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Universal recovery of the bright-exciton level-degeneracy in quantum dots without structural symmetry

The lack of structural symmetry which usually characterizes semiconductor quantum dots lifts the energetic degeneracy of the bright excitonic states and hampers severely their use as high fidelity sources of entangled photons. We demonstrate experimentally and theoretically that it is always possible to restore the excitonic degeneracy by the simultaneous application of large strain and electric fields, despite the fact that this possibility has fundamentally been doubted. This is achieved by using one external perturbation to align the polarization of the exciton emission along the axis of the second perturbation, which then erases completely the energy splitting of the states. This result, which holds for any QD structure, highlights the potential of combining complementary external fields to create artificial atoms meeting the stringent requirements posed by scalable semiconductor-based quantum-technology.

Semiconductor quantum dots (QDs) are nanostructures made of several thousands of atoms which can self-assemble during epitaxial growth. In spite of the remarkable progress in QD fabrication [1,2] and even in the ideal case in which defects and impurities can be neglected, unavoidable fluctuations in the exact number of atoms, their arrangement in the host matrix and intermixing with the substrate and the cap material make each QD unique. As a result, the possibility to grow semiconductor QDs showing specific properties of symmetry is a mere theoretical construct [3,4]. This represents an enormous obstacle towards the exploitation of QDs in quantum technologies, where the precise control over the QD emission properties is a fundamental requirement [5]. In particular, the absence of structural symmetry usually occurring in real QDs induces a coherent coupling of the two bright excitonic states [6][7][8][9][10] which leads to an energetic separation between them -the so-called fine structure splitting (FSS). When the FSS is larger than the radiative linewidth of the transitions (of the order of 1 µeV), the fidelity of the entangled photon pairs emitted during the decay biexciton-excitonground state is strongly reduced [11][12][13][14]. The origin and magnitude of the FSS are mainly determined by the QD morphology, piezoelectricity, alloying and the symmetry of the underlying lattice [15][16][17][18]. For more than a decade researchers have struggled to find a reproducible way to suppress the FSS and, to date, some quantum optics experiments are still carried out with the rare QDs showing small FSS [19,20]. Recently, the possibility to use external perturbations such as electric [6,21], magnetic [14], and strain fields [7,22] to erase the FSS has been explored. Theoretical [23,24] and experimental [6,7] results have demonstrated that there is a lower bound of the FSS (usually larger than 1 µeV) caused by the coherent coupling of the two bright exciton states. On the other hand, there are a few works reporting values of the FSS close to zero [14,25], but it remains unclear whether a strictly zero FSS=0 can be universally achieved, especially considering that it is commonly believed that the excitonic degeneracy cannot be restored in QDs with low structural symmetry [23,26]. In this work, we demonstrate for the first time that the coupling between the bright excitons can be reversibly controlled and erased by the simultaneous application of large strain and electric fields, thereby showing that it is always possible to drive the excitons confined in arbitrary QDs towards a universal level crossing.
The device employed in this work is shown schematically in Fig. 1(a) (PMN-PT) piezoelectric actuators by gold-thermocompression bonding, as described elsewhere [27]. The thickness of the nanomembrane and the position of the QD layer were optimized to enhance the light-extraction-efficiency via a simple metal-semiconductor planar cavity [27]. The quantum emitters (InGaAs QDs grown on GaAs (001)) were grown by solid source molecular beam epitaxy and were placed in the centre of a 10 nm-thick GaAs/Al 0.4 Ga 0.6 As quantum well. This layer sequence reduces carrier ionization at high electric fields across the diode (F d ) and allows the QD emission lines to be shifted over a broad spectral range by applying a reverse bias (V d ) to the nanomembranes [6].
Simultaneously, a voltage V p (electric field F p ) applied to the PMN-PT induces anisotropic inplane (compressive or tensile) biaxial strains in the QD layer [28], which allow the energy of the QD emission lines to be controlled in a well-defined manner [27,29].
Micro-photoluminescence spectroscopy was used for the optical characterization of the device. The measurements were performed at low temperature (typically 4-10 K) in a helium flow cryostat and with a microscope objective with numerical aperture of 0.42. The device was excited with a 850 nm continuous-wave Ti:Sa laser and the signal was collected by the same microscope objective used for the excitation, spectrally analyzed by a single or double spectrometer with 0.75 m focal length and detected by a Si charge-coupled device.
Polarization-resolved measurements were performed by rotating a half lambda waveplate in combination with a linear polarizer, whose transmission axis was set along the [110] crystal direction of the nanomembranes with an accuracy of 10°. The FSS and the polarization angle of the excitonic emission were evaluated using the same procedure reported in Ref. 6, which ensures sub-microelectronvolt resolution.
The device shown in Fig. 1(a) combines the most powerful "tuning knobs" available to date (electric and strain fields) and allows the QD emission properties to be engineered on demand. This is for example demonstrated in Fig. 1(b), where the two knobs are used for tuning all the QD emission lines across a spectral range larger than 35 meV, which is one of the largest reported so far. Furthermore, control over the binding energies of biexciton (XX) and charged excitons (X + and X -) is also demonstrated in Fig. 1(b), as reported previously with either electric fields or strain [6,29]. We now show that the possibility to apply simultaneously two independent external fields allows a fundamental problem plaguing QDs -the presence of a coherent coupling of excitons -to be solved.
The coupling between the bright excitonic states can be highlighted by sweeping their energetic levels one across the other [6,7,23]. This is shown schematically in Fig. 2   values of F p . For tensile strain (F p <0), the value of Δ increases up to a maximum observed value of Δ max~2 5 μeV, while under compressive strain (F p >0) Δ first decreases to a minimum value (Δ min ) and then increases again. As can be observed more clearly in Fig. 3(b) where the FSS is plotted against the exciton energy (E x ), the value of Δ min is comparable with the experimental spectral resolution (~0.5 μeV). Similar results were obtained in all QDs chosen randomly in our device (see Fig. 3(c)), where it is always possible to tune the FSS down to the experimental spectral resolution. This evidence, along with the fact that even the FSS of a QD showing Δ max as large as 40 μeV can be tuned to values below 0.5 μeV, strongly suggests that electro-elastic fields enable us to "erase" the FSS of most of (if not all) the QDs in our sample.
Additional information can be found studying the behaviour of θ (see Fig. 4(a)) for the same QD reported in Fig. 3(a). When F p is increased, the θ curve shows sharper variations with F d and, after the critical point where Δ=Δ min , the direction of rotation is inverted, from anticlockwise to clockwise. This change in the handedness of θ has never been observed in the same QD [6], and suggests that the FSS=0 critical point has been crossed. Obviously, the finite spectral resolution of the experimental apparatus prevents FSS=0 to be proved but the fact that we observe systematically the same behaviour in all the QDs we measured indicates the existence of a universal behaviour of the FSS under the influence of strain and electric fields, despite the different properties of the quantum emitters. In order to check if FSS=0 can be achieved we now analyze the experimental observations theoretically.
The FSS originates from the breaking of the D 2d or C 4v symmetry occurring even in ideal QDs with C 2v symmetry [10]. Any additional disorder -we can safely assume that any realistic QD has only C 1 symmetry -implies that the polarization angle of the bright excitons further departs from the pure lens shaped ones [10,24,30]. Whether or how the FSS can be tuned to zero depends on the existence of external perturbations able to "universally" restore the level degeneracy, independent of the specific QD parameters. By considering the combined effect of a vertical electric field (F) applied along the [001] direction and anisotropic biaxial stresses [28]  σ are the usual Pauli matrices, η and k accounts for the QD structural asymmetry.
The parameters related to the external fields areα and γ (related to the elastic compliance constants renormalized by the valence band deformation potentials), and β (proportional to the difference of the exciton dipole moments). Diagonalization of the above Hamiltonian gives the following values of FSS and θ: Equations (1) In other words, there are always values of F p and F d such that FSS=0, regardless of the QD structure. It is obvious that in real experiments large enough tuning ranges are needed to access the values of critic p and critic F given above. This requirement is satisfied by our device, which allows us to tune systematically all the QDs we measure to FSS=0. It is also important to observe that the remarkable agreement between all the experimental data and the simple theory developed in this work indicates that higher order terms in p and F can be neglected.
The result is in line with those of Ref. [24], which found good agreement between this continuum theory and atomistic calculations accounting for the microscopic effects that lower the QD symmetry to C 1 (note that Ref. 24 considered only the effect of a single perturbing field (strain), while here two independent fields are addressed).
Finally, by combining the theoretical analysis with the experimental data a clear and intuitive picture emerges. Figure 4( [6,7,23,24]. The above discussion highlights the crucial importance of having at hand two independent and broad-range "tuning knobs" where we can use one perturbation (e.g. F p ) to align θ along a specific direction ([110]) and the other to tune FSS to 0.
In conclusion, we have presented a novel and all-electrically controlled device where the QD emission properties are engineered by large electro-elastic fields provided by diode-like nanomembranes integrated onto piezoelectric actuators. In particular, we have demonstrated that the simultaneous application of electric and elastic fields allow us for the first time to control and cancel the coherent coupling between the bright excitonic states -and hence the exciton fine structure splitting -in all the quantum dots we measure. The experimental observations are supported by a simple theoretical model which holds for every QD structure.
The tedious search for QDs suitable for quantum optics experiments can therefore be avoided, and a deterministic implementation of QDs into a scalable semiconductor-based quantuminformation technology seems much more likely in the foreseeable future.  The colours correspond to the F p values shown in Fig. 3(a). The data taken at F p = 12 kV/cm, not displayed in Fig. 3(a) for clarity purposes, are also reported (orange points connected by lines). The value of the minimum Δ is denoted Δ min , which coincides with the experimental spectral resolution. (c), Histogram of Δ max and Δ min for the different measured QDs.