Fundamental intrinsic lifetimes in semiconductor self-assembled quantum dots
Wen Xiong Jun-Wei Luo Xiulai Xu Ming Gong Shu-ShenLi Guang-CanGuo
cond-mat.mes-hall cond-mat.mtrl-sci quant-ph




Department of Physics, Chongqing University, Chongqing 401331, China

Email: jwluo@semi.ac.cn
State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences,  Beijing 100083, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China

Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China

Email: gongm@ustc.edu.cn
Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, People’s Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

State Key Laboratory of Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences,  Beijing 100083, China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing .100049, China

Key Laboratory of Quantum Information, University of Science and Technology of China, CAS, Hefei, 230026, People’s Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China



The self-assembled quantum dots (QDs) provide an ideal platform for realization of quantum information technology  because it provides on demand  single photons, entangled photon pairs from biexciton cascade process, single spin qubits, and so on. The fine structure splitting (FSS) of exciton is a fundamental property of QDs for thees applications. From the symmetry point of view, since the two bright exciton states belong to two different representations for QDs with C_2v symmetry, they should not only have different energies, but also have different lifetimes, which is termed exciton lifetime asymmetry. In contrast to extensively studied FSS, the investigation of the exciton lifetime asymmetry is still missed in literature. In this work, we carried out the first investigation of the exciton lifetime asymmetry in self-assembled QDs and presented a theory to deduce lifetime asymmetry indirectly from measurable qualities of QDs. We further revealed that intrinsic lifetimes and their asymmetry  are fundamental quantities of QDs, which determine the bound of the extrinsic lifetime asymmetries, polarization angles, FSSs, and their evolution under uniaxial external forces.  Our findings provide an important basis to deeply understanding properties of QDs.


78.67.Hc, 42.50.-p, 73.21.La
Fundamental intrinsic lifetimes in semiconductor self-assembled quantum dots
    Guang-Can Guo
    December 30, 2023
============================================================================



The self-assembled quantum dots (QDs) provide a promising platform for realizing on-demand entangled photon pairs from the biexciton-exciton-vacuum cascade process<cit.>,
which are essential for practical quantum communication<cit.>. However the major obstacle in realizing this goal comes from the
non-degeneracy of the two intermediate bright exciton states (see Fig. 1a), in which their energy difference, called fine structure splitting (FSS), is much larger than the homogeneous
broadening of the emission lines (Γ∼ 1 μeV<cit.>), thus the ”which-way” information is erased and only classically corrected photons
instead of maximally entangled photon pairs can be created from this cascade process. In the past decade, strenuous efforts have been devoted to eliminate this splitting by applying various experimental techniques, including thermal annealing<cit.>, electric field<cit.>, magnetic field<cit.> and external stress<cit.> etc.. However, none of them is efficient. In recent years, the entangled photon pairs were demonstrated in a way that first picking out the QDs with small FSS from a QDs ensemble after post annealing and then eliminating the FSS using magnetic fields (see more details in the first experiment by Stevenson et al. in 2006<cit.>). It is worth to note that only a tiny fraction of QDs in experimental grown samples can be used to achieve entangled photon pairs. Moreover,  these devices can only work at low temperature since the emitted exciton energies are very close to the emission lines from the wetting layer<cit.>. The mechanism underlying  the difficulty of  eliminating the FSS comes fundamentally from the low symmetry of self-assembled QDs. The higher symmetry of the bulk materials are impossible to be restored in QDs by the above mentioned techniques<cit.>. Gong et al. based on a proposed minimal two-band model  <cit.> uncovered that the FSS is impossible to be tuned to zero for a general QD with a single external force. However, the FSS can be eliminated by two independent external forces<cit.>. Employing three external forces, it is even possible to construct a wavelength tunnable entangled photon emitter<cit.>, opening the door for interfacing between QDs and other solid state systems and even between dissimilar QDs.  Recently, this proposal has indeed  been realized in experiments <cit.>, in which the wavelength of exciton can be tuned in the range of few meV.



From the symmetry point of view, since the two bright exciton states belong to two different representations for QDs with C_2v symmetry<cit.>, they should not only have different energies, but also have different lifetimes. The lifetime difference between two bright exciton states are termed as exciton lifetime asymmetry (see Fig. 1a). In contrast to extensively studied FSS, the investigation of this anisotropy effect is still missed in literatures. In this work, we carried out the first investigation of the exciton lifetime asymmetry in self-assembled QDs and presented a theory to deduce lifetime asymmetry indirectly from measurable qualities of QDs. We further revealed that the lifetime asymmetries are fundamental quantities of QDs and unraveled some exact relations between FSS, polarization angle and lifetime asymmetry for exciton and biexciton in QDs. These exact relations are verified by performing atomistic simulations of self-assembled QDs using the empirical pseudopotential method<cit.>.

Atomistic Simulation Method. We employ the empirical pseudopotential method<cit.> to simulate the electronic and optical properties of self-assembled QDs. We model the InGaAs/GaAs QDs by embedding it a much larger GaAs supercell with periodic boundary condition and minimize the total strain energy using the valence force field
model<cit.>. The single particle wave functions are determined by,

    (-1 2∇^2 + V_ps( r))ψ_i = E_iψ_i,

where V_ps = 𝒱_soc + ∑_iα𝒱_iα( r - R_iα) is the empirical pseudopotential and i is the site index,  R_iα is the position of the atom type α, and 𝒱_soc the spin-orbit coupling term. The position of each atom in the supercell is obtained by minimizing the total strain energy. The exciton and biexciton energies are then calculated by employing the configuration
interaction method taking into account the Coulomb interaction and exchange-correlation interaction <cit.>.


Theoretical Modelling. We carry out the symmetry analysis of QDs as given in Ref. Gong11 by decomposing the QD Hamiltonian H into two parts: H = H_2v + V_1, where H_2v is a predominant term having a C_2v symmetry and includes the kinetic energy, Coulomb interactions and all the potentials with C_2v symmetry.  The second term V_1 represents the remaining potentials lowering the symmetry of QDs from C_2v to C_1. V_1 can be treated as a perturbation to H_2v considering its weak effect on energy levels of QDs. Two bright states of the H_2v Hamiltonian are denoted as |3⟩ = |Γ_2 - iΓ_4⟩ and
|4⟩ = |Γ_2 + iΓ_4⟩, respectively,  where |Γ_i⟩ (i=1,2,3,4) are the irreducible representations of the C_2v point group (see Fig. <ref>b and the symmetry table in Koster63). We refer quantities of the H_2v Hamiltonian to intrinsic. Since these two bright exciton states belong to two different irreducible representations, they must have different energies with an energy separation of FSS, and different lifetimes (τ_3η and τ_4η, respectively) with a time difference termed as intrinsic lifetime asymmetry δτ_η. Hence,  τ_3η = τ_η + δτ_η/2 and τ_4η = τ_η - δτ_η/2; here, τ_η is averaged lifetime (hereafter η=x for exciton and η=xx for biexciton). Since the V_1 potential is inevitable and uncontrollable in experimental grown QDs, the intrinsic lifetimes and their asymmetry  can not be measured  directly  in experiments.


To simplify the effective model for total QD Hamiltonian H, we take advantage of two additional features. Firstly, the time-reversal symmetry for exciton, 𝒯^2 = +1 (excitons have integer spin), ensures that the wave functions of two bright exciton states can be real simultaneously<cit.>. Secondly, the spin selection rule forbids the mixture between dark exciton states (m = ± 2) and bright exciton states (m = ± 1) even in the presence of V_1 term.
The dark exciton states can only be probed by coupling to bright states through applying external (in-plane) magnetic fields<cit.>. We thus can safely neglect the dark states and construct the effective model based only on  two bright exciton states of the H_2v as following,

    H = E_0 + δσ_z + κσ_x

where E_0 is the mean energy of two bright exciton states, σ_x and σ_z are Pauli matrices acting on two bright states with eigenvalues of m = ± 1, and δ = ⟨ 3|H|3⟩ - ⟨ 4|H|4⟩
and κ = ⟨ 3|H|4⟩. The bright states of the total Hamiltonian H can be constructed as a linear combination of |3⟩ and |4⟩,

    [ ψ_3; ψ_4 ]
    = u(θ)
    [ |3⟩; |4⟩ ],   
    u(θ) =
    [  cos(θ)  sin(θ); -sin(θ)  cos(θ) ],

where θ is the polarization angle as schematic shown in Fig. 1c and u(θ) is a so(2) rotation matrix. We learn that tan(θ) = (δ + Δ/2)κ^-1, where Δ = 2√(δ^2 + κ^2)  the magnitude of exciton FSS. The values of κ and δ can be determined from experimental measurements of FSS and polarization angle. In an ensemble of QDs, the V_1 potential is usually a random potential, thus κ and δ can be treated as two independent random numbers<cit.>, for which reason the QDs with similar structural profile may have fairly different optical properties.



If two bright exciton states have measurable extrinsic lifetimes T_3η and T_4η, which are usually deduced experimentally by fitting the time-resolved photoluminescence spectrum to a single exponential decaying function. The difference in lifetimes gives rise to the extrinsic lifetime asymmetry δ T_η,

    δ T_η = T_3η - T_4η,    T_3η + T_4η = 2T_η,

where T_η is the mean extrinsic lifetime measured from off-resonant excitation. In this measurement, we have assumed that the spin information in the electron-hole pairs are totally lost during the relaxation from the wetting layer to the ground state of exciton, thus the two bright states are equally populated. However, those extrinsic lifetime asymmetries are not well defined for following reasons. Firstly, the time-resolved photoluminescence spectrum is in fact composed by two exponential decay functions with two slightly different lifetimes<cit.>, and thus T_3η and T_4η can only be obtained in the sense of best fitting. Secondly, the V_1 potential, which induces inter-state mixing (see Eq. <ref>), is not the major origin of lifetime asymmetries <cit.>. A more accurate description of lifetime asymmetries should be defined by the model of H_2v, instead of the total Hamiltonian H.

Because lifetime asymmetries are in general much smaller than the mean lifetimes, we are ready to obtain

    1 T_3η= cos^2(θ) τ_3η + sin^2(θ) τ_4η,
        1 T_4η= sin^2(θ) τ_3η + cos^2(θ) τ_4η.

To the leading term of δτ_η,

    T_3η = τ_η + cos(2θ)  2δτ_η,    T_4η = τ_η - cos(2θ)  2δτ_η.

The above results are identical to fitting the photoluminescence spectrum to a single exponential decaying function by minimizing the following functional,

    ℱ_n = ∫_0^∞ | c_n e^-t τ_3η + (1-c_n) e^-t τ_4η - e^-t  T_η|^2 dt,

where n=3, 4, with c_3 = cos^2(θ) for |3⟩ and c_4 = sin^2(θ) for |4⟩.  Assuming T_η = τ_η + x δτ_η (where |x| ≪ 1), we obtain ℱ_η = (-1 + 2c_n + 2x)^2 δτ^2/16τ + 𝒪(δτ/τ)^3. It is straightforward to obtain its solution, which is identical to ones given in Eq. <ref>.  We therefore see that definitions given in Eq. <ref> and Eq. <ref> are equivalent  in the small asymmetry limit.




  
The unusual effects of low symmetry perturbation potential.
(i) Lifetime sum rule: The weak C_1 potential will not alter the averaged exciton lifetime, which is determined as,

    T_3η + T_4η = τ_3η + τ_4η,   τ_η =T_η.

The equality relation τ_η = T_η indicates that the mean lifetime is independent of V_1 potential, which is manifested in the investigated QDs ensembles as shown in Fig. <ref>. (ii) Lifetime asymmetry relation: The extrinsic lifetime asymmetry is determined as

    δ T_η = T_3η - T_4η = cos(2θ)δτ_η≤ |δτ_η|.

We see that, to the leading term, the extrinsic lifetime asymmetries depends only on the intrinsic lifetime asymmetries δτ_η and the polarization angle θ, and is independent of the mean lifetimes τ_η and T_η. Interestingly, while the V_1 potential can enhance the FSS and polarization angle, it will unexpectedly suppress the magnitude of δ T_η, which is upper bounded by |δτ_η|. Moreover, when δτ_η = 0 (in QDs with high symmetries) or θ = ±π 4 (polarized along Γ_2 and Γ_4 directions), the change of low symmetry potential will never induce a finite extrinsic lifetime asymmetry.







To verify above predictions we carry out atomistic simulations for single QDs as well as QDs ensembles. The calculated results for various types of single pure InAs/GaAs QDs and alloyed In(Ga)As/GaAs QDs are summarized in Table <ref>. For pure  InAs/GaAs QDs, we find that, as expected since  V_1=0, the polarization angle θ = 0 (or π/2), and δτ_η = δ T_η,  following exactly  Eq. <ref>. The magnitude of both intrinsic and extrinsic lifetime asymmetries δ T_η and δτ_η are in range of 0.1 - 0.4 ns, and much smaller than the mean lifetimes T_η and τ_η, which are around 1.5-2.0 ns.
From Table I we see that the lifetime sum rule given in Eq . <ref> holds for all calculated QDs, including alloyed ones. Although the polarization angles θ in alloyed QDs deviate significantly away from the [110] and [11̅0] directions caused by wave functions mixing,  |δ T_η| ≤ |δτ_η| holds in all QDs. Therefore, we demonstrate that the low symmetry perturbation potential
can remarkably suppress the lifetime asymmetries along with the derived relation  given in Eq. <ref>.



We further consider the alloyed QDs ensembles, in which the FSSs, polarization angles, and mean lifetimes T_η fluctuate from dot to dot in a wide range. We arbitrarily choose an alloyed InGaAs/GaAs QD from Table <ref> (No. 9) and then generate 50 different replica with randomly placed In and Ga atoms for a specific composition of 60%, which mimics an experimentally grown QDs ensemble. The calculated results for this ensemble are shown in Fig. <ref>. As expected, the FSSs, polarization angles θ, mean lifetimes T_η as well as lifetime asymmetries δ T_η  fluctuate from dot to dot in a wide range within an ensemble. Because all modeled dots within the ensemble have the same shape, size and alloy composition, they share always the same H_2v Hamiltonian, which means their intrinsic properties should be similar. Indeed, the deduced intrinsic lifetime asymmetries δτ_η  are rather insensitive to alloy atom fluctuations in an ensemble. The observed fluctuations in extrinsic quantities of FSSs, θ, T_η, and δ T_η are attributed to alloy fluctuation induced change of the V_1 potential.  We find that |δ T_η| ≤ |δτ_η| for all dots (see Fig. <ref>d-e), and the ratios between extrinsic and intrinsic lifetime asymmetries fall on a cosine curve as shown in Fig. <ref>f, in  consistent with the prediction given in  Eq. <ref>. The alloy-induced V_1 perturbation potential remarkably reduces extrinsic lifetime asymmetries δ T_η to around zero in compared with their intrinsic lifetime asymmetry around 0.1 ns.





  
The optical anisotropy relying on the intrinsic lifetimes. The signals collected along ϕ direction after off-resonance excitation can be written as

    I_η(ϕ) ∝ 1 + cos(2θ) cos(2(ϕ-θ)) δτ_η 2τ_η,

where ϕ is the angle relative to [110] direction. Here, we also assume that both bright states are equally occupied from off resonance excitation like we developing Eq. <ref>.
 We gain the maximum degree of linear polarization ρ_η, max  when ϕ = θ or θ + π/2,

    ρ_η, max = I_max - I_min I_max + I_min = |cos(2θ) δτ_η 2τ_η| ≤|δτ_η 2τ_η |.

We learn that the degree of polarization is determined fully by polarizaton angle θ, lifetime  asymmetry δτ_η, and mean lifetime τ_η, considering that the transition between two bright exciton states are strictly forbidden by selection rule. The degree of polarization is upper bounded by |δτ_η/τ_η|/2. Regarding |δτ_η| ≪τ_η, the degree of polarization is restrict to small magnitudes despite the large varying from dot to dot. Note that the polarization discussed here should different from the linear polarization defined in some of experiments, in which the excitation and measurement are performed along two orthogonal direction, thus the degree of polarizations are generally in
the order of 80% - 90% out of the available experimental data<cit.> . In the latter case polarization is complicated since carrier scattering, spin flipping and lifetime asymmetry all contribute to its non-unity.

For the measurement along [110] and [11̅0] directions  (let ϕ = 0),

    ρ_η, × = I_[110] - I_[11̅0] I_[110] + I_[11̅0] = cos^2(2θ) δτ_η 2τ_η,

which is smaller than ρ_η,max by a factor of cos(2θ). We further prove straightforwardly that the degree of linear polarization along
the x and y directions is ρ_η,+ = sin(2θ)ρ_η,max. The relations between these degree of polarizations are presented in Fig. <ref>.
Although the small extrinsic lifetime asymmetry maybe hard to measure directly by fitting to two exponential decay functions, the relative polarization angle θ, the mean lifetime τ_η (see Eq. <ref>) as well as the degree of polarization ρ_η, i can be measured precisely, thus making the intrinsic lifetime asymmetries δτ_η rather feasible to get in experiments. Once we obtain intrinsic lifetime asymmetry δτ_η, the extrinsic lifetime asymmetry δ T_η   can be instead deduced using Eq. <ref>.






  
The optical polarization relying on the intrinsic  lifetimes Since exciton and biexciton states possess the same polarization angle θ, we obtain

    ρ_x,iρ_xx,i = δτ_xδτ_xx·τ_xxτ_x,   
        i = {max, ×, +},

which is also independent of the measured ϕ. The deduced relations of the degree of polarizations with intrinsic  lifetimes τ_η and lifetime asymmetry δτ_η of the H_2v term along various directions can be examined with the data of atomistic calculation shown in Fig. <ref>. For this QDs ensemble, δτ_η∼ 0.1 ns, τ_η∼ 1.25 - 1.35 ns, thus ρ_η, max⩽δτ_η/2τ_η∼ 4%, which is in good agreement with atomistic simulation predicted upper bound of the degree of polarization of ∼ 4% as shown in Fig. 3a. We may also calculate the ratio between the degree of polarizations for exciton and biexciton, which is independent of directions of optical measurements. We find this ratio being ρ_x,i/ρ_xx,i∼ 1.08. We expect this ratio becomes larger for QDs with  large shape asymmetry (such as  elongation). In principle, the evolution of linear polarization as a function of angle ϕ (Eq. <ref>) can be obtained by carefully calibrating the light path in experiments.



  
The Quadratic and Strong Nonlinear responses of the extrinsic lifetimes to external stress.
In the presence of weak external force F, the effective Hamiltonian becomes<cit.>,

    H = E_0 + (α F + δ)σ_z + (κ + β F) σ_x,

where, the extra perturbative term δ V = V_s F is responsible for the applied external force. The symmetry of δ V depends strongly on the directions and the ways the forces been applied (see Fig. <ref>). Here, we only consider the case with single external force. The additivity of δ V ensures that multiple forces can be treated in the same way as the single force.
Notice that two unperturbed (intrinsic) bright exciton states |3⟩ and |4⟩  are also functions of F due to the H_2v Hamiltonian depends on F.  For the stress applied along the [110] direction, the strained Hamiltonian still keeps the C_2v symmetry (see Fig. <ref>a, c), thus the stress induced coupling between the two bright states is forbidden. This is different from the case along the [100] direction, where the stress induces not only coupling of bright exciton states to highly excited states arising from other bands (termed as inter-band coupling),  but also coupling between two bright exciton states (termed as intra-band coupling) (see Fig. <ref>b, d). The linear coupling dominates the inter-band coupling due to the large energy separation between the ground s band and the excited p and d bands, while the nonlinear effect may become significant in intra-band coupling due to much smaller energy difference between these two bright exciton states. This difference has remarkable consequences to the lifetime asymmetries. According to perturbation theory, we rewrite the intrinsic lifetimes as,


    1 τ_i^η(F) = 1 τ_i^η + γ_i^η F + ξ_i^η F^2,

for i = 3, 4 and η = {x, xx}. Here, we have introduced γ_i^η and ξ_i^η to characterize the linear and quadratic dependence on external force, respectively. When the contributions of the second and third terms are small in compared with 1/τ_i^η, we obtain,

    τ_i^η(F) = τ_i^η - (τ_i^η)^2 [ γ_i^η F + ξ_i^η F^2 - (γ_i^η)^2τ_i^η F^2],

which is again a quadratic function of F. It it expected that τ_η = (τ_3^η + τ_4^η)/2 and δτ_η = (τ_3^η - τ_4^η)/2 are also linear and/or quadratic functions of F. With these defined intrinsic lifetime asymmetries, the extrinsic lifetimes and their asymmetries can be obtained via,

    δ T_η(F) = cos(2θ(F)) δτ_η(F),    T_η(F) = τ_η(F).

The striking consequence is that even under a weak force the extrinsic lifetime asymmetry δ T_η is not necessary to be a simple quadratic function of F due to the presence of the strong nonlinear cosine term. We can derive  more exact relations via a combination of current results and results in previous literatures<cit.>.

We next attempt to study the evolution of optical properties of QDs under external forces, which enables us to verify the last and the most intriguing prediction (Eq. <ref>) in this work. The properties of QDs fluctuate strongly from dot to dot, therefore the investigation in single QDs could provide more convincing evidences for our predictions. We consider the pure InAs/GaAs QD (No. 3 in Table I) and alloyed InGaAs/GaAs QD (No. 12) under uniaxial stress along [110] and [100] directions, respectively. Fig. <ref> shows calculated FSSs and polarization angles as functions of stress for these two QDs employing the atomistic method, accompanying fitted results using the two-level model. We demonstrate that we can tune the FSS to zero upon applying a stress along the [110] direction, which does not lower the QD symmetry (see Fig. <ref>), for the pure QDs, but it is impossible for alloyed QDs where the achievable minimum lower bound of FSS is significant larger than the spontaneous broadening of the spectra  Γ<cit.>. This large minimum FSS implies that we can not eliminate the FSS of alloyed QDs by using one single external force.



Fig. <ref> shows calculated intrinsic lifetimes and their asymmetries as functions of applied stress for both pure InAs/GaAs and alloyed InGaAs/GaAs QDs. For stress applied along the [110] direction, we find that τ_3x and τ_4x are linear functions of stress F, as shown in Fig. <ref>b-d. To further verify their linear feature, we fit these data to quadratic functions. We gain tiny coefficients of the quadratic terms for both pure InAs/GaAs and alloyed InGaAs/GaAs QDs (ξ_3,4^x∼ 10^-7 - 10^-8 ns/MPa^2, see fitted data in Fig. <ref>), which illustrate both intrinsic lifetimes and their asymmetries are linear against [110] stress. In striking contrast to [110] stress applied along the [100] direction gives rise to different response of the lifetime of τ_3x and τ_4x because it causes direct coupling between two bright exciton states (see mechanism in Fig. <ref>). From Fig. <ref> we see that both τ_3x and τ_4x exhibit quadratic relations against the applied [100] stress F.  As expected, the corresponding lifetime asymmetry δτ_x also displays a quadratic function of F. The similar features can also be found for transition from biexciton to exciton states.

The extrinsic lifetime asymmetries have more complicated response behaviours to applied stress and will exhibit strong nonlinearity even under weak force due to direct coupling between the two bright states
induced by the C_1 symmetry potential V_1. Fig. <ref> shows  atomistic calculated results. In the pure InAs/GaAs QD the V_1 potential is absent, and thus T_3x and T_4x should possess perfect linear functions of F for stress applied along the [110] direction and quadratic functions for stress applied along the [100] direction. The results shown in Fig. <ref> well support  the theoretical prediction. These stress-responses are identical to τ_η, as shown in Fig. <ref>b,d, in the sense that θ = 0. However, for alloyed InGaAs/GaAs QD, strong nonlinearity behaviours of both mean extrinsic lifetimes and extrinsic lifetime asymmetries are expected for stress applied along [100] and [110] directions. Since T_i(F) = τ_i(F) (see Eq. <ref>), we can observe the strong nonlinearity in both the extrinsic lifetimes T_iη and the extrinsic lifetime asymmetries δ T_η. Moreover, while the extrinsic lifetime asymmetries are generally very small due to wave function mixing effect, we find that the extrinsic lifetime asymmetries in Fig. <ref> c-d, and the intrinsic lifetime asymmetries in Fig. <ref>c-d, can be pronouncedly enhanced by external stress from tens of ps to 0.2 - 0.7 ns, that is, by at least one order of magnitude. This enhanced asymmetries may lead to direct measurement of lifetime asymmetries with only time-resolved photoluminescnce spectrum, in which the time-resolved spectrum should be fitted using two exponential decay functions. The similar strong nonlinearity effect has also been observed for transition from biexciton to exciton states in a reason discussed above. These results can be well described by the data in Fig. <ref> and are in fully accordance with our theoretical predictions.

Summary. We introduced lifetime asymmetry, which is a new concept,  into self-assembled QDs. We revealed that intrinsic lifetimes are fundamental quantities of QDs, which determine the bound of the extrinsic lifetime asymmetries, polarization angles, FSSs, and their evolution under uniaxial external forces. These predictions can be direct measured or extracted from experiments using the state-of-the-art techniques, such as the measured  linear polarization as well as the optical properties of QDs under external forces. We verified these predictions using atomistic simulations. We found that the intrinsic lifetime asymmetries can be of the order of few hundred picoseconds in pure InAs/GaAs QDs, but the extrinsic lifetime asymmetries can be much smaller in alloyed InGaAs/GaAs QDs. However, the lifetime asymmetries are susceptible to external forces and its directions, thus can be more conclusively verified by investigating their behaviours under external forces. These exact relations represent a complete description of the optical properties of QDs. Our findings provide an important basis to deeply understanding properties of QDs. 


Acknowledgement. M.G. is supported by the National Youth Thousand Talents Program (No. KJ2030000001), the USTC start-up funding (No. KY2030000053),
the national natural science foundation (NSFC) under grant No. GG2470000101). X. X. is supported by National Basic Research Program of China under No. 2014CB921003, 
the NSFC under Nos. 11721404, 91436101 and 61675228; the Strategic Priority Research Program of the Chinese Academy of Sciences under No. XDB07030200 and XDPB0803, and the CAS 
Interdisciplinary Innovation Team. J. L. is supported by NSFC under Nos. 61121491, 11474273, 11104264 and U1530401, and National Young 1000 Talents Plan. X. W. is supported 
by NSFC under No. 61404015 and fundamental research funds for the central universities (2015CDJXY300001). 

99
Beson00 Benson, O.; Santori, C.; Pelton, M.; Yamamoto, Y. Phys. Rev. Lett. 2000, 84, 2513.
Pan98Pan, J.-W.; Bouwmeester, D.; Weinfurter, H.; Zeilinger, A. Phys. Rev. Lett.  1998. 80, 3891.
Gisin02Gisin, N.; Ribordy, G.; Tittel, W.; Zbinden, H. Rev. Mod. Phys. 2002, 74, 145.
Duan01Duan, L.-M.; Lukin, M. D.; Cirac, J. I.; Zoller, P. Nature 2001, 414, 413.
JSXu10Xu, J.-S.; Li, C.-F.; Gong, M.; Zou, X.-B.; Shi, C.-H.; Chen, G.; Guo, G.-C. Phys. Rev. Lett. 2010, 104, 100502.
YFH11Huang, Y.-F.; Liu, B.-H.; Peng, L.; Li, Y.-H.; Li, L.; Li, C.-F.; Guo, G.-C. Nat. Commun. 2011, 2, 546.
Pan2016Yin, J.; Cao, Y.; Li, Y.-H.; Liao, S.-K.; Zhang, L.; Ren, J.-G.; Cai, W.-Q.; Liu W.-Y. et al. Science 2017, 356, 1140.
Gammon96Gammon, D.; Snow, E. S.; Shanabrook, B. V.; Katzer, D. S.; Park, D. Phys. Rev. Lett. 1996, 76, 3005.

Bayer02Bayer, M.; Ortner, G.; Stern, O.; Kuther, A.; Gorbunov, A. A.; Forchel, A.; Hawrylak, P.; Fafard, S.; Hinzer, K.; Reinecke, T. L.; Walck, S. N.; Reithmaier, J. P.; Klopf, F.; Schafer, F. Phys. Rev. B 2002, 65, 195315.

Seguin05Seguin, R.; Schliwa, A.; Rodt, S.; Potschke, K.; Pohl, U. W.; Bimberg, D. Phys. Rev. Lett. 2005, 95, 257402.
Langbein04Langbein, W.; Borri, P.; Woggon, U.; Stavarache, V.; Reuter, D.; Wieck, A. D. Phys. Rev. B 2004, 69, 161301(R).
Tark04Tartakovskii, A. I.; Makhonin, M. N.; Sellers, I. R.; Cahill, J.; Andreev, A. D.; Whittaker, D. M.; Wells, J-P. R.; Fox, A. M.; Mowbray, D. J.; Skolnick, M. S.; Groom, K. M.; Steer, M. J.; Liu, H. Y.; Hopkinson, M. Phys. Rev. B 2004, 70, 193303.
Ellis07Ellis, D. J. P.; Stevenson, R. M.; Young, R. J.; Shields, A. J.; Atkinson, P.; Ritchie, D. A. Appl. Phys. Lett. 2007, 90, 011907.
SeguinSeguin, R.; Schliwa, A.; Germann, T. D.; Rodt, S.; Pötschke, K.; Strittmatter, A.; Pohl, U. W.; Bimberg, D.; Winkelnkemper, M.; Hammerschmidt, T.; Kratzer, P. Appl. Phys. Lett. 2006, 89, 263109.
TartakovskiiTartakovskii, A. I.; Makhonin, M. N.; Sellers, I. R.; Cahill, J.; Andreev, A. D.; Whittaker, D. M.; Wells, J-P. R.; Fox, A. M.; Mowbray, D. J.; Skolnick, M. S.; Groom, K. M.; Steer, M. J.; Liu, H. Y.; Hopkinson, M. Phys. Rev. B 2004, 70, 193303.

Gerardot07Gerardot, B. D.; Seidl, S.; Dalgarno, P. A.; Warburton, R. J.; Granados, D.; Garcia, J. M.; Kowalik, K.; Krebs, O.; Karrai, K.; Badolato, A.; Petroff P. M. Appl. Phys. Lett. 2007, 90, 041101.
Kowalik05Kowalik, K.; Krebs, O.; Lemaître, A.; Laurent, S.; Senellart, P.; Voisin, P.; Gaj, J. A.  Appl. Phys. Lett. 2005, 85, 041907.
Vogel07Vogel, M. M.; Ulrich, S. M.; Hafenbrak, R.; Michler, P.; Wang, L.; Rastelli, A.; Schmidt, O. G. Appl. Phys. Lett. 2007, 91, 051904.
Ghali12 Ghali, M.; Ohtani, K.; Ohno, Y.; Ohno, H. Nat. Commun. 2012, 3, 661.
Bennett10Bennett, A. J.; Pooley, M. A.; Stevenson, R. M.; Ward, M. B.; Patel, R. B.; Boyer de la Giroday, A.; Sköld, A.; Farrer, I.; Nicoll, C. A.; Ritchie, D. A.; Shields, A. J. Nature Phys. 2010, 6, 947.
Xiulai1 Tang, J., Cao, S., Cao, Y., Sun, Y., Geng, W, Williams, D. A., Jin, K., Xu, X., Appl. Phys. Lett. 105, 041109 (2014).
Xiulai2 Mar, J. D. and Baumberg, J. J. and Xu, X. L. and Irvine, A. C. and Williams, D. A., Phys. Rev. B 93, 045316 (2016).
Xiulai3 Mar, J. D. and Xu, X. L. and Sandhu, J. S. and Irvine, A. C. and Hopkinson, M. and Williams, D. A.,  Applied Physics Letters, 97, 221108 (2010).
MrowinskiMrowiński, P.; Musiał, A.; Maryński, A.; Syperek, M.; Misiewicz, J.; Somers, A.; Reithmaier, J. P.; Höfling, S.; Sȩk, G. Appl. Phys. Lett. 2015, 106, 053114.
Stevenson06Stevenson, R. M.; Young, R. J.; See, P.; Gevaux, D. G.; Cooper, K.; Atkinson, P.; Farrer, I.; Ritchie, D. A.; Shields, A. J. Phys. Rev. B 2006, 73, 033306.
Puls99Puls, J.; Rabe, M.; Wunsche, H.-J.; Henneberger, F. Phys. Rev. B 1999, 60, 16303.
Stevenson06bStevenson, R. M.; Young, R. J.; Atkinson, P.; Cooper, K.; Ritchie, D. A.; Shields, A. J. Nature 2006, 439, 179.
Kuklewicz12Kuklewicz C. E.; Malein, R. N. E.; Petroff, P. M.; Gerardot, B. D. Nano Lett. 2012, 12, 3761.
Ding10Ding, F.; Singh, R.; Plumhof, J. D.; Zander, T.; Křápek, V.; Chen, Y. H.; Benyoucef, M.; Zwiller, V.; Dörr, K.; Bester, G.; Rastelli, A.; Schmidt, O. G. Phys. Rev. Lett. 2010, 104, 067405.
Plumhof11Plumhof, J. D.; Křápek, V.; Ding, F.;  Jöns, K. D.; Hafenbrak, R.; Klenovský, P.; Herklotz, A.; Dörr, K.; Michler, P.; Rastelli, A.; Schmidt, O. G. Phys. Rev. B 2011, 83, 121302(R).
Rastelli12Rastelli, A.; Ding, F.; Plumhof, J. D.; Kumar, S.; Trotta, R.; Deneke, Ch.; Malachias, A.; Atkinson, P.; Zallo, E.; Zander, T.; Herklotz, A.; Singh, R.; Křápek, V.; Schröter, J. R.; Kiravittaya, S.; Benyoucef, M.; Hafenbrak, R.; Jöns, K. D.; Thurmer, D. J.; Grimm, D.; Bester, G.; Dörr, K.; Michler, P.; Schmidt, O. G. Phys. Status Solidi B 2012, 249, 687.

Trotta12Trotta, R.; Atkinson, P.; Plumhof, J. D.; Zallo, E.; Rezaev, R. O.; Kumar, S.; Baunack, S.; Schröter, J. R.; Rastelli, A.; Schmidt, O. G. Adv. Mater. 2012, 24, 2668.
TrottaTrotta, R.; Zallo, E.; Ortix, C.; Atkinson, P.; Plumhof, J. D.; van den Brink, J.; Rastelli, A.; Schmidt, O. G. Phys. Rev. Lett. 2012, 109, 147401.
Plumhof13Plumhof, J. D.; Trotta, R.; Křápek, V.; Zallo, E.; Atkinson, P.; Kumar, S.; Rastelli, A.; Schmidt, O. G. Phys. Rev. B 2013, 87, 075311.
Kumar2014Kumar, S.; Zallo, E.; Liao, Y. H.; Lin, P. Y.; Trotta, R.; Atkinson, P.; Plumhof, J. D.; Ding, F.; Gerardot, B. D.; Cheng, S. J.; Rastelli, A.; Schmidt, O. G. Phys. Rev. B 2014, 89, 115309.
Trotta14Trotta, R.; Wildmann, J. S.; Zallo, E.; Schmidt, O. G.; Rastelli, A. Nano Lett. 2014, 14, 3439.
Trotta2015Trotta, R.; Martín-Sánchez, J.; Wildmann, J. S.; Piredda, G.; Reindl, M.; Schimpf, C.; Zallo, E.; Stroj, S.; Edlinger, J.; Rastelli, A. Nat. Commun. 2015, 7, 10375.
Hofer17Höfer, B.; Zhang, J.; Wildmann, J.; Zallo, E.; Trotta, R.; Rastelli, A.; Schmidt, O. G. Appl. Phys. Lett. 2017, 110, 151102.
Hafenbrak07Hafenbrak, R.; Ulrich, S. M.; Michler, P.; Wang, L.; Rastelli, A.; Schmidt, O. G. New J. Phys. 2007, 9, 315.

Gong11Gong, M.; Zhang, W.; Guo, G.-C.; He, L. Phys. Rev. Lett. 2011, 106, 227401.
Gong13Sapienza, L.; Malein, R. N. E.; Kuklewicz, C. E.; Kremer, P. E.; Srinivasan, K.; Griffiths, A.; Clarke, E.; Gong, M.; Warburton, R. J.; Gerardot, B. D. Phys. Rev. B 2013, 88, 155330.
WangJP12Wang, J.; Gong, M.; Guo G.-C.; He, L. Appl. Phys. Lett. 2012, 101, 063114.
WangJP15Wang, J.; Gong, M.; Guo G.-C.; He, L. Phys. Rev. Lett. 2015, 115, 067401.
Trotta15Trotta, R.; Martín-Sánchez, J.; Daruka, I.; Ortix, C.; Rastelli, A. Phys. Rev. Lett. 2015, 114, 150502.
ZhangJX2015Zhang, J.; Wildmann, J. S.; Ding, F.; Trotta, R.; Huo, Y.; Zallo, E.; Huber, D.; Rastelli, A.; Schmidt, O. G. Nat. Commun. 2015, 6, 10067.
ChenYan2016Chen, Y.; Zhang, J.; Zopf, M.; Jung, K.; Zhang, Y.; Keil, R.; Ding, F.; Schmidt, O. G. Nat. Commun. 2016, 7, 10387.
ZhangJX2016Keil, R.; Zopf, M.; Chen, Y.; Hoefer, B.; Zhang, J.; Ding, F.; Schmidt, O. G. Nat. Commun. 2016, 8, 15501.
ZhangJX2013Zhang, J.; Ding, F.; Zallo, E.; Trotta, R.; Höfer, B.; Han, L.; Kumar, S.; Huo, Y.; Rastelli, A.; Schmidt, O. G. Nano Lett. 2013, 13, 5808-5813.
HuangH17Huang, H.; Trotta, R.; Huo, Y.; Lettner, T.; Wildmann, J. S.; Martín-Sánchez, J.; Huber, D.; Reindl, M.; Zhang, J.; Zallo, E.; Schmidt, O. G.; Rastelli, A. ACS Photonics 2017, 4, 868.

Koster63Koster,G. F.; Dimmock, J. O.; Wheeler, R. G.; Statz, H. Properties of the Thirty-Two Point Groups (MIT Press, Cambridge, MA, 1963).
Wang99Wang, L. W.; Zunger, A. Phys. Rev. B 1999, 59, 15806.
Wang00Williamson, A. J.; Wang, L. W.; Zunger, A. Phys. Rev. B 2000, 62, 12963.

Keating66Keating, P. N. Phys. Rev. 1966, 145, 637.
Martin70Martin, R. M. Phys. Rev. B 1970, 1, 4005.
Franceschetti99Franceschetti, A.; Fu, H.; Wang, L. W.; Zunger, A. Phys. Rev. B 1999, 60, 1819.
SakuraiSakurai, J. J.; Modern Quantum Mechanics, San Fu Tuan (Benjamin/Cummings, Menlo Park, CA, 1985), p. 276.
Gong14Gong, M.; Hofer, B.; Zallo, E.; Trotta, R.; Luo, J.-W.; Schmidt, O. G.; Zhang, C.  Phys. Rev. B 2014, 89, 205312.
Mukai1996Mukai, K.; Ohtsuka, N.; Shoji, H.; Sugawara, M. Phys. Rev. B 1996, 54, R5243.
Inokuma90Inokuma, T.; Arai, T.; Ishikawa, M. Phys. Rev. B 1990, 42, 11093.
Wilfried2002van Sark, W. G. J. H. M.; Frederix, P. L. T. M.; Bol, A. A.; Gerritsen, H. C.; Meijerink, A. ChemPhysChem 2002, 3, 871.
Singh09Singh, R.; Bester, G. Phys. Rev. Lett. 2009, 103, 063601.
Dup11Dupertuis, M. A.; Karlsson, K. F.; Oberli, D. Y.; Pelucchi, E.; Rudra, A.; Holtz, P. O.; Kapon, E. Phys. Rev. Lett. 2011, 107, 127403.
Kulakovskii1999Kulakovskii, V. D.; Bacher, G.; Weigand, R.; Kümmell, T.; Forchel, A.; Borovitskaya, E.; Leonardi, K.; Hommel, D. Phys. Rev. Lett. 1999, 82, 1780.
Ulrich2003Ulrich, S. M.; Strauf, S.; Michler, P.; Bacher. G.; Forchel, A. Appl. Phys. Lett. 2003, 83, 1848.
Astakhov2006Astakhov, G. V.; Kiessling, T.; Platonov, A. V.; Slobodskyy, T.; Mahapatra, S.; Ossau, W.; Schmidt, G.; Bruuner, K.; Molenkamp, L. W. Phys. Rev. Lett. 2006, 96, 027402.
Dzhioev1998Dzhioev, R. I.; Zakharchenya, B. P.; Ivchenko, E. L.; Korenev, V. L.; Kusraev, Y. G.; Ledentsov, N. N.; Ustinov, V. M.; Zhukov,  A. E.; Tsatsulnikov, A. F. Phys. Solid State 1998, 40, 790.







































