March, N H

CERN Document Server

Available at: http://www.ictp.trieste.it/~pub−off IC/2004/63United Nations Educational Scientific and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSEXQUISITIVELY SENSITIVE MATTER-WAVE INTERFEROMETRY USEDTO MEASURE THE CHANGE IN VELOCITY OF A MOVING MAGNETTRAVERSING AN ELECTRIC FIELDN.H. MarchDepartment of Physics, University of Antwerp,Groenenborgerlaan 171, B-2020 Antwerp, Belgium,Oxford University, Oxford, EnglandandThe Abdus Salam International Centre for Theoretical Physics,Trieste, Italy.AbstractMotivated by the quite recent interferometric experiment of Shinohara, Aoki and Morinaga(Phys. Rev. A66, 042106, 2002) in which the scalar Aharonov-Bohm effect was studied, we re-open the extension to neutral particles carrying a magnetic moment and passing through a regionof intense electric field, treated theoretically by Aharonov and Casher (AC) and independentlyby Anandan. An alternative interpretation of results on (a) neutrons and (b) T`F moleculesto that afforded by AC is shown to involve only (i) the de Broglie wavelength of matter wavesand (ii) the prediction from Maxwell’s equations for the change in velocity of a neutral movingmagnet as it enters or leaves an electric field. The exquisite sensitivity of experiment (b) allowsa fractional change in velocity of order 10−15 to be quantitatively determined.MIRAMARE – TRIESTEAugust 2003The motivation for this Brief Report comes from the recent study of Shinohara, Aoki andMorinaga [1] in this journal, using matter-wave interferometry. Their immediate concern waswith the scalar Aharanov-Bohm effect for ultracold atoms. As they pointed out, both scalar andvector Aharonov-Bohm [AB] effects [2] were extended to neutral particles carrying a magneticmoment by Aharonov and Casher [3: termed AC below] and independently by Anandan [4].As stressed in ref.[1], for neutral particles the first experiment bearing on the theoreticalstudies of [3] and [4] above employed slow neutrons [5] and used real-space interferometry. Thiswas complemented later by spin-space interferometry with neutral molecules [6-8], out of whichwe choose to discuss the specific experiment of Sangster et al. [6]. The authors selectively flippedthe magnetic moments of some of the F nuclei in a beam of T`F molecules which was traversinga strong electric field and this will be discussed further below. Later, in an experiment alsorelevant to the theory presented in [3] and [4], Yanagimachi et al. [9] successfully observed theAC phase for a relatively short measurement time by means of an atomic polarizing white colorinterferometer.While, of course, the AC formula for the phase shift experienced by a neutral particle withmagnetic moment µ travelling through an electric field E is both illuminating and elegant(see also [4]), there is an alternative physical, argument which, as we will expand on below,demonstrates very strikingly the exquisite sensitivity of the interferometric experiments detailedabove.In the light of the substantial current interest in matter waves, even involving huge organicmolecules such as the buckyball C60 [10], we emphasize first, and expand on below, the fact thatthe measured phase shift, as predicted by AC [3] and Anandan [4], can be interpreted in theexperiments [5] and [6] referred to specifically below, by invoking two very basic physical results.These are (i) the de Broglie relation h/p for the wavelength of the matter wave correspondingto momentum p and (ii) the change in velocity, denoted below by ∆v, of a moving magnet as itenters or leaves an electric field.To focus on the sensitivity required to detect ∆v/v, we stress first of all that under theconditions obtaining in the experiment of Sangster et al. [6] for the T`F molecular beam, theratio ∆v/v was ∼ 10−15, a value to be discussed a little further below. It is truly remarkableto the present writer that spin-space interferometry has reached the stage where it cannot justdetect, but also quantitatively measure, such an increment ∆v in the velocity v.The basic electromagnetism needed to calculate (ii) above can be found in various places (see,for example, [11-13]) but for completeness we summarize below, and quite briefly, the essentialpoints needed for the present purposes. Consider such a neutral moving magnet having momentµ and entering a region of electric field E . The initial velocity v, as the magnet enters the fieldE , is changed by a very small increment, ∆v as introduced already, due to the interaction of themoving dipole with the magnetic flux density given byB = v × E/c2 (1)2where c is the velocity of light. When such a magnet, having mass m, moves from a regionwhere E = 0 to an intense field region of strength E , the magnet gains or loses momentum ∆pgiven by∆p = µ× E/c2, (2)the change in velocity being ∆p/m. For common magnets, the fractional change of velocity∆v/v is incredibly small. But for, say, a beam of neutrons, which has, of course, accuratelymeasurable de Broglie wavelengths, minute changes in velocity are observable by interferometry.Taking therefore specifically the coherent neutron beam experiment of Cimmino et al. [5],their experimental conditions were such that∆v = µF/mc2 = 1.87 × 10−9ms−1 (3)But v = h/mλ where the de Broglie wavelength λ = 148 pm and v = 2680 ms−1. Hence fromeqn.(3) it follows that∆v/v = 7× 10−13 (4)The extent of the electric field region in the experiment of Cimmino et al. [5] was 0.0253 m andthus the neutrons experience the field for a time t = 9.4× 10−6s. The change of phase, ∆φ say,of the neutrons while traversing the electric field is given by∆φ = 2t∆v/(λ/2pi), (5)since the beam is split into two branches and the phase changes there are equal and opposite.Thus one finds, inserting t = 9.4 × 10−6s into eqn.(5), a phase shift ∆φ = 1.5 × 10−3 radians.This phase shift, and consequently from eqn.(5) the velocity change ∆v, is what is detected bythe delicate interferometric technique employed by Cimmino et al. [5]. These workers reportedan observed phase shift of 2.2 ± 0.5 m rad., which is quite close to embracing the value of1.5 × 10−3 rad. obtained above. Thus, the de Broglie relation for matter waves, plus standardelectromagnetism embodied in Maxwell’s equations, yield by simple physical arguments closeagreement with this neutron experiment [5] in which the unpolarized coherent beam was splitby diffraction in a Si crystal and the two branches then passed through opposite electric fieldshaving magnitudes E of +2.92 × 107V m−1 and −2.92 × 107V m−1.As a second numerical example, let us return to the interferometric experiment carriedout on thallium fluoride (T`F) molecules. Here Sangster et al. [6] employed as their movingmagnets the fluorine nuclei in a T`F molecular beam, again in a strong externally appliedelectric field of magnitude E = 0.5 − 3MV m−1. For numerical estimates, made again as forthe neutron experiment above, the T`F molecules have mass m = 3.71 × 10−25kg. For a fieldstrength E = 2.01 × 106v/m, Sangster et al. obtained a phase shift ∆φ of 2.47 × 10−3 rad.From eqn.(5) above, using eqn.(3) for ∆v, the values we obtained were ∆v = 3.39 × 10−13m/swith corresponding fractional change ∆v/v = 1.33×10−15, the order of magnitude of which was3already anticipated above. The corresponding phase shift from eqn.(5) agrees with the measuredphase shift to within a few percent.In summary, by means of the de Broglie relation for the wavelength h/p associated withmatter waves, plus the prediction from Maxwell’s equations for the change in velocity of a movingmagnet as it enters or leaves an electric field, one can quantitatively interpret the interferometricexperiments using both a beam of neutrons and molecular beam of T`F molecules. The exquisitesensitivity of the interferometry used is plain from the fractional changes in velocity, ∆v/v,which are for the neutron experiment ∼ 10−12 and for the T`F molecules 10−15. This in no waydenies the interest and elegance of the AC work in [2] and the study of Anandan in [4]; and inparticular their finding that the phase shift ∆φ calculated above can be represented in a precisepath-independent formula.AcknowledgmentsThe author wishes to acknowledge invaluable collaboration on the general area of this BriefReport with Professor G.R. Freeman. As mentioned above, the motivation for the writer’s re-opening the subject was the appearance, quite recently of ref.[1] in this Journal. The workwas brought to fruition during a stay at the Abdus Salam ICTP, Trieste, Italy, in 2004. NHMthanks Professor V.E. Kravtsov, Head of the Condensed Matter Group, for both the stimulatingenvironment in which the work was completed and for generous hospitality.References[1] K. Shinohara, T. Aoki and A. Morinaga, Phys. Rev. A66, 042106 (2002).[2] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).[3] Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 (1984).[4] J. Anandan, Phys. Rev. Lett. 48, 1660 (1982).[5] A. Cimmino, G.I. Opat, A.G. Klein, H. Kaiser, S.A. Werner, M. Arif and R. Clothier, Phys.Rev. Lett. 63, 380 (1989).[6] K. Sangster, E.A. Hinds, S.M. Barnett and E. Riis, Phys. Rev. Lett. 71, 3641 (1993).[7] A. Gorlitz, B. Schuh and A. Weis, Phys. Rev. A51, R4305 (1995).[8] K. Zeiske, G. Zinner, F. Riehle and J. Helmcke, Appl. Phys. B: Lasers Opt. 60, 205 (1995).[9] S. Yanagimachi, M. Kajiro, M. Machiya and A. Morinaga, Phys. Rev. A65, 042104 (2002).[10] A. Zeilinger et al., Science 296, 1189 (2002).4[11] P. Lorrain, D.R. Corson and F. Lorrain, Electromagnetic Fields and Waves, 3rd Edn. (NewYork: Freeman, 1988).[12] G.R. Freeman and N.H. March, Eur. J. Phys. 18, 290 (1997).[13] G.R. Freeman, L.D. Coulson and N.H. March, Mod. Phys. Lett. B12, 933 (1998).5

Exquisitively sensitive matter-wave interferometry used to measure the change in velocity of a moving magnet traversing an electric field

Abstract

Similar works

Full text

Available Versions

CERN Document Server