Quantum dot spin coherence governed by a strained nuclear environment

The interaction between a confined electron and the nuclei of an optically active quantum dot provides a uniquely rich manifestation of the central spin problem. Coherent qubit control combines with an ultrafast spin–photon interface to make these confined spins attractive candidates for quantum optical networks. Reaching the full potential of spin coherence has been hindered by the lack of knowledge of the key irreversible environment dynamics. Through all-optical Hahn echo decoupling we now recover the intrinsic coherence time set by the interaction with the inhomogeneously strained nuclear bath. The high-frequency nuclear dynamics are directly imprinted on the electron spin coherence, resulting in a dramatic jump of coherence times from few tens of nanoseconds to the microsecond regime between 2 and 3 T magnetic field and an exponential decay of coherence at high fields. These results reveal spin coherence can be improved by applying large magnetic fields and reducing strain inhomogeneity.

Parameters describing the quadrupolar broadening of nuclear spin species for an indium concentration of ∼ . . Included are the biaxiality parameter (η), the quadrupolar energy (νQ) and the width of its distribution (σνQ), the tilt of the quadrupolar axis (θk) and its corresponding width (σθk).

Supplementary Note 1: Free induction decay
To find the timescale of the free induction decay (cf. Fig. 2 in the main text) and determine its functional form, we employ a moving Fourier transform method. The data from

Supplementary Note 2: Suppression of bath polarisation in Hahn echo
We employ the same alternating pulse sequence used for free induction decay measurements to measure the Hahn echo of the electron spin. This is necessary to cancel the phase dependence in the average signal and suppress the build-up of nuclear spin polarisation.
In order to show this, we repeat the spin-echo measurement, however this time without flipping the initial spin state every second measurement. The comparison of the two methods is

Supplementary Note 3: Hahn-echo modelling
We calculate the evolution of the electron spin Hahn echo under the assumption that the Overhauser field can be treated as a classical effective field. The Hahn echo is calculated by introducing a noise spectrum and a Hahn-echo filter function following the approach taken in Cywinski et al. 1 . The noise spectrum determining the electron decoherence in our quantum dot is governed by the Larmor precession of the nuclear spin bath which is distorted and broadened by inhomogeneous strain-induced quadrupolar fields. In order to derive a noise spectrum for the effective Overhauser field we calculate the quantum evolution of a nucleus under a Hamiltonian which contains Zeeman and quadrupolar terms. We then sum over an ensemble of quadrupolar configurations and the different nuclear isotopes. The model is described in detail below.

Free induction decay and Overhauser field variance
In the free induction decay (FID) measurement presented Fig. 2 of the main manuscript, we observe the well-understood inhomogeneous dephasing ( FID ) of the elecstron spin due to slowly varying nuclear field fluctuations OH . This inhomogeneous dephasing can be related to the Overhauser field's standard deviation ( OH ): where FID is taken at the 1/ point of the Gaussian FID decay. From FID = 1.93 ns we estimate that the Overhauser field's standard deviation is 33 mT which is typical for this system.
The Overhauser field variance can be related to a number of nuclei ( ) interacting with the electron spin. This number is arbitrary in the sense that it depends on a choice for the electronic envelope wave-function, however it provides a synthetic way of normalizing the spectra in the model and will therefore be used later on. Throughout this work, we assume two atoms per unit cell (volume 0 ) and a flat electronic probability density in the quantum dot (| e | 2 = 2/( 0 )). If we neglect correlations between nuclei the variance of nuclear fluctuations becomes with , , representing the concentration, the hyperfine coupling constant and the spin of the nuclear isotope k. The values we take for , , are presented in Supplementary Table   1.

Nuclear precession in the presence of quadrupolar interactions
If we neglect pairwise interactions between nuclei and hyperfine coupling to the electron spin the evolution of a single nucleus is simply given by the Hamiltonian Here, ( , , ) is the coordinate set in the laboratory frame and ( ′, ′, ′) denotes the frame of the quadrupolar field, which is specific to the location of the nucleus " ". We reference the orientation of the quadrupolar frame relative to the lab using the Euler angles ( , , ). is the g-factor of the nucleus, is the nuclear magneton, is the external magnetic field, Q is the characteristic energy for the quadrupolar interaction the location of the nucleus " " and  is the biaxiality parameter.

Nuclear Spin correlator
The spin-correlator for the nucleus " ", along the axis = , , reads: If we take the infinite temperature density matrix = (2 + 1) ⁄ , and if we consider the eigenvalues and eigenvectors {ℏ , | ⟩} of nuc , the spin-correlator can formally be written: We emphasize that " " designates a an agreement between the spin-echo data at long delays and the model requires a bin-size smaller than 1 kHz which constitutes an upper bound for the quasi-static noise frequency spread. Physically, this contribution describes the 'frozen' nuclear spin bath evolving on very slow timescales due to nuclear-nuclear interactions, and is the main cause for the electron FID discussed earlier.
We then generalise the single spin description to a correlator for the nuclear isotope " ", crystal axis: The spin-correlator taken at = 0 is related to the variance of the nuclear field: This relation will be used to weight the noise power spectra introduced in the next paragraph.

Noise spectrum due to nuclear precession
Due to the large difference between electron and nuclei Zeeman energies, their dynamics can be decoupled, and to first order, the electron precesses around a total magnetic field tot = ext + OH . This semi-classical picture, originally introduced in Ref. 2, predicts a modification of the electronic level splitting by the Overhauser field following: The noise spectrum which affects the electron coherence can be decomposed into three power spectral densities: ∥ 0 ( ) which is a delta function at 0-frequency, with an amplitude controlled by the constant terms in Equation (5) where ( ) represents the Fourier transform of the spin-correlator for the species k introduced earlier: Using equations (5)-(9) and the fact that ∫ − ( ) = ( ( − ) + ( + )), we obtain: From equation (11), we see that the frequency components in the noise spectrum ∥ correspond to the Larmor frequencies of the different nuclear species. The amplitude of this noise depends on the matrix element ⟨ | | ⟩. In the absence of quadrupolar coupling, this matrix element is null for ≠ as commutes with the Hamiltonian ̂n uc . It is only the presence of quadrupolar coupling, along an axis which differs from the magnetic field that allows ∥ ( ) to differ from 0. The corresponding physical picture is that the quadrupolar field adds to the external field to create a precession which is tilted from ext . As this simple picture suggests, the amplitude of the first-order coupling will decrease with increasing magnetic field, as the quadrupolar term becomes negligible compared to the Zeeman term.
We note, however, that a different mechanism of first order coupling has recently been reported by Botzem et al. in gate-defined quantum dots 2 . In this case, the coupling is due to an anisotropy of the electron g-factor. Interestingly, this leads to a noise amplitude which is independent from the external B-field.
To calculate ⊥ ( ), it is convenient to use a convolution in the spectral domain. We find: The quadratic coupling to the transverse Overhauser field results in an auto-convolution in the spectral domain. This causes the broad-frequency shoulder around zero frequency that is so detrimental to the electron spin coherence.

Delay-dependence of the Hahn echo and the FID
Once the noise spectrum has been calculated, the Hahn echo or the FID can be computed using the filter function corresponding to the echo pulse sequence. For example, this approach is presented in detail in Ref.
where 0 is the maximum visibility, limited by pulse imperfections, and the function ( ) is expressed as a function of the total noise spectrum tot ( ): and

Model parameters
We base our mean-field calculation of the nuclear spin spectra, and hence the electron Hahn echo, on mean parameters found in Bulutay's atomistic calculation 4,5 . The indium concentration determines overall strain characteristics, affecting all relevant parameters. We perform a least-mean-squared analysis of the calculated spin echo to our data and find best agreement for an indium concentration between 0.4 and 0.7. All calculations in this paper use = 0.5 and the parameters in Supplementary Table 2.
We note a sum of two Gaussian distributions is used to describe . The relevant weights are 1 = 0.55 and 1 = 0.45. Further, to account for a mismatch of the direction of the applied magnetic field and the sample edges (along the [110] and [11 ̅ 0] crystal directions) the angle is offset by 10 ° from zero.

Spectra of nuclear spin species
Supplementary Figure 3 highlights the composition of the perpendicular and parallel spin spectra in terms of the four nuclear species for two magnetic fields. Indium plays the dominant part due to its large spin quantum number, followed by arsenic, which contributes the majority of atoms to the quantum dot. Gallium nuclei play a negligible part overall.
Sub-kHz narrow linewidths have been found in recent nuclear magnetic resonance measurements on individual neutral quantum dots 6 . While we expect significantly broadened linewidths for charged quantum dots due to hyperfine-mediated nuclear spin-spin interaction we find a negligible effect on the nuclear spectra for linewidths below 100 kHz, where quadrupolar broadening masks any effects of nuclear spin coherence.

Effect of linear and quadratic coupling of on spin coherence and magnetic field dependence
To provide some intuition on the role of the first-and second-order coupling to the nuclear spin dynamics we show deconstructed spin echo functions from Fig. 3, main text, in Supplementary Figure 4. Here, the two terms in the noise spectrum have been used individually to compute ( ) as input to the decoherence function ( ).
As outlined in the main text, the linearly coupled parallel component of is almost entirely responsible for the modulation of the spin echo signal. The collapse and revival of the spin coherence is strongly damped due to the complexity of the nuclear spin spectrum, which arises from both magnitude and inhomogeneity of the nuclear quadrupolar interaction.
The perpendicular component of couples to second order to the electron spin and provides a mostly smooth envelope to the echo function.
These Hahn-echo fitting coefficients correspond to an Overhauser field standard deviation ( ( OH )) of 40 mT for ∥ ( ) and 28 mT for ⊥ ( ). This is to be compared with a 33 mT standard deviation for ∥ 0 ( ) that we used to reproduce the FID decay with the same spectra but with a different filter function. The functional form of the decay approaches a −( 2 ⁄ ) 4 dependence as the magnetic field tends to zero. Interestingly, Press et al. reported a very similar magnetic field dependence in the range between 2 and 4 T 7 , followed by a saturation of coherence at higher fields. Lifetime limitations are expected to play a role only at the 10-T level and beyond 8,9 .
We do find the Hahn echo coherence at high magnetic fields to be very sensitive to experimental imperfections. Imperfect suppression of the readout laser during the long free evolution of the electron spin, for example, causes spin pumping which is equivalent to a 1 decay.
In comparison to the Hahn-echo data in Refs. 7 and 10, the pickup of spin coherence in our quantum dot sample takes place at a higher magnetic field and consequently coherence times at low and intermediate fields are shorter. In particular, in the low-field limit coherence is lost within ~ 20 ns in our case, while a similar measurement gives ~ 30 ns in Ref. 10. This points to a higher indium concentration and consequently a larger strain in our sample. We note that the Hahn-echo amplitude is rescaled in Refs. 7 and 10 while we use the measured fringe visibility. The normalisation has to be taken into account for a quantitative comparison.
Drawing quantitative conclusions about the structural composition of the quantum dots in Refs. 7 and 10 based on our measurements alone is challenging due to the complexity of the nuclear spin bath. This may be possible with nuclear bath spectra calculated from atomistic models, such as the ones of Bulutay 4,5 . With this caveat in mind, our model predicts that by reducing the indium content slightly (x=0.4), accompanied with the expected reduction in quadrupolar field strength (following Bulutay, Ref. 4) and an equivalent reduction in the inhomogeneity, the transition to high spin coherence takes place at a magnetic field of ~ 2 T instead of 3 T, and spin coherence at 4 T increases from ~1 μs to 1.5 μs. This is in line with the qualitative expectation that lower and more homogeneous strain reduces the effects of the second-order contributions of the hyperfine interaction.

Amplitude of Hahn-echo measurements
The amplitude in our Hahn-echo measurements corresponds to the visibility in the readout signal oscillations as the delay of the central pulse is scanned over several Larmor periods.
The sub-unity visibility at short storage times T is due to a non-equatorial rotation axis which prevents a complete spin inversion with the -pulse. A good estimate of the inversion fidelity can be found with the pulse scheme used in Fig. 3 (a, b) of the main text: the ratio of the readout visibility for the second half, where the electron spin is (imperfectly) inverted before the next /2 -pulse and the readout visibility obtained for the first half of the pulse