PONTIFICIA UNIVERSIDAD CATO´LICA DEL PERU´
ESCUELA DE POSGRADO
Polarimetric measurements of single-photon
geometric phases
A thesis in candidacy for the degree of Master of Science in Physics
presented by:
Omar H. Ortiz Cabello
Advisor:
Prof. Francisco De Zela
Jury:
Prof. Eduardo Massoni
Prof. Herna´n Castillo
Lima, 2017
Resumen
Reportamos medidas polarime´tricas de fases geome´tricas que son generadas por
evoluciones de fotones polarizados a lo largo de trayectorias no geode´sicas en la
esfera de Poincare`. La esencia de nuestro arreglo polarime´trico esta´ en las siete
la´minas retardadoras que son atravesadas por un haz de fotones individuales. Con
este arreglo, cualquier transformacio´n SU(2) puede ser realizada. Explotando la
invarianza Gauge de las fases geome´tricas bajo transformaciones locales U(1), anu-
lamos la contribucio´n dina´mica a la fase total, de este modo haciendo que la u´ltima
coincida con la fase geome´trica. Demostramos la insensibilidad de nuestro arreglo
bajo distintas fuentes de ruido. Esto hace del arreglo polarime´trico de fotones indi-
viduales una herramienta versa´til y prometedora para probar la robustez de la fase
geome´trica frente al ruido.
ii
Abstract
We report polarimetric measurements of geometric phases that are generated by
evolving polarized photons along nongeodesic trajectories on the Poincare` sphere.
The core of our polarimetric array consists of seven wave plates that are traversed by
a single-photon beam. With this array, any SU(2) transformation can be realized.
By exploiting the gauge invariance of geometric phases under U(1) local transfor-
mations, we nullify the dynamical contribution to the total phase, thereby making
the latter coincide with the geometric phase. We demonstrate our arrangement to
be insensitive to various sources of noise entering it. This makes the single-beam,
polarimetric array a promising, versatile tool for testing robustness of geometric
phases against noise.
iii
Acknowledgements
I would like to express my deepest gratitude to my mother Nelly Cabello Ormachea.
I am the proud son of a single mother whose efforts put me in this place and time.
I am the result of her teachings, everything I am today and everything I will be is
because of her.
I would also like to wish my thanks to Professor Francisco De Zela, who trusted
a group of students like me the privilege to work with him, his guidance has been
indispensable for this endeavor.
To my friends I am sincerely grateful for all the moments shared. There is not
enough space here to mention you all, but from physics I must mention Juan Pablo
Velasquez, Juan Carlos Loredo and Carlos Argu¨elles. I would also like to thank my
dear brother Roberto Rodriguez.
To my love and soon to be wife Jimena Vargas, I wish to thank you for all the
happiness you bring me, all those enriching conversations and most important I
wish to thank you for challenging me to be a better person every day. It is a
beautiful path the one we are walking together.
iv
Published Material in this Thesis
Partial and final results contained in this thesis work have been published in the
following journal articles and proceedings:
J. C. Loredo, O. Ort´ız, R. Weinga¨rtner and F. De Zela: Measurement of Pan-
charatnam’s Phase by Robust Interferometric and Polarimetric Methods Phys.
Rev. A 80 012113 (2009).
J. C. Loredo, O. Ort´ız, A. Ballo´n and F. De Zela: Measurement of Geometric
Phases by Robust Interferometric Methods J. Phys.: Conf. Ser. 274 012140
(2011).
O. Ort´ız, Y. Yugra, A. Rosario, J. C. Sihuincha, J. C. Loredo, M. V. Andre´s and
F. De Zela: Polarimetric measurements of single-photon geometric phases Phys.
Rev. A 89 012124 (2014).
v
PHYSICAL REVIEW A 89, 012124 (2014)
Polarimetric measurements of single-photon geometric phases
O. Ortı´z,1 Y. Yugra,1 A. Rosario,1 J. C. Sihuincha,1 J. C. Loredo,2 M. V. Andre´s,3 and F. De Zela1
1Departamento de Ciencias, Seccio´n Fı´sica, Pontificia Universidad Cato´lica del Peru´, Apartado 1761, Lima, Peru
2Centre for Engineered Quantum Systems, Centre for Quantum Computer and Communication Technology,
and School of Mathematics and Physics, University of Queensland, 4072 Brisbane, Queensland, Australia
3Departamento de Fı´sica Aplicada y Electromagnetismo, Universidad de Valencia, c/Dr. Moliner 50, Burjassot, Valencia, Spain
(Received 9 October 2013; published 24 January 2014)
We report polarimetric measurements of geometric phases that are generated by evolving polarized photons
along nongeodesic trajectories on the Poincare´ sphere. The core of our polarimetric array consists of seven wave
plates that are traversed by a single-photon beam. With this array, any SU(2) transformation can be realized. By
exploiting the gauge invariance of geometric phases under U(1) local transformations, we nullify the dynamical
contribution to the total phase, thereby making the latter coincide with the geometric phase. We demonstrate our
arrangement to be insensitive to various sources of noise entering it. This makes the single-beam, polarimetric
array a promising, versatile tool for testing robustness of geometric phases against noise.
DOI: 10.1103/PhysRevA.89.012124 PACS number(s): 03.65.Vf, 03.67.Lx, 42.65.Lm
I. INTRODUCTION
Even though experiments testing different properties of ge-
ometric phases are continuously reported, theoretical develop-
ments can expand at such an accelerated pace that experimental
testing can be temporarily left behind. This seems to be the case
with the subject of geometric phases. Since Berry’s seminal
work [1], which brought to light the appearance of geometric
phases in adiabatically evolving, cyclic quantum processes,
there have been considerable generalizations of the subject.
From Hannay angles in the classical domain [2] to geometric
phases in mixed quantum states subjected to nonunitary and
noncyclic evolutions [3–7], the original concept of geometric
phases has been widely expanded. Experimental testing is
required not only because of fundamental reasons lying at the
basis of all empirical sciences, but because experimental input
can help us to find the answer to open questions. Notably,
the question about a proper, self-consistent definition of a
geometric phase for nonunitary evolutions still remains open
[8–13]. Similarly, the kind of robustness that geometric phases
might have against decohering mechanisms is also an open
question of utmost importance, particularly in the realm of
quantum computation [14]. It is thus useful to explore as
many experimental techniques as possible. One should not
refrain from mirroring experiments already performed with
one technique and conduct similar experiments based on
another independent technique. This can provide not only
new insights, but an enlarged versatility as well. Geometric
phases are particularly well suited for such an approach, as
they notoriously appear in the evolution of two-level systems.
Such systems can be realized under manifold situations, i.e.,
quantal and classical ones. The drawbacks of one technique
could then be replaced by some advantages of the other.
For example, the physical realization of the qubit as a spin
one-half particle, e.g., a neutron, has its counterpart in the
realization of the qubit as a polarized photon. While as
a source of the former, one needs a nuclear reactor, as a
source of the latter, a diode-laser suffices. On the other
hand, the versatility reached in experiments with neutrons
can outperform that reached with their optical counterparts.
A challenge is thereby put on the latter as to how to improve
their versatility. We have addressed such a challenge in the
present work. We report on experiments performed with single
photons, which to some extent mirror previous experiments
that were conducted with neutrons [15–18]. Our experiments
put under test theoretical predictions about SU(2) evolutions
along nongeodesic paths. Using neutrons, experiments along
these lines have been conducted by exploiting the advantages
offered by polarimetric techniques. In contrast to interfero-
metric techniques [19], polarimetric ones have an intrinsic
robustness because they require a single beam [20]. The
challenge posed here, however, is how to manipulate two
coherently superposed states that are not spatially separated.
In interferometry, the (binary) path degree of freedom can
be used together with an “internal” degree of freedom, e.g.,
the spin, that is carried along by the particle. In polarimetry,
instead, there is only one path. One must then figure out how
to deal with this restriction and nevertheless reach a versatility
that is comparable to that of interferometry. The latter offers,
for example, the possibility of spin-path entanglement. In
neutron polarimetry, energy-polarization entanglement and
even a tripartite energy-polarization-momentum entanglement
have been achieved [21]. Although an all-optical version of
the latter seems difficult to implement, there are other features
that can be exploited with advantage in optical polarimetry.
We show here how to exploit the invariance of geometric
phases under local gauge transformations [22] in order to
nullify the dynamical part of the total (Pancharatnam) phase
[23], thereby making this phase coincide with the geometric
phase. What is meant by gauge invariance is the invariance
under the change |ψ(s)〉 → |ψ ′(s)〉 = exp [iα(s)] |ψ(s)〉 of an
unitarily evolving state |ψ(s)〉. By exploiting this invariance,
one can nullify the dynamical contribution to the total
phase, P = arg〈ψ(s1)|ψ(s2)〉, between an initial and a final
state, |ψ(s1)〉 and |ψ(s2)〉, respectively. What remains after
elimination of the dynamical part is the purely geometric
contribution g to the total phase, P = g + dyn. The
SU(2) evolutions we have addressed are those of the type
given by Un(θ,ϕ,s) = exp [−isn(θ,ϕ) · σ/2]. Here, n is a
unit vector, σ is the triple of Pauli matrices, and s is the
1050-2947/2014/89(1)/012124(8) 012124-1 ©2014 American Physical Society
O. ORT´IZ et al. PHYSICAL REVIEW A 89, 012124 (2014)
rotation angle (on the Bloch or Poincare´ sphere). We could
generalize our approach so as to deal with unit vectors that
depend on s, but we have focused on cases with a fixed n.
We also restricted ourselves to deal with pure single-photon
states. These restrictions are justified in view of the extension
already achieved by considering the production of geometric
phases in systems subjected to transformations Un(θ,ϕ,s) of
the above type. Previous experimental tests were restricted to
particular trajectories that a system follows when subjected to
some special transformations [15,17,18]. The cases we address
here let us study what happens when we lift these restrictions.
In such a case, a series of features shows up that is worthwhile
to analyze before undertaking a systematic investigation of,
say, the sensitivity of geometric phases to environmental
influences. A main motivation of the present work was to
analyze and explain the appearance of the aforementioned
features. This opens the way for using this array as a basic
component for testing the impact of decohering mechanisms.
II. POLARIMETRY
The standard procedure to exhibit the relative phase
between two states is to make them interfere and then record
the intensity of the interfering pattern by varying the relative
phase. An archetypical setup for doing this is a Mach-Zehnder
interferometer. Expressed in the language of quantum gates
[24], such a device consists of two Hadamard gates—i.e.,
two beam splitters—and a phase shifter. A Hadamard gate
can be represented in terms of Pauli matrices as UH =
(σx + σz)/
√
2, while the phase shifter can be represented
as Uφ = exp(−iφσz/2). Hereby, we establish a one-to-one
correspondence between the eigenvectors |±〉 of σz and the two
paths of the interferometer. The action of the interferometer
on an input state |+〉 is thus given by |+〉 → UHUφUH |+〉.
The output intensity that is recorded at, say, a |+〉 detector
reads I = |〈+|UHUφUH |+〉|2 = (1 + cosφ)/2. Now, instead
of assigning the states |±〉 to the two possible paths of the inter-
ferometer, we can make them correspond to the horizontal and
vertical polarization states of a single light beam. We thereby
change from interferometry to polarimetry. In the latter, the
action of Uφ and UH can be realized with the help of quarter-
wave (Q) and half-wave (H ) plates. Indeed, we have that Uφ =
Q(π/4)H ((φ − π )/4)Q(π/4) and UH = −iH (π/8). The ar-
guments in H and Q refer to the angles made by the plate’s
major axis and the vertical direction. Up to a global phase,
the action of the Mach-Zehnder interferometer can then be
mirrored in polarization space by letting a polarized light beam
traverse a gadget that consists of a couple of aligned retarders.
In the present case, such an array is given by Q(π/2)H ((2π −
φ)/4)Q(π/2). This last expression is obtained by us-
ing Q(α)H (β) = H (β)Q(2β − α) and Q(α)H (β)H (γ ) =
Q(α + π/2)H (α − β + γ − π/2). Hence, by setting a hor-
izontal polarizer before a detector and recording the intensity
as a function of φ, we get a pattern that looks the same as
the interferogram produced with the Mach-Zehnder device.
Polarimetry has the great advantage of being largely insensitive
to those perturbations that in the case of interferometry lead to
random phase shifts. On the other hand, the states |±〉 cannot
be individually addressed, as they are no longer spatially sep-
arated from one another, as occurs in interferometry. We must
then find a way to extract the desired information by adequately
projecting the manipulated states before detection. In the case
of geometric phases, this is indeed possible, as we show next.
Following a similar procedure as the one introduced
by Wagh and Rakhecha [20]—thereby extending to single
photons some techniques already employed with classical
light [25–27]—we consider an initial, horizontally polarized
state |h〉 and submit it to a π/2 rotation around the x axis.
This produces a circularly polarized state (|h〉 − i |v〉) /√2.
By submitting this state to the transformation exp (−iφσz/2),
we get V |h〉 ≡ exp (−iφσz/2) exp (−iπσx/4) |h〉, which is
the state (|h〉 − ieiφ |v〉)/√2, up to a global phase. Hence,
we have generated a relative phase shift φ − π/2 between
|h〉 and |v〉. If we now apply U ∈ SU(2), we then ob-
tain UV |h〉 = (e−iφ/2U |h〉 − ieiφ/2U |v〉)/√2. We are inter-
ested inUn(θ,ϕ,s) = exp [−isn(θ,ϕ) · σ/2] and the geometric
phase that this transformation generates. We recall that the
geometric phase is given by [22]
g(C) = arg〈ψ(0)|ψ(s)〉 − Im
∫ s
0
〈ψ(s ′)| ˙ψ(s ′)〉ds ′, (1)
for a path C joining the initial state |ψ(0)〉 with the fi-
nal state |ψ(s)〉. As already said, g is invariant under
local gauge transformations. We exploit this property in
order to nullify the dynamical contribution to g . That
is, we choose a gauge transformation |ψ(s)〉 → |ψ ′(s)〉 =
exp [iα(s)] |ψ(s)〉 so that 〈ψ ′(s)| ˙ψ ′(s)〉 = 0. In other words,
instead of applying Un(θ,ϕ,s), we apply exp [iα(s)]Un(θ,ϕ,s)
and measure the total phase arg〈ψ(0)|ψ(s)〉. In the present
case, this can be achieved by setting α(s) = s 〈+| n · σ |+〉 /2.
That is, we seek to implement the transformation |h〉 →
UnV |h〉 = (e−iγ /2Un |h〉 − ieiγ /2Un |v〉)/
√
2, where γ (s) =
φ − α(s). We can realize this with the help of wave plates.
To begin with, Un can be implemented with a gadget proposed
by Simon and Mukunda [28], which is given by
Un(θ,ϕ,s) = Q
(
π + ϕ
2
)
Q
(
θ + ϕ
2
)
×H
(−π + θ + ϕ
2
+ s
4
)
Q
(
θ + ϕ
2
)
Q
(ϕ
2
)
.
(2)
The rotation axis is here given by n =
(sin θ cosϕ, sin θ sinϕ, cos θ ) and the Pauli matrices are de-
fined according to the convention that is commonly employed
in optics. That is, the diagonal matrix in the basis {|h〉,|v〉}
of horizontally and vertically polarized states is σx . The other
two Pauli matrices follow from cyclically completing the
change σz → σx . With this choice, our gauge is given by
α(s) = s
2
sin θ cosϕ. (3)
On the other hand, V (γ ) = e−iγ σz/2e−iπσx/4 can be
implemented as V (γ ) = Q(π/4)H ((γ − π )/4)H (π/4).
The total transformation is thus
Utot ≡ V †UnV = H
(
−π
4
)
H
(
γ + π
4
)
Q
(
−π
4
)
×Un(θ,ϕ,s)Q
(π
4
)
H
(
γ − π
4
)
H
(π
4
)
. (4)
012124-2
POLARIMETRIC MEASUREMENTS OF SINGLE-PHOTON . . . PHYSICAL REVIEW A 89, 012124 (2014)
Applying as before relations such as Q(α)H (β) = H (β)Q(2β − α), Q(α)H (β)H (γ ) = Q(α + π/2)H (α − β + γ − π/2),
etc., we reduce the above array to one that consists of seven plates:
Utot(θ,ϕ,φ,s) = Q
[
π
4
− γφ(s)
2
]
Q
[
−π − ϕ
2
− γφ(s)
2
]
Q
[
π − θ − ϕ
2
− γφ(s)
2
]
H
[−θ − ϕ
2
− s
4
− γφ(s)
2
]
×Q
[
π − θ − ϕ
2
− γφ(s)
2
]
Q
[
π − ϕ
2
− γφ(s)
2
]
Q
[
−π
4
− γφ(s)
2
]
, (5)
where γφ(s) = φ − α(s). We use this notation to emphasize
that γ depends on both φ and s. Note that by going from
Eq. (4) to Eq. (5), the gauge-fixing role—originally played
by the plates implementing V (γ )—turns to be shared by
all seven plates of the final array. The path followed by the
polarization state subjected to Utot can be represented on the
Poincare´ sphere by a circular arc; see Fig. 1. This arc is
fixed by n(θ,ϕ), by the initial polarization state, and by s.
The latter fixes the angle by which the initial state is rotated.
Once we have fixed n and the initial state, we record the
geometric phase as a function of s. This is done by varying
the registered intensity as a function of γφ(s), which plays a
double role. First, it contains the phase shift φ that is required
to implement the polarimetric version of the Mach-Zehnder
interferometer, as discussed above. Second, it contains the
gauge shift α(s) that is required to make the total phase
coincide with the geometric phase. In order to extract this
geometric phase, we project the state UnV (γ ) |h〉 onto the state
V (γ ) |h〉 = e−iγ /2(|h〉 − ieiγ |v〉)/√2. The recorded intensity
is thus given by I = |〈h|V †(γ )UnV (γ )|h〉|2. As we shall see,
after having fixed θ , ϕ, and s, we can let γ (viz., φ) vary so
as to generate an intensity pattern I (φ), whose maxima and
minima determine the value of the geometric phase at (s,θ,ϕ).
This value can be compared with the theoretical one, which is
FIG. 1. (Color online) Path followed on the Poincare´ sphere by
the Stokes vector that corresponds to an initial state |h〉 being
submitted to a transformation exp(−isn · σ/2). The rotation axis
n has polar angles θ = π/3,ϕ = π/4. The dynamical contribution
to the total phase P is gauged-away all along the curve, so that
P = g holds at each value of s.
given by g = P − dyn, where
P = arg〈ψ(0)|ψ(s)〉 = arg 〈h|Un(s) |h〉
= − arctan
[
sin θ cosϕ tan
(
s
2
)]
, (6)
dyn = Im
∫ s
0
〈ψ(s)| ˙ψ(s)〉ds
= Im
∫ s
0
〈h|U †n(s)(−in · σ )Un(s) |h〉 ds
= − s
2
〈h| n · σ |h〉 . (7)
The theoretical expression for the geometric phase thus reads
thg = − arctan
[
sin θ cosϕ tan
(
s
2
)]
+ s
2
sin θ cosϕ. (8)
On the other hand, a straightforward calculation of the intensity
I = |〈h|V †[φ − α(s)]Un(θ,ϕ,s)V [φ − α(s)]|h〉|2 gives
I = cos2
(
s
2
)
+ sin2
(
s
2
)
{cos θ cos[α(s) − φ]
+ sin θ sinϕ sin[α(s) − φ]}2. (9)
We then have
Imin(s) = cos2
(
s
2
)
, (10)
Imax(s) = cos2
(
s
2
)
+ sin2
(
s
2
)
[cos2 θ + (sin θ sinϕ)2],
(11)
where we have used that the maximum of f (α) = a cosα +
b sinα is given by
√
a2 + b2. From the above equations, we get
1 − Imax
1 − Imin = sin
2 θ cos2 ϕ, (12)
1 − Imax
Imin
= sin2 θ cos2 ϕ tan2
(
s
2
)
. (13)
We can thus express thg in terms of the experimentally
accessible quantities Imin and Imax as
g(s) =
√
1 − Imax(s)
1 − Imin(s) arccos[
√
Imin(s)]
− arctan
[√
1 − Imax(s)
Imin(s)
]
for −π < s < π,
(14)
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O. ORT´IZ et al. PHYSICAL REVIEW A 89, 012124 (2014)
g(s) =
√
1 − Imax(s)
1 − Imin(s) arccos[−
√
Imin(s)]
+ arctan
[√
1 − Imax(s)
Imin(s)
]
±π for π < s < 3π.
(15)
Note that g is undefined for s = π ; cf. Eq. (8).
The ±π that appears in g(s > π ) comes from
the Pancharatnam contribution, arg 〈h|Un(s) |h〉,
that is contained in thg . Indeed, 〈h|Un(s) |h〉 =
cos(s/2)[1 − i sin θ cosϕ tan(s/2)], so that arg 〈h|Un(s) |h〉 =
arg [cos(s/2)] − arctan[sin θ cosϕ tan(s/2)]. For π < s
< 3π, we have that arg [cos(s/2)] = ±π .
III. EXPERIMENTAL PROCEDURE AND ANALYSIS
OF RESULTS
A sketch of our experimental arrangement is shown in
Fig. 2. Its core is the array of seven plates that realize the
transformation Utot(θ,ϕ,φ,s), as given in Eq. (5). Our single-
photon source was a beta barium borate (BBO) crystal pumped
by a cw diode laser (measured central wavelength 400 nm;
spectral linewidth lies between 0.5 and 1 nm at operating
temperatures; output power 37.5 mW). Two photon beams
were produced in the BBO crystal by type-I spontaneous para-
metric down-conversion, with each beam having a wavelength
of 800 nm. One beam, the idler or heralding one, was directed
towards an avalanche photodetector. The other, signal beam,
was directed towards the array of seven plates. Coincidence
SPCM FPGA PC
BBO
H
CLF
PUtot
L
F
M
P
FIG. 2. Polarimetric array. The set of seven wave plates shown
at the bottom can be oriented so as to realize the desired SU(2)
transformation (Utot) in polarization space. Polarized photons enter
this array after having been produced in a nonlinear, beta barium
borate (BBO) crystal that is fed by a diode laser (L) that emits 400 nm
light whose polarization is fine tuned with a λ/2 plate (H ) placed
before the crystal. Polarizers (P ) set before and after the retarders
project the photon’s polarization as required (see text). Signal photons
are recorded in coincidence with their heralding twins in a single-
photon counting module (SPCM). Other components are M: mirrors;
CL: converging lenses; F: filters, FPGA: field programmable gate
array; and PC: personal computer.
counts (I ) of idler and signal beams made up our raw data,
with coincidences being defined within a time window of
10.42 ns. Our photon-counting module was a Perkin-Elmer
SPCM-AQ4C, with a dark count rate of 500 ± 10 cps. Photons
were collected with the help of converging lenses that focused
them into multimode fiber-optic cables having fiber-coupling
connectors at both ends. The recorded coincidences were
obtained according to the following procedure. For given
values of θ , ϕ, and s, the seven plates were oriented as
prescribed in Eq. (5), with γ = φ − s sin θ cosϕ/2. The angle
φ was varied from 0◦ to 360◦ in steps of 40◦. Coincidence
counts were recorded as a function of φ and then normalized
to obtain the intensity I (φ). Theoretically, I (φ) is given by
Eq. (9), with s, θ , and ϕ kept fixed. By repeated measurements,
we sampled 30 points for each value of φ. The parameter s took
values si from 40◦ to 320◦ in steps of 40◦. After averaging the
recorded coincidence counts for each φ, we obtained a series of
points I (φi). A best fit I (φ) to these points was found, where
I (φ) is a sinusoidal function whose parameters were fixed
by the least-squares method. Figure 3 shows the so-obtained
curves for θ = π/2, ϕ = π/3 and different values of s. From
these curves, we determined Imax and Imin. By entering Imax and
Imin in Eqs. (14) and (15), the experimental values ofg(s,θ,ϕ)
can be obtained and compared with the ones predicted by
Eq. (8). Figure 4 shows our experimental results together
with the corresponding theoretical predictions. As can be
seen, two of the three cases seem to reflect a systematic
departure of our experimental findings from the theoretical
predictions. We will come back to this point below. As for
the single-photon production, it was checked by the standard
procedure [29,30] of measuring the degree of second-order
coherence, g(2), between the output fields of a beam splitter,
i.e., the reflected (R) and transmitted (T) beams. Detections at
gates T and R were conditioned upon detection at a third gate G.
In such a case, g(2) = PGTR/(PGT PGR), where the Pa denote
probabilities for simultaneous detection at gates specified by
label a. In terms of photo counts Na , the degree of coherence
can be expressed as [31] g(2) = NGTRNG/(NGT NGR). It has
a value that is less than 1 for nonclassical light. We obtained
g(2) = 0.187 ± 0.011 in our experiments.
Several sources of experimental error could be identified.
The main source of error came from the accuracy with which
our plates could be oriented, i.e., approximately ±1◦. Another
possible source of error came from our photons having a
wavelength of 800 nm instead of the 808 nm that would be
required for optimal performance of our wave plates. These are
zero-order plates whose effective retardances at the produced
wavelength made them slightly differ from being λ/2 and λ/4
plates. However, the corresponding departures (0.505λ instead
of λ/2 and 0.253λ instead of λ/4) were small enough to be
neglected as a sensible source of error. Accidental coincidence
counts were also estimated to be too small (contribution to g(2)
less than 0.19) for them to have a noticeable influence on the
departures of our experimental findings from the theoretically
predicted values when s > π (see Fig. 4, middle and right
panels). As illustrated in Fig. 4, left panel, the agreement
between the theoretical predictions and measured values was
very good. However, we also observed slight departures that
occasionally increased. The dashed curves in Fig. 4, middle
and right panels, correspond to the targeted geometric phase
012124-4
POLARIMETRIC MEASUREMENTS OF SINGLE-PHOTON . . . PHYSICAL REVIEW A 89, 012124 (2014)
0 Π
2
Π 3 Π
2
2 Π
0
0.5
1
Μ
I
s40°
0 Π
2
Π 3 Π
2
2 Π
0
0.5
1
Μ
I
s80°
0 Π
2
Π 3 Π
2
2 Π
0
0.5
1
Μ
I
s120°
0 Π
2
Π 3 Π
2
2 Π
0
0.5
1
Μ
I
s160°
0 Π
2
Π 3 Π
2
2 Π
0
0.5
1
Μ
I
s200°
0 Π
2
Π 3 Π
2
2 Π
0
0.5
1
Μ
I
s240°
0 Π
2
Π 3 Π
2
2 Π
0
0.5
1
Μ
I
s280°
0 Π
2
Π 3 Π
2
2 Π
0
0.5
1
Μ
I
s320°
FIG. 3. (Color online) The geometric phase is experimentally fixed by the maxima and minima of the measured curves Iexp(φ). The plotted
curves correspond to θ = π/3, ϕ = π/3.
g(s,θ,ϕ). Large departures seemed to reflect a drift of the
measured values with respect to the assumed theoretical curve,
rather than random fluctuations around this curve. In what
follows, we substantiate our claim that the ±1◦ accuracy
in the orientation of our plates does explain occasional,
systematic departures of experimental measurements from
theoretical predictions. Depending on the measured quantity,
rotation errors of this magnitude can give rise to inaccuracies
of various sorts, such as those recently reported in [32].
It is important to identify error sources and their effects,
especially when one’s ultimate goal is to have a good
understanding of how the geometric phase behaves in a noisy
environment.
Let us denote by δi the departure of the ith plate’s
orientation from its nominal value. For a quarter-wave plate,
we must then set Q(x + δ) instead of Q(x) in Eq. (5). To
first order in δ, we get dQ(x) = Q(x + δ) − Q(x) = √2iδRx ,
with
Rx =
(
sin(2x) −cos(2x)
−cos(2x) −sin(2x)
)
. (16)
Similarly, for a half-wave plate, we obtain dH (x) = H (x +
δ) − H (x) = 2iδRx . If we now replace the operators Q(x)
and H (x) in Eq. (5) by Q(x) + dQ(x) and H (x) + dH (x),
respectively, and then expand the result to first order in the δi ,
we obtain
Uδtot = Utot +
7∑
i=1
Uδi , (17)
whereUδi reads likeUtot [see Eq. (5)], except that its ith factor is
replaced by dH (x) when i = 4 and by dQ(x) otherwise. Uδtot is
then a function of all δi=1,...,7. From the amplitude 〈h|Uδtot |h〉,
we can calculate the total intensity Iδ =
∣∣〈h|Uδtot |h〉∣∣2, once
again to first order in the δi . With this expression, by choosing
different values for the δi , we can study how much Iδ(φ)
differs from the I (φ) given in Eq. (9). We have found that
the departures from I can be very sensitive to a change from,
say, δi ≈ +1◦ to δi ≈ −1◦, keeping fixed all the other δj =i .
The values of Imax and Imin can be calculated using I (φ) and
Iδ(φ) in order to assess the sensitivity of the array to changes
δi ≈ ±1◦ in the setting of the plates. The values of Imax and
0 4 Π
9
8 Π
9
4 Π
3
16 Π
9
1.5
1
0.5
0
s
g
2, 3
0 4 Π
9
8 Π
9
4 Π
3
16 Π
9
1.5
1
0.5
0
s
g
3 4
0 4 Π
9
8 Π
9
4 Π
3
16 Π
9
1.5
1
0.5
0
s
g
3 3
FIG. 4. (Color online) Geometric phase g(s,θ,ϕ) as a function of parameter s for three choices of (θ,ϕ). Curve g(s,π/2,π/3) closely
matches experimental results. However, g(s,π/3,π/4) and g(s,π/3,π/3) seem to systematically deviate from the measured values. By
properly identifying the actual values of (θ,ϕ), the theoretical curves do match experimental results. Dashed curves correspond tog(s,π/3,π/4)
(middle panel) and tog(s,π/3,π/3) (right panel). Full curves correspond tog(s,π/3 + δθ,π/4 + δϕ) with δθ = 3◦π/180◦, δϕ = −7◦π/180◦
(middle panel) and tog(s,π/3 + δθ,π/3 + δϕ) with δθ = 5◦π/180◦, δϕ = −4◦π/180◦ (right panel). Most error bars are smaller than symbols.
012124-5
O. ORT´IZ et al. PHYSICAL REVIEW A 89, 012124 (2014)
Imin that correspond to Iδ(φ) show that inaccuracies δi ≈ ±1◦
can explain the observed differences between recorded phases
and theoretically predicted ones; cf. Eqs. (14) and (15).
The last claim can be confirmed by the following, in-
dependent approach. Inaccuracies δi ≈ ±1◦ should translate
into a departure of θ and ϕ from their nominal values. Let
us then assume that our array does not realize the trans-
formation Un(θ,ϕ,s) = exp [−isn(θ,ϕ) · σ/2], but instead
exp [−isn(θ + δθ,ϕ + δϕ) · σ/2], with δθ ≈ ±7◦ ≈ δϕ. The
actual values of δθ and δϕ can be obtained by the following
procedure. From Eq. (12), we see that Imax(si) and Imin(si)
corresponding to targeted values θ and ϕ should satisfy
y(si) ≡ 1 − Imax(si)1 − Imin(si) = sin
2 θ cos2 ϕ ≡ f (θ,ϕ). (18)
The above equation can be used to determine the actual
values of θ and ϕ, i.e., θ + δθ and ϕ + δϕ, by the least-
squares method. To this end, we evaluate the right-hand
side of Eq. (18) in the sought-after values, expand it
to first order, i.e., we set f (θ + δθ,ϕ + δϕ) = f (θ,ϕ) +
(sin 2θ cos2 ϕ)δθ − (sin2 θ sin 2ϕ)δϕ, and then determine δθ ,
δϕ as (
δθ
δϕ
)
= (AT WA)−1AT Wb. (19)
Here, (·)−1 means the Moore-Penrose pseudoinverse, b is
the column vector [y(si) − f (θ,ϕ)]T , with i = 1, . . . ,n (n
is the number of recorded points), A is the n × 2 matrix
whose rows are all equal to (sin 2θ cos2 ϕ,−sin2 θ sin 2ϕ),
and W is the inverse of the covariance matrix, i.e., W =
diag(σ−21 , . . . ,σ−2n ). The latter corresponds to statistically
uncorrelated measurements having different variances σi at
different values si . We have assessed these variances in two
different ways: first by fitting a Gaussian to the distributions of
measured points (cf. Fig. 3), which gives us σi for each value
I (φj ) and hence for Imin, Imax, and g by error propagation.
Second, from our raw data, which consists of 30 values for
each φi – with s, θ , ϕ being kept fixed, we randomly chose
10 values for each φi and calculated g as we did when
using the 30 values. By iterating this procedure several times
(≈40), we got a series of values for each g(s,θ,ϕ). From
each series, we obtained a mean value and its corresponding
maximal and minimal departures. These departures constitute
our error bars. Such an estimation is justified by the statistical
independence of our measurements. Thus, randomly sampling
10 out of 30 measured values amounts to having recorded
10 values in each run of the experiment, while repeating
it several times (≈40). From the two methods, we observe
that our measured values σi span a range that goes from a
minimum of 1.3 × 10−4 to a maximum of 0.12. The plotted
error bars (cf. Fig. 4) are mostly smaller than the symbols and
can barely be seen only in cases for which σi ≈ 0.1. Now,
the above-mentioned application of the least-squares method
holds whenever inaccuracies δsi of the si can be neglected.
In our case, the nominal value of s enters in the orientations
of our wave plates and the inaccuracies of these orientations
are precisely the assumed main source of errors. Nevertheless,
the above application of the least-squares method is justified.
Indeed, we can assess the values of the δsi by using Eq. (10).
That is, we set δsi ≈ |si − 2 arccos (
√
Imin(si))| as an estimator
of the inaccuracies of the si . These inaccuracies turn out to
be negligible in comparison to our σi—besides, if they were
not, they would modify the above results only to higher order
than the first in (δθ,δϕ) because our y(si) do not depend
on s, as Eq. (18) shows. The least-squares method can thus
be iteratively applied to find successive values of δθ and
δϕ, until thg (s,θ + δθ,ϕ + δϕ) of Eq. (8) eventually matches
experimental results. In the present case, however, it proved
more practical to seek the right choice of θ and ϕ by hand, i.e.,
by trial and error when plottingthg (s,θ + δθ,ϕ + δϕ) together
with its measured values. Indeed, by doing so in the cases
of Fig. 4, middle and right panels, we quickly found values
δθ ≈ ±7◦ ≈ δϕ for which the theoretical curves very closely
approximate our experimental results. Figure 4 shows the
curves obtained with δθ = 3◦, δϕ = −7◦ (middle panel) and
δθ = 5◦, δϕ = −4◦ (right panel). Such a result is consistent
with the assumed errors δi ≈ ±1◦, which may accumulate so as
to produce inaccuracies δθ ≈ ±7◦ ≈ δϕ. Thus, departures of
θ and ϕ from their targeted values do explain our experimental
findings. We have thereby assessed the amount by which the
theoretically predicted value thg (s,θ,ϕ) might differ from the
experimentally realized one. Such a difference should be taken
into account when assessing, with the help of a polarimetric
array, the robustness of g against decohering mechanisms.
Finally, let us point out the following feature of our array.
As can be seen from Eqs. (14) and (15), the geometric phase we
produce depends on θ and ϕ only through |sin θ cosϕ|. This
means that we can fix the actually realized values of θ and
ϕ only up to changes (θ,ϕ) → (θ ′,ϕ′) that leave |sin θ cosϕ|
invariant. Instead of seing this as a weakness of our approach,
such a feature can be helpful when seeking to exploit the
robustness of g against decoherence. Indeed, if one is able to
confine decohering effects to those regions in the plane (θ,ϕ)
for which the variations in |sin θ cosϕ| are sufficiently small,
then g will vary also within acceptable limits. Of course,
these limits will depend on the application one has in mind
and on the decohering mechanisms, which should be studied
in detail. Such an endeavor goes beyond the scope of the
present paper and is deferred to future work.
IV. CONCLUSIONS
Our polarimetric setup proved to be a versatile tool for
testing geometric phases. The main part of it, an array made
of one λ/2 and six λ/4 plates, allows us to realize geometric
phases that are associated to nongeodesic paths on the Poincare´
sphere. Although we have limited ourselves to study circular
trajectories, our approach can be extended to deal with
arbitrary paths. Our experimental results fit very closely with
the theoretical predictions once we have accurately identified
the trajectory on the Poincare´ sphere that has been actually
realized by our setting. The end product of such a setting is
a geometric phase g that is nontrivially related to various
parameters entering our setup. Indeed, coincidence counts
must be optimized by adjusting the laser polarization, the
acquisition window for photon counts must also be properly
fixed, and the wave plates must be repeatedly set to their
nominal orientations when recording the data from which g
can be extracted. Not only because of the photon-counting
statistics but mainly because of our ±1◦ accuracy in the
012124-6
POLARIMETRIC MEASUREMENTS OF SINGLE-PHOTON . . . PHYSICAL REVIEW A 89, 012124 (2014)
setting of the plates, one could expect experimental results
falling within some region around the theoretical curves, as
reported, e.g., in [32]. If that were the case, our polarimetric
array would have proven to be inappropriate for studying the
robustness of geometric phases against noise. However, our
array does produce geometric phases that are in accordance
with theoretical expressions. Occasionally, these expressions
must be evaluated a posteriori, thereby identifying the actually
realized values of the parameters fixing g . Once the value
of g has been fixed, our array can be used for assessing
the robustness of this g against noise. To this end, the
array must be complemented so as to simulate different
kinds of noise. For instance, one can replace the single-
crystal photon’s source and use instead polarization-entangled
photons produced by parametric down-conversion in a two-
crystal geometry [33,34]. This produces variable entangled
polarization states. After tracing over the polarization of one
of these photons, its twin photon is brought into a mixed
polarization state ρ = (1 + rn · σ )/2, with r ∈ [0,1] being
the degree of polarization. Such a state can be submitted
to a polarimetric array similar to the one discussed in this
paper. Now, ρ can be written in the form ρ = λ+|n+〉〈n+| +
λ−|n−〉〈n−|, with λ± = (1 ± r)/2 and n · σ |n±〉 = ±|n±〉.
Applying to |n±〉 the techniques of the present work, one
can get the corresponding (pure-state) geometric phases ±g .
This is all one needs [35] to obtain the geometric phase
of the mixed state ρ, thereby assessing the effect of noise.
Experiments along these lines have already been performed
in neutron polarimetry [17,18]. The kind of noise studied in
[17] translated into a Stokes vector r = rn of the restricted
form r = (0,−r,0), and the explored paths on the Bloch
sphere originated from unitary transformations that depended
on two of the three Euler angles [17]. By appropriate choice
of these two angles, one can generate purely geometric, purely
dynamical, or combinations of both phases. However, once
this choice is made, one cannot freely address different paths
on the Bloch sphere. Nevertheless, these results represented a
considerable extension of previous ones [15], which dealt with
Pancharatnam’s phase only. Further progress in assessing the
robustness of geometric phases was achieved by addressing
adiabatic evolutions [18]. Here, the dynamical contribution to
the total phase was eliminated by spin-echo techniques, which
impose some restrictions on the class of paths being explored.
Our all-optical setting offers some advantages compared to
neutron polarimetry. It allows choosing arbitrary paths on
the Poincare´ sphere, as well as different kinds of noise
to be explored in conjunction with the chosen path. The
aforementioned remote state preparation of mixed states is
not the only choice. One can also employ interferometric
techniques to produce an enlarged family of mixed states
[32,36]. By applying interferometry for input-state preparation
and polarimetry for state manipulation, one has the possibility
of studying the resilience of purely geometric phases to various
types of noise.
ACKNOWLEDGMENTS
This work was partially financed by DGI-PUCP (Grant No.
2013-0130). J.C.S. thanks E. J. Galvez for support and advice
during his stay at Colgate University.
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012124-8
Measurement of geometric phases by robust interferometric methods
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Measurement of geometric phases by robust
interferometric methods
J. C. Loredo, O. Ort´ız, A. Ballo´n and F. De Zela
Abstract. We present a novel interferometric arrangement that makes it possible to measure
with great versatility geometric phases produced in polarization states of classical light. Our
arrangement is robust against thermal and mechanical disturbances and can be set up in a Mach-
Zehnder, a Michelson or a Sagnac configuration. We present results concerning the geometric
phase as an extension of previous measurements of the Pancharatnam, or total phase. The
geometric phase is obtained by compensating the dynamical contribution to the total phase,
so as to extract out of it a purely geometric phase. This can be achieved over trajectories on
the Poincare´ sphere that are not necessarily restricted to be great circles (geodesics). We thus
demonstrate the feasibility of our method for dynamical extraction of the geometric contribution
to the total phase, a prerequisite for building geometric quantum gates. Although our results
correspond to polarization states of classical light, the same methodology could be applied in
the case of polarization states of single photons.
Pontificia Universidad Cato´lica del Peru´, Av. Universitaria 1801, San Miguel, Lima, Peru
E-mail: fdezela@pucp.edu.pe
1. Introduction
In a previous paper [1], we have demonstrated the versatility of an interferometric array that
allows accurate measurements of the Pancharatnam phase [2] for two arbitrary polarization
states. The array was shown to be robust to thermal and mechanical disturbances and could be
set up in different configurations, like those corresponding to a Mach-Zehnder, a Sagnac, or a
Michelson interferometer. Its robustness was shown to be similar to that of a polarimetric array
[3]. Alternative, robust interferometric setups have been recently demonstrated [4]. Although
such interferometers were used for other purposes, they could represent a promising alternative
to our setup, in case one aims at scaling the latter down, so as to construct a compact device.
Besides being interesting on its own as a feature which exposes a common root underlying
different phenomena in quantum and classical physics, the geometric phase could also be a
useful tool for quantum computation. This is due to the fact that it is largely immune to
those disturbances that usually cause decoherence. As is well known, decoherence is one of the
central problems precluding the construction of a quantum computer. Indeed, computational
tasks on a qubit should be performed in a time that is short compared to the decoherence
time, and this is usually difficult to achieve. A promising route towards the construction of
fault-tolerant quantum computers requires having at one’s disposal some devices with the help
of which one can move a qubit around a parameter space. In this way one could implement
quantum gates based on exploiting the capability of manipulating the phase acquired by the
qubit as it completes a closed path on the chosen parameter space. This is called geometric
(or holonomic) quantum computation. Its theoretical foundations were laid by Zanardi and
XVII Reunión Iberoamericana de Óptica & X Encuentro de Óptica, Láseres y Aplicaciones IOP Publishing
Journal of Physics: Conference Series 274 (2011) 012140 doi:10.1088/1742-6596/274/1/012140
Published under licence by IOP Publishing Ltd 1
Rasetti [5] some years ago. Experimental demonstrations of geometric quantum gates have
been reported [6] using nuclear magnetic resonance. Different experimental scenarios have been
explored over the last years, like semiconductor quantum dots [7], ion traps [8], and others.
The photonic version of geometric gates is a promising candidate because the corresponding
setups are relatively cheap and easy to mount and manipulate. They allow testing different
configurations that can afterwards be implemented by using other means. We have thus focused
on implementing an all-optical array that allows generating and measuring geometric phases
connected to, in principle, arbitrary circuits on the Poincare´ sphere. This is a distinctive feature
of our approach, as compared to previous ones. Indeed, geometric phases have been exhibited
almost exclusively using paths consisting on a series of geodesic segments, i.e., great circles on
the Poincare´ or Bloch spheres. In other cases, special circuits have been proposed, which were
chosen so that the dynamical contribution to the total (Pancharatnam) phase vanished, leaving
only the geometrical contribution [9]. This approach imposes, of course, undesirable restrictions
to the realization of geometrical phases and also puts technical challenges for its implementation.
In our case, we are able to nullify the dynamical contribution to the total phase for any chosen
path on the Poincare´ sphere. Thus, a great versatility is achieved, offering the possibility of
choosing the most appropriate path in parameter space for implementing a geometric quantum
gate. In particular, this path could be chosen short enough to meet the requirements set by the
need of counteracting decoherence effects. Though geometric phases should be largely immune
to decoherence, this is a point in need of further investigation [10], something for which our
approach also naturally lends itself.
2. The interferometric method for measuring the geometric phase
In what follows, we use Dirac’s notation of bras and kets to stress that our results are equally
valid for the classical and for the quantum case. A ket |ψ〉 will thus denote a polarization state,
whose general form can be parametrized as
|ψ〉 =
(
cos θ
eiϕ sin θ
)
. (1)
Pancharatnam’s phase ΦP between two non-orthogonal states, |i〉 and |f〉, is defined as
ΦP = arg 〈i|f〉. It can be exhibited through interferometry by applying a phase-shift φ to
one of the states, so that the resulting intensity pattern is given by
I =
∣∣∣eiφ |i〉+ |f〉∣∣∣2 = 2 + 2 |〈i|f〉| cos (φ− arg 〈i|f〉) . (2)
By noting that the maxima of I occur for φ = arg 〈i|f〉 = ΦP , Pancharatnam gave an operational
definition for the phase between two arbitrary – though non-orthogonal – polarization states. We
are interested in the case where |f〉 = U |i〉, with U being a unitary transformation. Because the
states we consider are formally spinors (whose global phase is physically irrelevant), U ∈ SU(2).
Among the different parametrizations of U , the following one is particularly well suited for
extracting Pancharatnam’s phase:
U(β, γ, δ) = exp
(
i(
δ + γ
2
)σz
)
exp (−iβσy) exp
(
i(
δ − γ
2
)σz
)
, (3)
with σi=x,y,z being the Pauli matrices. Indeed, taking |i〉 = |+〉z, i.e., the eigenstate of σz that
belongs to the eigenvalue +1, and setting |f〉 = U |+〉z we have
〈i|f〉 = z〈+|U(β, γ, δ) |+〉z = eiδ cosβ. (4)
From ΦP = arg 〈i|f〉 we obtain ΦP = δ + arg(cosβ). Because cosβ can take on positive and
negative real values, arg(cosβ) equals 0 or pi, and ΦP is thus obtained modulo pi. In principle,
XVII Reunión Iberoamericana de Óptica & X Encuentro de Óptica, Láseres y Aplicaciones IOP Publishing
Journal of Physics: Conference Series 274 (2011) 012140 doi:10.1088/1742-6596/274/1/012140
2
then, we could obtain ΦP (modulo pi) with the help of an interferometric array by comparing two
interferograms, one of them obtained for ΦP = 0, i.e., U = I, and serving us as a reference, and
the second interferogram corresponding to the applied U . The relative shift of the latter with
respect to the reference interferogram gives us ΦP . Now, the unitary transformations that we
can implement using common optical devices like quarter-wave plates (Q) and half-wave plates
(H) are of the form
U(ξ, η, ζ) = exp
(
−i ξ
2
σy
)
exp
(
i
η
2
σz
)
exp
(
−iζ
2
σy
)
. (5)
They can be realized with the following gadget:
U(ξ, η, ζ) = Q
(−3pi + 2ξ
4
)
H
(
ξ − η − ζ − pi
4
)
Q
(
pi − 2ζ
4
)
. (6)
Q
Q
H
P1
P2
P
E
BS
BS
L
MM
M
Y
X
Figure 1. Interferometric setup for measuring the geometric phase (P : polarizer, BS: beam-
splitter, M : mirror, E: beam expander, Q: λ/4 waveplate, H: λ/2 waveplate). The gadget
QHQ on the left arm produces the desired unitary transformations, while that on the right arm
nullifies the dynamic contribution to the total phase. Two interferograms are simultaneously
produced and recorded. As the inset shows, one interferogram corresponds to the vertically
polarized part, while the other corresponds to the horizontally polarized part of the expanded
beam. From the relative shift between these interferograms one can obtain the geometric phase.
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3
The corresponding interferogram has an intensity pattern given by
IV =
∣∣∣∣ 1√2
(
eiφ |+〉z + U(ξ, η, ζ) |+〉z
)∣∣∣∣2 = (7)
=
1
2
[
1− cos
(η
2
)
cos
(
ξ + ζ
2
)
cos (φ)− sin
(η
2
)
cos
(
ξ − ζ
2
)
sin (φ)
]
.
IV refers to an initial state |+〉z that is vertically polarized. From the relationship connecting
the two parametrizations, U(ξ, η, ζ) and U(β, γ, δ), of the same U ∈ SU(2), one can show that
IV can also be written as
IV =
1
2
[1− cosβ cos (φ− δ)] . (8)
R0
Figure 2. The geometric phase can be extracted from the relative fringe-shift between the
upper and lower parts of the interferogram. The left panels show column averages of the fringes
obtained after applying a low-pass filter to get rid of noise features. The column average is
performed after selecting an evaluation area R0, as illustrated on the right panel.
Pancharatnam’s phase ΦP = δ is thus given by the shift of the interferogram whose
intensity pattern is IV , with respecto to a reference interferogram whose intensity pattern is
I = [1− cosβ cosφ] /2. By recording one interferogram after the other one could measure their
relative shift. However, thermal and mechanical disturbances make it difficult to record stable
reference patterns, thereby precluding accurate measurements of ΦP . A way out of this situation
follows from observing that the intensity pattern that corresponds to an initial, horizontally
polarized state |−〉z is given by
IH =
1
2
[1− cos (β) cos (φ+ δ)] . (9)
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4
Hence, the relative shift between IV and IH is just twice Pancharatnam’s phase. This suggests
dividing the laser beam into a vertically polarized part and a horizontally polarized part,
something that can be achieved with the help of a beam-displacer prism. By so doing, we
have the two parts of the laser beam being subjected to the same disturbances and we can
record two interferograms in a single shot. The relative shift is thus easily measurable, being
robust to thermal and mechanical disturbances. With such an array we were able to measure
Pancharatnam’s phase for different unitary transformations. Our results were reported in [1].
Our aim here is to measure a geometric phase Φg. Given a curve C0 in parameter space, Φg
relates to ΦP by
Φg(C0) = ΦP (C0)− Φdyn(C0), (10)
with
ΦP (C0) = arg〈ψ(s1)|ψ(s2)〉, (11)
Φdyn(C0) =
∫ s2
s1
Im〈ψ(s)|ψ˙(s)〉ds (12)
While ΦP (C0) and Φdyn(C0) depend on the curve C0 described by |ψ(s)〉 in parameter
space, which in our case is the Poincare´ sphere spanned by (θ, ϕ) in Eq.(1), Φg(C0) turns out
to depend only on the curve C0 that is described by |ψ(s)〉 〈ψ(s)|. This object and hence
also the curve C0 are “gauge-invariant”, i.e., invariant under parameter-dependent changes
of the phase: |ψ(s)〉 → exp(iα(s)) |ψ(s)〉. This is what makes Φg(C0) a geometrical object.
Exploiting such a gauge freedom we can choose an appropriate phase factor exp(iα(s)), so as
to make Φdyn(C0) = 0 along a given curve C0 : |ψ(s)〉 , s ∈ [s1, s2] which is traced out by
our polarization states |ψ(s)〉 as a result of applying to an initial state |ψ(0)〉 some unitary
transformation U(s): |ψ(s)〉 = U(s) |ψ(0)〉. Any U(s) can be realized by making one or
more of the parameters appearing in U(ξ, η, ζ) (see Eq.(6)) functions of s while keeping the
other ones fixed. Hence, any desired curve on the Poincare´ sphere can be realized in this
way. Setting the corresponding QHQ-gadget on one arm of the interferometer we make the
polarization state |ψ(s)〉 follow the prescribed curve. A second QHQ-gadget can be set on the
other arm of the interferometer in order to produce with its help the factor exp(iα(s)) that is
needed to nullify Φdyn(C0). This is achieved by observing that under the gauge transformation
|ψ(s)〉 → |ψ′(s)〉 = exp (iα(s)) |ψ(s)〉 the integrand entering the definition of Φdyn(C0), Eq.(12),
changes according to Im〈ψ(s)|ψ˙(s)〉 −→ Im〈ψ′(s)|ψ˙′(s)〉 = Im〈ψ(s)|ψ˙(s)〉 + ·α(s). Solving
Im〈ψ(s)|ψ˙(s)〉+ ·α(s) = 0 for α(s) we can fix the QHQ-gadget that makes Φdyn(C0) = 0.
Our interferometric setup is shown in Fig.(1). It is of the Mach-Zehnder type; but a Sagnac-
like and a Michelson-like interferometer could be used as well. With the help of this array
we could measure geometric phases stemming from non-geodesic trajectories on the Poincare´
sphere. Our results will be discussed in the next section.
3. Experimental results
We carried out our measurements employing a 30 mW , cw He-Ne laser (632.8 nm) to feed
the interferometric array shown in Fig.(1). The interferograms were recorded with the help of
a CCD camera (1/4′′ Sony, video format of 640 × 480 pixels, frame rate adjusted to 30 fps)
and digitized with a computer. The upper and lower halves of the interferograms showed a
small relative shift stemming from surface irregularities and tiny misalignments. We used a
first interferogram taken without phase-shifting (U = I) to gauge all the successive ones that
correspond to transformations U(ξ, η, ζ) 6= I. They were evaluated using an algorithm that
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5
1 2 3 4 5 6
s
-80
-60
-40
-20
Fg
n = H
3
5
,
4
5
,0L Β = 53o
Figure 3. Geometric phase
for a non-geodesic trajectory
on the Poincare´ sphere. The
trajectory is a circle, which
is the intersection of a cone
with the Poincare´ sphere. It is
specified by giving n, the axis
of the cone, and β, its aperture
angle.
n = H
3
5
,
4
5
,0L Β = 53o
-1.0
-0.50.0 0.5
1.0
x
-1.0 -0.5
0.0 0.5 1.0
y
-1.0
-0.5
0.0
0.5
1.0
z
Figure 4. The trajectory de-
scribed on the Poincare´ sphere
as the polarization is continu-
ously changed with simultane-
ous cancelling of the dynamic
contribution.
1 2 3 4 5 6
s
20
40
60
80
100
Fg
n = H-
1
2
,
3
2
,0L Β = 120o
Figure 5. Geometric phase
for a non-geodesic trajectory.
n = H-
1
2
,
3
2
,0L Β = 120o
-1.0
-0.5
0.0
0.5
1.0
x
-1.0 -0.5 0.0 0.5 1.0
y
-1.0
-0.5
0.0
0.5
1.0
z
Figure 6. The trajectory on
the Poincare´ sphere.
works as follows. First, by optical inspection one selects (defining pixel numbers) a common
region R0 of the images that the algorithm should work with (see Fig. (1)). The algorithm
performs a column average of each half of the interferogram and the output is then submitted
to a low-pass filter to get rid of noisy features. For each pair of curves the algorithm searches for
relative minima and compares their locations. It gives as a first output the relative shifts between
the minima. After averaging these relative shifts the algorithm produces a final output for each
pair of curves. We repeated this procedure for a series of regions: R0 . . . R3, that were defined by
their pixel numbers, in order to estimate the accuracy of our results. We applied this procedure
to a whole set of interferograms corresponding to different choices of U(ξ, η, ζ). Our results are
shown in Figs. (3) to (8). As can be seen, our experimental results are in very good agreement
with theoretical predictions. As expected (retarders and polarizers could be oriented to within
10), the experimental values were within 6% in accordance with the theoretical predictions.
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6
1 2 3 4 5 6
s
-80
-60
-40
-20
Fg
n = H
1
2
,
3
2
,0L Β = 60o
Figure 7. Geometric phase
for a non-geodesic trajectory.
n = H
1
2
,
3
2
,0L Β = 60o
-1.0
-0.5
0.0
0.5
1.0
x
-1.0 -0.5 0.0
0.5 1.0
y
-1.0
-0.5
0.0
0.5
1.0
z
Figure 8. The trajectory on
the Poincare´ sphere.
4. Conclusions and outlook
We have implemented a robust interferometric array to measure geometric phases. Our setup
allows producing and measuring geometric phases with great versatility, without the restriction
of having to move polarization states along paths that are composed of geodesic segments. Given
a path, we could submit the polarization state to a transformation that nullifies the dynamical
contribution to its total (Pancharatnam) phase, so as to obtain a purely geometric phase. Our
results represent a proof-of-principle that could be applied to different arrays. These arrays
ought to be capable of implementing unitary transformations on two qubits that are carried
along by a single beam, as it occurs in our case, where a laser beam was divided into two halves.
One half was vertically, and the other half horizontally polarized. In this way one can get rid of
the commonly encountered instabilities of interferometric arrays. The robustness of our setup
is similar to the one achieved using a polarimetric approach. The results obtained by using the
latter will be reported elsewhere.
As a next step, we plan to submit our polarization states to random disturbances in order to
test the robustness of the geometric phase against decoherence. Thereafter, the single-photon
version of our experiments should be implemented, in order to prove that the same array can
be used to construct geometric quantum gates.
References
[1] Loredo J C, Ort´ız O, Weinga¨rtner R and De Zela F 2009 Phys. Rev. A 80 0121113
[2] Pancharatnam S 1956 Proc. Indian Acad. Sci. A 44 247
[3] Wagh A G and Rakhecha V C 1995 Phys. Lett. A 197 112
[4] Ferrari J A and Frins E M 2002 Appl. Opt. 41, 5313, Ferrari J A and Frins E M 2007 Opt. Commun. 279 235
[5] Zanardi P and Rasetti M 1999 Phys. Lett. A 264 94
[6] Jones J A, Vedral V, Ekert A and Castagnoli G 2000 Nature 403 869
[7] Solinas P, Zanardi P, Zangh’i N and Rossi F 2003 Phys. Rev. B 67 121307
[8] Duan L M, Cirac J I and Zoller P 2001 Science 292 1695
[9] Ota Y, Goto Y, Kondo Y and Nakahara M 2009 Phys. Rev. A. 80 052311
[10] Blais A and Tremblay A -M S 2003 Phys. Rev. A. 67 012308
XVII Reunión Iberoamericana de Óptica & X Encuentro de Óptica, Láseres y Aplicaciones IOP Publishing
Journal of Physics: Conference Series 274 (2011) 012140 doi:10.1088/1742-6596/274/1/012140
7
Measurement of Pancharatnam’s phase by robust interferometric and polarimetric methods
J. C. Loredo,1 O. Ortíz,1 R. Weingärtner,1,2 and F. De Zela1
1Departamento de Ciencias, Sección Física, Pontificia Universidad Católica del Perú, Apartado, Lima 1761, Peru
2Department of Materials Science 6, University of Erlangen-Nürnberg, Martensstrasse 7, 91058 Erlangen, Germany

Received 17 April 2009; published 24 July 2009
We report on theoretical calculations and experimental observations of Pancharatnam’s phase originating
from arbitrary SU
2 transformations applied to polarization states of light. We have implemented polarimetric
and interferometric methods, which allow us to cover the full Poincaré sphere. As a distinctive feature, our
interferometric array is robust against mechanical and thermal disturbances, showing that the polarimetric
method is not inherently superior over the interferometric one, as previously assumed. Our strategy effectively
amounts to feeding an interferometer with two copropagating beams that are orthogonally polarized with
respect to each other. It can be applied to different types of standard arrays, such as a Michelson, a Sagnac, or
a Mach-Zehnder interferometer. We exhibit the versatility of our arrangement by performing measurements of
Pancharatnam’s phases and fringe visibilities that closely fit the theoretical predictions. Our approach can be
easily extended to deal with mixed states and to study decoherence effects.
DOI: 10.1103/PhysRevA.80.012113 PACS number
s: 03.65.Vf, 03.67.Lx, 42.65.Lm
I. INTRODUCTION
As is well known, Pancharatnam’s phase was originally
introduced to deal with the relative phase of two polarized
light beams 1. It anticipated geometrical phases that are
nowadays intensively studied both theoretically and experi-
mentally. Among all geometrical phases, Berry’s phase 2
has played a major role in prompting the upsurge of a vast
amount of investigations dealing with topological phases in
quantum and classical physics. Berry’s phase was originally
introduced by considering the adiabatic evolution of a quan-
tum state subjected to the action of a parameter-dependent
Hamiltonian. However, the first experiments aiming at exhib-
iting such a phase were performed with classical states of
light, using cw lasers 3. It was soon realized that the phase
tested in such experiments differed from Berry’s phase, as it
was larger than the latter by a factor of 2. The reason for this
was that the experimentally studied phase 3 arose from
SO
3 instead of SU
2 transformations. Indeed, Tomita and
Chiao 3 let polarized light pass a coiled optical fiber and
measured the phase originated from the adiabatic change suf-
fered by the propagation direction of a light beam. Thus, the
corresponding parameter space being explored—the sphere
of directions—differed from the parameter space that was
involved in Berry’s original phase. The latter was Bloch
sphere, on which any spin-1/2 state can be represented. An-
other two-state system formally equivalent to a spin-1/2 state
is a polarized light, in which, e.g., vertically 
V and hori-
zontally 
H polarized states constitute the counterparts of
the spin-up and spin-down quantum states. Polarization
states can be represented on the Poincaré sphere, which is
equivalent to the Bloch sphere. An early experiment testing
the appearance of Pancharatnam’s phase in polarization
states describing closed paths on the Poincaré sphere was the
one performed by Bhandari and Samuel 4. This interfero-
metric test was however restricted to a limited set of SU
2
transformations and, moreover, some of the transformations
used by the authors were nonunitary, as they employed linear
polarizers to bring the polarization back to its initial value.
Thus, Chyba et al. 5 performed alternative tests by employ-
ing only unitary transformations to exhibit Pancharatnam’s
phase, although such transformations were still restricted to
cover a limited SU
2 range. In spite its original formulation
in terms of polarization states of light, Pancharatnam’s phase
has not been fully exhibited in optical implementations, in
contrast to more recent experiments based on neutron spin
interferometry 6–8. Some years ago, Wagh and Rakhecha
9,10 proposed two alternative methods to measure Pan-
charatnam’s phase. One method is based on a polarimetric
procedure, while the other is an interferometric one. Both
procedures have been tested and compared against one an-
other in experiments using neutrons 6,7. The conclusion
drawn from these experiments was that the polarimetric
method is inherently superior over the interferometric
method. This is so mainly because the polarimetric method is
insensitive to mechanical and thermal disturbances that usu-
ally plague interferometric methods. Neutron interferometry,
in particular, is also limited through spatial constraints that
are imposed by the geometry of the monocrystals used to
construct the interferometers. In order to explore a large
range in the parameter space of the geometric phase, people
contrived to realize some regions of this space by electrically
inducing phase changes that were beyond the range acces-
sible through rotation of a flipper. However, such a procedure
prompted some criticisms 11 concerning the parameter
spaces that were involved in the two phase evolutions, as one
of them was physically obtained by the rotation of a flipper
and the other by electrical means. On the other hand, the
allegedly more accurate polarimetric method allows phase
measurements only modulo 
 and is therefore unable to
verify certain features such as the anticommutation of Pauli
matrices, e.g., xy =−yx, which is something that was
beautifully done with the interferometric method 6.
To the best of our knowledge, the two methods referred to
above have not yet been tested against each other in all-
optical experiments being capable of exploring the full pa-
rameter range of the Poincaré sphere. We have thus endeav-
ored to compare both methods of measuring Pancharatnam’s
phase by using all-optical setups. In this work we present a
PHYSICAL REVIEW A 80, 012113 
2009
1050-2947/2009/80
1/012113
9 ©2009 The American Physical Society012113-1
robust interferometric arrangement that makes the full range
of SU
2 polarization transformations accessible. Further-
more, we have also implemented a polarimetric array with a
similar coverage, so that both methods could be compared
against each other. As we shall see, our interferometric ar-
rangement is insensitive to mechanical and thermal distur-
bances. This represents an important improvement, as com-
pared to conventional interferometric arrangements. The
latter are usually set up as a variant of a Michelson, a Mach-
Zehnder, or a Sagnac interferometer. Our method works with
any of these variants, so that one could choose the most
appropriate arrangement. For example, one could explore de-
coherence effects by measuring geometric phases in polar-
ization single-photon mixed states using Mach-Zehnder in-
terferometers, similarly to recently reported experiments
12. In such a case, the fringe contrast 
visibility of the
interferometric pattern also conveys information about the
geometric phase. Although our work deals with pure states
only, we have also tested the visibility of our patterns as a
function of SU
2 transformations, obtaining very good
agreement with theoretical predictions.
Our experiments, in addition to test Pancharatnam’s phase
with great versatility, serve the purpose of showing a com-
mon ground for classical and quantum manifestations of to-
pological phases. Indeed, although our tests have been per-
formed with classical states of light, they could be
straightforwardly extended to experiments with single pho-
tons. Our theoretical discussion has thus been couched in a
quantum-mechanical language, so that, e.g., the polarization
states of classical light are represented by kets like V and
H. It should thus be clear that the features under study are
not of an intrinsic classical or quantum-mechanical nature.
Instead, it is the topological aspect that manifests itself as a
common ground for both classical and quantum phenomena.
The paper is organized as follows. In Sec. II we review
the interferometric and the polarimetric methods for measur-
ing Pancharatnam’s phase and derive theoretical results that
apply in our case. In Sec. III we describe our experimental
arrangements and present our results, comparing them with
our theoretical predictions. Finally, we present in Sec. IV our
conclusions.
II. INTERFEROMETRIC AND POLARIMETRIC
METHODS
Given two states, i and f, their Pancharatnam’s relative
phase P is defined as P=argi  f. A very direct way to
exhibit P is through interferometry. Indeed, consider two
interfering nonorthogonal states i and f, with i
 f. If
we apply a phase shift  to one of the states, the resulting
intensity pattern is given by
I = eii + f2 = 2 + 2ifcos
 − argif . 
1
The maxima of I are thus attained at =argi  f=P. We are
interested in exhibiting P in two-level systems and when
Pancharatnam’s phase arises as a consequence of having sub-
mitted an initial state i to an arbitrary transformation U

SU
2 that converts it into a final state f=Ui. The in-
tensity measurement for which Eq. 
1 applies can be imple-
mented with the help of, say, a Mach-Zehnder interferometer.
Alternatively, one could employ polarimetric methods. We
will discuss both methods in what follows. But before, and
for later reference, let us introduce the two parametrizations
of U
SU
2 that we shall use in our analysis. We call them
the YZY and the ZYZ forms for obvious reasons: the first one
is given by
U
,, = exp− i 2y	expi2z	exp− i 2y	 , 
2
while the second form is given by
U
,	,
 = exp
i
 + 	2 	zexp
− iyexp
i
 − 	2 	z
=  ei
 cos  − ei	 sin 
e−i	 sin  e−i
 cos  	 . 
3
To pass from one form of U to the other, one needs to
connect the respective parameters. The corresponding equa-
tions of transformation involve, generally, trigonometric for-
mulas, so that the different parameters are not connected to
one another through algebraic relations 16. The representa-
tion of Eq. 
3 is particularly well adapted to exhibit Pan-
charatnam’s phase. Indeed, taking as initial state i= +z,
i.e., the eigenstate of z that belongs to the eigenvalue of +1,
and setting f=U+z we have
if = z+ U
,	,
+ z = ei
 cos  . 
4
From the definition of Pancharatnam’s phase, i.e., P
=argi  f, we obtain P=
+arg
cos , for 
 
2n
+1
 /2. Because cos  can take on positive and negative
real values, arg
cos  equals 0 or 
. Hence, P is defined
modulo 
. In any case, the parametrization U
 ,	 ,
 of Eq.

3 is seen to be most appropriate to exhibit P=
 
modulo

. On the other hand, for the optical implementation of U,
the parametrization of the YZY form is more appropriate.
Indeed, it is well known 13 that with the help of three
retarders, viz., two quarter-wave plates and one half-wave
plate, it is possible to implement any U
SU
2 in the po-
larization space of, e.g., horizontally and vertically polarized
states of light: H , V
. This requires that one represents U
in the form given by Eq. 
2, i.e., the YZY form, because of
the following relationship involving the Euler angles
1 ,2 ,3 
see, e.g., 14:
exp− i
3 + 3
/4yexpi
1 − 22 + 3z
expi
1 − 
/4y = Q
3H
2Q
1 . 
5
Here, Q means a quarter-wave plate and H means a half-
wave plate. The arguments of the retarders are the angles of
their major axes to the vertical direction. In the case of a U
given by Eq. 
2, by applying Eq. 
5 we obtain
U
,, = Q− 3
 + 24 	H  −  −  − 
4 	Q
 − 24 	 .

6
Having discussed the two parametrizations of U
SU
2 that
are useful for our purposes, we turn now to the implementa-
LOREDO et al. PHYSICAL REVIEW A 80, 012113 
2009
012113-2
tion of the experimental arrangements that allow us to ex-
hibit Pancharatnam’s phase.
A. Interferometric arrangement: Mach-Zehnder and Sagnac
In general, with an interferometric array Pancharatnam’s
phase can be drawn from intensity measurements that are
essentially described by Eq. 
1. If we introduce U as given
in Eq. 
2 into Eq. 
1, we obtain the intensity as
I =  12 ei+ z + U
,,+ z
2
= 1 − cos2 	cos  + 2 	cos

− sin2 	cos  − 2 	sin
 . 
7
From Eqs. 
2 and 
3 it follows that the parameters of
these two representations of U
SU
2 are related through
tan

=tan
2 cos

−
2  /cos

+
2 . Hence, I can be written as
I = 1 − cos2 	cos  + 2 	sec

cos

 −  , 
8
making it evident that an interferometric method for exhib-
iting P would require measuring the shift induced by U on
the intensity pattern by an angle 
=P 
modulo 
. Now, the
expression for I as given in Eq. 
8 is somewhat inconve-
nient, because it mixes 
 with parameters of a representation
to which it does not belong. By expressing Eq. 
8 with the
parameters of U
 ,	 ,
, we obtain
I = 1 − cos
cos

 −  , 
9
thus rendering clear that the visibility v
Imax− Imin / 
Imax
+ Imin is given by v=cos , i.e., it is independent of Pan-
charatnam’s phase. In terms of the parameters  , , the
square of the visibility is given by
v2
,, = 12 1 + cos  cos  − cos  sin  sin  . 
10
For experimental tests, it will be useful to write the vis-
ibility in terms of the angles of the retarders as follows:
v2
1,2,3 =
1
2
1 + cos3
 + 432 	cos
 − 412 	
− cos
21 − 42 + 23
sin3
 + 432 	sin
 − 412 	 . 
11
Let us now refer specifically to a Mach-Zehnder interfer-
ometer. In order to calculate its output intensity, let us rep-
resent light beams as a superposition of polarization states

H , V
 and momentum 
or “which way,” i.e., spatial
states 
X , Y
. These last states denote the two-way alter-
native that can be ascribed to the Mach-Zehnder interferom-
eter. Let us consider first that our initial state is taken to be a
vertically polarized state that enters the first beam splitter
along the X direction 
e.g., the beam passing polarizer P1 in
Fig. 1. It is represented by VXV
 X. The actions of
beam splitters, mirrors, and phase shifters are represented by
operators in the two-qubit space with basis
VX , VY , HX , HY
. They act on the X , Y states, leav-
ing the polarization states H , V unchanged. The actions of
a 50:50 beam splitter and a mirror are given, respectively, by
14
UBS = 1P 

1
2 
XX + YY + iXY + iYX ,

12
Umirr = 1P 
 − i
XY + YX , 
13
where 1P means the identity operator in polarization space.
Let us stress that the above expressions for the actions of a
beam splitter and a mirror hold true irrespective of the fact
that the spatial qubits are realized by classical or by quantum
fields 
see, e.g., 15. Working with classical fields, the us-
age of kets 
and bras is just a useful mathematical means to
represent field amplitudes 16. Accordingly, a phase factor
in one or in the other arm of the interferometer can be rep-
resented by UX
=1P
 
exp
iXX+ YY and UY

=1P
 
XX+exp
iYY, respectively. If we mount an
Q
Q
H
P1
P2
P
E
BS
BS
L
MM
M
Y
X
FIG. 1. 
Color online Interferometric arrangement for testing
Pancharatnam’s phase P. Light from a He-Ne laser 
L passes a
polarizer 
P and enters a beam expander 
E, after which half of
the beam goes through one polarizer 
P1 and the other half goes
through a second polarizer 
P2, orthogonally oriented with respect
to the first. The two collinear beams feed the same Mach-Zehnder
interferometer 
BS: beam splitter; M: mirror in one of whose arms
an array of three retarders has been mounted 
Q:quarter-wave plate;
H: half-wave plate, so as to realize any desired SU
2 transforma-
tion. This transformation introduces a Pancharatnam’s phase P
=
 on one half of the beam and an opposite phase P=−
 on the
other perpendicularly polarized half, so that the relative phase of the
two halves equals 2
. From the relative shift between the upper and
the lower halves of the interferogram that is captured by a CCD
camera set at the output of the array, one can determine P. Any
instability of the array affects both halves of the interferogram in
the same way, so that the relative shift of 2
 is insensitive to
instabilities.
MEASUREMENT OF PANCHARATNAM’S PHASE BY… PHYSICAL REVIEW A 80, 012113 
2009
012113-3
array of retarders on, say, arm X of the interferometer, its
action would be represented by UP
X
=U
 XX+1P
 YY,
where U
SU
2 means the respective polarization transfor-
mation that the retarders produce, in our case the one given
in Eq. 
2. Similarly, UP
Y
=1P
 XX+U
 YY. For the
arrangement shown in Fig. 1 we obtain
UT = UBSUmirrUX
UP
YUBS. 
14
This U acts on the initial state VX and the intensity
measured at one of the output ports of the final beam splitter
is obtained by projecting the resulting state, UTVX, with the
appropriate projectors: VXVX and HXHX, thereby ob-
taining the vectors VXVXUTVX and HXHXUTVX,
respectively. Squaring the respective amplitudes and sum-
ming up we get the intensity as IV= VXUTVX2
+ HXUTVX2. A straightforward calculation yields
IV =
1
2
1 − cos2 	cos  + 2 	cos

− sin2 	cos  − 2 	sin
 . 
15
As already shown, this can be written as
IV =
1
2 1 − cos
cos
 − 
 . 
16
Using the above result, a direct measurement of Pan-
charatnam’s phase 
=P 
modulo 
 becomes possible: all
one needs to do is to measure the fringe shift between two
interferograms, with one of them serving as the reference


=0 and the other being obtained after applying the U
transformation. The practical problem with this method is the
instability of the interferometric array. Minute changes in
any component of the interferometer preclude an accurate
determination of 
. Different strategies can be applied to
overcome this kind of shortcomings. Mechanical and thermal
isolations of the arrangement is the most direct one, but it
makes measurements rather awkward. Damping instabilities
by a feedback mechanism is another possibility; but it makes
the arrangement more involved and difficult to operate. A
third option would be to use a Sagnac instead of a Mach-
Zehnder interferometer. In a Sagnac-like interferometer one
can make the two beams pass the same optical elements, so
that any instability would affect both beams equally. One
should then design the interferometer in such a way that the
U transformation acts on one beam alone, so that the other
can serve as the reference beam. In our case, for reasons
explained in detail in Sec. III, we turned to a different option
that is based on the following observations.
Equation 
16 holds for an initial state that is vertically
polarized. When the initial state is horizontally polarized,
then the intensity is given by
IH =
1
2 1 − cos
cos
 + 
 . 
17
We observe that intensities IV and IH are shifted with respect
to each other by 2
. Thus, we can exploit this fact for mea-
suring 
. To this end, we polarize one half—say the upper
half—of the laser beam vertically and the lower half hori-
zontally. With such a beam we feed our interferometer. It can
be mounted either in a Mach-Zehnder or in a Sagnac con-
figuration. In both cases we can capture at the output an
interferogram, half of which corresponds to IV and the other
half corresponds to IH. The upper fringes of the output will
be shifted with respect to the lower ones by 2
. As both
halves of the beam pass the same optical elements, they will
be equally affected by whatever perturbations. The array is
therefore insensitive to instabilities. We thus need only to
accurately measure the relative fringe shift in each interfero-
gram in order to obtain 
. By applying this method, we have
measured Pancharatnam’s phase with an accuracy that is
similar to that reached by the polarimetric method, on which
we turn next.
B. Polarimetric arrangement
The optical setup for the polarimetric method, as pro-
posed by Wagh and Rakhecha 9, is somewhat more de-
manding as compared to the interferometric method. At first
sight, however, the polarimetric method could appear to be
the simpler of the two options, because it requires a single
beam, from which one extracts phase information. It is not
obvious that phase information can be extracted from a
single beam. However, the polarimetric method is in fact
based on an analogous principle as the interferometric one,
and in a certain sense polarimetry could be seen as “virtual
interferometry.” Let us briefly discuss how it works.
Consider an initial polarized state i= +z and submit it to
the action of a 
 /2 rotation around an axis perpendicular to
the polarization axis 
z, e.g., a rotation around the x axis. As
a result, we obtain the state 
+z− i−z /2. If we now phase
shift this state by applying to it the operator exp
−z /2,
we obtain the state
V+ z  exp
− iz/2exp
− i
x/4+ z
= 
e−i/2+ z − iei/2− z/2
= e−i/2
+ z − iei− z/2.
We have thus generated a relative phase  between the
states +z and −z, which is something analogous to what is
achieved in an interferometer by changing the length of one
of the two optical paths. Subsequently, we let U act and as a
result we obtain the state UV+z= 
e−i/2U+z
− iei/2U−z /2
++ 
−. It is from this last state that
we can extract Pancharatnam’s phase by intensity measure-
ments. In order to accomplish this goal, we project 
+
+ 
−
 on the state V+z, i.e., the phase-shifted split state we
prepared before applying U. The corresponding intensity we
measure is thus given by
I = z+ V†

+ + 
−2. 
18
Let us write V+z= 
e−i/2+z− iei/2−z /2++ +
and take U as given by U
 ,	 ,
 of Eq. 
3. Calculating the
amplitude z+V†

++ 
−= 
++ −

++ 
−, we
obtain, using  
=exp
i
cos
 /2 and  

= i expi
	+sin
 /2, that 
++ −

++ 
−
=cos
cos

+ i sin
cos
	+ and, hence, that the inten-
sity amounts to
I = cos2
cos2

 + sin2
cos2
	 +  . 
19
LOREDO et al. PHYSICAL REVIEW A 80, 012113 
2009
012113-4
Equation 
19 contains Pancharatnam’s phase 
=P

modulo 
 in a form that allows its extraction through in-
tensity measurements. Indeed, we observe from Eq. 
19 that
the minimal and the maximal intensities are given by Imin
=cos2
cos2

 and Imax=cos2
cos2

+sin2
, respec-
tively, so that
cos2

 =
Imin
1 − Imax + Imin
, 
20
which is the expression on which the polarimetric method is
finally based.
A concrete experimental arrangement requires that we
implement V and U with the help of retarders. To begin with,
exp
−i
x /4=Q

4  and exp
−iz /2=Q

4 H
−
4 Q

4 .
Using Q2

4 =H

4  and exp
+iz /2=Q
−
4 H
+
4 
Q
−
4 , we obtain
Utot  V†UV
= H− 
4 	H + 
4 	Q− 
4 	
UQ
4 	H − 
4 	H
4 	 . 
21
As for U, it is convenient to employ the form U
 , , of
Eq. 
2, a form which can be directly translated into an ar-
rangement with retarders, according to Eq. 
6, i.e., an ar-
rangement of the form QHQ. Inserting this QHQ for U into
Eq. 
21, we end up with an arrangement that consists of nine
plates. In order to reduce the number of plates, we apply
relations such as, e.g., Q
H
=H
Q
2−,
Q
H
H
	=Q
+
 /2H
−+	−
 /2. The final re-
duction gives an array that consists of five retarders:
Utot = Q− 3
4 − 2 	Q− 5
 + 24 − 2 	
Q− 9
 + 2
 + 4 − 2 	H− 7
 +  +  − 4 − 2 	
Q− 
4 − 2 	 . 
22
Note that such an arrangement could be implemented by
mounting five plates having a common rotation axis, so that
all the plates can be rotated simultaneously by the same
angle  /2. The intensity that we should measure at the de-
tector depends on , , and  according to the following
expression:
I = z+ Utot+ z2
= cos22 	cos2  + 2 	
+ 
cos2 	sin  + 2 	cos

+ sin2 	sin  − 2 	sin
2. 
23
From this intensity we can extract Pancharatnam’s phase,
as given by Eq. 
20. We have tested this theoretical predic-
tion under restricted conditions by manually rotating the re-
tarders. Thus, we fixed  to 2
, so that cos2

= Imin
1
− Imax+ Imin−1=cos2
 /2 for all . In such a case, Pancharat-
nam’s phase 
modulo 
 should be given by P= /2. For
=2
 the arrangement that realizes the corresponding Utot
reduces to the following expression:
Utot
=2

= Q
Q− 2 + 	H − 4 + 	 , 
24
in which we have redefined the rotation angle  according to

−3
−2 /4→. If we instead fix =−
, it still remains
true that cos2

= Imin
1− Imax+ Imin−1=cos2
 /2, this time
for all , so that P= /2 
modulo 
 as before. The corre-
sponding arrangement of retarders is now given by
Utot
=−

= Q3
 + 2 − 24 	H− 4
 +  +  − 24 	
Q− 
 − 24 	 . 
25
It is worth noting that the intensity in this case is given by
I = cos2 2	cos2 − 22 	 + sin2 2	cos22 	 . 
26
Setting =0, =
, the intensity has a constant value, which
is useful for adjusting the arrangement. The results of our
measurements, including those corresponding to the full ar-
ray with five retarders, are shown in Fig. 2. As one can see,
they confirm the theoretical predictions within experimental
errors.
0 Π2 Π
0
0.5
1
Η
co
s2


P

Η


1 Ξ  2Π, Ζ  0
0 Π2 Π
0
0.5
1
Η
co
s2


P

Η


2 Ξ  Π, Ζ  0
0 Π2 Π
0
0.5
Η
co
s2


P

Η


3 Ξ  3 Π
18
, Ζ  7 Π
18
FIG. 2. Experimental results from a polarimetric measurement
of Pancharatnam’s phase. The upper graphs correspond to an array
that consists of three retarders set in the forms QQH 
left and QHQ

right. Parameter values are as indicated and cos2
P was mea-
sured as a function of . The lower curve corresponds to the full
array of five retarders set in the form QQQHQ.
MEASUREMENT OF PANCHARATNAM’S PHASE BY… PHYSICAL REVIEW A 80, 012113 
2009
012113-5
III. EXPERIMENTAL PROCEDURES AND RESULTS
A. Polarimetric measurements
We have carried out measurements of the Pancharatnam
phase by applying the polarimetric and the interferometric
methods presented in the previous sections. In both cases the
light source was a 30 mW cw He-Ne laser 
632.8 nm. The
polarimetric arrangement shown in Fig. 3 could have been
designed so that the five retarders see Eq. 
22 could be
simultaneously rotated by the same amount. If one aims at
systematically measuring Pancharatnam’s phase with the po-
larimetric method, this would require having a custom-made
apparatus on which one can mount the five plates with any
desired initial orientation and then submit the whole assem-
bly to rotation. As our aim was to simply exhibit the versa-
tility of the method and to compare its accuracy with that of
the interferometric method, we mounted a simple array of
five independent retarders, so that each one of them could be
manually rotated. With such an approach it takes some hours
of painstaking manipulation to record all necessary data,
whenever the experiment is performed with the full array of
five retarders. For this reason, we initially restricted our tests
to three retarders. This could be achieved by lowering the
degrees of freedom, i.e., by fixing one of the three Euler
angles, as explained in the previous section see Eqs. 
24
and 
25. Having made measurements with three plates, we
performed an additional run of measurements with the full
arrangement of five retarders. Our results are shown in Fig.
2. They correspond to intensity measurements obtained with
a high-sensitivity light sensor 
Pasco CI-6604, Si PIN pho-
todiode with spectral response in the range 320–1100 nm.
As expected 
retarders and polarizers could be oriented to
within 10, the experimental values are within 3–6 % in ac-
cordance with the theoretical predictions, depending on the
number of retarders being employed.
B. Interferometric measurements
We used two interferometric arrangements. One of them
was a Mach-Zehnder interferometer and the other was a Sa-
gnac interferometer. We started by mounting both interfer-
ometers in the standard way, but adding an array of three
retarders on one arm for implementing any desired U

SU
2. Usually, phase shifts , as appearing in Eq. 
9,
originate from moving one mirror with, e.g., a low-voltage
piezotransducer. One can then record the interference pattern
by sensing the light intensity with a photodiode set at one of
the output ports of the exiting beam splitter. Alternatively,
one can capture the whole interference pattern with a charge-
coupled device 
CCD camera. The Mach-Zehnder interfer-
ometer is easier to mount in comparison to the Sagnac inter-
ferometer. However, it has the disadvantage of being more
unstable against environmental disturbances, thus requiring
the application of some stabilizing technique such as, e.g., a
feedback system. In contrast, the Sagnac interferometer is
very stable with respect to mechanical and thermal distur-
bances. Nevertheless, mounting a Sagnac interferometer can
be difficult for geometrical reasons. By using one or the
other method, one can obtain two interferograms—one with
and the other without the retarders in place. In our case,
capturing the whole interference pattern with a CCD
camera—instead of sensing it with a photodiode—proved to
be the most convenient approach with both arrangements,
Mach-Zehnder and Sagnac. When working with the Mach-
Zehnder array, we first implemented a feedback system in
order to stabilize the reference pattern. One of the two paths
followed by the laser beam was used for feedback. The feed-
back system should allow us to compensate the jitter and
thermal drifts of the fringe patterns that preclude a proper
measurement of the phase shift. This requires an electronic
signal, after proportional-integral amplification, to be fed to a
piezotransducer in a servoloop, so as to stabilize the interfer-
ometer, thereby locking the fringe pattern. Although we suc-
ceeded in locking the fringe pattern, the geometry of our
array severely limited the parameter range we could explore.
We thus turned to a different option, i.e., the one based on
Eqs. 
16 and 
17. It required polarizing one half of the laser
beam in one direction and the other half in a direction per-
pendicular to the first one.
In order to exhibit the feasibility of our interferometric
method, we performed experiments with both Mach-Zehnder
and Sagnac arrays. In both cases we obtained similar prelimi-
nary results. However, the systematic recording of our results
corresponds to the Mach-Zehnder array shown in Fig. 1, as it
was the simpler one to mount and manipulate. As shown in
the figure, the initially polarized laser beam was expanded,
so that its upper half passed through one polarizer P1 and its
lower half through a second polarizer P2 orthogonally ori-
ented with respect to the first. Each run started by setting the
retarders so as to afford the identity transformation
Q

 /4H
−
 /4Q

 /4=1P; the corresponding interfero-
gram was captured with a CCD camera 
1 /4
 Sony CCD,
video format of 640480 pixels, and frame rate adjusted to
30 fps and digitized with an IBM-compatible computer. The
upper and the lower halves of this interferogram showed a
small relative shift stemming from surface irregularities and
tiny misalignments. The initial interferogram served to gauge
all the successive ones that correspond to transformations
U
 , ,
1P. Each interferogram was evaluated with the
help of an algorithm that works as follows. First, by optical
inspection of the whole set of interferograms—
corresponding to a given U
 , ,—one selects 
by pixel
numbers a common region R0 of the images the algorithm
should work with 
see Fig. 4. Having this region as its input
Q Q Q H Q


P P
FIG. 3. Polarimetric arrangement for testing Pancharatnam’s
phase P. With an array of five retarders 
Q:quarter-wave plate; H:
half-wave plate and two polarizers 
P, a relative phase  between
two polarization components  z can be introduced, on which any
desired SU
2 transformation can be applied. The five retarders are
simultaneously rotated, thereby varying , and the intensity I
 is
recorded. From the maximum and the minimum values of I one can
determine P, according to cos2
P= Imin / 
1− Imax+ Imin.
LOREDO et al. PHYSICAL REVIEW A 80, 012113 
2009
012113-6
the algorithm performs a column average of each half of the
interferogram—thereby obtaining the mean profile of the
fringes—and the output is then submitted to a low-pass filter

Savitzky-Golay filter to get rid of noisy features. The result
is a pair of curves like those shown in Fig. 4. The algorithm
then searches for relative minima in each of the two curves
and compares their locations so as to output the relative
shifts between the minima of the curves. After averaging
these relative shifts the algorithm produces its final output
for each pair of curves. We repeated this procedure for a
series of regions 
fixed by pixel numbers R0 , . . . ,R3, so that
we could estimate the uncertainty of our experimental val-
ues. No attempt was made to automate the selection of the
working regions. Visual inspection proved to be effective
enough for our present purposes. Some series of interfero-
grams showed limited regions that were clearly inappropriate
for being submitted to evaluation, as they reflected inhomo-
geneities and other features that stemmed from surface ir-
regularities of the optical components. We applied the com-
plete procedure to a whole set of interferograms
corresponding to different choices of U
 , ,. Our results
are shown in Fig. 5. As can be seen, our experimental results
are in very good agreement with theoretical predictions.
A second independent, algorithm was also used to check
the above results. This algorithm was developed as a variant
of some commonly used procedures in image processing.
Like in the previous approach, the algorithm first constructs
the mean profiles of the fringes and submits them to a low-
pass filter. But now, instead of searching for relative minima,
the algorithm does the following. First, it determines the
dominant spatial carrier frequency k0 by Fourier transform-
ing curves like those shown in Fig. 4. Let us denote these
curves by iˆup
x and iˆlow
x, corresponding, respectively, to
the upper and the lower halves of the interferogram. The
Fourier transforms are denoted by iup
k and ilow
k. The goal
is to determine the relative shift r=2
 between iˆup
x and
iˆlow
x. It can be shown 17 that r=up−low
R0
FIG. 4. 
Color online Pancharatnam’s phase can be extracted from the relative fringe shift between the upper and the lower parts of the
interferogram. The relative shift equals twice the Pancharatnam’s phase. The left panels show the result of performing a column average of
the fringes plus the application of a Savitzky-Golay filter to get rid of noise features. The column average is performed after selecting the
evaluation area R0 on the interferogram, as illustrated on the right panel. The reported shifts are mean values obtained from four different
selections R0 , . . . ,R3 of the evaluation area.
Ξ  0
0 Π2 Π
0
0.5
1
Η
co
s2

Φ

Ξ  Π
18
0 Π2 Π
0
0.5
1
Η
co
s2

Φ

Ξ  Π
9
0 Π2 Π
0
0.5
1
Η
co
s2

Φ

Ξ  Π
6
0 Π2 Π
0
0.5
1
Η
co
s2

Φ

Ξ  2 Π
9
0 Π2 Π
0
0.5
1
Η
co
s2

Φ

0Π
18Π
9Π
62 Π
9
Ξ
Π
4 Π
2 3 Π
4 Π
Η
0
0.5CosΦ2
FIG. 5. 
Color online Experimental results from the interfero-
metric measurement of Pancharatnam’s phase. We plot cos2
P as
a function of  and , with  being held fixed to zero. In the upper
panels we plot the single curves that are highlighted on the surface
shown on the lower panel. Dots correspond to experimental values,
some of which fall below and some fall above the surface.
MEASUREMENT OF PANCHARATNAM’S PHASE BY… PHYSICAL REVIEW A 80, 012113 
2009
012113-7
 Imlniup
k0
−Imlnilow
k0
, up to a constant phase
offset that is the same for all the interferograms pertaining to
a given U
 , ,. The above expression for r comes from
observing that both iup
k0 and ilow
k0 have the structure
i
k0=a
k0+b
0exp
i+b
2k0exp
−i, so that i
k0
b
0exp
i whenever b
0 b
2k0 , a
k0. Thus, the
accuracy of the approximation for r depends on how well
one can separate the Fourier components of i
k0. In the
present case we applied this procedure only for the sake of
checking our results. An attempt to systematize this method
would be worth only if one’s goals require an automated
phase-retrieval method. In our case, as we were interested in
giving a proof of principle only, the method of choice was
not a fully automated one, but a partially manual method
which was envisioned to demonstrate the feasibility of our
approach.
Another series of tests was devoted to measuring the vis-
ibility v as given in Eq. 
11. The quantity v
1 ,2 ,3 was
submitted to test by fixing two of its three arguments. Our
results are shown in Fig. 6. The left panels correspond to
v
1 ,2 ,3 as a function of 2 and 3, that is, the surface
obtained by fixing 1 as indicated. In the right panels we
compare the theoretical predictions against our measure-
ments of v
1 ,2 ,3, whereby two of the three arguments
have been held fixed. The interferograms were evaluated fol-
lowing a procedure similar to the one already explained.
However, in this case it was not the full cross section of the
beam that was submitted to evaluation, but a manually cho-
sen region of the images corresponding to a part of the input
beam having almost uniform intensity. This had to be so,
because Eq. 
11 presupposes a uniform profile of the input
beam. In order to test the visibility of the whole cross section
of the beam, Eq. 
11 should be modulated with a Gaussian
envelope. Such a refinement was however unnecessary for
our scopes. In any case, the experimental value of the vis-
ibility, viz., 
Imax− Imin / 
Imax+ Imin, was obtained by choos-
ing in each interferogram several maxima and minima, so as
to assess the accuracy of our measurements. Thus, the error
bars in the figures take proper account of the tiny variations
in the chosen region of the input-beam profile. As can be
seen, the experimental values closely fit the theoretical pre-
dictions.
IV. CONCLUSIONS
We have carried out theoretical calculations and the cor-
responding measurements of Pancharatnam’s phase by ap-
plying the polarimetric and the interferometric methods. Our
interferometric array is robust against thermal and mechani-
cal disturbances. It can be implemented with a Michelson, a
Sagnac, or a Mach-Zehnder interferometer. We have com-
pared our measurements with those obtained in a polarimet-
ric array, finding similar results in both cases. Our polarimet-
ric array consisted of five wave plates and two polarizers.
Five plates are necessary to realize an arbitrary SU
2 trans-
formation with the polarimetric array. As well known, three
plates are instead required for realizing an arbitrary SU
2
transformation with an interferometric array. The whole
Poincaré sphere of polarization states could be explored with
both our polarimetric and interferometric arrays. Thus, any
two given polarization states could be connected by the ap-
propriate SU
2 transformation. The associated relative Pan-
charatnam’s phase would thereby be realized. This phase can
be decomposed as a sum of dynamical and geometrical
phases. By appropriately choosing the path connecting two
given states on the Poincaré sphere, one can study different
aspects of both the dynamical and the geometrical phases.
We have also tested theoretical predictions concerning
fringe visibility when applying the interferometric method.
Our experimental findings were in very good agreement with
theoretical predictions. This is interesting not only on its
own, but also in view of extracting Pancharatnam’s phase
from visibility measurements in the case of mixed states.
Indeed, it has been proved 18 that, for mixed states, fringe
visibility is a simple function of Pancharatnam’s phase.
ACKNOWLEDGMENTS
We wish to thank E. J. Galvez for his technical advice and
for kindly lending us some optical equipments. This work
was partially supported by DAI-PUCP 
Contract No. DAI-
2009-0010. R.W. acknowledges financial support from the
Deutsche Forschungsgemeinschaft 
Contracts No. WI 393/
20-1 and No. WI 393/21-1.
Θ1
Π
4
0
Π
2 Π
Θ20
Π
2
Π
Θ3
0
0.5
1
v
0 Π
8
Π
4
3 Π
8
Π
2
0.6
0.7
0.8
0.9
1
Θ2
v
Θ3
Π
4
Θ10

Π
2
0
Π
2
Θ2
Π
2
0
Π
2
Θ3
0
0.5
1
v
0 Π
8
Π
4
3 Π
8
Π
2
0.6
0.7
0.8
0.9
1
Θ3
v
Θ20
FIG. 6. Interferometric measurement of the visibility
v
1 ,2 ,3. The left panels show the surfaces obtained by fixing
one of the three angles, 1, as indicated. The right panels show the
experimental results that correspond to the curves highlighted on
the surfaces. The upper curve is obtained by fixing 3 besides 1,
while the lower curve is obtained by fixing 2 and 1. In the upper
curve all experimental values fall below the predicted 
maximal
visibility of 1. This is because Imin is never zero, as required to
obtain v=1. By subtracting the nonzero average of Imin the experi-
mental points would fall above and below the theoretical curve, as
it occurs for the lower curve, which corresponds to v1.
LOREDO et al. PHYSICAL REVIEW A 80, 012113 
2009
012113-8
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MEASUREMENT OF PANCHARATNAM’S PHASE BY… PHYSICAL REVIEW A 80, 012113 
2009
012113-9
Pancharatnam and Geometric Phases
Omar Ortiz
PUCP
2016
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 1 / 47
Pancharatnam’s phase
Pancharatnam prescribed how to determine when two polarizations states
were in phase
phase difference between |A〉 and |B〉 = phase of 〈A|B〉
〈A|B〉 = | 〈A|B〉 | exp(iφ)
φ = Pancharatnam′s phase = φP
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 2 / 47
Pancharatnam’s phase: Interferometry
Intensity at the output can be written as
I ∝ 1 + v cos (φ− ΦP)
visibility : v = |〈i |f 〉|
Pancharatnam′s phase : ΦP = arg 〈i |f 〉
where
|f 〉 = U |i〉
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 3 / 47
Pancharatnam’s phase: Interferometry
Instabilities in the interferometer overshadow Pancharatnam’s phase. We
explore solutions using Sagnac interferometry and close-loop phase-locked
interferometer.
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 4 / 47
Pancharatnam’s phase: Interferometry
Final version of our setup
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 5 / 47
Pancharatnam’s phase: Interferometry
Vertical and horizontal polarized beams produce interferograms associated
with the following expressions for intensity
IV =
1
2
[1− cos(β) cos(φ− δ)]
IH =
1
2
[1− cos(β) cos(φ+ δ)]
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 6 / 47
Pancharatnam’s phase: Interferometry
Considering the final state |f 〉 = Uˆzyz |i〉 for i = H,V resulting from
applying a SU(2) transformation parametrized as
Uˆzyz(β, γ, δ) = exp(i
δ + γ
2
σˆz) exp(iβσˆy ) exp(i
δ − γ
2
σˆz)
it can be proved that for i = H
φP = δ
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 7 / 47
Pancharatnam’s phase: Interferometry
Vertical and horizontal polarized beams produce interferograms associated
with the following expressions for intensity
IV =
1
2
[1− cos(β) cos(φ−δ)]
IH =
1
2
[1− cos(β) cos(φ+δ)]
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 8 / 47
Pancharatnam’s phase: Interferometry
Pancharatnam’s phase can be measured by comparison of the two
interferograms
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 9 / 47
Pancharatnam’s phase: Interferometry
Final state |f 〉 = Uˆyzy |i〉 for i = H,V results from applying SU(2)
transformation:
Uˆyzy (ξ, η, ζ) = exp(−i ξ
2
σˆy ) exp(i
η
2
σˆz) exp(−i ζ
2
σˆy )
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 10 / 47
Pancharatnam’s phase: Interferometry
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 11 / 47
Pancharatnam’s phase: Polarimetry (virtual interferometry)
An analogy is possible between operators acting on momentum dof and
polarization dof
UˆpolBS = H(pi/8) = −
i√
2
(
1 1
1 −1
)
Uˆpolφ = Q(pi/4)H(−
φ+ pi2
2
)Q(pi/4) =
(
e iφ 0
0 e−iφ
)
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 12 / 47
Pancharatnam’s phase: Polarimetry (virtual interferometry)
With this result we can build our virtual interferometer as
Uˆint = Uˆ
pol
BS Uˆzyz Uˆ
pol
φ Uˆ
pol
BS
then
I = | 〈H| Uˆint |H〉 | = cos2(β) cos2(δ) + sin2(β) cos2(γ + φ)
which allows us to measure Pancharatnam phase using only the maximun
and minimun values of intensity
cos2(ΦP) = cos
2(δ) =
Imin
1− Imax − Imin
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 13 / 47
Pancharatnam’s phase: Polarimetry, virtual interferometry
Our final setup looks as follows
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 14 / 47
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 15 / 47
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 16 / 47
Geometric phase
Kinematic definition of geometric phase (Mukunda and Simon)
φg = arg 〈Ψ(si )|Ψ(s)〉︸ ︷︷ ︸
φP
−
∫ s
si
Im 〈Ψ(s)|Ψ˙(s)〉 ds︸ ︷︷ ︸
φdyn
φg is invariant under local phase changes
|Ψ′(s)〉 = e iα(s) |Ψ(s)〉
φ′g = φg
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 17 / 47
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 18 / 47
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 19 / 47
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 20 / 47
Polarimetric measurements of single-photon geometric
phases
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 21 / 47
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 22 / 47
Conclusions
We have implemented robust interferometric arrays for measuring
Pancharatnam and geometric phases.
We have exhibited inherently robust and very versatile polarimetric
arrays for generating Pancharatnam and geometric phases.
Our setups work with both classical light (laser) and quantum light
(single photons)
Omar Ortiz (PUCP) Pancharatnam and Geometric Phases 2016 23 / 47