Download Observing the Average Trajectories of Single

Observing the
Average Trajectories of Single Photons in
a Two-Slit Interferometer
Danielle Boddy
Durham University – Atomic & Molecular Physics group
Scientific paper
Sacha Kocsis, et al., Science 332, 1170 (2011)
Journal Club Seminar 08-02-12
Outline
Introduction:
-
Heisenberg uncertainty principle
-
Double slit: Classical picture
-
Double slit: Quantum picture
-
The problem so far…
-
Strong measurement
-
Weak measurement
-
Clever theorists
-
Polarization pointer
-
Making the measurement
-
Pictorial view of the set-up
-
What do you see on the screen?
-
Mathematical view
-
What do we actually see?
-
Is the measurement weak?
Experimental set-up:
-
Basic experimental set-up
Results:
-
Interference patterns
-
Trajectories
-
Intensity distributions
-
Questions?
Conclusion:
Journal Club Seminar 08-02-12
Introduction
In classical physics, dynamics of a particle’s
evolution are governed by its position and
velocity.
To simultaneous know the particle’s position and
velocity is to know its past, present, and future.
v(t’)
x(0) x(t’)
Used with great success in the macroscopic world.
Journal Club Seminar 08-02-12
x(t)
Heisenberg uncertainty principle
Experiment cannot simultaneously determine the exact value of a
component of momentum, px, of a particle and also the exact value of its
corresponding coordinate, x
ΔpxΔx ≥ ħ/2
This restriction is not on the accuracy to which px or x can be measured,
but on the product of ΔpxΔx in a simultaneous measurement of both.
e.g. if Δpx = 0, then Δx = ∞
Journal Club Seminar 08-02-12
Double slit: Classical picture
A
B
screen
Journal Club Seminar 08-02-12
Double slit: Quantum picture
xA
A
B
X
xB
screen
Journal Club Seminar 08-02-12
Double slit: Quantum picture
A
B
X
screen
Journal Club Seminar 08-02-12
The problem so far…
In a von Neumann measurement, an observable of a system is coupled
to a measurement apparatus or ‘pointer’ via its momentum.
0
1
Measurement
Induces a measurement shift
0
1
Journal Club Seminar 08-02-12
Strong measurement
0
1
Measurement
Induces a measurement shift
0
1
Hˆ I  g (t ) xˆPˆp
Determining which slit (position) the photon passed through induced a large
uncertainty in photon momentum.
Journal Club Seminar 08-02-12
Weak measurement
0
1
Measurement
Induces a measurement shift
0
1
Measurement of
yields little information
Journal Club Seminar 08-02-12
Clever theorists
There is a limit in which you find out everything without disturbing the
system
Follow the weak measurement with just the
right strong measurement
Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)
Journal Club Seminar 08-02-12
Polarization pointer
Can’t use the momentum as a measurement pointer if we want
to measure it.
Need a pointer that commutes with both the momentum and
position.
Set the polarization as a pointer
Journal Club Seminar 08-02-12
Making the measurement
Use calcite to perform both the weak and strong measurement.
Calcite is birefringent.
Phase shift is induced between the
components of polarization.
Weak measurement of the photon momentum
Strong measurement of the photon position
Journal Club Seminar 08-02-12
Figure taken from Wikipedia
Pictorial view of the set-up
x
1
2
( +
)
1
2
( +
)
1
2
( +
)
A


B
Set polarization
Weak
measurement
Strong
measurement
xB
Post-select
Journal Club Seminar 08-02-12
What do we see on the screen?
Journal Club Seminar 08-02-12
Mathematical view
Set polarization
After the double slits the polarizer sets the polarization to
D 
1
2
H

 V
Initial transverse two-slit wave function
  D
pol

path
Journal Club Seminar 08-02-12
Mathematical view
Weak Measurement
We wish to weakly observe the transverse momentum

ˆ
ˆ
Pp 
 S1   H
2
H V V

Interaction Hamiltonian
ˆ  gkˆ Sˆ
H
I
x 1
After the measurement the state evolves as
'  e
 iHˆ I t 
 e
 igkˆx Sˆ1t 
Journal Club Seminar 08-02-12
D 
Mathematical view
Weak Measurement
But because interaction is weak →Taylor expand the Hamiltonian
e
 igkˆx Sˆ1t 
 1  igkˆx Sˆ1t 
Such that the state evolves as
igt ˆ
'  D  
kx 
2
Initial state
A
where
A 
1
H  V
2

i.e. the state can be written in terms of the initial state and the
weak measurement
Journal Club Seminar 08-02-12
Mathematical view
Strong Measurement & Post-selection
In order to measure the final position of the photon, we must measure the
rotation of the polarization.
We measure the rotation of the pointer by performing a strong
measurement.
Project polarization into the circular basis to get
L 
1
2
H
i V

R 
Journal Club Seminar 08-02-12
1
2
H
i V

Mathematical view
Strong Measurement & Post-selection
At a specific position, xf , we can find the weak momentum value
xf
igt
'  x f  D 
x f kˆx 
2

 x f  
e
2 

igt ˆ
kx
2
w
H e

igt ˆ
kx
2
w
A

V 

Phase shift between polarization components  k x  tells us about
kˆx
Where


w
k

sin
1
 IR  IL 

I I 

L 
 R
is the coupling strength of the calcite to the system
Journal Club Seminar 08-02-12
k̂ x
w
What do we actual see?
 k x 
The bottom pattern is undeviated by the strong measurement, but the
top pattern suffers a phase shift  k x 
Journal Club Seminar 08-02-12
Is the measurement ‘weak’?
Journal Club Seminar 08-02-12
How do you know if the measurement is ‘weak’?
In each square we can detect a photon
Δ
Δ
The width Δ of the square must be smaller
than the fringe spacing
Can treat the weak value as constant over the width of the pixel
Journal Club Seminar 08-02-12
Outline
Introduction:
-
Heisenberg uncertainty principle
-
Double slit: Classical picture
-
Double slit: Quantum picture
-
The problem so far…
-
Strong measurement
-
Weak measurement
-
Clever theorists
-
Polarization pointer
-
Making the measurement
-
Pictorial view of the set-up
-
What do you see on the screen?
-
Mathematical view
-
What do we actually see?
-
Is the measurement weak?
Experimental set-up:
-
Results:
-
Interference patterns
-
Trajectories
-
Intensity distributions
-
Questions?
Conclusion:
Basic experimental set-up
Journal Club Seminar 08-02-12
Basic experimental set-up
Polarizer
Single
photons
from
quantum
dot
Calcite QWP
50:50
beam
splitter
Polarizing
beam splitter
CCD
g2(0) = 0.17 ± 0.04
D 
1
2
H
 V

H ,V
Components pick up a
relative phase shift
 k x 
 k x  depends on angle of the crystal’s optic axis, the length of crystal, incident
angle of photons
Journal Club Seminar 08-02-12
Basic experimental set-up
Polarizer
Single
photons
from
quantum
dot
Calcite QWP
50:50
beam
splitter
Polarizing
beam splitter
CCD
i
  k x 
1   2i  k x 
2
e
H e
V
2 




Crystal parameters are chosen to induce a small momentum-dependent
polarization rotation
Journal Club Seminar 08-02-12
Basic experimental set-up
Polarizer
Calcite QWP
Polarizing
beam splitter
L 
Single
photons
from
quantum
dot
1
2
H
50:50
beam
splitter
i V

CCD
R 
1
2
H
i V
To measure how much the pointer has rotated, project polarization into
circular basis using the QWP
Journal Club Seminar 08-02-12

Basic experimental set-up
Polarizer
Calcite QWP
Polarizing
beam splitter
L 
Single
photons
from
quantum
dot
1
2
H
50:50
beam
splitter
i V

CCD
R 
Weak momentum value
kˆx

w
 IR  IL 

sin 1 



 IR  IL 
k
Journal Club Seminar 08-02-12
1
2
H
i V

Basic experimental set-up
Polarizer
Single
photons
from
quantum
dot
Calcite QWP
50:50
beam
splitter
Polarizing
beam splitter
(PBS)
CCD
To measure trajectory, increase the separation between the calcite and
polarizing beam splitter.
Calcite remains in a fixed position.
Journal Club Seminar 08-02-12
Basic experimental set-up
Polarizer
Single
photons
from
quantum
dot
Calcite QWP
Polarizing
beam splitter
(PBS)
50:50
beam
splitter
CCD
Measurement result is not affected since
Hˆ , Hˆ   Hˆ Hˆ
I
free
I
free
ˆ
ˆ
H
free H I  0
Trajectories are reconstructed over the range (2.75 ± 0.05) to (8.2 ± 0.1) m
Journal Club Seminar 08-02-12
Outline
Introduction:
-
Heisenberg uncertainty principle
-
Double slit: Classical picture
-
Double slit: Quantum picture
-
The problem so far…
-
Strong measurement
-
Weak measurement
-
Clever theorists
-
Polarization pointer
-
Making the measurement
-
Pictorial view of the set-up
-
What do you see on the screen?
-
Mathematical view
-
What do we actually see?
-
Is the measurement weak?
Experimental set-up:
-
Basic experimental set-up
Results:
-
Interference patterns
-
Trajectories
-
Intensity distributions
-
Questions?
Conclusion:
Journal Club Seminar 08-02-12
Results: Interference patterns
Pixel on
CCD where
each photon
is detected
corresponds
to the
photon’s x
position.
26 μm pixel
width sets
the precision
Journal Club Seminar 08-02-12
Results: Interference patterns
Can extract
each value of
kx at each
pixel using
 I  I L 
kx
1

  sin 1  R

I

I
k
L 
 R

Journal Club Seminar 08-02-12
Results: Trajectories
Repeat
measurement for
many imaging
planes along z
41 imaging planes
80 trajectories
Journal Club Seminar 08-02-12
Results: Trajectories
Photons are not
constrained to
follow these
precise trajectories
Represent the
average behaviour
Journal Club Seminar 08-02-12
Results: Trajectories
Trajectories
originating from
one slit do not
cross the central
line.
Trajectories cross
over dark fringes
at steep angles.
Separation of
planes sets the
scale over which
features in the
trajectories can be
observed.
Journal Club Seminar 08-02-12
Results: Intensity distribution
Overlay
trajectories on top
of the measured
intensity
distribution.
Trajectories
reproduce the
global interference
pattern well.
Journal Club Seminar 08-02-12
Outline
Introduction:
-
Heisenberg uncertainty principle
-
Double slit: Classical picture
-
Double slit: Quantum picture
-
The problem so far…
-
Strong measurement
-
Weak measurement
-
Clever theorists
-
Polarization pointer
-
Making the measurement
-
Pictorial view of the set-up
-
What do you see on the screen?
-
Mathematical view
-
What do we actually see?
-
Is the measurement weak?
Experimental set-up:
-
Basic experimental set-up
Results:
-
Interference patterns
-
Trajectories
-
Intensity distributions
Conclusion:
-
Questions?
Journal Club Seminar 08-02-12
Conclusion
Observed trajectories provide an intuitive picture of the way in which a
single particle interferes with itself.
Information has been gained about the average momentum of the particle
at each position within the interferometer
Exact interpretation of these observed trajectories will require continued
investigation
Using power of weak measurements, a new prospective on the double-slit
experiment was provided.
Journal Club Seminar 08-02-12
Questions?
Thanks for listening, any questions?
Sacha Kocsis, et al., Science 332, 1170 (2011)
Y. Aharonov, D. Z. Albert, L. Vaidman,
Phys. Rev. Lett. 60, 1351 (1988)
Journal Club Seminar 08-02-12