Experimental verification and generalization of the geometric-phase–which-path complementarity relation of the photon

Pillinen, Halder, Friberg, Setälä, and Norrman [Phys. Rev. A 111, 063709 (2025); preprint arXiv:2310.20273] proved theoretically that a single photon in a two-slit interferometer obeys the complementarity relation $|\Phi_{G}^{\prime}|+d\le 1$, where $\Phi_{G}^{\prime}=\Phi_{G}/\pi$ is the $\pi$-normalized geometric (Pancharatnam–Berry) phase accumulated by the photon's polarization state across the observation plane and $d=D_{0}/D_{S}$ is the ratio of intensity predictability to polarization distinguishability of the two slits, with the exact formula $\Phi_{G}=\arctan(\sin\phi/(\eta+\cos\phi))-\phi(1-d)/2$, $\eta=(1+d)/(1-d)$, and a tight identity $|\Phi_{G}^{\prime}|+d=1$ for cyclic evolution. The prediction remains untested: no experiment with prepared single-photon states has measured $\Phi_{G}$ and $d$ in one setup, the explicit map between the observation-plane coordinate $\Delta r$ and the Poincaré-sphere azimuth $\phi$ (stated to be generally nonlinear) has not been determined, and no analogous geometric-phase–which-path relation has been derived for any other quantum system. The problem is to establish the relation experimentally and to determine its range of validity across quantum systems.

Background

Two foundational concepts of quantum physics meet in this problem. The first is the geometric phase: when a quantum state evolves along a path in its state space, it can acquire a phase that depends only on the geometry of that path, not on how fast it is traversed (Berry, 1984). For light this is the Pancharatnam–Berry phase: as the polarization state traces a curve on the Poincaré sphere — the sphere whose points represent pure polarization states, with coordinates given by the normalized Stokes parameters $S_{1},S_{2},S_{3}$ — the field acquires a phase equal to minus half the solid angle enclosed by the curve; for open, noncyclic curves the endpoints are connected by a geodesic (the geodesic rule). Even a single photon carries this phase, as demonstrated by Kwiat and Chiao (1991). The second concept is wave–particle duality, quantified since Englert (1996) by inequalities of the form $V^{2}+D^{2}\le 1$: the fringe visibility $V$ (wave behavior) and the which-path information (particle behavior) available in a two-path interferometer cannot be simultaneously maximal.

For polarization-modulated two-slit interference, Norrman, Blomstedt, Setälä, and Friberg (2017) introduced quantitative which-path measures: the intensity (path) predictability $D_{0}=|S_{0}(\mathbf{r}_{1})-S_{0}(\mathbf{r}_{2})|/(S_{0}(\mathbf{r}_{1})+S_{0}(\mathbf{r}_{2}))$, quantifying how well the photon's slit can be guessed from unequal slit intensities, and the polarization distinguishability $D_{S}=|\mathbf{S}(\mathbf{r}_{1})-\mathbf{S}(\mathbf{r}_{2})|/(S_{0}(\mathbf{r}_{1})+S_{0}(\mathbf{r}_{2}))$, quantifying how well the slit can be discriminated from the polarization state at the slits; $S_{0}$ is the intensity and $\mathbf{S}=(S_{1},S_{2},S_{3})$ the polarization part of the Stokes vector. For polarized light $D_{0}\le D_{S}$, so the ratio $d=D_{0}/D_{S}\in[0,1]$ measures the mixture of the two which-path species. On the experimental side, Hannonen et al. [Optica 7, 1435 (2020)] measured the Pancharatnam–Berry phase in two-beam interference of classical polarized light, and Leinonen et al. [Commun. Phys. 6, 132 (2023)] measured its noncyclic form in the same geometry, where the polarization state varies periodically across the observation plane.

Against this background, Pillinen et al. proved the single-photon counterpart. A photon prepared in the four-mode one-photon state $|\Psi\rangle=c_{1x}|1,0,0,0\rangle+c_{1y}|0,1,0,0\rangle+c_{2x}|0,0,1,0\rangle+c_{2y}|0,0,0,1\rangle$ (a superposition of $x$- and $y$-polarized one-photon states at the two slits) produces in the far field a periodic intensity-and-polarization fringe pattern whose polarization state is fully polarized and evolves along a circle of constant polar angle $\theta$ on the Poincaré sphere as the observation point is translated across the screen, the azimuth $\phi$ running over one spatial period. Using the quantum-kinematic (Mukunda–Simon) formulation of the geometric phase — the kinematic expression of the phase for arbitrary, generally open and noncyclic state evolutions in terms of the Pancharatnam geodesic rule — they obtained $\Phi_{G}=\arctan[\sin^{2}(\theta/2)\sin\phi/(\cos^{2}(\theta/2)+\sin^{2}(\theta/2)\cos\phi)]-\phi(1-\cos\theta)/2$, connected the polar angle to which-path information through $d=\cos\theta$, and derived the exact formula $\Phi_{G}=\arctan(\sin\phi/(\eta+\cos\phi))-\phi(1-d)/2$ with $\eta=(1+d)/(1-d)$, valid for cyclic ($\phi=2\pi$) and noncyclic ($\phi<2\pi$) evolution. From it follows the complementarity relation $|\Phi_{G}^{\prime}|+d\le 1$ with $\Phi_{G}^{\prime}=\Phi_{G}/\pi$, saturated as the identity $|\Phi_{G}^{\prime}|+d=1$ for every cyclic evolution: the geometric phase in the observation plane (wave aspect) and the which-path ratio at the slits (particle aspect) are strictly exclusive under cyclic evolution — a wave–particle duality quantifier built from the geometric phase rather than from fringe visibility.

What the work does not do: it reports no experiment; it states that the map between the physical path-difference coordinate $\Delta r$ on the observation screen and the Poincaré-sphere azimuth $\phi$ is in general not linear but leaves its explicit form undetermined; and it offers only an expectation ("we expect that similar features... exist in other quantum systems as well") that analogous relations hold beyond the photon polarization qubit.

Problem Statement

Under the single objective of establishing the geometric phase as an operational quantifier of wave–particle duality beyond its present status as one untested theoretical derivation, resolve the following linked directions left open by Pillinen, Halder, Friberg, Setälä, and Norrman:

(1) Experimental verification. Demonstrate (or refute) with prepared single-photon states in a concrete two-slit/two-pinhole interferometric setup the complementarity relation $|\Phi_{G}^{\prime}|+d\le 1$, the cyclic identity $|\Phi_{G}^{\prime}|+d=1$, and the exact formula $\Phi_{G}(\phi,d)=\arctan(\sin\phi/(\eta+\cos\phi))-\phi(1-d)/2$ with $\eta=(1+d)/(1-d)$, covering nontrivial which-path content $0<d<1$ and both cyclic and noncyclic polarization evolution. This includes determining the explicit functional mapping between the observation-plane path-difference coordinate $\Delta r$ and the Poincaré-sphere azimuth $\phi$ (stated in the source to be generally nonlinear) — at minimum for the chosen arrangement, which is what allows measured spatial fringe data to be compared with the $\phi$-parametrized theory, the source leaving the explicit form of this mapping undetermined for general optical arrangements. Since no measurement of $\Phi_{G}$ and $d$ in a single prepared-single-photon setup is available, a demonstration may be given by an explicit quantitative treatment of a concrete setup — a fully specified experimental design with predicted values of the measurable quantities, or a numerical simulation of the corresponding spatial fringe and polarization data — and does not require a new laboratory measurement.

(2) Generalization. Determine whether geometric-phase-versus-which-path complementarity relations analogous to $|\Phi_{G}^{\prime}|+d\le 1$ hold in other quantum systems beyond the single-photon polarization qubit (for example matter-wave or neutron interferometry with internal spin states, or atom- and solid-state-qubit interferometers), formulating for each candidate system a geometric-phase measure (cyclic and noncyclic) and which-path measures parallel to $D_{0}$, $D_{S}$, and $d$, and either deriving the corresponding relation together with its saturation conditions or proving that no such relation holds. At minimum, one such system beyond the single-photon polarization qubit must be treated.

References

1. Elvis Pillinen, Atri Halder, Ari T. Friberg, Tero Setälä, Andreas Norrman, "Geometric phase and wave-particle duality of the photon", Physical Review A 111, 063709 (2025), DOI 10.1103/s3hf-2zyr, https://doi.org/10.1103/s3hf-2zyr; preprint arXiv:2310.20273, https://arxiv.org/abs/2310.20273
2. Diego S. Starke, Jonas Maziero, Marcos L. W. Basso, Tabish Qureshi, "Bohr's complementarity", arXiv:2605.26375 (2026), https://arxiv.org/abs/2605.26375
3. Naofumi Abe, Keiichi Edamatsu, "Wave–particle duality of unpolarized photons", Optics Express 34, 19013 (2026), DOI 10.1364/OE.582177, https://doi.org/10.1364/OE.582177; preprint arXiv:2605.25738, https://arxiv.org/abs/2605.25738
4. Martti Hanhisalo, Atri Halder, Tero Setälä, Andreas Norrman, "Coherence squeezing in optical interference", arXiv:2602.22921 (2026), https://arxiv.org/abs/2602.22921
5. Antti Hannonen, Henri Partanen, Aleksi Leinonen, Janne Heikkinen, Tommi K. Hakala, Ari T. Friberg, Tero Setälä, "Measurement of the Pancharatnam–Berry phase in two-beam interference", Optica 7, 1435 (2020), DOI 10.1364/optica.401993, https://doi.org/10.1364/optica.401993
6. Aleksi Leinonen, Antti Hannonen, Henri Partanen, Janne Heikkinen, Tero Setälä, Ari T. Friberg, Tommi K. Hakala, "Noncyclic continuous Pancharatnam–Berry phase in dual-beam interference", Communications Physics 6, 132 (2023), DOI 10.1038/s42005-023-01249-2, https://doi.org/10.1038/s42005-023-01249-2
7. Berthold-Georg Englert, "Fringe Visibility and Which-Way Information: An Inequality", Physical Review Letters 77, 2154 (1996), DOI 10.1103/PhysRevLett.77.2154, https://doi.org/10.1103/PhysRevLett.77.2154
8. Andreas Norrman, Kasimir Blomstedt, Tero Setälä, Ari T. Friberg, "Complementarity and Polarization Modulation in Photon Interference", Physical Review Letters 119, 040401 (2017), DOI 10.1103/PhysRevLett.119.040401, https://doi.org/10.1103/PhysRevLett.119.040401
9. Paul G. Kwiat, Raymond Y. Chiao, "Observation of a nonclassical Berry's phase for the photon", Physical Review Letters 66, 588 (1991), DOI 10.1103/PhysRevLett.66.588, https://doi.org/10.1103/PhysRevLett.66.588
10. Michael Victor Berry, "Quantal phase factors accompanying adiabatic changes", Proceedings of the Royal Society of London A 392, 45 (1984), DOI 10.1098/rspa.1984.0023, https://doi.org/10.1098/rspa.1984.0023