Optical Detection of Single-Electron Tunneling into a Semiconductor Quantum Dot

The maximum information of a dynamic quantum system is given by real-time detection of every quantum event, where the ultimate challenge is a stable, sensitive detector with high bandwidth. All physical information can then be drawn from a statistical analysis of the time traces. We demonstrate here an optical detection scheme based on the time-resolved resonance fluorescence on a single quantum dot. Single-electron resolution with high signal-to-noise ratio ($4\sigma$ confidence) and high bandwidth of 10 kHz make it possible to record the individual quantum events of the transport dynamics. Full counting statistics with factorial cumulants gives access to the nonequilibrium dynamics of spin relaxation of a singly charged dot (${\gamma }_{\uparrow \downarrow }=3{\mathrm{ms}}^{-1}$), even in an equilibrium transport measurement.

Figure 1

Optical monitoring of electron tunneling and resonance fluorescence (RF) of the quantum dot (QD). (a) The resonance fluorescence of the exciton transition is monitored, while an electron tunnels between the QD and a charge reservoir at a fixed gate voltage. The electron tunneling is directly observed in the time-resolved RF intensity as a random telegraph signal (depicted in the upper left-hand inset). (b) Time-averaged resonance fluorescence of the exciton transition at zero magnetic field for a laser frequency of 325.305 THz. In a gate-voltage sweep, the fine structure of the exciton transition is resolved with a linewidth of $1.5\mu \mathrm{eV}$ and a fine-structure splitting of $28\mu \mathrm{eV}$.

Figure 2

Time-resolved RF random telegraph signal, induced by single-electron tunneling. (a) Resonance-fluorescence intensity of the exciton transition in a magnetic field of 10 T (laser frequency of 325.803 THz). Resonance is achieved over a broad range of gate voltages due to the so-called nuclear spin dragging. The Fermi distribution in the electron reservoir is observed in the RF counts between $V_G=0.33$ and 0.42 V, where the chemical potential is energetically lifted from below the QD level at point 1 to above the QD level at point 3. (b) Schematic picture of the underlying model, where the quantum dot can be in three states [uncharged (0) or charged by either a spin-up ($\uparrow$) or a spin-down ($\downarrow$) electron]. In case of a fast spin-relaxation rate ${\gamma }_{\uparrow \downarrow }$, the model effectively reduces to a two-state configuration with states 0 and 1. (c) The resonance-fluorescence signal of the exciton transition measured for gate voltages 1–3 in (a). The magnetic field is $B=10\mathrm{T}$ and the time resolution is $100\mu \mathrm{s}$.

Figure 3

Analysis of the quantum-jump counting statistics. (a)–(c) Ordinary normalized cumulants of the electron tunneling at $B=10\mathrm{T}$ compared with simulations assuming slow (dashed lines) or fast (solid lines) spin relaxation. The quality of the optically obtained data allows us to well resolve the cumulants as a function of time [(a),(b) with asymmetry $a=-0.9$] and asymmetry [(c) in the long-time limit for a time interval $t=20\mathrm{ms}$]. The dependencies, including the oscillations of the higher-order cumulants (b), are in agreement with the two-state model and clearly disagree with the three-state model; see Fig. 2.

Figure 4

Interplay between charge and spin dynamics. (a)–(c) Factorial cumulants for an applied magnetic field of 2 T (a), 4 T (b), and 10 T (c) and excitation energies $ϵ/k_{B}T$ of $-1.7$ (green), $-2.0$ (blue), $-2.3$ (purple), and $-2.6$ (red). The black lines are obtained from simulation with spin-relaxation rate ${\gamma }_{\uparrow \downarrow }=\infty$ (solid), ${\gamma }_{\uparrow \downarrow }=3{\mathrm{ms}}^{-1}$ (dot-dashed), ${\gamma }_{\uparrow \downarrow }=1{\mathrm{ms}}^{-1}$ (dotted), and ${\gamma }_{\uparrow \downarrow }=0$ (dashed). Decreasing the magnetic field yields a transition from fast to slow spin relaxation. For 10 T, the data are reproduced by assuming fast spin relaxation ${\gamma }_{\uparrow \downarrow }=\infty$, at 4 T best agreement is obtained for ${\gamma }_{\uparrow \downarrow }=3{\mathrm{ms}}^{-1}$, and at 2 T we have to assume slow spin relaxation ${\gamma }_{\uparrow \downarrow }=0$.