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Journal of Applied Physics

Quantum interference visibility in an oscillating macroscopic mirror

Klaus K. Zhang, Ashok Puri

Published Sep 15, 2012• 0 citations
Publisher

Abstract

This paper studies the problem of quantum entanglement in an oscillating macroscopic mirror consisting of a modified Michelson interferometer in which one of the mirrors is free to oscillate about its center of mass. Previous works determined the interference visibility and the decoherence rate, among other things, assuming that the mirror starts in a normally distributed thermal state. Recommendations for design parameters were made based on their results. However, the initial thermal state could be non-Gaussian. The purpose of this investigation was to determine the interference visibility for initial thermal states of the mirror described by modified Gaussian distributions. This information was used to qualitatively determine the decoherence rate. The results of this study could be useful in quantum error correction to accurately determine the amount of environmentally induced decoherence, which negatively affects the computational results of quantum computers.

DOI
10.1063/1.4749399

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This paper studies the problem of quantum entanglement in an oscillating macroscopic mirror consisting of a modified Michelson interferometer in which one of the mirrors is free to oscillate about its center of mass. Previous works determined the interference visibility and the decoherence rate, among other things, assuming that the mirror starts in a normally distributed thermal state. Recommendations for design parameters were made based on their results. However, the initial thermal state could be non-Gaussian. The purpose of this investigation was to determine the interference visibility for initial thermal states of the mirror described by modified Gaussian distributions. This information was used to qualitatively determine the decoherence rate. The results of this study could be useful in quantum error correction to accurately determine the amount of environmentally induced decoherence, which negatively affects the computational results of quantum computers.
We show that the generation of entanglement of two heavily macroscopic mirrors is feasible with state of the art techniques of high-precision laser interferometry. The basis of such a demonstration would be a Michelson interferometer with suspended mirrors and simultaneous homodyne detections at both interferometer output ports. We present the connection between the generation of entanglement and the standard quantum limit (SQL) for a free mass. The SQL is a well-known reference limit in operating interferometers for gravitational-wave detection and provides a measure of when macroscopic entanglement can be observed in the presence of realistic decoherence processes.
Reflection of a microscopic particle from a mesoscopic/macroscopic `mirror' generates two-body correlated interference from the incident and reflected particle substates and their associated mirror substates. The microscopic momentum exchanged generates two mirror substates which interfere to produce fringes which do not vanish as the mirror mass increases. The small displacement between these mirror states can yield negligible environmental decoherence times. Mirror coherence lengths impose constraints on the extent of this interference, which are mitigated using interference of the two-body states associated with the particle reflecting from both of the two surfaces of a slab of matter in a manner analogous to the classical interference of a pulse of light reflecting from a `thin film'. This two-body correlated interference is modeled as a particle traversing a finite well with both the particle and well treated quantum mechanically. Such a treatment predicts the expected `thin-film' interference but only as a special case of a more general result. It is also shown that measurements on only the reflected particle (yielding a marginal probability density function) can act as a probe to reveal the quantum state of the macroscopic reflector. For equal masses, coherence of the particle substate is transferred to the mirror substate, a quantum manifestation of a familiar classical result.
We propose an experiment for creating quantum superposition states involving of the order of 10(14) atoms via the interaction of a single photon with a tiny mirror. This mirror, mounted on a high-quality mechanical oscillator, is part of a high-finesse optical cavity which forms one arm of a Michelson interferometer. By observing the interference of the photon only, one can study the creation and decoherence of superpositions involving the mirror. A detailed analysis of the requirements shows that the experiment is within reach using a combination of state-of-the-art technologies.
We investigate how the temporal coherence interference properties of light in a Michelson-Morley interferometer (MMI), using only a single-photon detector, can be understood in a quantum-optics framework in a straightforward and pedagogical manner. For this purpose we make use of elementary quantum field theory and Glaubers theory for photon detection in order to calculate the expected interference pattern in the MMI. If a thermal reference source is used in the MMI local oscillator port in combination with a thermal source in the signal port, the interference pattern revealed by such an intensity measurement shows a distinctive dependence on the differences in the temperature of the two sources. The MMI can therefore be used in order to perform temperature measurements. A related method was actually used to carry out high precision measurements of the cosmic micro-wave background radiation on board of the COBE satellite. The theoretical framework allows us to consider any initial quantum state. The interference of single photons as a tool to determine the angular peak-frequency of a one-photon pulse interfering with a single-photon reference pulse is, e.g., considered. A similar consideration for coherent laser pulses leads to a different response in the detector. The MMI experimental setup is therefore in a sense an example of an optical device where one can exhibit the difference between classical and quantum-mechanical light using only intensity measurements.
For the purpose of understanding the quantum behavior such as quantum decoherence, fluctuations, dissipation, entanglement and teleportation of a mesoscopic or macroscopic object interacting with a general environment, we derive here a set of exact master equations for the reduced density matrix of N interacting harmonic oscillators in a heat bath with arbitrary spectral density and temperature. For the case N = 2, we also give an influence functional treatment of this model and provide explicit expressions for the evolution operator of the reduced density matrix which are useful for the study of decoherence and disentanglement issues. We show a simple application of this master equation by examining the decoherence and disentanglement of two harmonic oscillators due to their interaction with a common environment under Markovian approximation. Two classes of problems of interest to us which these equations can be usefully applied to are that of the quantum dynamics of nanoelectromechanical oscillators and the entanglement evolution of multi-partite macroscopic states such as quantum superposition of mirrors in a high Q cavity. To address a key conceptual issue for macroscopic quantum phenomena we examine the conditions for an assumption often implicitly made in these studies to be valid, namely, that the quantum behavior of a macroscopic object in an environment can be accurately represented by only treating the dynamics of its center-of-mass variable. We also mention how these results can be used to calculate the uncertainty principle governing a macroscopic object at finite temperature.
This paper is aimed at investigating the feasibility of generating quantum conditional entanglement between macroscopic mechanical mirrors in optomechanical systems while under continuous measurement and feedback control. We consider the squeezing of the states of the mechanical common and the differential motions of the mirrors by the action of measuring the common and the differential output light beams in the Fabry-P\'erot-Michelson interferometer. We carefully derive a covariance matrix for the mechanical mirrors in a steady state, employing the Kalman filtering problem with dissipative cavities. We demonstrate that Gaussian entanglement between the mechanical mirrors is generated when the states of the mechanical common and differential modes of the mirrors are squeezed with high purity in an asymmetric manner. Our results also show that quantum entanglement between 7-mg mirrors is achievable in the short term.