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 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)

Outline Photon statistics Classical light Non-classical light 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 The spectral and photon number distributions depend on the system Blackbody/thermal radiation Luminescence/fluorescence Lasers Parametric processes

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

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

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

General correlation functions 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 For weak fields g(2) is the ratio of coincident click probability to the product of the single click probabilities 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 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

General correlation functions 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 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 (𝑘) = 𝑎 † 𝑘 𝑎 𝑘 𝑛 𝑘 A 1 2

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

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

Other non-classicality signatures Squeezing: reduction of noise in one quadrature Δ 𝐸 𝜑 2 <1/4 𝐸 𝜑 = 1 2 𝑎 𝑒 −𝑖𝜑 + 1 2 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 2 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

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

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

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 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 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

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 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

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 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 + 1 𝑛 𝑝𝑎𝑖𝑟𝑠 Phase-matching controls the direction of the output

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

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

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

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

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

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

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