Non-classical light and photon

murmerlastUrban and Civil

Nov 16, 2013 (3 years and 8 months ago)

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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


?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