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

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What is light? 17 th -19 th century – particle: Corpuscular theory (Newton) dominates over wave theory (Huygens). 19th century – wave: Experiments support wave theory (Fresnel, Young), Maxwells 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? 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? Laser Lamp (spoiler alert: yes)

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

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Most light is from statistical processes in macroscopic systems The spectral and photon number distributions depend on the system Blackbody/thermal radiation Luminescence/fluorescence Photon statistics 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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A B 50/50 beamsplitter Photo-detectors Auto-correlation functions A B

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A B 50/50 beamsplitter Photo-detectors Auto-correlation functions 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 A 1 2

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A 1 2

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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 << 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 Photodetection

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Cauchy-Schwarz inequality

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Other non-classicality signatures

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Types of light Classical light Coherent states – lasers Thermal light – pretty much everything other than lasers Non-classical light Collect light from a single emitter – one at a time behavior Exploit nonlinearities to produce photons in pairs

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|α||α| ϕ

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Thermal light

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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 BS D1D1 D2D2

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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) Desired single photon properties

20 Weak 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)<

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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)

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Pair sources 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 Parametric processes in χ (2) and χ (3) nonlinear media Atomic ensembles Single emitters 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 Cascade Statistics: one pair at a time

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

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Heralded single photons Heralding detector Single photon output Heralded statistics of one arm of a thermal source

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Properties of heralded sources 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 Heralding detector Single photon output

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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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EOM Neat example of storage + heralding: Nearly Fock states EOM Number resolving idler detector Pump Output PBS Signal Idler Control output switching Type-II SPDC 13…8

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Source M, µ M Source 3, µ 3 Source 2, µ 2 Source 1, µ 1 Mode reconstruction …… # p total (n) or g (k) µ 1, µ 2, … µ M Photon number resolving detector µ = mean photon number p(n) = probability of detecting n photons g (k) = zero time intensity correlation 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

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