Experimental observation of fractional topological phases with photonic qudits

Abstract

Geometrical and topological phases play a fundamental role in quantum theory. Geometric phases have been proposed as a tool for implementing unitary gates for quantum computation. A fractional topological phase has been recently discovered for bipartite systems. The dimension of the Hilbert space determines the topological phase of entangled qudits under local unitary operations. Here we investigate fractional topological phases acquired by photonic entangled qudits. Photon pairs prepared as spatial qudits are operated inside a Sagnac interferometer and the two-photon interference pattern reveals the topological phase as fringes shifts when local operations are performed. Dimensions d = 2, 3 and 4 were tested, showing the expected theoretical values.Geometrical phases were introduced long ago in a seminal work on polarization transformations in classical optics [1,2]. Later, geometrical and topological features were demonstrated to play a fundamental role in the realm of quantum theory [3,4]. Tomita and Chiao verified experimentally the existence of Berry's phase and its topological properties at the classical level in helically wound optical fiber [5]. Geometric phases were measured by using coincidence detection of photon pairs produced in parametric down-conversion in conjunction with a Michelson interferometer [6,7], by changing adiabatically the polarization state of the photon pairs [8], with single photons in a mixed state of polarization by using a Mach-Zenhder interferometer [9] or in a polarization pure state by using a polarimetric technique [10]. More recently, they have been proposed as a robust tool for implementing unitary gates for quantum computation [11,12]. From this perspective, the geometric phase on entangled bipartite systems and the role of entanglement in its topological nature have been discussed both theoretically [13][14][15] and experimentally [16][17][18] for twoqubit systems. Later, they were generalized to pairs of qudits of any dimension [19,20] and to multiple qubits [21], showing that the dimension of the Hilbert space plays a crucial role in determining fractional topological phases in both cases. A recent review on this rich subject can be found in Ref. [22]. Although experimental schemes for demonstrating these fractional values have been proposed [23, 24], they have not been implemented so far. In a broader context, fractional phases may be revealed in quantum Hall systems, related to different homotopy classes in the configuration space of anyons. This nontrivial topology has been conjectured to be a possible resource for fault tolerant quantum computation [25].In this work we present experimental results of fractional topological phase measurements on entangled qu-dits. A qudit is a quantum state that belongs to a d-dimensional Hilbert space (d > 2). A quantum system in a general qudit state can be written in terms of d states that form a basis in the d-dimensional Hilbert space. Photonic qudits with dimensions d = 3 and 4 and qubits (d = 2) were encoded on the transverse ...

Fractional topological phase measurement with a hyperentangled photon source

“…Another important issue is the improvement of the signal to noise ratio with respect to the measurements in ref. 21 . We can estimate this ratio taking C /Δ C for a typical coincidence count, lets say approximately the maximum in Fig.…”

“…In the experiment, this is achieved by means of a spatial light modulator (SLM). The SLM performs local unitary transformations in the spatial qudits 21 , producing…”

“…The advantage of this experimental scheme as compared with the interferometer of ref. 21 is that here we do not need to post select this superposition state from a more general state. Our technique already prepares the necessary state for measuring the topological phase.…”

“…Different works appeared since then, defining the topological phase for multiqubit systems 16 , proposing experiments for photonic qudits 17 and multiqubits 18 , generalizing the theory for two qudits with different dimensions 19 , and investigating the entanglement dependency for two qubits 20 . Experimental works have been devoted to measuring fractional topological phases in two-photon interference 21 . However, the visibilities obtained were below the limit allowed by classical correlations and the quantum nature of the interference fringes could not be ascertained.…”

“…These are important advancements brought by the present scheme as compared to the previous one published in ref. 21 .…”