PONTIFICIA UNIVERSIDAD CATÓLICA DEL PERÚ 
ESCUELA DE POSGRADO 
 
 
Polarimetric measurements of single‐photon geometric phases 
Artículo publicable para optar el título de Magíster en Física que presenta: 
Yonny Daniel Yugra Carcasi 
 
Asesor: 
Prof. Francisco de Zela 
Jurado: 
Prof. Eduardo Massoni  
Prof. Hernán Castillo 
Lima, 2015 
RESUMEN 
Nombre del graduando: Yonny Daniel Yugra Carcasi 
Posgrado en: Física 
Título del artículo publicado: 
Polarimetric measurements of single‐photon geometric phases 
Este artículo ha sido publicado en: 
PHYSICAL REVIEW A, Volume 89, Issue 1, 2014 
DOI: http://dx.doi.org/10.1103/PhysRevA.89.012124 
Resumen:  Se  presenta  mediciones  polarimétricas  de  fases  geométricas 
generadas  en  la  evolución  de  la  polarización  de  fotones  a  lo  largo  de 
trayectorias  no‐geodésicas  en  la  esfera  de  Poincaré.El  núcleo  del  arreglo 
polarimétrico  consiste  de  siete  placas  retardadoras.  Este  arreglo  permite 
realizar  cualquier  transformación unitaria del  grupo  SU(2) en el espacio de 
polarización. Haciendo uso de  la  invariancia gauge de  las  fases geométricas 
bajo  transformaciones  locales  U(1),  es  posible  anular  la  contribución 
dinámica a  la fase total, con  lo cual se  logra que esta última coincida con  la 
fase  geométrica.  Como  la  fase  total  es  accesible  a  las  mediciones 
experimentales,  la  fase  geométrica  se  torna  así  también  accesible  a  las 
mismas. Se demuestra que nuestro dispositivo es  robusto  frente a diversas 
perturbaciones  que  usualmente  afectan  arreglos  interferométricos,  ya  que 
utilizamos un solo haz de fotones. Nuestro arreglo polarimétrico de muestra 
ser  una  herramienta  sumamente  versátil  que  podría  ser  utilizada  para 
someter a prueba la robustez de las fases geométricas frente a varias fuentes 
de decoherencia. 
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)
012124-3
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 eachφ, 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
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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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