Formation of nonequilibrium steady states in interacting double quantum dots: When coherences dominate the charge distribution

We theoretically investigate the full time evolution of a nonequilibrium double quantum dot structure from initial conditions corresponding to different product states (no entanglement between dot and lead) to a nonequilibrium steady state. The structure is described by a two-level spinless Anderson model where the levels are coupled to two leads held at different chemical potentials. The problem is solved by a numerically exact hierarchical master equation technique and the results are compared to approximate ones obtained from Born-Markov theory. The methods allow us to study the time evolution up to times of order ${10}^4$ of the bare hybridization time, enabling eludication of the role of the initial state on the transient dynamics, coherent charge oscillations and an interaction-induced renormalization of energy levels. We find that when the system carries a single electron on average the formation of the steady state is strongly influenced by the coherence between the dots. The latter can be sizeable and, indeed, larger in the presence of a bias voltage than it is in equilibrium. Moreover, the interdot coherence is shown to lead to a pronounced difference in the population of the dots.

Figure 1

(a) Graphical representation of a double quantum dot system. The two quantum dots (QD) are coupled to a left (L) and a right electrode (R). The corresponding coupling matrix elements are denoted by ${\nu }_{K,m}$ with $K\in \{\mathrm{L},\mathrm{R}\}$ and $m\in \{a,b\}$. The interdot coupling is denoted by $\alpha$. In this work, we focus on a serial coupling configuration where ${\nu }_{\mathrm{R},a}={\nu }_{\mathrm{L},b}=0$ and a branched configuration where ${\nu }_{\mathrm{R},b}={\nu }_{\mathrm{L},b}=0$. (b) Level structure considered in this work where the single-particle levels are below and the levels associated with double occupancy are above the chemical potentials in the leads. This situation corresponds to a double quantum dot structure that is operated in the nonresonant transport regime.

Figure 2

Population of the doubly occupied state, the single-particle levels in dots $a$ and $b$ and the real part of the coherence ${\sigma }_{a,b}$ as functions of time, starting with the unpopulated system [${\sigma }_{00,00}\left(t\right)=1$]. The left and the right columns show these functions for the unbiased systems SERIAL and BRANCHED, respectively.

Figure 3

Population of the doubly occupied state, the single-particle levels in dots $a$ and $b$ and the real part of the coherence ${\sigma }_{a,b}$ as functions of time, starting with an electron in dot $a$ (${\sigma }_{a,a}\left(t\right)=1$). The left and the right columns show these functions for the unbiased systems SERIAL and BRANCHED, respectively.

Figure 4

Decay times of the coherent charge oscillations in junction SERIAL (left plot) and junction BRANCHED (right plot) as a function of the applied bias voltage, starting from the asymmetric initial state ${\sigma }_{a,a}(t=0)=1$ (which we used, because coherent charge oscillations are quenched in the SERIAL configuration if the symmetric initial state is used, cf. the discussion of Figs. 2 and 3). To this end, we fitted the oscillation amplitude in ${\sigma }_{b,b}\left(t\right)$ to an exponential decay.

Figure 5

Time scale to reach the steady state in junction SERIAL (left plot) and junction BRANCHED (right plot) as a function of the applied bias voltage, starting from the symmetric initial state ${\sigma }_{00,00}(t=0)=1$ (which we used to avoid ambiguities due to the presence of coherent charge oscillations in the SERIAL configuration). To determine this scale, we use the time where the real part of the coherence deviates 0.5% from the steady-state value. Note that the oscillatory behavior originates from dynamical phases and is, therefore, most pronounced when the steady state is reached on short time scales.

Figure 6

Population of the doubly occupied state, the single-particle levels in dots $a$ and $b$ and the real part of the coherence ${\sigma }_{a,b}$ as functions of time, starting with the unpopulated system [${\sigma }_{00,00}\left(t\right)=1$]. The left and the right columns show these functions for the systems SERIAL and BRANCHED, respectively, where a bias voltage of $\Phi =0.1$ V is applied.

Figure 7

Population of the doubly occupied state, the single-particle levels in dots $a$ and $b$ and the real part of the coherence ${\sigma }_{a,b}$ as functions of time, starting with an electron in dot $a$ (${\sigma }_{a,a}\left(t\right)=1$). The left and the right columns show these functions for the systems SERIAL and BRANCHED, respectively, where a bias voltage of $\Phi =0.1$ V is applied.

Figure 8

Difference in the population of the dots $a$ and $b$ and the real part of the coherence ${\sigma }_{a,b}$ as a function of the bias voltage $\Phi$ applied to junction SERIAL. Note that the bias voltage is to be compared with the width of the transport resonances, which, in the present context, is given predominantly by the temperature scale, $k_{\mathrm{B}}T\approx 25$ meV.

Figure 9

Difference in the population of the dots $a$ and $b$ and the corresponding real part of the coherence ${\sigma }_{a,b}$ as a function of the energy level position ${\epsilon }_0$ in junction SERIAL at bias voltage $\Phi =0.1$ V. The scale of the level position ${\epsilon }_0$ is, similar to the bias voltage, determined by the temperature $k_{\mathrm{B}}T\approx 25$ meV.

Figure 10

Difference in the population of the dots $a$ and $b$ and the corresponding real part of the coherence ${\sigma }_{a,b}$ as a function of the energy level difference $\delta \epsilon$ in junction SERIAL at bias voltage $\Phi =0.1$ V. The width of the dip structure is determined by the interdot coupling strength $\alpha =0.5$ meV.

Figure 11

Normalized amplitude of the coherent charge oscillations in junction SERIAL (left plot) and junction BRANCHED (right plot) as a function of the applied bias voltage, starting from the asymmetric initial state ${\sigma }_{a,a}(t=0)=1$. To this end, a Fourier analysis of ${\sigma }_{a,a}\left(t\right)$ has been employed.