Simulating the exchange of Majorana zero modes with a photonic system

Abstract

The realization of Majorana zero modes is in the centre of intense theoretical and experimental investigations. Unfortunately, their exchange that can reveal their exotic statistics needs manipulations that are still beyond our experimental capabilities. Here we take an alternative approach. Through the Jordan-Wigner transformation, the Kitaev's chain supporting two Majorana zero modes is mapped to the spin-1/2 chain. We experimentally simulated the spin system and its evolution with a photonic quantum simulator. This allows us to probe the geometric phase, which corresponds to the exchange of two Majorana zero modes positioned at the ends of a three-site chain. Finally, we demonstrate the immunity of quantum information encoded in the Majorana zero modes against local errors through the simulator. Our photonic simulator opens the way for the efficient realization and manipulation of Majorana zero modes in complex architectures.

Figure 1. The braiding of Majorana zero modes and the mapping between the fermionic and spin models.

The spheres with numbers at their centres represent the Majorana fermions, γ_j, for j=1,…,6, at the corresponding sites. A pair of Majorana fermions bounded by an enclosing ring represents a normal fermion. The wavy lines between different sites represent the interactions between them. The interactions illustrated in a,b,c and d represent the Hamiltonians , , and , respectively. The letters A and B in each pane represent the corresponding isolated Majorana zero modes. The mapping between the Kitaev chain model (KCM) and the transverse-field Ising model (TFIM) through the JW transformation is shown in e. m_3f and k_3s are bases of the ground-state spaces defined in the KCM and TFIM (m and k=0 or 1), respectively.

Figure 2. Experimental setup for the simulation of the exchange of Majorana zero modes.

The process follows the logical diagram provided in the pane enclosed by the black solid line, denoted by Logi. The state of the three-spin-1/2 system can be expressed in the eight-dimensional space with the basis denoted by |l〉 (l=1 to 8), which are encoded as the spatial modes of photons. C_l are the corresponding amplitudes. For the initial Hamiltonian H₀, the space is expanded by the basis of . The polarization of single photons is rotated using HWPs and quarter-wave plate (QWPs), and the spatial modes are separated by BDs, each with a beam displacement of either 3.0 mm (BD30) or 6.0 mm (BD60). The state preparation is illustrated in the pane labelled Pre. The basis rotations BR1, BR2 and BR3 are used to express the input states in terms of the eigenstates of H₁, H₂ and H₀, respectively. The dissipative evolutions DE0, DE1 and DE2 drive the input states to the ground states of H₀, H₁ and H₂, respectively. Some of the detailed basis representations of the spatial modes are given in the right column. The solid magenta rings represent the preserved optical modes and the dashed magenta rings represent the discarded optical modes. The states indicated near the optical modes represent the corresponding preserved basis in the eight-dimensional space. Two types of measurements are performed, that is, TM and FM. Beam splitters (BSs) are used to send the photons to different measurement instruments. H, V, R and D represent the horizontal polarization, vertical polarization, right-hand circular polarization and diagonal polarization, respectively. Finally, photons are detected using single-photon detectors (SPDs).

Figure 3. Experimental results on simulating the braiding evolution.

(a) The six experimental initial states after the first dissipative evolution DE0 with the dark green dot labelled as 1, cyan dot labelled as 2, magenta dot labelled as 3, dark yellow dot labelled as 4, violet dot labelled as 5 and navy dot labelled as 6 in the Bloch sphere. (b) The corresponding experimental final states after the second DE0 with the dark green dot labelled as 1′, cyan dot labelled as 2′, magenta dot labelled as 3′, dark yellow dot labelled as 4′, violet dot labelled as 5′ and navy dot labelled as 6′ in the Bloch sphere. The black dots in the poles of the Bloch spheres represent the corresponding theoretical predictions with the states |xxx〉 (+Z direction), (−Y direction), (−Z direction), (+X direction, |0_3s〉), (+Y direction) and (−X direction, |1_3s〉), respectively. Owing to the experimental errors, the coloured dots (experimental results) are slightly separated from the corresponding black dots. The final states are shown to be rotated along the X axis by π/2 from the initial states. (c) Real (Re) and (d) imaginary (Im) parts of the exchange operator in the basis of {I(identity), X(σ^x), Y(σ^y), Z(σ^z)}, with a fidelity of 94.13±0.04%.

Figure 4. Experimental results on simulating local noises immunity.

(a) Real (Re) and (b) imaginary (Im) parts of the flip-error protection operator, with a fidelity of 97.91±0.03%. (c). Real (Re) and (d) imaginary (Im) parts of the phase-error protection operator, with a fidelity of 96.99±0.04%. The basis are expressed as {I(identity), X(σ^x), Y(σ^y), Z(σ^z)}.