Non-classical light and photon statistics
Elizabeth GoldschmidtJQI 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?
• 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
• Heisenberg uncertainty requires • For light with phase independent noise this manifests as photon
number fluctuations
(spoiler alert: yes)
Outline• Photon statistics
– Correlation functions– Cauchy-Schwarz inequality
• Classical light• Non-classical light
– Single photon sources– Photon pair sources
• 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
Frequency
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Frequency
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Frequency
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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
Frequency
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Photon number
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A
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50/50 beamsplitter
Photo-detectors
Auto-correlation functions• Second-order intensity auto-correlation
characterizes photon number fluctuations
- Attenuation does not affect
• 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
𝑔 (2 ) (𝜏 )=⟨:�̂� (𝑡 )�̂� (𝑡+𝜏 ): ⟩
⟨�̂� ⟩2
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(arb. units)g(2
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Photo-detectors
Auto-correlation functions• Second-order intensity auto-correlation
characterizes photon number fluctuations
- Attenuation does not affect
• 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
𝑔 (2 ) (𝜏 )=⟨:�̂� (𝑡 )�̂� (𝑡+𝜏 ): ⟩
⟨�̂� ⟩2
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
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• Accurately measuring g(k)(τ=0) requires timingresolution 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
Photodetection
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Cauchy-Schwarz inequality
• Classically, operators commute:
• With quantum mechanics:
• Some light can only be described with quantum mechanics
⟨ 𝑨𝑩 ⟩𝟐≤ ⟨ 𝑨𝟐 ⟩ ⟨𝑩𝟐 ⟩
, no anti-bunched light
⇒𝑔(2) (𝜏 )≤𝑔 ( 2) (0 )
⇒𝑔(2)𝑐𝑟𝑜𝑠𝑠≤√𝑔 (2 )
𝑎𝑢𝑡𝑜, 1(0)𝑔(2 )𝑎𝑢𝑡𝑜, 2(0)
𝑔 (2 )1,2❑ =
⟨ : �̂�1 �̂�2: ⟩⟨�̂�1 ⟩ ⟨ �̂�2 ⟩
=⟨ �̂�†1 �̂�†2 �̂�1�̂�2 ⟩
⟨�̂�1 ⟩ ⟨�̂�2 ⟩
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
Types of lightClassical 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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ThermalAttenuatedsingle photonPoissonianPairs
Coherent states • Laser emission• Poissonian number statistics:
, • Random photon arrival times• for all τ
• Boundary between classical and quantum light• Minimally satisfy both Heisenberg uncertainty
and Cauchy-Schwarz inequality
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• 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
Thermal light
Photon number
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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
• … …
Some single photon applicationsSecure 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
D1
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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
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)<<p(1)
• Perfect indistinguishability
LaserAttenuator
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
• 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
χ(2) or χ(3) Nonlinear medium/
atomic ensemble/
etc.
Pump(s)
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
• Heralded single photons
• Entangled photon pairs
• Entangled images
• Cluster states
• Metrology
• … …
Some pair source applications
Heralding detector
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
Single photon output
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<n>=1.2
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Perfect Heralding
Heralded statistics of one arm of a thermal source
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Heralding with loss
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
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
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