Charge-Qubit Operation of an Isolated Double Quantum Dot

We have investigated coherent time evolution of pseudomolecular states of an isolated (leadless) silicon double quantum dot, where operations are carried out via capacitively coupled elements. Manipulation is performed by short pulses applied to a nearby gate, and measurement is performed by a single-electron transistor. The electrical isolation of this qubit results in a significantly longer coherence time than previous reports for semiconductor charge qubits realized in artificial molecules.

Figure 1

(a) Scanning electron micrograph of the sample. The device was fabricated by high-resolution electron-beam lithography and reactive-ion etching to “trench isolate” the elements. Thermal oxidation was carried out to reduce the dimensions of the silicon regions and to passivate the surface-charge trap states. The device was mounted below the mixing chamber of a dilution refrigerator operating at base temperature ($\sim 20\mathrm{mK}$). (b) Material profile, showing the 35 nm-thick active region between the insulating layers. The dopant (phosphorous) concentration was $5.7\times {10}^{19}{\mathrm{cm}}^{-3}$. (c) Schematic view of the IDQD after oxidation, with the active region surrounded by the oxide layer. Localized charge configurations $|L⟩$ and $|R⟩$ of the electron at the highest occupation level are illustrated schematically.

Figure 2

Scheme for qubit initialization, manipulation, and measurement. (a) Energy diagram illustrating the localized electron states for an uncoupled system $|L⟩$ and $|R⟩$ (dashed lines) with eigenenergies $E_L$ and $E_R$, respectively. The interdot coupling energy $\Delta$ results in new stationary eigenstates $|0⟩$ and $|1⟩$, with eigenenergies $E_0$ and $E_1$, respectively (solid lines). Far from resonance $(|\epsilon |\gg 0)$ the qubit eigenstates are well approximated by $|L⟩$ and $|R⟩$, but near resonance $(\epsilon \simeq 0)$ the eigenstates are maximally delocalized. (b) Gates 1, 2, and 3 maintain an effective detuning $\epsilon =E_L-E_R$ and coupling $\Delta$. The pulse is switched on at time $t_{\mathrm{p}}$, and the qubit coherently evolves for the duration $\Delta t$. The lower graph shows the pulse shape for $\Delta t=5\mathrm{ns}$ measured at the output of the pulse generator. At time $t_{\mathrm{p}}+\Delta t$ the pulse is switched off, which stops the manipulation. The resulting quantum superposition determines the SET measurement outcome. The qubit relaxes to the ground state $|0⟩$ after a time $\sim {\tau }_1$. The cycle is repeated approximately ${10}^4$ times. The SET current corresponds to the ensemble average of the measurement outcomes. The SET measurement integration time was 100 ms.

Figure 3

Measured responses for different experimental parameters. (a) Color plot of the SET current $I$ as a function of $\Delta t$ and $V_{\mathrm{g}3}$, with $V_P=500\mathrm{mV}$, $T_r=100\mu \mathrm{s}$, $V_{\mathrm{g}1}=0\mathrm{V}$, $V_{\mathrm{g}2}=-9\mathrm{V}$, $V_{\mathrm{g}4}=-0.7\mathrm{V}$, and $V_{\mathrm{sd}}=450\mu \mathrm{V}$. Measurement of the qubit state is clearly visible. The oscillation frequency $\omega$ can be changed continuously by the gate voltage $V_{\mathrm{g}3}$. The pulse also induces a time-averaged (dc) bias, manifested as a small change in $\omega$ as $\Delta t$ is increased. (b) Pulse-height dependence of the coherent oscillations, with $T_r=10\mu \mathrm{s}$, $V_{\mathrm{g}1}=-6\mathrm{V}$, $V_{\mathrm{g}2}=4.5\mathrm{V}$, $V_{\mathrm{g}3}=-6\mathrm{V}$, $V_{\mathrm{g}4}=-0.5\mathrm{V}$, and $V_{\mathrm{sd}}=450\mu \mathrm{V}$. For increasing $V_P$, the oscillation frequency $\omega$ also increases, and the response shifts towards positive effective $V_{\mathrm{g}2}$, reducing ${\tau }_2$. (c) $T_r$ dependence, with $V_P=300\mathrm{mV}$, $V_{\mathrm{g}1}=-6\mathrm{V}$, $V_{\mathrm{g}2}=4.5\mathrm{V}$, $V_{\mathrm{g}3}=-6\mathrm{V}$, $V_{\mathrm{g}4}=-0.5\mathrm{V}$, and $V_{\mathrm{sd}}=450\mu \mathrm{V}$. A small dependence on $T_r$ is observed in $\omega$, attributed to the change in the time-averaged induced bias on the qubit for different $T_r$. Data sets in (b) and (c) are offset by 140 pA and 70 pA, respectively.

Figure 4

(a) The gate-compensated method, whereby the SET operating point is maintained by proportionally sweeping $V_{\mathrm{g}4}$ simultaneously with $\Delta t$. The maximum oscillation amplitude is $\sim 5\%–10\%$ of the maximum SET current. The inset shows two SET conductance oscillations due to Coulomb blockade, as a function of $V_{\mathrm{g}4}$. The arrow indicates the position where the SET was set up as a highly sensitive electrometer. (b) Ramsey-interference experiment showing the free-evolution dephasing of the qubit. The experimental parameters were $V_P=0.6\mathrm{V}$, $\Delta t(\pi /2)=17\mathrm{ns}$, $V_{\mathrm{g}2}=0.3\mathrm{V}$. The data fit to an exponentially damped sine function (solid line) with a characteristic decay time of ${\tau }_2\approx 200\mathrm{ns}$. The small peak at $\delta =220\mathrm{ns}$ is possibly due to a quantum level which is interfering with the qubit evolution. Since the pulse widths are constant, $\epsilon$, $\Delta$, and $U$ are unchanged, which eliminates the need for compensation by the gates.