Cavity Photons as a Probe for Charge Relaxation Resistance and Photon Emission in a Quantum Dot Coupled to Normal and Superconducting Continua

Microwave cavities have been widely used to investigate the behavior of closed few-level systems. Here, we show that they also represent a powerful probe for the dynamics of charge transfer between a discrete electronic level and fermionic continua. We have combined experiment and theory for a carbon nanotube quantum dot coupled to normal metal and superconducting contacts. In equilibrium conditions, where our device behaves as an effective quantum dot-normal metal junction, we approach a universal photon dissipation regime governed by a quantum charge relaxation effect. We observe how photon dissipation is modified when the dot admittance turns from capacitive to inductive. When the fermionic reservoirs are voltage biased, the dot can even cause photon emission due to inelastic tunneling to/from a Bardeen-Cooper-Schrieffer peak in the density of states of the superconducting contact. We can model these numerous effects quantitatively in terms of the charge susceptibility of the quantum dot circuit. This validates an approach that could be used to study a wide class of mesoscopic QED devices.

Popular Summary

Quantum dots are garnering increasing interest because of their potential applications in quantum information processing. A common trend is to reduce the coupling between quantum dots and their environment to obtain effective atoms characterized by long coherence times. In this spirit, cavity quantum electrodynamics architectures embedding quantum dot circuits have recently been developed. However, quantum dots can also be viewed as test beds for condensed matter problems when the coupling between the dots and their environment is increased. Here, we combine experiment and theory for a quantum dot tunnel coupled to various types of fermionic reservoirs embedded in a microwave cavity.

We focus on a carbon nanotube quantum dot in the limit of strong dot-reservoir coupling. Our setup consists of coupling the dot to (i) a normal metal reservoir and (ii) a normal metal reservoir and a superconducting reservoir. We simultaneously study the dot conductance and the cavity microwave transmission, and we detect, with high sensitivity, charge relaxation in the dot and photon-assisted dot-superconductor tunneling. We obtain an unprecedented agreement between experiment and theory for our hybrid quantum system. We thereby validate an approach applicable to a wide class of multisite nanocircuits with normal, superconducting, and ferromagnetic reservoirs.

Our work establishes a new path for the study of charge transfer dynamics in quantum dot circuits and, more generally, mesoscopic systems. We expect that our findings will be instrumental for the study of topical phenomena such as Cooper-pair splitting, Kondo physics, and Majorana quasiparticles.

Figure 1

Panels (a) and (b): Scanning electron micrograph of the microwave resonator and the quantum dot circuit. Panel (c): Principle of our setup. The dot level is tunnel coupled to the N and S reservoirs and modulated by the cavity electric field. Panel (d): Current through the S contact versus the effective gate voltage $V_g$ and the bias voltage $V_b$.

Figure 2

Panels (a–d): Measured phase shift $\mathrm{\Delta }\phi$ (blue dots) and reduced amplitude shift $\mathrm{\Delta }A/A_0$ (red dots) of the microwave signal transmitted by the cavity versus the energy ${ϵ}_d$ of the dot orbital, for $V_b=0$ and different dot orbitals (for clarity, we have plotted the opposite of these signals). The red and blue lines show the predictions given by Eqs. (5) and (6) for values of ${\mathrm{\Gamma }}_N$ and $g$ given in the different panels, and $T=60$ $\mathrm{mK}\simeq 0.19{\omega }_0$.

Figure 3

Top panel: Ratio ${\theta }_0=\theta ({ϵ}_d=0)$ versus the tunnel rate ${\mathrm{\Gamma }}_N$, calculated from Eqs. (5), (8), and (b16) for $T=0$ (black dashed line) and $T=60\mathrm{mK}$ (gray solid line). The crosses correspond to fitted values of ${\theta }_0$, calculated from Eqs. (5) and (8) for the different resonances in Figs. 2 and 7. Bottom panels: Comparison between the experimental $\mathrm{\Delta }{\mathrm{\Lambda }}_0$ and $(\mathrm{\Delta }{\omega }_0{)}^2$, using the scaling factor $\alpha =\pi {\omega }_0/2{\theta }_0{g}^2$, with ${\theta }_0$ indicated with arrows in the top panel. We use ${\mathrm{\Gamma }}_N/{\omega }_0=0.16$, 1.23, 2.33, and 2.86 from left to right and top to bottom panels. We also show, as blue and red solid lines, the calculated $\mathrm{\Delta }{\mathrm{\Lambda }}_0$ and $(\mathrm{\Delta }{\omega }_0{)}^2$.

Figure 4

Panels (a), (c), and (e): Measured linear conductance $G$, phase shift $\mathrm{\Delta }\phi$, and total amplitude $A$ of the transmitted microwave signal versus the dot gate voltage $V_g$ and the bias voltage $V_b$. Panels (b), (d), and (d): Predictions from Eqs. (3), (6), and (b17), for ${\mathrm{\Gamma }}_N/2\pi =0.6\mathrm{GHz}$, ${\mathrm{\Gamma }}_S/2\pi =65\mathrm{MHz}$, ${\mathrm{\Gamma }}_n/2\pi =8\mathrm{GHz}$, $g/2\pi =99\mathrm{MHz}$, $\mathrm{\Delta }=0.17\mathrm{meV}$, $T=90\mathrm{mK}$, ${\omega }_0/2\pi =6.65\mathrm{GHz}$, $A_0=6.1\mathrm{mV}$, and ${\mathrm{\Lambda }}_0/2\pi =0.259\mathrm{MHz}$. The white color corresponds to $A=A_0$ in panels (e) and (f). Panels (1), (2), and (3): Electric potential configuration corresponding to the black points in panel (d).

Figure 5

Conductance $G$ and microwave amplitude $A$ versus $V_g$ (red dots), measured along the dashed line in Fig. 4 for $V_b=0.336\mathrm{mV}$, and theory using the same parameters as in Fig. 4 and ${\mathrm{\Gamma }}_n/2\pi =8\mathrm{GHz}$ (red lines) or ${\mathrm{\Gamma }}_n/2\pi =1\mathrm{GHz}$ (blue lines). The theoretical $G$ for ${\mathrm{\Gamma }}_n/2\pi =1\mathrm{GHz}$ has been multiplied by 0.2. Panels (1), (2), and (3) illustrate the transport regimes corresponding to the gray dashed lines. In panel (2), the dot orbital is resonant with a BCS peak in the DOS of the S reservoir. In panels (1) and (3), an electron can pass from the dot orbital to a BCS peak by absorbing or emitting a cavity photon.

Figure 6

Current $I$ through the quantum dot (left panel) and cavity signal $\mathrm{\Delta }\phi$ (right panel) versus $V_g^L$ and $V_g^R$, in the area corresponding to the data of Figs. 3 and 3.

Figure 7

Cavity signals $\mathrm{\Delta }\phi$ (blue dots) and $\mathrm{\Delta }A/A_0$ (red dots) versus ${ϵ}_d$ for $V_b=0$ and different dot orbitals with decreasing tunnel rates ${\mathrm{\Gamma }}_N$ from top to bottom and left to right panels. The red and blue lines show the predictions given by Eqs. (5) and (6) for the values of ${\mathrm{\Gamma }}_N$ and $g$ given in the different panels and $T=60\mathrm{mK}$. When a dot-reservoir resonance is already shown in the main text, we indicate the corresponding figure number in pink.

Figure 8

Top panel: Measured amplitude $A$ versus $V_b$ and $V_g$, already shown in Fig. 4. Bottom panels: Measured conductance $G$ (black dots), and cavity signals $\mathrm{\Delta }\phi$ (blue dots) and $A$ (red dots) versus $V_g$, along the dashed lines in the top panel, for $V_b=0.25$, 0.303, and 0.336 mV from left to right. The solid red, blue, and black lines show the predictions given by Eqs. (3), (6), and (b17), for the same parameters as in Fig. 4. The areas in the gray rectangles are enlarged in Fig. 5 of the main text.