interferenza in meccanica quantistica -...

interferenza in meccanica

quantistica

fisica interdisciplinaregiuseppe 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

• Mitragliatrice

• Parete con due fessure

• Rilevatore di proiettili

Apparato Sperimentale


• Chiudiamo la fessura 2

• Otteniamo P1



• Otteniamo la curva P2


• Apriamo le due fessure

• Otteniamo la curva

• Verifichiamo che

• I proiettili arrivano sempre in blocchi

• “No Interferenza”

P12

P12 = P1 + P2

Esperimento con onde d’acqua

• Sorgente d’onde circolari

• Parete con due fessure

• Rilevatore di intensità

Apparato sperimentale



• Onda primaria diffratta

• Intensità onda fessura 1 I1


• Chiudiamo fessura 2

• Onda primaria diffratta

• Intensità onda fessura 2 I2

Esperimento con onde d’acqua: Risultati

• Misuriamo Intensità

• No granularità

• Le due onde interferiscono

I12

I12 != I1 + I2

Interferenza costruttiva e distruttiva

• Interferenza distruttiva

• ddc=

• Interferenza costruttiva

• ddc=

(2n + 1)!/2

n!

Matematica della interferenza

• Altezza istantanea dell’onda al detector proveniente dalla fessura 1 (2)

• Altezza istantanea quando le due fessure sono aperte

• Intensità dell’onda

•

h1ei!t

!h2e

i!t"

h1ei!t + h2e

i!t

I12 = | (h1 + h2) |2 = |h1|2 + |h2|2 + 2|h1||h1|cos!

I12 = I1 + I2 + 2!

I1I2cos!

Esperimento con Elettroni

• Elettroni emessi da valvola termoionica

• Accelerati verso la parete

• Parete di metallo con due fessure

• Rilevatore (electron multiplier)

Apparato Sperimentale

Esperimento con elettroni: Risultati

• Gli elettroni arrivano sempre in granuli identici tra loro

•

P12

Gli elettroni interferiscono!

P12 != P1 + P2



•

P1 P12


P12 != P1 + P2



•

P1 P2 P12


P12 != P1 + P2

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

• Aumentando la lunghezza d’onda della luce non sappiamo più da dove passano gli elettroni e riotteniamo la figura di interferenza

P12 = P1 + P2

P12 != P1 + P2

Riassunto dei nostri esperimenti

• La probabilità di un evento è data dal quadrato di una quantità complessa chiamata ampiezza di probabilità

• Se un evento può accadere in diversi modi alternativi la ampiezza di probabilità per l’evento è

• Se è possibile determinare a priori se una o l’altra alternativa è scelta realmente

P12 = P1 + P2

!

P = |!|2

P = |!1 + !2|2

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”

• 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!

|!! =1"2

(|e, 1!|f, 1!+ |e, 2!|f, 2!)

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"2

(|a!1|b!2 + |a!!1|b!!2)


• 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) e perchè entangled con e

|a!!1|a!1|b!2 |b!!2


• 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"2

(|a!1|a!!2 + |b!1|b!!2)


• Se la posizione della particella che decade è incerta molto più di 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 , 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

d >> !/" !k/k << "

Distinguibilità dei Paths (Zou et al. 1991)

effect, enhance or sup-press the atomic emissionprocess in the crystal bysmall movements of mir-rors that are several feetaway!

The quantum eraserFigure 5 shows an experi-mental arrangement firstused by Alley and Shih,6

and recently exploited byPaul Kwiat and coworkersat Berkeley17 to demon-strate Marian Scully's no-tion of a "quantum eraser."Others have called it"haunted" or "phantom"measurement.18 To ap-preciate the basic idea,first suppose that only thebeam splitter is in the pathof the two beams emergingfrom the down-convertercrystal. This arrangementproduces an interestingand very basic two-particleeffect: Both particles must end up in the same detector.19

The reason is simple. To get coincident counts at thetwo detectors, either particle 1 (the photon that ultimatelylands in detector 1) takes route a and particle 2 takes routeb, or vice versa. In the former alternative, both photonsare reflected by the beam splitter, each reflection contrib-uting a 90° phase shift to the overall amplitude. The latteralternative, by contrast, involves no phase-shifting reflec-tions. Thus the two alternative amplitudes are 180° out ofphase with each other, and they cancel.

So if both beams are, for example, horizontally po-larized, they will remain so, and there will be no coinci-dent counts in detectors 1 and 2. But if we now inserta 90° polarization rotator into beam b, it will becomevertically polarized and the two amplitudes will no longerinterfere because one can tell which path a photon tookto its detector by measuring its polarization. Thus there

Manipulating one photon can

alter the interference patternof another. The arrangement16

sketched in a can produce aninterference pattern atdetector D, when the phaseshifter P is varied. Anentering ultraviolet photon ais split at beam splitter A sothat both down-conversioncrystals (X, and X,) areilluminated. One of theresulting pair of down-conversion photons can reachD, by way of beam path d orh. If one could monitorbeams e and k separately, onewould know in which crystalthe down-conversionoccurred, and there would beno interference. But mergingbeams e and k in thisconfiguration lets thealternative paths of the otherphoton interfere. A new variantof this scheme, shown in b,uses only one crystal (at thecenter). The ambiguity hereis created by reflecting theoriginating beam (blue, fromthe top of the photo) and itsdown-converted progeny (redand green) back through thecrystal from mirrors (at thebottom of the photo), so onecan't know on which pass thedown-conversionoccurred. Figure 4

will now be no superpositionof the two amplitudes, andtherefore coincidence countsbetween the two detectorswill be observed.

But it is still possible to"erase" this path informationand recover the interferenceby placing a linear polarizerin each beam, as shown inthe figure. If both polarizersare either horizontal or ver-tical there will be path infor-

mation present. But if they are oriented at 45° to thehorizontal, either route, a or b, can now lead to eitherdetector. The coherence is restored and there are, onceagain, no coincident counts.

The idea here is that one can seem to destroy infor-mation about a system without actually doing so. Theinformation remains subliminally present; it can be re-captured. Or, as Chiao has put it, "Nothing has reallybeen erased here, only scrambled!" Helmut Rauch'sVienna group performed a conceptually identical experi-ment in 1982 with a neutron interferometer, but thatinvolved only single-particle interference.17

Generalization to many statesExperiments with entangled particles have generallybeen confined to two-state systems, either spin-V2 parti-cles or photons. One way to generate superpositions

PHYSICS TODAY AUGUST 1993 2 7

1/!

2 (|d"1|e"2 + |h"1|k"2)

1/!

2 (|d"+ |h")1 |k"2

Esperimento di Zou

• Il secondo fotone non è necessario che sia rilevato

• Se il fascio e è bloccato le frange in scompariranno perchè adesso è possibile (potenzialmente) rilevando un fotone in k ci potrebbe dire che è stato creato in

• i fasci e e k non sono nei paths alternativi che portano A a D

D1

X2

Herzog et al. (1993)

effect, enhance or sup-press the atomic emissionprocess in the crystal bysmall movements of mir-rors that are several feetaway!

The quantum eraserFigure 5 shows an experi-mental arrangement firstused by Alley and Shih,6

and recently exploited byPaul Kwiat and coworkersat Berkeley17 to demon-strate Marian Scully's no-tion of a "quantum eraser."Others have called it"haunted" or "phantom"measurement.18 To ap-preciate the basic idea,first suppose that only thebeam splitter is in the pathof the two beams emergingfrom the down-convertercrystal. This arrangementproduces an interestingand very basic two-particleeffect: Both particles must end up in the same detector.19

The reason is simple. To get coincident counts at thetwo detectors, either particle 1 (the photon that ultimatelylands in detector 1) takes route a and particle 2 takes routeb, or vice versa. In the former alternative, both photonsare reflected by the beam splitter, each reflection contrib-uting a 90° phase shift to the overall amplitude. The latteralternative, by contrast, involves no phase-shifting reflec-tions. Thus the two alternative amplitudes are 180° out ofphase with each other, and they cancel.

So if both beams are, for example, horizontally po-larized, they will remain so, and there will be no coinci-dent counts in detectors 1 and 2. But if we now inserta 90° polarization rotator into beam b, it will becomevertically polarized and the two amplitudes will no longerinterfere because one can tell which path a photon tookto its detector by measuring its polarization. Thus there

Manipulating one photon can

alter the interference patternof another. The arrangement16

sketched in a can produce aninterference pattern atdetector D, when the phaseshifter P is varied. Anentering ultraviolet photon ais split at beam splitter A sothat both down-conversioncrystals (X, and X,) areilluminated. One of theresulting pair of down-conversion photons can reachD, by way of beam path d orh. If one could monitorbeams e and k separately, onewould know in which crystalthe down-conversionoccurred, and there would beno interference. But mergingbeams e and k in thisconfiguration lets thealternative paths of the otherphoton interfere. A new variantof this scheme, shown in b,uses only one crystal (at thecenter). The ambiguity hereis created by reflecting theoriginating beam (blue, fromthe top of the photo) and itsdown-converted progeny (redand green) back through thecrystal from mirrors (at thebottom of the photo), so onecan't know on which pass thedown-conversionoccurred. Figure 4

will now be no superpositionof the two amplitudes, andtherefore coincidence countsbetween the two detectorswill be observed.

But it is still possible to"erase" this path informationand recover the interferenceby placing a linear polarizerin each beam, as shown inthe figure. If both polarizersare either horizontal or ver-tical there will be path infor-

mation present. But if they are oriented at 45° to thehorizontal, either route, a or b, can now lead to eitherdetector. The coherence is restored and there are, onceagain, no coincident counts.

The idea here is that one can seem to destroy infor-mation about a system without actually doing so. Theinformation remains subliminally present; it can be re-captured. Or, as Chiao has put it, "Nothing has reallybeen erased here, only scrambled!" Helmut Rauch'sVienna group performed a conceptually identical experi-ment in 1982 with a neutron interferometer, but thatinvolved only single-particle interference.17

Generalization to many statesExperiments with entangled particles have generallybeen confined to two-state systems, either spin-V2 parti-cles or photons. One way to generate superpositions

PHYSICS TODAY AUGUST 1993 2 7

Quantum Eraser

three-particle decay at the center.(Alternatively one could start with athree-particle down-conversion.) As-sume for simplicity that the three de-cay particles all have the same massand the same energy. Then they will,of course, come off 120° apart in thedecay plane. A suitable placement ofscreens with holes restricts them totwo possible states: |a6c> or \a'b'c'y.The coherent superposition will be

\a'b'c'>)

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

information and recovers the coherence that prevents coincidence

counts. Figure 5

involving more than two states is to use systems of higherspin. An easier alternative is to generalize the beamsplitter to provide more than two paths for each photon.We call such systems "multiports."

20 The half-silvered

mirror that serves as a beam splitter is a four-port; ithas two input ports and two output ports. The outputbeams in such devices are mathematically related to theinput beams by a unitary transformation.

A simple generalization of the beam splitter, shownin figure 6, is the six-port, with three input beams andthree output beams. We call it a "tritter." (An eight-port would be called a "quitter.") If the beam splittersA, B and C in the figure are chosen to have reflectivities1/V2~, 1/VjT, 1/V2~, respectively, and the phase shiftersa, ji and y axe chosen appropriately, this device willyield three equally intense output beams if any one ofthe three input ports is illuminated. That's a straight-forward generalization of the symmetric beam splitter.But by varying all these parameters one can induce arange of unitary transformations. Multiports provide apractical method for investigating such transformationsin an A^-dimensional Hilbert space.

Three-particle interferometryFigure 7 shows an idealized three-particle interferome-ter.

21 No such device has been been built, but a number

of models have been proposed.22

This particular configu-ration was inspired by David Mermin's Reference Framecolumn in PHYSICS TODAY, June 1990, page 9. Imagine a

The beams a', b' and c' pass throughthe phase shifters with phase shifts a,(3 and y, after which the appropriatebeams are recombined at beam split-ters A, B and C. Beyond the beamsplitters are the three pairs of count-ers. One records only triplets of si-multaneous counts at G or G', H or H'and K or K'. If one assigns a +1 (or-1) to a count in an unprimed (orprimed) counter, then the probabilitythat a triplet of counts will give theproduct +1 (for example, GHK orGH'K) will be (l + sinA)/2, whereA = a + fi + y. The probability for a

product -1 (for example, G'H'K or G'HK) will be(1 - sinA)/2. Then the expectation value over a large num-ber of counts is simply sinA.

This result is remarkably similar to the case of thetwo-particle interferometer, but the implications are en-tirely different. In the three-particle case a perfect cor-relation occurs when A = 77/2 or 3T7/2. Then if one knowsat which counter two of the particles have landed, onecan predict with certainty the counter at which the thirdparticle will land, without having disturbed it at all.Hence there exists what Einstein, Podolsky and Rosencalled an "element of reality" associated with the pathfrom the beam splitter to the counter. Elements of realityare those entities to which, from the Einstein-Podolsky-Rosen point of view, the concept of an objective realitymost clearly applies. They are natural entities for dis-cussions of local, realistic descriptions of quantum events.

In the two-particle interferometer, measurements in-volving only perfect correlations are uninteresting, in thesense that they cannot violate Bell's inequality. (Thisfamous inequality is a statement of the limits of corre-lation allowable between separated events in any theorythat preserves local reality.) But from the Einstein-Podolsky-Rosen viewpoint these perfect correlations areessential for the introduction of elements of reality.

In the three-particle interferometer, however, if oneassumes the existence of these elements of reality, onealready runs into a contradiction even if the correlationsare perfect.

21 If the particles are perfectly correlated in

the two-particle case, their spin directions are preciselyopposite. But perfect correlation in the three-particlecase yields a continuum of possibilities, a condition thatis impossible to satisfy within the classical restrictions.

A six-port, or 'tritter,' is made with three beam splitters. If

the reflectivities of the beam splitters A, B and C and the

phase angles a, /3 and y of the phase shifters (black) are

properly chosen, this configuration yields three output

beams of equal intensity when any one of the three input

ports is illuminated. Figure 6

28 PHYSICS TODAY AUGUST 1993

© 1999 Macmillan Magazines Ltd

Fig. 2a, can be achieved by allowing for a gaussian variation of theslit widths over the grating, with a mean open gap width centred ats0 ! 38 nm with a full-width at half-maximum of 18 nm. That best-fit value for the most probable open gap width s0 is significantlysmaller than the 55 ! 5 nm specified by the manufacturer (T. A.Savas and H. Smith, personal communication). This trend isconsistent with results obtained in the diffraction of noble gasesand He clusters, where the apparently narrower slit was interpretedas being due to the influence of the van der Waals interaction withthe SiNx grating during the passage of the molecules15. This effect isexpected to be even more pronounced for C60 molecules owing totheir larger polarizability. The width of the distribution seems alsojustified in the light of previous experiments with similar gratings:both the manufacturing process and adsorbents could accountfor this fact (ref. 16, and T. A. Savras and H. Smith, personalcommunication). Recently, we also observed interference of C70

molecules.Observation of quantum interference with fullerenes is interest-

ing for various reasons. First, the agreement between our measuredand calculated interference contrast suggests that not only thehighly symmetric, isotopically pure 12C60 molecules contribute tothe interference pattern but also the less symmetric isotopomericvariants 12C59

13C and 12C5813C2 which occur with a total natural

abundance of about 50%. If only the isotopically pure 12C60

molecules contributed to the interference, we would observe amuch larger background.

Second, we emphasize that for calculating the de Broglie wave-length, l ! h=Mv, we have to use the complete mass M of the object.Thus, each C60 molecule acts as a whole undivided particle during itscentre-of-mass propagation.

Last, the rather high temperature of the C60 molecules impliesbroad distributions, both of their kinetic energy and of their internalenergies. Our good quantitative agreement between experiment andtheory indicates that the latter do not influence the observedcoherence. All these observations support the view that each C60

molecule interferes with itself only.

In quantum interference experiments, coherent superpositiononly arises if no information whatsoever can be obtained, even inprinciple, about which path the interfering particle took. Interac-tion with the environment could therefore lead to decoherence. Wenow analyse why decoherence has not occurred in our experimentand how modifications of our experiment could allow studies ofdecoherence using the rich internal structure of fullerenes.

In an experiment of the kind reported here, ‘which-path’ infor-mation could be given by the molecules in scattering or emissionprocesses, resulting in entanglement with the environment and aloss of interference. Among all possible processes, the following arethe most relevant: decay of vibrational excitations via emission ofinfrared radiation, emission or absorption of thermal blackbodyradiation over a continuous spectrum, Rayleigh scattering, andcollisions.

When considering these effects, one should keep in mind thatonly those scattering processes which allow us to determine the pathof a C60 molecule will completely destroy in a single event theinterference between paths through neighbouring slits. Thisrequires l p d; that is, the wavelength l of the incident or emittedradiation has to be smaller than the distance d between neighbour-ing slits, which amounts to 100 nm in our experiment. When thiscondition is not fulfilled decoherence is however also possible viamulti-photon scattering7,8,17.

At T ! 900 K, as in our experiment, each C60 molecule has onaverage a total vibrational energy of Ev ! 7 eV (ref. 18) stored in 174vibrational modes, four of which may emit infrared radiation atlvib ! 7–19 "m (ref. 10) each with an Einstein coefficient ofAk ! 100 s # 1 (ref. 18). During its time of flight from the gratingtowards the detector (t ! 6 ms) a C60 molecule may thus emit onaverage 2–3 such photons.

In addition, hot C60 has been observed19 to emit continuousblackbody radiation, in agreement with Planck’s law, with a mea-sured integrated emissivity of e ! 4:5 " ! 2:0# $ 10 # 5 (ref. 18). Fora typical value of T ! 900 K, the average energy emitted during thetime of flight can then be estimated as only Ebb ! 0:1 eV. Thiscorresponds to the emission of (for example) a single photon atl ! 10 "m. Absorption of blackbody radiation has an even smallerinfluence as the environment is at a lower temperature than themolecule. Finally, since the mean free path for neutral C60 exceeds100 m in our experiment, collisions with background molecules canbe neglected.

As shown above, the wavelengths involved are too large for singlephoton decoherence. Also, the scattering rates are far too small toinduce sufficient phase diffusion. This explains the decoupling ofinternal and external degrees of freedom, and the persistence ofinterference in our present experiment.

A variety of unusual decoherence experiments would be possiblein a future extension of the experiment, using a large-area inter-ferometer. A three-grating Mach–Zehnder interferometer6 seems tobe a particularly favourable choice, since for a grating separation ofup to 1 m we will have a molecular beam separation of up to 30 "m,much larger than the wavelength of a typical thermal photon. In thiscase, the environment obtains ‘which-path’ information eventhrough a single thermal photon, and the interference contrastshould thus be completely destroyed. The parameters that could becontrolled continuously in such an experiment would then be theinternal temperature of the fullerenes, the temperature of theenvironment, the intensity and frequency of external laser radiation,the interferometer size, and the background pressure of variousgases.

An improved interferometer could have other applications. Forexample, in contrast to previous atom-optical experiments20–22

which were limited to the interaction with only a few lines in thewhole spectrum, interferometry with fullerenes would enable us tostudy these naturally occurring and ubiquitous thermal processesand wavelength-dependent decoherence mechanisms for (we

letters to nature

NATURE | VOL 401 | 14 OCTOBER 1999 | www.nature.com 681

200

400

600

800

1,000

1,200

b

a

–100 –50 0 50 1000

50

100

150

200

Cou

nts

in 1

sC

ount

s in

50

sPosition (µm)

Figure 2 Interference pattern produced by C60 molecules. a, Experimental recording(open circles) and fit using Kirchhoff diffraction theory (continuous line). The expectedzeroth and first-order maxima can be clearly seen. Details of the theory are discussed inthe text. b, The molecular beam profile without the grating in the path of the molecules.


.................................................................Wave–particle dualityof C60 moleculesMarkus Arndt, Olaf Nairz, Julian Vos-Andreae, Claudia Keller,Gerbrand van der Zouw & Anton Zeilinger

Institut fur Experimentalphysik, Universitat Wien, Boltzmanngasse 5,A-1090 Wien, Austria

.................................. ......................... ......................... ......................... ......................... ........

Quantum superposition lies at the heart of quantum mechanicsand gives rise to many of its paradoxes. Superposition of deBroglie matter waves1 has been observed for massive particlessuch as electrons2, atoms and dimers3, small van der Waalsclusters4, and neutrons5. But matter wave interferometry withlarger objects has remained experimentally challenging, despitethe development of powerful atom interferometric techniques forexperiments in fundamental quantum mechanics, metrology andlithography6. Here we report the observation of de Broglie waveinterference of C60 molecules by diffraction at a material absorp-tion grating. This molecule is the most massive and complexobject in which wave behaviour has been observed. Of particularinterest is the fact that C60 is almost a classical body, because of itsmany excited internal degrees of freedom and their possiblecouplings to the environment. Such couplings are essential forthe appearance of decoherence7,8, suggesting that interferenceexperiments with large molecules should facilitate detailedstudies of this process.

When considering de Broglie wave phenomena of larger andmore complex objects than atoms, fullerenes come to mind assuitable candidates. After their discovery9 and the subsequentinvention of efficient mass-production methods10, they becameeasily available. In our experiment (see Fig. 1) we use commercial,99.5% pure, C60 fullerenes (Dynamic Enterprises Ltd, Twyford, UK)which were sublimated in an oven at temperatures between 900 and1,000 K. The emerging molecular beam passed through twocollimation slits, each about 10 !m wide, separated by a distanceof 1.04 m. Then it traversed a free-standing nanofabricated SiNx

grating11 consisting of nominally 50-nm-wide slits with a 100-nmperiod.

At a further distance of 1.25 m behind the diffraction grating, theinterference pattern was observed using a spatially resolving detec-tor. It consisted of a beam from a visible argon-ion laser (24 W alllines), focused to a gaussian waist of 8 !m width (this is the sizerequired for the light intensity to drop to 1/e2 of that in the centre ofthe beam). The light beam was directed vertically, parallel both tothe lines of the diffraction grating and to the collimation slits. Byusing a suitable mirror assembly, the focus could be scanned withmicrometre resolution across the interference pattern. Theabsorbed light then ionized the C60 fullerenes via heating andsubsequent thermal emission of electrons12. The detection region

was found to be smaller than 1 mm in height, consistent with a fullRayleigh length of 800 !m and the strong power dependence of thisionization process. A significant advantage of the thermionicmechanism is that it does not detect any of the residual gasespresent in the vacuum chamber. We could thus achieve dark countrates of less than one per second even under moderately highvacuum conditions (5 " 10 # 7 mbar). The fullerene ions werethen focused by an optimized ion lens system, and accelerated toa BeCu conversion electrode at !9 kV where they induced theemission of electrons which were subsequently amplified by aChanneltron detector.

Alignment is a crucial part of this experiment. In order to be ableto find the beam in the first place, our collimation apertures aremovable piezo slits that can be opened from 0 to 60 !m (in the caseof the first slit) and from 0 to 200 !m (for the second slit). Thevacuum chamber is rigidly mounted on an optical table togetherwith the ionizing laser, in order to minimize spatial drifts.

The effect of gravity also had to be considered in our set-up. Forthe most probable velocity (220 m s!1), the fullerenes fall by 0.7 mmwhile traversing the apparatus. This imposes a constraint on themaximum tilt that the grating may have with respect to gravity. As atypical diffraction angle into the first-order maximum is 25 !rad,one can tolerate a tilt angle of (at most) about one mrad beforemolecules start falling from one diffraction order into the trajectoryof a neighbouring order of a different velocity class. Theexperimental curves start to become asymmetric as soon as thegrating tilt deviates by more than 500 !rad from its optimumvertical orientation.

The interference pattern of Fig. 2a clearly exhibits the centralmaximum and the first-order diffraction peaks. The minimabetween zeroth and first orders are well developed, and are due todestructive interference of C60 de Broglie waves passing throughneighbouring slits of the grating. For comparison, we show in Fig. 2bthe profile of the undiffracted collimated beam. The velocitydistribution has been measured independently by a time-of-flightmethod; it can be well fitted by f !v" # v3exp! # !v # v0"2=v2

m", withv0 # 166 m s # 1 and vm # 92 m s # 1 as expected for a transitionbetween a maxwellian effusive beam and a supersonic beam13. Themost probable velocity was v # 220 m s # 1, corresponding to a deBroglie wavelength of 2.5 pm. The full-width at half-maximum wasas broad as 60%, resulting in a longitudinal coherence length ofabout 5 pm.

The essential features of the interference pattern can be under-stood using standard Kirchhoff diffraction theory14 for a gratingwith a period of 100 nm, by taking into account both the finitewidth of the collimation and the experimentally determined veloc-ity distribution. The parameters in the fit were the width of thecollimation, the gap width s0 of a single slit opening, the effectivebeam width of the detection laser and an overall scaling factor. Thismodel, assuming all grating slits to be perfect and identical,reproduces very well the central peak of the interference patternshown in Fig. 2a, but does not fit the ‘wings’ of this pattern.

Agreement with the experimental data, including the ‘wings’ in

letters to nature

680 NATURE | VOL 401 | 14 OCTOBER 1999 | www.nature.com

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Scanning photo-ionization stage

Figure 1 Diagram of the experimental set-up (not to scale). Hot, neutral C60 moleculesleave 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 afterthe second slit, and are detected via thermal ionization by a laser 1.25 m behind thegrating. The ions are then accelerated and directed towards a conversion electrode. Theejected electrons are subsequently counted by a Channeltron electron multiplier. Thelaser focus can be reproducibly scanned transversely to the beam with 1-!m resolution.




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letters to nature


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letters to nature


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letters to nature


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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

• Lunghezza d’onda associata

• 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

! =h

p

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