1. Non-classical light and photon statistics
Elizabeth Goldschmidt
JQI tutorial
July 16, 2013

2. 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.

3. What is non-classical light and why do we need it?
Heisenberg uncertainty requires Δ 𝐸 𝜑 Δ 𝐸 𝜑+𝜋/2 ≥1/4
For light with phase independent noise this manifests as photon number fluctuations Δ 𝑛 2 ≥ 𝑛
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)

4. Outline Photon statistics Classical light Non-classical light
Correlation functions
Cauchy-Schwarz inequality
Classical light
Non-classical light
Single photon sources
Photon pair sources

5. Photon statistics
Most light is from statistical processes in macroscopic systems
The spectral and photon number distributions depend on the system
Blackbody/thermal radiation
Luminescence/fluorescence
Lasers
Parametric processes

6. Photon statistics
Most light is from statistical processes in macroscopic systems
Ideal single emitter provides transform limited photons one at a time

7. Auto-correlation functionsB
50/50 beamsplitter
Photo-detectors A
Second-order intensity auto-correlation characterizes photon number fluctuations
Attenuation does not affect 𝑔 2
𝑔 2 𝜏 = : 𝑛 𝑡 𝑛 𝑡+𝜏 : 𝑛 2 B
Hanbury Brown and Twiss setup allows simple measurement of g(2)(τ)
For weak fields and single photon detectors
𝑔 (2) =𝑝(𝐴,𝐵)/(𝑝 𝐴 𝑝 𝐵 )≈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 𝐴 ×𝐼 𝐵 / 𝐼 𝐴 × 𝐼 𝐵

8. Auto-correlation functionsB
50/50 beamsplitter
Photo-detectors
Second-order intensity auto-correlation characterizes photon number fluctuations
Attenuation does not affect 𝑔 2
𝑔 2 𝜏 = : 𝑛 𝑡 𝑛 𝑡+𝜏 : 𝑛 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

9. General correlation functions
Correlation of two arbitrary fields: 𝑔 ,2 = : 𝑛 1 𝑛 2 : 𝑛 𝑛 2 = 𝑎 † 𝑎 † 2 𝑎 1 𝑎 𝑛 𝑛 2
For weak fields g(2) is the ratio of coincident click probability to the product of the single click probabilities
𝑔 ,1 is the zero-time auto-correlation 𝑔
𝑔 ,2 for different fields can be:
Auto-correlation 𝑔 2 𝜏≠0
Cross-correlation between separate fields
k single photon counters can measure the
kth order zero-time auto-correlation g(k)
A photon number resolving detector can
measure g(k) without beamsplitters A 1 2

10. General correlation functions
Correlation of two arbitrary fields: 𝑔 ,2 = : 𝑛 1 𝑛 2 : 𝑛 𝑛 2 = 𝑎 † 𝑎 † 2 𝑎 1 𝑎 𝑛 𝑛 2
𝑔 ,1 is the zero-time auto-correlation 𝑔
𝑔 ,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 (𝑘) = 𝑎 † 𝑘 𝑎 𝑘 𝑛 𝑘 A 1 2

11. Photodetection 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

12. Cauchy-Schwarz inequality
𝑨𝑩 𝟐 ≤ 𝑨 𝟐 𝑩 𝟐
𝑔 ,2 = : 𝑛 1 𝑛 2 : 𝑛 𝑛 2 = 𝑎 † 𝑎 † 2 𝑎 1 𝑎 𝑛 𝑛 2
Classically, operators commute: 𝑔 ,2 = 𝑛 1 𝑛 𝑛 1 𝑛 2
𝑔 ,1 = 𝑛 𝑛 2 ≥1
𝑔 ,2 ≤ 𝑔 ,1 𝑔 ,2
With quantum mechanics: 𝑔 ,1 = 𝑛 2 − 𝑛 𝑛 2
𝑔 ,1 ≥1− 1 𝑛 𝑔 ,2 ≤ 𝑔 , 𝑛 𝑔 , 𝑛 2
Some light can only be described with quantum mechanics
⇒ 𝑔 2 (𝜏=0)≥1, no anti-bunched light
⇒ 𝑔 2 𝜏 ≤ 𝑔
⇒ 𝑔 2 𝑐𝑟𝑜𝑠𝑠 ≤ 𝑔 2 𝑎𝑢𝑡𝑜,1 (0) 𝑔 2 𝑎𝑢𝑡𝑜,2 (0)

13. Other non-classicality signatures
Squeezing: reduction of noise in one quadrature
Δ 𝐸 𝜑 2 <1/4 𝐸 𝜑 = 1 2 𝑎 𝑒 −𝑖𝜑 𝑎 † 𝑒 𝑖𝜑
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(𝛼) or Wigner function 𝑊 𝛼
𝜌 = 𝑃 𝛼 𝛼 𝛼 𝑑 2 𝛼 𝑊 𝛼 = 2 𝜋 𝑃(𝛼) 𝑒 −2 𝛼−𝛽 𝑑 2 𝛽
Useful for tomography of Fock, kitten, etc. states
Higher order zero time auto-correlations: 𝑔 (𝑙) 𝑔 (𝑚) ≤𝑔 (𝑙+𝑘) 𝑔 (𝑚−𝑘) , 𝑙≥𝑚
Non-classicality of pair sources by auto-correlations/photon statistics

14. Types of light Non-classical light 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

15. Coherent states 𝛼 Laser emission Poissonian number statistics:
𝑝 𝑛 = 𝑒 − 𝑛 𝑛 𝑛 𝑛! , 𝑛 = 𝛼 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

16. Thermal light Also called chaotic light Blackbody sources
Fluorescence/spontaneous emission
Incoherent superposition of coherent states (pseudo-thermal light)
Number statistics: p 𝑛 = 𝑛 𝑛 +1 𝑛 1 𝑛 +1
Bunched: 𝑔 =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
p 𝑛 = 𝑒 −𝑛ℏ𝜔/ 𝑘 𝐵 𝑇 𝑛 𝑒 −𝑛ℏ𝜔/ 𝑘 𝐵 𝑇 = 1− 𝑒 −ℏ𝜔/ 𝑘 𝐵 𝑇 𝑒 −𝑛ℏ𝜔/ 𝑘 𝐵 𝑇
𝑛 = 1 𝑒 ℏ𝜔/ 𝑘 𝐵 𝑇 −1

17. 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
… …

18. 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

19. 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)

20. Weak laser Laser Easiest “single photon source” to implement
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

21. 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.

22. Two-mode squeezing/pair sources
χ(2) or χ(3) Nonlinear medium/ atomic ensemble/ etc.
Pump(s)
Photon number/intensity identical in two arms, “perfect beamsplitter”
Cross-correlation violates the classical Cauchy-Schwarz inequality 𝑔 2 𝑐𝑟𝑜𝑠𝑠 = 𝑔 2 𝑎𝑢𝑡𝑜 𝑛 𝑝𝑎𝑖𝑟𝑠
Phase-matching controls the direction of the output

23. Parametric processes in χ(2) and χ(3) nonlinear media
Pair sources
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
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)
Often high spectrally multi-mode
Single emitters
Cascade
Statistics: one pair at a time

24. Some pair source applications
Heralded single photons
Entangled photon pairs
Entangled images
Cluster states
Metrology
… …
Heralding detector
Single photon output

25. Heralded single photons
Heralding detector
Single photon output
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) ℎ𝑒𝑟𝑎𝑙𝑑𝑒𝑑 / 𝑔 (2) 𝑢𝑛ℎ𝑒𝑟𝑎𝑙𝑑𝑒𝑑 ≥(1− 𝑝 𝑢𝑛ℎ𝑒𝑟𝑎𝑙𝑑𝑒𝑑 0 )
Heralded statistics of one arm of a thermal source

26. Properties of heralded sources
Heralding detector
Single photon output
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

27. 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

28. 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

29. Neat example of storage + heralding: Nearly Fock states
Type-II SPDC
Signal
Pump
Idler
EOM
EOM
PBS
Output
PBS … 3 8 1
Number resolving idler detector
Control output switching

30. Photon number resolving detector
Mode reconstruction
µ = mean photon number
p(n) = probability of detecting n photons
g(k) = zero time intensity correlation
: 𝑛 𝑘 : / 𝑛 𝑘
Photon number resolving detector
Source 1, µ1 #
ptotal(n) or g(k)
Source 2, µ2
µ1, µ2, … µM
Source 3, µ3
Multiple sources add together randomly (g(k)(0) approaches 1)
m orders to reconstruct m modes/sources (up to one poissonian source)
g(2) only provides quantitative information about up to 2 modes ……
Source M, µM