Broken symmetry and quantum entanglement of an exciton in ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum dot molecules

The ability of a quantum dot to confine photogenerated electron-hole pairs created interest in the behavior of such an exciton in a “dot molecule,” being a possible register in quantum computing. When two quantum dots are brought close together, the quantum state of the exciton may extend across both dots. The exciton wave function in such a dot molecule may exhibit entanglement. Atomistic pseudopotential calculations of the wave function for an electron-hole pair in a dot molecule made of two identical ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ dots reveal that the common assumption of single-particle wave functions forming bonding and antibonding states is erroneous. The true behavior of single-particle electrons and holes leads to symmetry-broken excitonic two-particle wave functions, dramatically suppressing entanglement. We find that at large interdot separations, the exciton states are built from heteronuclear single-particle states while at small interdot separations the exciton is derived from heteronuclear hole states and homonuclear electron states. We calculate the entanglement of the excitons and find a maximum value of 80% at an interdot separation of $8.5\mathrm{nm}$ and very small values for larger and smaller distances.

Figure 1

Single-particle energies assumed in models 1 and 2 [panel (a)] and results from our pseudopotential calculations [panel (b)]. The reference energy for our results is set to the unstrained valence-band maximum of GaAs.

Figure 2

(Color online) Exciton energies as a function of the interdot separation for two different models (models 1 and 2) and for our pseudopotential-CI results (actual results). The circles on the excitonic lines of the lower panel are proportional to the oscillator strength of the transitions.

Figure 3

Schematic representation of the excitonic wave functions obtained from our pseudopotential CI calculations (left) and in the simple model presented in the introduction (right). The symbols are + (hole), − (electron), or ± (exciton). The two spheres denote top and bottom dots. The value of the critical distance is $8.5\mathrm{nm}$ for our specific case.

Figure 4

(Color online) Square of the single-particle electron wave functions $e_0$ and $e_1$ for different interdot separations. The shape of the dots is given in light grey and the two isosurfaces with two different tones of dark gray (red online) contain 75% and 40% of the state densities.

Figure 5

(Color online) Square of the single-particle hole wave functions $h_0$, $h_1$, $h_2$, $h_3$, $h_4$ and $h_5$, for different interdot separations. The shape of the dot is given in light grey and the two isosurfaces with two different tones of dark gray (blue online) contain 75% and 40% of the state densities.

Figure 6

Formation of single-particle “molecular orbitals” (MO’s) [in the central part of panels (a), (b), (c), (d)] from the single-particle “atomic orbitals” [on the left and right sides of panels (a), (b), (c), (d)]. For large interdot separation the electrons and the hole form heteronuclearlike MO’s. For small separation the electrons form bonding-antibonding combinations like homonuclear MO’s.

Figure 7

Strain-modified confining potential for holes along the growth direction (001) (given in the inset of the top panel) for three dot molecules with a base-to-base separations of 5.1, 11.3, and $22.6\mathrm{nm}$. Each data point is an average over the results obtained in the the (001) plane. The size of the circles is proportional to the weight of the heavy-hole contributions; the sizes of the triangles pointing upward [downward] are proportional to the weight of the $J\left(xy\right)$ $\left[J\left(z\right)\right]$ contributions.

Figure 8

Qualitative picture of the strain in a truncated-cone and in a truncated-cone molecule. The deformation of the unit cell at the base and the apex of the dot is schematically given with the corresponding strained bulk band structure. The base of the single dot and the base of the bottom dot (for the dot molecule) is shown to be more favorable for heavy holes.

Figure 9

Analysis of the first hole state $h_0$ in terms of the axial angular momentum $J_z$ (see text) and the orbital character of the envelope functions as a function of the interdot separation. Only the heavy-hole contributions with orbital $S$ character and $J\left(z\right)$ with orbital $P$ character are shown.

Figure 10

Upper panels: localization of the first four exciton states numbered with increasing energy as ∣1⟩, ∣2⟩, ∣3⟩, ∣4⟩, as a function of the interdot distance. On each panel, four lines describe the occupation probability to find the electron and the hole both on the bottom dot $\left(e_B{h}_B\right)$, both on the top dot $\left(e_T{h}_T\right)$, the electron on the top and the hole on the bottom $\left(e_T{h}_B\right)$, and the electron on the bottom and hole on the top dot $\left(e_B{h}_T\right)$. Lower panel: entropy of entanglement as a function of the base-to-base dot separation.

Figure 11

Excitonic spectrum for the first two exciton states ∣1⟩ and ∣2⟩ as a function of the interdot separation $d$.

Figure 12

Differences between single-particle electron and hole energies: $\mid TT⟩=e_0-h_1$, $\mid BB⟩=e_1-h_0$, $\mid BT⟩=e_1-h_1$, and $\mid TB⟩=e_0-h_0$. The denomination $\mid TT⟩$, $\mid BB⟩$, $\mid TB⟩$, and $\mid BT⟩$, where $T$ stands for top and $B$ for bottom, is only meaningful outside the shaded area, for large interdot separations, since the single-particle electron states at short base-to-base separations are neither top nor bottom.

Figure 13

Effective parameters for the two-site Hamiltonian $H$ [Eq. (3)], distilled from our many-body pseudopotential calculation. The lines are a parametrized fit to our data points, listed in Table .