Although electronic interactions are generally negligible in metals, confinement of electrons in a quantum dot, such as a carbon nanotube, enhances them. A striking consequence of electronic interactions in a metal is the Kondo effect, where the spin of magnetic impurities is screened by conduction electrons at low enough temperature. This same effect emerges for a quantum dot coupled to metallic electrodes: the spin of the quantum dot is screened by the conduction electrons of the electrodes resulting in a singlet state of spin zero (Fig. 1A). Then, any properties of the system are described by a single characteristic scale, the Kondo temperature TK, which is the binding energy of the singlet.

Thanks to a collaboration between the LPS and Osaka University (Japan), it was possible to measure the shot noise, i.e. the fluctuations in the backscattered current due to the variations in the number of charges reflected by the quantum dot. Our experiment was realized in a carbon nanotube connected to metallic electrodes. An additional gate electrode allows to tune the confining potential. The noise was extracted with a very good accuracy from the amplitude of the resonance of an LC circuit connected in parallel (Fig. 1B).

Two signatures of the Kondo state emerged. First, at low current the delocalized state transmits perfectly the current through the dot, without any backscattering. The quantum dot is thus totally silent as seen in Fig. 1C where the shot noise displays a plateau at low current.

*A) Continuous line: extension of the Kondo state between the quantum dot and electrodes. When a current flows (dashed line), a pair of electrons can be generated by interaction (yellow wave). B) Experimental set-up: a nanotube is connected to an LC resonator with a resonance frequency of 2.58 MHz to measure the current noise. Distance between electrodes is 400 nm. C) Conductance G (blue) and noise Si (red) as a function of current. A zero current, the slope of the noise Si(Isd) is zero denoting an absence of backscattering. Si non-linearly increases at high current (out of equilibrium). D) The effective charge is the slope of Si as a function of the backscattered current IK. The values e*>1, indicate the appearance of backscattered pairs. The probability to generate a pair depends only on the symmetry of the Kondo state SU(2) or SU(4). Those two symmetries are obtained for different gate voltages*

At high current, noise increases rapidly and non-linearly (Fig. 1C), which indicates a fluctuating backscattered current. Transmission is no longer perfect, as confirmed by the decrease in conductance. However, the non-linear shot noise (Fig. 1D) is the only quantity which measures the effective charge of the backscattered carriers. Our measurement of the effective charge, e*=5/3 ± 5%, is in perfect agreement with theory, and demonstrates the appearance of electrons pairs. Moreover we have shown that this value is universal for any spin ½ Kondo system independently of TK.

Finally, the peculiar electronic band structure of carbon nanotubes, which presents a doubly degenerate valley, provides an additional degree of freedom for electrons. According to theory, it modifies the effective charge. Hence, for certain gate voltages, electronic states in the dot are four-fold degenerate resulting in a new symmetry of the Kondo state: SU(4) instead of SU(2). We have shown that the effective charge decreases then to e*=3/2 in agreement with theory (Fig. 1D).

This measurement of e* allows to determine the symmetry of the ground state and amplitude of electronic interactions renormalized in the Kondo state. More generally, we confirm experimentally the extension in the non-equilibrium regime of the Fermi-liquid theory, which provides a description of the gas of fermion (such as electrons in a metal) in the presence of interactions.

### Reference

*Universality of non-equilibrium fluctuations in strongly correlated quantum liquids*

Meydi Ferrier, Tomonori Arakawa, Tokuro Hata, Ryo Fujiwara, Raphaëlle Delagrange, Raphaël Weil, Richard Deblock, Rui Sakano, Akira Oguri & Kensuke Kobayashi

*Nature Physics* **12**, 230–235 (2016)

doi:10.1038/nphys3556