100 Years of Light Quanta Roy J. Glauber Harvard University Max - PowerPoint PPT Presentation

100 Years of Light Quanta Roy J. Glauber Harvard University

Max Planck: October 19, 1900 Interpolation formula for thermal radiation distribution – a brilliant success December 14, 1900: Model: Ensemble of 1-dimensional charged harmonic oscillators exchanging energy with radiation field – reached “correct” equilibrium distribution only if oscillator energy states were discrete E n = nhv

Albert Einstein: 1905 Found two suggestions that light is quantized - Structure of Planck’s entropy for high frequencies - The photoelectric effect He noted in later studies – hv - Momentum of Quantum (1909) c - New Derivation of Planck’s law (1916) A = Spontaneous radiation probability B = Induced radiation rate

Compton effect: 1923 Completed picture of particle-like behavior of quanta - soon known as photons (1926) L. de Broglie, W. Heisenberg, E. Schrödinger: 1924-26 - told all about atoms But radiation theory was still semi-classical until P. Dirac devised Quantum Electrodynamics in 1927

Split real field into two complex conjugate terms + − = + ( ) ( ) E E E * − + = ( ) ( ) E ( E ) − ω ( + i t ) ~ e E contains only positive frequencies ( − e ω ) i t E ~ contains only negative frequencies ( ± ) E physically equivalent ( classically )

Define correlation function − + = 〈 〉 ( 1 ) ( ) ( ) G ( r t r t ) E ( r t ) E ( r t ) 1 1 2 2 1 1 2 2 Young’s 2-pinhole experiment measures: G (1) (r 1 t 1 r 1 t 1 ) + G (1) (r 2 t 2 r 2 t 2 ) + G (1) (r 1 t 1 r 2 t 2 ) + G (1) (r 2 t 2 r 1 t 1 ) Coherence maximizes fringe contrast

Let x = (r,t) Schwarz Inequality: 2 ≤ ( 1 ) ( 1 ) ( 1 ) G ( x x ) G ( x x ) G ( x x ) 1 2 1 1 2 2 Optical coherence: 2 = ( 1 ) ( 1 ) ( 1 ) G ( x x ) G ( x x ) G ( x x ) 1 2 1 1 2 2 Sufficient condition: G (1) factorizes G (1) (x 1 x 2 ) = E * (x 1 ) E (x 2 ) i.e. ~ also necessary: Titulaer & G. Phys. Rev. 140 (1965), 145 (1966)

Quantum Theory: ( ± ) E Are operators on quantum state vectors E + ( ) ( rt ) Annihilation operator, lowers n ⎥ n 〉 → ⎥ n -1 〉 E − ( ) ( rt ) Creation operator, raises n ⎥ n 〉 → ⎥ n +1 〉 Lowering n stops with n = 0 , vac. state + = ( ) E ( rt ) vac . 0

Ideal photon counter: - point-like, uniform sensitivity ( + ) f E ( rt ) i Transition Amplitude f - Square, sum over f - Use completeness of ∑ = f f 1 (unit op.) f Total transition probability ~ = ∑ ∑ 2 + − + = − + ( ) ( ) ( ) ( ) ( ) ( rt ) f E ( nt ) i i E ( rt ) f f E ( rt ) i i E ( rt ) E ( f f = 〈 i ⎥ E (-) ( rt ) E (+) ( rt ) ⎥ i 〉

i Initial states random i Take ensemble average over ρ = { | i 〉〈 i | } Average Density Operator: ~ Then averaged counting probability is {〈 i | E (-) ( rt ) E (+) ( rt ) | i 〉} Av. = Trace { ρ E (-) ( rt ) E (+) ( rt ) }

To discuss coherence we define G (1) ( r 1 t 1 r 2 t 2 ) =Trace { ρ E ( - ) ( r 1 t 1 ) E (+) ( r 2 t 2 ) } ( 1 ) G - obey same Schwarz Inequality as classical Upper bound attained likewise by factorization, G (1) ( r 1 t 1 r 2 t 2 ) = E * ( r 1 t 1 ) E ( r 2 t 2 ) = − ( 1 ) ( 1 ) G G ( t t ) Statistically steady fields: 1 2 - If optically coherent, G (1) ( t 1 - t 2 ) = E * ( t 1 ) E ( t 2 ) E ( t ) ~ e -i ω t for ω > 0 The only possibility is:

R. Hanbury Brown and R. Q. Twiss Intensity interferometry Two square-law detectors D 1 M D 2 Signal

Ordinary (Amplitude) interferometry measures ′ ′ ′ ′ − + ≡ ( 1 ) ( ) ( ) G ( rt r t ) E ( rt ) E ( r t ) Ave . Intensity interferometry measures ′ ′ ′ ′ ′ ′ ′ ′ − − + + = ( 1 ) ( ) ( ) ( ) ( ) G ( rt r t r t rt ) E ( rt ) E ( r t ) E ( r t ) E ( rt ) The two photon dilemma!

Hanbury Brown and Twiss ‘56 D 1 D 2 MULT. Pound and Rebka ‘57 Coincidence rate 1 Delay time

Define higher order coherence ( e.g. second order) G (2) ( x 1 , x 2 , x 3 , x 4 ) = 〈 E (-) ( x 1 ) E (-) ( x 2 ) E (+) ( x 3 ) E (+) ( x 4 ) 〉 = E * ( x 1 ) E * ( x 2 ) E ( x 3 ) E ( x 4 ) � Joint count rate factorizes 2 E ( x 2 ) 2 (2) ( x 1 , x 2 , x 2 , x 1 ) = E ( x 1 ) G � Wipes out HB-T correlation n th order coherence, n � ∞

What field states factorize all G (n) ? Recall normal ordering - Sufficient to have: E (+) ( rt ) ⏐ 〉 = E ( rt ) ⏐ 〉 ~ defines coherent states Convenient basis for averaging normally ordered products All G (n) can factorize � Full coherence

Any classical (i.e., predetermined) current j radiates coherent states ~ R.G. Phys . Rev. 84, ’51 What is current j for a laser? j r ∂ r P P = j Strong oscillating polarization current j ∂ t Quantum Optics = Photon Statistics

Quantum Field Theory – for bosons Field oscillation modes ↔ harmonic oscillators For harmonic oscillator: a ⏐ n 〉 = √ n ⏐ n - 1 〉 a lowers excitation † a † ⏐ n 〉 = √ n + 1 ⏐ n + 1 〉 a raises excitation − = † † aa a a 1

α = α α a Special states: α = any complex number α ∞ 1 n 2 − α ∑ α = 2 e n n ! = n 0 α 2 n 2 − α = P ( n ) e , Poisson distribution n ! 〈 n 〉 = ⏐ α ⏐ 2 ~ single mode coherent states

Superposition of coherent excitations : α 1 Source #1 � α 2 Source #2 � * ) α 1 + α 2 1 2 ( α 1 * α 2 − α 1 α 2 Sources #1 and #2 � e Combined density operator : ρ = α 1 + α 2 α 1 + α 2 n ∑ ρ = α α , α = α j With n sources j=1

α j For n � ∞ , ’s random Sum α has a random-walk probability distribution – Gaussian − α 2 1 α 2 P ( α ) = π α 2 e But α 2 AV . = n , mean quantum number

e.g. Gaussian distribution of amplitudes { α n } Single-mode density operator: α 2 - ∫ 1 ρ chaotic = α α d 2 α n e π n ⎛ ⎞ j ∞ ∑ n 1 = ⎜ ⎟ j j 1 + n 1 + n ⎝ ⎠ j = 0

Two-fold joint count rate: ( ) = G (1) x 1 x 1 ( ) G (1) x 2 x 2 ( ) + G (1) x 1 x 2 ( ) G (1) x 2 x 1 ( ) G (2) x 1 x 2 x 2 x 1 HB-T Effect Note for x 2 � x 1 : G (2) ( x 1 x 1 x 1 x 1 ) = 2 [ G (1) ( x 1 x 1 ) ] 2

If the density operator for a single mode can be written as: ∫ ρ = α α α α 2 P ( ) d 〈 a † n a m 〉 = Tr ( ρ a † a m ) = ∫ P ( α ) α * n α m d 2 α Then Operator averages become integrals P( α ) = quasi-probability density Scheme works well for pseudo-classical fields, but is not applicable to some classes of fields e.g. “squeezed” fields, (no P-function exists).

One mode excitation: P (| α |) P �� d �� laser chaotic � Α � Photocount distributions ( w = average count rate) p ( n ) = ( wt ) n coherent p(n) n ! e − wt Coherent state: chaotic ( wt ) n p ( n ) = Chaotic state: (1 + wt ) n + 1 n

Distribution of time intervals until first count: P ( t ) = we − wt Coherent: w P ( t ) = Chaotic: (1 + wt ) 2 Given count at t = 0, distribution of intervals until next count: P (0 | t ) = we − wt 2w P (0 |t ) 2 w Coherent: 2 w w w P ( t ) P (0 | t ) = Chaotic: (1 + wt ) 3 t t

Quasi-probability representations for quantum state ρ Define characteristic functions: s 2 λ 2 ( ) = Trace { ρ e λ a † − λ a } e χ x λ , s Family of quasi-probability densities: ∫ W ( α , s ) = 1 e αλ *- α * λ χ ( λ , s ) d 2 λ e αλ * − α * λ x ( λ , s ) d 2 λ π W( α ,1) = P( α ) s = 1 P-rep. W( α ,0) = w( α ) s = 0 Wigner fn. α , − 1) = 1 π α ρα s = -1 Q-rep. W(

Later Developments: • Measurements of photocount distributions ~ Arecchi, Pike, Bertolotti… • Photon anti-correlations ~ Kimble, Mandel • Quantum amplifiers • Detailed laser theory ~ Scully, Haken, Lax • Parametric down-conversion – entangled photon pairs • Application to other bosons • bosonic atoms (BEC) • H.E. pion showers • HB-T correlations for He* atoms • Statistics of Fermion fields ~ with K.E. Cahill • • • •