Measurement and Control of Photon Spatial Wave...

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



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



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



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


Laser

A

B

BS2

BS1

y

xE

x

y

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+ = 0

=−

Laser

A

B

BS2

BS1

y

xE

x

y

E


+ = 0

Laser

A

B

BS2

BS1

y

xE

x

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“Odd” modes exit port A


●

● 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

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y


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














MY RESEARCH(THEORY AND EXPERIMENT)














MY RESEARCH(EXPERIMENTAL)

Measurement and Control of Photon Spatial Wave...

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