Radio frequency pulsed-gate charge spectroscopy on coupled quantum dots

Time-resolved electron dynamics in coupled quantum dots is directly observed by a pulsed-gate technique. While individual gate voltages are modulated with periodic pulse trains, average charge occupations are measured with a nearby quantum point contact as detector. A key component of our setup is a sample holder optimized for broadband radio frequency applications. Our setup can detect displacements of single electrons on time scales well below a nanosecond. Tunneling rates through individual barriers and relaxation times are obtained by using a rate equation model. We demonstrate the full characterization of a tunable double quantum dot using this technique, which could also be used for coherent charge qubit control.

Figure 1

(Color online) (a) A scanning electron micrograph of a device nominally identical to the investigated one. Details in the main text. (b) The radio frequency sample holder with four impedance-matched microstrip lines and 22 low bandwidth connections. The central gold square is electrically shorted to the copper plated backside of the circuit board. The sample would be mounted on the central gold square and wire bonded.

Figure 2

Charge stability diagrams of the double QD. Plotted in gray scale as a function of gate voltages $V_{\beta }$ and $V_{\gamma }$ [see Fig. 1a] is the transconductance $d{I}_{\text{QPC}}/d{V}_{\beta }$, which has been obtained by numerical differentiation of the dc current $I_{\text{QPC}}$. $d{I}_{\text{QPC}}/d{V}_{\beta }\simeq 0$ at the gray background, $d{I}_{\text{QPC}}/d{V}_{\beta }<0$ on darker lines and $d{I}_{\text{QPC}}/d{V}_{\beta }>0$ at brighter regions. Pairs of numerals 0 or 1 depict the number of electrons charging the QDs B and C in the ground-state configuration. (a) No pulses applied. (b) $V_{\beta }$ is modulated with a symmetric continuous square wave signal, for which the waiting time $t_{\text{w}}$ between two pulses equals the pulse duration $t_{\text{p}}$, at a period of $f_0^{-1}=t_{\text{w}}+t_{\text{p}}=554\text{ps}$ and an amplitude of $V_{\text{p}}=5\text{mV}$ (see inset and main text).

Figure 3

(Color online) (a) Reflection coefficient (S11 in decibel) of a single open-ended lead of an rf sample holder (black solid line), optimized for minimal cross-talk and impedance matching, compared to that of a nonoptimized sample holder (blue dashed line) as a function of frequency. The (red) closed circles are a numerical simulation of the rf sample holder using the commercial software SONNET. (b) Transmission (S12 in decibel) through the two sample holders as a function of frequency. The transmission is measured between two adjacent but electrically isolated gates. The inset shows the calculated Fourier transform of a pulse train with $t_{\text{p}}=t_{\text{w}}=277\text{ps}$ as a function of frequency, where $P$ is the power in arbitrary units. (c) A section of the double QD stability diagram, as in Fig. 2b, but using a nonoptimized sample holder while a symmetric pulse train with a repetition frequency of $f_0=300\text{MHz}$ is applied at gate $\gamma$. (d) Measurement as in (c) but at a slightly larger repetition frequency of $f_0=400\text{MHz}$ (nonoptimized sample holder). The two frequencies are marked in (b) by vertical arrows.

Figure 4

Charge stability diagrams of the double QD as in Fig. 2 but under application of an asymmetric pulse train $\left(t_{\text{w}}⪢t_{\text{p}}\right)$. $V_{\beta }$ and $V_{\gamma }$ are both modulated in opposite directions. A square pulse of duration $t_{\text{p}}=1\text{ns}$ is periodically applied after the waiting time of $t_{\text{w}}=50\text{ns}$. Effective amplitude and direction of the pulses are indicated by white arrows. The pulse amplitude $V_{\text{p}}$ is increased from (a) to (c).

Figure 5

(Color online) (a) Qualitative explanation of the electron dynamics for a asymmetric pulse train. (a-i) Chemical potentials of the QDs and leads $\left({\mu }_{\text{l}}\right)$ in the middle of the black-white line in Figs. 4a, 4b, 4c. Vertical lines symbolize tunnel barriers. In the ground-state configuration 0,1, one electron occupies QD C. (a-ii) Situation while the double QD is pulsed onto (the middle of) the charge reconfiguration line. The chemical potentials of both QDs are degenerate, which allows the electron to tunnel between the QDs with rate ${\Gamma }_1$. (a-iii) After the pulse, the electron occupies either QD B (iii-b) or is back in QD C (iii-a). (a-iv) Two possible relaxation channels (combined rate ${\Gamma }_2$) from the excited state (with QD B occupied) into the ground state. One channel (iv-a) features an interdot tunneling process in combination with energy relaxation. The other channel (iv-b) involves two resonant QD-lead tunneling processes and energy relaxation in the leads. (b) The sketch shows the steady-state probability of the (unpulsed) ground state being unoccupied, $P\left(t\right)$, as a function of time for arbitrarily chosen ${\Gamma }_1$ and ${\Gamma }_2$. See main text for details. The periodic pulse train is also indicated where $t_{\text{p}}$ is the pulse duration and $t_{\text{w}}$ is the waiting time between pulses.

Figure 6

(Color online) Charge stability diagrams of a double QD for various QD-lead and interdot tunnel couplings (device and measurements as for Figs. 2, 4). Sketches: relevant charge transition processes (arrows) during pulses and between pulses. Crossed out arrows: energetically favored transitions which are very slow and, hence, result in nonequilibrium occupations. Vertical lines depict tunnel barriers (smaller width equals larger tunnel coupling). [(a) and (b)] Asymmetric QD-lead tunnel couplings as indicated in the corresponding sketches. In (a) $t_{\text{p}}=0.5\text{ns}$ and $t_{\text{w}}=10\text{ns}$. In (b) comparable parameters except larger interdot tunnel coupling and $t_{\text{w}}=20\text{ns}$. Within the triangle marked in (a) the transition $0,1\to 1,1$ might happen during pulses while the opposite transition $1,1\to 0,1$ is unlikely to happen between pulses. (c) Weak QD-lead tunnel couplings on both sides suppress the transition $0,1\to 1,1$ ($t_{\text{p}}=0.5\text{ns}$ and $t_{\text{w}}=25\text{ns}$). (d) Weak QD-lead tunnel couplings on both sides combined with a very strong interdot tunnel coupling ($t_{\text{p}}=0.5\text{ns}$ and $t_{\text{w}}=20\text{ns}$).

Figure 7

(Color online) (a) Charge stability diagram of the same double QD as in Figs. 4a, 4b, 4c but for different QD-lead and interdot tunnel couplings. Plotted is the dc current $\delta I_{\text{QPC}}$ after subtraction of a linear background. A pulse train with $t_{\text{p}}=0.6\text{ns}$ and $t_{\text{w}}=25\text{ns}$ is applied. (b) $\delta I_{\text{QPC}}$ along the cut through the stability diagram indicated in Fig. 7a as a white dotted line (red triangles). The (black) filled circles show the measurement without pulsing. $I_{10}$, $I_{01}$, and $I_{00}$ are the current values expected for the charge configurations 1,0 and 0,1 and 0,0, respectively. $I_{\text{peak}}$ marks the pulse-induced maximum of the detector current. (c) Relative detector current $\Delta I/\Delta I_{\text{max}}$ (where $\Delta I=I_{\text{peak}}-I_{01}$ and $\Delta I_{\text{max}}=I_{00}-I_{01}$) as a function of $t_{\text{p}}$ at $t_{\text{w}}=25\text{ns}$ for three different values of the gate voltage $V_{t2}$ (as indicated). The lines are model curves from Eq. (2) (see main text). (d) Filled circles are measured values of $\Delta I/\Delta I_{\text{max}}$ as a function of $t_{\text{p}}$ and $t_{\text{w}}$. The meshed surface is a model curve from Eq. (2) for the fit parameters ${\Gamma }_1^{-1}=0.74\text{ns}$ and ${\Gamma }_2^{-1}=12.9\text{ns}$. Here we assume $\Delta I/\Delta I_{\text{max}}=\overline{P}$ (details in main text).