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**Non-classical light and photon statistics**

Elizabeth Goldschmidt JQI tutorial July 16, 2013

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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.

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**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)

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**Outline Photon statistics Classical light Non-classical light**

Correlation functions Cauchy-Schwarz inequality Classical light Non-classical light Single photon sources Photon pair sources

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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

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

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**Auto-correlation functions**

B 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

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**Auto-correlation functions**

B 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

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**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

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**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

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**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

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**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)

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**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

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**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

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**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

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**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

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**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 … …

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**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

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**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)

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**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

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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.

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**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

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**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

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**Some pair source applications**

Heralded single photons Entangled photon pairs Entangled images Cluster states Metrology … … Heralding detector Single photon output

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**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

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**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

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**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

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**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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**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

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**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

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