Git repo for MSci thesis
This commit is contained in:
Wojciech Kozlowski
commit
8eceb64521
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Images/EntanglementMaximisationResults/EntanglementMax.dat
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Images/EntanglementMaximisationResults/entanglement.eps
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||||||
|
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
|
||||||
|
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||||||
|
/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
|
||||||
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BIN
Images/entanglement.pdf
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40
README.rst
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40
README.rst
Normal file
@ -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