Vacuum-field-induced THz transport gap in a carbon nanotube quantum dot

The control of light-matter interaction at the most elementary level has become an important resource for quantum technologies. Implementing such interfaces in the THz range remains an outstanding problem. Here, we couple a single electron trapped in a carbon nanotube quantum dot to a THz resonator. The resulting light-matter interaction reaches the deep strong coupling regime that induces a THz energy gap in the carbon nanotube solely by the vacuum fluctuations of the THz resonator. This is directly confirmed by transport measurements. Such a phenomenon which is the exact counterpart of inhibition of spontaneous emission in atomic physics opens the path to the readout of non-classical states of light using electrical current. This would be a particularly useful resource and perspective for THz quantum optics.


I. SUPPLEMENTARY NOTE 1: MAPS FOR DEVICES A,B AND C
In this section, we present V sd − V g maps of the conductance of samples A, B and C. It also shows one extra G(V sd ) curve for sample B. Supplementary Fig. 1 shows the V sd − V g conductance map of Sample A already shown in Figure 3a, but on a larger scale. Supplementary Fig. 2 and 3 show the V sd − V g conductance maps of Samples B and C whose G(V sd ) cuts are shown in panels 3b and 3c respectively. The arrows on the top of these three figures indicate the position of the G(V sd ) cuts shown in Figure 3b and 3c of the main text and Supplementary Fig. 4. The dashed horizontal lines indicate the position of the conductance steps/resonances expected at eV sd = −2hf cav , eV sd = −hf cav , eV sd = hf cav , and eV sd = 2hf cav . The line eV sd = 0 is also shown in Supplementary Fig. 3. For Sample A (Supplementary Fig. 1), we have shifted vertically these lines at V g 200 mV to take into account an offset in V sd .
For sample A, a gap delimited by eV sd = ±hf cav is clearly visible in the conductance along the whole gate voltage range of Supplementary Fig. 1, as well as a conductance variation for at eV sd = ±2hf cav . For Sample B, a gap delimited by eV sd = ±hf cav is visible only in the range 20 mV V g 20 mV. This gap is less visible for the other gate voltage ranges.
One can also guess the presence of slight conductance steps at eV sd = ±2hf cav in some areas of the the figures. The variations in the visibility of these features can be attributed to the fact that when the dot gate voltage is varied, the spatial profile of the electronic Green's function changes, and consequently, the amplitude of the electron/photon coupling changes notably. Note that for sample B, negative differential resistance is indicated in green. Such a feature can already happen in the absence of electron/photon coupling and is often due to electronic interaction effects, but its occurrence may also be influenced/modified by the presence of the light/matter interaction. In sample C, the low voltage conductance gap is clearly delimited by a step at eV sd = hf cav , but the step at eV sd = −hf cav is missing and seems to be replaced by a smooth limit at eV sd = 0. In fact, a Kondo conductance ridge at eV sd = 0 is even visible for 50 mV V g 65 mV. Therefore one can say that there is only a half gap in the data at 0 mV V sd hf cav /e, in agreement with the data shown in Fig.   3c of the main text.
Supplementary Fig. 4 shows one extra G(V sd ) curve for sample B, for V g = 13.8 mV. An area with negative differential resistance (G < 0) is visible in this curve.
where V cav is the mode volume and ε 0 is the vacuum permittivity. This is a simplified expression which holds for atoms in vacuum and this is why we used the COMSOL and HFSS softwares to estimate E zpf and the corresponding V zpf in our case. The result of the HFSS simulation is shown in Supplementary Fig. 6. However, we can get interesting insights about the basic ingredient which help to boost the vacuum field fluctuations by inspecting the above formula. As shown in the main text, one can bridge between circuit QED to cavity QED and a reliable estimate of the coupling strength in our case is g l ≈ eE zpf d, where d is the typical size of the dipole which spans from below the THz gate to the source electrode, and e is the elementary charge. The quantity C cav−matter = ε 0 V cav /d 2 has the dimension of the a capacitance which is characteristic of how the dipole fits in the mode volume once both system are assembled. Defining the joint charging energy E cav−matter = e 2 /C cav−matter , we arrive at an insightful expression : This expression should of course take into account the electron-electron interactions which are expected to modify both the mode volume and the size of the dipole due to their role in the screening of the electric field. For typical sizes of dipoles and cavity modes, this charging energy can be very large, as large as the one measured for the nanotube i.e. in the several meV range. This suggests that the deep strong coupling limit may be reached for nanoscale conductors with large interaction effects. In our case, both COMSOL and HFSS show that already without the nanotube, V zpf can be in the 200 − 400µV range. Electron-electron interactions play a crucial role in screening processes. Our findings suggest that they could boostg further into the deep strong coupling regime.