Observation of the frozen charge of a Kondo resonance


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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Figure 1: Hybrid quantum dot–cavity set-up.
Figure 2: Nature of the electron–photon coupling.
Figure 3: Transparent Kondo resonance.
Figure 4: Temperature dependence of conductance and compressibility.


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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.).

Author information

M.M.D. set up the experiment, made the devices and carried out the measurements with the help of T.K. M.M.D. performed the analysis of the data with input from T.K. J.J.V., M.C.D., L.E.B. and M.R.D. contributed through early experiments and development of the nanofabrication process. M.C.D. developed the data acquisition software. M.L. and M.S.C. carried out the NRG calculations. M.M.D., T.K. and A.C. carried out the semi-classical theory of dot–cavity coupling. T.K., A.C. and M.M.D. co-wrote the manuscript with input from all authors.

Correspondence to T. Kontos.

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Reviewer Information Nature thanks K. Murch and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Figure 1 Microwave cavity characterization.

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.

Extended Data Figure 2 Coulomb blockade regime.

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.

Extended Data Figure 3 Phase in the Kondo regime.

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.

Extended Data Figure 4 Dual conductance/compressibility measurements for other Kondo ridges.

ao, 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).

Extended Data Figure 5 Systematics for the Kondo regime.

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.

Extended Data Figure 6 Temperature dependence for other Kondo ridges.

a, Conductance (top panel) and phase (bottom panel) as a function of temperature for the second Kondo ridge of Fig. 3a. b, As a but for the fourth Kondo ridge of Fig. 3a.

Extended Data Figure 7 Kondo peak for temperature dependence.

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).

Extended Data Figure 8 Control experiment for calibration of electron–photon coupling.

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) doi:10.1038/nature21704

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