next up previous contents
Next: Entanglement Up: Two quantum dots in Previous: Introduction   Contents


Two quantum dots in a microcavity

Figure 6.1: QD levels as compared to the cavity mode $ \omega_a$ for Hamiltonian 6.1.
\includegraphics[width=0.8\linewidth]{chap6/2P/Levels/levels-no-biexc.eps}

We consider that the two QDs of Chapter 4 are not directly coupled but each of them interacts separately with the same cavity mode, with coupling strengths $ g_1$ and $ g_2$. In general, the modes are not at resonance but detuned by a small quantity $ \Delta_i=\omega_a-\omega_i$, $ i=1$, 2, from the cavity mode $ \omega_a$. These four parameters $ g_1\neq g_2$ and $ \Delta_1\neq\Delta_2$ represent the experimental inhomogeneity present in the sample. The total Hamiltonian in this case is the sum of two Jaynes-Cummings Hamiltonians, a particular case of the so-called Dicke Hamiltonian for only two emitters:

$\displaystyle H= \omega_a \ud{a}a + \sum_{i=1,2} [\omega_i \ud{\sigma_i}\sigma_i + g_i \big( a\ud{\sigma_i} + \ud{a} \sigma_i \big) ] .$ (6.1)

The QD levels as compared with the cavity mode are plotted in Fig. 6.1. Together with this ``localized'' QD basis of states, that we already introduced in Chapter 4, one can refer to the Dicke states, corresponding to the triplet states:

$\displaystyle \{ \ket{\mathrm{T}_{-1}}=\ket{G},\ \ket{\mathrm{T}_0}=\frac{1}{\sqrt{2}}(\ket{E1}+ \ket{E2}),\ \ket{\mathrm{T}_{1}}=\ket{B} \}\,,$ (6.2)

and to the singlet state

$\displaystyle \{\ket{S}=\frac{1}{\sqrt{2}}(\ket{E1}-\ket{E2}) \}\,.$ (6.3)

In the Dicke basis, the interaction part of $ H$ becomes:

$\displaystyle H_\mathrm{int} = g \big(\ket{\mathrm{T}_0}\!\bra{\mathrm{T}_{-1}}... ...\mathrm{T}_{-1}}-\ket{\mathrm{T}_1}\!\bra{\mathrm{S}} \big)2 + \mathrm{h.c.}\,,$ (6.4)

where $ g=(g_1+g_2)/ \sqrt{2}$ and $ \delta g=(g_1-g_2)/ \sqrt{2}$. Fig. 6.2 shows the corresponding level scheme, up to two excitations in the Dicke basis, with all the coherent and incoherent couplings, represented by curved and straight arrows respectively. Note that when $ g_1=g_2$, the singlet state decouples from the other ones and becomes a dark state. This will play an important role in what follows.
Figure 6.2: Levels of the system up to two excitations in the Dicke basis. Only the case with common pumping bath ( $ P_\mathrm{ind}=0$) is presented here in order to illustrate the discussion to come in Sec 6.2.1. Solid lines represent the common pump which only affects the triplet subspace $ \ket{\mathrm{T}_{-1},n},\ket{\mathrm{T}_0,n},\ket{\mathrm{T}_1,n}$ with $ 2P_\mathrm{com}$, increasing the dot excitations without changing the number of photons $ n$. Dotted lines stand for the cavity photon decay. Dashed lines take account of the leaky modes affecting all QD levels. Finally, curved arrows show coherent couplings ($ g$ and $ \delta g$) between levels.
\includegraphics[width=\linewidth]{chap6/fig_1.eps}

The master equation of the system includes decay for both the cavity mode and the QDs. The QD leaky parameters are set to $ \gamma_1=\gamma_2=5\times10^{-3}g$, a value much smaller than the typical cavity decay rates $ \gamma_a$ considered and experimentally achievable in general. We neglect pure dephasing of the QDs for simplicity. The pumping scheme is the most important ingredient in this work. We consider two physically different situations that we explain in what follows, deriving the appropriate Lindblad terms from the microscopic approach.

First, the case where the two QDs are distinguishable in a classical (as opposed to ``quantum'') way for the pump excitation. Such a situation arises when both QDs are far enough from each other to be resolved and pumped independently or have very different excitation energies. This is the case when the collection areas around each dot--the areas of the wetting layer where free carriers are captured by the dot--are completely separated. In the following we denote by $ A$ the collection areas of the dots (considered equal for simplicity) and $ A_c$ their overlapping area. Each of the QDs couple independently to each element of its own reservoir (electrons $ e_{R_i}$, holes $ h_{R_i}$ and phonons $ f_{R_i}$), with coupling strengths  $ \delta_{R_i}$. This is the situation encountered with atoms and that has been more systematically explored. The Hamiltonian of such a coupling reads:

$\displaystyle H_\mathrm{pump} = \sum_{R_1} \big[ \, \delta_{R_1} \ud{\sigma_1} ... ...elta_{R_2} \ud{\sigma_2}\, e_{R_2} h_{R_2}\ud{f_{R_2}} + \mathrm{h.c.} \big]\,.$ (6.5)

Applying the method and approximations described in Chapter 2, Sec. 2.4, one arrives to two independent Lindblad terms of the form:

$\displaystyle \frac{P_\mathrm{ind}}{2} \sum _{i=1,2} \big( 2 \ud{\sigma_i} \rho \sigma_i - \sigma_i \ud{\sigma_i} \rho - \rho \sigma_i \ud{\sigma_i} \big)$ (6.6)

with parameters that we expect to be proportional to the collection areas $ A$ through the average injection efficiency per unit area and unit time $ \eta$. This magnitude is proportional to the number of carriers in the wetting layer that actually create an exciton in the dot. Here it is considered approximately the same for both dots. The rate can then be expressed as: $ P_\mathrm{ind}= \eta A $.
Figure 6.3: Schema of the two QDs with their associated collection areas $ A$ which can be deformed by applying electric field as was explained by Holtz (2007). The overlapping area is called $ A_c$. When it is nonzero and comparable to the coherence length of the excitation, cross-terms of the pump operators appear in the master equation.
\includegraphics[width=0.4\linewidth]{chap6/fig_2.eps}

On the other hand, when, e.g., two identical dots are close to each other, if the coherence length of the excitation is larger than the distance between the two dots, the final state is a quantum superposition of the excited states of the QDs. This second situation of a common excitation bath has been considered by Braun (2002), keeping the coherent nature of the couplings to the bath. An analogous scheme of a common reservoir has been developed by Ficek & (2002) and by Akram et al. (2000), but for a common squeezed vacuum. It requires the QDs to be indistinguishable for the pumping mechanisms (with equal excitation energies $ \omega_1=\omega_2$), the reservoir excitations--electron-hole pairs with high energy and phonons--to have a large enough coherence length to be shared by both dots, and the two collection areas to be fully overlapped ($ A=A_c$). With these characteristics, there is only one common reservoir and, given that we consider equal efficiencies for the dots (the excitation always affects both dots in the same way), only symmetrical states can be pumped. The Hamiltonian now reads:

$\displaystyle H_\mathrm{pump} = \sum_{R} \big[ \,\delta_{R}\big( \, \ud{\sigma_1} + \ud{\sigma_2}\,\big) e_{R} h_{R} \ud{f_{R}} + \mathrm{H.c.}\big]\,.$ (6.7)

Taking into account the fact that the reservoir is common, we obtain two different contributions to the master equation:

  1. The first is an incoherent contribution to the dynamics given by a Lindblad term:

    $\displaystyle \frac{P_\mathrm{com}}{2} \sum_{i,j=1,2} \big( 2 \ud{\sigma_i} \rho \sigma_j - \sigma_j \ud{\sigma_i} \rho - \rho \sigma_j \ud{\sigma_i} \big)\,,$ (6.8)

    with rate $ P_\mathrm{com} = \eta A_c$.
  2. The second is a direct coupling between the QDs which appears as a coherent coupling in the Hamiltonian, $ H_{12}=g_{12} \big[ \, \ud{\sigma_1} \sigma_2^- + \mathrm{H.c.} \, \big]$ with $ g_{12}$ of the order of magnitude of the common pumping $ g_{12} \approx 2P_\mathrm{com}$. In the Dicke basis this coupling detunes the state $ \ket{\mathrm{T}_0}$ from $ \ket{\mathrm{S}}$.

In a more general and realistic case, the collection areas overlap partially in the region $ A_c$, which contributes to the common pumping with a rate $ P_\mathrm{com} = \eta A_c$, while the rest of the areas $ A-A_c$ contribute to the excitation of each of the QDs separately with rates $ P_\mathrm{ind}=\eta(A-A_c)$ (see Fig. 6.3). We define the degree of common pumping as the fraction $ C=A_c/A$. Varying it between 0 and 1 interpolates between the two extreme cases of independent and common pumping. The Lindblad term of the total pumping is separated in two parts, one specific to each dot which depends on  $ \ud{\sigma_i}$ and another one which is invariant under QDs exchange (creates symmetrical states) and that can be expressed in terms of the operator  $ \ud{J}=\ud{\sigma_1}+\ud{\sigma_2}$. The total master equation of the system is now complete:

\begin{subequations}\begin{align}\frac{d \rho}{dt} =& i \lbrack \rho, H \rbrack ... ...frac{P_\mathrm{com}}{2} \mathcal{L}^{\ud{J}}\rho\,.\end{align}\end{subequations}

As a summary, the first line (6.9a) describes the coherent dynamics of the two dots and the cavity including the direct QD coupling created by the common excitation bath ($ H_{12}$). The second line (6.9b) describes in the usual way the losses of cavity photons and QD excitations. The third line (6.9c) describes the incoherent pumping written here to set apart clearly the two schemes which play an important role in our analysis: first the pumping of each dot regardless of the other, at rate  $ P_\mathrm{ind}$, and then the joint pumping which distributes the excitation among the two dots as a symmetrical quantum superposition, at rate  $ P_\mathrm{com}$. As proved in the next Section, one would expect this common pumping mechanism to create new correlations and coherent superposition between the dots. Taking advantage of this situation, we will show how to build up entanglement between the QD excited states. On the other hand, we find the incoherent independent pumping--that cannot increase coherence between dots--more suitable for lasing properties.

Here, we are interested in the properties of the steady state of Eq. (6.9) in the limit of strong coupling between cavity and QDs. This state was obtained in two independent and equivalent ways. First we solved the set of linear equations for the density matrix elements resulting from setting the time derivative to zero  $ d \rho/dt=0$. Second, we time-integrated the master equation and waited a time long enough to reach the steady state. The solution is unique for a given set of parameters, regardless of the initial state, and both methods agreed exactly except when a singularity arises (as detailed in the next Section) which can only be reached asymptotically with the time-integrated approach.


Subsections
next up previous contents
Next: Entanglement Up: Two quantum dots in Previous: Introduction   Contents
Elena del Valle ©2009-2010-2011-2012.