Volume 143, number 8 PHYSICS LETFERS A 29 January 1990

WEAK MEASUREMENT OF PHOTON POLARIZATION

James M. KNIGHT and Lev VAIDMAN Department ofPhysics and Astronomy, University ofSouth Carolina, Columbia, SC 29208, USA

Received 12 October 1989; revised manuscript received 27 November 1989; accepted for publication 28 November 1989 Communicated by J.P. Vigier

An optical procedure for measuringthe recently introduced weak value ofa photon polarization variable is suggested and some of its experimental aspects are discussed. The deflection of a beam passing through a birefringent prism is considered. It is shown that if the beam also passes through preselection and postselection polarization filters, it may have a deflection which is greater than the deflection of the beam without the filters. The magnitude of the deflection is give by weak measurement theory.

The concept of the weak value of a quantum me- measurement of photon polarization”. The optical chanical variable A has recently been introduced by polarization filters employed provide an especially Aharonov, Albert, and Vaidman (AAV) [1] (see also precise means of performing the preselection and ref. [2]), and discussed by other authors [3—71.This postselection which play an essential role in weak value is the statistical result of the standard meas- measurement. In addition to the interest of this ex- uring procedure carried out on a preselected and periment as an example ofweak measurement, it may postselected ensemble when the coupling of the sys- suggest practical optical applications of the under- tern with the measuring device is sufficiently weak, lying AAV interference phenomenon. a procedure called weak measurement. The weak A weak measurement is carried out by allowing a valueobtained from such a measurement can lie out- quantum system to pass successively through a pair side the range of the eigenvalues of A, in sharp con- of filters which select states I W1> and I W2>’ A mea- trast to the expectation value. surement of some variable A is performed, with poor Such surprising outcomes of weak measurement resolution, when the system is between the filters. As represent a peculiar interference phenomenon of shown in AAV, this provides a convenient method quantum wave functions. We shallshow that the same for exploring the properties of the system that are phenomenon occurs in electromagnetic waves, and compatible with both selected states. The result of that it can be readily observed in the optical region. this measurement, the weak value of A, is given by In making such a measurement, we are, in fact, mak- / A \ ing a weak measurement of a photon polarization A,,, = ‘~~ ~“ / . (1) variable. Since considerable experimental difficulty can be expected inany attempt todetermine the weak When the two selected states are nearly orthogonal, valueof a spin component of a spin-i particle, as de- this value can be much greater than the largest ei- scribed in ref. [11, the optical measurements de- genvalue of A. scribed here provide the simplest means ofverifying For a spin-i particle, a large weak value of o~can the feasibility of such a determination, be obtained by selecting states I a> and b> which The observations we discuss are to be made on a have spins lying in the x—z plane and making an an- classical intense optical beam, but since they are mo- gle ~x— 0 with the z-axis. The situation is illustrated tivated by a consideration of the properties of a sin- in fig. 1, where the states of a spin-i particle are rep- gle photon, and since they directly reflect those prop- resented as points on the surface of a spatial sphere erties, it is proper to refer to them as “weak with orthogonal states at opposite ends of a diame-

357 Volume 143, number 8 PHYSICS LETTERS A 29 January 1990

C

a~ Fig. 2. Schematic diagram ofthe system proposedfor weak mea- x 0 surementIi’>

lected by filter 1, and the photons are then deflected by the bire- fringent prism P with axes oriented in the x andydirections. The —z filter 2 postselects the polarization state before the photons pass through a narrow slit in the screen S to be observed at D. Fig. 1. Spin states involved in weak measurement ofa~fora spin- particle. States arerepresented as points on the surface of a unit spatial sphere. States at opposite ends of a diameter are orthogo- surement by adding filters topreselect and postselect nal. Values ofthe spin variables in the directions Oa and Ob are states I w~>and I W2>, and by providing a small ap- fixed to be + by preselection and postselection. Oc represents erture in order to give the required poor resolution. the weak value of ~ After passing through the first filter, a planewave is normally incident on the front face ofthe prism and ter. The components of a satisfy the geometrical re- is deflected at the second face, the normal to which lation Oa+ ab= 2a~sin 0. Since an operator relation makes an angle x with the incident beam direction. of the form A = B+ C insures that the weak values The angle of deflection depends on the index of re- satisfy A,.,= B,., + C,,,,,, and since the weak values of ~ fraction of the material, and is different for the two and ab are both equal to + 1, the value selected by orthogonal states ofpolarization. The two beams with the filters, this relation shows that (a~)~has the value slightly different directions then pass through a sec- 1/sin 0. This value is much greater than unity for ond filter and through a slit of width a. In a con- small 0. ventional Stern—Gerlach type experiment, where the The experiment we discuss is a weak measurement second filter is not present, and where a is large of a two-valued photon polarization variable anal- enough to resolve the two peaks, the angular posi- ogous toa spin component of a spin-i particle ~‘. The tions of the centers ofthe two patterns are correlated variable Q in question can be either of the following: with the polarization state of the photon, and thus Q 1 = Ix>

Q~=IL> and I y> are orthogonal states of linear polarization, and IL> and IR> are left- and right- where circularly polarized states. We examine the condi- ~ = ~~ (n, sin x) — sin ~(n2 sin x)] . (4) tions under which the weak value of Q is well outside its eigenvalue range. Here n1 and n2 are the indices of refraction for the The proposed measuring configuration, fig. 2, is two polarization states. If a is now made narrow, the an optical model of the Stern—Gerlach apparatus, peaks become broad and overlap, but their centers employing a birefringent or optically active prism to remain in the same location. The intensity on the give different deflections to states of different po- screen withthe secondfilter absent is the sum of the larization. It is adapted for the purpose of weak mea- intensities ofthe peaks. If the second filter is present, however, the intensity is determined by a superpo- ~“ A somewhat different optical realization of weak measure- sition of the two amplitudes. This pattern can take ments was suggested in ref. [3]. the form of a broad peak centered at the weak value

358 Volume 143, number 8 PHYSICS LETTERS A 29 January 1990 of Q, far outside the interval (— 1, + 1) in which the Then the weak value of the polarization operator is expectation value of Q lies. Fig. 3 illustrates the choice of states for preselec- (Q 1 )~= <~‘~I (Ix> = . 12 tion and postselection. The states are represented as <~~2I Wi> 6 points on the surface of the Poincaré sphere. In fig. 3a, where Q~is measured, states a and b are ellipt- The amplitude at a distance L from the slit arising ically polarized states obtained by superposing L with from a single coherent plane wave passing through x and iy, respectively (cf. fig. 1). If the operator Q~ a conducting slit of width a is [81 is measured (fig. 3b), then the states a and b are states of linear polarization obtained by superposing ~ = e x and x ±y. Measurement of linear polarization ap- pears more attractive because ofthe simplicity oflin- / sin ( ~kasin 0) . cos (fka sin 0)\ ear polarization filters. We therefore confine our Xli,, sin(0/2) ±1 cos(0/2) ) (7) considerations to this case for the rest of this paper. Let the crystal be cut in such a way that the x-axis where ±corresponds to polarizations parallel and lies in the plane of fig. 2, and the y-axis is parallelto perpendicular to the slit. This amplitude yields the both faces of the prism. The two linear polarization following intensity distribution: states selected by the filters (fig. 3b) lie in the XJ’ r / 2 plane, perpendicular to the direction ofpropagation dI= ~ I (2 sin(~kasin 0)) cos e of the incident beam, and make an angle i,~t—Owith 27t L \ ka sin 0 / the x-axis: 2 1Ide. (8) IW +[kacos(0/2)] 1>=cos(~.~~—O)Ix>+sin(~—O)IY>,

Iw2>=cos(~—0)lx>—sin(~—O)IY>. (5) Each polarization separately yields the pattern (8) shifted through an angle O~=±c(eq. (3)). In our arrangement, the amplitude will be a superposition C of two amplitudes of the form (7), shifted by ±C: W(0)=a1u(O—c)+a2u(0+c), (9) in which a1 and a2 are given by 2(~t—Ø) a1== cos L x a 2(~—O). (10) 2==—5in a b a The second term in the amplitude (7) gives rise to x - xy - - ‘x+Y a small 0-dependent elliptical polarization, produced 0 by diffraction at the slit. Placement of the second fil- ter before the slit, as in fig. 2, minimizes the effect A of this term by causing the postselection to occur when the polarization is purely linear. Otherwise, if (a) (b) the quantity l/ka were appreciable, this term could mask the effect we are seeking. With the choice of Fig. 3. Polarization states involved in weak measurement of (a) parameters described below, with ka 100, this term

Q~=IL>

359 Volume 143, number 8 PHYSICS LETTERSA 29 January 1990 of angle x= 0.01 rad. Light of wavelength 632.8 nm altering the theoretical amplitudes given by eq. (10). was assumed, and the slit width was taken to be 0.01 Propagation through the birefringent crystal pro- mm. The angle 0 defining the orientation of the two duces a phase difference between x and y polariza- filters had a value of 0.05 radians. If the light inten- tions, and a corresponding relative phase between sity incident on the slit is 1.0 kW/m2, the maximum the amplitudes a 2. state of elliptical1polarizationand a2 in eq.will(9).notTheproduceresultingthe Thisintensitycorresponds1 m fromtotheaboutslit will1 03bephotons1.5 X l0—~perkW/msecond desired shifted diffraction pattern. This phase dif- passing through a small observation slit of width 0.1 ference can be eliminated by careful lateral place- mm and height 2.0 mm. ment of the slit so that rays corresponding tothe two The results are shown as the solid curve in fig. 4. different polarization states traverse the same opti- The dashed curves in this figure represent the pat- cal path length. The correct position can be found by terns produced whenpurex and pure y polarizations moving the slit so as to obtain a local minimum in are incident on the slit. Thesetake the form ofbroad the overall intensity of the pattern. peaks centered at 0— ±c. The pattern arising from The absolute values of a 1 and a2 will also be af- the superposition is, indeed, of nearly the same form fected by the different transmission coefficients at as the pure polarization patterns. However, it is the faces of the prism, but a rotation of the initial shifted from the center by an amount which is larger polarization filter can compensate for this effect. If by a factor 1/sin 2Ø~10, as expected from eq. (1). J. and]; denote the products of the transmission am- The calculated position of the peak agrees with the plitudes at the two faces, then the appropriate angle prediction of eqs. (1) and (3) to within 3%. of rotation y is determined by the equation Care must be taken in setting up the experiment tan(—Ø—y)=(f/f;)tan(~it—Ø). (11) to prevent transmission and propagation effects from This rotation leavesthe relation (10) unchanged ex-

______cept for a small modification of the total amplitude. We have presented a simple experiment which 7 .~\ demonstrates that the weak value of a quantum me- / ,..‘ \. chanical variable can greatly exceed its largest eigen- / ,... .. ~, value. Although the motivation for performing this

/ ,•‘ ..,. \. experiment stems from the quantum mechanics of / .‘~...‘ photonpolarization, the collective action of the pho- / ..~:.‘ tons is to produce a new classical interference effect, / ~ \~\ the shift of the pattern by passage through a filter.

______This effectmight havepractical applications. For ex- —60 —40 —20 0 20 40 60 ample, the deflection of the beam when it passes Theta (mrad) through both filters is proportional to c, eq. (4), which is proportional to n1 — n2 when that quantity Fig. 4. The solid curve shows the intensity distribution, in arbi- is small. The shift is large compared to the shift ob- trary units, predicted in weak measurement of the photonpolar- tamed by varying the polarization of the beam pass- ization variable Qi = Ix> and lY>, giving Q’ = + 1 and Q1 = —1, respec- small differences in the indices of refraction. tively. The scale of the dashed curves is approximately 50 times greaterthan that of the solid curve. The peak ofthe solid curve is clearly shifted outside the interval (— 1,+ 1). The curves were We would like to acknowledge helpful discussions calculated for values of the parameters appropriate to calcite: with Y. Aharonov. = 1.658, n2 = 1.486. Other parameters (see text for definition) were taken to be: X=0.Ol, 0=0.05, and a=0.01 mm. The dashed curves are split by 1.72 mrad, and the solid curve is shifted from the center ofthe patternby 8.29 mrad. This corresponds to a weak value (Q~)~=9.7.

360 Volume 143, number 8 PHYSICS LETTERS A 29 January 1990

References [S]R. Golub and R. Gähler, Phys. Lett. A 136 (1989) 178. [6] A. Peres, Phys. Rev. Lett. 62 (1989) 2325;

[1] Y. Aharonov, D.Z. Albert and L. Vaidman, Phys. Rev. ~ A.J. Leggett, Phys. Rev. Lett. 62 (1989) 2326; 60(1988)1351. A. Aharonov and L. Vaidman, Phys. Rev. Lett. 62 (1989) [2] Y. Aharonov,D.Z. Albert, A. Casher and L. Vaidman, Phys. 2327. Lett. A 124 (1987) 199. [7] Y. Aharonov and L. Vaidman, Phys. Rev. A, to be published. [3] I.M. Duck, P.M. Stevenson and E.C.G. Sudarshan, Phys. Rev. [8] L.D. Landau and E.M. Lifshitz, Electrodynamics of D 40 (1989) 2112. continuous media (Pergamon, Oxford, 1984) pp. 328, 329. [4] P. Busch, Phys. Lett. A 130 (1988) 323.

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