proposed interferometry test of noncyclic geometric phase
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16 July 2001
Physics Letters A 286 (2001) 46www.elsevier.com/locate/pla
Proposed interferometry test of noncyclic geometric phaseErik Sjqvist
Department of Quantum Chemistry, Uppsala University, Box 518, Se-751 20 Uppsala, SwedenReceived 2 May 2001; accepted 1 June 2001
Communicated by P.R. Holland
An interferometric test of the geometric phase in noncyclic spin-1/2 precession is proposed. It is shown that the resultinginterference shift depends only on the noncyclic geometric phase for any opening angle on the Bloch sphere as well as for anydegree of polarisation. 2001 Published by Elsevier Science B.V.
PACS: 03.65.Vf; 03.75.Dg; 07.60.Ly
Weinfurter and Badurek  were first to announcean experiment of the noncyclic geometric phase  inneutron polarimetry. However, it was later pointed out that this experiment verified the amount of preces-sion of the neutron spin and did not constitute a mea-surement of phase. A phase measurement scheme wasrecently implemented in noncyclic spin-1/2 evolu-tion using neutron interferometry . This experimenttested the Pancharatnam phase difference , be-ing related to the noncyclic geometric phase g as
(1)g = ,where is the accumulated local phase change alongthe path of the spin state. The experiment  veri-fied the noncyclic geometric phase only in the veryparticular case where the neutron spin evolved alongthe equator of the Bloch sphere, for which = 0.However, experimental test of the noncyclic geomet-ric phase in spin precession for other opening angleson the Bloch sphere is still an open issue.
E-mail address: firstname.lastname@example.org (E. Sjqvist).
In this Letter, we suggest a feasible interferometryexperiment that in precession for any opening angleon the Bloch sphere eliminates and thereby tests thenoncyclic geometric phase. We further demonstratethat the set up could be used to test the geometric phasefor mixed states discovered in . The experiment issketched in Fig. 1. It is designed for spin-polarisedneutrons, but could in principle be modified for anytwo-level quantal system.
The set up is built up in two steps. First, the beampair of a balanced triple-Laue neutron interferome-ter is exposed to a homogeneous time-independentmagnetic field b0 = b0z pointing in the interferome-ter plane. Incident neutrons spin-polarised in the z di-rection are split coherently at the first crystal plate. Asthe magnetic field is aligned with the spin, neither anychange of state nor any field-dependent relative phaseoccur. Thus, by adding a U(1) phase to one of thebeams we expect the spin-independent intensity varia-tion
(2)IO 1 + cosin the O-beam.
0375-9601/01/$ see front matter 2001 Published by Elsevier Science B.V.PII: S0375-9601(01) 00 38 7- 5
E. Sjqvist / Physics Letters A 286 (2001) 46 5
Fig. 1. Triple-Laue neutron interferometer set up to test the geo-metric phase in noncyclic spin-1/2 evolution. A beam of neutronspolarised in the plane of the interferometer is split at the first crystalplate. The beam pair is exposed to a time-independent homogeneousmagnetic field b0 parallel to the initial spin direction. An additionalmagnetic field b1 perpendicular to the interferometer plane is addedto the I-beam, but shielded from the II-beam. The interference shiftin the O-beam obtained by varying the U(1) phase in the II-beamdepends only on the noncyclic geometric phase of the spin state ac-quired in the I-beam.
The second step consists of adding to the I-beam an-other homogeneous time-independent magnetic fieldb1 = b1x, now perpendicular to the interferometerplane. The II-beam is shielded from b1. Incident neu-trons are again spin-polarised in the z direction andsplit coherently at the first crystal plate. The spin inthe I-beam precesses around the magnetic field B =b0 + b1 yielding a noncyclic geometric phase (see,e.g., Ref. )
(3)g = arctan(
where cos = b0/B defines the opening angle of thespin precession on the Bloch sphere and = 2B/h is the corresponding change of azimuthal angle(see Fig. 2). Here, B = |B|, is the neutron magneticmoment, and is the precession time assumed tobe equal in the two beams. The geometric phase (3)equals minus half the solid angle enclosed by thespin path and the shortest geodesic connecting its endpoints on the Bloch sphere . The output intensity inthe O-beam now reads 
(4)IO 1+ cos[ ]
Fig. 2. Evolution of the spin state r on the Bloch sphere. The inputstate at the north pole precesses around the direction (sin ,0, cos )defined by the magnetic field B = b0 + b1. The precession angle is proportional to the magnitude of the magnetic field and the timespent by the neutrons in the field region.
with Pancharatnam phase difference
(5) = argII|Iand visibility
(6) = II|I,I and II being the input spin states to the thirdcrystal plate. Explicitly, we have
|I = exp(i
(7)|II = exp(i 0
where 0 = 2b0/h, n = B/B , = (x, y, z)are the standard Pauli matrices, and |+z is theeigenstate of z with eigenvalue +1. Inserting Eq. (7)into Eq. (5) yields
= arg+z| exp(i02z
(8)= arctan[nz tan(/2) tan(0/2)
1+ nz tan(/2) tan(0/2)].
Using the identity arctan[(xy)/(1+xy)] = arctanx arctany , we may write
(9) = arctan(nz tan
6 E. Sjqvist / Physics Letters A 286 (2001) 46
From nz = b0/B = cos , 0 = b0/B = cos ,and comparing with Eq. (3), we finally obtain
(10) = g.Thus, the accumulated local change of phase in thetwo beams cancel and only the noncyclic geometricphase acquired in the I-beam appear in the Pancharat-nam phase difference that could be tested by vary-ing the U(1) phase .
Consider now the case of partially spin-polarisedneutrons. The degree of polarisation is given by thelength r of the Bloch vector r that determines the spin-1/2 state
(11) = 12(1 + r ).
Perfectly polarised states define the unit Bloch spherer = 1 containing the partially polarised states r < 1.The output of the interferometer is characterised bythe visibility V and phase shift given by
(12)Vei = 1 + r2
ei + 1 r2
being the incoherent weighted average of pure stateinterference profiles . In the present set up, we havethat r = r z and for r = 0 we may insert Eq. (10) intoEq. (12) yielding
(13) = arctan(r tang),which is the noncyclic mixed state geometric phasediscovered in . Thus, the mixed state geometric
phase as well as the g-dependence of the visibility
cos2 g + r2 sin2 gmay be tested in the present set up.
To conclude, we have shown that the geometricphase in noncyclic spin-1/2 evolution can be observedas a Pancharatnam phase difference in triple-Laueneutron interferometry. We have further demonstratedhow to test the mixed state geometric phase for par-tially spin-polarised neutrons. It would be interestingto develop analogous interferometric tests of the non-cyclic geometric phase for other two-level quantal sys-tems.
This work was supported by the Swedish NaturalScience Research Council (NFR).
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