Git repo for MSci thesis
This commit is contained in:
Wojciech Kozlowski
commit
8eceb64521
22
Images/EntanglementMaximisationResults/EntanglementMax.dat
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% 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
|
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BIN
Images/entanglement.pdf
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README.rst
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README.rst
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@ -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
|
||||
17
abstract.tex
Normal file
17
abstract.tex
Normal file
@ -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}
|
||||
190
appendix.tex
Normal file
190
appendix.tex
Normal file
@ -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.
|
||||
46
conclusions.tex
Normal file
46
conclusions.tex
Normal file
@ -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.
|
||||
101
decoherence.tex
Normal file
101
decoherence.tex
Normal file
@ -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.
|
||||
131
experiment.tex
Normal file
131
experiment.tex
Normal file
@ -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}.
|
||||
|
||||
76
introduction.tex
Normal file
76
introduction.tex
Normal file
@ -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}.
|
||||
|
||||
207
model.tex
Normal file
207
model.tex
Normal file
@ -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}
|
||||
|
||||
BIN
nnScatt.pdf
Normal file
BIN
nnScatt.pdf
Normal file
Binary file not shown.
59
numerical.tex
Normal file
59
numerical.tex
Normal file
@ -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.
|
||||
125
report.tex
Normal file
125
report.tex
Normal file
@ -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}
|
||||
212
results.tex
Normal file
212
results.tex
Normal file
@ -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.
|
||||
|
||||
Reference in New Issue
Block a user