantum state is steerable with a photodetection efficiency as low as 50%, approaching the ideal limit shown by Wiseman and Gambetta [Phys. Rev. Lett. {\bf 108}, 220402 (2012)]. We also show that requirement for steerability is the same as those for achieving sub-Heisenberg state tomography using the same experimental setup.

DOI
10.48550/arxiv.1211.4315

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We show that optomechanical systems in the quantum regime can be used to demonstrate EPR-type quantum entanglement between the optical field and the mechanical oscillator, via quantum-state steering. Namely, the conditional quantum state of the mechanical oscillator can be steered into different quantum states depending the choice made on which quadrature of the out-going field is to be measured via homodyne detection. More specifically, if quantum radiation pressure force dominates over thermal force, the oscillator's quantum state is steerable with a photodetection efficiency as low as 50%, approaching the ideal limit shown by Wiseman and Gambetta [Phys. Rev. Lett. {\bf 108}, 220402 (2012)]. We also show that requirement for steerability is the same as those for achieving sub-Heisenberg state tomography using the same experimental setup.
We consider a generic optomechanical system and investigate quantum entanglement between the outgoing light fields and the mechanical motion of the mirror. In contract to traditional approach, we do not consider quantum correlations between the intracavity optical mode and the mechanical motion of a mirror, rather the correlations between the continuum of optical fields in the outgoing light wave and the mechanical degree of freedom. This entanglement is shown to be robust to thermal decoherence and depend on the ratio of measurement strength (light power) and decoherence rate ∝ T/Qm. Furthermore, we describe a feasible way to demonstrate quantum state steering, usinging time-variable homodyne detection scheme that allows to evade back-action. An intimate connection between optomechanical steerability and quantum state tomography [Phys. Rev. A 81, 012114 (2010)] is demonstrated.
Einstein-Podolsky-Rosen (EPR) steering is a quantum nonlocal effect which is intrinsically distinct from Bell nonlocality and quantum entanglement. In this paper, we investigate in detail the properties of steady-state light-mechanical Gaussian steerable correlations in a generic cavity optomechanical system. When considering the steering between the intracavity field and the mechanical oscillator, we find that under blue-detuned driving, the steady-state steering via optomechanical parametric downconversion is present in only one direction and, moreover, the steering direction is determined merely by the relative dissipation strength of the cavity to the mechanics. Furthermore, when considering the steering between the cavity output field and the mechanical oscillator, we reveal that under red-detuned driving, strong steering can be achieved in the sideband-unresolved regime for a filtered output field with given central frequency and bandwidth. This steering with the output field can also be present in one way by adjusting the driving strength and exists up to the environment temperature $T\ensuremath{\approx}10$ K for the parameters close to those in the recent experiments. Finally, we show that the achieved strong light-mechanical correlations can be explored to realize macroscopic EPR steering of two distant optomechanical oscillators in the regime of unresolved sidebands via entanglement swapping.
Abstract In this paper, we propose a continuous measurement and feedback scheme to achieve strong Einstein–Podolsky–Rosen (EPR) steering and Bell nonlocality of two macroscopic mechanical oscillators in cavity optomechanics. Our system consists of two optomechanical cavities in which two cavity fields are coupled to each other via nondegenerate parametric downconversion. The two cavity output fields are subject to continuous Bell-like homodyne detection and the detection currents are fed back to drive the cavity fields. It is found that when the feedback is absent, the two mechanical oscillators can only be prepared in steady weakly entangled states which however do not display EPR steering and Bell nonlocality, due to the so-called 3 dB limit. But when the feedback is present, it is found that the mechanical entanglement is considerably enhanced such that strong mechanical steering and Bell nonlocality can be obtained in the steady-state regime. We analytically reveal that this is because the feedback drives the mechanical oscillators into a steady approximate two-mode squeezed vacuum state, with arbitrary squeezing in principle. It is shown that the feedback can also obviously improve the purity of the nonclassical mechanical states. The dependences of the mechanical quantum nonlocality on the feedback strength and thermal fluctuations are studied, and it is found that Bell nonlocality is much more vulnerable to thermal noise than EPR steerable nonlocality.
Einstein-Podolsky-Rosen (EPR) steering is a quantum nonlocal effect that characterizes the ability to remotely control quantum states of one system via local measurements on another distant system entangled with the controlled one. Entanglement swapping is an effective way to realize distant entanglement, which is useful for building quantum networks. Previous studies show that the presence of entanglement of subsystems in mixed states is merely necessary for achieving entanglement swapping. In this paper, we find that for Gaussian entanglement swapping with two pairs of entangled modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{j}\phantom{\rule{4pt}{0ex}}(j=1,2)$, the existence of directional EPR steering from the modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{j}$ under homodyne detection to the other modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}$ is a sufficient and necessary condition for achieving swapped entanglement between modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{2}$, and moreover this achievement is independent of the reverse steering from modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{j}$ to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}_{j}$. We further reveal that this is because the steering in that direction enables the amplitude and phase squeezing of the two composite modes of modes ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{1}$ and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{2}$ via homodyne detection, which is necessary and sufficient for swapped entanglement. As a concrete example, we next investigate light-mechanical steering in a dispersively or dissipatively coupled optomechanical system, and we consider the generation of entanglement between two distant optomechanical oscillators via entanglement swapping by utilizing steering. It is interesting to find that even without the constraint that the cavity linewidth should be smaller than the mechanical frequency, robust mechanical entanglement can be obtained in the steady-state regime.
In this paper, we first consider the generation of mechanical squeezing in a dispersively or dissipatively coupled optomechanical system by continuously homodyning the output field of the optomechanical cavity. It is found that strong steady-state mechanical squeezing beyond the 3 dB limit can be achieved in both of the optomechanical systems. The properties of the squeezing are quite different for the two types of systems and the reasons for the differences are analyzed. We next consider the achievement of optomechanical steering in the dispersive or dissipative optomechanical system via continuously monitoring the position of the mechanical oscillator. It is revealed that the monitoring can lead the steady-state optomechanical entanglement to be enhanced considerably such that strong steerable correlations can be achieved between the mechanical oscillator and the cavity field. The effects of thermal phonons are studied and it is shown that the generated squeezing and steering are quite robust against the thermal fluctuations.
In the past several years, the field of optomechanics has progressed from proof-of-principle experiments to the realization of mechanical oscillators and measurements in the quantum regime. Mechanical oscillators are of great interest because they can have small dissipation rates, can couple to many different systems of interest, and are the fundamental elements of ultrasensitive force detectors. Coupling these mechanical oscillators to microwave or optical fields provides a twofold advantage. Firstly, information about mechanical position can be encoded in the interrogating field, enabling sensitive readout of the mechanical oscillator. Secondly, the radiation pressure force of that field can be used to control the state of the mechanical oscillator. Including a high-quality microwave or optical cavity enhances both of these effects, as the field strength is resonantly increased.\nThe major questions in the field of optomechanics in the last several years have dealt with using mechanical oscillators for ultrasensitive measurements and as tools for quantum information. Both of these goals have the prerequisite that we be able to read out the motion of the mechanical oscillator in a quantum efficient manner. To that end, we developed a nearly shot-noise limited microwave interferometer capable of measuring mechanical motion with an imprecision below that at the standard quantum limit. This achievement is not only a critical improvement for the electromechanical experiments we do, but is also an important tool for any experiment that encodes the information of interest in microwave fields. In order to use mechanical oscillators as tools for quantum information, the mechanical oscillator must also be cooled into the quantum regime and fully controllable by the interrogating fields. To this end, we used the radiation pressure of microwave fields to cool our macroscopic mechanical oscillator to less than one phonon. We also demonstrated coherent transfer between itinerant microwave states and the mechanical oscillator, even for incident fields with less than one photon of energy.\nThese accomplishments have set the foundation for further experiments to extend the quantum information abilities of optomechanical systems, couple diverse quantum systems via a mechanical intermediary, and potentially explore the foundations of quantum mechanics at macroscopic scales.