Topological phase for spin-orbit transformations on a laser beam
Topological phase for spin-orbit transformations on a laser beam
C. E. R. Souza†, J. A. O. Huguenin†, P. Milman††, and A.Z. Khoury†
Instituto de F́ısica, Universidade Federal Fluminense, 24210-340 Niterói - RJ, Brasil. and
Laboratoire Matériaux et Phenomènes quantiques CNRS UMR 7162,
Université Denis Diderot, 2 Place Jussieu 75005 Paris cedex.
We investigate the topological phase associated with the double connectedness of the SO(3)
representation in terms of maximally entangled states. An experimental demonstration is provided
in the context of polarization and spatial mode transformations of a laser beam carrying orbital
angular momentum. The topological phase is evidenced through interferometric measurements and
a quantitative relationship between the concurrence and the fringes visibility is derived. Both the
quantum and the classical regimes were investigated.
PACS numbers: PACS: 03.65.Vf, 03.67.Mn, 07.60.Ly, 42.50.Dv
The seminal work by S. Pancharatnam [1] introduced
for the first time the notion of a geometric phase acquired
by an optical beam passing through a cyclic sequence
of polarization transformations. A quantum mechanical
parallel for this phase was later provided by M. Berry
[2]. Recently, the interest for geometric phases was re-
newed by their potential applications to quantum compu-
tation. The experimental demonstration of a conditional
phase gate was recently provided both in nuclear mag-
netic ressonance [3] and trapped ions [4]. Another optical
manifestation of geometric phase is the one acquired by
cyclic spatial mode conversions of optical vortices. This
kind of geometric phase was first proposed by van Enk
[5] and recently found a beautiful demonstration by E. J.
Galvez et al [6].
The Hilbert space of a single qubit admits an useful ge-
ometric representation of pure states on the surface of a
sphere. This is the Bloch sphere for spin 1/2 particles or
the Poincaré sphere for polarization states of an optical
beam. A Poincaré sphere representation can also be con-
structed for the first order subspace of the spatial mode
structure of an optical beam [7]. Therefore, in the quan-
tum domain, we can attribute two qubits to a single pho-
ton, one related to its polarization state and another one
to its spatial structure. Geometrical phases of a cyclic
evolution of the mentioned states can be beautifully in-
terpreted in such representations as being related to the
solid angle of a closed trajectory. However, in order to
compute the total phase gained in a cyclic evolution, one
should also consider the dynamical phase. When added
to the geometrical phase, it leads to a total phase gain
of π after a cyclic trajectory. This phase has been put
into evidence for the first time using neutron interfer-
ence [8]. The appearence of this π phase is due to the
double connectedness of the three dimensional rotation
group SO(3). However, in the neutron experience, only
two dimensional rotations were used, and this topologi-
cal property of SO(3) was not unambiguously put into
evidence, as explained in details in [9, 10].
As discussed by P. Milman and R. Mosseri [9, 11], when
the quantum state of two qubits is considered, the mathe-
matical structure of the Hilbert space becomes richer and
the phase acquired through cyclic evolutions demands a
more careful inspection. The naive sum of independent
phases, one for each qubit, is applicable only for prod-
uct states. In this case, the two qubits are geometrically
represented by two independent Bloch spheres. When
a more general partially entangled pure state is consid-
ered, the phase acquired through a cyclic evolution has
a more complex structure and can be separated in three
contributions: dynamical, geometrical and topological.
Maximally entangled states are solely represented on the
volume of the SO(3) sphere which has radius π and its di-
ametrically opposite points identified. This construction
reveals two kinds of cyclic evolutions, each one mapped
to a different homotopy class of closed trajectories in the
SO(3) sphere. One kind is mapped to closed trajecto-
ries that do not cross the surface of the sphere (0−type)
and the other one is mapped to trajectories that cross
the surface (π−type). The phase acquired by a maxi-
mally entangled state is 0 for the first kind and π for the
second one.
In the present work we demonstrate the topological
phase associated to polarization and spatial mode trans-
formations of an optical vortice. This phase appears first
in the classical description of a paraxial beam with ar-
bitrary polarization state and has its quantum mechan-
ical counterpart in the spin-orbit entanglement of a sin-
gle photon, which constitutes one possible realization of
a two-qubit system and the topological phase discussed
in Ref.[9]. However, it is interesting to observe that,
like the Pancharatnam phase, the two-qubit topological
phase also admits a classical manifestation, since it can
be implemented on the classical amplitude of the opti-
cal field. This is also the first experiment unambiguously
showing the double connectedness of the rotation group
SO(3). The optical modes used in our experiment have
a mathematical structure analog to the one of entangled
states, so that the geometrical representation developped
in [10] also applies and the results of Ref.[9, 11] can be
experimentally demonstrated. When excited with single
photons, these modes give rise to single particle entangled
http://arxiv.org/abs/0704.0893v1
states and provide a more direct relationship with the
ideas put forward in Refs.[9, 10, 11]. This regime is also
investigated in the present work. There are a number of
quantum computing protocols that can be implemented
with single particle entanglement and will certainly ben-
efit from our results.
Let us now combine the spin and orbital degrees of
freedom in the framework of the classical theory in order
to build the same geometric representation applicable to
a two-qubit quantum state. Consider a general first order
spatial mode with arbitrary polarization state:
E(r) = αψ+(r)êH + βψ+(r)êV + γψ−(r)êH
+ δψ−(r)êV , (1)
where êH(V ) are two linear polarization unit vectors
along two orthogonal directions H and V , and ψ±(r)
are the normalized first order Laguerre-Gaussian pro-
files which are orthogonal solutions of the paraxial wave
equation [12]. We may now define two classes of spatial-
polarization modes: the separable (S) and the nonsepa-
rable (NS) ones. The S modes are of the form
E(r) = (α+ψ+(r) + α−ψ+(r)) (βH êH + βV êV ) . (2)
For these modes, a single polarization state can be
atributted to the whole wavefront of the paraxial beam.
They play the role of separable two-qubit quantum
states.
For nonseparable (NS) paraxial modes, the polariza-
tion state varies across the wavefront. As for entangle-
ment in two-qubit quantum states, the separability of a
paraxial mode can be quantified by the analogous defi-
nition of concurrence. For the spin-orbit mode described
by Eq.(1), it is given by:
C = 2 | αδ − βγ | . (3)
Let us first consider the maximally nonseparable
modes (MNS) of the form
E(r) = αψ+(r)êH + βψ+(r)êV − β∗ψ−(r)êH
+ α∗ψ−(r)êV . (4)
For these modes C = 1. It is important to mention that
the concept of entanglement does not applies to the MNS
mode, since the object described by Eq.(4) is not a quan-
tum state, but a classical amplitude. However, we can
build an SO(3) representation of the MNS modes as it
was done in Refs.[11, 13]. Let us define the following
normalized MNS modes:
E1(r) =
[ψ+(r)êH + ψ−(r)êV ] ,
E2(r) =
[ψ+(r)êH − ψ−(r)êV ] , (5)
E3(r) =
[ψ+(r)êV + ψ−(r)êH ] ,
E4(r) =
[ψ+(r)êV − ψ−(r)êH ] .
Laser
HWP-A HWP-B
HWP-1
HWP-2
HWP-3
Pol-V Pol-H
QWP-2
QWP-1
FIG. 1: Experimental setup.
The SO(3) sphere is then constructed in the following
way: mode E1 is represented by the center of the sphere,
while modes E2, E3, and E4 are represented by three
points on the surface, connected to the center by three
mutually orthogonal segments. Each point of the SO(3)
sphere corresponds to a MNS mode. Following the recipe
given in Ref.[13], the coefficients α and β of Eq.(4) are
parametrized to:
α = cos
− i kz sin
β = −(ky + i kx) sin
, (6)
where (kx, ky , kz) = k is a unit vector, and a is an angle
between 0 and π. With this parametrization, each MNS
mode is represented by the vector ak in the sphere.
In order to evidence the topological phase for cyclic
transformations, we must follow two different closed
paths, each one belonging to a different homotopy class,
and compare their phases. The experimental setup is
sketched in Fig.(1). First, a linearly polarized TEM00
laser mode is diffracted on a forked grating used to gen-
erate Laguerre-Gaussian beams [14]. The two side orders
carrying the ψ+(r) and ψ−(r) spatial modes are trans-
mitted through half waveplates HWP-A and HWP-B, fol-
lowed by two orthogonal polarizers Pol-V and Pol-H, and
finally recombined at a beam splitter (BS-1). Half wave-
plates HWP-A and HWP-B are oriented so that their
fast axis are paralell. This allows us to adjust the mode
separability at the output of BS-1 without changing the
corresponding output power, what prevents normaliza-
tion issues.
Experimentally, an MNS mode is produced when both
HWP-A and HWP-B are oriented at 22.5o , so that the
FIG. 2: Interference patterns for a-) a maximally nonsepara-
ble, and b-) a separable mode. From left to right the images
were obtained with QWP-2 oriented at −45o, 0o, and 45o,
respectively.
setup prepares mode E1 located at the centre of the
sphere. Other MNS modes can then be obtained by uni-
tary transformations in only one degree of freedom. Since
polarization is far easier to operate than spatial modes
we choose to implement the cyclic transformations in the
SO(3) sphere using waveplates. The MNS mode E1 is
first transmitted through three waveplates. The first one
(HWP-1) is oriented at 0o and makes the transformation
E1 → E2, the second one (HWP-2) is oriented at −45o
and makes the transformation E2 → E3, and the third
one (HWP-3) is oriented at 90o and makes the transfor-
mation E3 → E4. Finally, two alternative closures of
the path are performed in a Michelson interferometer.
In one arm a π−type closure is implemented by dou-
ble pass through a quarter-waveplate (QWP-1) fixed at
−45o. In the other arm, either a 0−type or a π−type
closure is performed by a double pass through another
quarter-waveplate (QWP-2) oriented at a variable angle
between −45o (π−type) and 45o (0−type). These tra-
jectories are analogous to spin rotations around different
directions of space [13]. They evidence the topological
properties of the three dimensional rotation group.
In order to provide spatial interference fringes, the in-
terferometer was slightly misaligned. The interference
patterns were registered with either a charge coupled
device (CCD) camera or a photocounter (PC), depend-
ing on the working power. First, we registered the
interference patterns obtained when an intense beam
is sent through the apparatus. The images shown in
Fig.(2a) demonstrate clearly the π topological phase
shift. The phase singularity characteristic of Laguerre-
Gaussian beams can be easily identified in the images
and is very useful to evidence the phase shift. When
both arms perform the same kind of trajectory in the
SO(3) sphere (QWP-1 and QWP-2 oriented at −45o), a
bright fringe falls on the phase singularity. When QWP-
2 is oriented at 45o, the trajectory performed in each arm
belongs to a different homotopy class and a dark fringe
falls on the singularity, what clearly demonstrates the π
topological phase shift.
In order to discuss the role played by mode separa-
bility, it is interesting to observe the pattern obtained
when QWP-2 is oriented at intermediate angles, which
correpond to open trajectories in the SO(3) sphere. We
observed that during the phase shift transition, the in-
terference fringes are deformed and finally return to its
initial topology with the π phase shift. This is clearly il-
lustrated by the intermediate image displayed in Fig.(2a),
which corresponds to QWP-2 oriented at 0o . Notice that,
despite the deformation, the interference fringes display
high visibility.
As we mentioned above, the mode preparation settings
can be adjusted in order to provide a separable mode. For
example, when we set HWP-A and B both at 45o , the
output of BS-1 is the separable mode ψ+(r)êH , which
can be represented in the Poincaré spheres for spatial
and polarization modes. The same π phase shift can
be observed when QWP-2 is rotated, but the transition
is essentially different. The intereference pattern is not
topologically deformed, but its visibility decreases until
it completelly vanishes at 0o , and then reappears with
the π phase shift. This transition is clearly illustrated
by the three patterns displayed in Fig.(2b). In this case,
the π phase shift is of purely geometric nature, since the
spatial mode is kept fixed while the polarization mode is
turned around the equator of the corresponding Poincaré
sphere.
The relationship between mode separability and
fringes visibility can be clarified by a straightforward cal-
culation of the interference pattern. Therefore, let us
consider that HWP-A and B are oriented so that the
output of BS-1 is described by
Eǫ(r) =
ǫ ψ+(r)êH +
1− ǫ ψ−(r)êV , (7)
where ǫ is the fraction of the ψ+(r)êH mode in the out-
put power. Now, let us consider that QWP-2 is oriented
at 0o and suppose that the two arms of the Michelson
interferometer are slightly misaligned so that the wave
vectors difference between the two outputs is δk = δk x̂ ,
orthogonal to the propagation axis. Taking into account
the passage through the three half waveplates, and the
transformation performed in each arm of the Michelson
interferometer, we arrive at the following expression for
the interference pattern:
I(r) = 2 |ψ(r)|2
1 + 2
ǫ(1− ǫ) sin 2φ sin (δk x)
, (8)
where φ = arg(x + iy) is the angular coordinate in
the transverse plane of the laser beam, and |ψ(r)|2 is
the doughnut profile of the intensity distribution of a
Laguerre-Gaussian beam. It is clear from Eq.(8) that the
visibility of the interference pattern is 2
ǫ(1− ǫ), which
is precisely the concurrence of Eǫ(r) as given by Eq.(3).
Therefore, the fringes visibility is quantitatively related
to the separability of the mode sent through the setup.
However, the numerical coincidence with the concurrence
0 2 4 6 8 10
Displacement (mm)
FIG. 3: Interference patterns measured in the photocount-
ing regime for ǫ = 1/2 . Empty and full circles correspond
to QWP-2 oriented at −45o and 45o, respectively. Solid and
dashed lines are theoretical fits with sinusoidal functions mod-
ulated by a Laguerre-Gaussian envelope. The phase shift
given by the fits is 3.14 rad .
is restricted to modes of the form given by Eq.(7). In fact,
it is important to stress that the fringes visibility can-
not be regarded as a measure of the concurrence for any
nonseparable mode, but for our purposes it evidences the
topological nature of the phase shift implemented by the
experimental setup. A detailed discussion on the mea-
surement of the concurrence is available in Ref.[15].
Next, we briefly discuss the quantum domain. When
a partially nonseparable mode like Eǫ(r) is occupied by
a single photon, this leads to partially entangled single
particle quantum states of the kind
|ϕǫ〉 =
ǫ |+H〉+
1− ǫ | − V 〉 . (9)
Experimentally, we attenuated the laser beam down to
the single photon regime, and scanned a photocounting
module across the interference pattern. First, HWP-A
and B were set at 22.5o (ǫ = 1/2) in order to evidence
the topological phase in this regime. Fig.(3) displays the
interference patterns obtained with QWP-2 oriented at
−45o and 45o. The π phase shift is again clear.
The relationship between the fringes visibility and the
state separability was evidenced by fixing QWP-2 at
0o and rotating HWP-A and B by an angle θ so that
ǫ = cos2 2θ . Fig.(4) shows the experimental results for
the fringes visibility for several values of ǫ . The solid
line corresponds to the analytical expression of the con-
currence, showing a very good agreement with the exper-
imental values.
As a conclusion, we demonstrated the double con-
nected nature of the SO(3) rotation group and the topo-
logical phase acquired by a laser beam passing through
a cycle of spin-orbit transformations. We investigated
both the classical and the quantum regimes and com-
0,0 0,2 0,4 0,6 0,8 1,0
cos2(2
FIG. 4: Fringes visibility as a function of ǫ. The solid line is
a theoretical fit with C = 2
ǫ(1− ǫ) .
pared the separability of the mode travelling through the
apparatus with the visibility of the interference fringes.
Our results may constitute an useful tool for quantum
computing and quantum information protocols.
The authors are deeply grateful to S.P. Walborn and
P.H. Souto Ribeiro for their precious help with the photo-
counting system and for fruitful discussions. Funding was
provided by Coordenação de Aperfeiçoamento de Pes-
soal de Nı́vel Superior (CAPES), Fundação de Amparo
à Pesquisa do Estado do Rio de Janeiro (FAPERJ-BR),
and Conselho Nacional de Desenvolvimento Cient́ıfico e
Tecnológico (CNPq).
[1] S. Pancharatnam, Proc. Ind. Acad. Sci. 44, 247 (1956).
[2] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).
[3] J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Na-
ture (London) 403, 869 (2000).
[4] L.-M. Duan, J. I. Cirac, and P. Zoller, Science 292, 1695
(2001).
[5] S.J. van Enk, Opt. Comm. 102, 59 (1993).
[6] E. J. Galvez et al, Phys. Rev. Lett. 90, 203901 (2003).
[7] M. J. Padgett and J. Courtial, Opt. Lett. 24, 430 (1999).
[8] S. A. Werner et al, Phys. Rev. Lett. 35, 1053 (1975).
[9] P. Milman, and R. Mosseri, Phys. Rev. Lett. 90, 230403
(2003).
[10] R. Mosseri, and R. Dandoloff, J. Phys. A 34, 10243
(2003).
[11] P. Milman, Phys. Rev. A 73, 062118 (2006).
[12] A. Yariv, ”Quantum Electronics”, John Wiley & Sons,
third ed. (1988).
[13] W. LiMing, Z. L. Tang, and C. J. Liao, Phys. Rev. A 69,
064301 (2004).
[14] N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G.
White, Opt. Lett. 17, 221 (1992); G.F. Brand, Am. J. of
Phys. 67, 55 (1999).
[15] S. Walborn, P. H. Souto Ribeiro, L. Davidovich, F.
Mintert, and A. Buchleitner, Nature 440, 1022 (2006).
