77 lines
4.4 KiB
TeX
77 lines
4.4 KiB
TeX
\section{Introduction}
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A significant amount of experimental effort has gone into detecting
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gravitational waves since the 1960s. They were first predicted by
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Einstein in 1916 as a consequence of general relativity. Mass (or
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energy) warps spacetime and changes in the shape or position of such
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objects will cause distortions which propagate as waves at the speed
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of light. Gravitational waves have still not been observed
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directly. However, the study of the period of the binary pulsar
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discovered by Hulse and Taylor in 1974 provides strong indirect
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evidence for their existence \cite{pulsar}. The search for direct
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evidence of gravitational waves has resulted in a number of
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large-scale experiments such as the Laser Interferometer
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Gravitational-Wave Observatory (LIGO) and the Laser Interferometer
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Space Antenna (LISA). The main difficulty of direct detection of
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gravitational radiation is its small effect on a detector, distortions
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from equilibrium on Earth due to astrophysical sources are predicted
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to be no larger than one part in $10^{21}$ \cite{hobson}. To observe
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such a small effect an extremely sensitive apparatus is necessary. The
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LIGO and LISA experiments use laser interferometry as a means to
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detect such tiny changes. However, the fragile nature of entanglement
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could provide an alternative for an experiment to detect this effect.
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Entanglement, a phenomenon unique to quantum mechanics, allows for
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stronger correlations between separate components of a composite
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system than are possible with classical statistics. This ``spooky
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action at a distance'' led Einstein to dismiss the theory as an
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incomplete description of reality \cite{epr}. However, in 1964 John
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Bell showed that no physical theory of local hidden variables, as
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suggested by Einstein, can reproduce the predictions of quantum
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mechanics. A number of experiments have been performed to test Bell's
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theorem and all of them provide strong evidence for the validity of
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quantum mechanics. The first definitive experiment was performed by
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Alain Aspect in 1982 \cite{aspect}.
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Experiments have shown that quantum entanglement is not only real, but
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that it can also be used as a resource. The idea that it can be
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generated and manipulated like any other physical property of a system
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gave rise to the field of quantum information. Entanglement has
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allowed us to exceed limits imposed by classical mechanics in
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computing, cryptography and data transmission \cite{steane}. However,
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in reality quantum entanglement is a fragile resource which is very
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difficult to control. Decoherence, the loss of quantum coherences due
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to coupling to the environment, occurs on time scales much shorter
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than the rate at which we can manipulate the systems experimentally
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\cite{zurek}. This sensitivity to environmental effects is one of the
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main obstacles in developing quantum technologies and is a subject of
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active research. However, this fragility could potentially be used to
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measure very small effects that require extremely sensitive
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detectors.
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We propose and investigate numerically the possibility of performing
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an experiment to detect gravitational waves using the entanglement
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between a pair of neutrons initially localized on either side of a
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harmonic potential in a multilayer. Entanglement is generated in
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collisions due to the particles' natural motion \cite{edmund}. By
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working in the weak-field limit of general relativity we combine the
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effect of gravitational waves with the Schr\"{o}dinger equation. The
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resulting equation is then investigated numerically and we demonstrate
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that entanglement amplifies the effect of a gravitational wave, but
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the effect is too small to detect using conventional, easily
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accessible techniques originally envisaged for this
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experiment. However, the results show that entanglement can be a
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useful mechanism for detecting high frequency waves.
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In section \ref{sec:experiment}, we present the experimental setup
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that will be investigated and the mechanism for entanglement
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generation. In section \ref{sec:model}, we describe the effect of
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gravitational waves, the neutron-neutron interaction and how these
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elements are combined in a two-particle Hamiltonian. We also address
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various concerns that arise when combining general relativistic
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effects with quantum mechanics. Section \ref{sec:numerical} presents
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the numerical simulations of the modified Schr\"{o}dinger equation and
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their implications for the feasability of the suggested experiment. We
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conclude in section \ref{sec:conclusions}.
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