Frequency and time... dispersion-cancellation, etc. (AKA: An old experiment of mine whose interpretation helps illustrate this collapse-vs-correlation business, and which will serve as a segué into time & phase msmt) • Dispersion cancellation in an HOM interferometer – (more "collapse versus correlations") – (useful for time measurements) • What are time measurements? – (no time operator) – (indirect measurements) – (energy-time "uncertainty relation") • States of an electromagnetic mode – (number-phase "uncertainty relation") – (homodyne measurements, et cetera) – Phase of a single photon... 31 Jan 2012
Entangled photon pairs (spontaneous parametric down-conversion) The time-reverse of second-harmonic generation. A purely quantum process (cf. parametric amplification) Each energy is uncertain, yet their sum is precisely defined. (For a continuous-wave pump!) Each emission time is uncertain, yet they are simultaneous. (What does this remind you of?)
Hong-Ou-Mandel interferometer Re Reme membe mber: r: if if you you de dete tect t only only one one ph photon oton, , the the oth other ph photon oton "knows" "knows" wher where yours yours cam came e from. from. He Hence there there is is no no interf interference erence (each (each de dete tector tor sees sees 1/ 1/2 2 of of the ph the photon otons, , irresp irrespective ective of of any any ph phases or or pa path th-length gth di diffe ffere rences). But: But: if if you you de dete tect t both both ph photon otons, , there there is is no no way way to to tell tell whet whether her both both wer were reflec reflected ted or or both both wer were e transmitted. transmitted. r 2 +t +t 2 = = (i (i 2 + + 1 2 )/ )/2 2 = = 0. 0. (any (any lossless lossless symmetric symmetric be beam m sp splitter litter has has a a π /2 /2 ph phase shif shift t be betw tween r r and and t.) t.) CA CAVEAT: : there there must must be be no no way way to to tell tell which which oc occurred. urred. If If the the pa path ths aren't aren't aligned aligned righ right, t, no no interf interference erence oc occurs. urs. If If one one ph photon oton reac reaches hes the the be beam m sp splitter litter be before fore the the other, other, no no interf interference erence oc occurs. urs. How How long long is is a a ph photon oton?
the famous dip RATE NCE RA NCIDENCE In In every every exp experiment eriment to to da date te, , the the COINCI width wid h of of this this fe feature ture is is limited limited only only COI by the by the ba bandw dwidth dth of of the the ph photon otons; in in other other wo words, s, the the ph photon otons are are as as tightly tightly co correlat elated ed as as they they co could ld pos possibl bly y be be given given their their ow own uncerta un rtainty ty in in time time ( Δ t t > > 1/ 1/2 Δω Δω ). ). DE DELAY AY TIME TIME (f (fs) s) Instead Instead of of an an amp amplitu litude e for for each each fre freque uency y co component nent of of each each be beam, there there is is an an amp amplitu litude e for for each each fre freque uency-c y-corre orrelate ted d pa pair r of of ph photon otons. Energy-entangled Energy-entangled state. state.
What's the speed of a photon? Silly Silly qu quest estio ions ns ab abou out t group group velo velocity city, ph phase velo velocity city, "c "collec ollective" tive" natu nature re of of the the ind index ex of of refrac refraction, tion, pre precurs ursors ors, , et et cet ceter era. a. Mo More e seriou serious s qu quest estio ions ns ab abou out t ho how w to to qu quant antiz ize e electro electromagnetic magnetic fi fields ds in in a a di dispe pers rsive medium. medium. Lon Longs gsta tandi ding g de deba bate te ab abou out t "su "superlu erluminal" minal" tunneling. tunneling.
Problem with propagation measurements Quadratic Quadratic term term lead leads s to to group-veloc group-velocity ity di dispe pers rsion on, broa broade dening, g, chir chirp.
But wait!
So, what can we predict? Det Detect ectors s are are "infinitely "infinitely slow" slow" (ns (ns = = 10 10 6 fs fs);... ;... Coinc. Coi . proba probabi bility ty is is an an integral integral ov over all all times times at at which which D1 D1 and and D2 D2 co could ld fi fire re: UGH! UGH! Each Each E E is is an an integral integral ov over r its its ow own fre freque uency! y! But... But... if if T T goes goes to to inf infinity, inity, lif life e simp simplif lifies: ies: integrate a † a † aa We We have have only only to to integrate aa over over all all ω 1 , , ω 2 . The The ph phys ysical meaning: meaning: calcu calculat late e the the proba probabi bility ty for for each each pa pair r of of fre freque uencies which which might might reac reach h the the two two de dete tectors tors, , and and then then integrate. integrate. Wh Why? y? No No interf interference erence be betw tween pa path ths lead leading ing to to di diffe ffere rent t fre freque uencies at at the the de dete tectors tors, be becaus use in in pri principl ple one one co could ld go go ba back k and and measu measure re how how mu much h energy energy had had be been ab absorb sorbed ed. Note Note: : it it took took a a long long time-integral time-integral to to enf enforce rce this. this. If If the the de dete tector tor had had be been ope open on only ly for for 1 1 fs fs, , it it wo would ld be be imp impossib ossible le to to tell tell what what fre freque uency y it it had had seen. seen.
No, we're not done yet... The The proba probabi bility ty of of de dete tecti ting g a a given given fre freque uency y pa pair: r: (one (one can can always always insert insert a a co complet lete e set set of of states.) states.) But But we we only only started started wit with h 2 2 ph photon otons, , so so if if we we annihilate annihilate 2, 2, there there will will be be 0 0 lef left: t: (the (the squ square are of of the the tw two-photon o-photon amp amplitu litude) e) the the ph phase di diffe ffere rence is is all all that that co concer ncerns ns us us: Note Note 0: 0: all all that that math math di did d no nothing thing but but get get us us ba back k to to Feynman Feynman’s ’s rule! rule! Note Note 1: 1: the the GVD GVD terms terms (qu (quad adratic ratic in in fre freq.) .) cancelled cancelled out. out. Note Note 2: 2: the the pa patte ttern rn moves moves at at the the group group velo velocity city. Note Note 3: 3: the the shap shape e is is the the Fourier Fourier transform transform of of f( f( ω ), ), like like the the pul pulse itself itself.
Nonlocal cancellation of dispersion (O (Oh, h, and and by by the the way way... yeah, single-p yeah, single-photon hoton pul pulses travel travel at at the the group group velocity.) velocity.)
The Feynman path picture REMEM REM EMBER: BER: interf interference erence only on ly oc occurs urs be betw tween tw two pa path ths which which yi yield d the the same same ω 1 and and the the same same ω 2 . If in If in the the TT TT pa path th, , a a "blue" "blue" ph photon oton is is de dete tecte ted d at at D1 D1 (and a (and a "red" "red" at at D2), D2), then then I must must co compar are e wit with h the the ph phase of an of an RR RR pa path th wit with h the the same same outcome... outc ome... The The tw two o pa path ths are are ind indistingu istinguishab ishable le even even th though ough in in each each case case on one of of th the pul pulses was was broa broade dened. d. Perfect interf Perfect interference erence oc occurs. urs. Note Note: : this this is is on only ly pos possibl ble be becaus use we we have have assu assumed med a a cw cw (co (continu ntinuous) s) pum pump; p; oth otherw rwise ise, , the the time time of of de dete tecti tion on of of a a bl blue ue ph photon oton wo would ld tell tell us us whet whether her it it had had travelled travelled along along R R or or T
The collapse picture Af After ter D1 D1 fi fire res, , it it proj projects ts the the light light in in arm arm 2 2 into into a a su superp erposition osition of of tw two id identical entical chir chirped ed wave wave pa packets ts-- these these tw two o pa packe kets ts exhib exhibit it pe perfe rfect interf interference. erence.
NOTE: the slow detectors were important! Fast Fast de dete tector tor -> -> broa broad d fe feature ture in in the the pre presence of of di dispe pers rsion on (to (too mu much h inf information) ormation) Slow Slow de dete tector tor -> -> narrow narrow fe feature ture in in the the pre presence of of di dispe pers rsion on
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