From 8eceb645211a72edf22976b44d9af36176c234d2 Mon Sep 17 00:00:00 2001 From: Wojciech Kozlowski Date: Sun, 19 Mar 2017 00:02:11 +0000 Subject: [PATCH] Git repo for MSci thesis --- .../EntanglementMax.dat | 22 + .../EntanglementMaxResult.dat | 1 + .../entanglement.eps | 711 +++++++++++++++ .../entanglement.pdf | Bin 0 -> 3758 bytes Images/EntanglementSingleWave.pdf | Bin 0 -> 4876 bytes Images/FidelityHundredWaves.pdf | Bin 0 -> 6100 bytes Images/FidelityThousandWaves.pdf | Bin 0 -> 6496 bytes Images/Freqs.pdf | Bin 0 -> 6500 bytes Images/HighFreq.pdf | Bin 0 -> 6848 bytes Images/LowFreq.pdf | Bin 0 -> 7753 bytes Images/NWaves.pdf | Bin 0 -> 3546 bytes Images/Uplot.pdf | Bin 0 -> 6311 bytes Images/WaveMags.pdf | Bin 0 -> 5084 bytes Images/collision.eps | 825 ++++++++++++++++++ Images/collision.pdf | 109 +++ Images/entanglement.pdf | Bin 0 -> 3758 bytes Images/rings.gif | Bin 0 -> 8651 bytes Images/rings.pdf | Bin 0 -> 13125 bytes README.rst | 40 + abstract.tex | 17 + appendix.tex | 190 ++++ conclusions.tex | 46 + decoherence.tex | 101 +++ experiment.tex | 131 +++ introduction.tex | 76 ++ model.tex | 207 +++++ nnScatt.pdf | Bin 0 -> 267843 bytes numerical.tex | 59 ++ report.tex | 125 +++ results.tex | 212 +++++ 30 files changed, 2872 insertions(+) create mode 100644 Images/EntanglementMaximisationResults/EntanglementMax.dat create mode 100644 Images/EntanglementMaximisationResults/EntanglementMaxResult.dat create mode 100644 Images/EntanglementMaximisationResults/entanglement.eps create mode 100644 Images/EntanglementMaximisationResults/entanglement.pdf create mode 100644 Images/EntanglementSingleWave.pdf create mode 100644 Images/FidelityHundredWaves.pdf create mode 100644 Images/FidelityThousandWaves.pdf create mode 100644 Images/Freqs.pdf create mode 100644 Images/HighFreq.pdf create mode 100644 Images/LowFreq.pdf create mode 100644 Images/NWaves.pdf create mode 100644 Images/Uplot.pdf create mode 100644 Images/WaveMags.pdf create mode 100644 Images/collision.eps create mode 100644 Images/collision.pdf create mode 100644 Images/entanglement.pdf create mode 100755 Images/rings.gif create mode 100644 Images/rings.pdf create mode 100644 README.rst create mode 100644 abstract.tex create mode 100644 appendix.tex create mode 100644 conclusions.tex create mode 100644 decoherence.tex create mode 100644 experiment.tex create mode 100644 introduction.tex create mode 100644 model.tex create mode 100644 nnScatt.pdf create mode 100644 numerical.tex create mode 100644 report.tex create mode 100644 results.tex diff --git a/Images/EntanglementMaximisationResults/EntanglementMax.dat b/Images/EntanglementMaximisationResults/EntanglementMax.dat new file mode 100644 index 0000000..4dcc6c1 --- /dev/null +++ b/Images/EntanglementMaximisationResults/EntanglementMax.dat @@ -0,0 +1,22 @@ +0.000000 1.692726e-10 +0.250000 5.098522e-02 +0.645910 2.286679e-01 +0.654509 2.331342e-01 +1.309017 5.739241e-01 +2.875042 9.860267e-01 +3.123199 9.980069e-01 +3.322563 1.000717e+00 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0 S12 stroke grestore} bind def +/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def +/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def +/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def +/DiaE {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke} def +/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke} def +/TriUE {stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V closepath stroke} def +/TriDE {stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V closepath stroke} def +/PentE {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore} def +/CircE {stroke [] 0 setdash + hpt 0 360 arc stroke} def +/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def +/DiaW {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V Opaque stroke} def +/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V Opaque stroke} def +/TriUW {stroke [] 0 setdash vpt 1.12 mul add M + hpt neg vpt -1.62 mul V + hpt 2 mul 0 V + hpt neg vpt 1.62 mul V Opaque stroke} def +/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M + hpt neg vpt 1.62 mul V + hpt 2 mul 0 V + hpt neg vpt -1.62 mul V Opaque stroke} def +/PentW {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + Opaque stroke grestore} def +/CircW {stroke [] 0 setdash + hpt 0 360 arc Opaque stroke} def +/BoxFill {gsave Rec 1 setgray fill grestore} def +/Density { + /Fillden exch def + currentrgbcolor + /ColB exch def /ColG exch def /ColR exch def + /ColR ColR Fillden mul Fillden sub 1 add def + /ColG ColG Fillden mul Fillden sub 1 add def + /ColB ColB Fillden mul Fillden sub 1 add def + ColR ColG ColB setrgbcolor} def +/BoxColFill {gsave Rec PolyFill} def +/PolyFill {gsave Density fill grestore grestore} def +/h {rlineto rlineto rlineto gsave closepath fill grestore} bind def +% +% PostScript Level 1 Pattern Fill routine for rectangles +% Usage: x y w h s a XX PatternFill +% x,y = lower left corner of box to be filled +% w,h = width and height of box +% a = angle in degrees between lines and x-axis +% XX = 0/1 for no/yes cross-hatch +% +/PatternFill {gsave /PFa [ 9 2 roll ] def + PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate + PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec + gsave 1 setgray fill grestore clip + currentlinewidth 0.5 mul setlinewidth + /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def + 0 0 M PFa 5 get rotate PFs -2 div dup translate + 0 1 PFs PFa 4 get div 1 add floor cvi + {PFa 4 get mul 0 M 0 PFs V} for + 0 PFa 6 get ne { + 0 1 PFs PFa 4 get div 1 add floor cvi + {PFa 4 get mul 0 2 1 roll M PFs 0 V} for + } if + stroke grestore} def +% +/languagelevel where + {pop languagelevel} {1} ifelse + 2 lt + {/InterpretLevel1 true def} + {/InterpretLevel1 Level1 def} + ifelse +% +% PostScript level 2 pattern fill definitions +% +/Level2PatternFill { +/Tile8x8 {/PaintType 2 /PatternType 1 /TilingType 1 /BBox [0 0 8 8] /XStep 8 /YStep 8} + bind def +/KeepColor {currentrgbcolor [/Pattern /DeviceRGB] setcolorspace} bind def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke} +>> matrix makepattern +/Pat1 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 8 8 L 0 8 M 8 0 L stroke + 0 4 M 4 8 L 8 4 L 4 0 L 0 4 L stroke} +>> matrix makepattern +/Pat2 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 0 M 0 8 L + 8 8 L 8 0 L 0 0 L fill} +>> matrix makepattern +/Pat3 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 8 M 8 -4 L + 0 12 M 12 0 L stroke} +>> matrix makepattern +/Pat4 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -4 0 M 8 12 L + 0 -4 M 12 8 L stroke} +>> matrix makepattern +/Pat5 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 8 M 4 -4 L + 0 12 M 8 -4 L 4 12 M 10 0 L stroke} +>> matrix makepattern +/Pat6 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop -2 0 M 4 12 L + 0 -4 M 8 12 L 4 -4 M 10 8 L stroke} +>> matrix makepattern +/Pat7 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 8 -2 M -4 4 L + 12 0 M -4 8 L 12 4 M 0 10 L stroke} +>> matrix makepattern +/Pat8 exch def +<< Tile8x8 + /PaintProc {0.5 setlinewidth pop 0 -2 M 12 4 L + -4 0 M 12 8 L -4 4 M 8 10 L stroke} +>> matrix makepattern +/Pat9 exch def +/Pattern1 {PatternBgnd KeepColor Pat1 setpattern} bind def +/Pattern2 {PatternBgnd KeepColor Pat2 setpattern} bind def +/Pattern3 {PatternBgnd KeepColor Pat3 setpattern} bind def +/Pattern4 {PatternBgnd KeepColor Landscape {Pat5} {Pat4} ifelse setpattern} bind def +/Pattern5 {PatternBgnd KeepColor Landscape {Pat4} {Pat5} ifelse setpattern} bind def +/Pattern6 {PatternBgnd KeepColor Landscape {Pat9} {Pat6} ifelse setpattern} bind def +/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def +} def +% +% +%End of PostScript Level 2 code +% +/PatternBgnd { + TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse +} def +% +% Substitute for Level 2 pattern fill codes with +% grayscale if Level 2 support is not selected. +% +/Level1PatternFill { +/Pattern1 {0.250 Density} bind def +/Pattern2 {0.500 Density} bind def +/Pattern3 {0.750 Density} bind def +/Pattern4 {0.125 Density} bind def +/Pattern5 {0.375 Density} bind def +/Pattern6 {0.625 Density} bind def +/Pattern7 {0.875 Density} bind def +} def +% +% Now test for support of Level 2 code +% +Level1 {Level1PatternFill} {Level2PatternFill} ifelse +% +/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont +dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall +currentdict end definefont pop +/MFshow { + { dup 5 get 3 ge + { 5 get 3 eq {gsave} {grestore} ifelse } + {dup dup 0 get findfont exch 1 get scalefont setfont + [ currentpoint ] exch dup 2 get 0 exch R dup 5 get 2 ne {dup dup 6 + get exch 4 get {Gshow} {stringwidth pop 0 R} ifelse }if dup 5 get 0 eq + {dup 3 get {2 get neg 0 exch R pop} {pop aload pop M} ifelse} {dup 5 + get 1 eq {dup 2 get exch dup 3 get exch 6 get stringwidth pop -2 div + dup 0 R} {dup 6 get stringwidth pop -2 div 0 R 6 get + show 2 index {aload pop M neg 3 -1 roll neg R pop pop} {pop pop pop + pop aload pop M} ifelse }ifelse }ifelse } + ifelse } + forall} def +/Gswidth {dup type /stringtype eq {stringwidth} {pop (n) stringwidth} ifelse} def +/MFwidth {0 exch { dup 5 get 3 ge { 5 get 3 eq { 0 } { pop } ifelse } + {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont + 6 get Gswidth pop add} {pop} ifelse} ifelse} forall} def +/MLshow { currentpoint stroke M + 0 exch R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/MRshow { currentpoint stroke M + exch dup MFwidth neg 3 -1 roll R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/MCshow { currentpoint stroke M + exch dup MFwidth -2 div 3 -1 roll R + Blacktext {gsave 0 setgray MFshow grestore} {MFshow} ifelse } bind def +/XYsave { [( ) 1 2 true false 3 ()] } bind def +/XYrestore { [( ) 1 2 true false 4 ()] } bind def +end +%%EndProlog +gnudict begin +gsave +doclip +50 50 translate +0.050 0.050 scale +0 setgray +newpath +(Helvetica) findfont 140 scalefont setfont +1.000 UL +LTb +686 448 M +63 0 V +6198 0 R +-63 0 V +stroke +602 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0)] +] -46.7 MRshow +1.000 UL +LTb +686 1001 M +63 0 V +6198 0 R +-63 0 V +stroke +602 1001 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.2)] +] -46.7 MRshow +1.000 UL +LTb +686 1554 M +63 0 V +6198 0 R +-63 0 V +stroke +602 1554 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.4)] +] -46.7 MRshow +1.000 UL +LTb +686 2107 M +63 0 V +6198 0 R +-63 0 V +stroke +602 2107 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.6)] +] -46.7 MRshow +1.000 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zVypu3KHf*72u32w>{ST%DrF84Wlm>3PG@zFqkkNlL(fFCe^p3il?P|SgNx)^jO46R z1&W(nQcK=|#um*_`p*x$6ARMI&TAb+Iui zTn;arsh6muAfiWbB1^PCho>rsl{CPbX>+1K_RyKgSw-Z+$#dZy1s7ilZWo9J*a*e% z5!btIhmZfkj7S)B45}RHrN(5S*tXKfUoWm6Qq{Ff`h37{X+s1sZYEuua~m4$tX$5Lk#9n z?HD6!YKM3L{traq@rNeI2r@B=GBH^*F*&q4#-Zae4qcB?;p)2P>U!vNj6@%wIv<}| zAD_dZV?+f7-3|)s3kn(u3ObBB#^V8<{vUkusz)9A|DY+V9bG8+2Te)k$hv>h)Q+D1f6$Z_j=JLiK~quq551}? zibwYc{wtsA(YpVGri%He>`}|1@;~i6qN)DV7L2?K`smieUpj(`&fdQ6#6NDmSbIkv tweKGtIT-tS_{0Bf 0$ and $A$ is a normalization constant. To be spatially distinct, the +single particle wave functions must satisfy $\int_{-\infty}^0 \! \psi_R(x) \, +\mathrm{d} x \to 0$ and $\int^{\infty}_0 \! \psi_L(x) \, \mathrm{d} x +\to 0$. We can then write the initial two-particle state as +\begin{equation} +\begin{array} {lcl} +|\Psi(t=0)\rangle & = & \left( \int \! +\psi_L(x_1)a^\dagger_{\sigma_1}(x_1) \, \mathrm{d} x_1 \right) \left( +\int \! \psi_R(x_2)a^\dagger_{\sigma_2}(x_2) \, \mathrm{d} x_2 \right) +|0\rangle \\\\ & \equiv & \left( \int\int\Psi(x_1, x_2; +t=0)a^\dagger_{\sigma_1}(x_1)a^\dagger_{\sigma_2}(x_2) \mathrm{d} x_1 +\mathrm{d} x_2 \right) |0\rangle \\\\ & \equiv & +|L_{\sigma_1}R_{\sigma_2}\rangle +\end{array} +\end{equation} +where $a^\dagger_{\sigma_i}(x_i)$ are the creation operators for a +neutron at $x_i$ with spin $\sigma_i$. We only consider neutrons which +are introduced into the trap with opposite spins, $\sigma_1 \neq +\sigma_2$, hence we can treat them as distinguishable. Therefore, the +creation operators commute. + +The two-particle Hamiltonian for the two neutrons in the same harmonic +potential is given by +\begin{equation}\label{eq:hamiltonian} +\hat{H} = -\frac{\hbar^2}{2m}\left( \frac{\partial^2}{\partial x_1^2} ++ \frac{\partial^2}{\partial x_2^2} \right) + +\frac{1}{2}m\omega^2(x_1^2 + x_2^2) + V_{nn}(x_1-x_2) +\end{equation} +where $V_{nn}(x_1-x_2)$ is a translationally invariant potential between +the two neutrons and its exact form will be discussed in the next +section. + +In the parity-dependent harmonic approximation, which assumes that the +spectrum of the Hamiltonian in \eqref{eq:hamiltonian} is equal to the +spectrum of a quantum harmonic oscillator except for a +parity-dependent energy shift, the wave function at integer +half-periods has been shown to be +\begin{equation} +|\Psi(t=n\pi/\omega)\rangle \propto +\cos\theta|L_{\sigma_1}R_{\sigma_2}\rangle + +e^{i\chi}\sin\theta|R_{\sigma_1}L_{\sigma_2}\rangle, +\end{equation} +where $\theta = n\pi\phi/2$, $\phi\hbar\omega$ is the energy shift +to the even parity eigenstates and $\chi$ is some phase \cite{edmund}. + diff --git a/introduction.tex b/introduction.tex new file mode 100644 index 0000000..b253fc0 --- /dev/null +++ b/introduction.tex @@ -0,0 +1,76 @@ +\section{Introduction} + +A significant amount of experimental effort has gone into detecting +gravitational waves since the 1960s. They were first predicted by +Einstein in 1916 as a consequence of general relativity. Mass (or +energy) warps spacetime and changes in the shape or position of such +objects will cause distortions which propagate as waves at the speed +of light. Gravitational waves have still not been observed +directly. However, the study of the period of the binary pulsar +discovered by Hulse and Taylor in 1974 provides strong indirect +evidence for their existence \cite{pulsar}. The search for direct +evidence of gravitational waves has resulted in a number of +large-scale experiments such as the Laser Interferometer +Gravitational-Wave Observatory (LIGO) and the Laser Interferometer +Space Antenna (LISA). The main difficulty of direct detection of +gravitational radiation is its small effect on a detector, distortions +from equilibrium on Earth due to astrophysical sources are predicted +to be no larger than one part in $10^{21}$ \cite{hobson}. To observe +such a small effect an extremely sensitive apparatus is necessary. The +LIGO and LISA experiments use laser interferometry as a means to +detect such tiny changes. However, the fragile nature of entanglement +could provide an alternative for an experiment to detect this effect. + +Entanglement, a phenomenon unique to quantum mechanics, allows for +stronger correlations between separate components of a composite +system than are possible with classical statistics. This ``spooky +action at a distance'' led Einstein to dismiss the theory as an +incomplete description of reality \cite{epr}. However, in 1964 John +Bell showed that no physical theory of local hidden variables, as +suggested by Einstein, can reproduce the predictions of quantum +mechanics. A number of experiments have been performed to test Bell's +theorem and all of them provide strong evidence for the validity of +quantum mechanics. The first definitive experiment was performed by +Alain Aspect in 1982 \cite{aspect}. + +Experiments have shown that quantum entanglement is not only real, but +that it can also be used as a resource. The idea that it can be +generated and manipulated like any other physical property of a system +gave rise to the field of quantum information. Entanglement has +allowed us to exceed limits imposed by classical mechanics in +computing, cryptography and data transmission \cite{steane}. However, +in reality quantum entanglement is a fragile resource which is very +difficult to control. Decoherence, the loss of quantum coherences due +to coupling to the environment, occurs on time scales much shorter +than the rate at which we can manipulate the systems experimentally +\cite{zurek}. This sensitivity to environmental effects is one of the +main obstacles in developing quantum technologies and is a subject of +active research. However, this fragility could potentially be used to +measure very small effects that require extremely sensitive +detectors. + +We propose and investigate numerically the possibility of performing +an experiment to detect gravitational waves using the entanglement +between a pair of neutrons initially localized on either side of a +harmonic potential in a multilayer. Entanglement is generated in +collisions due to the particles' natural motion \cite{edmund}. By +working in the weak-field limit of general relativity we combine the +effect of gravitational waves with the Schr\"{o}dinger equation. The +resulting equation is then investigated numerically and we demonstrate +that entanglement amplifies the effect of a gravitational wave, but +the effect is too small to detect using conventional, easily +accessible techniques originally envisaged for this +experiment. However, the results show that entanglement can be a +useful mechanism for detecting high frequency waves. + +In section \ref{sec:experiment}, we present the experimental setup +that will be investigated and the mechanism for entanglement +generation. In section \ref{sec:model}, we describe the effect of +gravitational waves, the neutron-neutron interaction and how these +elements are combined in a two-particle Hamiltonian. We also address +various concerns that arise when combining general relativistic +effects with quantum mechanics. Section \ref{sec:numerical} presents +the numerical simulations of the modified Schr\"{o}dinger equation and +their implications for the feasability of the suggested experiment. We +conclude in section \ref{sec:conclusions}. + diff --git a/model.tex b/model.tex new file mode 100644 index 0000000..c58932b --- /dev/null +++ b/model.tex @@ -0,0 +1,207 @@ +\section{The Model}\label{sec:model} + +\subsection{Gravitational Waves} + +Any reasonable signal possible to detect on Earth will be of +astrophysical origin. Hence, we can assume that the source is far +enough to treat the waves as plane waves. Therefore, in order for our +model to be able to account for several gravitational waves we have to +expand the formalism for the plane wave solutions presented in +appendix \ref{sec:appendix} to a superposition of multiple waves. We +adopt the viewpoint that $h_{\mu\nu}$ is a symmetric tensor field +(under global Lorentz transformations) defined in quasi-Cartesian +coordinates on a flat Minkowski background spacetime. Since we work +with linearised general relativity we can easily obtain the solution +by superposing the single plane wave solutions +\begin{equation}\label{eq:manywaves} +\bar{h}^{\mu\nu} = \sum_j(A_j)^{\mu\nu}\exp(i(k_j)_\rho x^\rho), +\end{equation} +where $A_j^{\mu\nu}$ are constant components of symmetric tensors and +$(k_j)_\mu$ are the constant, real components of vectors. We assume the +summation convention, but explicitly state the summation over the +index $j$ as it does not run over the coordinate indices, but over all +plane waves present. + +We consider a guage transformation of the same form as the transverse +traceless gauge used in appendix A which is defined as +\begin{equation}\label{eq:manyTT} +\bar{h}^{0a}_{TT} \equiv 0 \,\,\,\,\,\, \text{and} \,\,\,\,\,\, \bar{h}_{TT} \equiv 0, +\end{equation} +where latin alphabet indices run over the spatial dimensions +only. Furthermore the Lorenz gauge condition gives the constraints +\begin{equation} +\partial_0\bar{h}^{00}_{TT} = 0 \,\,\,\,\,\, \text{and} \,\,\,\,\,\, \partial_a\bar{h}^{ab}_{TT} = 0. +\end{equation} +Whilst we cannot consider this gauge to be transverse anymore since we +are considering waves travelling in different directions, we will keep +the labels since we are using exactly the same definition. This +generalisation is straightforward, because of the linear nature of the +field equations in a weak gravitational field. The field tensor +transforms as +\begin{equation}\label{eq:transformation} +\bar{h}^{\mu\nu}_{TT} = \bar{h}^{\mu\nu} - \partial^{\mu}\xi^{\nu} - +\partial^{\nu}\xi^{\mu} + \eta^{\mu\nu}\partial_\rho\xi^\rho, +\end{equation} +where $\xi^\mu$ are four functions that define the gauge +transformation and which must satisfy $\Box^2\xi^\mu$ to preserve the +Lorenz gauge. The solution in \eqref{eq:manywaves} is a linear +superposition of waves with different wavevectors and Each of the +components can be transformed into the TT gauge using some set of +functions $\xi^\mu_j$ (see appendix \ref{sec:appendix} for +details). Therefore, we can transform \eqref{eq:manywaves} using +$\xi^{\prime\mu} = \sum_j \xi^\mu_j$ since the transformation +\eqref{eq:transformation} is linear in $\xi^\mu$. Applying this +transformation gives us the same constraints on the constant +$A_j^{\mu\nu}$ components separately for all values of $j$. Therefore +in the new gauge the coefficients must satisfy +\begin{equation} +(A_j)^{0a}_{TT} = 0 \,\,\,\,\,\, \text{and} \,\,\,\,\,\, ((A_j)_{TT})^\mu_\mu = 0 +\end{equation} +and the Lorenz gauge conditions require that +\begin{equation} +(A_j)^{00}_{TT} = 0 \,\,\,\,\,\, \text{and} \,\,\,\,\,\, (A_j)^{ab}_{TT}k_b = 0. +\end{equation} +Using these conditions we can construct a traceless tensor which +satisfies the definition in \eqref{eq:manyTT}. It is a superposition +of waves of the form \eqref{eq:manywaves} +\begin{equation}\label{eq:manyh} +\bar{h}^{\mu\nu}_{TT} = \sum_j(A_j)_{TT}^{\mu\nu}\exp(i(k_j)_\rho x^\rho), +\end{equation} +where $(A_j)_{TT}^{0\mu} = (A_j)_{TT}^{\mu 0} = 0$, the spatial +components are given by +\begin{equation} +(A_j)^{ab}_{TT} = ((P_j)^a_c(P_j)^b_d - \frac{1}{2}(P_j)^{ab}(P_j)_{cd})(A_j)^{cd}, +\end{equation} +$(P_j)_{ab} \equiv \delta_{ab} - (n_j)_a(n_j)_b$ and $(n_j)_a = +(\hat{k}_j)_a$. + +\subsection{Neutron-neutron Interaction} + +Nucleon-nucleon scattering is fundamentally a many body problem +governed by quantum chromodynamics. However, the strong interaction +has a very short range ($\sim$ a few fm) and $kR \ll 1$, where $k$ is +the neutron wave vector and $R$ the range of the potential. This means +that we only have to consider s-wave scattering and we can approximate +the scattering potential with a delta function $V_{nn} = +U\delta(\vec{r}_1 - \vec{r}_2)$. To obtain the value of $U$ we use the +fact that for s-wave scattering the cross-section is given by $\sigma += 4\pi a_s^2$, where $a_s$ is the scattering length which can be +measured experimentally. Therefore, in the Born approximation we +obtain +\begin{equation}\label{eq:nn} +U = \frac{2 \pi a_s \hbar^2}{m_n}, +\end{equation} +where $m_n$ is the mass of the neutron. Gravitational waves cause the +distance between the two neutrons, $|\vec{r}_1 - \vec{r}_2|$, to +oscillate. However, because the inter-particle interaction is in the +form of a contact potential it remains unaffected in the presence of a +passing wave. The physical spatial separation will only be zero if the +coordinate separation vector will also be zero. + +\subsection{The Schr\"{o}dinger Equation} + +The gravitational wave background is predicted to have no significant +components at wavelengths shorter than a few km which is much larger +than the lengthscales we are considering for the +experiment. Therefore, combined with the fact that we are only +considering weak gravitational fields, we can assume that in order to +model the effect of gravitational waves on the quantum state of two +neutrons in a harmonic trap we do not need to resort to a full quantum +description of the interaction. + +In order to incorporate the effect of passing waves on the quantum +state we begin by considering the two-particle Schr\"{o}dinger +equation for the Hamiltonian given in \eqref{eq:hamiltonian} +\begin{equation}\label{eq:schrodinger} +i\hbar\frac{\partial \Psi(x_1, x_2; t)}{\partial t} = \left[ + -\frac{\hbar^2}{2m}\left( \frac{\partial^2}{\partial x_1^2} + + \frac{\partial^2}{\partial x_2^2} \right) + + \frac{1}{2}m\omega^2(x_1^2 + x_2^2) + V_{nn}(x_1-x_2) + \right]\Psi(x_1, x_2; t). +\end{equation} +We have already shown that the form of $V_{nn}$ is unaffected. The +potential due to the interaction with the harmonic trap also remains +unchanged as it does not depend on the physical separation between two +points. A huge benefit of the gauge we have chosen to work in is the +fact that only the spatial components of the metric are affected by +the gravitational wave which means that the time derivative on the +left hand side does not have to be modified. Only the spatial +derivatives have to be considered in this situation. + +For a general metric the Laplacian of a scalar field is given by +\begin{equation}\label{eq:laplacian} +\nabla^2\phi = \frac{1}{\sqrt{|g|}}\partial_a \left( \sqrt{|g|} g^{ab} +\partial_b\phi \right), +\end{equation} +where $g$ is the determinant of the metric tensor which in the TT +gauge, to first order in $h$, is equal to unity. We now want to reduce +the problem to only one spatial dimension to simplify the model. As +this is only an investigation into the feasibility of such an +experiment such a simplification is desirable as it reduces the +necessary computational time required to obtain qualitatively the same +results. The gravitational waves are very weak so we assume that +confinement in the $y$ and $z$ directions is strong enough that the +wavefunction remains in its ground state along those axes at all +times. This means that we can ignore all second derivative terms apart +from $\frac{\partial^2 \Psi}{\partial x^2}$ since this is the only +significant kinetic energy term. Furthermore, we will have terms of +the form $\partial_a g^{ab} \partial_b \phi$. These terms can also be +ignored, because $\partial_a g^{ab} \propto k_a$ and for any realistic +setup the wavelength of the waves will be much larger than the size of +the expriment making these terms insignificant. Therefore the only +relevant terms that we are left with are +\begin{equation}\label{eq:kinetic} +\nabla^2 \Psi = g^{xx}\frac{\partial^2 \Psi}{\partial x^2}, +\end{equation} +where the effective form of $g^{xx}$ is obtained from \eqref{eq:manyh} +\begin{equation} +g^{xx} = 1 + \sum_i A^{xx}_i \cos(\Omega_i t + \phi_i), +\end{equation} +$\Omega_i$ is the frequency of the $i$th wave, $\phi_i$ is the $i$th +wave's phase at $x=0$ and we have ignored phase variation along the +trap axis, $k_x x$, since we consider wavelengths much larger than the +dimensions of the well. + +The most general form of the one-dimensional Laplacian in the TT gauge +that preserves probability when used in the Sch\"{o}dinger equation is +\begin{equation}\label{eq:general} +\nabla^2 \Psi = g^{xx}\frac{\partial^2 \Psi}{\partial x^2} + +\frac{\partial g^{xx}}{\partial x} \frac{\partial \Psi}{\partial x}. +\end{equation} +This imposes some additional restrictions on the components of the +metric tensor. We again ignore second derivative terms due to +confinement along the other axes. However, we must now require $g^{xy} += g^{xz} = 0$ for equation \eqref{eq:general} to be correct. In order +to investigate gravitational waves with shorter wavelengths we want to +be able to use this equation for the Laplacian. However, under the TT +and Lorenz gauge conditions the additional constraints also require +$g^{xx} = 0$ unless the wavevectors are perpendicular to the +$x$-axis. This in turn implies $\partial_x g^{xx} = 0$ since $k_x = +0$. Therefore, in our investigation we always use equation +\eqref{eq:kinetic}, but for wavelengths comparable to or smaller than +the trap dimensions this limits us to the study of waves perpendicular +to the $x$-axis. + +Neutrons are spin-$\frac{1}{2}$ particles and so also we have to +address the question of how the intrinsic angular momentum of the +particles couples to the gravitational waves. The theoretical +possibility of such coupling was first considered by Kobzarev and Okun +in 1963 \cite{kobzarev}. If we consider a Newtonian gravitational +potential $\phi$ then the coupling to spin would take the form +$H_{int}=A\vec{\sigma}\cdot\nabla\phi$, where $A$ is an amplitude +and $\vec{\sigma}$ is the particle spin. However, this violates the +equivalence principle which states that the trajectory of a point mass +in a gravitational field depends only on its initial position and +velocity, and is independent of its composition. Therefore, we do not +have to take into account any spin-gravity coupling since we are +working in the weak field limit. + +The final form of the two-particle Hamiltonian that we +will be investigating is +\begin{equation}\label{eq:main} +\hat{H} = - g^{xx} \frac{\hbar^2}{2m} \left( + \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial + x_2^2} \right) + \frac{1}{2}m\omega^2(x_1^2 + x_2^2) + +V_{nn}(x_1-x_2). +\end{equation} + diff --git a/nnScatt.pdf b/nnScatt.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b7dcf7aecf420da419e1d3bed9d90118fcceab5a GIT binary patch literal 267843 zcma&sLzFH|@Fw`OZSyVLwr$(CZQZhM+qPYG%hoO1)_nh-=`)Mo^d>V;Y-165e({hg zii*=Q(X+ykjbFtV!LSiA5;+)K!|?Jl$eP((xLOji|5s6F5Vx{*HFG9n5VtjQH4`;6 zaWFOG=ZA4|bv84ygYnE6)zxvqA4Q&P-iHJ@c=J+(rR9FgD))d<3jYx#sw$zJ2%*K^ z7)-OZg(`bET;fa5&D+T1>&o0Oy2xqnb~!(vUNd;TIKJnF_ToM4e>nJhU-&fWJe<~$ zc)X{$`#oNmCKaIiV$d(opU)@!Jx8-g9CyyENqu>H>Lx{Ih%%vwmvc-GCd?=FYyNZ$ 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+\section{Numerical Simulations}\label{sec:numerical} + +\subsection{The Algorithm} + +We solve the two-particle Schr\"{o}dinger equation for the Hamiltonian +\eqref{eq:main} via the explicit staggered method \cite{numerics}. We +choose our units such that $\hbar = 1$ and the neutron mass $m_n = +1$. The wave function is evaluated on a grid of discrete values for +the independent variables +\begin{equation} +\Psi(x_1, x_2, t) = \Psi(x_1 = l\Delta x, x_2 = m\Delta x, t = n\Delta +t) \equiv \Psi^n_{l, m}, +\end{equation} +where $l$, $m$ and $n$ are integers. We separate it into real and +imaginary parts, +\begin{equation} +\Psi^n_{l,m} = u^{n+1}_{l,m} + iv^{n+1}_{l,m}, +\end{equation} +which allows us to solve the Schr\"{o}dinger equation via a finite +difference method with a pair of coupled equations: +\begin{equation} +\begin{array} {lcl} +u^{n+1}_{l,m} & = & u^{n-1}_{l,m} - \left\{ \alpha \left[ v^n_{l+1,m} + + v^n_{l-1,m} + v^n_{l,m+1} + v^n_{l,m-1} - 4v^n_{l,m} \right] - 2 +\Delta t V_{l,m}v^n_{l,m} \right\} , +\end{array} +\end{equation} +\begin{equation} +\begin{array} {lcl} +v^{n+1}_{l,m} & = & v^{n-1}_{l,m} + \left\{ \alpha \left[ u^n_{l+1,m} + + u^n_{l-1,m} + u^n_{l,m+1} + u^n_{l,m-1} - 4u^n_{l,m} \right] - 2 +\Delta t V_{l,m}u^n_{l,m} \right\}, +\end{array} +\end{equation} +where +\begin{equation} +\alpha = g^{xx}\frac{\Delta t}{\Delta x^2}, +\end{equation} +and $V_{l,m}$ is the combined value of the potential due to the +harmonic well and neutron-neutron interaction on the discretised +grid given by +\begin{equation} +V_{l,m} = \frac{1}{2}\omega^2(l^2 + m^2)\Delta x^2 + \nu\delta_{l,m}. +\end{equation} +In order to evaluate the value of the real part at step $n+1$ we need +to know the values of the real part at $n-1$ and the imaginary part at +$n$. The same is true for evaluating the imaginary part. Therefore, we +do not have to calculate both parts at every time step and we compute +the real and imaginary parts at slightly different (staggered) +times. The form of the discretised equations is identical to those +presentied in \cite{numerics}. However, the value of $\alpha$ is now +scaled by the oscillating metric tensor component $g^{xx}$. This novel +modification does not lead to instabilities and simulations have been +confirmed to conserve probability to the same precision as before. + +For the more general Laplacian \eqref{eq:general}, the equations have to +be modified even further and new terms are introduced for the +wavefunction's first derivative terms present. This too, does not +cause instabilities and conserves probability very well. diff --git a/report.tex b/report.tex new file mode 100644 index 0000000..e128ace --- /dev/null +++ b/report.tex @@ -0,0 +1,125 @@ +\documentclass[12pt]{article} +\usepackage{a4} +\usepackage{hyperref} +\usepackage{color} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{amssymb,amsmath} +\usepackage{bm} +\usepackage[lofdepth,lotdepth]{subfig} + +\relpenalty=10000 +\binoppenalty=10000 + +% Set page size: + +\textwidth 160mm +\textheight 235mm +\oddsidemargin 0mm +\topmargin -15mm +\baselineskip 13pt +\parskip 0pc +\parindent 1pc + +% Headers and footer using fancyhdr: + +\definecolor{darkred}{rgb}{0.80,0.00,0.05} +\definecolor{blue}{rgb}{0.00,0.00,0.95} +\definecolor{darkgreen}{rgb}{0.00,0.60,0.0} +\definecolor{gray}{rgb}{0.95,0.9,0.9} + +\title{Theoretical and Numerical Study of Models of Entanglement for + Neutrons} + +\date{14 May, 2012} + +\author{ Candidate Number: 8221T \\ Supervisor: Dr Crispin Barnes } + +\begin{document} + +\maketitle + +\input{abstract} + +\input{introduction} + +\input{experiment} + +\input{model} + +\input{results} + +\input{conclusions} + +\newpage + +\input{appendix} + +\input{decoherence} + +\begin{thebibliography}{9} + +\bibitem{pulsar} J.M. Weisberg, J. H. Taylor, \emph{Relativistic + Binary Pulsar B1913+16: Thirty Years of Observations and Analysis}, + arXiv:astro-ph/0407149v1 (2004). + +\bibitem{hobson} M. P. Hobson, G. Efstathiou, A. N. Lasenby, + \emph{General Relativity: An Introduction for Physicists}, + (ch. 17-18), Cambridge University Press, 2009. ISBN-13 + 978-0-521-82951-9. + +\bibitem{epr} + A. Einstein, B. Podolsky, N. Rosen, + \emph{Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?} + Phys. Rev. 47, 777–780 + (1935). + +\bibitem{aspect} A. Aspect, J. Dalibard, G. Roger, \emph{Experimental + Test of Bell's Inequalities Using Time-Varying Analyzers} + Phys. Rev. Lett. 49, 1804-1807 (1982). + +\bibitem{steane} A. Steane, \emph{Quantum Computing}, + arXiv:quant-ph/9708022v2 (1997). + +\bibitem{zurek} W. H. Zurek, \emph{Decoherence and the transition from + quantum to classical -- REVISITED} arXiv:quant-ph/0306072v1 (2003). + +\bibitem{edmund} E. T. Owen, M. C. Dean, C. H. W. Barnes, + \emph{Generation of Entanglement between Qubits in a One-Dimensional + Harmonic Oscillator}, Phys. Rev. A 85, 022319 (2012). + +\bibitem{rings} + http://www.johnstonsarchive.net/relativity/pictures.html + +\bibitem{blundell} S. J. Blundell, J. A. C. Bland, \emph{Polarized + Neutron Reflection as a Probe of Magnetic Films and Multilayers}, + Phys. Rev. B 46, 3391-3400 (1992). + +\bibitem{kobzarev} I.Y. Kobzarev, L.B. Okun, \emph{Gravitational + Interaction of Fermions}, Soviet Journal of Experimental and + Theoretical Physics, Vol. 16, p.1343 (1963). + +\bibitem{decoherence} T. Fujisawa, T. Hayashi, H. D. Cheong, + Y. H. Jeong, Y. Hirayama, \emph{Rotation and Phase-shift Operations + for a Charge Qubit in a Double Quantum Dot}, Phys. Rev. Lett 91 + 226804-1 (2003). + +\bibitem{numerics} J. J. V. Maestri, R. H. Landau, M. J. Paez, + \emph{Two-Particle Schroedinger Equation Animations of + Wavepacket-Wavepacket Scattering (revised)}, + arXiv:physics/9909042v1 [physics.comp-ph] (2008). + +\bibitem{feynman} R. P. Feynman, F. L. Vernon Jr., \emph{The theory of + a general quantum system interacting with a linear dissipative + system}, Annals Phys. 24, 118-173 (1963). + +\bibitem{caldeira} A. O. Caldeira, A. J. Leggett, \emph{Path Integral + Approach to Quantum Brownian Motion}, Physica 121A, 587-616 (1983). + +\bibitem{master} W. G. Unruh, W. H. Zurek, \emph{Reduction of a wave + packet in quantum Brownian motion}, Phys. Rev. D. 40, 1071-1094 + (1984). + +\end{thebibliography} + +\end{document} diff --git a/results.tex b/results.tex new file mode 100644 index 0000000..78b6a07 --- /dev/null +++ b/results.tex @@ -0,0 +1,212 @@ +\section{Numerical Simulations}\label{sec:numerical} + +We want to study the effect of gravitational waves on the entanglement +and distinguishability of the final state from one unaffected by the +radiation. The von Neumann entropy of entanglement, +\begin{equation} +S = -\text{Tr}(\rho_1 \log_2 \rho_1) = -\text{Tr}(\rho_2 \log_2 +\rho_2), +\end{equation} +where $\rho_i$ is the reduced density matrix of particle $i$, is a +standard measure of entanglement of two particles. The fidelity of +quantum states is a suitable measure of their distinguishability. It +is given by +\begin{equation} +\mathcal{F} = | \langle \psi_0 | \psi_{gw} \rangle |, +\end{equation} +where $| \psi_{gw} \rangle$ is the state calculated in the presence of +a gravitational wave background and $| \psi_0 \rangle$ is the state +calculated in their absence. $\mathcal{F}^2$ is then just the +probability of observing the outgoing particle in the state predicted +for a flat spacetime and will always be equal to unity if $| \psi_{gw} +\rangle = | \psi_0 \rangle $. + +\begin{figure}[htbp] + \begin{center} + \subfloat[][\label{fig:HighFreq}]{\includegraphics[width=75mm]{Images/HighFreq.pdf}} + \qquad + \subfloat[][\label{fig:LowFreq}]{\includegraphics[width=75mm]{Images/LowFreq.pdf}} + \caption{\label{fig:Fidelity} Fidelities of the wavefunction in + the presence of a single gravitational wave at different + frequencies to the unaffected state. \subref{fig:HighFreq} + $\Omega = 100\omega$. The large drop at $t=\pi/\omega$ is + evidence that interaction enhances the effect of the + gravitational waves. \subref{fig:LowFreq} $\Omega = + 10\omega$. We see no evidence of interaction at $t=\pi/\omega$ + as it has no effect in this regime. } + \end{center} +\end{figure} + +We can distinguish two regimes for gravitational waves interacting +with a two particle quantum system in a harmonic trap based on their +relation to the interaction time, $\tau$, which is the length of time +during which the wave packets overlap is significant. Firstly, there +are high frequency waves for which $\Omega \gg \tau^{-1}$. Figure +\ref{fig:HighFreq} shows the evolution of fidelity over a single +collision for a single wave of frequency $\Omega = 100 \omega$. The +biggest change in fidelity occurs during the collision itself when the +wave packets overlap, interact and entangle. Outside the collision, +when the overlap is small the fidelity simply oscillates with an +amplitude that increases with the particle's speed. We clearly see +that the effect of the gravitational wave is amplified in the +interaction which leads to a permanent change in the value of +$\mathcal{F}$. The change in fidelity is dominated by the change in +the entanglement of the two particles due to the radiation. The low +frequency regime is defined as $\Omega \ll \tau^{-1}$. Figure +\ref{fig:LowFreq} clearly shows that such waves have a larger effect +on the fidelity, but it is independent of the interaction. This effect +is classical and the low fidelity is due to a change of the classical +trajectory and final position of the particle in the harmonic +potential. We will not consider low frequency waves in this +investigation anymore since if we were intending on studying such +waves we would not need to resort to a quantum mechanical model to +describe their effect on the particles. + +We see from figure \ref{fig:Freqs} that the high frequency regime +begins beyond $10 \omega$ as the inter-particle interaction starts +having a visible effect on the fidelity of the final state. These +results show that entanglement may be more useful in detecting high +frequency gravitational waves rather than a general stochastic +background since the effect of lower frequencies, which affect the +classical behaviour of the system, are much larger. Figure +\ref{fig:Freqs} shows that in the high frequency regime the quantum +effect of the waves on the fidelity can be even $10^6$ larger than the +effect due to the change in the classical trajectory. This +amplification is likely to be even stronger for higher frequencies +when we would no longer be able to ignore the first derivative terms +in equation \eqref{eq:laplacian}. Unfortunately, the gravitational +wave background is predicted not to have any significant components +above \mbox{100 kHz}, which will usually be smaller than or comparable +to the value of $\omega$ for a multilayer trap. Therefore, +entanglement is not a very useful mechanism for detecting background +radiation. + +\begin{figure}[htbp] + \begin{center} + \includegraphics[width=120mm]{Images/Freqs.pdf} + \caption{\label{fig:Freqs} The fidelity change of the wavefunction + in the presence of a single gravitational wave to an unaffected + state for two different interaction strengths. We observe + resonance at $\omega$ which is in the classical low frequency + regime. The interplay between the gravitational waves and + entanglement begins beyond $\Omega = 10 \hbar \omega$.} + \end{center} +\end{figure} + +Figure \ref{fig:EntanglementSingle} shows the von Neumann entropy of +entanglement for a single collision. The form of the entropy does not +change at all as a function of gravitational radiation. However, its +final value does vary in the presence of waves, but the effect is very +small compared to the change due to the particle dynamics in the +collision itself. In the high frequency regime the waves' effect on +the entanglement is larger than their effect on the trajectories of +the particles, therefore, the fidelity of the final state to the +unaffected wavefunction is a better measure of the change in +entanglement of the system than entropy itself. + +\begin{figure}[htbp] + \begin{center} + \includegraphics[width=120mm]{Images/EntanglementSingleWave.pdf} + \caption{\label{fig:EntanglementSingle} The von Neumann entropy of + entanglement of the wavefunction in the presence of a single + gravitational wave. Oscillations are present, but they are + significantly smaller than the change in entropy due to the + interaction.} + \end{center} +\end{figure} + +The values of the $S$ and $\mathcal{F}$ are plotted over a suitable +range of interaction strengths in figure \ref{fig:Uplot} for a single +high frequency wave. Whilst the largest change in fidelity does not +coincide with the case where the final state is maximally entangled, +the fidelity changes the most when entanglement is produced. This +suggests that the amplification of the effect is due to the wave's +influence on the system's entanglement. Following the fidelity and von +Neumann entropy for multiple collisions confirms the correlation +between fidelity and entropy. The dynamics of particles will cause the +entanglement to increase and decrease during the collisions and the +change in fidelity follows the same trend. + +\begin{figure}[htbp] + \begin{center} + \includegraphics[width=120mm]{Images/Uplot.pdf} + \caption{\label{fig:Uplot} The values of the von Neumann entropy + of entanglement and fidelity of the state affected by waves to + the unaffected state after a single collision. Entanglement + amplifies the effect of the gravitational wave on the + wavefunction, but the maximum value of entropy does not coincide + with the minimum value of fidelity.} + \end{center} +\end{figure} + +To compare the effects of different radiation strengths on the quantum +state we define the intensity of the signal to be $I = \sum_j +(A^{xx}_j)^2$ which is proportional to the energy flux of the wave +\cite{hobson}. All of the results produced so far were calculated for +the largest possible value, $I = 0.01$. Higher intensities are not +investigated since in those cases the weak field approximation is no +longer valid. However, simulating realistic values of strain is beyond +even machine double precision. In order to estimate the size of the +effect of gravitational waves on the quantum state the value of +fidelity is evaluated for different intensities that are within +machine precision. The results in figure \ref{fig:WaveMags} show that +the change in the final value of fidelity is proportional to the +intensity of the waves. This is a very useful relationship as it lets +us easily estimate the size of the gravitational waves' effect for +different intensities. The square of the fidelity $\mathcal{F}^2$ is +just the probability of observing the final state to be that predicted +for a flat spacetime. Therefore, if we measure the final state +(e.g. by measuring the spin of outgoing neutrons) several times then +on average $1 - \mathcal{F}^2$ of the time we should obtain a +different result than predicted by the dynamics of the particles alone +in the absence of radiation. From the results in figure +\ref{fig:WaveMags} we find that $1 - \mathcal{F} = 0.5I$. Therefore, +the probability, $p$, of measuring a different state than expected can +be shown to be +\begin{equation}\label{eq:prob} +p = 1 - \mathcal{F}^2 \approx I. +\end{equation} +Different wave combinations will lead to different numerical +prefactors as the relationship between $1 - \mathcal{F}$ and $I$ is +linear regardless of the frequency regime and number of waves. The +highest intensity we can investigate in the weak field limit and high +frequency regime gives $p=0.01$. This is a small value, but +potentially measurable. However, extrapolating to the expected +detectable values of strain gives $p \approx 10^{-40}$. Assuming +Poissonian statistics in the spin counting process this requires +$\sim10^{27}$ measurements in order to make the error smaller than the +signal which shows that such an experiment is impractical. + +\begin{figure}[htbp] + \begin{center} + \includegraphics[width=120mm]{Images/WaveMags.pdf} + \caption{\label{fig:WaveMags} The change in fidelity $\Delta = 1 - + \mathcal{F}$ plotted against the total intensity. This change is + proportional to the total intensity $\Delta = 0.5I$ over a very + large range of values. The relationship is linear regardless of + the number of waves or the frequency components present. } + \end{center} +\end{figure} + +We can now answer the question whether gravitational waves can be +detected using neutrons in the suggested experiment. We have already +discussed the applicability of the proposed scheme to detecting the +gravitational wave background and concluded that the experiment's +frequency range is too high and the classical effect of low frequency +waves is much larger anyway. We have also shown that the probability +of measuring an effect due to gravitational waves scales linearly with +the wave intensity which means that for realistic wave amplitude +values we will observe a different state only $\sim 10^{-40}$ of the +time. Additionally, it can be shown that for typical multilayer +dimensions and potentials the value of the neutron interaction in +\eqref{eq:nn} will be $U \approx 10^7 \hbar\omega$ which corresponds +to a point beyond the plotted range in figure \ref{fig:Uplot}. This +means that two neutrons introduced in coherent states on either side +of the well will not entangle in the collision and hence the quantum +effect of even high frequency gravitational waves will be minimal. The +neutrons will simply bounce of each other and behave classically. The +most significant effect the waves have on such a system is to cause +the physical separation of the two particles to oscillate and in this +case two neutrons in a multilayer are not the most optimal system for +detecting gravitational waves. +