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

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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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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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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

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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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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