Application in a three QD transport experiment

Before moving on to other properties of this system, we will discuss an application for the effect that we have just analyzed. It is also possible to create entanglement between the two dots when instead of being coupled to a cavity mode, they are both coupled to a third QD. In this case, instead of self assembled QD in a microcavity, we rather have in mind electrostatically defined QDs in the vacuum. Experiments on transport through these kind of QDs have recently experienced such a development that it is now possible to reproduce many of the phenomena that the field of quantum optics, involving atoms, has been exploring for many years, as argued for instance by Brandes (2005). In particular, state preparation and manipulation of one or more QDs is nowadays feasible by controllable external means such as gate and bias potentials plus either continuous or AC electric and/or magnetic fields. Again, each QD can be considered as a qubit with the lower state $ \vert\rangle$ corresponding to a neutral QD and the upper state $ \vert 1\rangle$ to having one extra electron in the QD. Due to Coulomb blockade, charging the QD with more than one electron requires an energy that can be considered infinite for any practical purpose, reducing the Hilbert space to that spanned by the two mentioned states. Entanglement effects in transport through double QDs have been extensively studied, for instance by Hayashi et al. (2003), Marquardt & (2003), Vorrath & (2003), Brandes (2005), Michaelis et al. (2006), Dias da Silva (2006) or Lambert et al. (2007). In a similar way, ours is a proposal for experimentally preparing and measuring a charge-entangled state of two QDs. Our starting point is the pioneering ideas of Michaelis et al. (2006), where the constraint of having no more than one electron in the whole system allows the population trapping in a dark-entangled state. The same configuration does not give the desired results within a regime more experimentally accessible (more than one electron in total) as we will show below. We obtain interesting results (in experimentally accessible conditions) when we add cross-terms in the incoherent pumping of the two QDs as we have seen in the previous Section. In the framework of transport, the role of the cavity is played by a third QD and both incoherent pumping of the QDs and cavity photon emission find their counterpart for the transport realization in the tunneling processes produced by the application of a bias. We will show here that, apart from all the analogies, there is an important advantage in doing transport: entanglement could be easily detected in the same setup that prepares the equivalent to the quasi-dark (entangled) state.

Figure 6.8: (a) Schema of the proposed setup, with a two-dimensional electron gas depleted by eight gate potentials $ V_i$. $ V_3$, $ V_4$ and $ V_7$ control the levels of the QDs. $ V_2$ and $ V_8$ control $ \Gamma_\mathrm{p}$ and $ \Gamma_\kappa$. $ V_5$ and $ V_6$ control $ g_{AC}$ and $ g_{BC}$. Switching $ V_1$ from ``on'' to ``off'' tunes the pumping of $ A$ and $ B$ from distinguishable to indistinguishable quantum mechanically. The current is induced by a bias from left to right. (b) Schema of the dynamics in the Hilbert space spanned by $ \vert n_A,n_B,n_C\rangle$. We simplify the plot by setting detunings to zero.
\includegraphics[width=.45\linewidth]{chap6/3QDs/fig1a.eps} \includegraphics[width=.45\linewidth]{chap6/3QDs/fig1b.eps}

The three QD system is presented in Fig. 6.8(a). A two-dimensional electron gas is depleted in some regions by means of a series of gate potentials. A bias applied from the left to the right lead produces two tunneling processes: incoherent population of QDs $ A$ and $ B$ as well as electron current from QD $ C$ to the right lead. QDs $ A$ and $ B$ are coherently coupled to QD $ C$ (acting as the cavity in the previous Section). The gate-potentials $ V_3$, $ V_4$ and $ V_7$ are designed to control the levels of the three qubits, $ V_2$ and $ V_8$ control the in- ( $ \Gamma_\mathrm{p}$) and out- ( $ \Gamma_\kappa$) tunneling rates while the gates $ V_5$ and $ V_6$ control the coherent couplings $ g_{AC}$ and $ g_{BC}$ respectively. The crucial novelty with respect to the configuration of Michaelis et al. (2006) is the gate $ V_1$. Switching on and off the $ V_1$ gate, one can experimentally tune from a quantum mechanically distinguishable ($ V_1$ on) to an indistinguishable ($ V_1$ off) pumping of the two QDs $ A$ and $ B$. Let us first see how this appears in the quantum mechanical description of the transport through the whole system and later what are the physical effects that can be tuned in this setup.

As opposed to the case with photons, where the truncation can be arbitrarily high, Coulomb blockade on each QD limits the Hilbert space to that spanned by a basis of 8 states $ \vert n_A,n_B,n_C\rangle$ [depicted in Fig. 6.8(b)] with $ n_{A,B,C}=0,1$. Cross term effects only occur for electrons with the same spin. We therefore consider the system under the action of an in-plane magnetic field and neglect the spin. Both intra-dot Coulomb blockade and spin polarization stand within an experimentally accessible regime. To reduce the Hilbert space to the lowest four states in Fig. 6.8(b), as is done by Michaelis et al. (2006), would require an extremely high inter-dot Coulomb repulsion, something unreasonable in a system such as the one shown in Fig. 6.8(a). The coherent part of the dynamics is controlled by the Hamiltonian:

$\displaystyle H=\sum_{i=A,B} \Delta_i \ud{\sigma_i} \sigma_i+ g_{iC} (\ud{\sigma_i} \sigma_C+\ud{\sigma_C} \sigma_i)\,,$ (6.15)

where $ \sigma_i$, $ \ud{\sigma_i}$ are, this time, annihilation and creation operators of an electron in QD $ i$. We have taken the level of the QD $ C$ as the origin of energies so that only the detunings $ \Delta_A$, $ \Delta_B$ and the couplings $ g_{AC}$, $ g_{BC}$ are relevant. The master equation is, in analogy with Eq. (6.9):
\begin{subequations}\begin{align}\frac{d \rho}{dt}=&i \lbrack \rho,H \rbrack + i...
...\sigma_i} \rho-\rho \sigma_j \ud{\sigma_i}\big)\,, \end{align}\end{subequations}

where  $ \mathcal{P}_{i}=\vert n_A,n_B,n_C\rangle\langle n_A,n_B,n_C\vert$ ( $ i=1,\hdots,8$) is the general projector for the eight possible states in the system. In this configuration, instead of pump and decay, the incoherent part of the master equation features the in- ( $ \Gamma_\mathrm{p}$) and out- ( $ \Gamma_\kappa$) tunneling processes [Eq. (6.16a)] and pure dephasing at rate $ \gamma_\mathrm{d}$ [Eq. (6.16b)]. There are also the two pump cross terms [Eq. (6.16c)] and their coherent direct coupling that accompanies them [Eq. (6.16a)]. They appear when the QDs $ A$ and $ B$ are pumped from the same reservoir (left lead) in a complete indistinguishable (quantum) way. In such a case, the corresponding rate of pumping is given by $ \Gamma_{AB}=\Gamma_\mathrm{p}$. This corresponds to switching off the gate $ V_1$. By smoothly switching it on, the upper and lower parts of the left lead separate from each other. $ \Gamma_{AB}$ is reduced down to the situation in which the two left reservoirs become completely independent and $ \Gamma_{AB}=0$. Cross terms in the pumping are therefore experimentally controllable. The new Förster-like direct coupling in Eq. (6.16a) contributes to enhancing the entanglement between QDs $ A$ and $ B$. The coupling parameter is $ g_{AB}=2\Gamma_{AB}$ as derived by Ficek & (2002) in analogy with the Lamb shift of a single 2LS. Once again, one can experimentally control this mechanism from switching on $ V_1$, which gives $ g_{AB}=0$, to switching it off, which gives $ g_{AB}=2\Gamma_\mathrm{p}$.

The current flowing through the system is an easily measurable experimental quantity that plays the role of the cavity emission intensity from the previous configuration. The current is simply given by $ I=\Gamma_\kappa\langle\ud{\sigma_C}\sigma_C\rangle $. In the stationary limit, the master equation (6.16) simplifies to a set of 64 linear equations, plus the normalization condition $ \mathrm{Tr}(\rho)=1$. In this finite Hilbert space, the tangle is $ \tau = [ \max \{0,
2(\vert\widetilde{\rho}_{10,01}\vert-\sqrt{\widetilde{\rho}_{00,00}\widetilde{\rho}_{11,11}})\}]^2$, in terms of the matrix elements of $ \widetilde{\rho}=\mathrm{Tr}_C(\rho)$.

Figure 6.9: Current intensity $ I$ (empty symbols) and tangle $ \tau$ (full symbols) as a function of detuning $ \Delta$ when $ V_1$ is so large that pumpings to $ A$ and $ B$ are distinguishable from each other, i.e. $ \gamma_{AB}=0$. Two different values for the dephasing are considered: $ \gamma_\mathrm{d}=0.001$ (in solid-red) and $ \gamma_\mathrm{d}=1$ (in dashed-blue). $ \Gamma_\mathrm{p}=\Gamma_\kappa=2$ and $ g_{BC}=1$. Energies and rates in units of $ g_{AC}=1$. Strong Coulomb blockade is considered either inter-QD (maximum charge of 1 electron in the whole system, plotted with squares) or just intra-QD (maximum charge of 1+1+1=3 electrons, plotted with circles). Red and blue full circles coincide to zero (as the tangle in the three-electron case for all dephasing rates is zero) and therefore they appear superimposed in the plot. Also the current in the case of 1 electron and negligible dephasing ( $ \gamma_\mathrm{d}=0.001$) is zero.
\includegraphics[width=0.7\linewidth]{chap6/3QDs/fig2.ps}

Let us discuss the results to be expected from the setup we propose. We consider that the QDs $ A$ and $ B$ are equal, $ \Delta_A
=\Delta_B=\Delta$, what can be achieved by adjusting independently the gates $ V_3$ and $ V_4$. The entanglement can be manipulated by having different couplings $ g_{AC}$ and $ g_{BC}$ as controlled by the gates $ V_5$ and $ V_6$. Hereafter, all the couplings and rates will be given in units of $ g_{AC}=1$.

First of all, we analyze the adequacy of truncating the Hilbert space basis to only four states as it was done by Michaelis et al. (2006). For this purpose we consider the simple case of neglecting cross-terms and estimate the total mean number of excitations in the system. For a system that is pumped with a total rate $ P_{tot}$, decays with $ \gamma_{tot}$ and that has a saturation limit $ S$, one finds [generalizing the single 2LS result in Eq. (2.52)] that the mean total number of excitations in the very strong coupling regime is given by  $ \langle n\rangle =SP_\mathrm{tot}/(P_\mathrm{tot}+\gamma_\mathrm{tot})$. In our case where  $ P_{tot}=2\Gamma_p$, $ \gamma_{tot}=\Gamma_\kappa$ and $ S=3$ (maximum of three electrons in the system), and in the symmetric case, $ \Gamma_{P}=\Gamma_{\kappa}=\Gamma$, that we consider in what follows, the total average excitation is further simplified into $ \langle n\rangle = 3\times 2 \Gamma/(2\Gamma+\Gamma)=2$. This result, that agrees with numerical calculations, is the first indication of the inadequacy of truncating the Hilbert space to only one electron. Moreover, we find that the relevant magnitudes under study ($ I$ and $ \tau$) depend strongly on the truncation. Fig. 6.9 shows $ I$ and $ \tau$ as a function of the detuning for two different dephasing rates, $ \gamma_\mathrm{d}=1$ and $ \gamma_\mathrm{d}=0.001$, both with truncation (up to one electron in the system) and without truncation (up to three electrons in the system). When truncation is not imposed, $ \tau$ is always zero and entanglement is not achieved.

Figure 6.10: Current intensity $ I$ (in red) and tangle $ \tau$ (in blue) as a function of $ \Gamma_{AB}$ (in logarithmic scale). This is experimentally controlled by varying $ V_1$ from a large value (giving $ \Gamma_{AB}=0$) to zero (giving $ \Gamma_{AB}=\Gamma_\mathrm{p}$). $ \Delta =4$, $ \Gamma_\mathrm{p}=\Gamma_\kappa=2$, $ \gamma_\mathrm{d}=0.001$ and $ g_{BC}=0.7$ with energies and rates in units of $ g_{AC}=1$. Fast rising of $ \tau$, detectable by fast quenching of $ I$, is due to the switching on of a quantum mechanically indistinguishable pumping. Only intra-QD Coulomb blockade is considered (maximum charge of 3 electrons).
\includegraphics[width=0.7\linewidth]{chap6/3QDs/fig3.ps}

The approximation of keeping just one electron in the whole system that was made by Michaelis et al. (2006), forces the steady state of QDs $ A$ and $ B$ to be a singlet $ \vert\mathrm{S~0}\rangle=(\vert 100\rangle -
\vert10\rangle)/\sqrt{2}$ (with QD $ C$ in the vacuum). This implies a tangle of one and no current passing through the system if the dephasing is negligible. This is a new example of a trapping mechanism. The pumping is populating both the singlet and its symmetric counterpart, the triplet state $ \vert\mathrm{T~0}\rangle=(\vert 100\rangle + \vert10\rangle)/\sqrt{2}$. However, when the couplings are equal $ g_{AC}=g_{BC}$, the singlet is dark, does not couple to other states and finally stores all the excitation of the system in the steady state. Therefore, when more than one electron is allowed, this trapping mechanism breaks as also the states $ \vert 11n_C\rangle$ become pumped. In the absence of cross terms, the tangle drops to zero and there is a current through the system. A negative result to be drawn from Fig. 6.9 is that without cross terms, in the actual case of more than one electron, there is no entanglement to be expected experimentally.

Figure 6.11: Current intensity $ I$ (a) and tangle $ \tau$ (b) in density plots as a function of $ g_{BC}$ and $ \Delta$ in the quantum mechanically indistinguishable case $ \Gamma_{AB}=\Gamma_\mathrm{p}$. $ \Gamma_\mathrm{p}=\Gamma_\kappa=2$ and $ \gamma_\mathrm{d}=0.001$. Energies and rates are in units of $ g_{AC}=1$. Bright areas correspond to maximum values of current and tangle and the dark ones to zero.
\includegraphics[width=0.4\linewidth]{chap6/3QDs/fig4a.eps} \includegraphics[width=0.4\linewidth]{chap6/3QDs/fig4b.eps}

Our main finding here is the entanglement induced by cross terms in the dynamics, enhanced by the coherent coupling between $ A$ and $ B$. Hereafter we consider the general case, i.e., without truncation to only one electron. Fig. 6.10 shows $ I$ and $ \tau$ as a function of $ \Gamma_{AB}$ for the larger detuning $ \Delta =4$ and the lowest dephasing $ \gamma_\mathrm{d}=0.001$ considered in Fig. 6.9. An important fact is that now the couplings $ g_{AC}$, $ g_{BC}$ must be slightly different (for instance $ g_{BC}=0.7$) so that the singlet is not completely dark, but a quasi-dark state weakly coupled to the rest of the system (with a coupling given by $ \vert g_{AC}-g_{BC}\vert/\sqrt{2}$). When the gate $ V_1$ is completely switched on, $ \Gamma_{AB}=0$ and, as it happened in Fig. 6.9, there is a current larger than $ I=0.2$, implying no entanglement. Increasing $ \Gamma_{AB}$ by quenching the gate $ V_1$ does not affect the behavior of the system until the regime where cross terms apply fully is reached. Here, when $ \Gamma_{AB}$ tends to $ \Gamma_\mathrm{p}$, adding cross terms in Eq. 6.16 translates in pumping only the symmetric states (under QDs $ A$, $ B$ exchange). Therefore the incoherent pump with cross terms neither excites the singlet nor induces decoherence of it. This fact, together with the weak link between the singlet state and the other levels, results in a slow coherent transfer of population to the singlet $ \vert\mathrm{S~0}\rangle$, which can be described as a quasi-dark state free of decoherence, as we already showed in the previous Section. This trapping mechanism is enhanced strongly by the direct coupling $ g_{AB}$, also induced by the cross pump. In this case, the tangle becomes close to its highest possible value of 1. The detectable manifestation is a sharp reduction of the current through the system, as QD $ C$ is practically empty. This means a clear way of entangled state preparation between QDs $ A$ and $ B$ as well as a straightforward measurement associated to its occurrence (drop of the current).

Finally, we want to show how entanglement induced by cross terms depends on the coherent part of the dynamics controlled by $ H$. For this purpose, Fig. 6.11 presents current $ I$ (a) and tangle $ \tau$ (b) as a function of the detuning $ \Delta$ and the coherent coupling $ g_{BC}$ (always in units of $ g_{AC}$). The tangle plot shows that detuning is needed to generate a high degree of entanglement. As we explained, also slightly different couplings $ g_{AC}$ and $ g_{BC}$ are necessary to create the quasi-dark state. In Fig. 6.10 we were giving results for the situation with highest tangle ($ \tau=0.85$), that is $ g_{BC}=0.7$ and $ \Delta =4$, corresponding also to lowest current ($ I=0.01$). On the other hand, for the symmetric case $ g_{AC}=g_{BC}$, the singlet is completely dark and therefore there is no entanglement, as we also showed in Fig. 6.9. In this case the current is nonzero. The correlation between high tangle and negligible current and vice versa is clear from Fig. 6.11.

Elena del Valle ©2009-2010-2011-2012.