Nonclassical light and photon statistics Elizabeth Goldschmidt JQI

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). • 19 th century – wave: Experiments support wave theory (Fresnel, Young), Maxwell’s equations describe propagating electromagnetic waves. • 1900 s – ? ? ? : Ultraviolet catastrophe and photoelectric effect explained with light quanta (Planck, Einstein). • 1920 s – wave-particle duality: Quantum mechanics developed (Bohr, Heisenberg, de Broglie…), light and matter have both wave and particle properties. • 1920 s-50 s – 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? 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 • The spectral and photon number distributions depend on the system • Blackbody/thermal radiation • Lasers • Luminescence/fluorescence • 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 50/50 beamsplitter AA Photo-detectors B

Auto-correlation functions 50/50 beamsplitter A Photo-detectors • 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 B

General correlation functions 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

Other non-classicality signatures

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

Thermal light

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 • N 00 N 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 D 1 BS D 2

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.

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 Heralding detector Heralded statistics of one arm of a thermal source

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