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**Geometrical Qubit and non-Abelian gauge field
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The symmetry of the Hilbert space is the key feature from which drives remarkable physical properties of a quantum system. U(1) symmetry leads to abelian transformation, whereas a larger symmetry, such as SU(2), leads to non-Abelian gauge field sensitive to paths ordering. We study an adiabatic non-Abelian transformation using a four-level resonant tripod scheme on a laser cooled fermionic strontium gas. We make sure that the transformation is limited to a subspace made of two degenerated dark state. As a main result, we show the non-Abelian character of the transformation proceeding the same close loop circulation swapping the path ordering. Furthermore we introduce a new thermometry of the gas based on the interferometric displacement of the atoms.

*Experimental implementation*

We implemented a tripod scheme on the F_{g} = 9/2 → F_{e} = 9/2 intercombination line of Strontium-87 at 689 nm using an cold gas containing around 10^{5} atoms. The cold sample is prepared in a crossed optical dipole trap where atoms are optically pumped in the stretched m = F_{g} magnetic sub-state and Doppler cooled at temperatures around 0.5 μK. A magnetic bias field isolates a particular tripod scheme among the excited and ground Zeeman substate manifolds. The three coupling laser beams are set on resonance with their common |m = 7/2, F_{g} = 9/2> excited state and the relative phases can be varied using two electro-optic modulators (EOM).

*figure*: (a) black arrow: tripod beam, red arrow: spin-sensitive imaging system, purple arrow: Doppler cooling in the far-off resonant trap. (b) arrangement of the tripod beam.

*Ballistic expansion and thermonetry*

As a benchmark of the gauge fields generated by optical fields, it comes from photons redistribution among the planes waves modes. So a gauge field would be characterized by a dispersion law that exhibited non-trivial properties for energies lower or comparable to the photon recoil energy. As a consequence, the temperature of the gas should be lower that the recoil temperature. In our system the temperature, obtained by Doppler cooling, is larger than the recoil temperature, which prevents observation of gauge field features. Nevertheless, we show that, regardless the temperature of the gas and as far as the adiabatic condition is fulfilled, the internal state population dynamic is still imposed by the gauge field (see example on figure below). We then find a new interferometric thermometry method which is derived form the measurement of the Fourier transformation of the velocity distribution of the gas.

*Figure*: Ground state population evolution during Ballistic expansion of a cold gas in presence of the non-Abelian gauge field. The red, green and blue colors correspond respectively to the |m=9/2>, |m=7/2> and |m=5/2>. The temperature of the gas can be extracted from evolution of the state populations.

*Geometric quantum gate and path ordering sensitivity*

We investigated the adiabatic geometric operation in the dark-state manifold by sweeping the relative phases of the tripod beams into a close loop. We prepare the system into a given dark state and we observe a transfer of population after the close loop (see figure below). In contrast with a U(1) system which is characterized by a geometrical phase (the Berry phase), our underlying symmetry is SU(2) and the transformation is characterized by a 2×2 unitary geometric operator. We were able to reconstruct this unitary operator and check the non-Abelian character of the transformation, i.e. the path ordering sensitivity.

*Figure: *(a) close loop of the laser phase. (b) evolution of the dark state population and azimuthal angle in the dark state subspace. Dots: experimental date. Plain curve: A pinned atom. Dashed curve: cold gas at finite temperature. (c) reconstruction of the geometric unitary operator for two different path ordering of the close loop. Green: experiment, red: cold gas at finite temperature and blue: pinned atom. The difference between the two close loop is due to the non-Abelian character of the transformation.