commit 8eceb645211a72edf22976b44d9af36176c234d2 Author: Wojciech Kozlowski Date: Sun Mar 19 00:02:11 2017 +0000 Git repo for MSci thesis 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 +3.322563 1.000717e+00 +3.322563 1.000717e+00 +3.322563 1.000717e+00 +3.322564 1.000717e+00 +3.322564 1.000717e+00 +3.322565 1.000717e+00 +3.322604 1.000717e+00 +3.317032 1.000715e+00 +3.324693 1.000717e+00 +3.398753 1.000352e+00 +4.244425 9.593967e-01 +4.244425 9.593967e-01 +6.058607 7.835530e-01 +8.994015 5.339690e-01 diff --git a/Images/EntanglementMaximisationResults/EntanglementMaxResult.dat b/Images/EntanglementMaximisationResults/EntanglementMaxResult.dat new file mode 100644 index 0000000..ea56b01 --- /dev/null +++ b/Images/EntanglementMaximisationResults/EntanglementMaxResult.dat @@ -0,0 +1 @@ +3.322563e+03 -1.000717e+00 diff --git a/Images/EntanglementMaximisationResults/entanglement.eps b/Images/EntanglementMaximisationResults/entanglement.eps new file mode 100644 index 0000000..80b0706 --- /dev/null +++ b/Images/EntanglementMaximisationResults/entanglement.eps @@ -0,0 +1,711 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: entanglement.eps +%%Creator: gnuplot 4.4 patchlevel 3 +%%CreationDate: Mon Apr 9 00:36:34 2012 +%%DocumentFonts: (atend) +%%BoundingBox: 50 50 410 302 +%%EndComments +%%BeginProlog +/gnudict 256 dict def +gnudict begin +% +% The following true/false flags may be edited by hand if desired. +% The unit line width and grayscale image gamma correction may also be changed. +% +/Color true def +/Blacktext false def +/Solid false def +/Dashlength 1 def +/Landscape false def +/Level1 false def +/Rounded false def +/ClipToBoundingBox false def +/TransparentPatterns false def +/gnulinewidth 5.000 def +/userlinewidth gnulinewidth def +/Gamma 1.0 def +% +/vshift -46 def +/dl1 { + 10.0 Dashlength mul mul + Rounded { currentlinewidth 0.75 mul sub dup 0 le { pop 0.01 } if } if +} def +/dl2 { + 10.0 Dashlength mul mul + Rounded { currentlinewidth 0.75 mul add } if +} def +/hpt_ 31.5 def +/vpt_ 31.5 def +/hpt hpt_ def +/vpt vpt_ def +Level1 {} { +/SDict 10 dict def +systemdict /pdfmark known not { + userdict /pdfmark systemdict /cleartomark get put +} if +SDict begin [ + /Title (entanglement.eps) + /Subject (gnuplot plot) + /Creator (gnuplot 4.4 patchlevel 3) + /Author (wojtek) +% /Producer (gnuplot) +% /Keywords () + /CreationDate (Mon Apr 9 00:36:34 2012) + /DOCINFO pdfmark +end +} ifelse +/doclip { + ClipToBoundingBox { + newpath 50 50 moveto 410 50 lineto 410 302 lineto 50 302 lineto closepath + clip + } if +} def +% +% Gnuplot Prolog Version 4.4 (August 2010) +% +%/SuppressPDFMark true def +% +/M {moveto} bind def +/L {lineto} bind def +/R {rmoveto} bind def +/V {rlineto} bind def +/N {newpath moveto} bind def +/Z {closepath} bind def +/C {setrgbcolor} bind def +/f {rlineto fill} bind def +/g {setgray} bind def +/Gshow {show} def % May be redefined later in the file to support UTF-8 +/vpt2 vpt 2 mul def +/hpt2 hpt 2 mul def +/Lshow {currentpoint stroke M 0 vshift R + Blacktext {gsave 0 setgray show grestore} {show} ifelse} def +/Rshow {currentpoint stroke M dup stringwidth pop neg vshift R + Blacktext {gsave 0 setgray show grestore} {show} ifelse} def +/Cshow {currentpoint stroke M dup stringwidth pop -2 div vshift R + Blacktext {gsave 0 setgray show grestore} {show} ifelse} def +/UP {dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def + /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def} def +/DL {Color {setrgbcolor Solid {pop []} if 0 setdash} + {pop pop pop 0 setgray Solid {pop []} if 0 setdash} ifelse} def +/BL {stroke userlinewidth 2 mul setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +/AL {stroke userlinewidth 2 div setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +/UL {dup gnulinewidth mul /userlinewidth exch def + dup 1 lt {pop 1} if 10 mul /udl exch def} def +/PL {stroke userlinewidth setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +3.8 setmiterlimit +% Default Line colors +/LCw {1 1 1} def +/LCb {0 0 0} def +/LCa {0 0 0} def +/LC0 {1 0 0} def +/LC1 {0 1 0} def +/LC2 {0 0 1} def +/LC3 {1 0 1} def +/LC4 {0 1 1} def +/LC5 {1 1 0} def +/LC6 {0 0 0} def +/LC7 {1 0.3 0} def +/LC8 {0.5 0.5 0.5} def +% Default Line Types +/LTw {PL [] 1 setgray} def +/LTb {BL [] LCb DL} def +/LTa {AL [1 udl mul 2 udl mul] 0 setdash LCa setrgbcolor} def +/LT0 {PL [] LC0 DL} def +/LT1 {PL [4 dl1 2 dl2] LC1 DL} def +/LT2 {PL [2 dl1 3 dl2] LC2 DL} def +/LT3 {PL [1 dl1 1.5 dl2] LC3 DL} def +/LT4 {PL [6 dl1 2 dl2 1 dl1 2 dl2] LC4 DL} def +/LT5 {PL [3 dl1 3 dl2 1 dl1 3 dl2] LC5 DL} def +/LT6 {PL [2 dl1 2 dl2 2 dl1 6 dl2] LC6 DL} def +/LT7 {PL [1 dl1 2 dl2 6 dl1 2 dl2 1 dl1 2 dl2] LC7 DL} def +/LT8 {PL [2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 4 dl2] LC8 DL} def +/Pnt {stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore} def +/Dia {stroke [] 0 setdash 2 copy vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke + Pnt} def +/Pls {stroke [] 0 setdash vpt sub M 0 vpt2 V + currentpoint stroke M + hpt neg vpt neg R hpt2 0 V stroke + } def +/Box {stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke + Pnt} def +/Crs {stroke [] 0 setdash exch hpt sub exch vpt add M + hpt2 vpt2 neg V currentpoint stroke M + hpt2 neg 0 R hpt2 vpt2 V stroke} def +/TriU {stroke [] 0 setdash 2 copy 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 + Pnt} def +/Star {2 copy Pls Crs} def +/BoxF {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 fill} def +/TriUF {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 fill} def +/TriD {stroke [] 0 setdash 2 copy 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 + Pnt} def +/TriDF {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 fill} def +/DiaF {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath fill} def +/Pent {stroke [] 0 setdash 2 copy gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore Pnt} def +/PentF {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath fill grestore} def +/Circle {stroke [] 0 setdash 2 copy + hpt 0 360 arc stroke Pnt} def +/CircleF {stroke [] 0 setdash hpt 0 360 arc fill} def +/C0 {BL [] 0 setdash 2 copy moveto vpt 90 450 arc} bind def +/C1 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + vpt 0 360 arc closepath} bind def +/C2 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C3 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C4 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C5 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc + 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc} bind def +/C6 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C7 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C8 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C9 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 450 arc closepath fill + vpt 0 360 arc closepath} bind def +/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill + 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C11 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C12 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C13 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C14 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 360 arc closepath fill + vpt 0 360 arc} bind def +/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto + neg 0 rlineto closepath} bind def +/Square {dup Rec} bind def +/Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def +/S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def +/S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def +/S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def +/S3 {BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare} bind def +/S4 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def +/S5 {BL [] 0 setdash 2 copy 2 copy vpt Square fill + exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def +/S6 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare} bind def +/S7 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill + 2 copy vpt Square fill Bsquare} bind def +/S8 {BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare} bind def +/S9 {BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare} bind def +/S10 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill + Bsquare} bind def +/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill + Bsquare} bind def +/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def +/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy vpt Square fill Bsquare} bind def +/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def +/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def +/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def +/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def +/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def +/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def +/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def +/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def +/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def +/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def +/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def +/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def +/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def +/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def +/D12 {gsave translate 45 rotate 0 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 +686 588 M +63 0 V +6198 0 R +-63 0 V +stroke +602 588 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0)] +] -46.7 MRshow +1.000 UL +LTb +686 1302 M +63 0 V +6198 0 R +-63 0 V +stroke +602 1302 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.2)] +] -46.7 MRshow +1.000 UL +LTb +686 2016 M +63 0 V +6198 0 R +-63 0 V +stroke +602 2016 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.4)] +] -46.7 MRshow +1.000 UL +LTb +686 2730 M +63 0 V +6198 0 R +-63 0 V +stroke +602 2730 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.6)] +] -46.7 MRshow +1.000 UL +LTb +686 3443 M +63 0 V +6198 0 R +-63 0 V +stroke +602 3443 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.8)] +] -46.7 MRshow +1.000 UL +LTb +686 4157 M +63 0 V +6198 0 R +-63 0 V +stroke +602 4157 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 1)] +] -46.7 MRshow +1.000 UL +LTb +686 4871 M +63 0 V +6198 0 R +-63 0 V +stroke +602 4871 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 1.2)] +] -46.7 MRshow +1.000 UL +LTb +686 588 M +0 63 V +0 4220 R +0 -63 V +stroke +686 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0)] +] -46.7 MCshow +1.000 UL +LTb +1382 588 M +0 63 V +0 4220 R +0 -63 V +stroke +1382 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 1)] +] -46.7 MCshow +1.000 UL +LTb +2077 588 M +0 63 V +0 4220 R +0 -63 V +stroke +2077 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 2)] +] -46.7 MCshow +1.000 UL +LTb +2773 588 M +0 63 V +0 4220 R +0 -63 V +stroke +2773 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 3)] +] -46.7 MCshow +1.000 UL +LTb +3469 588 M +0 63 V +0 4220 R +0 -63 V +stroke +3469 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 4)] +] -46.7 MCshow +1.000 UL +LTb +4164 588 M +0 63 V +0 4220 R +0 -63 V +stroke +4164 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 5)] +] -46.7 MCshow +1.000 UL +LTb +4860 588 M +0 63 V +0 4220 R +0 -63 V +stroke +4860 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 6)] +] -46.7 MCshow +1.000 UL +LTb +5556 588 M +0 63 V +0 4220 R +0 -63 V +stroke +5556 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 7)] +] -46.7 MCshow +1.000 UL +LTb +6251 588 M +0 63 V +0 4220 R +0 -63 V +stroke +6251 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 8)] +] -46.7 MCshow +1.000 UL +LTb +6947 588 M +0 63 V +0 4220 R +0 -63 V +stroke +6947 448 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 9)] +] -46.7 MCshow +1.000 UL +LTb +1.000 UL +LTb +686 4871 N +686 588 L +6261 0 V +0 4283 V +-6261 0 V +Z stroke +LCb setrgbcolor +112 2729 M +currentpoint gsave translate -270 rotate 0 0 moveto +[ [(Helvetica) 140.0 0.0 true true 0 (von Neumann Entropy of Entanglement)] +] -46.7 MCshow +grestore +LTb +LCb setrgbcolor +3816 238 M +[ [(Helvetica) 140.0 0.0 true true 0 (U \(units of )] +XYsave +[(Times-Italic) 240.0 36.4 true true 0 (-)] +XYrestore +[(Times-Italic) 140.0 0.0 true true 0 (h)] +[(Helvetica) 140.0 0.0 true true 0 ( )] +[(Symbol) 140.0 0.0 true true 0 (w)] +[(Helvetica) 140.0 0.0 true true 0 (\))] +] -92.1 MCshow +LTb +1.000 UP +1.000 UL +LTb +% Begin plot #1 +1.000 UL +LT0 +/Helvetica findfont 140 scalefont setfont +686 588 M +860 770 L +275 634 V +6 16 V +456 1216 V +2686 4107 L +173 43 V +138 10 V +-3 0 V +5 0 V +51 -2 V +589 -146 V +4901 3385 L +6943 2494 L +% End plot #1 +stroke +LTb +686 4871 N +686 588 L +6261 0 V +0 4283 V +-6261 0 V +Z stroke +1.000 UP +1.000 UL +LTb +stroke +grestore +end +showpage +%%Trailer +%%DocumentFonts: Symbol Times-Italic Helvetica diff --git a/Images/EntanglementMaximisationResults/entanglement.pdf b/Images/EntanglementMaximisationResults/entanglement.pdf new file mode 100644 index 0000000..b4ae925 Binary files /dev/null and b/Images/EntanglementMaximisationResults/entanglement.pdf differ diff --git a/Images/EntanglementSingleWave.pdf b/Images/EntanglementSingleWave.pdf new file mode 100644 index 0000000..4f05e98 Binary files /dev/null and b/Images/EntanglementSingleWave.pdf differ diff --git a/Images/FidelityHundredWaves.pdf b/Images/FidelityHundredWaves.pdf new file mode 100644 index 0000000..81c4f65 Binary files /dev/null and b/Images/FidelityHundredWaves.pdf differ diff --git a/Images/FidelityThousandWaves.pdf b/Images/FidelityThousandWaves.pdf new file mode 100644 index 0000000..f446aa2 Binary files /dev/null and b/Images/FidelityThousandWaves.pdf differ diff --git a/Images/Freqs.pdf b/Images/Freqs.pdf new file mode 100644 index 0000000..353c46b Binary files /dev/null and b/Images/Freqs.pdf differ diff --git a/Images/HighFreq.pdf b/Images/HighFreq.pdf new file mode 100644 index 0000000..05c0321 Binary files /dev/null and b/Images/HighFreq.pdf differ diff --git a/Images/LowFreq.pdf b/Images/LowFreq.pdf new file mode 100644 index 0000000..405a162 Binary files /dev/null and b/Images/LowFreq.pdf differ diff --git a/Images/NWaves.pdf b/Images/NWaves.pdf new file mode 100644 index 0000000..812837a Binary files /dev/null and b/Images/NWaves.pdf differ diff --git a/Images/Uplot.pdf b/Images/Uplot.pdf new file mode 100644 index 0000000..a48fbd8 Binary files /dev/null and b/Images/Uplot.pdf differ diff --git a/Images/WaveMags.pdf b/Images/WaveMags.pdf new file mode 100644 index 0000000..7737397 Binary files /dev/null and b/Images/WaveMags.pdf differ diff --git a/Images/collision.eps b/Images/collision.eps new file mode 100644 index 0000000..75126a7 --- /dev/null +++ b/Images/collision.eps @@ -0,0 +1,825 @@ +%!PS-Adobe-2.0 EPSF-2.0 +%%Title: collision.eps +%%Creator: gnuplot 4.4 patchlevel 3 +%%CreationDate: Mon Apr 9 23:05:33 2012 +%%DocumentFonts: (atend) +%%BoundingBox: 50 50 410 302 +%%EndComments +%%BeginProlog +/gnudict 256 dict def +gnudict begin +% +% The following true/false flags may be edited by hand if desired. +% The unit line width and grayscale image gamma correction may also be changed. +% +/Color true def +/Blacktext false def +/Solid false def +/Dashlength 1 def +/Landscape false def +/Level1 false def +/Rounded false def +/ClipToBoundingBox false def +/TransparentPatterns false def +/gnulinewidth 5.000 def +/userlinewidth gnulinewidth def +/Gamma 1.0 def +% +/vshift -46 def +/dl1 { + 10.0 Dashlength mul mul + Rounded { currentlinewidth 0.75 mul sub dup 0 le { pop 0.01 } if } if +} def +/dl2 { + 10.0 Dashlength mul mul + Rounded { currentlinewidth 0.75 mul add } if +} def +/hpt_ 31.5 def +/vpt_ 31.5 def +/hpt hpt_ def +/vpt vpt_ def +Level1 {} { +/SDict 10 dict def +systemdict /pdfmark known not { + userdict /pdfmark systemdict /cleartomark get put +} if +SDict begin [ + /Title (collision.eps) + /Subject (gnuplot plot) + /Creator (gnuplot 4.4 patchlevel 3) + /Author (wojtek) +% /Producer (gnuplot) +% /Keywords () + /CreationDate (Mon Apr 9 23:05:33 2012) + /DOCINFO pdfmark +end +} ifelse +/doclip { + ClipToBoundingBox { + newpath 50 50 moveto 410 50 lineto 410 302 lineto 50 302 lineto closepath + clip + } if +} def +% +% Gnuplot Prolog Version 4.4 (August 2010) +% +%/SuppressPDFMark true def +% +/M {moveto} bind def +/L {lineto} bind def +/R {rmoveto} bind def +/V {rlineto} bind def +/N {newpath moveto} bind def +/Z {closepath} bind def +/C {setrgbcolor} bind def +/f {rlineto fill} bind def +/g {setgray} bind def +/Gshow {show} def % May be redefined later in the file to support UTF-8 +/vpt2 vpt 2 mul def +/hpt2 hpt 2 mul def +/Lshow {currentpoint stroke M 0 vshift R + Blacktext {gsave 0 setgray show grestore} {show} ifelse} def +/Rshow {currentpoint stroke M dup stringwidth pop neg vshift R + Blacktext {gsave 0 setgray show grestore} {show} ifelse} def +/Cshow {currentpoint stroke M dup stringwidth pop -2 div vshift R + Blacktext {gsave 0 setgray show grestore} {show} ifelse} def +/UP {dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def + /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def} def +/DL {Color {setrgbcolor Solid {pop []} if 0 setdash} + {pop pop pop 0 setgray Solid {pop []} if 0 setdash} ifelse} def +/BL {stroke userlinewidth 2 mul setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +/AL {stroke userlinewidth 2 div setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +/UL {dup gnulinewidth mul /userlinewidth exch def + dup 1 lt {pop 1} if 10 mul /udl exch def} def +/PL {stroke userlinewidth setlinewidth + Rounded {1 setlinejoin 1 setlinecap} if} def +3.8 setmiterlimit +% Default Line colors +/LCw {1 1 1} def +/LCb {0 0 0} def +/LCa {0 0 0} def +/LC0 {1 0 0} def +/LC1 {0 1 0} def +/LC2 {0 0 1} def +/LC3 {1 0 1} def +/LC4 {0 1 1} def +/LC5 {1 1 0} def +/LC6 {0 0 0} def +/LC7 {1 0.3 0} def +/LC8 {0.5 0.5 0.5} def +% Default Line Types +/LTw {PL [] 1 setgray} def +/LTb {BL [] LCb DL} def +/LTa {AL [1 udl mul 2 udl mul] 0 setdash LCa setrgbcolor} def +/LT0 {PL [] LC0 DL} def +/LT1 {PL [4 dl1 2 dl2] LC1 DL} def +/LT2 {PL [2 dl1 3 dl2] LC2 DL} def +/LT3 {PL [1 dl1 1.5 dl2] LC3 DL} def +/LT4 {PL [6 dl1 2 dl2 1 dl1 2 dl2] LC4 DL} def +/LT5 {PL [3 dl1 3 dl2 1 dl1 3 dl2] LC5 DL} def +/LT6 {PL [2 dl1 2 dl2 2 dl1 6 dl2] LC6 DL} def +/LT7 {PL [1 dl1 2 dl2 6 dl1 2 dl2 1 dl1 2 dl2] LC7 DL} def +/LT8 {PL [2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 4 dl2] LC8 DL} def +/Pnt {stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore} def +/Dia {stroke [] 0 setdash 2 copy vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath stroke + Pnt} def +/Pls {stroke [] 0 setdash vpt sub M 0 vpt2 V + currentpoint stroke M + hpt neg vpt neg R hpt2 0 V stroke + } def +/Box {stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M + 0 vpt2 neg V hpt2 0 V 0 vpt2 V + hpt2 neg 0 V closepath stroke + Pnt} def +/Crs {stroke [] 0 setdash exch hpt sub exch vpt add M + hpt2 vpt2 neg V currentpoint stroke M + hpt2 neg 0 R hpt2 vpt2 V stroke} def +/TriU {stroke [] 0 setdash 2 copy 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 + Pnt} def +/Star {2 copy Pls Crs} def +/BoxF {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 fill} def +/TriUF {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 fill} def +/TriD {stroke [] 0 setdash 2 copy 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 + Pnt} def +/TriDF {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 fill} def +/DiaF {stroke [] 0 setdash vpt add M + hpt neg vpt neg V hpt vpt neg V + hpt vpt V hpt neg vpt V closepath fill} def +/Pent {stroke [] 0 setdash 2 copy gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath stroke grestore Pnt} def +/PentF {stroke [] 0 setdash gsave + translate 0 hpt M 4 {72 rotate 0 hpt L} repeat + closepath fill grestore} def +/Circle {stroke [] 0 setdash 2 copy + hpt 0 360 arc stroke Pnt} def +/CircleF {stroke [] 0 setdash hpt 0 360 arc fill} def +/C0 {BL [] 0 setdash 2 copy moveto vpt 90 450 arc} bind def +/C1 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + vpt 0 360 arc closepath} bind def +/C2 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C3 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C4 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C5 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc + 2 copy moveto + 2 copy vpt 180 270 arc closepath fill + vpt 0 360 arc} bind def +/C6 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C7 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 270 arc closepath fill + vpt 0 360 arc closepath} bind def +/C8 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C9 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 270 450 arc closepath fill + vpt 0 360 arc closepath} bind def +/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill + 2 copy moveto + 2 copy vpt 90 180 arc closepath fill + vpt 0 360 arc closepath} bind def +/C11 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 180 arc closepath fill + 2 copy moveto + 2 copy vpt 270 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C12 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C13 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 0 90 arc closepath fill + 2 copy moveto + 2 copy vpt 180 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/C14 {BL [] 0 setdash 2 copy moveto + 2 copy vpt 90 360 arc closepath fill + vpt 0 360 arc} bind def +/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill + vpt 0 360 arc closepath} bind def +/Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto + neg 0 rlineto closepath} bind def +/Square {dup Rec} bind def +/Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def +/S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def +/S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def +/S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def +/S3 {BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare} bind def +/S4 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def +/S5 {BL [] 0 setdash 2 copy 2 copy vpt Square fill + exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def +/S6 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare} bind def +/S7 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill + 2 copy vpt Square fill Bsquare} bind def +/S8 {BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare} bind def +/S9 {BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare} bind def +/S10 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill + Bsquare} bind def +/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill + Bsquare} bind def +/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def +/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy vpt Square fill Bsquare} bind def +/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill + 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def +/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def +/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def +/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def +/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def +/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def +/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def +/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def +/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def +/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def +/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def +/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def +/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def +/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def +/D12 {gsave translate 45 rotate 0 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 UL +LTb +686 2660 M +63 0 V +6198 0 R +-63 0 V +stroke +602 2660 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.8)] +] -46.7 MRshow +1.000 UL +LTb +686 3212 M +63 0 V +6198 0 R +-63 0 V +stroke +602 3212 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 1)] +] -46.7 MRshow +1.000 UL +LTb +686 3765 M +63 0 V +6198 0 R +-63 0 V +stroke +602 3765 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 1.2)] +] -46.7 MRshow +1.000 UL +LTb +686 4318 M +63 0 V +6198 0 R +-63 0 V +stroke +602 4318 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 1.4)] +] -46.7 MRshow +1.000 UL +LTb +686 4871 M +63 0 V +6198 0 R +-63 0 V +stroke +602 4871 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 1.6)] +] -46.7 MRshow +1.000 UL +LTb +686 448 M +0 63 V +0 4360 R +0 -63 V +stroke +686 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0)] +] -46.7 MCshow +1.000 UL +LTb +1312 448 M +0 63 V +0 4360 R +0 -63 V +stroke +1312 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.1)] +] -46.7 MCshow +1.000 UL +LTb +1938 448 M +0 63 V +0 4360 R +0 -63 V +stroke +1938 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.2)] +] -46.7 MCshow +1.000 UL +LTb +2564 448 M +0 63 V +0 4360 R +0 -63 V +stroke +2564 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.3)] +] -46.7 MCshow +1.000 UL +LTb +3190 448 M +0 63 V +0 4360 R +0 -63 V +stroke +3190 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.4)] +] -46.7 MCshow +1.000 UL +LTb +3817 448 M +0 63 V +0 4360 R +0 -63 V +stroke +3817 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.5)] +] -46.7 MCshow +1.000 UL +LTb +4443 448 M +0 63 V +0 4360 R +0 -63 V +stroke +4443 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.6)] +] -46.7 MCshow +1.000 UL +LTb +5069 448 M +0 63 V +0 4360 R +0 -63 V +stroke +5069 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.7)] +] -46.7 MCshow +1.000 UL +LTb +5695 448 M +0 63 V +0 4360 R +0 -63 V +stroke +5695 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.8)] +] -46.7 MCshow +1.000 UL +LTb +6321 448 M +0 63 V +0 4360 R +0 -63 V +stroke +6321 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 0.9)] +] -46.7 MCshow +1.000 UL +LTb +6947 448 M +0 63 V +0 4360 R +0 -63 V +stroke +6947 308 M +[ [(Helvetica) 140.0 0.0 true true 0 ( 1)] +] -46.7 MCshow +1.000 UL +LTb +1.000 UL +LTb +686 4871 N +686 448 L +6261 0 V +0 4423 V +-6261 0 V +Z stroke +LCb setrgbcolor +112 2659 M +currentpoint gsave translate -270 rotate 0 0 moveto +[ [(Helvetica) 140.0 0.0 true true 0 (von Neumann Entropy of Entanglement)] +] -46.7 MCshow +grestore +LTb +LCb setrgbcolor +3816 98 M +[ [(Helvetica) 140.0 0.0 true true 0 (Time \(in units of )] +[(Symbol) 140.0 0.0 true true 0 (p)] +[(Helvetica) 140.0 0.0 true true 0 (/)] +[(Symbol) 140.0 0.0 true true 0 (w)] +[(Helvetica) 140.0 0.0 true true 0 (\))] +] -46.7 MCshow +LTb +1.000 UP +1.000 UL +LTb +% Begin plot #1 +1.000 UL +LT0 +/Helvetica findfont 140 scalefont setfont +686 448 M +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +62 1 V +63 1 V +63 3 V +62 11 V +63 25 V +62 57 V +63 114 V +62 208 V +63 332 V +63 474 V +62 598 V +63 661 V +62 631 V +63 510 V +62 325 V +63 117 V +63 -70 V +62 -203 V +63 -261 V +62 -255 V +63 -206 V +63 -142 V +62 -85 V +63 -45 V +62 -21 V +63 -9 V +62 -3 V +63 -1 V +63 0 V +62 -1 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +63 0 V +62 0 V +63 0 V +62 0 V +63 0 V +% End plot #1 +stroke +LTb +686 4871 N +686 448 L +6261 0 V +0 4423 V +-6261 0 V +Z stroke +1.000 UP +1.000 UL +LTb +stroke +grestore +end +showpage +%%Trailer +%%DocumentFonts: Symbol Helvetica diff --git a/Images/collision.pdf b/Images/collision.pdf new file mode 100644 index 0000000..b589a81 --- /dev/null +++ b/Images/collision.pdf @@ -0,0 +1,109 @@ +%PDF-1.4 +%쏢 +5 0 obj +<> +stream +xMO1+|C߾VB=#D"wQ}㱝R+̻O Tn:9er}22X#b7`j~o J_;u(L B[G)b:N7RȪjZ2!Jpdz!4x\bY3{f=-;j +\Q@)0 7t\EGs%it]m"̆%9Fދ&~O>:{;SJf~5no,;@[8>媫E#WH(;5 M?,2yRT+1K2b[Pe_n'R\+1n䭹}%drj ZcM;|tK(aehX8$۩UElZVt̓*CCc4xi@zL;cAx{C;A!b]y;h] ZsXSh*ޢ*``UsM>ȋy@~L 2^A"~ŏjG> +/Contents 5 0 R +>> +endobj +3 0 obj +<< /Type /Pages /Kids [ +4 0 R +] /Count 1 +>> +endobj +1 0 obj +<> +endobj +7 0 obj +<>endobj +10 0 obj +<> +endobj +11 0 obj +<> +endobj +8 0 obj +<> +endobj +9 0 obj +<> +endobj +12 0 obj +<> +endobj +13 0 obj +<>stream + + + + + +Mon -Ap-r T 9: 2:3:05::3 +Mon -Ap-r T 9: 2:3:05::3 +gnuplot 4.4 patchlevel 3 + +collision.epswojtekgnuplot plot + + + + + +endstream +endobj +2 0 obj +<>endobj +xref +0 14 +0000000000 65535 f +0000001202 00000 n +0000003170 00000 n +0000001143 00000 n +0000000992 00000 n +0000000015 00000 n +0000000973 00000 n +0000001267 00000 n +0000001377 00000 n +0000001441 00000 n +0000001308 00000 n +0000001338 00000 n +0000001518 00000 n +0000001613 00000 n +trailer +<< /Size 14 /Root 1 0 R /Info 2 0 R +/ID [<90FA299A9EFC1FBEBCD9D659175EC8A1><90FA299A9EFC1FBEBCD9D659175EC8A1>] +>> +startxref +3390 +%%EOF diff --git a/Images/entanglement.pdf b/Images/entanglement.pdf new file mode 100644 index 0000000..b4ae925 Binary files /dev/null and b/Images/entanglement.pdf differ diff --git a/Images/rings.gif b/Images/rings.gif new file mode 100755 index 0000000..5d8c409 Binary files /dev/null and b/Images/rings.gif differ diff --git a/Images/rings.pdf b/Images/rings.pdf new file mode 100644 index 0000000..7db938a Binary files /dev/null and b/Images/rings.pdf differ diff --git a/README.rst b/README.rst new file mode 100644 index 0000000..1587134 --- /dev/null +++ b/README.rst @@ -0,0 +1,40 @@ +Theoretical and Numerical Study of Models of Entanglement for Neutrons +====================================================================== + +Part III project submitted as part of the Experimental and Theoretical +Physics course of the Natural Sciences Tripos at the University of +Cambridge. + +Abstract +-------- + +We propose and investigate a scheme for detecting gravitational waves +using the entanglement generated by the dynamics of a pair of neutrons +trapped in a harmonic well. We develop a model that combines the +effects due to plane gravitational wave solutions of the linearised +field equations in the weak-field limit of general relativity with the +time-dependent Schrödinger equation. Numerical simulations show that +entanglement amplifies the effect of high frequency gravitational +waves on the quantum state. In the proposed experiment, for realistic +wave amplitudes and frequencies, the final state is practically +indistinguishable from one unaffected by gravitational +radiation. However, the results also show that quantum entanglement +could be used in the future for high frequency gravitational wave +detection. + +Download +-------- + +You can obtain the final PDF version from https://wojciechkozlowski.eu/dphil/files/msci_thesis.pdf + +You can obtain the source files by either: + +- cloning the git repository with ``git clone https://gitlab.wojciechkozlowski.eu/wojtek/msci-thesis.git`` + +- downloading them as a zip archive from https://gitlab.wojciechkozlowski.eu/wojtek/msci-thesis/repository/archive.zip?ref=master + +- downloading them as a tar.gz archive from https://gitlab.wojciechkozlowski.eu/wojtek/msci-thesis/repository/archive.tar.gz?ref=master + +- downloading them as a tar.bz2 archive from https://gitlab.wojciechkozlowski.eu/wojtek/msci-thesis/repository/archive.tar.bz2?ref=master + +- downloading them as a tar archive from https://gitlab.wojciechkozlowski.eu/wojtek/msci-thesis/repository/archive.tar?ref=master diff --git a/abstract.tex b/abstract.tex new file mode 100644 index 0000000..405ba85 --- /dev/null +++ b/abstract.tex @@ -0,0 +1,17 @@ +\begin{abstract} + +We propose and investigate a scheme for detecting gravitational waves +using the entanglement generated by the dynamics of a pair of neutrons +trapped in a harmonic well. We develop a model that combines the +effects due to plane gravitational wave solutions of the linearised +field equations in the weak-field limit of general relativity with the +time-dependent Schr\"{o}dinger equation. Numerical simulations show +that entanglement amplifies the effect of high frequency gravitational +waves on the quantum state. In the proposed experiment, for realistic +wave amplitudes and frequencies, the final state is practically +indistinguishable from one unaffected by gravitational +radiation. However, the results also show that quantum entanglement +could be used in the future for high frequency gravitational wave +detection. + +\end{abstract} diff --git a/appendix.tex b/appendix.tex new file mode 100644 index 0000000..fb4fd3d --- /dev/null +++ b/appendix.tex @@ -0,0 +1,190 @@ +\appendix +\section{\large Gravitational Waves in Linearised General Relativity}\label{sec:appendix} + +In general relativity we consider pseudo-Riemannian manifolds in which +the interval is given by (assuming the summation convention) +\begin{equation} +ds^2=g_{\mu\nu}(x)dx^{\mu}dx^{\nu} +\end{equation} +where $g_{\mu\nu}$ are the components of the metric tensor field in +the chosen coordinate system. A weak gravitational field corresponds +to a region of spacetime that is only slightly curved. Thus, there +exist coordinate systems $x^{\mu}$ in which the spacetime metric takes +the form +\begin{equation}\label{eq:linear} +g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} +\end{equation} +where $|h_{\mu\nu}| \ll 1$, the partial derivatives of $h_{\mu\nu}$ +are also small and $\eta_{\mu\nu}$ are the components of the metric +tensor in flat Minkowski spacetime. + +It can be shown that for infinitesimal general coordinate +transformations of the form +\begin{equation} +x^{\prime\mu} = x^{\mu}+\xi^{\mu}(x), +\end{equation} +working to first order in small quantities, the metric transforms as +follows: +\begin{equation} +g^{\prime}_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu} - \partial_\mu\xi_\nu +- \partial_\nu\xi_\mu. +\end{equation} +We note that $g^{\prime}_{\mu\nu}$ is of the same form +as \eqref{eq:linear}, hence the new metric perturbation functions are +related to the old ones via +\begin{equation}\label{eq:gauge} +h^{\prime}_{\mu\nu} = h_{\mu\nu} - \partial_\mu\xi_\nu +- \partial_\nu\xi_\mu. +\end{equation} + +We consider $h_{\mu\nu}$ to be a tensor field defined on the flat +Minkowski background spacetime, hence \eqref{eq:gauge} can be +considered as analogous to a gauge transformation in +electromagnetism. If $h_{\mu\nu}$ is a solution to the linearised +gravitational field equations then the same +physical situation is desribed by \eqref{eq:gauge}. However, in this +viewpoint \eqref{eq:gauge} is viewed as a gauge transformation rather +than a coordinate transformation. We are still working in the same set +of coordinates $x^\mu$ and have defined a new tensor whose components +in this coordinate system are given by \eqref{eq:gauge}. Here 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. + +The linearised field equations of general relativity can be written as +\begin{equation} +\Box^2\bar{h}^{\mu\nu} = -2\kappa T^{\mu\nu}, +\end{equation} +where $T^{\mu\nu}$ is the energy-momentum tensor, +$\bar{h}^{\mu\nu} \equiv h^{\mu\nu} - \frac{1}{2}\eta^{\mu\nu}h$, $h = +h^\mu_\mu$, provided that the $\bar{h}^{\mu\nu}$ components satisfy +the Lorenz gauge condition +\begin{equation} +\partial_\mu\bar{h}^{\mu\nu} = 0. +\end{equation} +In vacuo, where $T^{\mu\nu} = 0$, the general solution of the +linearised field equations may be written as a superposition of +plane-wave solutions of the form +\begin{equation}\label{eq:wave} +\bar{h}^{\mu\nu} = A^{\mu\nu}\exp(ik_\rho x^\rho), +\end{equation} +where $A^{\mu\nu}$ are constant components of a symmetric tensor and +$k_\mu$ are the constant, real components of a vector. The Lorenz gauge +condition is satisfied provided +\begin{equation} +A^{\mu\nu}k_\nu = 0. +\end{equation} +Physical solutions may be obtained by taking the real part +of \eqref{eq:wave}. + +Further gauge transformations will preserve the Lorenz gauge condition +provided that the four functions $\xi^\mu(x)$ satisfy $\Box^2\xi^\mu = +0$. A common choice for plane gravitational waves is the +transverse-traceless gauge defined by choosing +\begin{equation}\label{eq:TT} +\bar{h}^{0i}_{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}\label{eq:lorenzTT} +\partial_0\bar{h}^{00}_{TT} = 0 \,\,\,\,\,\, \text{and} \,\,\,\,\,\, \partial_i\bar{h}^{ij}_{TT} = 0. +\end{equation} +We note that the first condition in \eqref{eq:lorenzTT} implies that +for non-stationary perturbations $\bar{h}^{00}_{TT}$ also +vanishes. Therefore, in this gauge only the spatial components +$\bar{h}^{ij}_{TT}$ are non-zero. + +For a particular case of an arbitrary plane gravitational wave of the +form \eqref{eq:wave} the conditions \eqref{eq:TT} imply that +\begin{equation} +A^{0i}_{TT} = 0 \,\,\,\,\,\, \text{and} \,\,\,\,\,\, (A_{TT})^\mu_\mu = 0 +\end{equation} +and the Lorenz gauge conditions in \eqref{eq:lorenzTT} require that +\begin{equation} +A^{00}_{TT} = 0 \,\,\,\,\,\, \text{and} \,\,\,\,\,\, A^{ij}_{TT}k_j = 0. +\end{equation} +Under these conditions the $A^{\mu\nu}_{TT}$ components are given by +\begin{equation} +A^{ij}_{TT} = (P^i_kP^j_l - \frac{1}{2}P^{ij}P_{kl})A^{kl}, +\end{equation} +where $P_{ij} \equiv \delta_{ij} - n_in_j$ and $n_i = \hat{k}_i$ \cite{hobson}. + +\section{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{samepage} +\begin{equation} +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{equation} +\begin{equation} +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{equation} +\end{samepage} +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 +\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] \right. \\\\ & & \left. + \beta \left[ v^n_{l+1,m} - + v^n_{l-1,m} + v^n_{l, m+1} - v^n_{l, m-1} \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] \right. \\\\ & & \left. + \beta \left[ + u^n_{l+1,m} - u^n_{l-1,m} + u^n_{l, m+1} - u^n_{l, m-1} \right] - 2 +\Delta t V_{l,m}u^n_{l,m} \right\}, +\end{array} +\end{equation} +where +\begin{equation} +\beta = \frac{\partial g^{xx}}{\partial x} \frac{\Delta t}{2 \Delta x}. +\end{equation} +The two components can still be evaluated at staggered times. This +further modification does not cause instabilities and conserves +probability very well. diff --git a/conclusions.tex b/conclusions.tex new file mode 100644 index 0000000..c1ba1f5 --- /dev/null +++ b/conclusions.tex @@ -0,0 +1,46 @@ +\section{Conclusions}\label{sec:conclusions} + +We have proposed and investigated the feasibility of an experiment to +detect gravitational waves using the entanglement of two neutrons +trapped in a harmonic well. The quantum dynamics of the two particles +lead to entanglement which is affected by the presence of +gravitational radiation. We have shown that entanglement amplifies the +effect of high frequency gravitational waves on the +wavefunction. However, for realistic values of the wave amplitudes the +effect is too small to be measured in a device with the dimensions of +a typical multilayer. + +The effects of gravitational waves were combined with the quantum +dynamics of the two neutrons in the weak-field limit, where the +linearised field equation could be used. The effects of the +oscillating metric were combined with the Schr\"{o}dinger equation by +modifying the kinetic energy term. The potential due to the harmonic +well and the inter-nucleon interaction remained unchanged. + +Numerical solutions to the modified time-dependent Schr\"{o}dinger +equation were obtained with the explicit staggered method. It was +shown that there are two different behaviour regimes. For low +frequency waves there are no additional quantum effects and the +difference in the final quantum state is due to the different +classical particle trajectories. These waves were not investigated as +there are better ways of detecting them. However, the high frequency +regime couples with the particle interaction and the effect is +strongly dependent on its strength and is maximised close to the value +for which the particles maximally entangle. This is an interesting +result as it is a possible mechanism for detecting high frequency +gravitational waves. However, for any experimentally accessible values +we would not observe anything as the neutron-neutron interaction is +too strong for any entanglement to be generated via the system's +dynamics alone. + +This experiment is not solely limited by the size of the signal. One +other issue that would be difficult to resolve when building such an +experiment is the isolation of the system from all environmental +effects except for the gravitational waves. We have shown that the +effect of radiation on the quantum state is extremely small and that +entanglement is a key element, but with current technologies it is +difficult to even maintain entangled resources for times longer than a +few nanoseconds \cite{decoherence} unless we are working with trapped +particles in ultra-high vacuum. Limiting undesirable decoherence is +difficult with current technology and so any effects due to +gravitational waves would be unobservable. diff --git a/decoherence.tex b/decoherence.tex new file mode 100644 index 0000000..65dd1ce --- /dev/null +++ b/decoherence.tex @@ -0,0 +1,101 @@ +\section{Decoherence}\label{sec:decoherence} + +Decoherence is the process through which quantum correlations disperse +information throughout the degrees of freedom that are effectively +beyond the observer's control. It has been suggested as the mechanism +through which a measurement of a quantum system is effectively made +\cite{zurek}. In general, any interaction with the environment will +eventually lead to decoherence and this is one of the main obstacles +in generating and maintaining entangled resources. At the beginning of +this investigation we proposed that entanglement may be sensitive +enough to be affected by gravitational waves. In our approach we have +only investigated the dynamics of two neutrons in a harmonic well +subject to gravitational radiation by solving the Schr\"{o}dinger +equation for the modified Hamiltonian \eqref{eq:main}. This has +allowed us to study the effects of the waves on the wavefunction of +neutrons entangling in a harmonic potential, but the fact that we keep +track of all of the degrees of freedom meant that it was impossible to +observe decoherence. + +In order to be able to describe decoherence we have to construct a +dynamical model of a system coupled to its environment. In the +Feynman-Vernon theory of the influence functional \cite{feynman} we +consider a system A interacting with a second system B (the reservoir) +described by the Hamiltonian +\begin{equation} +H = H_A + H_I + H_B, +\end{equation} +where system A consists of a single particle, the reservior consists of +N particles and $H_I$ is the interaction Hamiltonian. We can obtain +the reduced density operator for system A by tracing out all the +environment coordinates. Assuming that the initial density operator is +separable into a product of the density operators for A and B we get +\begin{equation} +\rho(x, y, t) = \int \mathrm{d} x^\prime \mathrm{d} y^\prime J(x, y, +t; x^\prime, y^\prime, 0)\rho_A(x^\prime, y^\prime, 0), +\end{equation} +where x and y are the position coordinates of the single particle, +\begin{equation} +J(x, y, t; x^\prime, y^\prime, 0) = \int \int \mathrm{D} x \mathrm{D} +y \exp \left[ i \frac{S_A[x]}{\hbar} \right] \exp \left[ -i + \frac{S_A[y]}{\hbar} \right] F[x, y], +\end{equation} +$F[x, y]$ is the so called influence functional given by +\begin{equation} +\begin{array} {lcl} +F[x, y] & = & \int \mathrm{d} \bm{R^\prime} \mathrm{d} \bm{Q^\prime} +\mathrm{d} \bm{R} \rho_B(\bm{R^\prime}, \bm{Q^\prime}, 0) \int \int +\mathrm{D} \bm{R} \mathrm{D} \bm{Q} \\\\ & & \times \exp \left[ + \frac{i}{\hbar} \left( S_I[x, \bm{R}] - S_I[y, \bm{Q}] + S_B[\bm{R}] + - S_B[\bm{Q}] \right) \right], +\end{array} +\end{equation} +$S_i$ is the action corresponding to the Hamiltonian $H_i$ and +$\bm{R}$, $\bm{Q}$ are vectors of the positions of the reservoir +particles. This has been solved exactly in \cite{feynman} in the limit +of a weak coupling, where we only have to consider the linear response +of the reservoir to the system and we can describe the environment in +the harmonic approximation. + +The gravitational wave background could potentially be modelled with a +collection of harmonic oscillators. However, as we have seen in +section 3 the interaction of the wave function with gravitational +waves is implemented through a modification of the kinetic energy term +and not via an effective potential and modelling the coupling with the +system with a linear response would be insufficient. The more +complicated interaction would lead to an intractable form for the +influence functional. Furthermore, there is no way of obtaining an +irreversible process such as decoherence using the above solution. We +do not consider a reservoir of infinite size and any information lost +by the system will return within a finite period of time. + +Caldeira and Leggett addressed the issue of irreversibility in 1983 +\cite{caldeira} in an attempt to solve the problem of quantum Brownian +motion. Instead of coupling the system A to a reservoir B they +replaced the system-environment coupling with an externally applied +classical force $F(t)$. The fluctuating force has to obey the standard +stochastic relations +\begin{equation} +\langle F(t) \rangle = 0, +\end{equation} +\begin{equation} +\langle F(t) F(t^\prime) \rangle = 2 \eta k_B T \delta (t - t^\prime), +\end{equation} +where $\eta$ is a damping constant, $k_B$ is Boltzmann's constant and +$T$ is the temperature. However, there are issues with applying the +Caldeira-Leggett model to our problem regarding the nature of the +effect produced by the waves. In the situation presented in figure +\ref{fig:rings} all of the test particles have constant spatial +coordinates, they are stationary. However, the measured separation +between the points changes according to $l^2 = -g_{ij} \xi^i +\xi^j$. Modelling this effect via a classical force is questionable. + +In order to describe decoherence a master equation approach is +necessary, but the most successful models for dissipation due to a +system's interaction with the environment are not appropriate for the +interaction of a quantum wave function with gravitational waves. The +environment has also been modelled with a quantum field \cite{master} +and for a certain class of interactions the approach is equivalent to +the Caldeira-Leggett model. If a quantum field theory of gravity was +developed then a similar approach could be used to derive a master +equation to model decoherence due to gravitational fields. diff --git a/experiment.tex b/experiment.tex new file mode 100644 index 0000000..ee99925 --- /dev/null +++ b/experiment.tex @@ -0,0 +1,131 @@ +\section{The Experiment}\label{sec:experiment} + +\subsection{Gravitational Waves} + +The strength of gravitational radiation on Earth from typical +astrophysical sources is very small. The largest strains one might +expect to measure are of order $10^{-21}$ \cite{hobson}. Therefore, we +make the physical assumption that the gravitational fields are +weak. Mathematically this corresponds to linearising the gravitational +field equations. The basic mathematical framework of gravitational +waves in linearised general relativity is summarised in appendix +\ref{sec:appendix}. It is possible to choose a gauge in which the +coordinate separation between the particles is constant at all times, +but this has no coordinate-invariant physical \mbox{meaning +\cite{hobson}}. However, the physical spatial separation $l$, which is +given by $l^2 = -g_{ij}\xi^i\xi^j$, will vary since the metric tensor +components are not constant in the presence of a gravitational +wave. The effect of a single passing gravitational wave on a cloud of +non-interacting test particles is illustrated in figure +\ref{fig:rings}. + +\begin{figure}[htbp] + \begin{center} + \includegraphics[width=150mm]{Images/rings.pdf} + \caption{\label{fig:rings} The effect of a passing gravitational + wave on a circle of test particles. The two different + polarisations are at $45^\circ$ to each other. Figure reproduced + with permission \cite{rings}.} + \end{center} +\end{figure} + +\subsection{The Experimental Setup} + +We consider a one-dimensional system where the two neutrons with +opposite spins interact in a quantum harmonic oscillator. The two +particles are introduced in coherent states at rest on either side of +the potential. They will oscillate in the harmonic trap, interact and +eventually tunnel out. Repeating this experiment several times for the +same initial condition will reveal information about the state and +entanglement generated within the multilayer. If all unwanted effects +are reduced to acceptable levels then the results will depend on the +presence of gravitational waves if they have an effect on the +entanglement of the two neutrons. + +Such a trap can be constructed using a multilayer of ultrathin +magnetic films of suitably chosen materials using molecular beam +epitaxy. The detailed behaviour of neutrons in such multilayers is +presented in \cite{blundell} where polarized neutron reflection was +used to study the magnetic properties of thin films. The potential +energy in the $\alpha$th region can be written as a sum of a nuclear +and magnetic term +\begin{equation}\label{eq:pot} +V_\alpha = \frac{\hbar^2}{2\pi m_n}\rho_\alpha b_\alpha - +\boldsymbol{\mu_n}\cdot\boldsymbol{B_\alpha}, +\end{equation} +where $\boldsymbol{\mu_\alpha}$, $b_\alpha$, $\boldsymbol{B_\alpha}$ +and $\rho_\alpha$ are the neutron moment, coherent nuclear scattering +length, magnetic field due to the magnetization in the region and +atomic density, respectively. The harmonic oscillator potential itself +can be formed from layers with no magnetization as long as the +thickness of a single film is much less than the particle wavelength +so the discrete nature of the material can be ignored. The second term +in \eqref{eq:pot} leads to different values for the potential +depending on the orientation of the neutron's spin relative to the +magnetic field. Hence, we can use magnetized layers as spin dependent +barriers. By placing such barriers at the two ends of the trap we +provide means for a spin measurement. + +It has been shown that under certain approximations the quantum +dynamics of such a system are sufficient to generate maximally +entangled states under appropriate conditions \cite{edmund}. Producing +maximally entangled states with high fidelity is not necessary in this +experiment, but the work by Owen et al. provides a useful framework to +describe and model the entanglement in a harmonic trap. + +The two particles are initialized separately such that their single +particle wave functions are spatially distinct at $t = 0$. We consider +neutrons, mass $m$, in a harmonic potential of the form $V = +\frac{1}{2}m \omega^2x^2$ initially in the coherent states +$\psi_{L/R}(x)$, where +\begin{equation} +\psi_R = A\exp\left( -\frac{m\omega}{2\hbar}(x - x_0)^2 \right) , +\end{equation} +\begin{equation} +\psi_L(x) = \psi_R(-x), +\end{equation} +$x_0 > 0$ 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 0000000..b7dcf7a Binary files /dev/null and b/nnScatt.pdf differ diff --git a/numerical.tex b/numerical.tex new file mode 100644 index 0000000..cd711a1 --- /dev/null +++ b/numerical.tex @@ -0,0 +1,59 @@ +\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. +