Non

classical light and photon
statistics
Elizabeth Goldschmidt
JQI tutorial
July 16, 2013
What is light?
•
17
th

19
th
century
–
particle
: Corpuscular theory
(Newton) dominates over wave theory (Huygens).
•
19th century
–
wave
: Experiments support wave theory
(Fresnel, Young), Maxwell’s equations describe
propagating electromagnetic waves.
•
1900s
–
???
: Ultraviolet catastrophe and photoelectric
effect explained with light quanta (Planck, Einstein).
•
1920s
–
wave

particle duality
: Quantum mechanics
developed
(Bohr, Heisenberg, de Broglie
…), light and
matter have both wave and particle properties.
•
1920s

50s
–
photons
: Quantum field theories developed
(Dirac,
Feynman), electromagnetic field is quantized,
concept of the photon introduced.
What is non

classical light and why do we
need it?
•
Metrology
: measurement uncertainty due to uncertainty in number
of incident photons
•
Quantum information
: fluctuating numbers of
qubits
degrade
security, entanglement, etc.
•
Can we reduce those fluctuations?
Laser
Lamp
•
Heisenberg uncertainty requires
Δ
𝐸
𝜑
Δ
𝐸
𝜑
+
/
2
≥
1
/
4
•
For light with phase independent noise this manifests as photon
number fluctuations
Δ
2
≥
(
spoiler alert:
yes)
Outline
•
Photon statistics
–
Correlation functions
–
Cauchy

Schwarz inequality
•
Classical light
•
Non

classical light
–
Single photon sources
–
Photon pair sources
•
Most light is from statistical processes in macroscopic systems
•
The spectral and photon number distributions depend on the system
•
Blackbody/thermal radiation
•
Luminescence/fluorescence
Photon statistics
•
Lasers
•
Parametric processes
Frequency
Radiant energy
Photon number
Probability
Frequency
Radiant energy
Frequency
Radiant energy
Photon number
Probability
Frequency
Radiant energy
Photon statistics
•
Most light is from statistical processes in macroscopic systems
•
Ideal single emitter provides transform
limited photons one at a time
Frequency
Radiant energy
Photon number
Probability
A
B
50/50
beamsplitter
Photo

detectors
Auto

correlation functions
•
Second

order intensity auto

correlation
characterizes photon number fluctuations

Attenuation does not affect
𝑔
2
•
Hanbury
Brown and
Twiss
setup allows
simple measurement of
g
(2)
(
τ
)
•
For weak fields and single photon detectors
𝑔
(
2
)
=
𝑝
(
,
)
/
(
𝑝
𝑝
)
≈
2𝑝
(
2
)
/
𝑝
(
1
)
2
•
Are coincidences more (g
(2)
>1) or less (g
(2)
<1) likely than expected for
random photon arrivals?
•
For classical intensity detectors
𝑔
(
2
)
=
𝐼
×
𝐼
/
𝐼
×
𝐼
𝑔
2
𝜏
=
:
𝑡
𝑡
+
𝜏
:
2
A
B
1
0
1
0
0.5
1
1.5
2
(arb. units)
g
(2)
(
)
1
0
1
0
0.5
1
1.5
2
(arb. units)
g
(2)
(
)
A
B
50/50
beamsplitter
Photo

detectors
Auto

correlation functions
•
Second

order intensity auto

correlation
characterizes photon number fluctuations

Attenuation does not affect
𝑔
2
•
g
(2
)
(0
)
=1
–
random
, no correlation
•
g
(2
)
(0
)
>1
–
bunching
, photons arrive together
•
g
(2
)
(
0)
<1
–
anti

bunching
,
photons “repel”
•
g
(2)
(τ)
→ 1 at
long
times for all fields
𝑔
2
𝜏
=
:
𝑡
𝑡
+
𝜏
:
2
General correlation functions
•
Correlation of two arbitrary fields:
𝑔
2
1
,
2
=
:
1
2
:
1
2
=
𝑎
†
1
𝑎
†
2
𝑎
1
𝑎
2
1
2
•
𝑔
2
1
,
1
is the zero

time auto

correlation
𝑔
2
0
•
𝑔
2
1
,
2
for different fields can be:
•
Auto

correlation
𝑔
2
𝜏
≠
0
•
Cross

correlation between separate fields
•
Higher order zero

time auto

correlations
can also be useful
𝑔
(
)
=
𝑎
†
𝑘
𝑎
𝑘
𝑘
A
1
2
•
Accurately measuring
g
(k)
(
τ
=0)
requires
timing
resolution better than the coherence time
•
Classical intensity detection: noise floor >> single photon
•
Can obtain
g
(k)
with
k
detectors
•
Tradeoff between sensitivity and speed
•
Single photon detection: click for one or more photons
•
Can obtain
g
(k)
with
k
detectors if
<n> << 1
•
Area of active research, highly wavelength dependent
•
Photon number resolved detection: up to some maximum
n
•
Can obtain
g
(k)
directly up to
k=n
•
Area of active research,
true PNR detection still rare
Photodetection
1
0
1
0
0.5
1
1.5
2
(arb. units)
g
(2)
(
)
Cauchy

Schwarz inequality
•
Classically, operators commute:
𝑔
2
1
,
2
=
1
2
1
2
𝑔
2
1
,
1
=
2
2
≥
1
𝑔
2
1
,
2
≤
𝑔
2
1
,
1
𝑔
2
2
,
2
•
With quantum mechanics:
𝑔
2
1
,
1
=
2
−
2
𝑔
2
1
,
1
≥
1
−
1
𝑔
2
1
,
2
≤
𝑔
2
1
,
1
+
1
1
𝑔
2
2
,
2
+
1
2
•
Some light can only be described with quantum mechanics
𝟐
≤
𝟐
𝟐
?œ
?C
2
(
𝜏
=
0
)
≥
1
, no anti

bunched light
⇒
𝑔
2
𝜏
≤
𝑔
2
0
⇒
𝑔
2
≤
𝑔
2
𝑎
,
1
(
0
)
𝑔
2
𝑎
,
2
(
0
)
𝑔
2
1
,
2
=
:
1
2
:
1
2
=
𝑎
†
1
𝑎
†
2
𝑎
1
𝑎
2
1
2
Other non

classicality signatures
•
Squeezing: reduction of noise in one quadrature
Δ
𝐸
𝜑
2
<
1
/
4
𝐸
𝜑
=
1
2
𝑎
−
𝑖
𝜑
+
1
2
𝑎
†
𝑖
𝜑
•
Increase in noise at conjugate phase
φ
+
π
/2 to satisfy
Heisenberg uncertainty
•
No quantum description required: classical noise can be perfectly zero
•
Phase sensitive detection (homodyne) required to measure
•
Negative P

representation
𝑃
(
)
or Wigner
function
𝑊
=
𝑃
2
𝑊
=
2
𝜋
𝑃
(
)
−
2
−
2
2
•
Useful for tomography of
Fock
, kitten, etc. states
•
Higher
order
zero time auto

correlations:
𝑔
(
)
𝑔
(
)
≤
𝑔
(
+
)
𝑔
(
−
)
,
≥
•
Non

classicality of pair sources by auto

correlations/photon statistics
Types of light
Classical light
•
Coherent states
–
lasers
•
Thermal light
–
pretty much
everything other than lasers
Non

classical light
•
Collect light from a single
emitter
–
one at a time
behavior
•
Exploit nonlinearities to
produce photons in pairs
0
1
2
3
4
5
6
0
0.2
0.4
0.6
0.8
1
Photon number
Probability
Photon number
Probability
Thermal
Attenuated
single photon
Poissonian
Pairs
Coherent states
•
Laser emission
•
Poissonian
number statistics
:
𝑝
=
−
𝑛
𝑛
!
,
=
2
•
Random photon arrival times
•
𝑔
2
𝜏
=
1
for all
τ
•
Boundary between classical and quantum light
•
Minimally satisfy both Heisenberg uncertainty
and Cauchy

Schwarz inequality

α

ϕ
Photon number
Probability
p
=
−
ℏ𝜔
/
𝐵
𝑇
−
ℏ𝜔
/
𝐵
𝑇
=
1
−
−
ℏ𝜔
/
𝐵
𝑇
−
ℏ𝜔
/
𝐵
𝑇
=
1
ℏ𝜔
/
𝐵
𝑇
−
1
•
Also called chaotic light
•
Blackbody sources
•
Fluorescence/spontaneous emission
•
Incoherent superposition of coherent states (pseudo

thermal light)
•
Number statistics:
p
=
+
1
1
+
1
•
Bunched:
𝑔
2
0
=
2
•
Characteristic coherence time
•
Number distribution for a
single
mode of thermal light
•
Multiple modes add randomly, statistics approach
poissonian
•
Thermal statistics are important for non

classical photon pair sources
Thermal light
Photon number
Probability
1
0
1
0
0.5
1
1.5
2
(arb. units)
g
(2)
(
)
Types of non

classical light
•
Focus today on two types of non

classical light
•
Single photons
•
Photon pairs/two mode squeezing
•
Lots of other types on non

classical light
•
Fock
(number) states
•
N00N states
•
Cat/kitten states
•
Squeezed vacuum
•
Squeezed coherent states
•
… …
Some single photon applications
Secure communication
•
Example: quantum key
distribution
•
Random numbers, quantum
games and tokens, Bell tests…
Quantum information
processing
•
Example: Hong

Ou

Mandel
interference
•
Also useful for metrology
BS
D
1
D
2
•
High rate and efficiency (p(1)≈1)
•
Affects storage and noise requirements
•
Suppression of multi

photon states (
g
(2)
<<1)
•
Security (number

splitting attacks) and fidelity
(entanglement and
qubit
gates)
•
Indistinguishable photons (frequency and bandwidth)
•
Storage and processing of
qubits
(HOM interference)
Desired single photon properties
Weak laser
•
Easiest “single photon source” to implement
•
No multi

photon suppression
–
g
(2)
= 1
•
High rate
–
limited by pulse bandwidth
•
Low efficiency
–
Operates with p(1)<<1 so that p(2)<<p(1)
•
Perfect
indistinguishability
Laser
Attenuator
Single emitters
•
Excite a two level system and collect the spontaneous photon
•
Emission into 4
π
difficult to collect
•
High NA lens or cavity enhancement
•
Emit one photon at a time
•
Excitation electrical, non

resonant, or strongly filtered
•
Inhomogeneous broadening and
decoherence
degrade
indistinguishability
•
Solid state systems generally
not
identical
•
Non

radiative
decay decreases HOM visibility
•
Examples: trapped atoms/ions/molecules, quantum dots, defect (NV)
centers in diamond, etc.
Two

mode squeezing/pair sources
•
Photon number/intensity identical in two arms, “perfect
beamsplitter
”
•
Cross

correlation violates the classical Cauchy

Schwarz
inequality
𝑔
2
=
𝑔
2
𝑎
+
1
𝑝𝑎𝑖
•
Phase

matching controls the direction of the output
χ
(2)
or
χ
(3)
Nonlinear
medium/
atomic
ensemble/
etc.
Pump(s)
Pair sources
•
Spontaneous parametric down conversion,
four

wave mixing, etc.
•
Statistics: from thermal (single mode
spontaneous) to
poissonian
(multi

mode
and/or seeded)
•
Often high spectrally multi

mode
Parametric processes in
χ
(2)
and
χ
(3)
nonlinear media
Atomic ensembles
Single emitters
•
Atomic cascade, four

wave mixing, etc.
•
Statistics: from thermal (single mode
spontaneous) to
poissonian
(multi

mode
and/or seeded)
•
Often highly spatially multi

mode
•
Memory can allow controllable delay
between photons
•
Cascade
•
Statistics: one pair
at a time
•
Heralded single photons
•
Entangled photon pairs
•
Entangled images
•
Cluster states
•
Metrology
•
… …
Some pair source applications
Heralding
detector
Single
photon
output
Heralded single photons
•
Generate photon pairs and use one to herald the other
•
Heralding increases <n> without changing p(2)/p(1)
•
Best multi

photon suppression possible with heralding:
𝑔
(
2
)
ℎ 𝑎
/
𝑔
(
2
)
ℎ 𝑎
≥
(
1
−
𝑝
ℎ 𝑎
0
)
Heralding
detector
Single
photon
output
0
1
2
3
4
0
0.2
0.4
0.6
0.8
1
Photon number
Probability
<n>=0.2
g
(2)
=2
No Heralding
0
1
2
3
4
0
0.2
0.4
0.6
0.8
1
Photon number
Probability
<n>=1.2
g
(2)
=0.33
Perfect Heralding
Heralded
statistics of one arm of a thermal source
0
1
2
3
4
0
0.2
0.4
0.6
0.8
1
Photon number
Probability
<n>=0.65
g
(2)
=0.43
Heralding with loss
Properties of heralded sources
•
Trade off between photon rate and purity (
g
(2)
)
•
Number resolving detector allows operation at a higher rate
•
Blockade/single emitter ensures one

at

a

time pair statistics
•
Multiple sources and switches can increase rate
•
Quantum memory makes source “on

demand”
•
Atomic ensemble

based single photon guns
•
Write probabilistically prepares source to fire
•
Read deterministically generates single photon
•
External quantum memory stores heralded photon
Heralding
detector
Single photon
output
Takeaways
•
Photon number statistics to characterize light
•
Inherently quantum description
•
Powerful, and accessible with state of the
art
photodetection
•
Cauchy

Schwarz inequality and the nature of
“non

classical” light
•
Correlation functions as a shorthand for
characterizing light
•
Reducing photon number fluctuations has
many applications
•
Single photon sources and pair sources
•
Single emitters
•
Heralded single photon sources
•
Two

mode squeezing
Some interesting open problems
•
Producing
factorizable
states
•
Frequency entanglement degrades other,
desired, entanglement
•
Producing indistinguishable photons
•
Non

radiative
decay common in non

resonantly pumped solid state single emitters
•
Producing exotic non

classical states
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