Figure 1

The device consists of a 20-nm thick, 60-nm wide silicon nanowire etched out of a silicon-on-insulator substrate and partially overlapped in its center by two facing polysilicon top gates (in gray). The silicon substrate acts as an additional back gate. The silicon nanowire is implanted with As donors (red spheres) creating degenerately doped source and drain leads, while the gate electrodes and the surrounding insulating spacer layers (green) protect the 40-nm long channel region from high-dose implantation. Different methods are used to implant a few donor atoms (As or P) in the channel region (see Supplemental Material [20]). Transport through these few dopants is accomplished by independently tuning their levels into the energy window set by the source-drain bias ($V_{\mathrm{SD}}$). This tuning is accomplished by means of the two top gate voltages ($V_{\mathrm{GL}}$, $V_{\mathrm{GR}}$) and a back-gate voltage ($V_{\mathrm{BG}}$). A dc current $I_{\mathrm{SD}}$ through the dopants is measured. For experiments presented in Figs. 3, 4 an additional microwave signal is applied to one of the top gates.Reuse & Permissions

Figure 2

dc transport measurement through two As donors connected in series. (a) Current map, $I_{\mathrm{SD}}$ ($V_{\mathrm{GL}}$, $V_{\mathrm{GR}}$), for a fixed $V_{\mathrm{SD}}=15\mathrm{mV}$. At 15 mK, the observed current features, are confined within two triangular-shaped regions, denoting the sequential tunneling of single electrons through a pair of As donors in series, labeled as $D_L$ and $D_R$. $V_{\mathrm{GL}}$ and $V_{\mathrm{GR}}$ allow for a full control of the dopant energy levels. ${ϵ}_0$ defines the level detuning between the donor ground states. ${ϵ}_0=0$ all along the bases of the triangles (green cross) where a current peak is observed due to resonant tunneling via the donor ground states (note that the lower edge of the base line is broken up into multiple lines due to offset charges). ${ϵ}_0=e{V}_{\mathrm{SD}}$ at the apex of each triangle, providing the scaling factor between gate voltage and energy. A second current ridge is observed parallel to the base line (orange cross). This resonance is due to the alignement of $D_L$’s ground state with $D_R$’s first excited state. Its distance from the base line corresponds to ${ϵ}_0=7.4\pm 0.4\mathrm{meV}$ yielding a direct measurement of the excitation energy for donor $D_R$. We note that no measurable current is detected between ground and excited-state lines, indicating a minor contribution of inelastic tunneling between the donor ground states. (b) and (c) illustrate the qualitative energy diagrams corresponding to the green and orange crosses in (a).Reuse & Permissions

Figure 3

Landau-Zener-Stückelberg interference pattern under microwave irradiation. (a) Time evolution, under microwave driving, of the two-level system (TLS) formed by the donor ground states. $|L⟩$ and $|R⟩$ refer to the uncoupled donor ground states and $\Delta$ to their tunnel coupling amplitude. At ${ϵ}_0=0$ the TLS eigenstates are even and odd combinations of $|L⟩$ and $|R⟩$; for $|{ϵ}_0|\gg \Delta$, they localize to $|L⟩$ and $|R⟩$. At every passage through ${ϵ}_0=0$ an “incoming” state $|L⟩$ (or $|R⟩$) splits into a superposition of “outcoming” $|L⟩$ and $|R⟩$ states. Staying in the same state requires a Landau-Zener transition across the gap $\Delta$. Multiple passages done within the dephasing time interfere with each other. (b) Measured $I_{\mathrm{SD}}(V_{\mathrm{MW}},{ϵ}_0)$ interference pattern for a microwave frequency $f_{\mathrm{MW}}=15\mathrm{GHz}$ and a source-drain bias $V_{\mathrm{SD}}=5\mathrm{mV}$. $V_{\mathrm{MW}}$, the amplitude of the voltage modulation between the two donors, is proportional to the externally applied microwave voltage (see below). The static detuning ${ϵ}_0$ depends linearly on the position in the ($V_{\mathrm{GL}}$, $V_{\mathrm{GR}}$) plane. The scaling factor between gate-voltage and energy is adjusted to have the horizonal current ridges spaced by the photon energy. This factor agrees within 2% with the one detetermined from the triangles in Fig. 2a. (c) Current peak integrals extracted from (b). Upward (downward) triangles refer to $n>0$ ($n<0$). The ensemble of 17 data sets is simultaneously fitted (black curves) using equation (1) integrated over peaks of order $n$, with three free parameters: $\Delta$, $\eta$, and the scaling factor for $V_{\mathrm{MW}}$ (see text).Reuse & Permissions

Figure 4

Current maps for microwave driving frequencies from well above to well below the dephasing rate $T_2^{-1}$. (a) For $f_{\mathrm{MW}}=10\mathrm{GHz}$ $\approx 3/T_2$ we observe sharply defined interference fringes that can be seen as the superposition of two interference patterns: one consisting of diagonal and antidiagonal stripes, issued from the interference between two consecutive passages through ${ϵ}_0=0$, and one consisting of horizontal stripes, issued from the constructive interference between consecutive pairs of passages through ${ϵ}_0=0$. In the latter case, the time delay between consecutive pairs equals $f_{\mathrm{MW}}^{-1}$ and constructive interference is equivalent to having a static detuning equal to an integer number of photon quanta. (b) For $f_{\mathrm{MW}}=3\mathrm{GHz}$ $\approx T_2^{-1}$, coherence is preserved on the time scale of just one microwave period. As a result, only the interference stripes due to two consecutive passages through ${ϵ}_0=0$ remain visible. (c) For $f_{\mathrm{MW}}=1\mathrm{GHz}$ $\approx (3T_2{)}^{-1}$, coherence is lost also between consecutive passages through ${ϵ}_0=0$ leading to a structureless current map.Reuse & Permissions