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Measurement of Pancharatnam's phase by robust interferometric and polarimetric methods

We report 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 Poincar\'e sphere. As a distinctive feature, our interferometric array is robust against mechanical and thermal disturbances, showing that the polarimetric method is not inherently superior to the interferometric one, as previously assumed. Our strategy effectively amounts to feed 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, like 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.


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 experimentally. 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 quantum state subjected to the action of a parameter-dependent Hamiltonian. However, the first experiments aiming at exhibiting 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 two. 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 suffered 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. Another two-state system formally equivalent to a spin-1/2 state is polarized light, in which, e.g., vertically (V ) and horizontally (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 interferometric 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 employing only unitary transformations to exhibit Pancharatnam's phase, though such transformations were still restricted to cover a limited SU (2) range. In spite of 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][7][8]. Some years ago, Wagh and Rakhecha [9,10] proposed two alternative methods to measure Pancharatnam'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 another in experiments using neutrons [6,7]. The conclusion drawn from these experiments was that the polarimetric method is inherently superior to the interferometric method. This is so mainly because the polarimetric method is insensitive to mechanical and thermal disturbances that usually plague interferometric methods. Neutron interferometry, in particular, is also limited through spacial 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 accessible 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 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 π, and is therefore unable to verify certain features like the anticommutation of Pauli matrices, e.g., σ x σ y = −σ y σ x , 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 parameter range of the Poincaré sphere. We have thus endeavor to compare both methods of measuring Pancharatnam's phase by using all-optical setups. In this work we present a robust interferometric arrangement that makes accessible the full range of SU (2) polarization transformations. Furthermore, 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 arrangement is insensitive to mechanical and thermal disturbances.
This represents an important improvement, as compared 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 decoherence effects by measuring geometric phases in polarization single-photon mixed-states using Mach-Zehnder interferometers, 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. Though 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 common ground for classical and quantum manifestations of topological phases. Indeed, although our tests have been performed with classical states of light, they could be straightforwardly extended to experiments with single photons. 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 what manifests itself as a common ground for both classical and quantum phenomena.
The paper is organized as follows. In Section II we review the interferometric and the polarimetric methods for measuring Pancharatnam's phase and derive theoretical results that apply in our case. In Section III we describe our experimental arrangements and present our results, comparing them with our theoretical predictions. Finally, we present in Section IV our conclusions.

II. THE INTERFEROMETRIC AND POLARIMETRIC METHODS
Given two states, |i and |f , their Pancharatnam's relative phase Φ P is defined as Φ P = arg i|f . A very direct way to exhibit Φ P is through interferometry. Indeed, consider two interfering, non-orthogonal states, |i and |f , with |i = |f . If we apply a phase shift φ to one of the states, the resulting intensity pattern is given by The maxima of I are thus attained at φ = arg i|f = Φ P . We are interested in exhibiting Φ P in two-level systems and when Pancharatnam's phase arises as a consequence of having submitted an initial state |i to an arbitrary transformation U ∈ SU (2) that converts it into a final state |f = U |i . The intensity measurement for which Eq. (1) applies can be implemented 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 Y ZY -form and the ZY Z-form, for obvious reasons: The first one is given by while the second form is given by To pass from one form of U to the other one needs to connect the respective parameters.
The corresponding equations of transformation involve, generally, trigonometric formulae, so that the different parameters are not connected to one another through algebraic relations.
The representation of Eq. (3) is particularly well adapted to exhibit Pancharatnam's phase.
From the definition of Pancharatnam's phase, i.e., Φ P = arg i|f , we obtain Φ P = δ + arg(cos β), for β = (2n + 1)π/2. Because cos β can take on positive and negative real values, arg(cos β) equals 0 or π, and Φ P is thus defined modulo π. In any case, the parametrization U (β, γ, δ) of Eq. (3) is seen to be most appropriate to exhibit Φ P = δ (modulo π). On the other hand, for the optical implementation of U the parametrization of the Y ZY -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 polarization-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 Y ZY -form, because of the following relationship involving the Euler angles θ 1 , θ 2 , θ 3 (see, e.g., [14]): Here, Q means a quarter-wave plate and H 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 Having discussed the two parametrizations of U ∈ SU (2) that are useful for our purposes, we turn now to the implementation of the experimental arrangements that allow us to exhibit 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 From Eqs. (2) and (3) it follows that the parameters of these two representations of U ∈ SU (2) are related through tan(δ) = tan η 2 cos ξ−ζ 2 / cos ξ+ζ 2 . Hence, I can be written as making it evident that an interferometric method for exhibiting Φ P would require measuring the shift induced by U on the intensity pattern by an angle δ = Φ P (modulo π). Now, the expression for I as given in Eq. (8) is somewhat inconvenient, because it mixes δ with parameters of a representation to which it does not belong. By expressing Eq. (8) with the parameters of U (β, γ, δ) we obtain thus rendering clear that the visibility v ≡ (I max − I min ) / (I max + I min ) is given by v = cos β, i.e., it is independent of Pancharatnam's phase. In terms of the parameters ξ, η, ζ the square of the visibility is given by For experimental tests, it will be useful to write the visibility in terms of the angles of the retarders: Let us now refer specifically to a Mach-Zehnder interferometer. In order to calculate its output intensity, let us represent 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 alternative that can be ascribed to the Mach-Zehnder interferometer. 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 with 1 P meaning 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 be realized by classical or by quantum fields (see, e.g., [15]).
Working with classical fields, the usage of kets (and bras) is just a useful mathematical means to represent field amplitudes. Accordingly, a phase factor in one or in the other arm of the interferometer can be represented by , respectively. If we mount an array of retarders on, say, arm X of the interferometer, its action would be represented by U X P = U ⊗|X X|+ 1 P ⊗ |Y Y |, where U ∈ SU (2) means the respective polarization transformation that the retarders produce, in our case the one given in Eq. (2). Similarly, For the arrangement shown in Fig. (1) we obtain This U acts on the initial state |V X and the intensity measured at one of the output ports of the final beam-splitter is obtained by projecting the resulting state, U T |V X , with the appropriate projectors: |V X V X| and |HX HX|, thereby obtaining the vectors |V X V X| U T |V X and |HX HX| U T |V X , respectively. Squaring the respective amplitudes and summing up we get the intensity as A straightforward calculation yields As already shown, this can be written as Using the above result, a direct measurement of Pancharatnam's phase δ = Φ P (modulo π) becomes possible: all one needs to do is to measure the fringe-shift between two interferograms, one of them serving as reference (δ = 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 δ. Different strategies can be applied to overcome this kind of shortcomings. A mechanical and thermal isolation 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 Section III, we turned to a different option that is based on the following observations.
Eq. (16) holds for an initial state that is vertically polarized. When the initial state is horizontally polarized, then the intensity is given by We observe that intensities I V and I H are shifted with respect to each other by 2δ. Thus, we can exploit this fact for measuring δ. To this end, we polarize one half -say the upper half -of the laser beam vertically, and the lower half horizontally. With such a beam we feed our interferometer, be it in a Mach-Zehnder or in a Sagnac configuration, so that we can capture at the output an interferogram, half of which corresponds to I V and the other half to I H . The upper fringes of the output will be shifted with respect to the lower ones by 2δ. 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 interferogram in order to obtain δ. 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 proposed by Wagh and Rakhecha [9], is somewhat more demanding, as compared to the interferometric method. At first sight, however, the polarimetric method could appear to be the simplest 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 π/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 ( we now phase-shift this state by applying to it the operator exp (−φσ z /2) we obtain We have thus generated a relative phase φ between the states |+ z and |− z , something analogous to what is achieved in an interferometer by changing the length of one of the two optical paths. Subsequently, we let U ∈ SU (2) act and as a result we obtain the state It is from this last state that we can extract Pancharatnam's phase by intensity measurements.
Eq. (19) contains Pancharatnam's phase δ = Φ P (modulo π) in a form that allows its extraction through intensity measurements. Indeed, we observe from Eq. (19) that the minimal and maximal intensities are given by I min = cos 2 (β) cos 2 (δ) and I max = cos 2 (β) cos 2 (δ) + sin 2 (β), respectively, so that which is the expression on which the polarimetric method is finally based.
For ζ = 2π the arrangement that realizes the corresponding U tot reduces to the following expression: in which we have redefined the rotation angle φ according to (−3π − 2φ)/4 → φ. If we instead fix ξ = −π, it still remains true that cos 2 (δ) = I min (1 − I max + I min ) −1 = cos 2 (η/2), this time for all ζ, so that Φ P = η/2 (modulo π), as before. The corresponding arrangement of retarders is now given by It is worth noting that the intensity in this case is given by Setting η = 0, ζ = π the intensity has a constant value, which is useful for adjusting the arrangement. The results of our measurements, including those corresponding to the full array with five retarders, are shown in Fig. (5). As one can see, they confirm the theoretical predictions within experimental errors.

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. (2) 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 polarimetric 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 assembly to rotation. As our aim was to simply exhibit the versatility 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, 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. (5). They correspond to intensity measurements obtained with a high-sensitivity light sensor (Pasco CI-6604, Si PIN photodiode with spectral response in the range 320 nm -1100 nm). As expected (retarders and polarizers could be oriented to within 1 0 ), the experimental values are within 3% to 6% in accordance 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 Sagnac interferometer. We started by mounting both interferometers in the standard way, but adding an array of three retarders on one arm for implementing any desired U ∈ SU (2). Usually, phase shifts φ, 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 CCD camera. The Mach-Zehnder interferometer is easier to mount in comparison to the Sagnac interferometer. However, it has the disadvantage of being more unstable against environmental disturbances, thus requiring the application of some stabilizing technique like, e.g., a feedback system. In contrast, the Sagnac interferometer is very stable with respect to mechanical and thermal disturbances. 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 feedback 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 that an electronic signal, after proportional-integral amplification, be fed to a piezotransducer in a servo-loop, so as to stabilize the interferometer, thereby locking the fringe pattern. Although we succeeded 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 perpendicular to the first one.
In order to exhibit the feasibility of our interferometric method we performed experiments with both a Mach-Zehnder and a Sagnac array. In both cases we obtained similar preliminary 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 P 1 and its lower half through a second polarizer P 2 , orthogonally oriented with respect to the first. Each run started by setting the retarders so as to afford the identity transformation: Q(π/4)H(−π/4)Q(π/4) = 1 P , the corresponding interferogram was captured with a CCD camera (1/4 Sony CCD, video format of 640 × 480 pixels, frame rate adjusted to 30 fps) and digitized with an IBM -compatible computer. The upper and 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 (ξ, η, ζ) = 1 P . 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 (ξ, η, ζ) -one selects (by pixel numbers) a common region R 0 of the images the algorithm should work with (see Fig. (3)).
Having this region as its input 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. (3). 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): R 0 . . . R 3 , so that we could estimate the uncertainty of our experimental values. 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 interferograms showed limited regions that were clearly inappropriate for being submitted to evaluation, as they reflected inhomogeneities and other features that stemmed from surface irregularities of the optical components. We applied the complete procedure to a whole set of interferograms corresponding to different choices of U (ξ, η, ζ). Our results are shown in Fig. (4). 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 k 0 by Fourier transforming curves like those shown in Fig. (3). Let us denote these curves by i up (x) and i low (x), corresponding, respectively, to the upper and lower half of the interferogram. The Fourier transforms are denoted by i up (k) and i low (k). The goal is to determine the relative shift ∆ r = 2δ between i up (x) and i low (x). It can be shown , up to a constant phaseoffset that is the same for all the interferograms pertaining to a given U (ξ, η, ζ). The above expression for ∆ r comes from observing that both i up (k 0 ) and i low (k 0 ) have the structure Thus, the accuracy of the approximation for ∆ r depends on how well one can separate the Fourier components of i(k 0 ). 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 visibility v as given in Eq. (11). The quantity v(θ 1 , θ 2 , θ 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(θ 1 , θ 2 , θ 3 ) as a function of θ 2 and θ 3 , that is, the surface obtained by fixing θ 1 as indicated. In the right panels we compare the theoretical predictions against our measurements of v(θ 1 , θ 2 , θ 3 ), whereby two of the three arguments have been held fixed. The interferograms were evaluated following 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 chosen 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 visibility, viz., (I max − I min )/(I max + I min ), was obtained by choosing 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 predictions.

IV. CONCLUSIONS
We have carried out theoretical calculations and the corresponding measurements of Pan- 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 [19] that, for mixed states, fringe visibility is a simple function of Pancharatnam's phase. of the beam goes through one polarizer (P 1 ) and the other half goes through a second polarizer (P 2 ), 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) transformation. This transformation introduces a Pancharatnam phase Φ P = δ on one half of the beam and an opposite phase Φ P = −δ on the other, perpendicularly polarized half, so that the relative phase of the two halves equals 2δ. From the relative shift between the upper and lower halves of the interferogram that is captured by a CCD camera set at the output of the array one can determine Φ P . Any instability of the array affects both halves of the interferogram in the same way, so that the relative shift 2δ is insensitive to instabilities.