Charge qubit in a triple quantum dot with tunable coherence

The energy landscape of a single electron in a triple quantum dot can be tuned such that the energy separation between ground and excited states becomes a flat function of the relevant gate voltages. These so-called sweet spots are beneficial for charge coherence since the decoherence effects caused by small fluctuations of gate voltages or surrounding charge fluctuators are minimized. We propose a new operation point for a triple quantum dot charge qubit, a so-called $C{Q}_3$-qubit, having a third-order sweet spot. We show strong coupling of the qubit to single photons in a frequency tunable high-impedance SQUID-array resonator. In the dispersive regime, we investigate the qubit linewidth in the vicinity of the proposed operating point. In contrast to the expectation for a higher-order sweet spot, we there find a local maximum of the linewidth. We find that this is due to a non-negligible contribution of noise on the quadrupolar detuning axis not being in a sweet spot at the proposed operating point. While the original motivation to realize a low-decoherence charge qubit was not fulfilled, our analysis provides insights into charge decoherence mechanisms relevant also for other qubits.

Figure 1

(a) Schematic diagram of the triple dot device. The left, middle and right quantum dots are indicated by $|\mathrm{L}\rangle$, $|\mathrm{M}\rangle$, and $|\mathrm{R}\rangle$, respectively. We can independently tune the chemical potentials and the interdot tunnel couplings $t_{\mathrm{L}}$ and $t_{\mathrm{R}}$ as well as the couplings to the reservoirs by electrostatic gating. A frequency tunable $\lambda /4$ SQUID array resonator is coupled to the left dot. We apply a drive tone to the qubit through the right dot plunger gate. (b) Spectrum of the Hamiltonian for equal tunnel couplings at $E_{\mathrm{M}}=E_{\mathrm{M}}^{\mathrm{Opt}}$ as a function of detuning $\delta$. Solid lines show the energy levels for $t/h=2.5\mathrm{GHz}$ and the dashed lines for $t=0$. (c) Energy differences $E_{01}$ and $E_{12}$ of the spectrum from (b). (d) Plot of the coupling strength of the resonator to the qubit as a function of detuning $\delta$.

Figure 2

(a) TQD energy level schematic. Qubit energy $E_{01}/h$ for $\left|t\right|/h=2.5\mathrm{GHz}$ as a function of $\delta /h$ and $E_{\mathrm{M}}/h$. The thin solid black lines indicate qubit energy contours. The black dashed line show the charge transition lines, separating the single electron charge states of the qubit. Red roman numbers show the quantum dot energy level configurations at the indicated points in the configuration space. (b) Two tone spectroscopy measurement. Measured $|S_{11}|$ as a function of $\delta$ and $E_{\mathrm{M}}$ for $t/h=2.5\mathrm{GHz}$. A drive-tone is applied to the right plunger gate at ${\nu }_{\mathrm{d}}=4.2\mathrm{GHz}$ to map where $E_{01}/h={\nu }_{\mathrm{d}}$. The resonator is probed at ${\nu }_{\mathrm{p}}=3.791\mathrm{GHz}$. (c) Calculated qubit energy as a function of $\delta$ and $E_{\mathrm{M}}$. The black line is the qubit energy contour for $E_{01}/h=4.2\mathrm{GHz}$. The three red lines and icons and symbols are referred to in Fig. 3. (d) Resonant interaction of the qubit and the resonator for ${\nu }_{\mathrm{r}}=4.2\mathrm{GHz}$ and $E_{\mathrm{M}}\approx E_{\mathrm{M}}^{\mathrm{Opt}}\approx -0.493\left|t\right|$. (e) Simulation of $|S_{11}|$ for the measurement in (d) using Input-Output theory. Both measurements (d) and (e) share the same colorbar. The simulation parameters are ${\nu }_{\mathrm{r}}=4.19\mathrm{GHz}$, ${\kappa }_{\mathrm{int}}/2\pi =14\mathrm{MHz}$, ${\kappa }_{\mathrm{ext}}/2\pi =1.3\mathrm{MHz}$, $|t_{\mathrm{L}}|/h=2.47\mathrm{GHz}$, and $|t_{\mathrm{R}}|/h=2.48\mathrm{GHz}$. For the decoherence ${\gamma }_2$ we use the noise model we developed to fit the data in Fig. 4. The black dashed line is a guide to the eye at 4.2 GHz.

Figure 3

[(a)–(c)] Two tone spectroscopy of the qubit dispersion as a function of detuning $\delta$. We plot the complex amplitude $\left|A-A_0\right|$ of the reflected signal as a function of the drive frequency ${\nu }_{\mathrm{d}}$ and the detuning $\delta$. The red dashed lines show the expected qubit dispersion. We take the dispersive shift and the detuning dependent coupling $g\left(\delta \right)$ into account. The icons in the upper right corner correspond to the values for $E_{\mathrm{M}}$ at which the linecuts were made in Fig. 2. (d) Plot of the qubit energy for three settings $E_{\mathrm{M}}=\left\{E_{\mathrm{M}}^{\mathrm{Opt}}/h+0.5\mathrm{GHz},E_{\mathrm{M}}^{\mathrm{Opt}}/h,E_{\mathrm{M}}^{\mathrm{Opt}}/h-0.5\mathrm{GHz}\right\}$ of the middle dot potential. The different line styles in (d) correspond to the lines showed in (a)–(c). However the coupling to the resonator is not considered in (d).

Figure 4

(a) Measurement of the half width half maximum squared vs applied probe power. The blue line is a fit to the measurement. We extract a qubit decoherence rate ${\gamma }_2/2\pi =53\pm 2\mathrm{MHz}$. (b) Measurement of the half width half maximum of the qubit resonance as a function of detuning $\delta$. The blue line shows the fit using a noise model taking charge noise and magnetic noise into account. (c) Simulated qubit linewidth from 1/$f$ charge noise of amplitude $1\mu \mathrm{eV}$ on either $\delta$ or $E_{\mathrm{M}}$. (d) Plot of $E_{01}/h$ as a function of $E_M$ for $\delta =0$ and $\left|\delta \right|/h\approx 6\mathrm{GHz}$. The red dashed line indicates the qubit operation point.

Figure 5

Scanning electron micrograph picture of the sample. The three dots are indicated by the red dashed circles. We control the electrochemical potentials of the dots by tuning the voltages $V_{\mathrm{L}}$, $V_{\mathrm{M}}$, $V_{\mathrm{R}}$ applied to the corresponding plunger gates. The inter dot tunnel couplings as well as the coupling to the reservoirs is controlled by the rest of the gates. The left plunger gate that directly overlaps the dot is connected to the $\lambda /4$ resonator keeping its potential grounded. To change the left dot potential we tune the potential $V_{\mathrm{L}}$. Changing $V_{\mathrm{QPC}}$ we tune the electrostatic potential of the charge sensing QPC.

Figure 6

Plot of the wave-function components of the three eigenstates of the Hamiltonian (1) as a function of $\delta$. The dotted line represent the $|\mathrm{L}\rangle$ components, the dashed lines the $|\mathrm{M}\rangle$ components and the solid lines the $|\mathrm{R}\rangle$ components.

Figure 7

Two tone spectroscopy measurement of the middle-right DDQ charge qubit. We keep $V_{\delta }$ constant while measuring the qubit dispersion along $V_{{\mathrm{E}}_{\mathrm{M}}}$. The red line shows a fit to the measurement. From this we extract the tunnel coupling $t_{\mathrm{R}}$ and the lever arm ${\alpha }_{\mathrm{MR}}^{\mathrm{M}}$.

Figure 8

Charge stability diagram showing the measured QPC current as a function of the two qubit detuning parameters $\delta$ and $E_{\mathrm{M}}$.

Figure 9

Schematic model of the TQD coupled to a resonator being fed by a input line. The input field $a_{\mathrm{in}}$ of the feed line at frequency ${\nu }_{\mathrm{p}}$ couples at rate ${\kappa }_{\mathrm{ext}}$ to the $\lambda /4$ resonator. The internal losses of the resonator are described by ${\kappa }_{\mathrm{int}}$. The resonator couples to the TQD with strength $g$. We summarize the losses of the TQD qubit with the rate $\gamma$

Figure 10

Vacuum Rabi cut at $\delta =0$ from the measurement presented in Fig. 2. We measured the line cut 40 times consecutively, blue points correspond to the averaged data points, the red area shows the standard deviation of the averages. The solid line is the corresponding linecut taken from Fig. 2.

Figure 11

AC-Stark shift measurement performed on the left-middle DQD charge qubit. We measure the dressed qubit frequency ${\tilde{\nu }}_{\mathrm{q}}$ as a function of applied resonator probe tone power. We observe a linear shift of ${\tilde{\nu }}_{\mathrm{q}}$. From this shift, we calculate the average photon number occupation in the resonator $n$ as indicated in the second $x$ axis.

Figure 12

Plot of the qubit energy $E_{01}/h$ as a function of $\delta$ and $E_{\mathrm{M}}$ for $\left|t\right|/h=|t_{\mathrm{L}}|/h=|t_{\mathrm{R}}|/h=2.5\mathrm{GHz}$. Red lines indicate the position of sweet spots in $E_{\mathrm{M}}$ and the black solid line indicates sweet spots in $\delta$. The black dashed line shows the $E_{\mathrm{M}}^{\mathrm{Opt}}\approx -0.493\left|t\right|$.

Figure 13

Simulated influence of magnetic noise arising from different Overhauser fields in different dots. We assume Gaussian distributed magnetic noise with a standard deviation of $\sigma =4\mathrm{mT}$ on either the left-middle or middle-right DQD. Note the different axis scales compared to Fig. 4.