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Phase Space Evolution Phase Space Evolution ChenYu Liu Indiana University Nov. 9, 2012 Neutron Lifetime Workshop (Santa Fe) 1

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Page 1: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Phase Space EvolutionPhase Space Evolution

Chen‐Yu LiuIndiana University

Nov. 9, 2012Neutron Lifetime Workshop (Santa Fe)

1

Page 2: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

UCN Trap Experiments

Measures the Storage Time

2Steyerl Walstrom

Page 3: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Material Bottle

A. Serebrov (2008)

D. Dubbers (2011)

Page 4: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

How well to measure each non‐beta‐decay loss?

Each loss is a new degree of freedom, so this is a vector sum.

If the trap storage time is long, so that the correction to the beta‐decay lifetime is 

12 days!

• need to measure τab at the 10% level of precision. 4

Page 5: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

NIST Lifetime Experiment

• Discover marginally trapped UCN.

Fi ld l d t• Field lowered to 30% of the max value (1.1 T).

• 50% of UCN was flushed out

5

flushed out.

Page 6: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Motion in Phase SpaceMotion in Phase Space

• Phase space trajectories don’t intersect, because of the p j ,deterministic mechanics.

• If the Hamiltonian is time‐independent (autonomous), the trajectories in phase space are invariant.

• Liouville’s Theorem: The Hamiltonian flow in phase space is incompressible. The density of particles is p p y pinvariant (if the particle number is conserved).       

• The phase‐space volume of an statistical ensemble i i i t H ith h th “ i d”is invariant.  However, with chaos, the “coarse‐grained” phase‐space volume is always increasing.

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Page 7: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Phase Space Evolution in UCN traps1. Fill : populate the phase space. 

– Is the filling uniform? – How long does it take to establish static equilibrium?

Young

How long does it take to establish static equilibrium? 

2. Spectrum Cleaning: define the trappable volume in the hphase‐space.

– How long does it take to clean sufficiently?Bowman

3. Storage: Phase space evolution– Additional loss (when the trappable volume morphs to intersect with loss boundaries)

Coakley

intersect with loss boundaries)

4. Detection:  a fast sink in the phase‐space.d f f d ff– In‐situ detection: Uniformity of detection efficiency

– UCN drain: detection efficiency changes?7Golub, Morris

Page 8: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Trap Lifetime Experiment viewed in Phase‐Space

pField Potential

p

MaterialMaterialAbsorber

absorber(bottom Of the Trap)

vv

xp)

To detector

v During StorageAb ti /• Absorption/upscattering

• Spin flip• Slow

g

8

Field Potential• Slow 

evolution• Heating

Page 9: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

UCN Dynamics in TrapsUCN Dynamics in Traps

• Integrable potentialg p– A Hamiltonian with N degrees of freedom is integrableif and only if there exist N independent isolating integrals (action variables: global invariants of theintegrals (action variables: global invariants of the motion).

– Produce regular orbits 

• Near‐Integrable potential– D = 1,2: mixed dynamics, with regular trajectoriesD   1,2: mixed dynamics, with regular trajectories separating regions of stochasticity.

– D > 2: “web” of stochastic regions. Arnold diffusion.

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Page 10: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Asymmetric Trap  Phase Space MixingL ( h i h fi ld i l ) i d i i• Low symmetry (together with field ripples) induces states mixing between circular orbits, through chaotic motion (or not).

• quick cleaning (~ seconds) of the quasi‐bound UCN with large tangential velocities. 

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Page 11: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Halbach Array (NdFeB Permanent Magnets)

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Page 12: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

UCN CleaningUCN Cleaning

6Emax = E0 + 6 neV

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Page 13: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Poincare’s Section of SurfaceTwo degrees of freedom

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Page 14: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Height vs. Transverse VelocityHeight vs. Transverse Velocity

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UCN motion becomes chaotic when the height reaches R/2.

Page 15: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Three degrees of Freedom=90

gDifferent azimuthal angels 

=0

I i Increasing 

15Increasing stochasticity with 

Page 16: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Trap GeometrySymmetric vs Asymmetric

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Strong perturbation, KAM curves are destroyed.Thin stochastic regions bounded by KAM curves.

Page 17: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Arnold Diffusion =90

=03 degrees of freedom

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Page 18: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Spatial non‐uniformity of detection efficiency (acceptance)efficiency (acceptance)

Page 19: Phase Space Evolution - Nc State Universityneutron.physics.ncsu.edu/LifetimeWorkshop/talks/Liu.pdfspace is incompressible. The density of particles is invariant (if the particle number

Summary• Although quasi‐bound UCN can be cleaned from most traps, the evolution of the phase space motion and the p , p peffect of the experiment (fill, empty, detection, leakage) has not been even seriously considered in most previous experiments.

• Chaotic d namics ( ork in progress)• Chaotic dynamics (work in progress)• Small perturbations of an integrable potential could have narrow stochastic regions near resonances Thehave narrow stochastic regions near resonances. The evolution could have a long evolution time.

• Large perturbations (with discontinuous derivatives) increase stochasticity.