102 lines
5.3 KiB
TeX
102 lines
5.3 KiB
TeX
\section{Decoherence}\label{sec:decoherence}
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Decoherence is the process through which quantum correlations disperse
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information throughout the degrees of freedom that are effectively
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beyond the observer's control. It has been suggested as the mechanism
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through which a measurement of a quantum system is effectively made
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\cite{zurek}. In general, any interaction with the environment will
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eventually lead to decoherence and this is one of the main obstacles
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in generating and maintaining entangled resources. At the beginning of
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this investigation we proposed that entanglement may be sensitive
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enough to be affected by gravitational waves. In our approach we have
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only investigated the dynamics of two neutrons in a harmonic well
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subject to gravitational radiation by solving the Schr\"{o}dinger
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equation for the modified Hamiltonian \eqref{eq:main}. This has
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allowed us to study the effects of the waves on the wavefunction of
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neutrons entangling in a harmonic potential, but the fact that we keep
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track of all of the degrees of freedom meant that it was impossible to
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observe decoherence.
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In order to be able to describe decoherence we have to construct a
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dynamical model of a system coupled to its environment. In the
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Feynman-Vernon theory of the influence functional \cite{feynman} we
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consider a system A interacting with a second system B (the reservoir)
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described by the Hamiltonian
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\begin{equation}
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H = H_A + H_I + H_B,
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\end{equation}
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where system A consists of a single particle, the reservior consists of
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N particles and $H_I$ is the interaction Hamiltonian. We can obtain
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the reduced density operator for system A by tracing out all the
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environment coordinates. Assuming that the initial density operator is
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separable into a product of the density operators for A and B we get
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\begin{equation}
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\rho(x, y, t) = \int \mathrm{d} x^\prime \mathrm{d} y^\prime J(x, y,
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t; x^\prime, y^\prime, 0)\rho_A(x^\prime, y^\prime, 0),
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\end{equation}
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where x and y are the position coordinates of the single particle,
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\begin{equation}
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J(x, y, t; x^\prime, y^\prime, 0) = \int \int \mathrm{D} x \mathrm{D}
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y \exp \left[ i \frac{S_A[x]}{\hbar} \right] \exp \left[ -i
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\frac{S_A[y]}{\hbar} \right] F[x, y],
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\end{equation}
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$F[x, y]$ is the so called influence functional given by
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\begin{equation}
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\begin{array} {lcl}
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F[x, y] & = & \int \mathrm{d} \bm{R^\prime} \mathrm{d} \bm{Q^\prime}
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\mathrm{d} \bm{R} \rho_B(\bm{R^\prime}, \bm{Q^\prime}, 0) \int \int
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\mathrm{D} \bm{R} \mathrm{D} \bm{Q} \\\\ & & \times \exp \left[
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\frac{i}{\hbar} \left( S_I[x, \bm{R}] - S_I[y, \bm{Q}] + S_B[\bm{R}]
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- S_B[\bm{Q}] \right) \right],
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\end{array}
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\end{equation}
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$S_i$ is the action corresponding to the Hamiltonian $H_i$ and
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$\bm{R}$, $\bm{Q}$ are vectors of the positions of the reservoir
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particles. This has been solved exactly in \cite{feynman} in the limit
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of a weak coupling, where we only have to consider the linear response
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of the reservoir to the system and we can describe the environment in
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the harmonic approximation.
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The gravitational wave background could potentially be modelled with a
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collection of harmonic oscillators. However, as we have seen in
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section 3 the interaction of the wave function with gravitational
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waves is implemented through a modification of the kinetic energy term
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and not via an effective potential and modelling the coupling with the
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system with a linear response would be insufficient. The more
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complicated interaction would lead to an intractable form for the
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influence functional. Furthermore, there is no way of obtaining an
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irreversible process such as decoherence using the above solution. We
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do not consider a reservoir of infinite size and any information lost
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by the system will return within a finite period of time.
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Caldeira and Leggett addressed the issue of irreversibility in 1983
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\cite{caldeira} in an attempt to solve the problem of quantum Brownian
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motion. Instead of coupling the system A to a reservoir B they
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replaced the system-environment coupling with an externally applied
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classical force $F(t)$. The fluctuating force has to obey the standard
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stochastic relations
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\begin{equation}
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\langle F(t) \rangle = 0,
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\end{equation}
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\begin{equation}
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\langle F(t) F(t^\prime) \rangle = 2 \eta k_B T \delta (t - t^\prime),
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\end{equation}
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where $\eta$ is a damping constant, $k_B$ is Boltzmann's constant and
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$T$ is the temperature. However, there are issues with applying the
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Caldeira-Leggett model to our problem regarding the nature of the
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effect produced by the waves. In the situation presented in figure
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\ref{fig:rings} all of the test particles have constant spatial
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coordinates, they are stationary. However, the measured separation
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between the points changes according to $l^2 = -g_{ij} \xi^i
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\xi^j$. Modelling this effect via a classical force is questionable.
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In order to describe decoherence a master equation approach is
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necessary, but the most successful models for dissipation due to a
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system's interaction with the environment are not appropriate for the
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interaction of a quantum wave function with gravitational waves. The
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environment has also been modelled with a quantum field \cite{master}
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and for a certain class of interactions the approach is equivalent to
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the Caldeira-Leggett model. If a quantum field theory of gravity was
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developed then a similar approach could be used to derive a master
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equation to model decoherence due to gravitational fields.
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