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arXiv:2608.19679

Geometric phase of open paths and a geodesic-selection rule at a level degeneracy

Hyeonseok Yang, Changsuk Noh

quant-phmath-phSubmitted Aug 20, 2026• Announced Aug 21, 2026• 0 citations
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Abstract

When the control field of a qubit, a polarization state, or a spin-$\tfrac12$ system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle $Ω[C]$ intrinsically, and displacing the degeneracy by $ε\uhat$ closes the path with enclosed solid angle $Ω(ε\uhat)=Ω[C]+2α+O(ε)$, where $α$ is the azimuth of the transverse part of $\uhat$ measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic---the great circle in the osculating plane ($α=0$)---supplied by the curvature at the degeneracy. Berry's $π$ invariant under reversal of the displacement and the values $\pmπ/2$ under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.

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Related papers

Geometric Phases of a Driven Qubit: Comparing Berry and Uhlmann Holonomies

Hyeonseok Yang, Changsuk Noh

The Berry phase arises naturally from the adiabatic dynamics of a pure quantum state, whereas the Uhlmann phase is usually formulated kinematically for a prescribed path of mixed states. For a qubit subject to a uniformly gapped conical drive, we obtain the Uhlmann connection and holonomy in closed form for the equilibrium Gibbs cycle. We then solve the corresponding Lindblad equation exactly in the rotating frame and show that its steady state forms a nonequilibrium limit cycle that lags the instantaneous Gibbs cycle. As the driving becomes slow, the Uhlmann phase approaches the equilibrium Uhlmann phase, which cooling then reduces to the Berry phase: in the joint adiabatic and low-temperature limit, the dynamical Uhlmann phase therefore converges to the Berry phase. This grounds the Uhlmann--Berry correspondence in the physical dynamics of an open system. We also characterize the finite-driving corrections, together with the discontinuous $π$ jump that the Uhlmann phase undergoes on the equatorial cycle as the temperature is varied. Finally, we examine an isolated transversal gap closing, where this zero-temperature correspondence can break down: the pure-state path becomes open, its closure is ambiguous, and gap-opening regularizations close it along a geodesic set by the direction of the bias field, splitting the Berry phase into a one-parameter family of values. The Gibbs path instead closes smoothly through the maximally mixed state, so the Uhlmann holonomy requires no regularization and remains unique and continuous at every finite temperature. In the low-temperature limit the Uhlmann phase selects a single member of the Berry family---the geodesic closure in the osculating plane, fixed by the velocity and acceleration of the driving field. The Uhlmann--Berry correspondence of the gapped regime thus survives the gap closing as a geodesic selection rule.

quant-pharXiv:2608.22919•Aug 25, 2026•0 citations

Evolution of geometric phase and explaining the geodesic rule

Nathan A. Hagen, Luis A. Garza-Soto

We use the recently developed wave model of geometric phase to track the continuous evolution of geometric phase as a wave propagates through optical elements and throughout an optical system. By working directly with the wave properties, we encounter a natural explanation of why the conventional Poincaré sphere solid angle method must use geodesic paths rather than the physical paths of the polarization state-the "geodesic rule"-and show that the existing rules for the solid angle algorithm are incomplete. Finally, we use the physical model to clarify the differences between the Pancharatnam connection and the geometric phase of a wave.

Journal of the Optical Society of America A•Sep 12, 2024•3 citations

Polarized light and geometric phase in qg units: Stokes parameters, Mueller matrices and Stokes' theorem

Vicente Humberto Monteverde

⟨ ⟩ The qang project describes qubits by measurement—level quantities, the qg values qgP = P [1]. This short note states two classical results in those units, implemented and checked in the qang library (pip install qang, version 0.6.0). First, a polarization state of light is a qubit: the normalized Stokes parameters are exactly the qg values, the degree of polarization P is the length of the qg vector, and a lossless Mueller matrix is the qg gate rule qg′ = qgU†PU . Malus's law becomes I = I0(1 + qgZ cos 2θ + qgX sin 2θ)/2 for any input state. Second, the — geometric phase of a closed loop on the Bloch (Poincaré) sphere is minus half the enclosed solid angle, which is Stokes' theorem for the Berry curvature. For a loop at constant qgZ the phase is (qgZ 1)/2 of a turn: it is fixed by qgZ alone. Nothing here is new physics. The contribution is the translation into qang's units and a set of 26 numerical checks, with the phase computed three independent ways.

•Oct 5, 2026•0 citations

Geometric Phase Realization of Q5 Transport Holonomy

Craig Edwin Holdway

This Theorem establishes an explicit geometric-phase realization of the Q5 observable coefficient law μ_obs = 4κ_adj + 2κ_non. A closed loop on the Bloch sphere is constructed from six constant-polar-angle azimuthal arcs of equal azimuthal sweep Δφ, four at polar angle θ_adj and two at θ_non. By the standard solid-angle contribution of constant-polar arcs (Ω = Δφ(1 − cosθ)), the total enclosed solid angle is: Ω = 4Δφ(1 − cosθ_adj) + 2Δφ(1 − cosθ_non) The Berry phase of a spin-½ eigenstate adiabatically transported along this loop is γ_Berry = −½Ω. Defining κ_adj := −½Δφ(1 − cosθ_adj) and κ_non := −½Δφ(1 − cosθ_non) gives γ_Berry = 4κ_adj + 2κ_non = μ_obs. The (4,2) multiplicity structure arises geometrically from the decomposition into four adjacent-class and two non-adjacent-class Bloch segments. Status: Solid angle lemma and Berry phase formula are standard results. This is a realization result: it establishes that μ_obs admits a geometric-phase realization. It does not assert that Gray-constrained transport uniquely induces this specific loop, nor that θ_adj and θ_non are fixed by Q5 structure. Uniqueness and intrinsic derivation are addressed in T41. Dependencies: T39.

•May 3, 2026•0 citations

Carrying a Polarisation Around a Closed Path Shifts Its Phase by Half the Enclosed Solid Angle ── enclosing a hemisphere gives a phase of pi and flips the sign, the same half as the Berry phase of a qubit ── the separator is whether one measures intensity alone or the phase by interference with a reference ── [Paper 906]

Yuuki Yamagishi

Passing light through a few polarisers and back to its original polarisation, it is easy to assume the light has returned to what it was. This paper follows the state of polarisation as a point on the Poincaré sphere and compares the phase left after a closed path with the solid angle the path encloses. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──that a closed path of polarisations acquires a phase of half the solid angle (Pancharatnam 1956), that this is the same as Berry’s phase (1984; Berry 1987), and that the state space of polarisation is a Hopf fibration are all known. No real optical system is claimed──the number, angles and amplitudes of the polarisers are model values, not those of any apparatus. Partial polarisation and loss are not entered──only projections of fully polarised light are treated; partial polarisation, differential absorption and non-adiabatic effects are not included. The origin of the phase is not entered──that the phase is the holonomy of a Hopf fibration is the matter of Paper 68; here only the numbers for light are given. Relation to earlier papers: Paper 68 showed that the qubit is a Hopf fibration and that the Berry phase is measurable as the holonomy -Omega/2 of its fibre──the two-component Jones vector used here has the same form as a spin-1/2 spinor, and the same function gives the same phase. Since the formula, not merely the number, is the same, this is written as the same root in the sense of Paper 300. Paper 207 showed that there are three tools for writing polarisation and that the Jones vector cannot write unpolarised light──here the closed path is followed on the third of them, the Poincaré sphere. Paper 08 showed that in the Hopf fibration (the map from S^3 to S^2 whose fibre is S^1) the area of the base times the circumference of the fibre is the volume of the whole──here the base is the Poincaré sphere and the fibre is the phase of the polarisation. Paper 18 separated three roots of 2pi and took one of them to be the solid angle of a hemisphere──here the hemisphere’s solid angle 2pi becomes a phase of pi. What is added is checking the phase of closed paths made of polariser projections against -Omega/2 on random triangles, showing in numbers that on the hemispherical path the amplitude returns while the sign does not, measuring the convergence as the steps are refined, and placing the separator on intensity versus interference. First, the map to the state of polarisation is the Hopf map──the Stokes vectors built from 2000 two-component complex vectors all lie on the unit sphere and do not move, to the order of 10^-15, when an overall phase is applied (Section 2). Second, and this is the core. The phase of a closed path is half the solid angle enclosed──returning through three polarisers A-> B-> C-> A, the phase arg( A|C)( C|B)( B|A) agreed with -Omega/2 of the enclosed solid angle within 10^-9 for five random triangles (for instance Omega=-5.580641 gives phase +2.790320). The reverse path gives +Omega/2, which equals -(4pi-Omega)/2 modulo 2pi (Section 2). Third, enclosing a hemisphere flips the sign──turning a linear polarisation through half a turn in N steps and back (once around the equator, Omega=2pi) gives a phase of pi for every N from 3 to 384 (Section 3). Fourth, the amplitude returns but the sign does not──the surviving amplitude is cos^N(pi/N), rising from 0.125000 at N=3 to 0.987231 at N=384, yet the intensity when overlaid in phase with a reference cancels to 0.000041 at N=384 (Section 3). Fifth, finer steps converge to the half on any path──around a circle of angular radius 10 degrees near circular polarisation (Omega=0.095456), the phase moves from -0.039621 at N=6 to -0.047728 at N=1536, within 1.30x10^-7 of -Omega/2 (Section 4). Sixth, the separator is whether one measures intensity alone or the phase by interference with a reference──by intensity alone, the finer the steps the more fully the light returns; overlaid with a reference, the hemisphere’s phase pi remains as darkness. The formula is that of the same two-component vector as a spin-1/2 qubit, and shares its root with the -Omega/2 of Paper 68 (Section 5). Carrying a polarisation around a closed path shifts its phase by half the solid angle enclosed, and enclosing a hemisphere flips the sign: the same half as the Berry phase of a qubit. The phases of triangular paths made of polariser projections agreed with -Omega/2 in all five cases, and on the path that turns a linear polarisation through half a turn and back the phase was pi independently of N──the surviving amplitude returned to 0.987231 at N=384, yet overlaid with a reference the intensity cancelled to 0.000041. On a small circle too the phase converged to -Omega/2 as the steps were refined, the reverse path gave +Omega/2, and the full sphere 4pi corresponded to a phase of 2pi──what returned was the state; the phase remembers half the sky enclosed. Placed among the earlier papers──the same formula as the qubit of Paper 68 (the same root) was counted with light, on the Hopf fibration of Paper 08 and the Poincaré sphere of Paper 207. To be honest──the apparatus values are a model, neither partial polarisation nor loss was included, and the origin of the phase was left to Paper 68. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: polarisation, Poincaré sphere, geometric phase, solid angle, Hopf map, interference. ----- 偏光子を何枚か通して元の偏光に戻せば、光は元に戻った、と思いやすい。本稿は、偏光の状態をポアンカレ球の点として追い、閉じた道を一周したあとに残る位相を、囲んだ立体角と比べる。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──偏光の閉じた道に立体角の半分の位相が付くこと(パンチャラトナム 1956)、それがベリー位相(1984)と同じであること(ベリー 1987)、偏光の状態空間がホップ束であることは、いずれも既知である。実在の光学系を主張しない──偏光子の枚数・角度・振幅は模型の値で、どの装置の値でもない。部分偏光と損失を入れない──完全偏光の射影だけを扱い、部分偏光、吸収の違い、非断熱の効果は入れない。位相の由来に入らない──位相がホップ束のホロノミーであることの説明は論文68 の側で、ここでは光での数だけを出す。既刊との関係:論文68 は、量子ビットがホップ束であり、ベリー位相がその繊維のホロノミー -Omega/2 として測れることを示した──ここで使う二成分のジョーンズ・ベクトルはスピン 1/2 のスピノルと同じ形で、同じ関数から同じ位相が出る。数の一致ではなく式が同じなので、論文300 の意味で同根と書く。論文207 は、偏光を書く道具が三つあり、ジョーンズ・ベクトルは偏光していない光を書けないことを示した──ここではその三つ目、ポアンカレ球の上で閉じた道を追う。論文08 は、ホップ束(S^3 から S^2 への写像で、繊維は S^1)で底空間の面積と繊維の周長の積が全空間の体積になることを示した──ここでは、その底空間がポアンカレ球、繊維が偏光の位相である。論文18 は、2pi の三つの別根を分け、その一つを半球の立体角とした──ここでは半球の立体角 2pi が、位相の pi になる。加えたのは、偏光子の射影で作った閉じた道の位相を無作為の三角形で -Omega/2 と突き合わせたこと、半球の道で振幅が戻っても符号が戻らないことを数で示したこと、段を細かくしたときの収束を測ったこと、分離子を「強度か、干渉か」に置いたことである。 第一に、偏光の状態への写像はホップ写像である──二成分の複素ベクトルから作ったストークス・ベクトルは 2000 個すべてで単位球に乗り、全体の位相を掛けても 10^-15 の桁で動かない(第2節)。 第二に、これが本稿の芯である。閉じた道の位相は、囲んだ立体角の半分である──偏光子三枚で A-> B-> C-> A と戻したときの位相 arg( A|C)( C|B)( B|A) は、五つの無作為の三角形で囲んだ立体角の -Omega/2 と 10^-9 以内で一致した(例:Omega=-5.580641 で位相 +2.790320)。逆回りは +Omega/2 で、-(4pi-Omega)/2 と 2pi を法として同じである(第2節)。 第三に、半球を囲めば、符号が反転する──直線偏光の向きを N 段で半回転させて戻す(赤道を一周・Omega=2pi)と、位相は N=3 から 384 のすべてで pi である(第3節)。 第四に、振幅は戻るのに、符号は戻らない──残る振幅は cos^N(pi/N) で、N=3 の 0.125000 から N=384 の 0.987231 まで 1 に近づくのに、参照光と同位相で重ねた強度は N=384 で 0.000041 に打ち消す(第3節)。 第五に、段を細かくすると、どの道でも半分に収束する──円偏光の近くの角半径 10 度の円(Omega=0.095456)を N 段で一周すると、位相は N=6 の -0.039621 から N=1536 の -0.047728 へ、-Omega/2 に 1.30x10^-7 まで近づく(第4節)。 第六に、分離子は、強度だけを測るか、参照と干渉させて位相を測るかである──強度だけなら N を増やすほど元に戻り、参照と重ねれば半球の位相 pi が暗さとして残る。式はスピン 1/2 の量子ビットと同じ二成分ベクトルのもので、論文68 の -Omega/2 と同根である(第5節)。 偏光を閉じた道で一周させると位相は囲んだ立体角の半分だけ動き、半球を囲めば符号が反転する──量子ビットのベリー位相と同じ半分である。偏光子の射影で作った三角形の道の位相は五つとも -Omega/2 と一致し、直線偏光の向きを半回転させて戻す道では位相が N に依らず pi だった──残る振幅は N=384 で 0.987231 まで戻るのに、参照光と重ねた強度は 0.000041 に打ち消す。小さい円でも段を細かくすれば -Omega/2 に収束し、逆回りは +Omega/2 で、全球 4pi が位相の 2pi に対応する──元に戻ったのは状態で、位相は囲んだ空の半分を覚えている。既刊との位置──論文68 の量子ビットと同じ式(同根)を光で数え、論文08 のホップ束と論文207 のポアンカレ球の上に置いた。正直に言えば──装置の値は模型で、部分偏光も損失も入れず、位相の由来は論文68 に任せた。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:偏光、ポアンカレ球、幾何学的位相、立体角、ホップ写像、干渉。

•Sep 17, 2026•0 citations

Geometric phase evolution

Nathan A. Hagen, Luis A. Garza-Soto

Geometric phase has historically been defined using closed cycles of polarization states, often derived using differential geometry on the Poincare sphere. Using the recently-developed wave model of geometric phase, we show that it is better to define geometric phase more generally, allowing every polarized wave to have a well-defined value at any point in its path. Using several example systems, we show how this approach provides more insight into the wave's behavior. Moreover, by tracking the continuous evolution of geometric phase as a wave propagates through an optical system, we encounter a natural explanation of why the conventional Poincare sphere solid angle method uses geodesic paths rather than physical paths of the polarization state.

arXiv:2407.19887•Jul 29, 2024•0 citations
When the control field of a qubit, a polarization state, or a spin-12\tfrac1221​ system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle Ω[C]Ω[C]Ω[C] intrinsically, and displacing the degeneracy by ε\uhatε\uhatε\uhat closes the path with enclosed solid angle Ω(ε\uhat)=Ω[C]+2α+O(ε)Ω(ε\uhat)=Ω[C]+2α+O(ε)Ω(ε\uhat)=Ω[C]+2α+O(ε), where ααα is the azimuth of the transverse part of \uhat\uhat\uhat measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic---the great circle in the osculating plane (α=0α=0α=0)---supplied by the curvature at the degeneracy. Berry's πππ invariant under reversal of the displacement and the values ±π/2\pmπ/2±π/2 under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.
The Berry phase arises naturally from the adiabatic dynamics of a pure quantum state, whereas the Uhlmann phase is usually formulated kinematically for a prescribed path of mixed states. For a qubit subject to a uniformly gapped conical drive, we obtain the Uhlmann connection and holonomy in closed form for the equilibrium Gibbs cycle. We then solve the corresponding Lindblad equation exactly in the rotating frame and show that its steady state forms a nonequilibrium limit cycle that lags the instantaneous Gibbs cycle. As the driving becomes slow, the Uhlmann phase approaches the equilibrium Uhlmann phase, which cooling then reduces to the Berry phase: in the joint adiabatic and low-temperature limit, the dynamical Uhlmann phase therefore converges to the Berry phase. This grounds the Uhlmann--Berry correspondence in the physical dynamics of an open system. We also characterize the finite-driving corrections, together with the discontinuous πππ jump that the Uhlmann phase undergoes on the equatorial cycle as the temperature is varied. Finally, we examine an isolated transversal gap closing, where this zero-temperature correspondence can break down: the pure-state path becomes open, its closure is ambiguous, and gap-opening regularizations close it along a geodesic set by the direction of the bias field, splitting the Berry phase into a one-parameter family of values. The Gibbs path instead closes smoothly through the maximally mixed state, so the Uhlmann holonomy requires no regularization and remains unique and continuous at every finite temperature. In the low-temperature limit the Uhlmann phase selects a single member of the Berry family---the geodesic closure in the osculating plane, fixed by the velocity and acceleration of the driving field. The Uhlmann--Berry correspondence of the gapped regime thus survives the gap closing as a geodesic selection rule.
We use the recently developed wave model of geometric phase to track the continuous evolution of geometric phase as a wave propagates through optical elements and throughout an optical system. By working directly with the wave properties, we encounter a natural explanation of why the conventional Poincaré sphere solid angle method must use geodesic paths rather than the physical paths of the polarization state-the "geodesic rule"-and show that the existing rules for the solid angle algorithm are incomplete. Finally, we use the physical model to clarify the differences between the Pancharatnam connection and the geometric phase of a wave.
⟨ ⟩ The qang project describes qubits by measurement—level quantities, the qg values qgP = P [1]. This short note states two classical results in those units, implemented and checked in the qang library (pip install qang, version 0.6.0). First, a polarization state of light is a qubit: the normalized Stokes parameters are exactly the qg values, the degree of polarization P is the length of the qg vector, and a lossless Mueller matrix is the qg gate rule qg′ = qgU†PU . Malus's law becomes I = I0(1 + qgZ cos 2θ + qgX sin 2θ)/2 for any input state. Second, the — geometric phase of a closed loop on the Bloch (Poincaré) sphere is minus half the enclosed solid angle, which is Stokes' theorem for the Berry curvature. For a loop at constant qgZ the phase is (qgZ 1)/2 of a turn: it is fixed by qgZ alone. Nothing here is new physics. The contribution is the translation into qang's units and a set of 26 numerical checks, with the phase computed three independent ways.
This Theorem establishes an explicit geometric-phase realization of the Q5 observable coefficient law μ_obs = 4κ_adj + 2κ_non. A closed loop on the Bloch sphere is constructed from six constant-polar-angle azimuthal arcs of equal azimuthal sweep Δφ, four at polar angle θ_adj and two at θ_non. By the standard solid-angle contribution of constant-polar arcs (Ω = Δφ(1 − cosθ)), the total enclosed solid angle is: Ω = 4Δφ(1 − cosθ_adj) + 2Δφ(1 − cosθ_non) The Berry phase of a spin-½ eigenstate adiabatically transported along this loop is γ_Berry = −½Ω. Defining κ_adj := −½Δφ(1 − cosθ_adj) and κ_non := −½Δφ(1 − cosθ_non) gives γ_Berry = 4κ_adj + 2κ_non = μ_obs. The (4,2) multiplicity structure arises geometrically from the decomposition into four adjacent-class and two non-adjacent-class Bloch segments. Status: Solid angle lemma and Berry phase formula are standard results. This is a realization result: it establishes that μ_obs admits a geometric-phase realization. It does not assert that Gray-constrained transport uniquely induces this specific loop, nor that θ_adj and θ_non are fixed by Q5 structure. Uniqueness and intrinsic derivation are addressed in T41. Dependencies: T39.
Passing light through a few polarisers and back to its original polarisation, it is easy to assume the light has returned to what it was. This paper follows the state of polarisation as a point on the Poincaré sphere and compares the phase left after a closed path with the solid angle the path encloses. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──that a closed path of polarisations acquires a phase of half the solid angle (Pancharatnam 1956), that this is the same as Berry’s phase (1984; Berry 1987), and that the state space of polarisation is a Hopf fibration are all known. No real optical system is claimed──the number, angles and amplitudes of the polarisers are model values, not those of any apparatus. Partial polarisation and loss are not entered──only projections of fully polarised light are treated; partial polarisation, differential absorption and non-adiabatic effects are not included. The origin of the phase is not entered──that the phase is the holonomy of a Hopf fibration is the matter of Paper 68; here only the numbers for light are given. Relation to earlier papers: Paper 68 showed that the qubit is a Hopf fibration and that the Berry phase is measurable as the holonomy -Omega/2 of its fibre──the two-component Jones vector used here has the same form as a spin-1/2 spinor, and the same function gives the same phase. Since the formula, not merely the number, is the same, this is written as the same root in the sense of Paper 300. Paper 207 showed that there are three tools for writing polarisation and that the Jones vector cannot write unpolarised light──here the closed path is followed on the third of them, the Poincaré sphere. Paper 08 showed that in the Hopf fibration (the map from S^3 to S^2 whose fibre is S^1) the area of the base times the circumference of the fibre is the volume of the whole──here the base is the Poincaré sphere and the fibre is the phase of the polarisation. Paper 18 separated three roots of 2pi and took one of them to be the solid angle of a hemisphere──here the hemisphere’s solid angle 2pi becomes a phase of pi. What is added is checking the phase of closed paths made of polariser projections against -Omega/2 on random triangles, showing in numbers that on the hemispherical path the amplitude returns while the sign does not, measuring the convergence as the steps are refined, and placing the separator on intensity versus interference. First, the map to the state of polarisation is the Hopf map──the Stokes vectors built from 2000 two-component complex vectors all lie on the unit sphere and do not move, to the order of 10^-15, when an overall phase is applied (Section 2). Second, and this is the core. The phase of a closed path is half the solid angle enclosed──returning through three polarisers A-> B-> C-> A, the phase arg( A|C)( C|B)( B|A) agreed with -Omega/2 of the enclosed solid angle within 10^-9 for five random triangles (for instance Omega=-5.580641 gives phase +2.790320). The reverse path gives +Omega/2, which equals -(4pi-Omega)/2 modulo 2pi (Section 2). Third, enclosing a hemisphere flips the sign──turning a linear polarisation through half a turn in N steps and back (once around the equator, Omega=2pi) gives a phase of pi for every N from 3 to 384 (Section 3). Fourth, the amplitude returns but the sign does not──the surviving amplitude is cos^N(pi/N), rising from 0.125000 at N=3 to 0.987231 at N=384, yet the intensity when overlaid in phase with a reference cancels to 0.000041 at N=384 (Section 3). Fifth, finer steps converge to the half on any path──around a circle of angular radius 10 degrees near circular polarisation (Omega=0.095456), the phase moves from -0.039621 at N=6 to -0.047728 at N=1536, within 1.30x10^-7 of -Omega/2 (Section 4). Sixth, the separator is whether one measures intensity alone or the phase by interference with a reference──by intensity alone, the finer the steps the more fully the light returns; overlaid with a reference, the hemisphere’s phase pi remains as darkness. The formula is that of the same two-component vector as a spin-1/2 qubit, and shares its root with the -Omega/2 of Paper 68 (Section 5). Carrying a polarisation around a closed path shifts its phase by half the solid angle enclosed, and enclosing a hemisphere flips the sign: the same half as the Berry phase of a qubit. The phases of triangular paths made of polariser projections agreed with -Omega/2 in all five cases, and on the path that turns a linear polarisation through half a turn and back the phase was pi independently of N──the surviving amplitude returned to 0.987231 at N=384, yet overlaid with a reference the intensity cancelled to 0.000041. On a small circle too the phase converged to -Omega/2 as the steps were refined, the reverse path gave +Omega/2, and the full sphere 4pi corresponded to a phase of 2pi──what returned was the state; the phase remembers half the sky enclosed. Placed among the earlier papers──the same formula as the qubit of Paper 68 (the same root) was counted with light, on the Hopf fibration of Paper 08 and the Poincaré sphere of Paper 207. To be honest──the apparatus values are a model, neither partial polarisation nor loss was included, and the origin of the phase was left to Paper 68. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: polarisation, Poincaré sphere, geometric phase, solid angle, Hopf map, interference. ----- 偏光子を何枚か通して元の偏光に戻せば、光は元に戻った、と思いやすい。本稿は、偏光の状態をポアンカレ球の点として追い、閉じた道を一周したあとに残る位相を、囲んだ立体角と比べる。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──偏光の閉じた道に立体角の半分の位相が付くこと(パンチャラトナム 1956)、それがベリー位相(1984)と同じであること(ベリー 1987)、偏光の状態空間がホップ束であることは、いずれも既知である。実在の光学系を主張しない──偏光子の枚数・角度・振幅は模型の値で、どの装置の値でもない。部分偏光と損失を入れない──完全偏光の射影だけを扱い、部分偏光、吸収の違い、非断熱の効果は入れない。位相の由来に入らない──位相がホップ束のホロノミーであることの説明は論文68 の側で、ここでは光での数だけを出す。既刊との関係:論文68 は、量子ビットがホップ束であり、ベリー位相がその繊維のホロノミー -Omega/2 として測れることを示した──ここで使う二成分のジョーンズ・ベクトルはスピン 1/2 のスピノルと同じ形で、同じ関数から同じ位相が出る。数の一致ではなく式が同じなので、論文300 の意味で同根と書く。論文207 は、偏光を書く道具が三つあり、ジョーンズ・ベクトルは偏光していない光を書けないことを示した──ここではその三つ目、ポアンカレ球の上で閉じた道を追う。論文08 は、ホップ束(S^3 から S^2 への写像で、繊維は S^1)で底空間の面積と繊維の周長の積が全空間の体積になることを示した──ここでは、その底空間がポアンカレ球、繊維が偏光の位相である。論文18 は、2pi の三つの別根を分け、その一つを半球の立体角とした──ここでは半球の立体角 2pi が、位相の pi になる。加えたのは、偏光子の射影で作った閉じた道の位相を無作為の三角形で -Omega/2 と突き合わせたこと、半球の道で振幅が戻っても符号が戻らないことを数で示したこと、段を細かくしたときの収束を測ったこと、分離子を「強度か、干渉か」に置いたことである。 第一に、偏光の状態への写像はホップ写像である──二成分の複素ベクトルから作ったストークス・ベクトルは 2000 個すべてで単位球に乗り、全体の位相を掛けても 10^-15 の桁で動かない(第2節)。 第二に、これが本稿の芯である。閉じた道の位相は、囲んだ立体角の半分である──偏光子三枚で A-> B-> C-> A と戻したときの位相 arg( A|C)( C|B)( B|A) は、五つの無作為の三角形で囲んだ立体角の -Omega/2 と 10^-9 以内で一致した(例:Omega=-5.580641 で位相 +2.790320)。逆回りは +Omega/2 で、-(4pi-Omega)/2 と 2pi を法として同じである(第2節)。 第三に、半球を囲めば、符号が反転する──直線偏光の向きを N 段で半回転させて戻す(赤道を一周・Omega=2pi)と、位相は N=3 から 384 のすべてで pi である(第3節)。 第四に、振幅は戻るのに、符号は戻らない──残る振幅は cos^N(pi/N) で、N=3 の 0.125000 から N=384 の 0.987231 まで 1 に近づくのに、参照光と同位相で重ねた強度は N=384 で 0.000041 に打ち消す(第3節)。 第五に、段を細かくすると、どの道でも半分に収束する──円偏光の近くの角半径 10 度の円(Omega=0.095456)を N 段で一周すると、位相は N=6 の -0.039621 から N=1536 の -0.047728 へ、-Omega/2 に 1.30x10^-7 まで近づく(第4節)。 第六に、分離子は、強度だけを測るか、参照と干渉させて位相を測るかである──強度だけなら N を増やすほど元に戻り、参照と重ねれば半球の位相 pi が暗さとして残る。式はスピン 1/2 の量子ビットと同じ二成分ベクトルのもので、論文68 の -Omega/2 と同根である(第5節)。 偏光を閉じた道で一周させると位相は囲んだ立体角の半分だけ動き、半球を囲めば符号が反転する──量子ビットのベリー位相と同じ半分である。偏光子の射影で作った三角形の道の位相は五つとも -Omega/2 と一致し、直線偏光の向きを半回転させて戻す道では位相が N に依らず pi だった──残る振幅は N=384 で 0.987231 まで戻るのに、参照光と重ねた強度は 0.000041 に打ち消す。小さい円でも段を細かくすれば -Omega/2 に収束し、逆回りは +Omega/2 で、全球 4pi が位相の 2pi に対応する──元に戻ったのは状態で、位相は囲んだ空の半分を覚えている。既刊との位置──論文68 の量子ビットと同じ式(同根)を光で数え、論文08 のホップ束と論文207 のポアンカレ球の上に置いた。正直に言えば──装置の値は模型で、部分偏光も損失も入れず、位相の由来は論文68 に任せた。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:偏光、ポアンカレ球、幾何学的位相、立体角、ホップ写像、干渉。
Geometric phase has historically been defined using closed cycles of polarization states, often derived using differential geometry on the Poincare sphere. Using the recently-developed wave model of geometric phase, we show that it is better to define geometric phase more generally, allowing every polarized wave to have a well-defined value at any point in its path. Using several example systems, we show how this approach provides more insight into the wave's behavior. Moreover, by tracking the continuous evolution of geometric phase as a wave propagates through an optical system, we encounter a natural explanation of why the conventional Poincare sphere solid angle method uses geodesic paths rather than physical paths of the polarization state.