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Non-classical light and photon statistics

Non-classical light and photon statistics. Elizabeth Goldschmidt JQI tutorial July 16, 2013. What is light?. 17 th -19 th century – particle : Corpuscular theory (Newton) dominates over wave theory (Huygens).

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Non-classical light and photon statistics

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

  2. 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 mechanicsdeveloped (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.

  3. What is non-classical light and why do we need it? • Heisenberg uncertainty requires • For light with phase independent noise this manifests as photon number fluctuations 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)

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

  5. 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

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

  7. Auto-correlation functions 50/50 beamsplitter A • Second-order intensity auto-correlation characterizes photon number fluctuations • Attenuation does not affect A Photo-detectors B B • Hanbury Brown and Twiss setup allows simple measurement of g(2)(τ) • For weak fields and single photon detectors • Are coincidences more (g(2)>1) or less (g(2)<1) likely than expected for random photon arrivals? • For classical intensity detectors

  8. Auto-correlation functions 50/50 beamsplitter • Second-order intensity auto-correlation characterizes photon number fluctuations • Attenuation does not affect A Photo-detectors B • 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

  9. General correlation functions • Correlation of two arbitrary fields: • is the zero-time auto-correlation • for different fields can be: • Auto-correlation • Cross-correlation between separate fields • Higher order zero-time auto-correlations • can also be useful A 1 2

  10. 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 kdetectors • 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

  11. Cauchy-Schwarz inequality • Classically, operators commute: • With quantum mechanics: • Some light can only be described with quantum mechanics , no anti-bunched light

  12. Other non-classicality signatures • Squeezing: reduction of noise in one quadrature • 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 or Wigner function • Useful for tomography of Fock, kitten, etc. states • Higher order zero time auto-correlations:, • Non-classicality of pair sources by auto-correlations/photon statistics

  13. 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

  14. Coherent states |α| ϕ • Laser emission • Poissoniannumber statistics: • , • Random photon arrival times • for all τ • Boundary between classical and quantum light • Minimally satisfy both Heisenberg uncertainty • and Cauchy-Schwarz inequality

  15. Thermal light • Also called chaotic light • Blackbody sources • Fluorescence/spontaneous emission • Incoherent superposition of coherent states (pseudo-thermal light) • Number statistics: • Bunched: • 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

  16. 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 • … …

  17. BS 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 D2

  18. 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)

  19. 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

  20. 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.

  21. 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 • Phase-matching controls the direction of the output

  22. Atomic ensembles Pair sources • 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

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

  24. Single photon output Heralded single photons • 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: Heralding detector Heralded statistics of one arm of a thermal source

  25. Single photon output Properties of heralded sources 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

  26. 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

  27. 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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