Superradiance-like electron transport through a quantum dot

We theoretically show that intriguing features of coherent many-body physics can be observed in electron transport through a quantum dot (QD). We first derive a master-equation-based framework for electron transport in the Coulomb-blockade regime which includes hyperfine (HF) interaction with the nuclear spin ensemble in the QD. This general tool is then used to study the leakage current through a single QD in a transport setting. We find that, for an initially polarized nuclear system, the proposed setup leads to a strong current peak, in close analogy with superradiant emission of photons from atomic ensembles. This effect could be observed with realistic experimental parameters and would provide clear evidence of coherent HF dynamics of nuclear spin ensembles in QDs.

Figure 1

Schematic illustration of the transport system: An electrically defined QD is tunnel-coupled to two electron reservoirs, the left and right lead, respectively. A bias voltage $V=({\mu }_L-{\mu }_R)/e$ is applied between the two leads in order to induce a current through the QD. An external magnetic field is used to tune the system into the sequential-tunneling regime and the QD effectively acts as a spin filter. The resulting spin blockade can be lifted by the HF interaction between the QD electron and the nuclear spins in the surrounding host environment.

Figure 2

Normalized leakage current through a QD in the spin-blockade regime for $N$ nuclear spins, initial nuclear polarization $p$, and external Zeeman splitting ${\omega }_0$ in units of the total HF coupling constant $A_{\mathrm{HF}}\approx 100\mu \mathrm{eV}$, summarized as $(N,p,{\omega }_0/A_{\mathrm{HF}})$. For homogeneous HF coupling the dynamics can be solved exactly (black dotted line). Compared to this idealized benchmark, the effects are reduced for realistic inhomogeneous HF coupling, but still present: The relative peak height becomes more pronounced for smaller detuning ${\omega }_0$ or higher polarization $p$ (solid red line compared to the blue dashed and green dash-dotted line, respectively). Even under realistic conditions, the relative peak height is found to scale linearly with $N$, corresponding to a strong enhancement for typically $N\approx {10}^5–{10}^6$.

Figure 3

Fermi function for finite temperature (dashed blue line) and in the limit $T=0$ (solid blue line). The absolute value of the derivative of the Fermi function $f^{\prime }\left(\epsilon \right)$ (dotted orange line for finite temperature) is maximized at the chemical potential $\mu$ and tends to a $\delta$ function in the limit $T\to 0$. The Markovian description is valid provided that the Fermi function is approximately constant around the resonances ${\epsilon }_{\sigma }\left(t\right)$ on a scale of the width of these resonances, schematically shown in red [solid line for ${\epsilon }_{\sigma }\left(t\right)<\mu$ and dashed line for ${\epsilon }_{\sigma }\left(t\right)>\mu$].

Figure 4

The electronic QD system in the local moment regime after the adiabatic elimination of the $|0\rangle$ level including the relevant dissipative processes. Within the effective system (box) we encounter an effective decay term and an effective pure dephasing term, with the rates $\gamma$ and $\Gamma$, respectively. This simplification is possible for fast recharging of the QD, that is, $\beta \gg \alpha$.

Figure 5

Typical time evolution of the normalized current for homogeneous coupling under dynamical compensation of the Overhauser field and a relative coupling strength of $\epsilon =0.5$, shown here for $N=60$ and $N=100$ nuclear spins. The characteristic feature of SR, a pronounced peak in the leakage current proportional to $N$, is clearly observed.

Figure 6

Ratio of the maximum current to the initial current $I_{\mathrm{coop}}/I_{\mathrm{ind}}$ as a function of the number of nuclear spins $N$ for homogeneous coupling and a relative coupling strength of $\epsilon =0.5$: Results for perfect compensation (dashed line) are compared to the case of dynamic compensation (dotted line) of the Overhauser field (OHC). Simulations without compensation of the Overhauser field set bounds for the enhancement of the leakage current, depending on the sign of the HF coupling constant $A_{\mathrm{HF}}$; solid and dash-dotted line for $A_{\mathrm{HF}}>0$ and $A_{\mathrm{HF}}<0$, respectively.

Figure 7

Typical time evolution of the normalized current for inhomogeneous coupling, shown here for up to $N={13}^2$ nuclear spins and a relative coupling strength $\epsilon =0.55$. Compared to the idealized case of homogeneous coupling, the SR effects are reduced, but still clearly present. A Gaussian spatial electron wave function has been assumed and the Overhauser field is compensated dynamically.

Figure 8

Ratio of the maximum current to the initial current $I_{\mathrm{coop}}/I_{\mathrm{ind}}$ as a function of the number of nuclear spins $N$ for relative coupling strengths $\epsilon =0.5$ and $\epsilon =0.55$: Results for inhomogeneous coupling. The linear dependence is still present when starting from a nuclear state with finite polarization $p=0.8$.

Figure 9

Total time until the observation of the characteristic SR peaking $t_{\max }$ for $\epsilon =0.5$ (blue dots) and $\epsilon =0.55$ (orange squares). Based on Eq. (44), logarithmic fits are obtained from which we estimate $t_{\max }$ for experimentally realistic number of nuclear spins $N\approx {10}^5$.

Figure 10

Fluctuations of the Overhauser field relative to the external Zeeman splitting ${\omega }_0$. In the limit of homogeneous HF coupling, strong fluctuations build up towards the middle of the emission process (red line, $\epsilon =0.5$). For inhomogeneous coupling this buildup of fluctuations is hindered by the dephasing between the nuclear spins, resulting in considerably smaller fluctuations: The value of the Overhauser fluctuations is shown at the time of the SR peak $t_{\max }$ for $\epsilon =0.5$ (orange squares) and $\epsilon =0.55$ (green diamonds). The Overhauser fluctuations reach a maximum value later than $t_{\max }$; see blue dots for $\epsilon =0.5$. For independent homogeneously coupled nuclear spins, one can estimate the fluctuations via the binominal distribution (black line).