Scaling laws of the Kondo problem at finite frequency

Driving a quantum system at finite frequency allows one to explore its dynamics. This has become a well-mastered resource for controlling the quantum state of two-level systems in the context of quantum information processing. However, this can also be of fundamental interest, especially with many-body systems which display an intricate finite-frequency behavior. In condensed matter, the Kondo effect epitomizes strong electronic correlations, but the study of its dynamics and the related scaling laws has remained elusive so far. Here, we fill this gap by studying a carbon-nanotube-based Kondo quantum dot at half filling driven by a microwave signal. Our findings not only confirm long-standing theoretical predictions but also allow us to establish a simple ansatz for the scaling laws on the Kondo problem at finite frequency. More generally, our technique opens a path for understanding the dynamics of complex quantum dot circuits in the context of quantum simulation of strongly correlated electron fluids.

Figure 1

(a) Scanning electron microscope picture of our device. It behaves as a single dot, which can be biased by applying a dc voltage $V_{\mathrm{SD}}$ on the source electrode, while the current $I$ is measured through the drain electrode. A dc gate voltage $V_G$ is applied on a side gate. A microwave excitation $V_{\mathrm{ac}}$ at frequency $f$ can be applied on the central top gate. (b) Conductance color plot versus gate and bias voltage showing characteristic Kondo ridges. (c) Example of second-order tunnel process which flips the dot spin. The constructive interference of such processes at all orders gives rise to the Kondo peak for $V_{\mathrm{SD}}=0$. (d) Plot of Kondo resonance conductance peak at equilibrium ($V_{\mathrm{ac}}=0$). The black dots are data and the solid green line is a Lorentzian fit giving $k_B{T}_K=32.7$ $\mu \mathrm{eV}$, $G_{\mathrm{OFF}}=0.281(2e^2/h)$, and $G_{\mathrm{BG}}=0.159(2e^2/h)$ [see Eq. (A1) in Appendix pp1].

Figure 2

Color plot of the conductance versus bias voltage $V_{\mathrm{SD}}$ and excitation voltage $V_{\mathrm{ac}}$ with excitation frequency (a) $f=12$ GHz and (b) $f=19$ GHz, corresponding to reduced frequencies (a) $hf/k_B{T}_K=1.8$ and (b) $hf/k_B{T}_K=2.8$. $k_B{T}_K$, $G_{\mathrm{OFF}}$, and $G_{\mathrm{BG}}$ for this data set differ slightly from the data shown in Fig. 1 and their values are given in Fig. 8 (file number 18). Linear cuts of the data shown in (c) panel (a) and (d) panel (b). The lines’ color encodes the excitation voltage, and the color code applies to both frequencies.

Figure 3

(a) Experimental averaged normalized conductance color plot $G_{\mathrm{ON}}/G_{\mathrm{OFF}}$ as a function of excitation scaled amplitude $e{V}_{\mathrm{ac}}/k_B{T}_K$ and scaled frequency $hf/k_B{T}_K$. (b) Calculated map of $G_{\mathrm{ON}}/G_{\mathrm{OFF}}$ using the ansatz formula (1). (c) Linear cuts at $hf/k_B{T}_K=0.69$ (red) and $hf/k_B{T}_K=2.45$ (orange). (d) Linear cuts at $e{V}_{\mathrm{ac}}/k_B{T}_K=1.1$ (blue) and $e{V}_{\mathrm{ad}}/k_B{T}_K=1.7$ (green). In (c) and (d), squares correspond to data points, while dots linked by solid lines correspond to the ansatz. The dashed black line indicates the equilibrium reference ($V_{\mathrm{ac}}=0$).

Figure 4

(a) Sketch of a quantum dot with an ac bias voltage $V_{\mathrm{ac}}\left(t\right)=V_{\mathrm{ac}}cos\left(2\pi ft\right)$. Due to this modulation, photons can trigger second-order tunnel processes, represented here by the pair of green arrows, which flip the dot spin and leave a high-energy electron in the reservoirs. The three colored solid and dotted lines represent $V_{\mathrm{ac}}\left(t\right)$ for different values of excitation frequency $f$ with the same $V_{\mathrm{ac}}$. This color code applies to all panels. (b) Plot of the phase difference $\phi \left(t\right)$ between the electronic states in the source and drain, as a function of time. (c) Representation of the circuit dynamics. The dot leaves the Kondo ground state with a rate ${\mathrm{\Gamma }}_e$ and relaxes to the Kondo ground state again with a rate $k_B{T}_K/\hslash$. (d) Time dependence of the dot conductance $g\left(t\right)$, which reaches $G_U$ in the Kondo state and is much smaller otherwise. The average conductance $G=〈g\left(t\right)〉$ measured in the experiment is closer to $G_U$ for large values of $f$ for which ${\mathrm{\Gamma }}_e$ is smaller.

Figure 5

(a) Plots of conductance versus bias voltage for narrow and broad Kondo resonances at equilibrium ($V_{\mathrm{ac}}=0$). The black dots are data and the solid lines are Lorentzian fits yielding $k_B{T}_K=$ $33.3\mu \mathrm{eV}$ for the narrow Kondo peak (blue) and $k_B{T}_K=95$ $\mu \mathrm{eV}$ for the broad Kondo peak (pink). Comparison of (b) scaled amplitude dependence and (c) scaled frequency dependence of $G_{\mathrm{ON}}/G_{\mathrm{OFF}}$ for broad (pink) and narrow (blue) Kondo peaks to investigate universality. Squares correspond to experimental data, while dots linked by solid lines correspond to the ansatz.

Figure 6

(a) The local slope of the conductance $G$ versus bias voltage gives $\frac{d^{2}I}{d{V}_{\mathrm{SD}}^2}$. Here around $V_{\mathrm{SD}}=0.207$ mV, $\frac{d^{2}I}{d{V}_{\mathrm{SD}}^2}=1.0\times {10}^{-1}$ ${\mathrm{A}/\mathrm{V}}^2$. (b) Knowing this value, the slope $p$ of the current $I$ as a function of the square source amplitude $V_{\mathrm{ac},\text{source}}$ in the linear regime yields the amplitude calibration coefficient $c\left(f\right)=\sqrt{4p/\frac{d^{2}I}{d{V}_{\mathrm{SD}}^2}}$. Here for $f=20$ GHz $p=0.372\mathrm{nA}/{\mathrm{V}}^2$; hence $c(f=20\text{GHz})=1.2\times {10}^{-4}$. (c) An iterative relative amplitude calibration. For each frequency the conductance is successively measured with and without microwave excitation. Calibration coefficients $c_n\left(f\right)$ are used to set a $V_{\mathrm{ac}}\approx 200$ $\mu \mathrm{V}$ for each frequency. The new coefficient set $c_{n+1}\left(f\right)$ is deduced from Eq. (B3) with $G$ instead of $I$. (d) The calibration coefficients $c\left(f\right)$ used to calculate the excitation amplitudes $V_{\mathrm{ac}}$ applied to the narrow (blue dots) and broad (pink dots) Kondo peaks.

Figure 7

(a) Conductance measurement at finite bias ($V_{\mathrm{SD}}=$ $203\mu \mathrm{V}$) and under ac excitation $V_{\mathrm{ac}}=100$ $\mu \mathrm{V}$. Curves at $f=2.9$, 11.8, and 19.9 GHz superpose at all gate voltages, while the $f=3.3$ GHz curve does not. (b) Conductance measurement at zero bias under ac excitation ($V_{\mathrm{ac}}=100$ $\mu \mathrm{V}$) for the same frequencies.

Figure 8

Table of Kondo equilibrium parameters associated with each data file.

Figure 9

(a) Experimental and (b) ansatz scaled amplitude-frequency normalized conductance map for the broad peak.