*Corresponding author. Tel.: #43-1-72701-229; fax: #43-1-72-89-220.
E-mail address: [email protected] (G. Badurek)1Presently at: Department of Physics, Texas A&M Univer-
sity, College Station, TX 77843-4242, USA.
Nuclear Instruments and Methods in Physics Research A 440 (2000) 562}567
Wave}particle duality and quantum erasure inpolarized}neutron interferometry
G. Badurek!,*, R.J. Buchelt!, B.-G. Englert",1 , H. Rauch#
!Institut fu( r Kernphysik, Technische Universita( t Wien, Stadionallee 2, A-1020 Wien, Austria"Max-Planck-Institut fu( r Quantenoptik, D-85748 Garching, Germany
#Atominstitut der O$ sterreichischen Universita( ten, A-1020 Wien, Austria
Abstract
An interference experiment with polarized neutrons is proposed to check the duality relation between the fringevisibility and the which-way information for massive fermions with macroscopic spatial separation of the interferingsub-beams. The connection between polarization-sensitive post-selection of sub-ensembles and the concept of quantumerasure is discussed. This neutron experiment will supplement those recently performed with photons andatoms. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Wave}particle duality; Quantum erasure; Neutron interferometry
1. Introduction
Wave}particle duality (WPD) dates back to thevery early days of quantum mechanics, to Einstein'sseminal paper on the photoelectric e!ect [1], and isa striking manifestation of Bohr's principle ofcomplementarity [2,3]. The familiar phrase `eachexperiment must be described either in terms ofparticles or in terms of wavesa emphasizes the ex-treme cases and disregards the intermediate situ-ations, in which particle aspects and wave aspectsare present simultaneously. Theoretical investiga-
tions [4,11], supplemented by a few experimentalstudies [12}14], have led to a quantitative formula-tion of WPD, namely the duality relation of Eq. (3)below. Very recently, the validity of the dualityrelation was tested with a Mach}Zehnder-typephoton interferometer [15] and an atom inter-ferometer [16,17] that mimics some of the essentialfeatures of a Young double-slit setup. The experi-ment we are proposing here will test the dualityrelation in the realm of neutron interferometry.
Since its invention in 1974 [18] perfect crystalneutron interferometry has become a powerful andin many respects unique tool to test and to demon-strate fundamental principles of quantum mechan-ics with massive particles on a macroscopicspace}time scale. This is particularly true if thespin-1
2property of the neutron is explicitly taken
into account as it will be the case in our proposal.The list of successfully performed experimentsbelonging to the spinor character of the neutron
0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 0 3 7 - 2
Fig. 1. Schematic sketch of the proposed neutron interferometerwith which-way marking (see text).
comprises topics like the "rst explicit veri"ca-tion of the 4p-periodicity of spinor wave func-tions [19}21], demonstration of the quantummechanical principle of linear superposition ofstates [22,23], macroscopic quantum beating ofthe neutron wave function [24], multiple photonexchange between the neutron and an appliedelectromagnetic "eld [25], neutron inter-ferometric separation of geometric and dynam-ical phases [26], as well as interferometric [27]and polarimetric [28] veri"cation of the topologi-cal character of the scalar Aharonov}Bohm ef-fect. A general state-of-the-art survey aboutneutron interferometry is found in Refs. [29,30].Unlike any other matter wave interference tech-nique developed thus far with either electrons,atoms or ions, perfect-crystal neutron inter-ferometry allows a coherent separation of quantumstates of massive particles over macroscopicdistances of several centimeters followed by sub-sequent coherent recombination. It is thereforeideally suited to test the validity of the dualityrelation.
For a quanti"cation of WPD we need quantitat-ive, measurable characteristics for the wave aspectsand the particle aspects. The former are naturallyquanti"ed by the visibility V of the observed inter-ference fringes. How about the latter? Another fam-iliar phrase, `a wave is in both arms of aninterferometer simultaneously, a particle is in onearm or the othera, gives a hint. The better we cantell which way has been taken by a particular neu-tron, the more pronounced are its particle aspects.Their quanti"cation is, accordingly, based on thelikelihood L of guessing the way right. Since a ran-dom guess gives a likelihood of L"1
2, whereas
L"1 indicates that we know the way with cer-tainty, the actual which-way (WW) knowledge K isgiven by
L"12#1
2K with 04K41. (1)
The value of K depends on the `betting strategyawe are employing; the optimal strategy maximizesK and this identi"es the distinguishability D of theways,
D"maxMKN. (2)
In an asymmetric interferometer, one way ismore likely than the other to begin with, sothat we would know something about the waybeforehand; WW information of this kind is "tting-ly called predictability (denoted by P), and the in-equalities P4K4D state an obvious hierarchybecause WW detection cannot decrease the bettingodds.
The central quantitative statement about WWDis the duality relation
D2#V241 (3)
that has been found recently [10,11]; there areexamples, in which the equality sign holds. Theimplied statement P2#V241 has been knownfor some time, implicitly or explicitly, in a variety ofphysical contexts [4}9,12}14]. All experimentaltests refer necessarily to the version
K2#V241 (4)
with the knowledge K optimized under the con-straints of the setup, hoping for K
.!9+D. In the
experiment we are proposing, we will usually haveP"0 and it should be possible to extract (almost)all the WW information that becomes potentiallyavailable, so that the acquired WW knowledgeK is expected to come very close to its theoreticalupper bound D.
G. Badurek et al. / Nuclear Instruments and Methods in Physics Research A 440 (2000) 562}567 563
SECTION 3.
Table 1
D`
D~
M1-way 1#e ) s
11!e ) s
1M
2-way 1#e ) s
21!e ) s
2
2. Scheme of the experiment
Fig. 1 shows a schematic sketch of the proposedexperiment. It is the neutron analog of the photonexperiment of Ref. [15]. A standard Mach}Zehnder-type neutron interferometer is supplementedby devices for (i) encoding WW information in theneutron's polarization and (ii) extracting WWknowledge later on. For the sake of brevity here weare concentrating on the general concepts and noton the actual technical realization.
In the setup of Fig. 1, polarized neutrons enterfrom the left at the beam splitter BS. After beingre#ected at one of the mirrors (M
1or M
2) the
amplitudes are recombined at the beam mergerBM. The neutrons exiting at the symmetric outputport are eventually counted by the detectorsD
`and D
~. Those emerging at the asymmetric
output are not used for the measurement itself, onlyfor collecting control data.
2.1. Determining the fringe visibility V
The phase shifter PS in the lower way(BS}M
2}BM) enables us to change the relative
phase / of the two partial amplitudes, and so wecan scan through the interference pattern. The totalcount rate of D
`and D
~will be proportional to
1#V cos/, where V is the fringe visibility. Sincewe are observing in the symmetric output, themaximal value V"1 is achievable if the partiallytransparent mirrors that are used for the BS andthe BM have identical properties.
2.2. Labeling the neutrons and acquiring which-wayknowledge K
In the upper way (BS}M1}BM) a spin #ipper SF
can change the initial polarization to another one,so that the neutrons acquire a polarization labelthat marks their way through the interferometer.This information is read out with the aid of thepolarizing beam splitter PBS that sorts the neu-trons, so that D
`counts the neutrons polarized in
the #e direction and D~
those polarized in the(orthogonal) !e direction. Here, the unit vectore characterizes the PBS.
The Pauli vector s0"SrT
0, which is the expec-
tation value of the spin vector operator r in theinitial state, speci"es the spin state of the neutronsentering the interferometer at the BS. Neutronscoming along the M
1-way pass through the (uni-
tary) SF which rotates their spin vector around theaxis n and so turns s
0into s
1; the neutrons of the
M2-way have s
2"s
0. If no depolarization occurs,
the lengths of s1
and s2
remain the same (identicalto the length of s
0) and they di!er only in their
direction.In general, the count rates of D
`and D
~will be
di!erent for the two ways. Their relative sizes aresummarized in Table 1.
The experimental values corresponding toTable 1 entries are obtained by measuring thecount rates in each row with the other wayblocked.
The betting odds for guessing the way right aremost favorable when we always bet on the way thatcontributes most to the probability of triggering thedetector that actually "red. This results in a mea-sured WW knowledge given by [10,11]
K"
Dr1`
!r2`
D#Dr1~
!r2~
Dr1`
#r2`
#r1~
#r2~
"
1
2De ) (s
1!s
2)D
(5)
where, for example, r1`
is the count rate of D`
forM
1-way neutrons. Clearly, theory predicts that
e should be chosen parallel to s1!s
2for maximal
WW knowledge, and that K"1 is only possible if(i) s
1"!s
2and (ii) both are unit vectors. In
operational term: The neutrons must be fully polar-ized in both ways and these polarization statesmust be orthogonal.
For the equal sign to hold in Eq. (3), condition (ii)su$ces. It is then possible to study the trade-o!between fringe visibility and WW knowledge underideal circumstances.
564 G. Badurek et al. / Nuclear Instruments and Methods in Physics Research A 440 (2000) 562}567
Fig. 2. Angular splitting of the two spin states upon transmis-sion through one (v) and two (m) magnetic "eld prisms. Thecentral curve (r) displays the resolution curve of the double-crystal small-angle instrument.
2.3. Post-selection, quantum erasure
The polarization-sensitive detection of the neu-trons sorts them into two sub-ensembles. Depend-ing on the choice for the characterizing unit vectore, di!erent sortings are possible. For example, theWW sorting just described is given by eE(s
1!s
2).
Inasmuch as the choice of sorting is undecided untilthe neutrons reach the PBS in the setup of Fig. 1,that is, until they have left the interferometerproper, the sub-ensembles are post-selected.
A sorting of a particularly interesting kind hasfeatures in common with quantum erasure (QE)[31,32]. It identi"es sub-ensembles that exhibit per-fectly visible fringes, even when the fringe visibilityV of the unsorted totality of counted neutrons isreduced, possibly to V"0. This is achieved bysetting e equal to the unit vector n that speci"es thedirection around which the SF rotates the neu-tron's spin vector.
It is clear that intermediate situations are realiz-ed by choosing e di!erent from both the WWchoice and the QE choice. This amounts to anexperimental realization of partial QE.
It is remarkable that the initial polarizations0
is irrelevant for QE. Irrespective of the lengthand the direction of s
0the sorting characterized by
e"n always identi"es two sub-ensembles thathave fringes with unit visibility. In the extreme caseof totally unpolarized neutrons (that is s
0"0) the
SF does not produce a WW label, but will reducethe fringe visibility to zero if the rotation angle is1803. The fringes are then lost, but no WW in-formation is available [D"V"0 in Eq. (3)].Nevertheless, one can do the QE sorting and re-cover the fringes. An experimental demonstrationof the latter property of QE would con"rm that theavailability of WW information is not a precondi-tion for successful QE.
3. Experimental essentials
There is no room to present the details of theexperimental setup that will be required to realizeour proposal. Hence, we completely omit a descrip-tion of the interferometer itself and focus instead onthe methods to prepare the input and to analyze the
"nal neutron state, since they are of particularimportance for such kind of experiment.
3.1. Input state preparation
The extremely narrow angular width of perfect-crystal Bragg or Laue re#ections enables us toexploit the spin-dependent birefringence of neu-trons upon passing through prismatically shapedmagnetic "elds. The angular separation of the twospin states of an initially unpolarized beam of neu-trons with energy E which passes at an asymmetryangle e (see Fig. 2) through a magnetic "eld prism ofapex angle U is given by [33]
d"2kB
E
sin(U)
cos(U)#cos(2e). (6)
Recently, we could develop for the "rst timea polarizer of this kind consisting of a permanentmagnetic yoke with prismatically shaped air gap of1 cm height and apex angle U"1163. Withoutproducing any heat a "eld of about 0.9 T isachieved. Fig. 2 shows the angular splitting mea-sured with a high-resolution double-crystal small-angle camera for symmetric passage through oneand two of these "eld prisms. Owing to the in#u-ence of stray "elds the experimentally observed
G. Badurek et al. / Nuclear Instruments and Methods in Physics Research A 440 (2000) 562}567 565
SECTION 3.
Fig. 3. (a) Ideal polarizing beam splitter (PBS) with 100% re#ec-tivity of the spin-up state. (b) Approximate realization of a PBSby placing a high performance spin #ipper between two polariz-ing mirrors with reduced re#ectivity
splitting is about 20% smaller than theoreticallyexpected. Nevertheless, it is su$ciently large com-pared to the re#ection width of perfect Si crystals toguarantee fully polarized beams inside the inter-ferometer.
3.2. Spin state analysis
WW detection crucially depends on the qualityof the polarization sensitive detection. In the idealcase a polarizing beam splitter sorts each neutronaccording to its spin state either in detector D
`or
D~
(Fig. 3). Evidently, the re#ectivity of such anideal beam splitter must reach a value very close to100% in order to avoid a contamination of thetransmitted beam with the `wronga spin state. Atleast for cold neutrons the performance of super-mirror polarizers is quite close to this ideal situ-ation. But even in the case of reduced re#ectivitya combination of a pair of such imperfect polarizersshould allow for an almost perfect spin-dependentneutron sorting, provided a spin-#ip device with ane$ciency close to 100% is placed between them, asindicated in Fig. 3. There is no particular di$cultyin fabricating neutron spin #ippers of the requiredquality, but it is essential that the two polarizersneither attenuate nor depolarize the transmittedbeam. Then a low re#ectivity only causes a poorneutron economy, but it has no in#uence on thesigni"cance of the spin state sorting.
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SECTION 3.
