Abstract
Interaction with nuclear spins leads to decoherence and information loss in solidstate electronspin qubits. One particular, ineradicable source of electron decoherence arises from decoherence of the nuclear spin bath, driven by nuclear–nuclear dipolar interactions. Owing to its manybody nature nuclear decoherence is difficult to predict, especially for an important class of strained nanostructures where nuclear quadrupolar effects have a significant but largely unknown impact. Here, we report direct measurement of nuclear spin bath coherence in individual selfassembled InGaAs/GaAs quantum dots: spinecho coherence times in the range 1.2–4.5 ms are found. Based on these values, we demonstrate that straininduced quadrupolar interactions make nuclear spin fluctuations much slower compared with latticematched GaAs/AlGaAs structures. Our findings demonstrate that quadrupolar effects can potentially be used to engineer optically active IIIV semiconductor spinqubits with a nearly noisefree nuclear spin bath, previously achievable only in nuclear spin0 semiconductors, where qubit network interconnection and scaling are challenging.
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Introduction
Quantum dots (QDs) in IIIV semiconductors have many favourable properties for applications in quantum information processing^{1,2,3,4}. Selfassembled dots are particularly promising because of their strong interaction with light offering excellent optical interfacing, manipulation at ultrafast speeds and advanced manufacturing technology^{5,6,7,8}. However, all atoms of groups III and V have nonzero nuclear magnetic moments. Thus, instead of an ideal twolevel quantum system, the spin of a single electron in a QD is described by the socalled ‘central spin’ problem^{9,10,11,12}, where the electron (central) spin is subject to magnetic interaction with an ensemble of 10^{4}–10^{6} nuclear spins. This hyperfine interaction results in decoherence, that is, decay of the phase information encoded in electron spin^{1,2,3,4,5,6,7,8,9,10,11,12}.
Hyperfineinduced decoherence can be greatly reduced by applying static magnetic field and refocusing control pulses inducing electron spin echo. With this technique, very long electron qubit coherence times of ~200 μs were demonstrated in latticematched GaAs/AlGaAs QDs^{1}. However, the effect of nuclei can not be eliminated completely because of the presence of nuclear–nuclear (dipole–dipole) magnetic interactions, which cause spin exchange flipflops of nuclei, that is, nuclear spin bath decoherence. Such flipflops induce quasirandom fluctuating magnetic fields acting on electron spin and causing its decoherence (‘spectral diffusion’ process^{10,11}). It is thus evident that understanding the nuclear spin coherence is crucial for predicting the coherence properties of the central spin. Furthermore, it is predicted that large nuclear quadrupolar interactions (QIs) present in strained selfassembled dots^{13} can suppress the nuclear flipflops resulting in extended electron spin coherence^{14}. However, this possibility is little explored^{15}, mainly due to the lack of reliable data on nuclear spin coherence in selfassembled dots.
Here we demonstrate pulsed nuclear magnetic resonance (NMR) of as few as 10^{4}–10^{5} quadrupolar spins in individual inhomogeneously strained InGaAs/GaAs QDs. We probe nuclear coherence by measuring spinecho decay times T_{2}, which are found to be a factor of ~5 longer compared with latticematched (unstrained or homogeneously strained) GaAs/AlGaAs structures—direct evidence of nuclear spin flipflop suppression induced by inhomogeneous QI. We then show that the nuclear flipflop times T_{2,ff} (relevant for electron spin decoherence) are larger than spinecho T_{2} times, but can be estimated using a firstprinciples model^{16,17}. We conclude that the flipflops of nuclei in spin states other than ±1/2 are completely frozen. For ±1/2 spin states, there is a difference between isotopes: while arsenic is frozen, the flipflops of gallium and indium are possible but with T_{2,ff}~5 ms, which is a factor of ~3–8 slower than in latticematched structures.
The unusual behaviour of arsenic is explained by additional inhomogeneous QI arising from random alloy mixing of gallium and indium atoms^{18}. Such atomicscale disorder opens a new prospect for using the excellent properties of III–V QDs to build nuclearspinnoise free solidstate qubits: this can now be done without resorting to materials with zero nuclear spin (for example, isotopically pure ^{28}Si and ^{12}C)^{19,20,21}, which have inferior optical properties, hampering onchip integration of a large number of qubits.
Results
Nuclear quadrupolar effects in selfassembled QDs
Our experiments were performed on individual neutral QDs in InGaAs/GaAs samples, grown by straindriven selfassembly using molecular beam epitaxy. The sample was placed in an optical heliumbath cryostat (T=4.2 K). Magnetic field B_{z} up to 8 T was applied parallel to the sample growth axis (Oz) and light propagation direction (Faraday geometry). The structures were investigated using optically detected NMR techniques, which extend the concepts reported in our recent work^{13,22}. Radiofrequency (rf) fields B_{rf} perpendicular to B_{z} are induced by a minicoil wound around the sample (see further details in Methods, Supplementary Figs 1 and 2, and Supplementary Note 1).
In this work, we study the four most abundant isotopes: ^{69}Ga, ^{71}Ga, ^{75}As (spin I=3/2) and ^{115}In (spin I=9/2), all possessing nonzero quadrupolar moments. The proportion of Ga/In in our dots is estimated as 0.76/0.24 (ref. 13). The energy level diagram of a quadrupolar nuclear spin is shown schematically in Fig. 1a for the case of I=3/2. Magnetic field B_{z} induces shifts proportional to ∝I_{z}, so that all dipoleallowed NMR transitions (ΔI_{z}=±1) appear at the same frequency ν_{Z}. Electric field gradients (described by a secondrank tensor V_{ij}, where V is the electrostatic potential) induce quadrupolar shifts proportional to to first order of perturbation^{23}. The resulting NMR frequency shifts are strongly inhomogeneous and are on the order of few MHz in InGaAs dots^{13,24}. The central transition (CT) −1/2↔+1/2 is an exception, as it is affected by QI only to second order resulting in much smaller shifts on the order of tens to hundreds of kHz (ref. 23). The relatively small linewidths greatly simplify the experiments; thus in what follows we focus on spectroscopy of CTs only. In particular, selective pulsed NMR of CTs can be conveniently implemented by choosing the rf amplitude B_{rf} so that (γ is the nuclear gyromagnetic ratio).
Figure 1b shows CT spectra of ^{75}As and ^{69}Ga measured using continuouswave inverse NMR techniques^{13}. At highfield B_{z}=8 T, the ^{69}Ga resonance consists of a narrow line (fullwidth at halfmaximum=~9 kHz). The arsenic resonance is broader (fullwidth at halfmaximum ~30 kHz) with additional asymmetric sidebands approximately 100 kHz broad. When magnetic field is reduced down to 2 T both resonances broaden and diminish in amplitude, as expected for a lineshape determined by secondorder quadrupolar shifts (ref. 23). The significantly larger broadening of the ^{75}As CT resonance is attributed to random intermixing of the groupIII Ga and In atoms creating additional lowsymmetry electric field gradients at arsenic sites^{13,18}. As we demonstrate below, such random quadrupolar shifts result in pronounced suppression of dipolar nuclear flipflops and extension of nuclear spin coherence times.
Pulsed NMR spectroscopy of quadrupolar nuclei
The quadrupolar broadening of NMR spectra in Fig. 1b is inhomogeneous in character. It obscures the much weaker homogeneous broadening induced by the nuclear–nuclear interactions, which determine the nuclear spin coherence. In order to access the nuclear spin coherence, we use timedomain (pulsed) NMR^{23}: the timing diagram of the pulsed NMR experiment is shown in Fig. 2 (further details on techniques can be found in Methods, Supplementary Figs 1 and 2, and Supplementary Note 1). We start with a Rabi nutation experiment where a single rf pulse of a variable duration τ is applied^{25}. Rabi oscillations of nuclear polarization are clearly seen in Fig. 3a for ^{71}Ga and ^{75}As enabling the calibration of 90° and 180° rotation pulses. The decay of Rabi oscillations is due to dephasing caused by inhomogeneous spectral broadening (Fig. 1b). Such dephasing can be reversed using the Hahn echo sequence 90°−τ_{0}−180°−τ−90°. The result of a measurement with a fixed delay τ_{0}=0.4 ms and a variable τ are shown for ^{75}As in Fig. 3b where as expected a pronounced spin echo is observed at τ=τ_{0}.
We then turn to the spinecho decay measurements (90°−τ−180°−τ−90° pulse sequence) where the evolution times τ before and after the 180° refocusing pulse are varied simultaneously. Figure 3c shows experimentally measured nuclear spinecho amplitudes (symbols) as a function of the total delay time 2τ for ^{71}Ga and ^{75}As isotopes at B_{z}=8 T. Experimental curves are well fitted by a Gaussian decay function (solid lines) with characteristic 1/e decay time T_{2}≈1.18 ms for ^{71}Ga and T_{2}≈4.27 ms for ^{75}As. The spinecho sequence removes the effect of inhomogeneous spectral broadening, with the echo decay caused solely by nuclear–nuclear dipolar interactions^{23}: T_{2} thus characterizes the coherence of the nuclear spin bath. We have repeated spinecho measurements for all four studied isotopes at different magnetic fields B_{z}. The resulting coherence times T_{2} (and corresponding decay rates 1/T_{2}) are plotted in Fig. 4 by the circles. In addition, we have verified the reproducibility of our results by measuring T_{2} of ^{75}As for another six individual dots from the same sample (see Supplementary Fig. 3 and Supplementary Note 2).
Quadrupolar suppression of nuclear spin bath fluctuations
To examine the effect of inhomogeneous QI on the nuclear spin bath coherence, we first compare our experimental T_{2} times with previous nuclear spin echo measurements on latticematched GaAs/AlGaAs quantum wells (QWs) and dots. The data available for ^{75}As (selective echo on CT in QWs^{26,27,28}) and ^{71}Ga (nonselective echo on QWs^{29} and QDs^{25}) are shown in Fig. 4 by the triangles. It can be seen that the echo decay times in inhomogeneously strained selfassembled QDs are a factor of ~5–7 larger compared with unstrained or homogeneously strained latticematched structures. Such increase in T_{2} is due to suppression of nuclear spin flipflops and provides direct evidence for the slow down of nuclear spin bath fluctuations in the presence of spatially inhomogeneous QI.
To quantify the effect of QI on the nuclear spin bath dynamics, we turn to more detailed analysis of our experimental results. At sufficiently large magnetic field B_{z}>>10 mT, the interaction between any two nuclear spins I and J is described by the truncated dipole–dipole Hamiltonian^{23}:
Where Î and Ĵ are spin operators, and the coupling strength ν_{dd} depends on nuclei type and mutual position (ν_{dd}≲200 Hz in frequency units for nearest neighbours in InGaAs, and scales as ∝r^{−3} with internuclear distance r). The (Î_{x}Ĵ_{x}+Î_{y}Ĵ_{y}) term enables spin exchange flipflops between nuclei I and J: (I_{z}, J_{z}) ↔ (I_{z}±1, J_{z}∓1), the process ultimately responsible for electron spin decoherence via spectral diffusion. A flipflop can only happen if I and J are no more than a few unit cells apart (due to ∝r^{−3} scaling of Ĥ_{dd}) and have similar Zeeman energies requiring them to be of the same isotope. If, however, these two nuclei are subject to significantly different quadrupolar shifts ν_{Q,I} and ν_{Q,J}, so that ν_{Q,I}−ν_{Q,J}>>ν_{dd} the flipflops will become energetically forbidden, resulting in a slow down of nuclear spin bath dynamics and potential increase in electron qubit coherence time^{14}.
Despite the very simple structure of the Hamiltonian of equation 1, the calculation of the nuclear spin bath dynamics in a crystal is a very difficult task because of the manybody nature of the problem (each nuclear spin interacts with all other spins). When arbitrary inhomogeneous QI is added, the problem becomes unsolvable in practice. However, for the limiting cases of very small and very large QI, the nuclear spin echo decay times can be calculated with ~25% accuracy from first principles using the method of moments^{16}. The details of the calculation techniques are discussed in the Methods and further in Supplementary Note 3; in what follows, we present the results of these calculations and use them to analyse the experimental data.
When quadrupolar shifts are much smaller than the dipolar interaction (unstrained structures), the nuclear flipflops are not affected by QI. The T_{2} times calculated for that case are shown in Fig. 4 with dashed lines for different isotopes. These calculated values are in good agreement with experiments on latticematched GaAs/AlGaAs structures, confirming the validity of the model employed. We note that the same T_{2} estimates are also valid for homogeneously strained structures^{16,23} (that is, where , but is spatially homogeneous).
In the opposite case of very strong and inhomogeneous QI, the dipole–dipole flipflops become energetically forbidden. This effect can be described by truncating the offdiagonal flipflop term in equation 1, leaving only the diagonal Î_{z}Ĵ_{z} term in the Hamiltonian. However, even for completely suppressed flipflops, the nuclear spinecho coherence time (denoted as T_{2,zz}) remains finite: the Î_{z}Ĵ_{z} term still causes the nuclear spin decoherence (an effect known as instantaneous diffusion^{30}). The T_{2,zz} sets an upper limit on the echo decay time T_{2}. The T_{2,zz} times calculated for the studied InGaAs dots^{16} are shown by the solid lines in Fig. 4.
We now turn to the question of how nuclear spin T_{2} measurements can be used to predict the effect of the nuclear spin bath fluctuations on electron spin coherence. The electron spin decoherence is caused solely by the nuclear flipflops^{9,10,11,12}—the diagonal nuclear–nuclear interaction Î_{z}Ĵ_{z} has no effect on the electron. If the flipflops were completely suppressed, the electron spin would experience only a static nuclear field, which cannot cause any irreversible electron spin decoherence. Thus, in order to predict the electron spin coherence, we need to determine the nuclear flipflop rates. As explained above, the experimental nuclear spinecho T_{2} is controlled by both the nuclear flipflops and the diagonal Î_{z}Ĵ_{z} term of equation 1. To exclude the contribution of the diagonal interaction Î_{z}Ĵ_{z}, we examine how close the experimental T_{2} value is to the calculated limit T_{2,zz}. For that, we introduce a characteristic nuclear spin flipflop time T_{2,ff} defined as
so that in an ideal QD with T_{2,ff} →∞ we would expect infinite electron spin coherence times.
It can be seen in Fig. 4 that T_{2}<T_{2,zz} for In and both Ga isotopes implying only a partial suppression of the nuclear spin flipflops. Using equation 2, we calculate T_{2,ff}~5 ms for all three of those isotopes at low fields (1≲B_{z}≲2 T). This T_{2,ff} is approximately three to eight times larger than it would have been in InGaAs/GaAs structures without inhomogeneous strain (T_{2,ff} values for this case are calculated to be T_{2,ff}≈0.6, 1.5, 1.1 ms for ^{115}In, ^{69}Ga and ^{71}Ga, see details in Supplementary Table 1 and Supplementary Note 3). We also note that the T_{2} of ^{71}Ga and ^{115}In decreases with increasing B_{z} in agreement with the fact that secondorder quadrupolar shifts depend on magnetic field as (ref. 23), so that large B_{z} reenables the nuclear flipflops between the I_{z}=±1/2 spin states. By contrast, no significant trend is observed for ^{69}Ga most likely due to its smaller gyromagnetic ratio and larger quadrupolar moment making T_{2} less dependent on B_{z}.
A very different picture is observed in Fig. 4 for arsenic nuclei: the values of T_{2} measured at B_{z}=2–8 T coincide with the calculated T_{2,zz} within the experimental error. The value of T_{2,ff} calculated according to equation 2 in that case diverges becoming infinitely large: we can conclude that T_{2,ff}>>5 ms for arsenic, implying very strong flipflop suppression. These findings are consistent with the spectroscopic data in Fig. 1b, where the CT spectra of ^{75}As are found to be ~10 times broader than for gallium nuclei, revealing much larger secondorder quadrupolar shifts of arsenic nuclei, which are responsible for the strong suppression of the nuclear spin exchange flipflops.
Discussion
Secondorder quadrupolar shifts appear whenever V_{ij} is not a cylindrically symmetric tensor with its main axis along B_{z} (ref. 23). One obvious reason for lowsymmetry V_{ij} in selfassembled QDs is nonuniaxial symmetry of the elastic strain tensor, or deviation of the strain main axis from B_{z}. Such a mechanism is likely to be the main cause of the CT inhomogeneous broadening of gallium and indium, resulting in the above increase in T_{2,ff} by a factor of ~3–8.
For the anion ^{75}As, the picture is different: the additional secondorder shifts are induced by random alloy mixing of the cationic Ga and In atoms. Each arsenic nucleus has four nearest neighbours, and unless all of them are of the same type (all gallium or all indium) a nonzero V_{ij} will appear^{18}. Furthermore, unlike the elastic strain fields that change gradually over many crystal unit cells, the configuration of the neighbouring atoms is random, so that even the nearest arsenic nuclei (which have the strongest dipolar coupling) can have very different . Such compositional disorder induces large spatially inhomogeneous CT frequency shifts, drastically suppressing the flipflops. We propose that such effects can be used to engineer QDs with a frozen nuclear spin bath. One possible approach is to substitute some of the arsenic nuclei with antimony and/or phosphorus: in such InGaAsSb(P) QDs gallium and indium spins will also experience large inhomogeneous shifts because of atomicscale alloy disorder, resulting in an overall slow down of the flipflops for all isotopes.
The increase of the nuclear spinecho T_{2} observed for the CTs is driven by the secondorder quadrupolar shifts , which are significantly smaller than the firstorder shifts ( compared with ). Therefore, we conclude that the flipflops of the nuclei in I_{z}>1/2 states (affected by ) are effectively frozen for all isotopes in strained selfassembled InGaAs dots. Consequently, electron spin decoherence in selfassembled dots is caused solely by the flipflops of the nuclei in I_{z}=±1/2 states.
As we have shown the nuclear T_{2} time increases when magnetic field is reduced down to B_{z}=2 T. We expect this trend to continue down to magnetic fields where the nuclear Zeeman frequency ν_{Z} becomes comparable to the firstorder quadrupolar shifts . For ^{75}As, we have and ν_{Z}/B_{z}≈7.33 MHz T^{−1}, whereas other isotopes have even larger ν_{Z}/B_{z} and smaller . Thus, our results are expected to be valid at least down to B_{z}≈1 T. At lower magnetic fields, the nuclear spin energy levels cross^{31}, which may significantly alter the nuclear spin flipflop dynamics—this regime requires further experimental investigation.
Finding an exact relation between the nuclear spin bath coherence times and the central spin coherence times is a complicated problem, in particular when QIs are involved^{15}. However, some general qualitative conclusions can be readily drawn. First, it has been shown that for QDs the spectral diffusion is well into the ‘slow bath’ regime (as opposed to ‘motional narrowing’)^{10}, thus the extended nuclear spin coherence times reported here are expected to result in longer central spin coherence times. Second, dynamic nuclear spin polarization^{32}, which enhances the occupancy of the I_{z}≈I states can be used to effectively depopulate and ‘dilute’ the I_{z}=±1/2 states resulting in further reduction of the nuclear spin fluctuations. Recently, electron spin coherence times >200 μs in latticematched electrostatic GaAs/AlGaAs QDs^{1} were observed. The selfassembled InGaAs/GaAs dots typically have a factor of ~30 fewer nuclear spins and hence a factor of ~5 stronger fluctuations. This is compensated by a factor of ~5 slowdown of nuclear flipflops observed in this work. Thus, we expect that electron coherence times of the similar scale (~100 μs) could be achieved in selfassembled dots, if other sources of decoherence such as charge noise and interaction with phonons^{5,6} can be eliminated.
In conclusion, we have demonstrated the first direct probing of the coherent nuclear spin bath dynamics in inhomogeneously strained QDs. We anticipate that electron(hole) spin qubits in selfassembled structures exhibiting large inhomogeneous QIs^{14,15} have a significant advantage over the latticematched counterparts^{1}. As an outlook, we note that pulsed NMR techniques employed here to study direct nuclear–nuclear interactions in neutral dots can be readily applied to charged dots. This will provide insight into coherent nuclear spin dynamics in the presence of the Knight field^{33} and indirect electron(hole)mediated nuclear–nuclear spin interactions, which were previously shown to be significant for longitudinal nuclear spin relaxation (with T_{1}~100 s)^{34}. Furthermore, our NMR techniques are not restricted to spinecho and can be easily extended to accommodate the whole variety of pulse sequences used in Fourier transform NMR, offering a powerful tool to explore the manybody physics of interacting nuclear spins in strained nanostructures.
Methods
QD sample structure
The InGaAs/GaAs sample consists of a single layer of nominally InAs QDs placed within a microcavity structure, which is used to select and enhance the photoluminescence from part of the inhomogeneous distribution of QD energies. The sample was grown by molecular beam epitaxy. The QDs were formed by deposition of 1.85 monolayers of InAs—just above that required for the nucleation of dots. As a result, we obtain a low density of QDs at the postnucleation stage. The cavity is formed between an asymmetric set of distributed Bragg reflector pairs, which uses 16 pairs of GaAs/Al_{0.8}Ga_{0.2}As below and 6 pairs above the cavity. The cavity Q factor is ~250 and the cavity has a low temperature resonant wavelength at around 920 nm. The luminescence of the QDs is further enhanced by a shortperiod GaAs/AlAs superlattice surrounding the QD layer.
Continuous wave NMR spectroscopy
The CT spectra of an individual InGaAs/GaAs QD shown in Fig. 1b were measured using inverse method which provides >8 times CT signal enhancement for I=3/2 nuclei^{13}. The NMR signal is calculated as the hyperfine shift of the QD Zeeman doublet divided by the spectral gap width, so that the values on the vertical scale give the spectral density of the distribution of the nuclear resonance frequencies. The spectral gap width (determining the spectral resolution) is 6 kHz for the ^{69}Ga spectra, and 16 kHz (32 kHz) for the B_{z}=8 T (B_{z}=2 T) spectrum of ^{75}As. For convenience, the spectra are plotted as a function of Δν=ν−ν_{Z}, where ν_{Z} is a constant proportional to the isotope gyromagnetic ratio: ν_{Z}/B_{z}≈7.33 MHz T^{−1} for ^{75}As and ν_{Z}/B_{z}≈10.3 MHz T^{−1} for ^{69}Ga.
Techniques for pulsed NMR measurements
We implement optically detected pulsed NMR techniques, which extend the techniques and are based on the results of our previous work of ref. 13. The timing diagram of one measurement cycle and the changes to nuclear spin polarization are shown schematically in Fig. 2 (spin I=3/2 is used as an example). The cycle starts with optical nuclear spin pumping^{22,32} using highpower σ^{+} circularly polarized laser [stage (a)]. Spin polarized electrons of highly excited and/or multiexcitonic states transfer their polarization to nuclear spins via the hyperfine interaction^{35}. The pump duration is chosen long enough (~3–7 s depending on magnetic field B_{z}) to achieve the steadystate nuclear spin polarization. Nuclear polarization degrees exceeding 50% are obtained, which means that a large portion of the nuclei is initialized into the I_{z}=−3/2 state. To make the NMR signal of the CT detectable, the population of the I_{z}=−1/2(+1/2) state must be maximized (minimized). This is done at stage (b) using population transfer technique^{36}: an rf field containing two frequency components is applied, the frequencies are swept over both satellite transition bands −3/2↔−1/2 and +1/2↔+3/2 resulting in adiabatic inversion of the populations of the −3/2 and −1/2 states as well as +1/2 and +3/2 states. Following that a sequence of rf pulses resonant with the CT is applied (stage (c)). Different sequences can be implemented, depending on the experiment: a single pulse of a variable duration is used for Rabioscillation measurements (Fig. 3a), whereas a threepulse sequence is used to measure either the spinecho (Fig. 3b, 90°−τ_{0}−180°−τ−90° sequence with τ_{0} fixed to 0.4 ms) or spinecho decay (Fig. 3c, 90°−τ−180°−τ−90° sequence). The rf amplitude is chosen to give 90° phase rotation for 3 to 8μslong pulses (depending on isotope). This corresponds to pulse bandwidths of ~100 kHz, and, as satellite transitions are shifted by the much bigger (~1–10 MHz) firstorder quadrupolar shifts, this ensures selective excitation of the CT. Finally (stage (d)), we probe the effect of the NMR pulse sequence by measuring the changes in the average nuclear spin polarization ‹I_{z}› on the single QD. This is achieved by exciting the dot with a short (~1–4 ms depending on B_{z}) probe laser pulse and measuring the hyperfine shifts in the QD photoluminescence spectrum^{13,22,32}. To improve the signaltonoise ratio, the experimental cycle is repeated 20–50 times for each parameter value (for example, for each value of 2τ in Fig. 3c). Further details of experimental techniques can be found in Supplementary Note 1.
Theoretical model
Nuclear spin decoherence is a result of nuclear–nuclear spin interactions: each individual nuclear spin has its own spin environment producing additional magnetic field, which changes the resonant frequency of that nucleus. Thus, the problem of calculating the nuclear spin decoherence is equivalent to the problem of calculating homogeneous NMR line broadening. In principle, this problem can be solved by diagonalizing the Hamiltonian of the nuclear–nuclear interactions. This, however, is practically impossible even for a system of few tens of spins, let alone the whole crystal. An insightful solution to this difficulty has been found by Van Vleck^{17,23}, who showed that the moments of the NMR lineshape can be expressed as traces of certain quantum mechanical operators. The key property of the trace is that it can be calculated in any wavefunction basis, hence diagonalization of the Hamiltonian is not needed. This technique does not allow an exact resonance lineshape to be calculated, but in most cases the second moment M_{2} (corresponding to the homogeneous NMR linewidth) contains sufficient information.
The nuclear spin coherence time can be estimated as . The calculation of M_{2} for a whole crystal is a straightforward but very tedious process involving summation of various matrix elements. If one wants to calculate the spinecho coherence time (as opposed to the freeinduction decoherence time), some of the matrix elements must be discarded from the summation. QI is also taken into account by further truncation of the sums. The details of these calculations can be found in refs 16, 17, 23 and are also outlined in Supplementary Note 3.
Additional information
How to cite this article: Chekhovich, E. A. et al. Suppression of nuclear spin bath fluctuations in selfassembled quantum dots induced by inhomogeneous strain. Nat. Commun. 6:6348 doi: 10.1038/ncomms7348 (2015).
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Acknowledgements
We are grateful to G. Burkard, E. Welander, L. Cywinski and K. V. Kavokin for fruitful discussion. This work has been supported by the EPSRC Programme Grant EP/J007544/1, ITN S^{3}NANO and the Royal Society. E.A.C. was supported by a University of Sheffield ViceChancellor's Fellowship.
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M.H. developed and grew the samples. E.A.C. developed the techniques, carried out the experiments and analysed the data. E.A.C., M.S.S. and A.I.T. wrote the manuscript.
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Supplementary Figures 13, Supplementary Table 1, Supplementary Notes 13, and Supplementary References (PDF 110 kb)
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Chekhovich, E., Hopkinson, M., Skolnick, M. et al. Suppression of nuclear spin bath fluctuations in selfassembled quantum dots induced by inhomogeneous strain. Nat Commun 6, 6348 (2015). https://doi.org/10.1038/ncomms7348
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DOI: https://doi.org/10.1038/ncomms7348
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