Observation of Entanglement-Dependent Two-Particle Holonomic Phase

Holonomic phases—geometric and topological—have long been an intriguing aspect of physics. They are ubiquitous, ranging from observations in particle physics to applications in fault tolerant quantum computing. However, their exploration in particles sharing genuine quantum correlations lacks in observations. Here, we experimentally demonstrate the holonomic phase of two entangled photons evolving locally, which, nevertheless, gives rise to an entanglement-dependent phase. We observe its transition from geometric to topological as the entanglement between the particles is tuned from zero to maximal, and find this phase to behave more resiliently to evolution changes with increasing entanglement. Furthermore, we theoretically show that holonomic phases can directly quantify the amount of quantum correlations between the two particles. Our results open up a new avenue for observations of holonomic phenomena in multiparticle entangled quantum systems.

Figure 1

(a) A cyclic Schmidt evolution with ${\theta }_{\mathrm{S}}=2\pi$. Left and middle spheres show trajectories traced out in the local Bloch spheres of each qubit defined in the Schmidt basis. Here, they evolve at a single point and, therefore, enclose no area, leading to no gain in holonomic phase. The right sphere represents the trajectory spanned by the evolution of $\alpha$ and $\beta$ in the Schmidt sphere. In contrast to the local trajectories, a holonomic phase still arises as a result of the area enclosed by this trajectory. Equivalent Schmidt evolutions of nonentangled states induce a zero holonomic phase. (b) In contrast, segmented evolutions defined by angles $s_a$ and $s_b$ induce holonomies appearing from both local and Schmidt spheres. The parameters $s_a$ and $s_b$ define the opening angles for the projected curves of ${\rho }_a$ and ${\rho }_b$ onto the local Bloch spheres. In the right sphere, a rotation angle $2(s_a+s_b)=\beta$ around $|n_a{m}_b⟩$ also contributes to the holonomy. Equivalent evolutions of nonentangled states induce a nonzero holonomic phase. (c) Depiction of the double-connected parameter space of maximally entangled states: $\mathrm{SO}(3)$ in ${\mathbb{R}}^3$ with a border at $S_{\pi }^2$. This border is a two sphere of radius $\pi$ with identified antipodal points. Blue trajectories represent arbitrary evolutions for one homotopy class along which no phase is gained. Red curves represent evolutions of the other homotopy class after which a $\pi$ phase appears on the wave function.

Figure 2

(a) Depiction of the experimental method. Two indistinguishable photons, encoding a two-qubit entangled state $|\psi ⟩$ in their polarization, pass through an interferometer and are detected in coincidence with avalanche-photo-diodes (APD) at the output. Because of the Hong-Ou-Mandel effect at the first beam splitter BS1, a coincidence signal $c$ after the exit of the interferometer through BS2, cannot distinguish between different paths travelled in the interferometer. The resulting holonomic phase gained by the evolved state $|{\psi }^{\prime }⟩={\mathrm{Ȗ}}^{†}\mathrm{Ŭ}\otimes {\mathrm{Ȗ}}^{†}\mathrm{Ŭ}|\psi ⟩$, as well as the phase $\varphi$, thus, appears as a modulation in the coincidence given by $c\propto |e^{2i\varphi }|\psi ⟩+|{\psi }^{\prime }⟩{|}^2$. (b) Experimental implementation of the method. Single photons are injected via single-mode fiber couplers (FC) into a displaced-Sagnac interferometer configuration, composed of one $50:50$ beam splitter (BS) and three mirrors $M1–M3$. Polarization evolution is performed using two common-path quarter wave plates QWP and two semicircular half wave plates HWP in separate optical paths. Mirror $M3$ is rotated via a microtranslation stage to control $\varphi$.

Figure 3

Experimental results. (a) In a cyclic color scale, predicted (left) and measured (right) results for ${\mathrm{\Phi }}_{\mathrm{h}}=-\arctan [\cos (\alpha )\tan (2s)]$ [equivalent to Eq. (4)]. Centers of the rectangular measured data blocks represent the corresponding ($\alpha$, $s$) coordinates. (b) The black data points and theoretical curves correspond to measurements of holonomic phases with an initial state $|HH⟩$ (solid curve) and $|VV⟩$ (dashed curve) for a fitted level of tangle, $\mathbb{T}=0.01\pm 0.01$. Blue data points and theoretical curves show the extent to which we observe the topological behavior of ${\mathrm{\Phi }}_{\mathrm{h}}$ for a highly entangled state. Here, the data are fitted to a state tangle of $\mathbb{T}=0.99\pm 0.01$, for the cases in which $|\psi (0)⟩$ is more populated by $|HH⟩$ ($\alpha =\pi /2-\epsilon$) shown by the solid curve, and $|VV⟩$ ($\alpha =\pi /2+\epsilon$) given by the dashed curve, for $0<\epsilon \ll 1$. The red curve corresponds to the theoretical ideal case of $\mathbb{T}=1$. Errors are calculated via Poissonian counting statistics. (c) Depiction of cyclic (I,V) and noncyclic (II,III,IV) evolutions of MESs in a plane of the double-connected $\mathrm{SO}(3)$ ball. Green, red, and blue colored curves denote the first, second, and third parts of each evolution, respectively. Evolution III for which $s=\pi /4$, marks the switch between two distinct homotopy classes, those that cross the $S_{\pi }^2$ border zero times (I and II) and one time (IV and V).