Cody LearyOberlin College, April 2012
Measurement and Control of Photon Spatial Wave Functions
● Measuring the spatial wave function of a single ph oton
● Manipulating the wave function of a single photon● Imparting a quantum of orbital angular momentum to a photon
● Interaction between wave functions of two “collidi ng” photons
Agenda for today
What is a photon?
• A photon is an oscillating electric field (wave) that propagates through space and time
• A photon makes a photon detector go “click!”
• A photon has 4 degrees of freedom (DOFs):
1.) It oscillates with a certain frequency (energy)
2.) It oscillates in a certain plane (polarization)
3.) Its intensity (& E-field) in the transverse x d irection has a certain spatial shape
4.) Its intensity (& E-field) in the transverse y d irection has a certain spatial shape
This talk concerns:
• The measurement and control of transverse spatial DOFs of photons
Transverse (2-D)photon “shape”
Photon detectingcamera
Oscillating electric field amplitude
y
xE
x
y
E
Transverse spatial modes
1.) Vertical “Coffee Bean”
x
E2.) Horizontal
“Coffee Bean”
=∣E∣2Intensity
● The electric field of one “lobe” is out of phase w ith the other
● Both modes have either even or odd parity in each dimension:● The vertical mode is odd under reflection in x● The horizontal mode is even under reflection in x
● Phase structure lies at the heart of photon quantu m mechanics
y
xE
x
y
E
Transverse spatial modes
1.) Vertical “Coffee Bean”
x
E2.) Horizontal
“Coffee Bean”
=∣E∣2Intensity
● Transverse spatial modes are analogous to spherica l harmonics in a hydrogen atom● Transverse spatial modes are thus the photon's wav efunction
y
xE
x
y
E
Transverse spatial modes
1.) Vertical “Coffee Bean”
x
E2.) Horizontal
“Coffee Bean”
=∣E∣2Intensity
+=
● An equal superposition of the vertical and horizon tal coffee bean modes results in a new mode: a “diagonal coffee bea n”.
ETOT x , y =EVert x , yE Hor x , y
What I show
What I mean
y
xE
x
y
E
Transverse spatial modes
1.) Vertical “Coffee Bean”
x
E2.) Horizontal
“Coffee Bean”
=∣E∣2Intensity
● An equal superposition of the vertical and horizon tal coffee bean modes results in a new mode: a “diagonal coffee bea n”.
● How to measure a photon's mode? (vertical, horizo ntal, diagonal)
+=
• A photon counting detector alone cannot measure the transverse spatial mode of a single photon
• Suppose a “vertical” photon impinges on such a detector:
At this point, the photon could be in either mode:
• It would take many identical photons to build up the full spatial intensity pattern
How not to measure the spatial mode of a single photon
• We need a “magic box” that:
• Accepts an unknown state as input
• Routes photons to one output
• Routes photons to a separate output
• Detector click yields the desired measurement
• Let's call it a “Sorter”
S
How to measure the spatial mode of a single photon
?
Laser
A
B
BS2
1-D parity sorting interferometer
BS1
y
xE
x
y
E
● Parity sorter is based on the superposition principle of the phase of an electric field
Laser
A
B
BS2
BS1
y
xE
x
y
E
1-D parity sorting interferometer
Laser
A
B
BS2
BS1
y
xE
x
y
E
1-D parity sorting interferometer
Laser
A
B
BS2
BS1
y
xE
x
y
E
1-D parity sorting interferometer
+ = 0
=−
Laser
A
B
BS2
BS1
y
xE
x
y
E
1-D parity sorting interferometer
+ = 0
Laser
A
B
BS2
BS1
y
xE
x
y
E
“Odd” modes exit port A
1-D parity sorting interferometer
1-D parity sorting interferometer
●
● Delaying the phase between the below superposed mo des results in evolution of the “diagonal coffee bean” mode to a “donut” mode
Well-defined orbital angular momentum
New kid on the block: the “donut” mode
+= = + i
No phase delay phase delayi=ei
2=90o
+cos t = cos t = +cos t sin t
+ =
Experimental results
Conclusion: the 1-D parity interferometer imparts o ne quantum of orbital angular momentum to a single pho ton!
Bloch Sphere
Arbitrary two-mode superposition represented by sph ere
Each point on Bloch sphere surface represents disti nct state
Solution: optical fiber crushing
Fiber stress breaks symmetry, imparts relative phas e
Data: Margaret Raabe
Fiber crushing simulation
Orbital angular momentum may be deterministically i mparted to a single photon
Experiment:
Theory:
Discovered by Hong, Ou, and Mandel in 1985It is commonly thought that photons MUST be indisti nguishable
Putting it all together:Two photon interference at a beam splitter
● Measurement:● 1-D parity interferometer “sorts” photons with even and odd
x-parity into different ports.
● Control:● Demonstrated phase control between even and odd mod es.● Imparted a quantum of orbital angular momentum to p hotons
in two ways● interferometer ● optical fiber.
● “Collisions”: ● Overlapping photons with nontrivial spatial structu re within a
1-D parity interferometer causes their wave functio ns to interact in surprising ways-- interfering photons can be dis tinguishable!
Conclusions: Measurement and Control of Photon Wave Functions
Movie: states that exhibit two-photon interference
ˆxΠ ˆ
yΠ
180ˆˆ
xy RΠ = o
(a)
(b)x x
y y
y
x
y
x
x
y
2-D parity sorting interferometer
Experimental results: 2-D parity sorting
• The effects occur analogously for electrons and pho tons!• Independent of mass, charge, magnetic moment, etc.
Controlling a photon's wavefunction with its polarization state
“orbit”-controlled “spin” rotation “spin”-controlled
“orbit” rotation
Electrons (Matter) Photons (Light)Classical Mechanics
-Galileo, Brahe, Kepler, NewtonClassical (Ray) Optics
-Hero, Ptolemy, Sahl, al-Haytham, Kepler, Newton
Electrons are waves! Light is a wave!
Quantum (Wave) Mechanics-- Bohr, De Broglie, Heisenberg, Schrodinger
Wave Optics-Hooke, Huygens, Young, Fresnel
Relativity!Relativity!
Relativistic Quantum (Wave) Mechanics-Dirac
Relativistic Wave Optics-Maxwell, Heaviside, Gibbs, Hertz
Particle creation!Particle creation!
Quantum Field Theory-Feynman, Tomonaga, Schwinger, Dyson
Quantum Optics-Dirac
A brief history of light and matter
Quantum Electrodynamics
Electrons (Matter) Photons (Light)Classical Mechanics
-Galileo, Brahe, Kepler, NewtonClassical (Ray) Optics
-Hero, Ptolemy, Sahl, al-Haytham, Kepler, Newton
Electrons are waves! Light is a wave!
Quantum (Wave) Mechanics-- Bohr, De Broglie, Heisenberg, Schrodinger
Wave Optics-Hooke, Huygens, Young, Fresnel
Relativity!Relativity!
Relativistic Quantum (Wave) Mechanics-Dirac
Relativistic Wave Optics-Maxwell, Heaviside, Gibbs, Hertz
Particle creation!Particle creation!
Quantum Field Theory-Feynman, Tomonaga, Schwinger, Dyson
Quantum Optics-Dirac
A brief history of light and matter
MY RESEARCH(THEORY AND EXPERIMENT)
Electrons (Matter) Photons (Light)Classical Mechanics
-Galileo, Brahe, Kepler, NewtonClassical (Ray) Optics
-Hero, Ptolemy, Sahl, al-Haytham, Kepler, Newton
Electrons are waves! Light is a wave!
Quantum (Wave) Mechanics-- Bohr, De Broglie, Heisenberg, Schrodinger
Wave Optics-Hooke, Huygens, Young, Fresnel
Relativity!Relativity!
Relativistic Quantum (Wave) Mechanics-Dirac
Relativistic Wave Optics-Maxwell, Heaviside, Gibbs, Hertz
Particle creation!Particle creation!
Quantum Field Theory-Feynman, Tomonaga, Schwinger, Dyson
Quantum Optics-Dirac
A brief history of light and matter
MY RESEARCH(EXPERIMENTAL)