This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 128.178.131.113This content was downloaded on 20/07/2015 at 22:02

Please note that terms and conditions apply.

How to measure magnetic flux with a single position measurement?

View the table of contents for this issue, or go to the journal homepage for more

2015 EPL 110 50004

(http://iopscience.iop.org/0295-5075/110/5/50004)

Home Search Collections Journals About Contact us My IOPscience

June 2015

EPL, 110 (2015) 50004 www.epljournal.orgdoi: 10.1209/0295-5075/110/50004

How to measure magnetic ux with a single positionmeasurement?

E. Cohen1,2, L. Vaidman1 and Y. Aharonov1,2

1 Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv UniversityTel-Aviv 6997801, Israel2 Schmid College of Science and Technology, Chapman University - Orange, CA 92866, USA

received 2 May 2015; accepted 5 June 2015published online 25 June 2015

PACS 03.65.Vf Phases: geometric; dynamic or topologicalPACS 03.65.Ta Foundations of quantum mechanics; measurement theory

Abstract Current methods for measuring magnetic ux are based on performing many mea-surements over a large ensemble of electrons. We propose a novel method based on wave functionrevival for measuring the ux modulo hc

2eusing only a single electron. A preliminary analysis

of the feasibility of the experiment is provided.

Copyright c EPLA, 2015

Introduction. In classical theory, physical variablescan, in principle, be found by a measurement which con-sists of just a single event. For example, to measure a eldin a particular place we can send a particle in a knowninitial state (position and velocity) and then deduce thevalue of the eld from the place on a screen which theparticle lands on. Uncertainty relations of quantum me-chanics put some constraints, however, by increasing thecoupling strength we can reach an arbitrary precision alsowhen we use a quantum probe, i.e. a quantum particle.This is a consequence of the correspondence principle.

Some quantum experiments do not have a correspond-ing classical counterpart. For example, the magneticAharonov-Bohm (AB) eect [1] allows to measure a uxof magnetic eld (modulo a constant) through a ring with-out actually measuring the magnetic eld inside the ring.In fact, this experiment has been performed in many sys-tems, from normal metals [2,3] to 2D electron gas wafer [4],topological insulator nano-wires [5] and quantum rings ingraphene [6,7].

In a ring, small enough so that the electron states arenot randomized by inelastic scattering during the traversalof the arms of the ring, electronic interference inuencesthe resistance, and thus oscillations of the magnetoresis-tance as a function of the ux through the ring can befound [3]. After calibration, measurement of resistancetells us the value of the ux modulo hc2e (see g. 1). Anymeasurement of resistance involves measurement of elec-tric current and thus involves measurement of a macro-scopic number of electrons. Quantum mechanics allows

Fig. 1: Illustration of the typical setup for measuring the ABeect. A coherent beam of electrons enters from the left, splitsaround the magnetic ux and recombines. The intensity ofthe current on the right reveals the modular ux [8].

a direct measurement of a magnetic ux (without mea-surement of the magnetic eld everywhere), but currentmethods require measurement on an ensemble of electronsencircling the ring with a ux.

Another way to use the AB eect for direct measure-ment of the ux is to observe the shift in the two-slit in-terference pattern if we introduce the ux between the twoslits. But again, we need an ensemble of measurements toobserve the shift of the interference pattern, and a singleelectron cannot tell us much about the value of the shift.The shift becomes observable in a single electron measure-ment if instead of a two-slit interference experiment weperform a multiple-slit experiment with identical uxes

50004-p1

E. Cohen et al.

Fig. 2: Interference of a single electron on a grating with iden-tical solenoids. The angle of the outcoming electron gives themodular ux.

between every two slits (see g. 2) [9]. If the slits arespaced a distance d = apart, where is the wave-length of the electron, the interference pattern will con-sist of two well-localized lines at angles corresponding tosin 1 = (/0) mod 1 and sin 2 = (/0) mod 1 1,where 0 = hc2e . Detection of the electron in any of thelines will reveal mod 0.

The question we want to answer in this letter is how tomeasure the ux of a single solenoid, performing a singleexperiment with a single electron.

The proposed method. In our experiment the elec-tron is free to move inside a ring of radius R. The widthis small so that we can consider the radius of the electronmotion to be well dened. We want to measure the mag-netic ux through the ring. Let us rst consider the casein which there is no ux.

We start with the electron well localized in a particu-lar place in the ring. Immediately after its preparation,the wave function of the electron will spread out every-where along the ring. But after a particular time therewill be a revival: it will be localized again in the originalplace. Indeed, within the ring, the angular part of theHamiltonian is

H =p2

2mR2, (1)

and the angular momentum of the electron is quan-tized with eigenstates |n |p = nh. The energyeigenstates are

En =h2n2

2mR2, (2)

so after time

T =4mR2

h, (3)

every angular-momentum eigenstate will acquire the samephase. Any state which can be decomposed into a super-position

| =n

an|n, |n 12

ein, (4)

Fig. 3: Measuring ux with a single electron. The electron,

initially localized around = 0, ends up at time T = 4mR2

h

localized at the angle = 2AB , from which the modular uxcan be extracted.

thus, after time T , every state will return to the originalstate (up to an overall phase).

It is of interest to consider the state of the electron aftertime t = T2 =

2mR2

h . For even n we obtain |n |n,while for odd n we have |n |n. This is exactlywhat happens if we rotate the state by , i.e. + .Hence, at time T/2 the electron should be found at = .Using the same reasoning, it can be shown that after timet = T/3 the particle is in a superposition of being local-ized around = 0, = 2/3 and = 4/3. After timet = T/4 it is in a superposition of being localized around = 0 and = and so on for every T/k, when thesuperposition at odd k instances is richer than the su-perposition at even k instances). The dynamics hence de-scribes a well localized wave packet at t = 0 which quicklyspreads over the whole ring and acquires specic positionsat certain times, until coming back to its starting point att = T .

This kind of dynamics is known as revival [10].Recently, fractional revival, like the one demonstratedabove, was also suggested [11]. However, the analysis wehereby propose is unique in utilizing this phenomenon forgathering information about the setup (magnetic ux inthis case) using a single particle.

Let us consider now the AB setup in which there is aux inside the ring, see g. 3. We can use the Coulombgauge such that the Hamiltonian is

H =(p + e2c)

2

2mR2, (5)

with energies

En =(hn + e2c)

2

2mR2. (6)

50004-p2

How to measure magnetic ux with a single position measurement?

At t = T = 4mR2

h each eigenstate |n will be multiplied by

eiEnT/h = eiconste2n2ie2n(e/ch)i, (7)

where the rst exponent is n-independent and the secondequals 1. Hence, apart from an overall phase, every eigen-state |n will be multiplied by e2inAB , where AB = echis the well-known AB phase [1]. This is equivalent to theapplication of the shift operator e2ipAB/h, and there-fore, after time T , the electron will be localized at angle = 2AB , from which the ux modulo hc2e can be found.This stands in contrast with the usual methods of mea-suring ux which requires measurements over a large en-semble, and enables to test the AB eect in a completelynew manner.

But how precise is the measurement? Ideally, we wouldhave wanted to use for the well-localized electronic wavepacket a delta function. However, to reach a nite energycost, we will use a Gaussian truncation with zero mean,

() =12

n=

en2/(n)2ein(0), (8)

The error in the angular position measurement is [12]

1n

, (9)

and the error in measuring the ux will be = ch2e .A relative error of 0.5%, with respect to one ux quantumhc2e , would require a lower bound on the initial uncertainty,n 10.

Let us now take into account relativistic corrections.The rst-order correction for the energy eigenvalues is [13]

En =(hn)4

8c2m3R4. (10)

After time T , the additional phase shift of the n-th levelequals

n =h2n4

2(cmR)2. (11)

Requiring n , for |n| 10, we nd a relativisticlimitation on the radius

R hmc

(n)5

2= 3 1010 m. (12)

For large |n|, another important factor is the centrifugalforce which changes the electron radial wave function. Theeective radius of motion might be dierent for dierentn and it will cause a shift in phase. For estimating thiseect one has to know the bounding potential. If, for ex-ample, the radial potential is linear, the shift in the radiusof motion can be analytically measured as was performedin [14], but it does not seem like a realistic model for ac-tual experiments. In realistic cases, the binding potentialbecomes very steep towards the edge, thereby providingapproximately a xed radius.

In recent experiments where the AB phase was mea-sured [4,15], the radius of motion was R 106 m whichcorresponds to T 107 s. It seems feasible to perform anexperiment with this time resolution. The reason for us-ing a small radius in a laboratory experiment is that themain limiting factor is the coherence length of the elec-trons moving in the ring. To reach high coherence lengththe temperature of the electron gas has to be very low(140mK was used in [4]).

An idea for a possible physical implementation, is to useedge states which have been recently employed to demon-strate the AB eect in quasi-1D nanoribbons [16] and in2D topological insulators [17]. We hope that we can seethe revival eect in these edge states. Suitable gates canprepare, and later nd, the initial and nal states. To pre-pare the electron in a localized state, we need to accountfor the chirality of edge states. Therefore, we have to pre-pare a wave packet with Gaussian truncation of large, pos-itive mean so that only positive momenta will contribute.The mean rotation eect can be eliminated by arrangingthat the electron will complete an integer number of lapsesduring time T .

Conclusions. We have proposed in this letter amethod which allows indirect measurement of a magneticux using the detection of a single electron experiencingthe AB eect. This stands in contrast with all other ex-periments performed to date, where magnetic-ux mea-surements were based on the AB eect of large ensembles.Our method utilizes the fact that due to the special formof the energy spectrum the wave function of the electronundergoes a revival, and thus returns to be well local-ized. Preliminary considerations suggest the feasibility ofan experimental realization of our proposal.

We are thankful toVictor Fleurov, Moti Heiblum,Yuval Ronen and Eran Sela for helpful commentsand discussions. This work has been supported by theIsrael Science Foundation Grant No. 1311/14, by theICORE Excellence Center Circle of Light and by DIP,the German-Israeli Project cooperation.

REFERENCES

[1] Aharonov Y. and Bohm D., Phys. Rev., 115 (1959) 485.[2] Azbel M. Y., Gefen Y. and Imry Y., Phys. Rev. Lett.,

52 (1984) 129.[3] Webb R. A., Washburn S., Umbach C. P. and

Laibowitz R. B., Phys. Rev. Lett., 54 (1985) 2696.[4] Chang D. I., Khym G. L., Kang K., Chung Y., Lee

H. J., Seo M., Heiblum M., Mahalu D. and UmanskyV., Nat. Phys., 105 (2008) 205.

[5] Bardarson J. H., Brouwer P. W. and Moore J. E.,Phys. Rev. Lett., 4 (2010) 156803.

50004-p3

E. Cohen et al.

[6] Russo S., Oostinga J. B., Wehenkel D., HeerscheH. B., Sobhani S. S., Vandersypen L. M. K. andMorpurgo A. F., Phys. Rev. B, 77 (2008) 085413.

[7] Nguyen V. H., Niquet Y. M. and Dollfus P., Phys.Rev. B, 88 (2013) 035408.

[8] Washburn S. and Webb R. A., Adv. Phys., 35 (1986)375.

[9] Aharonov Y., arXiv:1303.2137 (2013).

[10] Eberly J. H., Narozhny N. B. and Sanchez-Mondragon J. J., Phys. Rev. Lett., 44 (1980) 1323.

[11] Romera E. and de Los Santos F., Phys. Rev. Lett., 99(2007) 263601.

[12] Harter W. G., Phys. Rev. A, 64 (2001) 012312.[13] Landau L. D. and Lifshitz E. M., Quantum Mechan-

ics: Non-relativistic Theory, 3rd edition (Butterworth-Heinemann, Boston) 1977, Chapt. 6.

[14] Olendski O., arXiv:1502.03013 (2015).[15] Khatua P., Bansal B. and Shahar D., Phys. Rev.

Lett., 112 (2014) 010403.[16] Peng H., Lai K., Kong D., Meister S., Chen Y.,

Qi X. L., Zhang S. C., Shen Z. X. and Cui Y., Nat.Mater., 9 (2010) 225.

[17] Gusev G. M., Kvon Z. D., Shegai O. A., MikhailovN. N. and Dvoretsky S. A., Solid State Commun., 205(2014) 4.

50004-p4