Download Slides - Particle Physics

Weak measurement and its experimental
realisation for non-zero mass particles
This work is supported by the John Fetzer Memorial Fund
Robert Flack
HEP/AMOPP Groups
University College London
May 2015
Liverpool 2015
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The Team
Peter Barker
Basil Hiley
Robert Flack
Thanks to Jim Clarke at the Cockroft helping with magnet design
Pete
Edmunds
May 2015
Vincenzo Monachello
Liverpool 2015
Joel Morley
David Robson 2
Plan of the talk
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Principle of weak measurement.
How weak measurement has been exploited.
Weak measurement of spin.
Weak measurement of momentum.
A possible interpretation of weak values.
Focus on experimental not theoretical
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What is “weak measurement”?
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Confusion in the literature
Strong measurement (von Neumann):
• Often referred to as collapsing the wave function.
• It is where we unambiguously measure the full set of eigenvalues of an observable.
Weak measurement (Aharonov et al):
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This is a two stage process; one weak, one strong.
Each stage acting on non-commuting variables.
It does not observe or measure the full set of eigenvalues of an observable.
It measures a weak value.
I will give one possible interpretation of weak values.
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Original Stern-Gerlach experiment
Silver atoms were originally
used. They have one valence
electron around a filled core and
it behaved like a spin-1/2
particle.
The classical prediction was the
beam should spread out in a
continuous manner.
The observation is a beam split into two parts: spin-up, spin-down.
A STRONG (von Neumann) MEASUREMENT
von Neumann. Mathematical foundations of quantum mechanics. Springer Verlag,1932.
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Results from the original paper of Stern and Gerlach
Magnet off
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Magnet on
Der experimentelle Nachweisder Richtungs quantelung
im Magnetfeld.Liverpool
Z. Phys
2015 9 (1922) 349-352
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Principal of weak measurement
Strong measurement
Weak stage:
Change of phase
Wave packets are much further apart
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This gives an amplification
effect
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Amplification?
A. K. Pan and A. Matzkin, Phys. Rev. A85, 022122
Gaussian wave function in x-space.
Fourier transform to p-space.
Strong measurement
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2-stage
weak measurement.
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Exploitation of amplification effect
Dixon et al, Phys. Rev. Lett., 102 173601 (2009)
Measurement of the transverse deflection of an optical beam using
a weak measurement amplification technique. They report to have
measured the deflection to 400 ± 200 frads.
Hosten and Kwiat, Science, 319 787-790 (2008)
They report they have observed the optical spin-Hall effect using
the amplification effect of weak measurement.
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Weak Stage
How weak is weak?
The initial wave packet is a superposition of the possible eigenstates.
The action of the weak stage cannot spread the packet sufficient to
collapse the wave function.
Has
the effect
of producing
change
in the phase.
Ballentine:
Quantum
Mechanics, a amodern
development
Each particle is emitted as a wave packet which has a width equal to
the width of the beam.
The action of the weak stage cannot produce a divergence in the
beam greater than its original width.
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Weak measurement of spin: schematic
I. M. Duck, P. M. Stevenson and Sudarshan, Phys. Rev. D, 1989
The particles enter
along the y-axis with
spin oriented at an
angle  in the zx-plane.
The first S-G is
oriented along
the z-axis. Weak
The second S-G is
oriented along the
x-axis. Strong
x = -1
x = +1
Non-commuting variables
spin z-axis, spin x-axis
•Aw = μtan(/2), μ magnetic moment of the particle.
•Note what happens as  approaches , Aw gets very large.
•Need
three magnets.
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•What happens as  approaches ?
•Let  =  - .
•P = pz/λ,
•pz = transverse momentum, λ = wavelength of the particle
(a) (p) resembles a Gaussian.
(b) Close up shows the peak is at 5.
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(a) (p) is now antisymmetric.
(b) The resulting probability
distribution peaks at 70.
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Optical analogue of extended Stern-Gerlach apparatus
Weak stage is a thin birefringent
crystal introducing a small phase
change between the o-ray and the
e-ray.
Using the width of the
beam as reference for
the allowed separation.
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Realisation of the optical analogue
Ritchie et al. Phys. Rev. Lett., 66, 1107
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Results of the optical analogue
The wavelength of the light λ=0.64m
==/4, corresponding to aligned polarizers.
The centroid of the distribution is shifted by 20λ.
Aw= ±60λ.
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Observe and measure the weak value
of spin for a non-zero mass particle
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Why non-zero mass particles?
The theory is based on the non-relativistic quantum mechanics using
Schröedinger’s equation.
Most experimental work used photons, i.e. Maxwell’s equations.
Therefore we wanted to confirm the process using atoms
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Metastable helium, He*
K. Baldwin, Contemporary Physics, Vol. 46, No. 2, March–April 2005, 105 – 120
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This excited state can only be reached by collision.
It is doubly forbidden to decay electromagnetically.
Therefore a very long life-time: ~8,000s
Magnetic moment 2 Bohr magnetons
Triplet state: m = +1,Liverpool
0, -1
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Set up
MCP/CCD
camera
Helium input
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Production of He*
Halfmann et al, Meas. Sci. Technol. 11 (2000) 1510 - 1514
Input pressure = 2000 mbar
Pulse width = 200 μsec
Pulse frequency = 50 Hz
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Detection with an MCP/CCD
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Parameters of the beam
Y. Yamauchi, Meas. Sci. Technol. 9 (1998) 531–533.
Preliminary experiments on the beam:
•Estimate the flux – current from the MCP.
•Measure the density profile – laser excited transition 1083nm.
•Measure the momentum distribution – time of flight.
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Magnet design
Jim Clarke, Kiri Marinov Cockcroft Institute
Coils are present
on this side but
not shown
Quadrapole with
space for a 10mm
diameter pipe
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Magnet design
Predicted field
gradient of 113 T/m
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Longitudinal field plot
Field plots
Ben Shepherd, Cockroft Institute
600
500
field [mT]
400
300
200
Field
versus vertical position
100
(near
positive end)
Field versus horizontal position
(near positive end)
0
-200
-150
-100
-50
0
50
x [mm]
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More field plots
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Weak measurement of momentum
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Experimental proposal
Young’s 2-slit apparatus.
Metastable argon, Ar* from a Magneto-optical trap (MOT).
Weak value of the transverse momentum.
Detector is an MCP/CCD.
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Ar* atoms falling under gravity
Trapped Cold argon atoms
Slits
MCP/CCD camera
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Metastable argon
13.5
12.5
4p levels
[5/2]2
[5/2]3
13.0
Energy (eV)
• Need to excite argon to metastable
state by using an RF discharge.
• Slowed by using a Zeeman slower
and then captured and cooled in a
magneto-optical trap (MOT).
Cooling transition
811.5 nm
978.4 nm
Quench
801.4 nm
842.5 nm
12.0
[1/2]1
11.5
4s levels
[3/2]1
[3/2]2
106.6 nm
104.8 nm
11.0
6 1
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3p [ S0] ground state
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MOT
Interferometer
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Molecular
Optical
(1/e
radius) trap
(mm ) (MOT)
2.0x106
1.8x106
1.6x106
1.4x106
1.2x106
1.0x106
8.0x105
6.0x105
4.0x105
0
20
40
60
Expansion time2 (ms2)
80
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a)
b)
100
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2.0x105
0.0
Helmholtz coils
2
2
2
Laser beams
Source of metastable argon atoms, Ar*
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Interferometry
2 μm
6 μm
Metastable argon
Double slit
• Advantage of long de Broglie wavelength.
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Spacing of fringes ~200 microns. Liverpool 2015
MCP Fluorescent
screen
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Weak measurement of transverse
momentum
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Mapping trajectories of photons in 2-slit experiment
Kocsis et al, Science 332:1170, 2011
A quantum dot was used as a source of
single photons. This made sure that only
one photon was in the apparatus at a
time.
A 50:50 beam
splitter.
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The strong measurement of
position is carried out by
observing the fringe pattern at
a range of horizontal positions
Birefringent calcite is used to for the weak
measurement of momentum. It imparts a
Liverpool 2015 phase shift (p=hk).
small kx-dependent
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Reconstruction of the trajectories
The majenta line is after constant
background has been subtracted
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A possible interpretation of the weak
value of momentum
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An interpretation of weak values
H. Wiseman, New J. Phys., 9 (2007) 165-177; R. Leavens, Found. Phys., 35 (2005) 469-491
Howard Wiseman and Rick Leavens have both shown that there
is a direct relationship between the weak values of momentum
and kinetic energy and the corresponding variables given in the
Bohm interpretation of quantum mechanics.
Bohm, D., A Suggested Interpretation of the Quantum Theory in Terms of
Hidden Variables, I, Phys. Rev., 85 (1952);
Bohm, D., A Suggested Interpretation of the Quantum Theory in Terms of
Hidden Variables, II, Phys. Rev., 85 (1952), 180-193.
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Quantum potential for Young’s slits
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Conclusion
I have explained the difference between a weak and strong
measurement from a experimental standpoint.
Weak measurement has been observed using photons in:
- A Stern-Gerlach experiment.
- A Young’s 2-slit experiment.
We are constructing two experiments to observe and measure the
weak values of spin and transverse momentum for non-zero
mass particles.
Can the amplification effect be exploited to observe small signals?
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Back up slides
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Measurement by von Neumann
In the chapter on measurement in his book he gives two types of
measurement process:
1. Processes that collapse the wave function.
2. Processes that are described by the Schrödinger evolution. He calls these
'automatic changes that occur with the passage of time’.
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He* selection rules
"However, by exciting rare gas atoms in
electronic discharges, electron collisions can
populate high lying states that are forbidden
from decaying to the ground state by quantum
mechanical selection rules."
"Neither the singlet nor triplet S states can
decay to the ground 11S0 state via an electric
dipole transition due to the L = +/- 1 selection
rule (L being the orbital angular momentum
quantum number which is zero for S states).
However, decay of the 23S1 state is doubly
forbidden because a transition to the ground
state also requires that one of the electron
spins has to flip to create a ground singlet
state. "
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A pyramid MOT
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Ensemble or individual
Ballentine: Quantum Mechanics, a modern development
Einstein (1949)
“One arrives at very implausible theoretical
conceptions, if one attempts to maintain the thesis
that the statistical quantum theory is in principle
capable of producing a complete description of an
individual physical system. On the other hand, those
difficulties of theoretical interpretation disappear, if
one views the quantum-mechanical description as the
description of ensembles of systems.”
Each particle is emitted as a wave packet which has a
momentum spread equal to the
momentum spread of the beam. 50
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Remind ourselves about the measurement process
Von Neumann. Mathematical foundations of quantum mechanics. Springer Verlag,1932.
An experiment has a system, S, and detector, D, coupled in such a way as to make the
measurement. The Hamiltonian, H, is written as the sum of three parts,
H = Hs(x) + HD(y) + HI(x,y)
The third term, HI , produces a correlation between the system and the detector
and has the form:
, where A is the variable to be measured with
values an and q, p are generalised conjugate variables.
This term is switched on long enough to make the measurement and produce a
strong correlation but not too long, impulse measurement. An example is the
shutter of a camera. If it is left open for too long then the image is overexposed.
An ideal measuring device has well defined initial and final values
of p such that the pointer reading, pf - pi gives the value of A.
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Weak measurement
Sudarshan et al Phys. Rev. D, 1989
Aharonov et al. Phys. Rev. Lett., 60:1351, 1988.
In a non-ideal (real) measurement p has a width Dp
the wave function of the detector will have the form:
The intial state of the system is prepared thus:
After the measurement has taken place the final
wave function has the form:
Putting these all together gives a sum
of Gaussians one for each an.
A strong measurement depends on Dp being
smaller than the spacing of the an making each
Gaussian narrow and non-overlapping.
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Consider the situation where ∆p is much larger than the spacing
between the an. Instead of a set of widely separated Gaussians
with narrow ∆p we have a set of overlapping Gaussians with
wide ∆p.
This will approximate a single large Gaussian-like function
peaked at a mean value of A i.e. ⟨A⟩.
This measurement on its own gives little or no information.
If this type of measurement is carried out many times then it is
possible to map out the distribution and obtain an estimate of
the centroid. The accuracy depending only on the number of
measurements.
Is this what we mean as an effective measurement?
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