Non-classical light and photon statistics

Report
Non-classical light and photon
statistics
Elizabeth Goldschmidt
JQI tutorial
July 16, 2013
What is light?
• 17th-19th 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?
• Heisenberg uncertainty requires Δ 
Δ  + /2
≥ 1/4
• For light with phase independent noise this manifests as photon
number fluctuations Δ 2 ≥ 
Lamp
Laser
• 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? (spoiler alert: yes)
Outline
• Photon statistics
– Correlation functions
– Cauchy-Schwarz inequality
• Classical light
• Non-classical light
– Single photon sources
– Photon pair sources
Photon statistics
• Most light is from statistical processes in macroscopic systems
Probability
Radiant energy
• The spectral and photon number distributions depend on the system
• Blackbody/thermal radiation
• Lasers
• Luminescence/fluorescence
• Parametric processes
Frequency
Photon number
Photon statistics
• Most light is from statistical processes in macroscopic systems
Probability
Radiant energy
• Ideal single emitter provides transform
limited photons one at a time
Frequency
Photon number
Auto-correlation functions
50/50 beamsplitter
AA
• Second-order intensity auto-correlation
characterizes photon number fluctuations
:    +  :

 =
2
- Attenuation does not affect 
Photo-detectors
B
2
2
• Hanbury Brown and Twiss setup allows simple measurement of g(2)(τ)
• For weak fields and single photon detectors
(2) = (, )/(    ) ≈ 2(2)/(1)2
• Are coincidences more (g(2)>1) or less (g(2)<1) likely than expected for
random photon arrivals?
• For classical intensity detectors
(2) =   ×   /   ×  
Auto-correlation functions
50/50 beamsplitter
A
• Second-order intensity auto-correlation
characterizes photon number fluctuations
:    +  :

 =
2
- Attenuation does not affect 
Photo-detectors
B
2
2
22
• g(2)(0)=1 – random, no correlation
•
g(2)(0)<1
(2)
g(2)()
• g(2)(0)>1 – bunching, photons arrive together
1.5
1.5
11
0.5
0.5
– anti-bunching, photons “repel”
00
-1
-1
• g(2)(τ) → 1 at long times for all fields
0
 (arb. units)
1
General correlation functions
• Correlation of two arbitrary fields: 
• 
2
• 
2
2
1,2
is the zero-time auto-correlation 
1,1
1,2
=
:1 2 :
1 2
2
0
=
† 1 † 2 1 2
1 2
A1
for different fields can be:
• Auto-correlation  2  ≠ 0
• Cross-correlation between separate fields
• Higher order zero-time auto-correlations
can also be useful () =
†

 

2
Photodetection
1.5
(2)
g ()
• Accurately measuring g(k)(τ=0) requires timing
resolution better than the coherence time
2
1
0.5
0
-1
0
 (arb. units)
1
• 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
Cauchy-Schwarz inequality

2
1,2


2
1,1
1,2
=
≤
2
2

≥1 ⇒
2

1,1
2
2
2
1,1
≥1−
1

≤  
2
1,2
=
1 2
1 2
( = 0) ≥ 1, no anti-bunched light
⇒
2
 ≤
⇒
2
≤
2,2
• With quantum mechanics: 


: 1 2 :
†1 † 2 1 2
=
=
1 2
1 2
• Classically, operators commute: 
2

2

1,1
2
=
1,2
≤

2
0

2
(0)
,1
2
,2
(0)
2 − 
2

2
+
1,1
1
1

2
+
2,2
1
2
• Some light can only be described with quantum mechanics
Other non-classicality signatures
• Squeezing: reduction of noise in one quadrature
2
1
1
Δ 
< 1/4
  =  − + †  
2
2
• 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 () or Wigner function  
2
2
 =     2 
  =
() −2 −  2 

• Useful for tomography of Fock, kitten, etc. states
• Higher order zero time auto-correlations: () () ≤ (+) (−) ,  ≥ 
• Non-classicality of pair sources by auto-correlations/photon statistics
Types of light
Non-classical light
• Collect light from a single
emitter – one at a time
behavior
• Exploit nonlinearities to
produce photons in pairs
Classical light
• Coherent states – lasers
• Thermal light – pretty much
everything other than lasers
1
Thermal
Attenuated
single photon
Poissonian
Pairs
Probability
Probability
0.8
0.6
0.4
0.2
0
0
1
2
3
4
Photon number
5
6
Coherent states 
  =
−   
!
,  = 
2
• Random photon arrival times
• 
2
 = 1 for all τ
Probability
• Laser emission
• Poissonian number statistics:
Photon number
• Boundary between classical and quantum light
• Minimally satisfy both Heisenberg uncertainty
and Cauchy-Schwarz inequality
|α|
ϕ
Thermal light
• Also called chaotic light
• Blackbody sources
• Fluorescence/spontaneous emission
• Incoherent superposition of coherent states (pseudo-thermal light)
2
Probability
• Number statistics: p  =
 1

1.5
 +1
 +1
 −ℏ/ 
(2)
g ()
1
• Bunched:  2 0 = 2 p  =
−ℏ/



0.5
• Characteristic coherence time
= 1 −  −ℏ/   −ℏ/ 
0
1
-1
1
Photon 0
number
 =
 (arb. units)
 ℏ/  − 1
• 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
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
D1
BS
D2
Desired single photon properties
• 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)
Weak laser
Attenuator
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
Single emitters
• Excite a two level system and collect the spontaneous photon
• Emission into 4π 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
Pump(s)
χ(2) or χ(3)
Nonlinear
medium/
atomic
ensemble/
etc.
• Photon number/intensity identical in two arms, “perfect
beamsplitter”
• Cross-correlation violates the classical Cauchy-Schwarz
1
2
2
inequality   =   +
 
• Phase-matching controls the direction of the output
Pair sources
Parametric processes in χ(2)
and χ(3) nonlinear media
• Spontaneous parametric down conversion,
four-wave mixing, etc.
• Statistics: from thermal (single mode
spontaneous) to poissonian (multi-mode
and/or seeded)
Atomic ensembles
• 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
• Often high spectrally multi-mode
Single emitters
• Cascade
• Statistics: one pair
at a time
Some pair source applications
• Heralded single photons
• Entangled photon pairs
• Entangled images
• Cluster states
• Metrology
• ……
Single
photon
output
Heralding
detector
Heralded single photons
Single
photon
output
• Generate photon pairs and use one to herald the other
Heralding
detector
• Heralding increases <n> without changing p(2)/p(1)
• Best multi-photon suppression possible with heralding:
(2) ℎ /(2) ℎ ≥ (1 − ℎ 0 )
Heralded statistics of one arm of a thermal source
No Heralding
Heralding with loss
1
0.8
<n>=0.2
Probability
Probability
0.8
0.6
g(2)=2
0.4
0.2
0
1
0.8
<n>=0.65
Probability
1
Perfect Heralding
0.6
g(2)=0.43
0.4
0.2
0
1
2
3
Photon number
4
0
<n>=1.2
0.6
g(2)=0.33
0.4
0.2
0
1
2
3
Photon number
4
0
0
1
2
3
Photon number
4
Properties of heralded sources
Single photon
output
Heralding
detector
• 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
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 nonresonantly pumped solid state single emitters
• Producing exotic non-classical states

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