Geometric Phases

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I wrote this article as part of extending my undergraduate thesis. Its certainly not original but should be of benefit to someone.

1. Introduction

A considerable understanding of the formal description of quantum mechanics has been achieved after Berrys discovery{BerryPhase} of a geometric feature related to the motion of a quantum system. He showed that the wave function of a quantum object retains a memory of its evolution in its complex phase argument, which, apart from the usual dynamical contribution, only depends on the geometry of the path traversed by the system. Known as the geometric phase factor, this contribution originates from the very heart of the structure of quantum mechanics. It has been over 25 Years since Berry discovered his phase. An entire industry of problems have arisen since then. It has been shown in a various reviews that a better understanding of these problems comes from the usage of topological and geometric techniques. A unified approach to geometric phase problems comes from the usage of basic topological techniques The evolution of the electric field along the curved path of a light ray is described by the Fermi-Walker parallel transport law. For the polarized light the analog of magnetic flux is the solid angle subtended on the sphere of directions k, where k represents the direction of propagation of light, which changes as light passes through the twisted fiber. This spin redirection phase is known as the Rytov-Vlasimirski-Berry{rytov,Rytov_Review,Vladimirski} phase, as opposed to the Pancharatnam-Berry Phase. Since Geometric Phases are intrinsically related to fibre-bundle topological theories, a short reviews of Fiber bundle theories is provided. Deep understanding of geometric phases phases related to time development by the Schrodinger equation and interference effects in optics needs the use of analytical methods. We suspect that in Quantum Computation research, these topological techniques will become increasingly important. The main original work in this paper will be a model of the Manual Fiber Polarization controller(MPC) ~{MPC}.

2. Topological Phases in Optics

To elaborate further, there are two types of topological phase in optics. The first one concerns the propagation of light, in which the polarization state changes. As has been demonstrated, in this case the light acquires an addtional phase shift, whose value is determined trajectory of the polarization state on the Poincare Sphere. This phase shift is proportional to the solid angle subtended by the polarization trajectory and is proved to be a manifestation of the Berry Phase. A well know example is the twisted monomode fiber.

This can be explained as a manifestation of the Pancharatnam-Berry Phase. A second example of Berry’s phase has been provided by the study of light propagation in coiled monomode fibers. As has been both theoretically and experimentally demonstrated, coiling the fibre induces in it so-called geometric circular birefringence, which results in the rotation of the polarization plane of a light beam transmitted through the fibre. The present work examines how a type of polarization controller can be analyzed in terms of this spin-redirection Berry Phase, and especially examines how much geometric circular birefringence contributes to the working of such controllers. In the original papers it was suspected to be stress induced birefringence. however we believe that the geometric phase explanation is more powerful. \paragraph{} There is an analogy between the change in direction {(\hat{k}_{x} , \hat{k}_{y} , \hat{k}_{z} )} of the photon obtained by using externally slowly varying parameters and by adiabatic change in the direction of the magnetic field {(B_{x} , B_{y} , B_{z} )}, where the geometrical phase for the latter case is has been treated by Berry. If there is nothing to change the sign of the helicity of the photon, the helicity quantum number is an adiabatic invariant. When the photon propagates smoothly down a helical waveguide (or more pratically through a fibre) k is constrained to remain parallel to the local axis of the waveguide, since the momentum of the photon is in this direction. The geometry of a helical path of a waveguide (or fibre with a unity winding number) contrains k to trace out loop on the surface of a sphere in parameter space {(k_{x} , k_{y} , k_{z} )} where the origin of this space k = 0 is singular. As was noted in previous studies the simple geometric of a circular helix gives the solid angle equal to
{\Omega = 2\pi(1 − \cos \theta)} where \theta is the pitch angle of the helix, or the angle between the axis of the helix and the local axis of the optical fiber.

2.1. Geometric Phases

Quantum states are represented as vectors in a complex vector space. However these vectors are only defined upto a unit modulus complex number, or a phase in other words. So the two states {|{x_{i}>}} and {\exp^{i\mu}|{x_{i}>}} are indistinguishable as far as quantum mechanics is concerned – no set of measurements can discriminate them. It is interesting that although this statement looks fairly innocent at first sight, it can lead to some very profound conclusions. We shall take the well know route of considering the extra phase, geometrically in the complex plane. \newline Even though we tend to think of the amplitudes between two quantum states as being fundamental quantum entities, experimentally we can only measure probabilities. We denote the gap between two quantum states {|{\psi_{i}}} and {|{\psi_{f}}>} as the mod squared overlap {} The rotations of a vector r(t) in {\mathbb{R}^{3}} space are described by the SO(3) group. For a one-parameter group with angular velocity {\mathbf{\omega}} we get

\displaystyle \frac{d\mathbf{r}(t)}{dt}\arrowvert_{t=t_{0}} = \mathbf{\omega}\times\mathbf{r}(t_{0} \ \ \ \ \ (1)

Starting with a vector in {\mathbb{R}^{3}} given by (0,0,1), the development on this vector with the SO(3) will lead to a vector (x,y,z) with a norm 1 ({(x^{2}+y^{2}+z^{2}=1}). The tip of this vector is moving on the surface of a sphere called a Poincare Sphere. An example which is related to the SO(3) group is given by a two level system described by the wavefunction

\displaystyle |{\psi}>=C_{1}|{\psi_{1}}>+C_{2}|{\psi_{2}}> \ \ \ \ \ (2)

where {C_{1}} and {C_{2}} are complex numbers. The components of the Bloch Vector are given by

\displaystyle r_{1}=\frac{C_{1}\bar{C_{2}}+\bar{C_{1}}C_{2}}{\sqrt{2}};\;\; r_{2}=\frac{C_{1}\bar{C_{2}}+\bar{C_{1}}C_{2}}{\sqrt{2i}}; \\ r_{3} = C_{1}\bar{C_{1}}-\bar{C_{2}}C_{2}. \ \ \ \ \ (3)

{r_{1}} and{r_{2}} are defined as the components of a complex dipoe vector, while {r_{3}} is defined as the inversion. The interaction between the TLS and resonant em fields leads (usually in a rotated system) to rotation of the Bloch vector on the Poincare Sphere.

\paragraph{} As is well known, in the isotropic linear medium, the Maxwells equations can be written in the form~{Geo_Schrodinger}:

\displaystyle \frac{i}{c_{n}}\frac{\partial}{\partial t}\mathbf{\Phi}=\nabla \times \mathbf{\Phi},\; \nabla\cdot\mathbf{\Phi}=0, \ \ \ \ \ (4)

where {\mathbf{\Phi}=\sqrt{\epsilon}\mathbf{E}+i\mathbf{H}, c_{n}=\frac{1}{\sqrt{\mu \epsilon}}}, {\mu} and {\epsilon} are magnetic conductivity and dielectric constant, respectively. \paragraph{} In the case of an optical fiber with a step-index-type profile, under the weak guidance approximation, there are two solutions for equation (1).

\displaystyle \mathbf{\Phi_{\pm}}=\Phi_{0}a_{\pm}(s)exp[\mp i(ks-\omega t)][\mathbf{n}(s)+i\mathbf{b}(s)] \ \ \ \ \ (5)

where {\pm} correspond to the left and right circularly polarizations, respectively. Here we denote s as the arc length of the fiber curve, {\omega} is the frequency of the light, k={\omega/c_{n}} is the propagation constant, {\mathbf{n}} and {\mathbf{b}} are normal vector and binormal vector of the fiber curve, respectively, and {\Phi_{0}} is a constant.
On account of the Frenet Formula~{GeometryPhysics, Struik} with Eqs. (1) and (2) we have

\displaystyle a_{+}(s)=a_{-}(s)=exp[i\gamma(s)], \ \ \ \ \ (6)

\displaystyle \gamma(s)=\int \tau ds, \ \ \ \ \ (7)

where {\tau} is the torsion of the fiber curve. After some analysis, {\gamma(s)} is the rotated angle about the normal vector {\mathbf{n}} caused by the electric vector.

In the case of a Manual Polarization Controller (MPC) the model indicates that {\tau} becomes infinite at some point.

2.2. Propogation of a linearly polarized EM wave

Let us consider a linearly polarized electromagnetic wave propagating in the direction of a wave vector k(t) such that t(0)=t(T) for some T > 0, i.e., the initial and final directions of the fibre coincide. If the shape of the fiber is represented by a curve C, then under the Gauss map C is mapped onto a closed curve {\tilde{C}} on the sphere of directions. No, in the plane {\Sigma} perpendicular to t(0)=t(T) let us introduce an orthonormal basis {\mathbf{\epsilon}_{1},\mathbf{\epsilon}_{2}}. Suppose that {\mathbf{\epsilon}_{1}} is the initial polarization vector, that is, {\mathbf{\epsilon}_{0}=\mathbf{\epsilon}_{1}}. What is the final polarization {\mathbf{\epsilon}(T)}? Clearly, {\mathbf{\epsilon}(0)} and {\mathbf{\epsilon}(T)} differ by a {SO(2)}-rotation in the plane {\Sigma}, i.e.

\displaystyle \mathbf{\epsilon}(T)=R(\psi)\mathbf{\epsilon}(0) \ \ \ \ \ (8)

Introducing circular polarization vectors

\displaystyle \mathbf{\epsilon}_{+}=\frac{\mathbf{\epsilon}_{0} \pm i\mathbf{\epsilon}_{2}}{\sqrt{2}} \ \ \ \ \ (9)

one has

\displaystyle \mathbf{\epsilon}_{+}=\frac{\mathbf{\epsilon}_{+}+\mathbf{\epsilon}_{-}}{\sqrt{2}} \ \ \ \ \ (10)

that is, the initial linear polarization is a superposition of right (s=1) and left (s=-1) polarized waves. Now, the helicity eigenstates (with eigenvalues s = {\pm 1}) acquire a geometric phase {s\Omega(\tilde{C})} after passage through the optical fibre, where {\Omega(\tilde{C})} is the solid angle subtended by {\tilde{C}} on the sphere of directions. Hence, the final polarization is

\displaystyle \mathbf{\epsilon(T)}=\frac{1}{\sqrt{2}}\left(\exp^{-i\Omega(\tilde{C})}\mathbf{\epsilon}_{+} + \exp^{-i\Omega(\tilde{C})}\mathbf{\epsilon}_{-} \right)=\mathbf{\epsilon}_{1}\cos\Omega(\tilde{C})+\mathbf{\epsilon}_{2}\sin\Omega(\tilde{C}) \linebreak \\ = R(\Omega(\tilde{C}))\mathbf{\epsilon}(0) \ \ \ \ \ (11)

Thus, the geometric phase that appears for circularly polarized photons corresponds to rotation of the linear polarization vector {\mathbf{\epsilon}} by the angle

\displaystyle \psi=\Omega(\tilde{C}) \ \ \ \ \ (12)

which is the basic geometric phase relation.

\paragraph{} In accordance with the popular terminology, we use the Jones vector Jto express the electromagnetic vector in this case. In the local co-ordinate system it can be written as

\displaystyle \mathbf{J}(s)= \left( \begin{array}{ccc} \epsilon_{n}(s) \\ \epsilon_{b}(s) \end{array} \right)exp(iks) \newline = \left( \begin{array}{ccc} cos\gamma & sin\gamma \\ -sin\gamma & cos\gamma \end{array} \right)exp(iks)\mathbf{J}(0)\ \ \ \ \ (13)

satisfying

\displaystyle \frac{d}{ds}\mathbf{J}(s)=N(s)\mathbf{J}(s),\ \ \ \ \ (14)

\newline N(s)=\left({cc} ik & \tau
-\tau & ik \right). \paragraph{}

3. General theories of topological phases

3.1. Berry’s Phases for the Schrodinger equation under the adiabatic approximation

In the present section we summarize the main results obtained by Berry in his paper of 1984. Although there is no experimental evidence for the existence of magnetic charges or monopoles, the interest in such monopoles arose in the scientific area of Berry’s phases due to the fact that fomally certain Berry’s phase systems have the mathematical structure as the Dirac magnetic monopole. The main purpose is not to discuss mathematical superstructure, for mathematics sake, but to provide the computational and topological tools for further analysis of Topological and Geometric Phases. An excellent elaboration and review of the literature is provided in

3.2. Berry’s phases on a Poincare sphere

Let us give a simple explanation for the description of a {Poincar\acute{e}} Sphere using Stokes parameters which are equivalent to the Bloch vector. The em field of a monochromatic plane polarized light propagating in the z direction can be given as

\displaystyle E_{x}=a_{1}\cos(\tau+\delta_{1});\;\;\;\;\; E_{y}=a_{2}\cos(\tau+\delta_{2}) \ \ \ \ \ (15)

where {a_{1}} and {a_{2}} are the amplitudes in the x and y directions. {\tau} is the variable phase factor while {\delta_{1}} and {\delta_{2}} are constants. By using a simple algebra {PrinOptics} one obtains the equation for an ellipse:

\displaystyle \left(\frac{E_{x}}{a_{1}}\right)^{2}+\left(\frac{E_{y}}{a_{2}}\right)^{2}-2\left(\frac{E_{x}}{a_{1}}\right)\left(\frac{E_{y}}{a_{2}}\right)\cos\delta=\sin^{2}\theta \ \ \ \ \ (16)

where

\displaystyle \delta=\delta_{1}-\delta_{2} \ \ \ \ \ (17)

The ellipse will be reduced to linearly polarized light when {\delta=\delta_{2}-\delta_{1}} = m{\pi} {m=0,\pm1,\pm2,...}. The ellipse is reduced to a right-handed (left-handed) circularly polarizedwave for {\delta}=2m {\pi}+{\frac{\pi}{2}}(2m{\pi}{\frac{\pi}{2}}){PrinOptics}. Instead of using the real form of (15), it is more convenient to describe the em field as a complex Jones vector {OpticalElectronics}

\displaystyle E_{x}(complex)=a_{1}e^{i(\tau+\delta_{1}}=A_{1}; \ \ \ \ \ (18)

\displaystyle E_{y}(complex)=a_{2}e^{i(\tau+\delta_{2}}=A_{2} \ \ \ \ \ (19)

The Jones vector is described as a complex spinor vector {\left(\frac{A_{1}}{A_{2}}\right)}. The Stokes parameters of a plane monochromatic wave are described by the four quantities

\displaystyle \begin{eqnarray}{rcl} S=a^{2}_{1}+a^{2}_{2}=|A_{1}|^{2}+|A_{2}|^{2}, \\ S_{1}=a_{1}^{2}-a_{2}^{2}=|A_{1}|^{2}-|A_{2}|^{2}, \\ S_{2}=2a_{1}a_{2}\cos\delta=A*_{1}A_{2}+A_{1}A*_{2}, \§_{3}=2a_{1}a_{2}\sin\delta=\left(A_{1}A*_{2}-A*_{1}A_{2}\right)/i \end{eqnarray}

\newline The parameter S is proportional to the intensity of the wave and this parameter is normalized to 1 for unitary transformations. Then the Stokes parameters are equivalent mathematicallyto the Bloch vector components discussed above. For nonunitary transformations the magnitude of the Bloch vector, or equivalently the radius of the Poincar{\acute{e}} sphere is decreasing , then the parameter S becomes important. However, using the Pancharatnam Phase approach ~{Pancharatnam} the topological phases obtained for nonunitary transformations are similar to those of unitary transformations. Using Stokes Parameters, right-handed(left-handed) circular polarization is represented by points in the equatorial plane~{OpticalElectronics}. The Jones vectors for right and left polarized light waves are given by ~{OpticalElectronics}

\displaystyle \hat{R}=\frac{1}{\sqrt{2}}\left(1\right) \ \ \ \ \ (20)

These two circular polarizations are mutually orthogonal in the sense that {\hat{R}^{T}\hat{L}=0}. Any pair of othogonal Jones vectors can be used as a basis of the mathematical space spanned by the Jones Vector. In the next subsection we discus and elaborate some more on the Jones Vector formulation

The above result was first given by Ross~{Ross}, it has since become part of the literature. Let us consider that there is linear birefringence induced in the helically coiled monomode fiber and other curves. In fact it is known that birefringence appears whenever the circular symmetry of an ideal fiber is broken, hence this produces an anisotropic refractive-index distribution in the core region. In general in some cases the optical axis and the normal vector (i.e. {\mathbf{n}(s)} ) do not coincide to each other, so that there is an angle {\theta} between them. To analyze the combination of linear and circular retardation, the N matrix rotation of Jones~{OpticalElectronics} can be used. The N matrix is used to describe the movement of the electric field as light travels through an elemental section of the optical system. The N matrix can be extended by adding to the matrix for the specific circular linear birefringence. It yields in the local co-ordinate system:

\displaystyle N'(s)=\left( \begin{array}{cc} 0 & \tau \\ -\tau 0 \end{array} \right) +\left(\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right).\left(\begin{array}{cc} ik_{f} & 0 \\ 0 & ik_{s} \end{array} \right)\left(\begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array}\right) \ \ \ \ \ (21)

i.e.

\displaystyle \left( \begin{array}{cc} ik & \tau \\ -\tau & ik \end{array} \right) +\frac{i\beta}{2}\left(\begin{array}{cc} \cos(2\theta) & \sin(2\theta) \\ \sin(2\theta) & -\cos(2\theta) \end{array}\right)\ \ \ \ \ (22)

where {k-(k_{f} + k_{s})/2}, {\beta = k_{f}-k_{s}}; where {k_{f}} denotes fast wavevector, and {k_{s}} the slow one. Geometrically we can now understand that the diagonal part of the N amtrix undergoes a rotation caused by the birefringence. It is possible to compute the geometric phase from the above corrected N matrix, and other previous equations. Bhandari and Samuel observed Poincare topological phases by moving the polarization state of a monochromatic field on the {Poincar\acute{e}} sphere~{PolBhandari}. For this purpose they used quarter-wave plates(which produce a difference in phase which is a quarter of {2\pi} between two orthogonal polarizations) and optically active medium (which causes a certain rotation of the plane of polarized light).They also exchanged the optical active material by means of polaroids and for this wave have shown that the Pancharatnam geometric phase can be obtained also for nonunitary transformations. In either case the beam comes back with an added topological phase which is equal to half of the solid angle subtended by the area of the circuit on the {Poincar\acute{e}} sphere. In principle, due to a large dynamical phase it is very difficut to measure the topological phases. Therefore the real measurements{PolBhandari} were obtained as the difference in topological phase for two circuits which have the same dynamical phase. Since the MPC has a similar effect to quarter wave plates, we expect that similar phenomenoa will be observed. The next section will contain the proposed model for the MPC.

4. Manual Fiber Polarization Controller

In this section, a very simple model is proposed for modelling the Manual Fiber Polarization Controller. The modification of Polarization in Optical Fibers is increasingly important in communication research. The modification that suffers the state of polarization in an optical fiber as a result of the action of thermal and mechanical stresses or irregularities in the circular shape of the core was largely studied by several authors, a notable example is ~{Ulrich}. Ross’ concluded that the rotation of the polarization plane was due to geometric effects ~{Ross} and the first expression in the literature that extended this to noncoplanar curves was that of ~{RotPolarizationFrame} in this note we propose on the basis of a suggestion in {Mobius_Strip} an alternative model in this note. Let us first describe an elementary and somewhat primitive model of the MPC. \mathbf{r_{1}(t)}= (a\cos t,a\sin t,0),\; 0 < t < \pi \mathbf{r_{2}(t)}= (a\cos t,0, a\sin t),\; \pi < t < 2\pi Let us consider what happens when these two curves meet up. Let us call the point where they meet {t_{c}} \lim_{t\rightarrow t_{c}}(r_{1}(t)=\lim_{t\rightarrow t_{c}}(r_{2}(t)$ \lim_{t \rightarrow t_{c}}(r'_{1}(t)=\lim_{t\rightarrow t_{c}}(r'_{2}(t) Let us consider further derivatives

\displaystyle r(t) = (acost,asint,0),0<\epsilon<\pi ; (acost,0,asint),\pi < t < 2\pi \ \ \ \ \ (23)

\displaystyle r'(t) = (-asint,acost,0),0<t<\pi ; (-asint,0,acost),\pi < t < 2\pi \ \ \ \ \ (24)

\displaystyle r''(t) = (-acost,-asint,0),0<t<\pi ; (-acost,0,-asint),\pi < t < 2\pi \ \ \ \ \ (25)

\displaystyle r'''(t) = (asint,-acost,0),0<t<\pi ; \newline (asint,0,acost),\pi < t < 2\pi \ \ \ \ \ (26)

Lets examine what happens around the point {t_{c}}

Above we have outlined a piecewise description of the curve r = {r_{1} \cup r_{2}} . The curve is only once differentiable on the whole domain t {\in \left[0,2 \pi\right]} or in formal notation {\textbf{r} \in C^{1}\left(\left[0,2\pi\right)\right]}. In particular the second derivative of r has a discontinuity at the point {t = \pi} . Hence if we attempt to evaluate the second derivative at the point {t=\pi}.

r ”\left(\pi \right) = $latex \lim_{t \rightarrow \pi} \frac{r‘\left(t \right)- r‘\left( \pi \right)}{t-\pi} = \lim_{t \rightarrow \pi} \frac{r_{2}’\left(t \right)- r_{1}’\left( \pi \right)}{t-\pi}$

Hence we have

r ''\left(\pi \right) = \lim_{t \rightarrow \pi}\frac{(-asint,0,acost)- (0,a,0)}{t - \pi}

Numerator is non zero since the two terms do not approach each other at the point t={\pi} hence since the denominator is zero and so we have a singularity at this point. Then second derivative is infinite at this point {\pi} ( strictly speaking the derivative is actually undefined at this point). The torsion of a curve is given by

\tau = \frac{\left(r‘ \times \mathbf{r}”\right).r”’}{\left|\mathbf{r}’ \times \mathbf{r}”\right|^{2}}

Then we can see this will be infinite at t = {\pi} hence we can describe the torsion by the delta function.

\tau = \mathbf{\delta} \left( \pi \right)
This delta function models the change in polarization between left and right circularly polarized light. Further work could examine in more detail what happens at non right angles.

5. Discussion and Conclusion

We’ve discussed some of the background on the Optical Berry Phase, and provided a simple (perhaps even crude) model for a specific class of Polarization Controller. Further work could examine in more detail the limits of this model, and even examine the suggestion made in in regards whether twisted Mobius strips could be used in the modelling of Optical Control devices. We hope that this article inspires others to research the Geometric Phase. And highlights that the Optical Berry Phase isn’t a dead research area.

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