Elizabeth Goldschmidt
JQI tutorial
July 16, 2013

• 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.

• Heisenberg uncertainty requires Δ 𝐸 𝜑 + 𝜋/2 ≥ 1/4
• For light with phase independent noise this manifests as photon number fluctuations Δ 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)

•
– Correlation functions
– Cauchy-Schwarz inequality
•
•
– 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 •
• Lasers
Parametric processes

• Most light is from statistical processes in macroscopic systems
• Ideal single emitter provides transform limited photons one at a time
Frequency Photon number

• Second-order intensity auto-correlation characterizes photon number fluctuations 𝑔 2 𝜏 = 𝑛 2
- Attenuation does not affect 𝑔
2
50/50 beamsplitter
Photo-detectors
• 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

• Second-order intensity auto-correlation characterizes photon number fluctuations 𝑔 2 𝜏 = 𝑛 2
- Attenuation does not affect 𝑔
2
50/50 beamsplitter
A
B
Photo-detectors
• 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
0
-1
 
0
(arb. units)

• Correlation of two arbitrary fields: 𝑔 2
1,2
= 𝑛
1 𝑛
2
: 𝑛
1 𝑛
2
= 𝑎
†
1 𝑎
†
2 𝑎
1 𝑎
2 𝑛
1 𝑛
2
• 𝑔
2
1,1 is the zero-time auto-correlation 𝑔
2
• 𝑔
2
1,2 for different fields can be:
• Auto-correlation 𝑔 2 𝜏 ≠ 0
• Cross-correlation between separate fields
0
• Higher order zero-time auto-correlations can also be useful 𝑔 (𝑘) = 𝑎
† 𝑘 𝑎 𝑘 𝑛 𝑘
2
A1

• Accurately measuring g (k) (τ=0) requires timing resolution better than the coherence time
2
1.5
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

𝑔 2 = 𝑛
1 𝑛
2
:
= 𝑎 †
1 𝑎 †
2 𝑎
1 𝑎
2
1,2 𝑛
1 𝑛
2 𝑛
1 𝑛
2
𝟐
𝟐
• Classically, operators commute: 𝑔
2
1,2
= 𝑛
1 𝑛
2 𝑛
1 𝑛
2
𝟐 𝑔 2
1,1
= 𝑛
2 𝑛 2
≥ 1 𝑔 2
1,2
≤ 𝑔 2
1,1 𝑔
⇒ 𝑔
2
(𝜏 = 0) ≥ 1 , no anti-bunched light
2
2,2
⇒ 𝑔
2 𝜏 ≤ 𝑔
2
0
⇒ 𝑔 2 𝑐𝑟𝑜𝑠𝑠
≤ 𝑔 2 𝑎𝑢𝑡𝑜,1
(0)𝑔 2 𝑎𝑢𝑡𝑜,2
(0)
• With quantum mechanics: 𝑔 𝑔
2
1,1
≥ 1 −
1 𝑛
2
1,1
= 𝑛
2 𝑛 2 𝑔
2
1,2
≤ 𝑔 2
1,1
+
1 𝑛
1 𝑔 2
2,2
+
1 𝑛
2
• Some light can only be described with quantum mechanics

• Squeezing: reduction of noise in one quadrature
2
< 1/4 𝐸 𝜑 =
1 𝑎𝑒 −𝑖𝜑 +
1 𝑎 † 𝑒 𝑖𝜑
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 𝑃(𝛼) or Wigner function 𝑊 𝛼 𝜌 = 𝑃 𝛼 𝛼 𝛼 𝑑 2 𝛼 𝑊 𝛼 =
2
𝑃(𝛼)𝑒
−2 𝛼−𝛽 𝜋
• Useful for tomography of Fock, kitten, etc. states
2 𝑑 2 𝛽
• 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

•
Classical light
• Coherent states – lasers
Thermal light – pretty much everything other than lasers
1
0.8
0.6
Thermal
Attenuated single photon
Poissonian
Pairs
Non-classical light
• Collect light from a single emitter – one at a time behavior
• Exploit nonlinearities to produce photons in pairs
0.4
0.2
0
0 1 2 3 4 5 6

𝛼
• Laser emission
• Poissonian number statistics: 𝑝 𝑛 = 𝑒
− 𝑛 𝑛 𝑛
, 𝑛 = 𝛼 2 𝑛!
• 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
Photon number
|α| ϕ

• Also called chaotic light
• Blackbody sources
• Fluorescence/spontaneous emission
• Incoherent superposition of coherent states (pseudo-thermal light)
2
• Number statistics: p 𝑛 =
• Bunched: 𝑔 2 0 = 2 𝑛 𝑛 +1 𝑛 p 𝑛 = 𝑛
•
= 1 − 𝑒
1 𝑒
−𝑛ℏ𝜔/𝑘
𝐵 𝑒 −𝑛ℏ𝜔/𝑘
𝐵
−ℏ𝜔/𝑘
𝐵
𝑇 𝑒 −𝑛ℏ𝜔/𝑘
𝐵
𝑇 𝑛 = 𝑒
1 ℏ𝜔/𝑘
𝐵
𝑇 − 1
0
-1
• Number distribution for a single mode of thermal light

(arb. units)
• Multiple modes add randomly, statistics approach poissonian
• Thermal statistics are important for non-classical photon pair sources
1

• 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
• … …

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)

Attenuator
• 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

• 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.

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 inequality 𝑔 2 𝑐𝑟𝑜𝑠𝑠
= 𝑔 2 𝑎𝑢𝑡𝑜
+ 𝑛
1 𝑝𝑎𝑖𝑟𝑠
• Phase-matching controls the direction of the output

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
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
Single emitters
• Cascade
• Statistics: one pair at a time

• Heralded single photons
• Entangled photon pairs
• Entangled images
• Cluster states
• Metrology
• … …
Single photon output
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 )
Heralding detector
1
0.8
0.6
0.4
0.2
0
0
Heralded statistics of one arm of a thermal source
No Heralding Heralding with loss Perfect Heralding
<n>=0.2
g
(2)
=2
1 2 3
Photon number
4
1
0.8
0.6
0.4
0.2
0
0
<n>=0.65
g
(2)
=0.43
1 2 3
Photon number
4
1
0.8
0.6
0.4
0.2
0
0
<n>=1.2
g
(2)
=0.33
1 2 3
Photon number
4

Single photon output
Heralding
• Trade off between photon rate and purity (g (2) ) detector
• 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

• 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

• 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