— physics pp. 1115–1122
Pancharatnam geometric phase originating from successive partial projections
SOHRAB ABBAS∗and APOORVA G WAGH
Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India
∗Corresponding author. E-mail: abbas@barc.gov.in
Abstract. The spin of a polarized neutron beam subjected to a partial projection in another direction, traces a geodesic arc in the 2-sphere ray space. We delineate the geometric phase resulting from two successive partial projections on a general quantal state and derive the direction and strength of the third partial projection that would close the geodesic triangle. The constraint for the three successive partial projections to be identically equivalent to a net spin rotation regardless of the initial state, is derived.
Keywords. Geometric phase; Pancharatnam triangle phase; polarized neutrons; partial projection.
PACS Nos 03.65.Vf; 03.65.Ta; 42.25.Ja
1. Introduction
Pancharatnam [1,2] was the first to recognize that two successive phase-preserving projections on a quantal state yield a phase that depends solely on the geometry of the two geodesic arcs traversed in the ray space. This pure geometric phase equals minus half the solid angle enclosed by the triangle obtained by joining the final and initial rays with the shorter geodesic. Extending the result to the case of partial projections, Samuel and Sinha [3] showed that for a sequence of three partial projections to close, the axes of the three imperfect polarizers must lie on the same great circle on the Poincar´e sphere and that the sequence then is equivalent to a rotation about the axis transverse to the great circle. Here we formulate a partial projection and enunciate the geometric phase arising from two successive partial projections. Next, we characterize the third projection that would close the geodesic triangle when the initial state is either a given stateψ0, or its orthogonal state ¯ψ0
or both. Finally, we derive analytic conditions satisfied by each partial projection in terms of the other two under which the sequence of three partial projections is universally equivalent to anSU(2) rotation.
2. Partial projection
A normalized ray ψ0 representing a quantal two-state corresponds to the ‘spin’
directions0=ψ0†σψ0= Trρ0σ. Hereσdenotes the vector of Pauli spin operators,
ρ0=ψ0ψ†0= (1 +σ·s0)/2 is the pure state density operator and 1 represents the unity operator. An imperfect projection of this state onto a nonorthogonal rayψ1
with the spins1, attenuates the component ofψ0 orthogonal toψ1 only by a factor exp(−α1), say, without introducing any phase. This optically dichroic operation,
ρ1+ exp(−α1)(1−ρ1) = exp(−α1/2) cosh(α1/2)(1 +t1σ·s1)
=P(t1,s1), say, (1)
whereρ1=ψ1ψ1† takes the state ψ0= cos(θ1/2)ψ1+ sin(θ1/2) ¯ψ1 along the shorter geodesic 0 → 1 on the two-sphere only part of the way to ψp = cos(θ1/2)ψ1+ exp(−α1) sin(θ1/2) ¯ψ1(figure 1b). Hereθ1denotes the angle betweens1ands0and t1 = tanh(α1/2) represents the strength of the partial projection. The stateψp is attenuated by a factor{cos2(θ1/2) + exp(−2α1) sin2(θ1/2)}1/2and has the spinsp
at an angle θ0 from s1, given by tan(θ0/2) = exp(−α1) tan(θ1/2). Pancharatnam connection [1,2] dictates that ψp is in phase ψ0. The partial projection effects a spin rotationδ=θ1−θ0 abouts0×s1. Angles θ0 andθ1 are analogous to those of photon propagation, viz.
cosθ0 = cosθ1+p1
1 +p1cosθ1 and sinθ0=sinθ1
p1−p21
1 +p1cosθ1 , (2) in two inertial frames with their relative velocity cp1 [3]. Here p1 = tanhα1 = 2t1/(1 +t21), denotes the polarization efficiency of the partial projection, i.e. an unpolarized beam subjected to this partial projection acquires a polarization p1
alongs1. Inverting the relation (2), we get cosθ1= cosθ0−p1
1−p1cosθ0 ⇒cos(π−θ1) = cos(π−θ0) +p1
1 +p1cos(π−θ0) and
sin(π−θ1) =sin(π−θ0)p 1−p21
1 +p1cos(π−θ0) . (3)
Thus a partial projection P(t1,s1) on a state with the spin at an angle π−θ0 with s1 brings it to an angle π−θ1 withs1. This is an equivalent of the inverse partial projection P(t1,−s1) along ¯ψ1 returning the ray ψp to ψ0, apart from a state attenuation. The spin rotationδis given by
tanδ
2 = sinθ1
t−11 + cosθ1
= sin(π−θ0)
t−11 + cos(π−θ0) = sinθ0
t−11 −cosθ0. (4) Thus for initial spin angles θ1 and the corresponding π−θ0 (cf. eq. (2)), the same spin rotation δ occurs (figure 2). The two angles become equal at θ1m = π−cos−1(t1) = π−θ0m and the spin rotation becomes a maximum, viz. δm = 2 sin−1(t1). This maximum can be understood by expressingδas
sinδ
2 = sinθ1−θ0
2 =t1sinθ1+θ0
2 →δm= 2 sin−1(t1). (5)
For the initial or final spin angle ofπ/2, i.e. θ1 =π/2 (θ0 = cos−1(p1)) or θ0 = π/2(θ1 =π−cos−1(p1)), equivalent to mutually inverse operations (ignoring the state attenuations),
δπ/2= 2 tan−1(t1) = sin−1(p1). (6)
3. Two partial projections
A second partial projectionP(t2,s2) will bring the stateψp part of the way along the shorter geodesicp→2 (figure 1b) to ψf given by (1 +t2σ·s2)(1 +t1σ·s1)ψ0
up to a real multiplier. Here again,ψfis in phase withψpbut relative toψ0, has a pure geometric phase−Ω/2 [1,2], since Φ = arg Trρ0(1 +t2σ·s2)(1 +t1σ·s1), i.e.
tan Φ =− t1t2[s0s1s2]
1 +t1t2s1·s2+t1s0·s1+t2s2·s0
=−tanΩ
2, (7)
Ω symbolizing the solid angle of the geodesic triangle 0→p→f →0 (figure 1b).
For t1 = t2 = 1, Φ reduces to the well-known Pancharatnam triangle phase [1,2]
for full projections (figure 1a). On reversing the order of two partial projections, the phase just changes sign as in the full projection case, but the ray traverses a different geodesic triangle 0 → p0 →f0(→ 0), enclosing a solid angle −Ω (figure 1c).
Without any loss of generality, we may assume s0 = ˆz, i.e. ψ0 = |zi, so that spinss1 and s2 have polar angular coordinates (θ1,0) and (θ2, φ12), respectively (figure 1a). Then polar coordinates (θf,φf) of the final spin sf are given by
tanθf
2eiφf =ψ¯0†ψf
ψ0†ψf
= (t1s1+t2s2−it1t2s1×s2)·(ˆx+iˆy)
(1 +t1t2s1·s2+t1s0·s1+t2s2·s0−it1t2[s0s1s2]), (8a) so that
φf = arg{(t1s1+t2s2−it1t2s1×s2)·(ˆx+iˆy)}+ Ω/2 (8b) and tan(θf/2) equals the modulus of the RHS of eq. (8a).
4. Pancharatnam triangle closure
We seek the third partial projectionP(t3,s3) which will close the geodesic triangle by taking ψf back to ψ0. The spin s3 must then lie beyond 0 on the geodesic f → 0 of length π. Hence the azimuthal angle of s3, φ13 = φf +π and the polar angle θ3 is constrained to lie between 0 andπ−θf. The inverse projection P(t3,−s3)∝(1−t3σ·s3) onψ0 must yieldψf up to an attenuation, i.e.
Figure 1. Pancharatnam triangles for two successive full (a), partial ((b) and (c)) projections and three identical (t1 =t2 = t3 =t= [−(1 + 2 cosφ)]1/2) projections equivalent to anSU(2) rotation (d).
t3s3·(ˆx+iˆy)
1−t3s3·s0 =−tanθf
2eiφf, φ13=φf+π and
t3= sinθ2f
sin(θ3+θ2f), 0< θ3< π−θf. (9) At the centreθ3c = (π−θf)/2 of the allowed domain (0, π−θf) ofθ3, the projection strengtht3exhibits a minimum equal to sin(θf/2) and rises symmetrically on either side ofθ3c, approaching unity at both extremes of theθ3 domain (figure 3).
The successive partial projections P(t3,−s3) followed by P(t2,−s2) take ψ0 to the rayψp(figure 1b) resulting in the triangle phase Ω/2. Hence Ω2 = arg Trρ0(1− t2σ·s2)(1−t3σ·s3), i.e.
tanΩ
2 = t2t3[s0s2s3]
1 +t2t3s2·s3−t3s0·s3−t2s2·s0. (10)
Figure 2. Spin deflection in a partial pro- jection.
Figure 3. Variation oft3vs.θ3for several θf.
The triangle can now be closed by the partial projectionP(t1,−s1) provided t1s1·(ˆx+iy)ˆ
1 +t1s1·s0 = tanθp
2eiφp
=− (t2s2+t3s3−it2t3s2×s3)·(ˆx+iˆy) (1 +t2t3s2·s3−t2s2·s0−t3s3·s0+it2t3[s0s2s3])
→t1= sinθ2p
sin(θ1−θ2p), θp< θ1< π, (11) yielding, for a given choice oft3ands3, a range [sin(θp/2),1] of solutionst1as a func- tion ofθ1 over the allowed domain on the extended geodesic 0→p. We note that the state P(t1,s1)ψ0 bears a phase −Ω/2 with respect to the state P(t3,−s3)ψ0, i.e.
tanΩ
2 = t3t1[s0s3s1]
1−t3t1s3·s1+t1s1·s0−t3s3·s0
. (12)
Combining eqs (7), (10) and (12), we arrive at the relation tanΩ
2 = s0· {t1t2s1×s2+t2t3s2×s3−t3t1s3×s1} 1 +t1t2s1·s2+t2t3s2·s3+t3t1s3·s1
. (13)
Equation (13) implies that the operationsP(t1,s1)P(t3,−s3) andP(t3,−s3)P(t1, s1) onψ0also yield the triangle phases Ω/2 and−Ω/2 respectively, albeit traversing geodesic triangles different from 0→p→f or 0 →f →p. P(t3,−s3)P(t2,−s2) also traces another triangle yielding the phase−Ω/2. Likewise, if with an initial state ψ0 = | −zi, we seek the closure of the geodesic triangle by a third partial projectionP(t3,s3), we obtain
t3s3·(ˆx−iˆy) 1 +t3s3·s0
= cotθ0f 2e−iφ˜f
=− (t1s1+t2s2−it1t2s1×s2)·(ˆx−iˆy)
(1 +t1t2s1·s2−t1s1·s0−t2s2·s0−it1t2[s0s1s2]). (14)
For the third partial projection of strengtht3 and an azimuth angleφ13 to effect a triangle closure for both initial states|ziand| −zi, we derive the constraint
t1cosθ1=−t2cosθ2=t3cosθ3. (15) Solid angles enclosed by the geodesic triangles for initial states|ziand| −ziare then equal and opposite [4], viz.
tanΩ
2 =−tanΩ˜
2 = sinφ12
1−t21cos2θ1
t1t2sinθ1sinθ2 + cosφ12, φ13=π+Ω
2 + tan−1 sinφ12
t1sinθ1
t2sinθ2 + cosφ12
=π+φf=π+ ˜φf. (16)
5. SU(2) operation
The three successive partial projections are identically equivalent to a net SU(2) rotation about ˆzto within a real multiplier,P(t3,s3)P(t2,s2)P(t1,s1)∝e−iσzΩ/2 regardless ofψ0only if
t3(1 +t1t2cosφ12)ˆs3+t1(1 +t2t3cosφ23)ˆs1
+t2(1−t1t3cosφ13)ˆs2= 0 and
θ1=θ2=θ3=π/2, (17)
i.e. s1, s2 and s3 must lie in same plane normal to s0= ˆz, a result derived earlier [3] for the optical analogue of a Thomas rotation. For the third partial projection, we then have
t3eiφ13 =− t1+t2eiφ12
1 +t1t2e−iφ12, (18a)
t3=
pt21+t22+ 2t1t2cosφ12 p1 +t21t22+ 2t1t2cosφ12
, φ13=π+Ω
2 + tan−1 sinφ12 t1
t2 + cosφ12
(18b) and
tanΩ
2 = sinφ12
(t1t2)−1+ cosφ12. (18c)
Likewise, projections 1 and 2 can be characterized respectively in terms of the other two as
t1e−iφ13 =−t2e−iφ23+t3
1 +t2t3eiφ23 , (19a)
t1=
pt22+t23+ 2t2t3cosφ23 p1 +t22t23+ 2t2t3cosφ23, φ13=π+ tan−1
µ sinφ23
t2/t3+ cosφ23
¶ +Ω
2, (19b)
tanΩ
2 = sinφ23
(t2t3)−1+ cosφ23 (19c)
t2e−iφ12 =−t1+t3ei(Ω−φ13)
1 +t1t3ei(Ω−φ13) =−sinhα1+ sinhα3e−iφ13 coshα1+ coshα3
(20a) and
tanΩ
2 = sinφ13
−(t1t3)−1+ cosφ13. (20b)
With identical partial polarizers,t1=t2=t3=t6= 1, say, we obtain the relations (see figure 1d)
φ12=φ23=φ=φ13/2, (21a)
⇓
Ω = 3φ−2πsgn (φ) (22b)
and
t2=−(1 + 2 cosφ), |φ| ∈ µ2π
3 , π
¶
(22c) and a universal attenuation factor tan3[(π−b)/4],bdenoting the base of Pancharat- nam triangles in figure 1d. The results, obtained here for spin polarized neutrons, can also be tested experimentally for light with identical dichroic polarizers whose axes lie in the plane transverse to the direction of light propagation, the successive axes being misaligned byφ/2.
6. Conclusions
We have elucidated the Pancharatnam triangle geometric phase in the partial pro- jection case. We have derived analytic relations for the effected spin rotations, geometric phase, triangle closure conditions for initial states |zi, | −zi or both,
as well as the scenario of equivalence of three successive partial projections to an SU(2) rotation.
References
[1] S Pancharatnam,Proc. Indian Acad. Sci.A44, 247 (1956) [2] M V Berry,J. Mod. Opt.34, 1401 (1987)
[3] J Samuel and S Sinha,Pramana – J. Phys.48, 969 (1997) [4] A G Wagh and S Abbas,Solid State Phys. (India)50, 99 (2005)
