Interferenza in meccanica quantistica
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interferenza in meccanica quantistica fisica interdisciplinare giuseppe gennaro Il Problema • La meccanica quantistica descrive con successo i più disparati fenomeni in natura, dalle particelle elementari alla nascita dell’universo • Il problema è che essa implica un certo numero di nozioni e concetti totalmente controintuitivi • R.P. Feynman “Nessuno capisce realmente la meccanica quantistica” Esperimenti sui Fondamenti della MQ • Sin dai primi anni MQ suscita dibattiti sui suoi fondamenti (Bohr-Einstein, Heisenberg) • Molti esperimenti, in questi ultimi anni, ha di fatto confermato la peculiarità controintuitiva della meccanica quantistica • Applicazioni sperimentali dei fenomeni fondamentali: Crittografia quantistica • Quantum Teleportation and Computation Esperimento con proiettili Apparato Sperimentale • Mitragliatrice • Parete con due fessure • Rilevatore di proiettili Esperimento con proiettili • Chiudiamo la fessura 2 • Otteniamo P1 Esperimento con proiettili • Chiudiamo la fessura 1 • Otteniamo la curva P2 Esperimento con proiettili • Apriamo le due fessure • Otteniamo la curva P12 • Verifichiamo che P12 = P1 + P2 • I proiettili arrivano sempre in blocchi • “No Interferenza” Esperimento con onde d’acqua Apparato sperimentale • Sorgente d’onde circolari • Parete con due fessure • Rilevatore di intensità Esperimento con onde d’acqua • Chiudiamo la fessura 1 • Onda primaria diffratta • Intensità onda fessura 1 I1 Esperimento con onde d’acqua • Chiudiamo fessura 2 • Onda primaria diffratta • Intensità onda fessura 2 I2 Esperimento con onde d’acqua: Risultati • Misuriamo Intensità I12 I12 != I1 + I2 • No granularità • Le due onde interferiscono Interferenza costruttiva e distruttiva • Interferenza distruttiva • ddc= (2n + 1)λ/2 • Interferenza costruttiva • ddc= nλ Matematica della interferenza • Altezza istantanea dell’onda al detector! " iωt iωt h e h e proveniente dalla fessura 1 (2) 1 2 • Altezza istantanea quando le due fessure sono iωt iωt aperte h1 e + h 2 e • Intensità dell’onda I12 = | (h1 + h2 ) | = |h1 | + |h2 | + 2|h1 ||h1 |cosδ 2 • I12 = I1 + I2 + 2 2 ! 2 I1 I2 cosδ Esperimento con Elettroni Apparato Sperimentale • Elettroni emessi da valvola termoionica • Accelerati verso la parete • Parete di metallo con due fessure • Rilevatore (electron multiplier) Esperimento con elettroni: Risultati P12 • Gli elettroni arrivano sempre in granuli identici tra loro • P12 != P1 + P2 Gli elettroni interferiscono! Esperimento con elettroni: Risultati P1 P12 • Gli elettroni arrivano sempre in granuli identici tra loro • P12 != P1 + P2 Gli elettroni interferiscono! Esperimento con elettroni: Risultati P1 P2 P12 • Gli elettroni arrivano sempre in granuli identici tra loro • P12 != P1 + P2 Gli elettroni interferiscono! Esperimento doppia fenditura: elettroni Interpretazione • Gli elettroni arrivano in granuli, come particelle • La probabilità di arrivo di questi granuli è distribuita come la distribuzione della intensità di una onda • Un elettrone si comporta talvolta come una particella talvolta come una onda • E’ non vero che gli elettroni o passano dalla fessura 1 o dalla fessura 2 Osserviamo gli elettroni • Quando osserviamo da dove passano gli elettroni distruggiamo la figura di interferenza P12 = P1 + P2 • Aumentando la lunghezza d’onda della luce non sappiamo più da dove passano gli elettroni e riotteniamo la figura di interferenza P12 != P1 + P2 Riassunto dei nostri esperimenti • La probabilità di un evento è data dal quadrato di una quantità complessa φ chiamata ampiezza di 2 probabilità P = |φ| • Se un evento può accadere in diversi modi alternativi la ampiezza di probabilità per l’evento è P = |φ1 + φ2 |2 • Se è possibile determinare a priori se una o l’altra alternativa è scelta realmente P12 = P1 + P2 Interferenza quantistica: Criterio • Se è impossibile distinguere fra differenti percorsi alternativi dalla sorgente al detector, le ampiezze di probabilità si addizioneranno coerentemente. • L’assenza di ogni informazione sul reale percorso è il criterio essenziale perchè si manifesti la interferenza quantistica • Se tale informazione è accessibile dall’esperimento si distrugge la figura di interferenza Principio di Indeterminazione • “E’ impossibile progettare un apparato per determinare da quale fessura passa l’elettrone che allo stesso tempo non disturbi l’elettrone abbastanza da distruggere le frange di interferenza” (Heisenberg) ∆x∆p ≥ ! Entanglement • Lo stato del sistema elettrone-fotone scatterato è uno stato “entangled” 1 |ψ! = √ (|e, 1!|f, 1! + |e, 2!|f, 2!) 2 • Se i due stati per il fotone sono ortogonali non appare nessuna figura di interferenza • Se la misura sul fotone non è capace di dire da quale fessura l’elettrone è passato, interferenza! Interferenza getta luce sulla natura • Possiamo osservare caratteristiche ondose (interferenza) o particellare (seguendo il path che la particella sceglie) a seconda dell’esperimento che scegliamo • Interpretazione errata: si propagano onde e si rilevano particelle • Interferenza a due particelle sconfessa questa interpretazione Interferenza a due particelle • Una particella decade in due e colpisce uno schermo secondo due path differenti: particella 1 o in fascio a o a’ e particella 2 o in b o in b’ 1 |ψ! = √ (|a!1 |b!2 + |a! !1 |b! !2 ) 2 Interferenza a due particelle • Non osserviamo fenomeni di interferenza per la particella 1 • Anche in principio possiamo porre detectors nei fasci b e b’ e quindi conoscere indirettamente che percorso ha fatto la particella 1 • Formalmente non possiamo sovrapporre le ! ampiezze (coerentemente)|a!1 e |a !1 perchè entangled con |b!2 e |b! !2 Interferenza a due particelle • Una particella decade in due che colpiscono due schermi secondo due path differenti Entanglement • Se la particella originaria è ferma, le due particelle figlie avranno momenti approssimativamente uguali e opposti • Le particelle possono passare o dalle fessure A e A’ o da B e B’ • Lo stato delle due particelle 1 |ψ! = √ (|a!1 |a! !2 + |b!1 |b! !2 ) 2 Interferenza a due particelle • Se la posizione della particella che decade è incerta molto più di λ/θ non si osservano frange di interferenza sui singoli schermi • Ma se si monitorano le posizioni di arrivo in coincidenza, le due particelle è più probabile che atterrino dove i percorsi alternativi PAOA’P’ e PBOB’P’ differiscono per un multiplo intero di lunghezze d’onda Due interferenze mutuamente esclusive • Se d >> λ/θ , δk/k << θ e quindi non capace di illuminare le due fessure contemporaneamente, così non ci può essere interferenza di singola particella • Se l’indeterminazione è piccola, le due fessure possono essere illuminate e creare interferenza nei singoli schermi, ma non si può più dire che se il fotone passa da A, l’altro passerà da A’. Questo fatto distrugge lo stato entangled di due particelle Distinguibilità dei Paths (Zou et al. 1991) √ 1/ 2 (|d"1 |e"2 + |h"1 |k"2 ) √ 1/ 2 (|d" + |h")1 |k"2 The quantum eraser effect, enhance or suppress the atomic emission process in the crystal by small movements of mirrors that are several feet away! Figure 5 shows an experimental arrangement first used by Alley and Shih,6 and recently exploited by Paul Kwiat and coworkers at Berkeley17 to demonstrate Marian Scully's notion of a "quantum eraser." Manipulating one photon can alter the interference pattern of another. The arrangement16 sketched in a can produce an interference pattern at detector D, when the phase shifter P is varied. An entering ultraviolet photon a is split at beam splitter A so that both down-conversion crystals (X, and X,) are illuminated. One of the resulting pair of downconversion photons can reach D, by way of beam path d or h. If one could monitor beams e and k separately, one would know in which crystal the down-conversion occurred, and there would be no interference. But merging beams e and k in this configuration lets the alternative paths of the other photon interfere. A new variant of this scheme, shown in b, uses only one crystal (at the center). The ambiguity here is created by reflecting the originating beam (blue, from the top of the photo) and its down-converted progeny (red and green) back through the crystal from mirrors (at the bottom of the photo), so one can't know on which pass the down-conversion occurred. Figure 4 will now be no superposition of the two amplitudes, and Esperimento di Zou • Il secondo fotone non è necessario che sia rilevato • Se il fascio e è bloccato le frange in D1 scompariranno perchè adesso è possibile (potenzialmente) rilevando un fotone in k ci potrebbe dire che è stato creato in X2 • i fasci e e k non sono nei paths alternativi che portano A a D conversion photons can reach D, by way of beam path d or h. If one could monitor beams e and k separately, one would know in which crystal the down-conversion occurred, and there would be no interference. But merging beams e and k in this configuration lets the alternative paths of the other photon interfere. A new variant of this scheme, shown in b, uses only one crystal (at the center). The ambiguity here is created by reflecting the originating beam (blue, from the top of the photo) and its down-converted progeny (red and green) back through the crystal from mirrors (at the bottom of the photo), so one can't know on which pass the down-conversion occurred. Figure 4 Herzog et al. (1993) effect, enhance or suppress the atomic emission process in the crystal by small movements of mirrors that are several feet away! The quantum eraser Figure 5 shows an experimental arrangement first used by Alley and Shih,6 and recently exploited by Paul Kwiat and coworkers at Berkeley17 to demonstrate Marian Scully's notion of a "quantum eraser." Others have called it "haunted" or "phantom" measurement.18 To appreciate the basic idea, first suppose that only the beam splitter is in the path of the two beams emerging from the down-converter crystal. This arrangement produces an interesting and very basic two-particle effect: Both particles must end up in the same detector.19 The reason is simple. To get coincident counts at the two detectors, either particle 1 (the photon that ultimately lands in detector 1) takes route a and particle 2 takes route will now be no superposition of the two amplitudes, and therefore coincidence counts between the two detectors will be observed. But it is still possible to "erase" this path information and recover the interference by placing a linear polarizer in each beam, as shown in the figure. If both polarizers are either horizontal or vertical there will be path information present. But if they are oriented at 45° to the horizontal, either route, a or b, can now lead to either detector. The coherence is restored and there are, once again, no coincident counts. Quantum Eraser three-particle d (Alternatively o three-particle do sume for simplic cay particles all and the same en of course, come decay plane. A screens with ho two possible sta The coherent su Polarizers can serve as quantum erasers.17 A single ultraviolet photon entering a down-conversion crystal (gray) produces two optical photons that mix at a beam splitter (dashed blue line) before arriving at detectors D, and D2. If there are no polarizers (P) or polarization rotator (R) in the beams, both photons must end up in the same detector. Inserting a 90° rotator into beam b provides information that renders the beams incoherent and thus produces coincidence counts in the two detectors. Inserting the polarizers oriented 45° to the horizontal after the beam splitter erases that The beams a', b the phase shifter (3 and y, after beams are recom ters A, B and C splitters are the ers. One record multaneous coun and K or K'. If -1) to a count primed) counter, ................................................................. ................................................................. energies. Our good quantitative agreement between experiment and blac are then accelerated and directed towards a conversion electrode. The theory indicates that lattertodobenot influence observed surec wasthe found smaller than1the 1mm mm height, found to be smaller than ininheight, co are subsequently counted by a Channeltron electron multiplier. Thecoherence. All thesewas observations support the view that each C a typ 60 powe Rayleighlength lengthofof800 800!m !mand andthe thestrong strong Rayleigh power time molecule interferes with itself only. reproducibly scanned transversely to the beam with 1-!m resolution. ionization process. A significant advantage Counts in 50 s ionization process. A significant advantage corro mechanismisisthat thatititdoes doesnot notdetect detectany anylof! o mechanism presentininthe thevacuum vacuumchamber. chamber.We Wecould couldthus thus present influa of less less than thanone oneper persecond secondeven evenunder und mol arates of 1,200 rates Markus Arndt, Olaf Nairz, Julian Vos-Andreae, Claudia Keller, # 7 Markus Arndt, Olaf Nairz, Julian Vos-Andreae, Claudia Keller, mbar).The The f vacuum conditions conditions (5(5""1010# 7 mbar). 100 fu vacuum Gerbrand van der Zouw & Anton Zeilinger Gerbrand van der Zouw & Anton Zeilinger be n 1,000 thenfocused focusedbybyan anoptimized optimizedion ionlens lenssystem, system then A a BeCu BeCu conversion conversionelectrode electrodeatat!9!9kV kVwhere wher a NATURE VOL 401 | 14 OCTOBER 1999 | www.nature.com Institut Experimentalphysik, Universität Wien, Boltzmanngasse5,5, pho 800 Institut für für|Experimentalphysik, Universität Wien, Boltzmanngasse emission ofof electrons electronswhich whichwere weresubsequent subsequen emission A-1090 Austria indu A-1090 Wien, Wien, Austria Channeltron detector. inte 600 Channeltron detector. .................................. ......................... ......................... ......................... ......................... ........ Alignment is a crucial part of this experimen .................................. ......................... ......................... ......................... ......................... ........ Alignment is a crucial part of this experiment. inte Quantum superposition lies lies at at the the heart heart of of quantum quantummechanics mechanics tofind findthe thebeam beamininthe thefirst firstplace, place,our ourcollima collim A 400 to and gives rise to many of of its its paradoxes. paradoxes. Superposition Superposition of of de de movable movablepiezo piezoslits slitsthat thatcan canbebeopened openedfrom from0in0toat fero 200 of has been been observed observed for for massive massive particles particles Broglie matter waves11 has ofthe thefirst firstslit) slit)and andfrom from0 0toto200 200!m !m(for (for t the b 200 vacuum vacuumchamber chamberisisrigidly rigidlymounted mountedononananbe oa atoms and and dimers dimers33,, small small van van der der Waals Waals such as electrons22, atoms opt up t clusters44, and neutrons55.. But But matter matter wave wave interferometry interferometry with with with withthe theionizing ionizinglaser, laser,ininorder ordertotominimize minimize sp spa muc Theeffect effectofofgravity gravityalso alsohad hadtotobebeconsidered considere larger objects has remained remained experimentally experimentally challenging, challenging, despite despite 150 The case !1 !1 the development of powerful powerful atom atom interferometric interferometrictechniques techniquesfor for the themost mostprobable probablevelocity velocity(220 (220mms s ),),the thefullere fulle thro whiletraversing traversingthe theapparatus. apparatus.This Thisimposes imposes experiments in fundamental fundamental quantum quantum mechanics, mechanics,metrology metrologyand and shou a 100 while 6 report the the observation observation of of de de Broglie Brogliewave wave maximum lithography . Here we report maximumtilt tiltthat thatthe thegrating gratingmay mayhave havewith withcont re resp typicaldiffraction diffractionangle angleinto intothe thefirst-order first-orderinte m molecules by by diffraction diffraction at ataamaterial materialabsorpabsorp-50 typical interference of C60 molecules ma envi tion grating. This molecule molecule is is the the most most massive massive and and complex complex one onecan cantolerate tolerateaatilt tiltangle angleofof(at (atmost) most)about abo the moleculesstart startfalling fallingfrom fromone onediffraction diffractionorder ord object in which wave behaviour behaviour has has been been observed. observed.Of Ofparticular particular0 molecules gase is almost a classical body, because of its –100 –50 0 50 100 of a neighbouring order of a different ve is almost a classical body, because of its interest is the fact that C60 of a neighbouring order of a different 60 A Position (µm) experimentalcurves curvesstart starttotobecome becomeasymmet asymme many excited internal degrees degrees of of freedom freedom and and their their possible possible experimental exam couplings to the environment. essential for grating tilt more 500 f whic environment. Such Such couplings couplings are areFigure essential for pattern 2 Interference produced by Cdeviates molecules.by a, recording grating tilt deviates byExperimental morethan than 500!rad !rad 60 7,8 who circles) and fit usingvertical Kirchhoff diffraction theory (continuous line). The expected orientation. ,, suggesting interference vertical orientation. suggesting that that(open interference the appearance of decoherence decoherence7,8 stud zeroth and first-order maximaThe can beinterference clearly seen. Details of the theory discussed inclearly experiments with large detailed pattern ofofare Fig. 2a2aclearly e large molecules molecules should should facilitate facilitate detailed The interference pattern Fig. the text. b, The molecularmaximum beam profile without the the gratingfirst-order the path of thediffraction molecules. pep maximum and and the infirst-order diffractionand studies of this process. When considering de Broglie develop Broglie wave wave phenomena phenomena of of larger larger and and between betweenzeroth zerothand andfirst firstorders orders arewell well develo ©are 1999 Macmillan M NATURE | VOL 401 | 14 OCTOBER 1999 | www.nature.com more complex objects than destructiveinterference interferenceofofCC than atoms, atoms, fullerenes fullerenes come come to to mind mind asas destructive deBroglie Brogliewave wav 6060de 99 suitable candidates. After After their their discovery discovery and and the the subsequent subsequent neighbouring neighbouringslits slitsofofthe thegrating. grating.For Forcomparison compariso 10 invention of efficient mass-production mass-production methods methods10,, they they became became the the profile profile ofof the the undiffracted undiffractedcollimated collimatedbeb Counts in 1 s es Ltd Wave–particle duality Wave–particle duality of C60 molecules 60 #1 tures between 900 and v0 ¼ 166 vm ¼ 92 s #601 as expected via for heating a transition absorbed lightm sthenand ionized them C fullerenes and 13 passed through two betweenthermal a maxwellian effusive and12a. supersonic beam region . The subsequent emission of beam electrons The detection eparated by a distance most probable velocity was v ¼ 220 m s # 1 , corresponding to a de g nanofabricated SiNx Broglie wavelength of 2.5 pm. The full-width at half-maximum was de slits with a 100-nm as broad as 60%, resulting in a longitudinal coherence length of about 5 pm. 100 nm diffraction Scanning photodiffraction grating, the The essential features of the interference pattern can be undergrating ionization stage atially resolving detec- stood using standard Kirchhoff diffraction theory14 for a grating gon-ion laser (24 W all with a period of 100 nm, by taking into account both the finite Oven width (this is the size width of the collimation and the experimentally determined velocof that in the centre of ity distribution. The parameters in the fit were the width of the tically, parallel both to collimation, the gap width s0 of a single slit opening, the effective he collimation slits. By beam width of the detection laser and an overall scaling factor. This Ion could be scanned with model, assuming all grating slits to be perfectdetection and identical, 10 µm 10 µm erence pattern. The reproduces very well the central peak of the interference pattern unit enes via heating and shown in Collimation Fig. 2a, butslits does not fit the ‘wings’ of this pattern. Agreement with the experimental data, including the ‘wings’ in The detection region shown in Fig. Agreement w Figure 1 Diagram leave the oven thro (width " height " (width " height) se the second slit, and grating. The ions ar ejected electrons a laser focus can be Laser nning photozation stage Ion detection unit 680 © 1999 Macmillan Magazines Figure 1 Diagram of the experimental set-up (not to scale). Hot, neutral C60 molecules leave the oven through a nozzle of 0:33 mm " 1:3 mm " 0:25 mm (width " height " depth), pass through two collimating slits of 0:01 mm " 5 mm (width " height) separated by 1.04 m, traverse a SiNx grating (period 100 nm) 0.1 m after the second slit, and are detected via thermal ionization by a laser 1.25 m behind the grating. The ions are then accelerated and directed towards a conversion electrode. The ejected electrons are subsequently counted by a Channeltron electron multiplier. The laser focus can be reproducibly scanned transversely to the beam with 1-!m resolution. RIsultati esperimento • Il pattern di interferenza dovuto a singola molecola • Non esiste alcuna interferenza tra due molecole durante la loro evoluzione nell’apparato • Molecole possono emettere radiazione (eccitati termicamente) • Se lo sperimentatore individua la molecola emettente il fotone no figura di interferenza Perchè otteniamo sempre interferenza h • Lunghezza d’onda associata λ = p • Se lunghezza d’onda fotone più piccola del doppio della distanza tra due fessure possiamo localizzare la molecola e conoscerne il path • Lunghezza d’onda fotone 100 volte più grande • Stato fotone e stato molecola non entangled perchè gli stati fotone non sono ortogonali
