Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

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.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

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.

Similar content being viewed by others


  1. Goldhaber-Gordon, D. et al. Kondo effect in a single-electron transistor. Nature 391, 156–159 (1998)

    Article  CAS  ADS  Google Scholar 

  2. Nygard, J., Cobden, D. H. & Lindelof, P. E. Kondo physics in carbon nanotubes. Nature 408, 342 (2000)

    Article  CAS  ADS  Google Scholar 

  3. Roch, N. et al. Quantum phase transition in a single-molecule quantum dot. Nature 453, 633 (2008)

    Article  CAS  ADS  Google Scholar 

  4. Hewson, A. C. The Kondo Problem to Heavy Fermions (Cambridge Univ. Press, 1993)

  5. Lee, M., Lopez, R. Choi, M.-S., Jonckheere, T. & Martin, T. Many-body correlation effect on mesoscopic charge relaxation. Phys. Rev. B 83, 201304(R) (2011)

    Article  ADS  Google Scholar 

  6. Filippone, M., Le Hur, K. & Mora, C. Giant charge relaxation resistance in the Anderson model. Phys. Rev. Lett. 107, 176601 (2011)

    Article  ADS  Google Scholar 

  7. Iftikhar, Z. et al. Two-channel Kondo effect and renormalization flow with macroscopic quantum charge states. Nature 526, 233 (2015)

    Article  CAS  ADS  Google Scholar 

  8. Keller, A. J. et al. Universal Fermi liquid crossover and quantum criticality in a mesoscopic device. Nature 526, 237 (2015)

    Article  CAS  ADS  Google Scholar 

  9. Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13 (1996)

    Article  CAS  ADS  MathSciNet  Google Scholar 

  10. Ashoori, R. C. et al. Single-electron capacitance spectroscopy of discrete quantum levels. Phys. Rev. Lett. 68, 3088 (1992)

    Article  CAS  ADS  Google Scholar 

  11. Ilani, S., Donev, L. A. K., Kindermann, M. & McEuen, P. L. Measurement of the quantum capacitance of interacting electrons in carbon nanotubes. Nat. Phys. 2, 687 (2006)

    Article  CAS  Google Scholar 

  12. Deblock, R., Noat, Y., Bouchiat, H., Reulet, B. & Mailly, D. Measurements of flux dependent screening in Aharonov-Bohm rings. Phys. Rev. Lett. 84, 5379 (2000)

    Article  CAS  ADS  Google Scholar 

  13. Gabelli, J. et al. Violation of Kirchhoff’s laws for a coherent RC circuit. Science 313, 499 (2006)

    Article  CAS  ADS  Google Scholar 

  14. Bruhat, L. E. et al. Cavity photons as a probe for charge relaxation resistance and photon emission in a quantum dot coupled to normal and superconducting continua. Phys. Rev. X 6, 021014 (2016)

    Google Scholar 

  15. Viennot, J. J., Palomo, J. & Kontos, T. Stamping single wall nanotubes for circuit quantum electrodynamics. Appl. Phys. Lett. 104, 113108 (2014)

    Article  ADS  Google Scholar 

  16. Laird, E. et al. Quantum transport in carbon nanotubes. Rev. Mod. Phys. 87, 703 (2015)

    Article  CAS  ADS  MathSciNet  Google Scholar 

  17. Büttiker, M., Prêtre, A. & Thomas, H. Mesoscopic capacitors. Phys. Lett. A 180, 364 (1993)

    Article  ADS  Google Scholar 

  18. Prêtre, A., Thomas, H. & Büttiker, M. Dynamic admittance of mesoscopic conductors: discrete-potential model. Phys. Rev. B 54, 8130 (1996)

    Article  ADS  Google Scholar 

  19. Cottet, A., Kontos, T. & Douçot, B. Electron-photon coupling in mesoscopic quantum electrodynamics. Phys. Rev. B 91, 205417 (2015)

    Article  ADS  Google Scholar 

  20. Delbecq, M. R. et al. Coupling a quantum dot, fermionic leads and a microwave cavity on a chip. Phys. Rev. Lett. 107, 256804 (2011)

    Article  CAS  ADS  Google Scholar 

  21. Lehnert, K. W. et al. Quantum charge fluctuations and the polarizability of the single-electron box. Phys. Rev. Lett. 91, 106801 (2003)

    Article  CAS  ADS  Google Scholar 

  22. Frey, T. et al. Dipole coupling of a double quantum dot to a microwave resonator. Phys. Rev. Lett. 108, 046807 (2012)

    Article  CAS  ADS  Google Scholar 

  23. Petersson, K. D. et al. Circuit quantum electrodynamics with a spin qubit. Nature 490, 380 (2012)

    Article  CAS  ADS  Google Scholar 

  24. Viennot, J. J., Dartiailh, M. C., Cottet, A. & Kontos, T. Coherent coupling of a single spin to microwave cavity photons. Science 349, 408 (2015)

    Article  CAS  ADS  Google Scholar 

  25. Frey, T. et al. Quantum dot admittance probed at microwave frequencies with an on-chip resonator. Phys. Rev. B 86, 115303 (2012)

    Article  ADS  Google Scholar 

  26. Wilson, K. G. The renormalization group: critical phenomena and the Kondo problem. Rev. Mod. Phys. 47, 773 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  27. Costi, T. A. & Hewson, A. C. Transport coefficients of the Anderson model. J. Phys. Condens. Matter 5, L361 (1993)

    Article  ADS  Google Scholar 

  28. Costi, T. A., Hewson, A. C. & Zlatic, V. Transport coefficients of the Anderson model via the numerical renormalization group. J. Phys. Condens. Matter 6, 2519 (1994)

    Article  CAS  ADS  Google Scholar 

  29. Hofstetter, W. Generalized numerical renormalization group for dynamical quantities. Phys. Rev. Lett. 85, 1508 (2000)

    Article  CAS  ADS  Google Scholar 

  30. Weichselbaum, A. & von Delft, J. Sum-rule conserving spectral functions from the numerical renormalization group. Phys. Rev. Lett. 99, 076402 (2007)

    Article  ADS  Google Scholar 

  31. Yoshida, M., Whitaker, M. A. & Oliveira, L. N. Renormalization-group calculation of excitation properties for impurity models. Phys. Rev. B 41, 9403 (1990)

    Article  CAS  ADS  Google Scholar 

  32. Zitko, R. & Pruschke, T. Energy resolution and discretization artifacts in the numerical renormalization group. Phys. Rev. B 79, 085106 (2009)

    Article  ADS  Google Scholar 

Download references


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

Authors and Affiliations



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.

Corresponding author

Correspondence to T. Kontos.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

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

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).

PowerPoint slides

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Desjardins, M., Viennot, J., Dartiailh, M. et al. Observation of the frozen charge of a Kondo resonance. Nature 545, 71–74 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing