The ability to control electronic states at the nanoscale has contributed to our modern understanding of condensed matter. In particular, quantum dot circuits represent model systems for the study of strong electronic correlations, epitomized by the Kondo effect1,2,3. We use circuit quantum electrodynamics architectures to study the internal degrees of freedom of this many-body phenomenon. Specifically, we couple a quantum dot to a high-quality-factor microwave cavity to measure with exceptional sensitivity the dot’s electronic compressibility, that is, its ability to accommodate charges. Because electronic compressibility corresponds solely to the charge response of the electronic system, it is not equivalent to the conductance, which generally involves other degrees of freedom such as spin. Here, by performing dual conductance and compressibility measurements in the Kondo regime, we uncover directly the charge dynamics of this peculiar mechanism of electron transfer. The Kondo resonance, visible in transport measurements, is found to be ‘transparent’ to microwave photons trapped in the high-quality cavity, thereby revealing that (in such a many-body resonance) finite conduction is achieved from a charge frozen by Coulomb interaction. This freezing of charge dynamics4,5,6 is in contrast to the physics of a free electron gas. We anticipate that the tools of cavity quantum electrodynamics could be used in other types of mesoscopic circuits with many-body correlations7,8, providing a model system in which to perform quantum simulation of fermion–boson problems.
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We thank L. I. Glazman, H. Baranger and A. Clerk for discussions, L. C. Contamin, T. Cubaynes, Z. Leghtas and F. Mallet for reading the manuscript, and J. Palomo, M. Rosticher and A. Denis for technical support. The devices were made by the consortium Salle Blanche Paris Centre. We acknowledge support from Jeunes Equipes de l’Institut de Physique du Collège de France (JEIP). This work was supported by ERC Starting Grant CIRQYS and by the NRF of Korea (grant nos 2009-0069554 and 2011-0030046 to M.L. and 2015-003689 to M.S.C.).
The authors declare no competing financial interests.
Reviewer Information Nature thanks K. Murch and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data figures and tables
Left, phase and amplitude of the microwave signal (plotted as transmission) as a function of frequency showing the cavity resonance used to measure the compressibility. The linewidth of the cavity κ can be read out from this plot as indicated by the grey double arrow. Right, temperature dependence of the linewidth of the cavity, κ.
Phase and conductance (G) plotted on a wide scale in the Coulomb blockade regime. The observation of groups of four peaks in both the conductance and the phase contrast arises from the spin/valley degeneracy of the nanotube spectrum.
Colour-scale plot of phase in the Kondo regime corresponding to Fig. 3a. We observe tilted lines arising from single charge peaks, but no Kondo ridge. The tilted dotted black lines are guides to the eye. The four vertical dashed lines correspond to the position of the cuts presented in the main text (first, from left to right in this figure), and in the Methods section (third for Extended Data Fig. 7 left panel, and second and fourth for Extended Data Fig. 4). A spurious tilted blue line is also observed. It probably arises from an impurity level coupled to the cavity field.
a–o, Examples of 15 different Kondo ridges displaying the same phenomenona as in Fig. 3b. These data correspond to cuts indicated by vertical dashed lines in Extended Data Fig. 5. In particular, the Kondo peak apparent in the conductance (in blue) is always absent from the compressibility (in orange).
a, b, Conductance and phase as a function of source–drain bias and gate voltage for Kondo ridges different from the set presented in the main text. c, Conductance and phase as a function of source–drain bias and gate voltage on a wide scale in the Kondo regime. The measurements have been performed for a different cool-down (from 2 K to 250 mK) of our 3He single-shot cryostat and correspond to physical parameters different from those for a and b.
Left panel, bias dependence of conductance and phase for the Kondo ridge used to determine the temperature dependence of Fig. 4a. Right panel, corresponding gate dependence at base temperature (255 mK) and at high temperature (2.05 K). To get rid of the thermal drift of the phase, we compute the difference of the phase between a Coulomb peak (green arrow) and a Coulomb valley (blue arrow), where the Kondo ridge is. The phase at 2.05 K has been rescaled to take into account the decrease of the quality factor of the cavity with temperature (22,000 to 18,000).
Power dependence of Coulomb peaks for four different peaks (a, b, c and d), shown at left. Each peak height is plotted in the right panels versus the microwave modulation amplitude, which controls the number of photons inside the cavity. The open dots are data and the solid lines are fits using equation (13).
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Desjardins, M., Viennot, J., Dartiailh, M. et al. Observation of the frozen charge of a Kondo resonance. Nature 545, 71–74 (2017). https://doi.org/10.1038/nature21704
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