Quantum Optics. Jeff Lundeen. Dept. of Physics uottawa. I don t know anything about photons, but I know one when I see one - Roy J. Glauber.

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1 Quantum Optics Jeff Lundeen Dept. of Physics uottawa I don t know anything about photons, but I know one when I see one - Roy J. Glauber. University of Ottawa Summer School 2014

2 The Photon in a nutshell The Photon as a wave 1 st observation: Newton s Rings Archetype: Mach-Zehnder Interferometer 1,0> + 0,1> 1,0> + e i2πx/λ 0,1> λ 1 Path length diff, x Each photon is put into a superposition of the two paths

$diffraction, polarization$ Light is a wave!

3 Light is made up of waves. Light is corpuscular Huygens 1690 Newton 1704 But what about diffraction, polarization and interference? Light is a wave! Fresnell 1818 Light is an Electromagnetic Wave! Maxwell 1864

4 Evidence for the photon Blackbody Radiation Light is emitted in units of hω Photo-Electric Effect Photo-Electrons are emitted instantaneously Number of PEs is equal to number of photons The Lamb-Shift Proof of Quantum Electrodynamics (and vacuum fluctuations)

5 The Proof of Photon as a particle Momentum Compton Effect Discreteness Instantaneous photo-electron emission Hanbury-Brown and Twiss Interferometer Laser: Photon number distribution Kimble, Mandel PRL 1975

6 Is the photon really a particle? Newton, T.D.; Wigner, E.P. (1949). Rev. Mod. Phys No equivalent of the position operator, x Relativistic so it can not be localized. No mass ^ No Schrödinger Equation. No wavefunction? 6/36

7 A Fabricated History of the Photon Wavefunction I. Bialynicki-Birula, Progress in Optics 36, 245 (1996). J. Sipe, Physical Review A 52, 1875 (1995). B. J. Smith and M.G. Raymer, New Journal of Physics 9 (2007) 414.

8 Non-relativistic Photons The problem: With the Maxwell Wavefunction, the inner product is non-local. The Born rule (i.e. Prob(x) = Ψ(x) 2 ) is incorrect. In the paraxial limit (i.e. small angles) and with the slowly varying amplitude approximation (i.e. narrow bandwidth compared to the carrier frequency), the photons are analogous to non-relativistic particles. Ψ(x,y,z,σ) = E(x,y,z,σ), the scalar electric field Normalized (i.e non-relativisitic) Wavefunctions! with z/t = c and p z = hc/f and σ is polarization

9 Single Photon Wavefunction Simplified Helmholtz Equation for Scalar Electric Field, ε Schrodinger Equation The optical index Kinetic Energy (i.e. E-V)

10 The Perfect Photon 1. Likes to be alone: P(n>1)<<1 2. Looks nice: Convenient spatial and spectral modes (e.g. Gaussian). 3. Low noise: High degree of modal (spectraltemporal) quantum purity. 4. Is there when we want it: P(0)<<1

Quantum Dots NRC Institute for Microstructural

11 Quantum Dots NRC Institute for Microstructural Sciences InP InAs InP pyramids are grown on square templates. The top of the pyramids is 30nm x 30nm room for only one dot. Single quantum dots are optically pumped to deterministically produce telecom single photons at high repetition rates (>1GHz). SiO 2

12 Quantum Dots: Drawbacks Outcoupling efficiency is still low (<10%). Low noise (high quantum purity) photons have not been demonstrated. Background photons are emitted from surrounding dots or other dot levels. Slow progress towards deterministically creating identical quantum dots (e.g. same energy levels).

13 Spontaneous Parametric Downconversion Downconversion Pump A pump photon is spontaneously converted into two lower frequency photons in a material with a nonzero (2) i s 1 Momentum is conserved.. k s k i k PUMP 2π/L..as well as energy s PUMP i PUMP = s + i

14 Types of Light: Photon Picture τ c Antibunched (e.g. single photons) Poissonian (e.g. laser beams) Bunched (e.g. light bulb) Photon detections as a function of time for a) antibunched, b) random, and c) bunched light t Start g (2) Antibunched (e.g. single photons) Stop g (2) Bunched (e.g. light bulb) Poissonian (e.g. laser beams) t stop -t start τ/τ c t stop -t start τ/τ c

avoid other Picosecond avalanche photodiodes Silicon Photomuliplier Arrays Electron

15 Single Photon Detectors In amplified photodiodes the noise is 10 6 times the signal from one photon Need intrinsic amplification soon after the photon is absorbed to avoid other Picosecond avalanche photodiodes Silicon Photomuliplier Arrays Electron Multiplying CCDs Superconducting Nanowire Arrays Andreas Fiore, Nature Photonics 2, (2008)

16 Avalanche Photodiode Single Photon Detectors In very pure materials (e.g. silicon) one can reverse bias above a diode s breakdown voltage. In Gieger mode, impact ionization creates an avalanche which creates a milliamp current pulse for every photon Quantum Efficiency: 50%-70% (visible), 10%-20% (infrared) Noise: Dark Counts /s Jitter: Absorption region is thick for high efficiency 50ps timing resolution After pulsing: 1% of the time trapped carriers emit up to 100ns after click Deadtime: Avalanche must be quenched 50ns Speed: 20 MHz count rate for active quenching

17 Superconducting Nanowire Single Photon Detectors Top View 5 microns Side View

How Superconducting Detectors Work Hotspot formation resistive bridge voltage spike Quantum Efficiency: 1%-10%, 70%-90% (with cavity) - visible to infrared Noise:

18 How Superconducting Detectors Work Hotspot formation resistive bridge voltage spike Quantum Efficiency: 1%-10%, 70%-90% (with cavity) - visible to infrared Noise: Dark Counts 1/s - 50/s Speed: 50 MHz to 3 GHz Jitter: Absorption anywhere along wire ps timing resolution Deadtime: Pulse length 0.1 to 10ns Temperature: 3-10K

19 Phase Space A classical particle has definite position x and momentum p. This is denoted by a point in x-p phase space: A statistical ensemble of particles is described by a p 2d phase-space probability distribution, a Louiville Distribution, Pr(x,p) Δx x p 2 Δp x In Quantum Mechanics, Heisenberg s uncertainty principle implies a particle is not at a point (x,p)

20 Electric Field, E Electric Field Phase Space The Quadratures of the Electric field are analogous to x and p, E = x + ip time The electric field E at ω behaves like a particle in a harmonic oscillator with frequency ω

21 The Electric Field Wavefunction Valid for any quantum state of Electromagnetic Field (not just single photons) The electric field at ω behaves like a particle in a harmonic oscillator with frequency ω, with E = x + ip Harmonic Oscillator Schrodinger Equation Example 1: The wavefunction of a laser beam (coherent state) ψ laser (x) = π -1/4 exp(-½ (x-x 0 ) 2 + ip 0 x) Example 2: The wavefunction the vacuum

22 X X X P α P α = α sinθ α The Coherent State a) Δθ Phase θ ½ θ b) α X α = α cosθ X Phase θ c) Shot Noise! Sensitivity: 1/ N Phase θ Whatever their intensity (i.e. α 2 ), all coherent states have the same quadrature uncertainties

23 The Quantum Harmonic Oscillator Each mode (degree of freedom) of the electric field is a quantum harmonic oscillator. The nth excited state represents a mode with n photons in it 2 nd Excited State 1 st Excited State Ground state Two photons One photon No photons Raising (a ) and lowering (a) operators create and destroy photons in a mode e.g. 2 = (a ) 2 0 / 2 Observables can be written in terms of a and a e.g. x=e cos = (a + a )/ 2, p=e sin = i(a - a )/ 2

24 The EM Mode Quantum State The quantum state of an optical mode, e.g. a pulse of light, longitudinal mode of a optical cavity, frequency band in a continuous source Antibunched Quantum State in the photon number basis: ψ = Σ c n n The quantum state of a laser pulse (coherent state) Poisson Statistics Bunched The photon number statistics exhibit features that are impossible in classical optics

25 Photon Number Resolving Detectors N Parallel Single Photon Detectors Superconducting Nanowire Array Silicon Photomuliplier Arrays Andreas Fiore, Nature Photonics 2, (2008) Time multiplexed detector Fiber-assisted detection with photon number resolution, D. Achilles, et al., Optics Letters, 28, (2003).

26 Electron Multiplying CCD Quantum Efficiency: 75% raw, 5-30% thresholded Pixels: 512x512 Temperature: -90C Speed: 60 Hz Noise: Dark Noise + readout noise 1click/200pixels/frame

27 Transition Edge Superconducting Detectors Sae Woo Nam NIST NIST Quantum Efficiency: 30% Bare, 95% anti-reflection, UV to far IR Noise: Dark Counts 0.001/s Jitter: 300ps timing resolution Deadtime: Rethermalization 1 microsecond Speed: 200 KHz Temperature: mK

28 Detection Intensity detector (e.g. photodiode) Classical: Intensity = EE* Quantum: photon number =n=a a Single Photon detector Click = I Single Photon detector with efficiency η No Click =(1-η) n = e n ln(1-η) Photon number detector with efficiency η m Photons = :(ηn) m e (-ηn) /m!:

29 f(ω) The Photon Wavefunction in QED The Quantum Theory of Light Rodney Loudon The photon spectral amplitude Angular frequency, ω with z/t = c and ω = ħc/p z

Representations of generalized quantum states These incorporate classical

coherence between m and n photon amplitudes in the state light bulb laser Phase

e. x-p) space p x Single photon state The Wigner Function, W ρ An attempt to

30 Representations of generalized quantum states These incorporate classical noise: mixed states, decoherence The Density Matrix, ρ A matrix giving the coherence between m and n photon amplitudes in the state light bulb laser Phase Space Cartoon The location and Heisenberg uncertainty of the state in phase (i.e. x-p) space p x Single photon state The Wigner Function, W ρ An attempt to give the probability for a quantum particle to be at x and p The problem: It goes negative! p x

Hermann Weyl introduced the inverse transformation,

31 The Wigner Function 1932: Eugene Wigner introduced a phasespace representation of the Quantum State: 1927: Hermann Weyl introduced the inverse transformation, taking phase-space distribution g to a Hilbert Space operator G

32 Wigner Function Examples a) b) The vacuum state: 0> Single Photon (Fock State): 1> Negative values! c) 5 Photon State (Fock State): 5> Zero crossings = # of photons At x=0,p=0, for n odd W is negative, for n even W is positive

33 Thermal State Coherent State (e.g. laser) Thermal State (e.g. light bulb) A thermal state is equivalent to a statistical mixture of coherent states with weighting: Images: Alex Lvovsky U. of Calgary

34 Squeezed Coherent States Images: Alex Lvovsky U. of Calgary

35 Two-mode Squeezed Vacuum q + = q 1 + q 2, q - = q 1 - q with 2 p + = p 1 + p 2, p - = p 1 - p 2 Squeezing occurs in correlations q + and p -. EPR Entanglement! Let s find the marginals: q + p + q - p - And one mode by itself is thermal!:

36 Three Photon Downconversion: Star State Images: Alex Lvovsky U. of Calgary

37 Schrodinger Cat State ψ = α + -α -α α Image: Thomas Curtright, U. of Miami

Typically, the dynamics are NOT equal to the

38 Typically, the dynamics are NOT equal to the classical equivalent Image: Thomas Curtright, U. of Miami Dynamics of the Wigner Function Free Evolution is the classical Liouville Equation: For a harmonic oscillator potential (e.g. light), all states rotate rigidly around the origin Superposition: 0>+ 1>

39 Coherent State in Action P X Image: Thomas Curtright, U. of Miami

40 Finding the Wigner function of an unknown state Born Rule: P n = ψ n n ψ = Tr[ ψ ψ n n ] Generalized Born Rule: Detection Probability P n = Tr[ ρ П n ] Unknown quantum state Known measurements Use tomography

41 Quantum Homodyne Tomography All Electric fields exhibit inherent quantum noise Increase E LO so that ΔE LO / E LO = 1/ n 0 ^ ^ Diff = Q(θ) = x cos(θ) + p sin(θ) ^ Images: Alex Lvovsky U. of Calgary

42 Balanced Homodyne Detection Adds phase sensitivity to the quantum optical detector Input Mode, E in (unknown state) E a Difference 50:50 Beamsplitter Diff = E a 2 - E b 2 = E in E LO 2 /2 - E in + E LO 2 /2 = 2 Re[E in E LO ] = 2 E LO Re[E in e iθ ] E b Local Oscillator, E LO = E LO e iθ (Laser beam) Noiseless scaling factor θ = 0, Diff = Re[E in ] = x θ = π, Diff = Im[E in ] = p In general, Diff = x θ = x cos(θ) + p sin(θ) p x θ θ x Smithey et al, Phys. Rev. Lett, 70, 1244 (1993)

p=e cos Gerdenbach, Nature (2000) By taking shadowgrams of

43 Quantum State Tomography Quantum State x=e sin Homodyne Detection Find the Wigner Function from these shadowgrams. p=e cos Gerdenbach, Nature (2000) By taking shadowgrams of the Wigner function from all angles we can reconstruct it 43/35

diffused Thermal (made with a blow torch) States

44 Some more examples Partially phase diffused Coherent state Totally phase diffused Amplitude diffused Thermal (made with a blow torch) States can be constructed with statistical mixtures of coherent states

Other variables: time-frequency mode In signal analysis, a time-varying

frequency ω = 2πf, where f is the regular frequency.

45 Other variables: time-frequency mode In signal analysis, a time-varying electrical signal, optical signal, mechanical vibration, or sound wave are represented by a Wigner function. Here, x is replaced with the time and p/ħ is replaced with the angular frequency ω = 2πf, where f is the regular frequency. Seismic Analysis Laser Pulse Characterization Earthquake Wigner function Chirp in a laser pulse Engine Diagnostics Scooter engine vibration Wigner function

46 Other Quasi-Probability Distributions 1940: Kodi Husimi ( 伏見康治 ) (alt: Fushimi Koji) introduced the Q function, It is always positive 1963: Roy Glauber and George Sudarshan introduced the P function, writing the density matrix in a diagonal basis of coherent states A coherent state is a single point in phase space It is highly singular for most non-classical states

47 Mach-Zehnder with Quantum States The phase accumulated is proportional to the energy E of the state 0,1> and the time t, =E t/ħ=2πx/λ 1,0> + 0,1> 1,0> + e i2πx/λ 0,1> λ 1 Path length diff, x States such as N,0> + 0,N> ("high-noon" states, for N large) have N times the energy. Thus, N photons behave like one photon with wavelength=λ/n. λ/n N,0> + 0,N> N,0> + e inθ 0,N> 0,N><0,N + N,0><N,0 NOON N At the Heisenberg limit: n=1, =1/N John Dowling, Yablanovitch, Jim Franson, Carl Caves

48 Quantum interference Each photon then interferes only with itself. Interference between two different photons never occurs. Dirac (1930) Hong-Ou-Mandel Interference 1/ 2 x 1/ 2 + i/ 2 x i/ 2 = 0 The photon pair never results in a coincidence detection Photons stick together as if there was an attractive force between them. Amplitudes for events interfere (i.e. Feynman paths) not particles

Photon applications Quantum communication Quantum cryptography: Absolute secrecy based on physical laws Ultimate bandwidth limits: 1 bit per photon Quantum metrology Reaching the fundamental limit of

49 Photon applications Quantum communication Quantum cryptography: Absolute secrecy based on physical laws Ultimate bandwidth limits: 1 bit per photon Quantum metrology Reaching the fundamental limit of measurement Quantum computing Photonic Quantum Logic gate: The nonlinear sign shift 01 01, Factoring large numbers: Internet encryption Simulating quantum systems x High quality Hong-Ou-Mandel interference is critical for optical quantum logic gates. For this we need textbook photons!

Single photons. how to create them, how to see them. Alessandro Cerè

Single photons. how to create them, how to see them. Alessandro Cerè Single photons how to create them, how to see them Alessandro Cerè Intro light is quantum light is cheap let s use the quantum properties of light Little interaction with the environment We can send them