Physica.dev

Anomalous spin precession systematic effects in the search for a muon EDM using the frozen-spin technique

2024-03-12T00:00:00+02:00

At the Paul Scherrer Institut (PSI), we are developing a high-precision apparatus with the aim of searching for the muon electric dipole moment (EDM) with unprecedented sensitivity. The underpinning principle of this experiment is the frozen-spin technique, a method that suppresses the spin precession due to the anomalous magnetic moment, thereby enhancing the signal-to-noise ratio for EDM signals.

This increased sensitivity enables measurements that would be difficult to achieve with conventional $g-2$ muon storage rings. Given the availability of the 125 MeV/$c$ muon beam at PSI, the anticipated statistical sensitivity for the EDM after a year of data collection is $6\times 10^{-23} e\cdot$cm. To achieve this goal, it is imperative to do a detailed analysis of any potential spurious effects that could mimic EDM signals. In this study, we present a quantitative methodology to evaluate the systematic effects that might arise in the context of the frozen-spin technique utilised within a compact storage ring. Our approach involves the analytical derivation of equations governing the motion of the muon spin in the electromagnetic (EM) fields intrinsic to the experimental setup, validated through numerical simulations. We also illustrate a method to calculate the cumulative geometric (Berry’s) phase. This work complements ongoing experimental efforts to detect a muon EDM at PSI and contributes to a broader understanding of spin-precession systematic effects.

Link to paper

Animated 3D plot in spherical coordinates with gnuplot

2022-05-21T00:00:00+03:00

I am working on a physics experiment that involves muons that are stored in a magnetic field and are traveling along circular orbits. Their motion inside the magnetic field is quite complex and involves oscillations with different periods – essentially they are like a spinning top, but instead of precessing around only one axis, they can precess around three.

What I am interested in is to track the spin of the muons through time and in order to visualize things better I wanted to have an animated 3D plot of the vector of the spin as the muons experience the changing magnetic field. My favorite plotting tool for 3D graphs is gnuplot.

The code to generate this plot is a bit large and messy, but it may be useful to someone so I’m posting it here:

# Initial spin vector
Sx = -0.0
Sy = -1.0
Sz = -0.0

# Transform to spherical coordinates
phi_0 = atan2(Sx, Sy) # rad
theta_0 = atan2(sqrt(Sx**2+Sy**2), Sz)-pi/2 # rad

# Spin motion equations
Eex = 0.05
e_z = 2.956e-3*(1.0+Eex) # rad/ns
om_z = 2*pi/42000*Eex/0.05  # rad/ns
omega = 0.03264 # rad/ns

f(x) = (191.49*(Eex) + 178.05*0.06/27.94/omega)*(cos(omega*x))
l(x) = -(((sin(omega*x+phi_0)))*81.75e-3/omega)
p(x) =  phi_0 -om_z*x
if (om_z != 0) {
    e(x) = -e_z/om_z*sin(p(x))
} else {
    e(x) = e_z*x*cos(phi_0)
}
c =    (f(0))*sin(p(0)) + l(0)*cos(p(0)) + e(0)
g(x) = (f(x))*sin(p(x)) + l(x)*cos(p(x)) + e(x) + theta_0 * 1e6 - c
# End of spin motion equations

# Plotting
set key samplen 1 height 5 bottom
set view 70, 340, 1.6, 1.6
set view equal xyz
set parametric
unset xtics
unset ytics
unset ztics
set border 0
set xlabel 'p'
set ylabel 'z'
set isosamples 12
set xyplane -0.5
set grid x y
set xrange [-1:1]
set yrange [-1:1]
set zrange [-1:1]
set urange [0:2*pi]
set vrange [-pi:pi]


phi(x) = (g(x) - theta_0*1e6 + c) / 20 + theta_0 - c / 1e6
the(x) = p(x) + f(x) * sin(phi(x)) / 20
av_the(x) = p(x)
av_phi(x) = e(x) / 20

av_x(u, v) = v*cos(av_the(u))*sin(av_phi(u))
av_y(u, v) = v*cos(av_the(u))*cos(av_phi(u))
av_z(u, v) = v*sin(av_the(u))

fx(u,v) = v*cos(the(u))*sin(phi(u))
fy(u,v) = v*cos(the(u))*cos(phi(u))
fz(u,v) = v*sin(the(u))

# Parametric functions for the sphere
r = 1.0
sx(u,v) = r*cos(u)*sin(v)
sy(u,v) = r*cos(u)*cos(v)
sz(u,v) = r*sin(u)

# Plot axis vectors
set arrow 2 from 0,0,0 to 0,-1.2,0 lc rgb 'gray40' lw 2 back
set arrow 5 from 0,0,0 to -1.2,0,0 lc rgb 'gray40' lw 2 back
set arrow 6 from 0,0,0 to 0,0,-1.2 lc rgb 'gray40' lw 2 back

# Plot magnetic fields in the top right corner
o = 0.3
o0 = 1.5
set arrow 7 from o0,o0,o to o0,o0,(o+sin(omega*t)/10)
set arrow 8 from o0,o0,o to 1.2,o0,o
set arrow 9 from o0,o0,o to o0,(o0+sin(omega*t)/8),o
set label 1 'B_r' at o0+0.05, o0+0.08, o+0.05
set label 2 'B_z' at 1.2,o0,o-0.05
set label 3 'B_p' at 1.6, 1.6, o-0.15

set term pngcairo size 800,800 enhanced
outfile = sprintf('animation/3d_vector_plot%05.0f.png', t)
newtitle = sprintf('t = %i ns', t)
set title newtitle
set output outfile

set object 99 circle at 0,0,0 radius 0.025 front fc rgb "red" fs solid
set arrow 1 from fx(t,-1),fy(t,-1),fz(t,-1) to fx(t, 1), fy(t, 1), fz(t, 1) back lc rgb 'black' lw 2

set multiplot
# Plot the sphere, the equator and zero meridian
splot sx(u,v),sy(u,v),sz(u,0) lc 'black' dt 3 notitle,\
    sx(0, v), sy(0,v), sz(0, 0) lc rgb '#8888dd' notitle,\
    0, sy(pi,u), sz(u,0) lc rgb '#dd8888' notitle

# Plot an arrow pointing to the average movement
set arrow 3 from 0,0,0 to av_x(t, 0.5), av_y(t, 0.5), 0 lc rgb 'red' back

# Change the range
set urange [0:t]
splot av_x(u, 0.46), av_y(u, 0.46),av_z(0, 0.46)  lw 2 lc rgb 'red' notitle,\
      av_x(u, 1), av_y(u,1), av_z(u,1) lw 2 lc rgb 'red' t 'Average value'

# Plot the spin trajectory, but only a part of it
set key height 10 bottom
start = t - 400 > 0 ? t - 400 : 0
set urange [start:t]
splot fx(u, 1), fy(u, 1), fz(u, 1) t 'Spin position' lc rgb '#57a757'

unset multiplot

print 'Time step: ', t, ' ns'

t = t + step
unset arrow 1
if (t < t_end) reread;

To control the start time, step and end time set theese:

$ gnuplot
    > t = 0
    > t_end = 4000
    > step = 20
    >
    > load 'plot_3d_sphere.gp'

Note: be sure that you have a folder like animation/ to store all the files!

The Prophecy

2021-11-30T00:00:00+02:00

Here I wanted to experiment with something different. I drew a ‘normal’ picture using the Sketchbook app. Then I used GIMP to add a different character to it. First I used the ‘Sobel’ edge detection algorithm and then I applied some method that clusters the colors in the image. You can see below the end result and after that the original image.

Click on the images to download in full size.

The Elders

2021-11-08T00:00:00+02:00

Orion the hunter rising over the Great Pyramides.

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The big mantle of the white dwarf

2021-11-05T00:00:00+02:00

Drawing of the planetary nebula Messier 57 (Ring Nebula).

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Conquest of Paradise

2021-10-17T00:00:00+03:00

A ship sailing towards an unknown land - his path illuminated by a distant galaxy.

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Messier Thirty-One

2021-10-13T00:00:00+03:00

Drawing of the Andromeda galaxy. It was supposed to be part of a more complex picture, but I liked it the way it is.

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The Pleiades

2021-10-11T00:00:00+03:00

Drawing of the Pleiades visible in a drop of water fallen on a leaf.

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Red moon rising

2021-10-07T00:00:00+03:00

A drawing of a blood red moon over a sand dune. I drew it in anticpation of the new Dune movie. Inspired by the style of this artist.

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Mariner 10 passing near Venus

2021-10-05T00:00:00+03:00

A drawing of the Mariner 10 spacecraft passing near Venus on its long journey to Mercury.

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A spotted sun rising behind the hills

2021-10-02T00:00:00+03:00

Digital drawing of the Sun rising behind a hill.

This is a reimagined¹ version of an “Astronomy Picture of the Day” (APOD) photograph.

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Reimagined as in how Holywood “reimagines” the same movies every 10 years. Credit to Dr. Ivo Iliev for the term. ↩

Drawing of a sunny beach

2021-09-19T00:00:00+03:00

Drawing of palms on the beach during sunrise. It was heavily inspired by a drawing I saw on the PenUp app by this author.

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The Red Giant

2021-09-13T00:00:00+03:00

A view of a red gas giant planet near its ring.

I made also a variant without the lens flare.

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Drawing of Neptune seen from Triton

2021-09-09T00:00:00+03:00

A drawing of Neptune as seen from its moon Triton. Some natural phenomena specific to Triton are visible - liquid nitrogen geysers and lakes. Triton has a very thin atmosphere, but it should form nitrogen mist near the surface. Triton’s surface is new as it has tectonic activity, but there might be a few craters here and there, as the one seen on the drawing.

The drawing was made on the Galaxy Tab S6 Lite with the Sketchbook app.

PS: After discussing with a friend, it seems that the atmosphere of Triton could be such that it gives orange tint to sunlight, especially at sunrise or sunset. So I decided to make another variant of the drawing with more orange colors. I think I like it more that way.

What can also be seen on the picture are Uranus, Saturn and Jupiter, shining near the sun.

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The Buzludzha monument flying away

2021-09-09T00:00:00+03:00

A drawing of the Buzludzha monument flying away into space. The monument always reminded me of a sci-fi spaceship so I decided to draw it that way.

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Drawing of Jupiter and Io

2021-09-07T00:00:00+03:00

A drawing of Jupiter and its moon Io seen when the Juno spacecraft approaches them. Europa can be seen in the distance, casting its shadow over the gas giant. The drawing was made on the Galaxy Tab S6 Lite with the Sketchbook app.

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Fishing on the lake

2021-09-01T00:00:00+03:00

The drawing was made on the Galaxy Tab S6 Lite with the Sketchbook app.

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July Morning

2021-08-23T00:00:00+03:00

A drawing of a fantastical July morning at the Black Sea. The drawing was made on the Galaxy Tab S6 Lite with the Sketchbook app. It was heavily inspired by a drawing I saw on the PenUp app by this author.

Click on the image to download in full size.

Drawing of a hand holding a shiny crystal

2021-08-09T00:00:00+03:00

A drawing of a hand holding a blue kyber crystal used to construct a lightsaber. The drawing was made on the Galaxy Tab S6 Lite with the Sketchbook app.

There is a variant of the drawing with the crystal levitating.

There is also only the layer with the hand and lights and shadows.

I also made a mix between this pic and the Pleiades one.

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Drawing of cave paintings

2021-08-07T00:00:00+03:00

A drawing of the Altamira cave paintings lit by the fire.

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