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WEAK MEASUREMENT of PHOTON POLARIZATION James M. KNIGHT

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WEAK MEASUREMENT of PHOTON POLARIZATION James M. KNIGHT 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 <W2 I ~1~i> 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’> <Yl. Theof polarizationthe photon statepolarizationof photonsvariablein a planeQ=waveIx> <xIis Se—- 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> <xl — IY> <Yl give a measure of the quantity Q. Taking 0=0 at the midpoint between the two peaks, the relation is Q~=IL><LI—IR><RI , (2) O=cQ, (3) where x> 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> <xl —IY> <YI ) I Wl> = . 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=<i,u2Ix><xIW1>= cos L x a 2(~—O).
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