#### Transcript Non-classical light and photon statistics

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 Δ𝐸 𝜑 Δ𝐸 𝜑 + 𝜋/2 ≥ 1/4 • For light with phase independent noise this manifests as photon number fluctuations Δ𝑛 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 – 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 :𝑛 𝑡 𝑛 𝑡 + 𝜏 : 𝑔 𝜏 = 𝑛2 - Attenuation does not affect 𝑔 Photo-detectors B 2 2 • 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 50/50 beamsplitter A • Second-order intensity auto-correlation characterizes photon number fluctuations :𝑛 𝑡 𝑛 𝑡 + 𝜏 : 𝑔 𝜏 = 𝑛2 - Attenuation does not affect 𝑔 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: 𝑔 • 𝑔 2 • 𝑔 2 2 1,2 is the zero-time auto-correlation 𝑔 1,1 1,2 = :𝑛1 𝑛2 : 𝑛1 𝑛2 2 0 = 𝑎† 1 𝑎† 2 𝑎1 𝑎2 𝑛1 𝑛2 A1 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 𝑛𝑘 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 𝑔 2 1,2 𝑔 𝑔 2 1,1 1,2 = ≤ 𝑛2 𝑛2 𝑔 ≥1 ⇒𝑔 2 𝑔 1,1 2 2 2 1,1 ≥1− 1 𝑛 ≤ 𝑨𝟐 𝑩𝟐 2 1,2 = 𝑛1 𝑛2 𝑛1 𝑛2 (𝜏 = 0) ≥ 1, no anti-bunched light ⇒𝑔 2 𝜏 ≤𝑔 ⇒𝑔 2 ≤ 2,2 • With quantum mechanics: 𝑔 𝑔 𝟐 : 𝑛1 𝑛2 : 𝑎†1 𝑎† 2 𝑎1 𝑎2 = = 𝑛1 𝑛2 𝑛1 𝑛2 • Classically, operators commute: 𝑔 2 𝑨𝑩 2 𝑔 1,1 2 = 1,2 ≤ 𝑐𝑟𝑜𝑠𝑠 2 0 𝑔 2 (0)𝑔 𝑎𝑢𝑡𝑜,1 2 𝑎𝑢𝑡𝑜,2 (0) 𝑛2 − 𝑛 𝑛2 𝑔 2 + 1,1 1 𝑛1 𝑔 2 + 2,2 1 𝑛2 • Some light can only be described with quantum mechanics Other non-classicality signatures • Squeezing: reduction of noise in one quadrature 2 1 1 Δ𝐸 𝜑 < 1/4 𝐸 𝜑 = 𝑎𝑒 −𝑖𝜑 + 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 2 2 𝜌 = 𝑃 𝛼 𝛼 𝛼 𝑑2 𝛼 𝑊 𝛼 = 𝑃(𝛼)𝑒 −2 𝛼−𝛽 𝑑 2 𝛽 𝜋 • 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 • 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 𝛼 𝑝 𝑛 = 𝑒− 𝑛 𝑛 𝑛 𝑛! , 𝑛 = 𝛼 2 • Random photon arrival times • 𝑔 2 𝜏 = 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 𝑛 = 𝑛 1 𝑛 1.5 𝑛 +1 𝑛 +1 𝑒 −𝑛ℏ𝜔/𝑘𝐵 𝑇 (2) g () 1 • Bunched: 𝑔 2 0 = 2 p 𝑛 = −𝑛ℏ𝜔/𝑘 𝑇 𝐵 𝑛𝑒 0.5 • Characteristic coherence time = 1 − 𝑒 −ℏ𝜔/𝑘𝐵 𝑇 𝑒 −𝑛ℏ𝜔/𝑘𝐵 𝑇 0 1 -1 1 Photon 0 number 𝑛 = (arb. units) 𝑒 ℏ𝜔/𝑘𝐵 𝑇 − 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 𝑔 𝑐𝑟𝑜𝑠𝑠 = 𝑔 𝑎𝑢𝑡𝑜 + 𝑛 𝑝𝑎𝑖𝑟𝑠 • 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: 𝑔(2) ℎ𝑒𝑟𝑎𝑙𝑑𝑒𝑑 /𝑔(2) 𝑢𝑛ℎ𝑒𝑟𝑎𝑙𝑑𝑒𝑑 ≥ (1 − 𝑝𝑢𝑛ℎ𝑒𝑟𝑎𝑙𝑑𝑒𝑑 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