proposed interferometry test of noncyclic geometric phase

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16 July 2001 Physics Letters A 286 (2001) 4–6 www.elsevier.com/locate/pla Proposed interferometry test of noncyclic geometric phase Erik Sjöqvist Department of Quantum Chemistry, Uppsala University, Box 518, Se-751 20 Uppsala, Sweden Received 2 May 2001; accepted 1 June 2001 Communicated by P.R. Holland Abstract An interferometric test of the geometric phase in noncyclic spin-1/2 precession is proposed. It is shown that the resulting interference shift depends only on the noncyclic geometric phase for any opening angle on the Bloch sphere as well as for any degree of polarisation. 2001 Published by Elsevier Science B.V. PACS: 03.65.Vf; 03.75.Dg; 07.60.Ly Weinfurter and Badurek [1] were first to announce an experiment of the noncyclic geometric phase [2] in neutron polarimetry. However, it was later pointed out [3] 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 was recently implemented in noncyclic spin-1/2 evolu- tion using neutron interferometry [4]. This experiment tested the Pancharatnam phase difference ∆φ [5], be- ing related to the noncyclic geometric phase φ g as (1) φ g = ∆φ γ, where γ is the accumulated local phase change along the path of the spin state. The experiment [4] veri- fied the noncyclic geometric phase only in the very particular case where the neutron spin evolved along the equator of the Bloch sphere, for which γ = 0. However, experimental test of the noncyclic geomet- ric phase in spin precession for other opening angles on the Bloch sphere is still an open issue. E-mail address: [email protected] (E. Sjöqvist). In this Letter, we suggest a feasible interferometry experiment that in precession for any opening angle on the Bloch sphere eliminates γ and thereby tests the noncyclic geometric phase. We further demonstrate that the set up could be used to test the geometric phase for mixed states discovered in [6]. The experiment is sketched in Fig. 1. It is designed for spin-polarised neutrons, but could in principle be modified for any two-level quantal system. The set up is built up in two steps. First, the beam pair of a balanced triple-Laue neutron interferome- ter is exposed to a homogeneous time-independent magnetic field b 0 = b 0 ˆ z pointing in the interferome- ter plane. Incident neutrons spin-polarised in the z di- rection are split coherently at the first crystal plate. As the magnetic field is aligned with the spin, neither any change of state nor any field-dependent relative phase occur. Thus, by adding a U(1) phase χ to one of the beams we expect the spin-independent intensity varia- tion (2) I O 1 + cos χ in the O-beam. 0375-9601/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII:S0375-9601(01)00387-5

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Page 1: Proposed interferometry test of noncyclic geometric phase

16 July 2001

Physics Letters A 286 (2001) 4–6www.elsevier.com/locate/pla

Proposed interferometry test of noncyclic geometric phase

Erik SjöqvistDepartment of Quantum Chemistry, Uppsala University, Box 518, Se-751 20 Uppsala, Sweden

Received 2 May 2001; accepted 1 June 2001Communicated by P.R. Holland

Abstract

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 [1] were first to announcean experiment of the noncyclic geometric phase [2] inneutron polarimetry. However, it was later pointed out[3] 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 [4]. This experimenttested the Pancharatnam phase difference∆φ [5], be-ing related to the noncyclic geometric phaseφg as

(1)φg =∆φ − γ,

whereγ is the accumulated local phase change alongthe path of the spin state. The experiment [4] veri-fied the noncyclic geometric phase only in the veryparticular case where the neutron spin evolved alongthe equator of the Bloch sphere, for whichγ = 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: [email protected] (E. Sjöqvist).

In this Letter, we suggest a feasible interferometryexperiment that in precession for any opening angleon the Bloch sphere eliminatesγ and thereby tests thenoncyclic geometric phase. We further demonstratethat the set up could be used to test the geometric phasefor mixed states discovered in [6]. 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 fieldb0 = b0z pointing in the interferome-ter plane. Incident neutrons spin-polarised in thez 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) phaseχ to one of thebeams we expect the spin-independent intensity varia-tion

(2)IO ∝ 1+ cosχ

in the O-beam.

0375-9601/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0375-9601(01)00387-5

Page 2: Proposed interferometry test of noncyclic geometric phase

E. Sjöqvist / Physics Letters A 286 (2001) 4–6 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 fieldb0 parallel to the initial spin direction. An additionalmagnetic fieldb1 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) phaseχ 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 fromb1. Incident neu-trons are again spin-polarised in thez direction andsplit coherently at the first crystal plate. The spin inthe I-beam precesses around the magnetic fieldB =b0 + b1 yielding a noncyclic geometric phase (see,e.g., Ref. [3])

(3)φg = −arctan

(cosθ tan

ϕ

2

)+ ϕ

2cosθ,

where cosθ = b0/B defines the opening angle of thespin precession on the Bloch sphere andϕ = −2µ×Bτ/h is the corresponding change of azimuthal angle(see Fig. 2). Here,B = |B|, µ is the neutron magneticmoment, andτ 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 [2]. The output intensity inthe O-beam now reads [4]

(4)IO ∝ 1+ ν cos[χ −∆φ]

Fig. 2. Evolution of the spin stater on the Bloch sphere. The inputstate at the north pole precesses around the direction(sinθ,0,cosθ)defined by the magnetic fieldB = b0 + b1. The precession angleϕis proportional to the magnitude of the magnetic field and the timespent by the neutrons in the field region.

with Pancharatnam phase difference

(5)∆φ = arg〈ψII |ψI〉and visibility

(6)ν = ∣∣〈ψII |ψI〉∣∣,

ψI and ψII being the input spin states to the thirdcrystal plate. Explicitly, we have

|ψI〉 = exp

(−i ϕ

2n · σ

)|+z〉,

(7)|ψII 〉 = exp

(−i ϕ0

2σz

)|+z〉,

whereϕ0 = −2µb0τ/h, n = B/B, σ = (σx, σy, σz)

are the standard Pauli matrices, and|+z〉 is theeigenstate ofσz with eigenvalue+1. Inserting Eq. (7)into Eq. (5) yields

∆φ = arg〈+z|exp

(iϕ0

2σz

)exp

(−i ϕ

2n · σ

)|+z〉

(8)= −arctan

[nz tan(ϕ/2)− tan(ϕ0/2)

1+ nz tan(ϕ/2) tan(ϕ0/2)

].

Using the identity arctan[(x−y)/(1+xy)] = arctanx− arctany, we may write

(9)∆φ = −arctan

(nz tan

ϕ

2

)+ ϕ0

2.

Page 3: Proposed interferometry test of noncyclic geometric phase

6 E. Sjöqvist / Physics Letters A 286 (2001) 4–6

From nz = b0/B = cosθ , ϕ0 = ϕb0/B = ϕ cosθ ,and comparing with Eq. (3), we finally obtain

(10)∆φ = φg.

Thus, the accumulated local change of phaseγ in thetwo beams cancel and only the noncyclic geometricphase acquired in the I-beam appear in the Pancharat-nam phase difference∆φ that could be tested by vary-ing the U(1) phaseχ .

Consider now the case of partially spin-polarisedneutrons. The degree of polarisation is given by thelengthr of the Bloch vectorr that determines the spin-1/2 state

(11)ρ = 1

2(1+ r · σ ).

Perfectly polarised states define the unit Bloch spherer = 1 containing the partially polarised statesr < 1.The output of the interferometer is characterised bythe visibility V and phase shift∆Φ given by

(12)Vei∆Φ = 1+ r

2νei∆φ + 1− r

2νe−i∆φ,

being the incoherent weighted average of pure stateinterference profiles [6]. In the present set up, we havethat r = r z and forr �= 0 we may insert Eq. (10) intoEq. (12) yielding

(13)∆Φ = arctan(r tanφg),

which is the noncyclic mixed state geometric phasediscovered in [6]. Thus, the mixed state geometric

phase as well as theφg-dependence of the visibility

(14)V = ν

√cos2φg + r2 sin2φg

may be tested in the present set up.To conclude, we have shown that the geometric

phase 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.

Acknowledgement

This work was supported by the Swedish NaturalScience Research Council (NFR).

References

[1] H. Weinfurter, G. Badurek, Phys. Rev. Lett. 64 (1992) 1318.[2] J. Samuel, R. Bhandari, Phys. Rev. Lett. 60 (1988) 2339.[3] A.G. Wagh, V.C. Rakhecha, Phys. Lett. A 197 (1995) 112.[4] A.G. Wagh, V.C. Rakhecha, P. Fischer, A. Ioffe, Phys. Rev.

Lett. 81 (1998) 1992.[5] S. Pancharatnam, Proc. Indian Acad. Sci. A 44 (1956) 247.[6] E. Sjöqvist, A.K. Pati, A. Ekert, J.S. Anandan, M. Ericsson,

D.K.L. Oi, V. Vedral, Phys. Rev. Lett. 85 (2000) 2845.