The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects

In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M. V. Berry, 1984). The process was supposed to be adiabatic, meaning that the time scale of the system’s evolution was much shorter than the time scale of the changing Hamiltonian. Until Berry’s study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning. Such phases could be eliminated by redefining the quantum state through a “gauge” transformation of the form |ψ〉 → eiα |ψ〉. However, Berry discovered that besides the dynamical, there was an additional phase that could not be “gauged away” and whose origin was geometric or topological. It depended on the path that |ψ〉 describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence. Berry’s discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems. Berry’s phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems. For example, the Aharonov-Bohm phase could be understood as a geometric phase. The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase. Another example is the precession of the plane of oscillation of a Foucault pendulum. Berry’s original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes. A spin-1/2 system is a special case of a two-level system. Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases. In fact, the first experimental test of Berry’s phase was done using polarized, classical light (A. Tomita, 1986). Pancharatnam (S. Pancharatnam, 1956) anticipated Berry’s phase when he proposed, back in 1956, how to decide whether two polarization states are “in phase”. Pancharatnam’s prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity. In that case, the two polarized beams are said to be “in phase”. Such a definition is analogous to the definition of distant parallelism in differential geometry. Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not. The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects


Introduction
In 1984 Berry addressed a quantum system undergoing a unitary and cyclic evolution under the action of a time-dependent Hamiltonian (M.V. Berry, 1984).The process was supposed to be adiabatic, meaning that the time scale of the system's evolution was much shorter than the time scale of the changing Hamiltonian.Until Berry's study, it was assumed that for a cyclic Hamiltonian the quantum state would acquire only so-called dynamical phases, deprived of physical meaning.Such phases could be eliminated by redefining the quantum state through a "gauge" transformation of the form |ψ → e iα |ψ .However, Berry discovered that besides the dynamical, there was an additional phase that could not be "gauged away" and whose origin was geometric or topological.It depended on the path that |ψ describes in the parameter space spanned by those parameters to which the Hamiltonian owed its time dependence.Berry's discovery was the starting point for a great amount of investigations that brought to light topological aspects of both quantum and classical systems.Berry's phase was soon recognized as a special case of more general phases that showed up when dealing with topological aspects of different systems.For example, the Aharonov-Bohm phase could be understood as a geometric phase.The rotation angle acquired by a parallel-transported vector after completing a closed loop in a gravitationally curved space-time region, is also a geometric, Berry-like phase.Another example is the precession of the plane of oscillation of a Foucault pendulum.Berry's original formulation was directly applicable to the case of a spin-1/2 system evolving under the action of a slowly varying magnetic field that undergoes cyclic changes.A spin-1/2 system is a special case of a two-level system.Another instances are two-level atoms and polarized light, so that also in these cases we should expect to find geometric phases.In fact, the first experimental test of Berry's phase was done using polarized, classical light (A.Tomita, 1986).Pancharatnam (S. Pancharatnam, 1956) anticipated Berry's phase when he proposed, back in 1956, how to decide whether two polarization states are "in phase".Pancharatnam's prescription is an operational one, based upon observing whether the intensity of the interferogram formed by two polarized beams has maximal intensity.In that case, the two polarized beams are said to be "in phase".Such a definition is analogous to the definition of distant parallelism in differential geometry.Polarized states can be subjected to different transformations which could be cyclic or not, adiabatic or not, unitary or not.
And in all cases Pancharatnam's definition applies.Pancharatnam's phase bore therefore an anticipation and -at the same time -a generalization of Berry's phase.Indeed, Berry's assumptions about a cyclic, adiabatic and unitary evolution, turned out to be unnecessary for a geometric phase to appear.This was made clear through the contributions of several authors that addressed the issue right after Berry published his seminal results (Y.Aharonov, 1987;J. Samuel, 1988).Pancharatnam's approach, general as it was when viewed as pregnant of so many concepts related to geometric phases, underlay nonetheless two important restrictions.It addressed nonorthogonal and at the same time pure, viz totally, polarized states.Here again the assumed restrictions turned out to be unnecessary.Indeed, it was recently proposed how to decide whether two orthogonal states are in phase or not (H.M. Wong, 2005).Mixed states have also been addressed (A.Uhlmann, 1986;E. Sjöqvist, 2000) in relation to geometric phases which -under appropriate conditions -can be exhibited as well-defined objects underlying the evolution of such states.The present Chapter should provide an overview of the Pancharatnam-Berry phase by introducing it first within Berry's original approach, and then through the kinematic approach that was advanced by Simon and Mukunda some years after Berry's discovery (N.Mukunda, 1993).The kinematic approach brings to the fore the most essential aspects of geometric phases, something that was not fully accomplished when Berry first addressed the issue.It also leads to a natural introduction of geodesics in Hilbert space, and helps to connect Pancharatnam's approach with the so-called Bargmann invariants.We discuss these issues in the present Chapter.Other topics that this Chapter addresses are interferometry and polarimetry, two ways of measuring geometric phases, and some recent generalizations of Berry's phase to mixed states and to non-unitary evolutions.Finally, we show in which sense the relativistic effect known as Thomas rotation can be understood as a manifestation of a Berry-like phase, amenable to be tested with partially polarized states.All this illustrates how -as it has often been the case in physics -a fundamental discovery that is made by addressing a particular issue, can show afterwards to bear a rather unexpected generality and applicability.Berry's discovery ranks among this kind of fundamental advances.

The adiabatic and cyclic case: Berry's approach
Let us consider a non-conservative system, whose evolution is ruled by a time-dependent Hamiltonian H(t).This occurs when the system is under the influence of an environment.The configuration of the environment can generally be specified by a set of parameters (R 1 , R 2 ,...).For a changing environment the R i are time-dependent, and so also the observables of the system, e.g., the Hamiltonian: The evolution of the quantum system is ruled by the Schrödinger equation, or more generally, by the Liouville-von Neumann equation (in units of h = 1): Here, the density operator ρ is assumed to describe a pure state, i.e., to be of the form ρ(t)= |ψ(t) ψ(t)|.An "environmental process" is given by t → R(t), the curve described by the vector R in parameter space.To such a process it corresponds a curve described by |ψ(t) in the Hilbert space H to which it belongs, or by the corresponding curve ρ(t)=|ψ(t) ψ(t)| The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects 3 in the "projection space" P (H) to which ρ belongs.We will assume that for all R there is an orthonormal basis |n; R such that An environmental process R(t) is called periodic with period T, whenever R(T)=R( 0), , w e c a n change the eigenbasis according to |n; R → |n; R ′ = e iα n (R) |n; R , which is called a gauge transformation.When the adiabatic approximation was first studied, people assumed that it would be always possible to get rid of phase factors by simply performing a gauge transformation, if necessary (A.Bohm, 2003).Berry's discovery made clear that this is not always the case.The point is that we are not always totally free to choose the required phase factors when performing gauge transformations.Let us see why it is so.To this end, we consider first two simple cases in which phase factors appear that can be eliminated.
A first case is a conservative system (∂H/∂t = 0).The initial condition |ψ(0) = |n; R leads to |ψ(t) = exp(−iE n t) |n; R .In this case the phase factor can be gauged away.A second case is a non-conservative system whose Hamiltonian is such that [H(t), H(t ′ )] = 0 for all t and t ′ .I n t h i s c a s e |ψ(t) = exp(−i 0) and the phase factor can again be gauged away.Now, if [H(t), H(t ′ )] = 0 the evolution is given by |ψ 0) ,w h e r eT means the time-ordering operator.In this case, the phase-factor cannot generally be gauged away.To see why is this the case, let us first restrict ourselves to a slowly evolving Hamiltonian and to an approximate solution of Eq.( 1), the so-called adiabatic approximation: When R(t) describes a closed path (R(T)=R(0))s oa l s od o e sρ(t) under the adiabatic approximation, because the eigenprojectors are single-valued: |ψ(T) ψ(T)| ≈ |n; R(T) n; R(T)| = |n; R(0) n; R(0)|.However, the state |ψ(t) may acquire a phase.Note that |ψ(t) ψ(t)| ≈ |n; R(t) n; R(t)| cannot be upgraded to an equality.This follows from observing that H(R(t)) and |n; R(t) n; R(t)| commute, so that for |ψ(t) ψ(t)| = |n; R(t) n; R(t)| to satisfy Eq.( 1), it must be stationary.Let us see under which conditions the adiabatic approximation applies.Writing |ψ(t) = ∑ k c k (t) |k; R(t) , the adiabatic approximation means that |ψ(t) ≈ c n (t) |n; R(t) ,withc n (0)=1, because |ψ(0) = |n; R(0) .By replacing such a |ψ(t) in the Schrödinger equation one easily obtains the necessary and sufficient conditions for the validity of the adiabatic approximation (A.Bohm, 2003): Hence, the energy differences E n (R) − E k (R) -or correspondingly, the transition frequencies of the evolving system -set the time scale for which the variation of H(t) can be considered "adiabatic", and |ψ(t) ≈ c n (t) |n; R(t) a valid approximation.Next, we multiply Eq.( 4) by n; R(t)| and obtain 7 )   whose solution is is the geometric phase, which is defined modulo 2π.We see that it appears as an additional phase besides the dynamical phase Φ dyn .W eh a v et h u s , The geometric phase γ n can also be written in the following way, to make clear that it does not depend on the parameter s: The vector potential A (n) ≡ i n; R| ∇ |n; R is known as the Mead-Berry vector potential.Eq.( 11) makes clear that γ n depends only on the path defining the environmental process, i.e., the path joining the points R(0) and R(t) in parameter space.This highlights the geometrical nature of γ n .Now, one can straightforwardly prove that a gauge transformation |n; R → |n; R ′ = e iα n (R) |n; R causes the vector potential to change according to As a consequence, the geometric phase transforms as At first sight, gauge freedom seems to be an appropriate tool for removing the additional phase factor exp (iγ n ) in Eq.( 10).Indeed, we can repeat the calculations leading to Eq.( 10) but now using |n; R ′ = e iα n (R) |n; R instead of |n; R .We thus obtain an equation like Eq.( 10) but with primed quantities.We could then choose This is what V. Fock made when addressing adiabatic quantal evolutions (A.Bohm, 2003), thereby exploiting the apparent freedom one has for choosing α n (R) when defining the eigenvectors |n; R ′ = e iα n (R) |n; R .However, when the path C is closed, a restriction appears that limits our possible choices of phase factors.This follows from the fact that R(T)=R(0) implies that |n; R(T) = |n; R(0) , because eigenvectors are single-valued (something we can always assume when a single patch is needed for covering our whole parameter space; otherwise, trivial phase factors are required).The eigenvectors |n; R ′ are also single-valued, so that |n; R(T) ′ = e iα n (R(T)) |n; R(T) = e iα n (R(0)) |n; R(0) = |n; R(0) ′ = e iα n (R(0)) |n; R(T) .
We have thus the restriction exp (iα n (T)) = exp (iα n (0)), which translates into α n (T)= α n (0)+2πm,withm integer.Hence, because of Eq.( 13), a n dw ec o n c l u d et h a tγ n (T) is invariant,m o d u l o2 π, under gauge transformations.Thus, it cannot be gauged away, as initially expected.According to Eq.( 11) γ n is independent of the curve parameter (t), so that we should write γ n (C) instead of γ n (T).We have, finally, with This is Berry's result (M.V. Berry, 1984).The vector potential A (n) behaves very much like an electromagnetic potential.The phase factors exp(iα n (R)) belong to the group U(1),hencethe name "gauge transformations" given to the transformations |n; R → |n; R ′ = e iα n (R) |n; R .
As in electromagnetism, we can also here introduce a field tensor F (n) whose components are Geometrically, F (n) has the meaning of a "curvature".In differential geometry, where the language of differential forms is used, A (n) is represented by a one-form, and F (n) by a two-form.When the parameter space is three-dimensional, Eq.( 19) can be written as Eq.( 18) can then be written as with a single patch.One needs at least two of them, which requires introducing two vector potentials, one for each patch.They are related to one another by a gauge transformation, i.e., their difference is a gradient.The corresponding curvature three-vector F = ∇×A is given by F = −e r /2r 2 ,w i t he r the unit radial vector.We note in passing that F = −e r /2r 2 looks like a Coulomb field, while F = ∇×A looks like a magnetic field.This hints at a formal connection between Berry's phase and Dirac's magnetic monopoles.In this case, Fig. 1.A spin-1/2 subjected to a variable magnetic field B(t) that describes a closed trajectory.When the field changes slowly in the time scale of the spin dynamics, then the spin S can follow the field adiabatically.After a period, the spin state has accumulated a geometric phase in addition to the dynamical one.
Ω(C) being the solid angle enclosed by C.This important result can be generalized to arbitrary dimensions, as we shall see below.
We have introduced Berry's phase by considering a unitary, cyclic and adiabatic evolution.This was Berry's original approach.It was generalized to the non-adiabatic case by Aharonov and Anandan (Y.Aharonov, 1987), as already said, and by Samuel and Bhandari (J.Samuel, 1988) to the noncyclic case.A purely kinematic approach showed that it is unnecessary to invoke unitarity of the evolution.Such an approach was developed by Mukunda and Simon (N.Mukunda, 1993) and is the subject of the next Section.

The kinematic approach: total, geometric, and dynamical phases
Now, consider the initial |ψ(s 1 ) and the end point |ψ(s 2 ) of C 0 .Following Pancharatnam, we define the total phase between these states as From these properties it is easy to see that we can construct the following quantity, the "geometric phase", which is gauge-invariant: Besides being re-parametrization invariant, Φ g (C 0 ) is, most importantly, also gauge invariant.This means that despite being defined in terms of |ψ(s) and C 0 , Φ g effectively depends on equivalence classes of |ψ(s) and C 0 , respectively.Indeed, the set {|ψ ′ = exp (iα) |ψ } constitutes an equivalence class.The space spanned by such equivalence classes is called the "ray space" R 0 .Instead of working with equivalence classes we can work with projectors: |ψ ψ|.T h es e t{|ψ ′ = exp (iα) |ψ } projects onto the object |ψ ψ| by means of a projection map π : H 0 →R 0 .In particular, the curves C 0 , C ′ 0 which are interrelated by a gauge transformation, are also members of an equivalence class.Under π, they project onto a curve C 0 ⊂R 0 .What we have seen above is that Φ g is in fact a functional not of C 0 ,b u to fC 0 , the curve defined by |ψ(s) ψ(s)|.This is the reason why we call Φ g the "geometric phase" associated with the curve C 0 ⊂R 0 .We should then better write Φ g (C 0 ), though its actual calculation requires that we choose what is called a "lift" of C 0 ;thatis,anycurveC 0 such that π(C 0 )=C 0 .Thus,Φ g (C 0 ) is defined in terms of two phases, see Eq.( 24): Φ dyn (C 0 )=Im Φ tot (C 0 ) is, as already said, the total or the Pancharatnam phase of C 0 .It is the argument Later on, we will discuss the physical meaning of this phase in the context of polarized states, the case addressed by Pancharatnam.Φ dyn (C 0 ) is the dynamical phase of C 0 .We see that even though both Let us stress that this definition of the geometric phase does not rest on the assumptions originally made by Berry.Φ g (C 0 ) has been introduced in terms of a given evolution of state vectors |ψ(s) .This evolution does not need to be unitary, nor adiabatic.Furthermore, the path C 0 could be open: no cyclic property is invoked.Given a C 0 ⊂R 0 , we may choose different lifts to calculate Φ g (C 0 ) and exploit this freedom to express Φ g (C 0 ) according to our needs.For example, we can always make Φ tot (C 0 )=0, by properly choosing the phase of, say, |ψ(s 2 ) .I nt h a tc a s e ,Φ g (C 0 )=−Φ dyn (C 0 ).Alternatively, we can make Φ dyn (C 0 )=0, so that Φ g (C 0 )=Φ tot (C 0 ), by choosing a so-called "horizontal lift", one which satisfies K(s) is obviously gauge invariant; hence, Eq.( 29) holds also for gauge-transformed quantities.By choosing a horizontal lift, ψ(s)| ψ(s) = 0, Eq.( 29) reads The solution of Eq.( 30) can be formally given as a Dyson series: |ψ(s) = P exp s s 1 ρ(s)ds |ψ(s 1 ) ,withP the "parameter-ordering" operator: it rearranges a product of parameter-labelled operators according to, e. g., Eq.( 31) gives the desired expression of Φ g (C 0 ) in terms of ray-space quantities.C 0 is any smooth curve in ray space.If C 0 is closed, ρ(s 2 )=ρ(s 1 ),and|ψ(s 2 ) must be equal to |ψ(s 1 ) up to a phase factor: |ψ(s 2 ) = e iα |ψ(s 1 ) ,withα = arg ψ(s 2 )|ψ(s 1 ) .For the horizontal lift we are considering, α = arg ψ(s 2 )|ψ(s 1 ) = Φ g (C 0 ), and we can thus write |ψ(s 2 ) = P exp in accordance with our previous results.

Geodesics
We introduce now the concept of geodesics in both Hilbert-space and ray-space, with the help of Eq.( 29).Notice that Geodesics are defined as curves making L(C 0 ) extremal.By applying the tools of variational calculus one obtains (N.Mukunda, 1993) with f (s) an arbitrary, real function.Although Eq.( 34) depends on the lifted curve C 0 ,i t must be gauge and re-parametrization invariant, because it follows from Eq.( 33).We may therefore change both the lift and the parametrization in Eq.( 34).We choose a horizontal lift: . Furthermore, because of re-parametrization freedom we may take s such that ψ(s)| ψ(s) is constant along C 0 .T h i s fixes s up to linear inhomogeneous changes, i.e., up to affine transformations.Then, Eq.( 34) reads Now, by deriving twice the equation Thus, Eq.( 35) reads finally with ω 2 ≡ ψ(0)| ψ(0) .This equation holds for geodesics that are horizontal lifts from the geodesic C 0 in ray space, and with s rendering ψ(s)| ψ(s) constant.Eq.( 37) is thus of second order and its general solution depends on two vectors.It can be solved, e.g., for the initial conditions |ψ(0) = |φ 1 and | ψ(0) = ω|φ 2 , i.e., φ 1 |φ 1 = 1, φ 1 |φ 2 = 0, and We see that ψ(0)|ψ(s) = φ 1 |ψ(s) = cos (ωs).B e c a u s es has been fixed only up to an affine transformation, we can generally choose it such that cos (ωs) Eq.( 38) shows that geodesics are arcs of circles in a space with orthonormal basis {|φ 1 , |φ 2 }.

Bargmann invariants
Consider N points in ray space: ρ 1 , ρ 2 ,...,ρ N .As we have seen, each pair can be connected by a geodesic arc.Let us denote by C 0 the curve formed by the N − 1 geodesic arcs joining the N points.Let us assume that any two neighboring points are nonorthogonal.That is, for any lift |ψ 1 , |ψ 2 ,...,|ψ N ,itholds ψ i |ψ i+1 = 0, for i = 1,...,N − 1.The geometric phase Φ g (C 0 ) is given by (k,k+1) dyn is the dynamical phase for the geodesic joining |ψ k with |ψ k+1 .B e c a u s e and we can finally write Although Φ g (C 0 ) has been derived by joining |ψ 1 ,...,|ψ N with geodesic arcs, the final expression does not depend on these arcs, but only on the vectors they join.Quantities like ψ 1 |ψ 2 ψ 2 |ψ 3 ψ 3 |ψ 1 are called "Bargmann invariants".They generalize Quantities that are invariant under U(1) ⊗ U(1) ⊗ ... were introduced by Bargmann for studying the difference between unitary and anti-unitary transformations.The curve C 0 in Eq.( 45) was assumed to be open: ρ N = ρ 1 .However, we can close the curve to C 0 ,b yc o m p l e t i n gt h eN − 1-sided polygon C 0 with a geodesic arc connecting ρ N with ρ 1 .By repeating the steps leading to Eq.( 45), though taking into account that now ) is given again by Eq.( 46).In other words, Φ g ( C 0 )= Φ g (C 0 ).

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The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects www.intechopen.comStarting from Eq.( 46) it is possible to recover the results previously found for general open curves (N.Mukunda, 1993).One proceeds by approximating a given curve by a polygonal arc made up of N → ∞ geodesic arcs.By a limiting procedure one recovers then Φ g (C 0 )= Φ tot (C 0 ) − Φ dyn (C 0 ) with Φ tot (C 0 ) and Φ dyn (C 0 ) given by Eqs.( 25) and ( 26), respectively.Also Eq.( 31) can be recovered in a similar fashion (N.Mukunda, 1993).The quantity ψ 1 |ψ 3 ψ 3 |ψ 2 ψ 2 |ψ 1 , the three-vertex Bargmann invariant, can be identified as the basic building block of geometric phases.It can be seen as the result of two successive filtering measurements, the first projecting |ψ 1 on |ψ 2 , followed by a second projection on |ψ 3 .The phase of the final state with respect to the first one is Here, Ω △ p is the solid angle subtended by the spherical triangle formed by shorter geodesics between |ψ 2 , |ψ 3 and the projection |ψ 1 p of |ψ 1 on the subspace spanned by the other two vectors.Now, given a closed curve C 0 , by triangulation with infinitesimal geodesic triangles it is possible to express Φ g ( C 0 ) as (A.G. Wagh, 1999) thereby generalizing Eq.( 22).

Interferometric arrangement
We introduced the total phase, arg ψ 1 |ψ 2 , as a generalization of Pancharatnam's definition for the relative phase between two polarized states of light.According to Pancharatnam's definition, we can operationally decide whether two nonorthogonal states are "in phase".Consider two nonorthogonal polarization states, |i and | f = |i , and let them interfere.Due to the optical-path difference, there is a relative phase-shift φ giving rise to an intensity pattern The maxima of I occur for φ = arg i| f ≡ Φ tot , which is thereby operationally defined as the total (Pancharatnam) phase between |i and | f .I fa r g i| f = 0, the states are said to be "in phase".Polarization states are two-level systems.When they are submitted to the action of intensity-preserving optical elements, like wave-plates, their polarization transformations belong to the group SU(2) (modulo global phase factors).We can exhibit Φ tot by submitting |i to U ∈ SU( 2), thereby producing a state | f = U |i .Eq.( 48) applies to, say, a Mach-Zehnder array.Alternatively, one could employ polarimetric methods.We will discuss both methods in what follows.Among the different parameterizations of U, the following one is particularly well suited for extracting Pancharatnam's phase:  (β, γ, δ) |+ = e iδ cos β.
(50) Thus, Φ tot = arg i| f = δ + arg(cos β),forβ =(2n + 1)π/2.Because cos β takes on positive and negative real values, arg(cos β) equals 0 or π,a n dΦ tot is thus given by δ modulo π.I n principle, then, we could obtain Φ P (modulo π) by comparing two interferograms, one taken as a reference and corresponding to Φ P = 0( U = I), and the other corresponding to the application of U. Their relative shift gives Φ P .We can implement unitary transformations using quarter-wave plates (Q) and half-wave plates (H).These transformations are of the form U(ξ, η, ζ)=exp −iξσ y /2 exp (iησ z /2) exp −iζσ y /2 .They can be realized with the following gadget (R. Simon, 1990), in which the arguments of Q and H mean the angles of their major axes to the vertical direction: The corresponding interferogram has an intensity pattern given by I V refers to an initial state |+ z that is vertically polarized.This result follows from the parametrization of U given by U (ξ, η, ζ).By using the relationship between this parametrization and that of Eq.( 49), i.e., U(β, γ, δ), one can show that I V canbewrittenas Pancharatnam's phase Φ P = δ is thus given by the shift of the interferogram I V with respect to a reference interferogram I = [1 − cos β cos φ] /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 Φ P .A way out of this situation follows from observing that the intensity pattern corresponding to an initial, horizontally polarized state |− z is given by Hence, the relative shift between I V and I H is twice Pancharatnam's phase.If one manages to divide the laser beam into a vertically and a horizontally polarized part, the two halves of the laser beam will be subjected to equal disturbances and one can record two interferograms in a single shot.The relative shift would be thus easily measurable, being robust to thermal and mechanical disturbances.With such an array it is possible to measure Pancharatnam's phase for different unitary transformations.This approach proved to be realizable, using either a beam expander or a polarizing beam displacer (J.C. Loredo, 2009).
A similar approach can be used to measure the geometric phase One can exploit the gauge freedom and choose an appropriate phase factor exp(iα(s)),s o as to make Φ dyn (C 0 )=0a l o n gac u r v eC 0 : |ψ(s) , s ∈ [s 1 , s 2 ] which is traced out by polarization states |ψ(s) resulting from U(s): |ψ(s) = U(s) |ψ(0) .A n yU(s) can be realized 301 The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects www.intechopen.comby making one or more parameters in U(ξ, η, ζ) (see Eq.( 51)) functions of s.Setting the corresponding QHQ-gadget on one arm of the interferometer, one lets the polarization state |ψ(s) follow a prescribed curve.A second QHQ-gadget can be put on the other arm, in order to produce the factor exp(iα(s)) that is needed to make Φ dyn (C 0 )=0.To fix α(s),onesolves The corresponding interferometric setup is shown in Fig.
(2).It is of the Mach-Zehnder type; but a Sagnac and a Michelson interferometer could be used as well.With the help of this array one can generate geometric phases associated to non-geodesic trajectories on the Poincaré sphere (J.C. Loredo, 2011).In this way, one is not constrained to use special trajectories, along which the dynamical phase identically vanishes (Y.Ota, 2009).The geometric phase is nowadays seen as an important tool for implementing robust quantum gates that can be employed in information processing (E.Sjöqvist, 2008).It appears to be noise resilient, as recent experiments seem to confirm (S.Fillip, 2009) recorded with the help of a CCD camera and evaluated using an algorithm that performs a column average of each half of the interferogram.The output was 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.This procedure could be applied to a set

Polarimetric arrangement
Some years ago, Wagh and Rakhecha proposed a polarimetric method to measure Pancharatnam's phase (A.G. Wagh, 1995;b).Such a method is experimentally more demanding than the interferometric one, but it was considered more accurate because it requires a single beam.Both methods were tested in experiments with neutrons (A.G. Wagh, 1997;2000), whose spins were subjected to SU(2) transformations by applying a magnetic field.Now, it is not obvious that one can extract phase information from a single beam.As we shall see, polarimetry can be understood as "virtual interferometry", in which a single beam is decomposed in two "virtual" beams.

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Theoretical Concepts of Quantum Mechanics www.intechopen.comSimon, 1990), we can reduce the array from elf to seven retarders: with γ ≡ σ + α(s)=σ + s sin θ cos ϕ.For each fixed value of s -t h a ti s ,f o re a c h point on the chosen trajectory -one generates an intensity pattern through variation of σ, i.e., by rotating the whole array σ radians over some interval, which should be large enough for recording several maximal and minimal intensity values.From these values one can obtain Φ g (s).Indeed, the intensity is given by and it can be proved (J.C. Loredo, 2011) that in the present case I = cos 2 (s) + sin 2 (s)[cos (θ) cos (σ − α(s)) − sin (θ) sin (ϕ) sin (σ − α(s))] 2 .From this result one derives the following expression for the geometric phase (J.C. Loredo, 2011): This result has been tested for various trajectories (J.C. Loredo, 2011), confirming theoretical predictions with the expected accuracy.Though all these experiments were performed with a cw He-Ne laser, an alternative setting using single-photon sources should produce similar results.This is so because all the aforementioned results have topological, rather than classical or quantal character.

Geometric phase for mixed states
Up to this point, the geometric phase refers to pure states ρ = |ψ ψ|.It is natural to ask whether geometric phases can be defined for mixed states as well.Uhlmann addressed this question (A.Uhlmann, 1986) and introduced a phase based on the concept of parallel transport.When a pure state |ψ(s) evolves under parallel transport, it remains in phase with |ψ(s + ds) , i.e., the system does not suffer local phase changes.After completing a closed loop, a state may acquire a nontrivial phase, stemming from the curvature of the underlying parameter space.This notion can be extended to mixed states.To this end, Uhlmann considered so-called "purifications" of mixed states.That is, one considers a mixed state as being part of a larger system, which is in a pure state.There are infinitely many possible purifications of a given mixed state.Hence, to a given cyclic evolution there correspond infinitely many evolutions of the purifications.However, one of these evolutions can be singled out as the one which is "maximally parallel" (A.Uhlmann, 1986), and this leads to a definition of geometric phases for mixed states.An alternative approach was addressed more recently by Sjöqvist et al. (E. Sjöqvist, 2000).The starting point is Pancharatnam's approach; i.e., the interference between two states: |i , to which a phase-shift φ is applied, and | f = U|i ,withU unitary.The interference pattern is given by 305 The Pancharatnam-Berry Phase: Theoretical and Experimental Aspects www.intechopen.com with v = | i|U|i | being the visibility and Φ tot = arg i|U|i the total phase between |i and U|i .
Consider now a mixed state ρ = ∑ i w i |i i|,with∑ i w i = 1.The intensity profile will now be given by the contributions of all the individual pure states: We can write I in a basis-independent form as (E.Sjöqvist, 2000) It is then clear that v = |Tr (Uρ)| and that the total phase can be operationally defined as Φ tot = arg Tr (Uρ), which is the value of the shift φ at which maximal intensity is attained.As expected, such a definition reduces to Pancharatnam's for pure states ρ = |i i|.
Let us now address the extension of the geometric phase for mixed states.For pure states |ψ(s) the geometric phase equals Pancharatnam's phase whenever |ψ(s) evolves under parallel transport: ψ(s)| ψ(s) = 0. We can try to extend the notion of parallel transport for mixed states by requiring ρ(s) to be in phase with ρ(s + ds)=U(s + ds) ρ 0 U † (s + ds)=U(s + ds) U † (s)ρ(s)U(s)U † (s + ds).According to our previous definition, the phase difference between ρ(s) and ρ(s + ds) is given by arg Tr U(s + ds)U † (s)ρ(s) in this case.We say that ρ(s) and ρ(s + ds) are in phase when arg Tr U(s + ds)U † (s)ρ(s) = 0, i.e., Tr U(s + ds)U † (s)ρ(s) is a positive real number.Now, because Tr (ρ(s)) = 1andρ(s) † = ρ(s), the number Tr UU † ρ is purely imaginary.Hence, a necessary condition for parallel transport is However, such a condition is not sufficient to fix U(s) for a given ρ(s).Indeed, considering any N × N matrix representation of the given ρ, Eq.( 65) determines U only up to N phase factors.In order to fix these factors we must impose a more stringent condition: where ρ(s)=∑ k w k |k(s) k(s)|.This gives the desired generalization of parallel transport to the case of mixed states.We can now define a geometric phase for a state that evolves along the curve C : s → ρ(s)=U(s)ρ 0 U † (s),withs ∈ [s 1 , s 2 ] and U(s) satisfying Eqs.( 65) and (66).
The dynamical phase Φ dyn ≡− i s 2 s 1 dsTr U † (s) U(s)ρ(0) = 0 and we define the geometric phase Φ g for mixed states as Φ g is gauge and parametrization invariant and has been defined for general paths, open or closed.In special cases, Φ g can be expressed in terms of a solid angle, as it is the case with Berry's phase.For example, a two-level system can be described by 306 with − → n • − → n = 1andr constant for unitary evolutions.For pure states r = 1, while for mixed states r < 1.The unitary evolution of ρ(s) makes − → n (s) to trace out a curve C on the Bloch sphere.If necessary, we close C to C by joining initial and final points with a geodesic arc, so that C subtends a solid angle Ω.Then, the two eigenstates |±; The eigenvalues of ρ are w ± =(1 ± r)/2.The geometric phase thus reads and the visibility Eqs.( 69) and ( 70) reduce for r = 1t oΦ g = −Ω/2 and v = v 0 , respectively, the known expressions for pure states.For maximally mixed states, r = 0, we obtain Φ g = arg cos (Ω/2), v = |cos (Ω/2)|, and Eq.( 64) yields We see that for Ω = 2π there is a sign change in the intensity pattern.This was experimentally observed in early experiments testing the 4π symmetry of spin-1/2 particles (H.Rauch, 1975).Much later, theoretical results like those expressed in Eqs.(69,70) have been successfully put to experimental test (M.Ericsson, 2005).The above extensions of Pancharatnam's and geometric phases assume a unitary evolution |i →| f = U|i .A non-unitary evolution -reflecting the influence of an environmentcan be handled with the help of an ancilla; that is, by replacing the true environment by an environment simulator, a fictitious system being in a pure state |0 e 0 e |, which is appended to the given system.The system plus the environment simulator are then described by ρ = ρ ⊗|0 e 0 e | and evolve unitarily, ρ → ρ ′ = U ρU † ,i ns u c haw a yt h a tb yt r a c i n go v e rt h e environment we recover the change of ρ → ρ ′ = Tr e ρ ′ .Introducing an orthonormal basis {|k e } k=0,...,M for the environment, we can write Tr e ρ ′ = ∑ k K k ρK † k ,withK k ≡ k e |U|0 e being so-called Kraus operators (S.Haroche, 2007).Using these tools it is possible to extend total and geometric phases to non-unitary evolutions (J.G. Peixoto, 2002).

Thomas rotation in relativity and in polarization optics
In this closing Section we address a well-known effect of special relativity, Thomas rotation, and show its links to geometric phases.We recall that Thomas rotation is a rather surprising effect of Lorentz transformations.These transformations connect to one another the coordinates of two inertial systems, O and O ′ ,b yx μ → x ′μ = Λ We can show that Eq.( 77) is just of this form.To this end, we write Denoting by |f ± the eigenvectors of − → f • − → ρ ;t h a ti s , Using exp A = ∑ n exp a n |a n a n | with A = − − → f • − → ρ and observing that exp − − → f • − → ρ has eigenvectors |f ± and eigenvalues exp (∓z),weget It is easy to show from Eq.( 85) that a Lorentz transformation exp(− − → f • − → ρ ) can generally be written as a product of a boost by a rotation.It is clear from Eq.( 77) that a rotation is obtained when − → α = − → 0 and a boost when − → β = − → 0 .A general rotation U(ξ, η, ζ) ∈ SU(2) can be implemented with the help of three wave-plates, see Eq.(51).A general boost can be implemented with dichroic elements realizing Eq.( 82).The global factor there, exp (−α s /2), corresponds to an overall intensity attenuation.We can thus in principle realize any transformation of the form exp(− − → f • − → ρ ) by using optical elements like wave-plates and dichroic elements.In particular, by letting a polarization state pass through two consecutive dichroic elements -each corresponding to a boost -we could make appear a phase between initial and final states.This is a geometric phase rooted on Thomas rotation, which can thus be exhibited by using the tools of polarization optics.Thus, we have here another example showing the topological root shared by two quite distinct physical phenomena.

Conclusion
Berry's phase was initially seen as a surprising result, which contradicted the common wisdom that only dynamical phases would show up when dealing with adiabatically evolving states.But soon after its discovery it brought to light a plethora of physical effects sharing a common topological or geometrical root.Once the initial concept was relatively well understood, people could recognize its manifestation in previously studied cases, like the Aharonov-Bohm effect and the Pancharatnam's prescription for establishing whether two polarization states of light are in phase.Thanks to the contributions of a great number of researchers, Berry's phase has evolved into a rich subject of study that embraces manifold aspects.There are still several open questions and partially understood phenomena, as well as promising approaches to implement practical applications of geometrical phases, notably those related to quantum information processing.The present Chapter can give but a pale portrait and a limited view of what is a wide and rich subject.However, it is perhaps precisely out of these limitations that it could serve the purpose of awaking the reader's interest for studying in depth such a fascinating subject-matter.

L
e tu ss t a r tb yc o n s i d e r i n gaH i l b e r ts p a c eH.W ed e fi n eH 0 ⊂Has the set of normalized, nonzero vectors |ψ ∈H.AcurveC 0 in H 0 is defined through vectors |ψ(s) that continuously depend on some parameter s ∈ [s 1 , s 2 ].B e c a u s e |ψ(s) is normalized, ψ(s)| ψ(s) + ψ(s)|ψ(s) = 0.Then, Re ψ(s)| ψ(s) = 0, and 294 Theoretical Concepts of Quantum Mechanics www.intechopen.comThe Pancharatnam-Berry Phase: Theoretical and Experimental Aspects 7 cos β − e iγ sin β e −iγ sin β e −iδ cos β .as initial state |i = |+ z ≡ |+ , the eigenstate of σ z for the eigenvalue +1, and setting | f = U |+ , we obtain i| f = +| U

Fig. 2 .
Fig. 2. Mach-Zehnder array for measuring the geometric phase.Quarter (Q) and half (H) wave plates are used for realizing the SU(2) transformations.L:H e -N el a s e r ,P, P 1 , P 2 : polarizers, E:b e a me x p a n d e r ,BS: beam-splitter, M:m i r r o r .

302TheoreticalFig. 3 .
Fig.3.Geometric phase for a non-geodesic trajectory on the Poincaré sphere.The trajectory is a circle resulting from intersecting a cone with the Poincaré sphere.It is fixed by the axis n of the cone and its aperture angle β.

Fig. 4 .
Fig. 4. The trajectory described on the Poincaré sphere.The dynamical phase is simultaneously cancelled by means of a QHQ gadget.