Abstract
Combining external control with long spin lifetime and coherence is a key challenge for solid state spin qubits. Tunnel coupling with electron Fermi reservoir provides robust charge state control in semiconductor quantum dots, but results in undesired relaxation of electron and nuclear spins through mechanisms that lack complete understanding. Here, we unravel the contributions of tunnellingassisted and phononassisted spin relaxation mechanisms by systematically adjusting the tunnelling coupling in a wide range, including the limit of an isolated quantum dot. These experiments reveal fundamental limits and tradeoffs of quantum dot spin dynamics: while reduced tunnelling can be used to achieve electron spin qubit lifetimes exceeding 1 s, the optical spin initialisation fidelity is reduced below 80%, limited by Auger recombination. Comprehensive understanding of electronnuclear spin relaxation attained here provides a roadmap for design of the optimal operating conditions in quantum dot spin qubits.
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Introduction
Semiconductor quantum dots (QDs) offer excellent quantum optical properties and welldefined quantum states of individual spins—an attractive combination for quantum information processing devices^{1}. Recent proofofconcept demonstrations with QDs include heralded entanglement of two remote spins^{2}, generation of photonic cluster states^{3}, spincontrolled photon switching^{4} as well as implementation of electron–nuclear quantum spin interfaces^{5} and nuclear spin quantum computing^{6}. The stability of the spin states, measured by their lifetimes, is crucial in all these applications.
Quantum dot is described by a central spin of a single charge (electron or hole) coupled to N ≈ 10^{3}−10^{5} nuclear spins via hyperfine interaction^{7}. The lack of translational motion combined with the mismatch in electron and nuclear spin energies suppresses relaxation^{8}, providing long spin lifetimes required for spin qubits. However, thorough understanding of spin relaxation is complicated by the multitude and complexity of the residual environment couplings, which include electron–phonon interactions^{9,10,11,12}, quadrupolar coupling of nuclear spins to strain^{13}, nuclear spin diffusion, and electron cotunneling^{14,15} arising from proximity of the Fermi reservoir. Moreover, impurity charge traps^{16,17} adjacent to QDs degrade spin qubit lifetimes. Thus, it remains an open question to establish the maximum (intrinsic) spin lifetimes that can be achieved at any given magnetic field and temperature, as opposed to spin relaxation arising from QD device design and imperfections.
Phononassisted electron spin relaxation enabled by spin–orbit interaction is a dominant mechanism^{9,10,18} at high magnetic field B_{z} ≳ 2 T, but the limit to electron spin lifetime T_{1,e} at low fields remains unexplored. In case of nuclear spins, cotunnelingmediated relaxation was identified as dominant mechanism^{15}, while direct verification is lacking, since bias control of cotunneling is restricted to a narrow range compatible with singleelectron QD state.
Here we study a series of structures where electron cotunneling is controlled directly by the thickness t_{B} of the tunnel barrier separating the dot from the Fermi reservoir. We find that at B_{z} ≳ 2 T and temperatures θ > 4.2 K nuclear spin relaxation is dominated by a higherorder process assisted by phonons^{19,20} and noncollinear hyperfine interaction^{13}, rather than by cotunneling, which is dominant only at low fields B_{z} ≲ 2 T. Electron spin lifetimes exceeding T_{1,e} > 1 s are found at B_{z} ≈ 0.4 T, with a fundamental maximum T_{1,e} ≈ 20 s estimated for an isolated dot at θ = 4.2 K, bounded by phonon relaxation and direct hyperfine interaction at high and low magnetic fields, respectively. While coupling to Fermi reservoir degrades T_{1,e}, it is shown to play a crucial role in counteracting Auger recombination^{21} and enabling electron spin initialisation with nearunity fidelity^{22}.
Results
Quantum dot structures and experimental techniques
Figure 1a sketches conduction band energy profile which is controlled with external bias V_{S} to tune an InAs QD into Coulomb blockade regime, where it is charged deterministically^{23} with a single electron (1e). The trion state with two electrons and one hole can be accessed through resonant optical excitation, and observed in resonance fluorescence (ResFl). Magnetic field B_{z} along the sample growth axis splits the electron spinup \(\left\uparrow \right\rangle\) and spindown \(\left\downarrow \right\rangle\) energies (Fig. 1b), enabling selective excitation of the optically allowed transition between \(\left\uparrow \right\rangle\) electron and the trion \(\left\uparrow \downarrow \Uparrow \right\rangle\) with a spinup hole \(\left\Uparrow \right\rangle\). Weak recombination enabled by hyperfine and heavy–light hole mixing β ≪ 1 can ‘shelve’ the dot^{14} into \(\left\downarrow \right\rangle\) state, quenching ResFl intensity I_{ResFl} until electron returns to \(\left\uparrow \right\rangle\) through a spin flip with rate ξ_{↑↓}. Such shelving provides an efficient way both for initialisation and readout of the electron spin^{14,22}. Furthermore, hyperfine interaction \({\hat{{\mathcal{H}}}}_{{\rm{hf}}}\propto (\hat{{\bf{I}}}\hat{{\bf{s}}})\) of electron spin s with nuclear spins I enables electron–nuclear flip–flops, so that repeated electron spin initialisation creates a net nuclear spin polarisation P_{N}, which can be monitored through optically measured hyperfine shifts E_{hf} in the splitting of the \(\left\uparrow \right\rangle\) and \(\left\downarrow \right\rangle\) states^{7,17}.
Electron and nuclear spin dynamics are measured using a pumpdelayprobe protocol (Fig. 1c). The decay of optically pumped electron [nuclear] spin polarisation over dark period T_{Dark} is probed by measuring I_{ResFl} [E_{hf}]. Figure 1d shows an example of timeresolved ResFl, which is used to derive the residual electron spin polarisation P_{e} after a delay T_{Dark} (see the “Methods” section). Measurements of P_{e} at different T_{Dark} reveal electron spin relaxation (symbols in Fig. 1e), while examples of nuclear spin relaxation E_{hf}(T_{Dark}) are shown in Fig. 1f. Fitting (lines in Fig. 1e, f) is used to derive the intrinsic spinrelaxation rates of electron Γ_{e} = 1/T_{1,e} = 2ξ_{↑↓} and nuclei Γ_{N} = 1/T_{1,N}.
Effect of Auger recombination on spin initialisation
We make a systematic comparison of spin dynamics in a thinbarrier sample (t_{B} = 37 nm, Fig. 2a–c), similar to structures used previously^{10,14,15}, and a thickbarrier structure (t_{B} = 52 nm, Fig. 2d–f), approximating a QD isolated from the Fermi reservoir. Examining the bias dependence of continuous excitation resonance flourescence intensity I_{ResFl} in a thinbarrier sample, we observe a 1e plateau at B_{z} = 0 T (triangles in Fig. 2a), while at high B_{z} (circles and squares in Fig. 2a) ResFl is strongly suppressed, indicating spin ‘shelving’^{10,14}. A striking difference is observed in a thickbarrier sample (Fig. 2d), where fluorescence intensity and spin ‘shelving’ contrast are reduced (Fig. 2a), which may at first suggest the lack of electron spin initialisation. However, this is ruled out by timeresolved ResFl (e.g. Fig. 1d), which reveals spinpumping fluorescence pulses of similar intensity for all t_{B}. We ascribe the reduction in I_{ResFl} and the peculiar twostage electron spin decay in a thickbarrier sample (squares in Fig. 1e) to the Auger process^{21,24}, where electron–hole recombination ejects the second electron with a rate γ_{A} (Fig. 1b). Following the ejection, an empty QD does not contribute to ResFl, hence P_{e} ≈ 1 observed initially. During T_{Dark} an unpolarised electron can return from the Fermi reservoir with recharging rate r, giving rise to the fast component of the P_{e}(T_{Dark}) decay (squares in Fig. 1e at T_{Dark} < 100 μs), whereas the slow component corresponds to resident electron spin relaxation with rate Γ_{e}.
Using rate equation modelling (see details in the “Methods” section) of the fourlevel system shown in Fig. 1b, we find good description of the experiments (dashed line Fig. 1e) and derive r ≈ 1.26 × 10^{5} s^{−1}, \({\varGamma }_{{\rm{e}}}={T}_{1,{\rm{e}}}^{1}\approx 3.3\) s^{−1}. Importantly, the level P_{e} ≈ 0.77 reached after dot recharging (T_{Dark} ≈ 100 μs) gives a direct measure of the electron spin initialisation fidelity in a thickbarrier sample, revealing the fundamental limitations arising in an isolated (t_{B} → ∞) QD. In a thinbarrier sample, Auger recombination is counteracted by fast recharging: the resulting spin initialisation fidelity is higher, but can never reach unity. The maximum spin initialisation fidelity is an algebraic function of QD properties such as relaxation rates and heavylight hole mixing. (The exact expression can be found in Supplementary Note 2.) Analysis shows that fidelity is improved for faster recharging r, larger trion mixing β and slower spin flip ξ_{↑↓}. Conversely, in the limit of infinitely slow recharging r → 0 spin initialisation becomes impossible (P_{e} → 0), imposing a practical lower limit on the tunnel coupling with the Fermi reservoir.
Fundamental limits of electron spin lifetimes
Figure 2 shows that electron (Fig. 2b, e) and nuclear (Fig. 2c, f) spin relaxation rates are reduced at the centre of the 1e plateau^{15,25}. This Coulomb blockade regime is of most interest, as it corresponds to a stable electron spin qubit, and is examined in more detail in Fig. 3. The dependence of Γ_{e} on magnetic field is well described (solid lines in Fig. 3a) by
where for the fielddependent mechanism assisted by spin–orbit interaction and phonons we find \({\varGamma }_{{\rm{e}},{\rm{ph}}}\approx 2.27\pm 0.48\,{\mathrm {{s}}}^{1}\times {{\rm{T}}}^{{k}_{{\rm{ph}}}}\) and k_{ph} ≈ 4.1 ± 0.13 in both samples. The exponent is in good agreement with k_{ph} = 4 predicted^{26} and observed experimentally^{9} for this mechanism in hightemperature regime μ_{B}g_{e}B_{z} ≲ k_{B}θ, where phonon thermal occupation factor gives rise to an additional \(\propto\! {B}_{z}^{1}\) factor in Γ_{e}. This condition is well satisfied for our experiments at B_{z} ≤ 8 T, θ ≥ 4.5 K and typical gfactor values ∣g_{e}∣ ≈ 0.4. By contrast, in previous studies at B_{z} ≤ 12 T, θ ≈ 1 K the onset of lowtemperature regime was observed^{9}, where phonon thermal occupation factor is ≈1, resulting in k_{ph} = 5.
Cotunneling involves virtual injection of a second electron into the dot, followed by return of a spinflipped electron to the Fermi reservoir^{14,27}. The fitted cotunnelinginduced relaxation rate in a thinbarrier (t_{B} = 37 nm) sample Γ_{e,cotun} ≈ 532 ± 65 s^{−1} = (0.0019 ± 0.0002 s)^{−1} is larger than Γ_{e,cotun} ≈ (1.65 ± 0.21 s)^{−1} found for a thicker t_{B} = 52 nm. Since Γ_{e,cotun} is field independent, the increase in Γ_{e} at very low B_{z} ≲ 0.2 T (squares in Fig. 3a) is likely due to energyconserving electron–nuclear flip–flops, which become allowed when electron Zeeman energy is comparable to nuclear quadrupolar energy. By extrapolating the phonon (dotted line) and hyperfine (dashdotted line) mechanisms we roughly estimate the fundamental minimum of the electron spin relaxation rate in an isolated (t_{B} → ∞) QD as \({\varGamma }_{{\rm{e}},\min }\gtrsim {(20\,{\rm{s}})}^{1}\), expected to occur at B_{z} ≈ 0.4 T for θ ≈ 4.5 K. Similarly slow electron spin relaxation rates were reported in strainfree III–V QDs^{28}, although at lower temperatures θ < 0.1 K.
Figure 3b shows Γ_{e} measured in cotunnelingdominated lowfield regime in samples with different barriers. A considerable variation between individual dots for the thin barrier t_{B} = 37 nm can be due to random atomicscale positioning of the individual Si dopants^{29} and Si segregation^{30} at the interface between tunnel barrier and Fermi reservoir. By contrast, in a thickbarrier sample (t_{B} = 52 nm) the dot is coupled to a large number of dopants, smoothing out atomicscale variations and leading to consistent Γ_{e}.
In lowfield regime the barrier thickness t_{B} controls both the electron spin relaxation rate Γ_{e} and recharging rate r, but we find that r exceeds Γ_{e} by approximately five orders of magnitude, as exemplified in Fig. 1e. The recharging of an empty QD is a firstorder tunnelling process, whereas cotunneling in a charged QD is a secondorder process^{31,32}, which qualititatively explains the difference in rates. Moreover, the charge state of the dot itself may affect the conduction band energy profile, altering the tunnel coupling. An accurate firstprinciple quantitative description of the rates would require development of a detailed theoretical model.
Nuclear spin relaxation mechanisms
The marked difference in nuclear spin relaxation rate Γ_{N} (solid symbols in Fig. 3a) of the two samples at low magnetic fields suggest cotunneling as the dominant channel, whereas at B_{z} ≳ 2 T cotunneling is negligible. This is in contrast to previous studies under similar conditions (t_{B} = 35 nm, B_{z} = 5 T), which identified cotunneling and nuclear spin diffusion^{15} as dominant mechanisms. We examine diffusion by varying the spin pumping time (squares in Fig. 3c): Taking the difference in Γ_{N} at short and long pumping, the diffusion rate is estimated to be as small as Γ_{N,diff} ≲ 10^{−4} s^{−1} (at B_{z} = 8 T). Slow diffusion is due to quadrupolar freezing of nuclear spin flip–flops^{15,17}. Moreover, optical pumping through wetting layer states used in this work (as opposed to resonant optical pumping in ref. ^{15}), is likely to polarise nuclear spins not only in the dot but also in its vicinity, further suppressing the diffusion. Thus cotunneling and spin diffusion alone do not account for all the relevant nuclear spin relaxation mechanisms.
To explain the entire Γ_{N}(B_{z}) dependence, we treat the average spin of the QD electron as a random process. Uniquely for selfassembled QDs, noncollinear hyperfine interaction permits nuclear spin relaxation without electron spin flip^{13}—this mechanism is expected to be more efficient than direct electron–nuclear spin flips^{15}. The transition rate Γ_{N,ij} between states \(\lefti\right\rangle\) and \(\leftj\right\rangle\) of a single nuclear spin is proportional to spectral power density^{33} of the fluctuating electron spin s_{z}(t) at the nuclear spin transition frequency ν_{ij}. Using firstorder perturbation theory^{13} we have
where A_{hf} is the hyperfine constant, \({M}_{ij}=\langle i {\hat{I}}_{z} j\rangle\) is the matrix element of the nuclear spin operator \({\hat{I}}_{z}\), and electron spin correlation time is approximated by τ_{e} ≈ 1/Γ_{e}. This model describes a higherorder nuclear spin relaxation process, mediating by electron spin relaxation, which in turn is dominated by phonons at high fields or cotunneling at low fields. In the high field limit \( {M}_{ij}{ }^{2}\propto {B}_{z}^{2}\), \({\varGamma }_{{\rm{e}}}\propto {B}_{z}^{4}\) and \({\nu }_{ij}^{2}\propto {B}_{z}^{2}\) (see the “Methods” section) leading to \({\varGamma }_{{\rm{N}}}\propto {\rm{const}}\), which agrees with the weak field dependence of Γ_{N} observed for both samples at B_{z} ≳ 4 T (solid symbols in Fig. 3a). Moreover, temperature dependence at high field (Fig. 3d) is close to linear Γ_{N} ∝ θ at θ ≲ 15 K, matching the Γ_{e} ∝ θ dependence^{34} of the underlying phononmediated electron spin relaxation process. Superlinear growth of Γ_{N} at θ ≳ 15 K is likely due to twophonon processes^{26,35}, with scaling predicted to range between ∝θ^{2} and ∝θ^{9}.
For quantitative description (see the “Methods” section) we use experimentally measured Γ_{e} and estimate ν_{ij} and M_{ij} from magnetic resonance spectra^{36,37}. The results (dashed lines in Fig. 3a) are in good agreement for the thin barrier (t_{B} = 37 nm), where electron correlation time is short. The discrepancy for the thickbarrier sample (t_{B} = 52 nm) is most prominent at B_{z} ≲ 2.5 T, revealing the limitations of the electronspin fluctuation model (Eq. (2)) in the previously unexplored regime of a nearly isolated longlived electron spin. To examine the cause, we note that \({\tau }_{{\rm{e}}}^{2}{\nu }_{ij}^{2}\gg 1\) except for possible quadrupolar anticrossings of the nuclear spin levels^{37,38}, so that Eq. (2) can be rewritten as
The right side of this equation is a function of magnetic field and quantum dot structural properties, such as chemical composition and strain inhomogeneity, but it does not depend on tunnel coupling to Fermi reservoir. If τ_{e} = 1/Γ_{e}, the equation predicts independence of Γ_{N}/Γ_{e} on tunnel barrier. This is seen to be the case in Fig. 4 for samples with t_{B} = 37 and 42 nm, whereas the thick barrier sample (t_{B} = 52 nm) shows excessive Γ_{N}/Γ_{e}.
The exact reason for the increased Γ_{N}/Γ_{e} in the thick barrier sample is not clear. One likely possibility is additional nuclear spin relaxation mechanisms where hyperfine interaction fluctuates without electron spin flips, resulting in τ_{e} < 1/Γ_{e}. For example, modulation of the hyperfine interaction can occur through electron wavefunction density shifts, arising from fluctuating electric fields of the itinerant carriers in the Fermi reservoir^{33,39}, or charge traps^{16}. Charge noise in the studied structures is indeed present and evidenced, e.g. by fluctuating electron spin relaxation rates at the edges of 1e plateau (circles in Fig. 3e). Future experiments using, e.g. bias modulation spectroscopy may elucidate the roles of different nuclear spin relaxation mechanisms and lead to more accurate theoretical models. Further improvements to nuclear spin relaxation description can be sought through a microscopic model that takes into account quadrupolar anticrossings of the individual nuclear spin levels^{37,38}, which may accelerate relaxation and reenable frozen spin diffusion. A contribution of direct nuclear–phonon interaction^{33,40} is also possible, as its rate Γ_{N} ≈ 10^{−4}−10^{−3} s^{−1} is comparable to the lowest Γ_{N} observed here in electroncharged QDs.
Discussion
Experiments presented here establish a comprehensive picture of electron–nuclear spin relaxation in selfassembled QDs in a wide range of practically accessible conditions. Present experiments require B_{z} ≳ 0.15 T to initialise the spins^{22} and resolve the Zeemansplit optical transitions for spin probing. Extension to lower fields could shed light on the less explored regime where electron spin relaxation abruptly slows down from Γ_{e} ≈ 10^{9} s^{−1} at zero field^{8,41} to Γ_{e} ≈ 1 s^{−1} observed here at 0.15 T. For the practically interesting range B_{z} ≳ 0.15 T, electron spin relaxation is fundamentally limited by phonon coupling, which is similar in other types of QDs. Hence, electron spin lifetimes exceeding 1 s should be achievable in GaAs electrostatic^{28} and epitaxial^{42} QDs, as well as in II–VI QDs^{43,44}. By contrast, nuclear spin relaxation studied here is specific to selfassembled III–V QDs, and is governed by noncollinear hyperfine interaction. All experiments here were conducted in Faraday geometry, whereas noncollinear interaction is expected to be even stronger for magnetic field tilted away from the sample growth axis^{37}, which may lead to faster nuclear spin relaxation in Voigt geometry. The techniques employed here, can also be applied to establish the less explored fundamental limits of nuclear spin dynamics in electroncharged strainfree QDs^{28,42}, where noncollinear interaction will be small, but nuclear spin diffusion might be more prominent.
Methods
Samples and experimental techniques
The samples are lowdensity InAs selfassembled QDs (≲1 QD per μm^{2}) grown on a GaAs substrate. The dot layer is positioned at the centre of a λ/2 optical cavity formed by a bottom Bragg mirror consisting of 15 GaAs/AlAs pairs and a top reflector with 2 pairs (estimated quality factor Q ≈ 60). Cavity mode is centred at 950 nm, which matches the longwavelength tail of the QD wavelength distribution. The Fermi reservoir is formed by a doped GaAs layer (Si concentration of 1.1 × 10^{18} cm^{−3}, thickness ≈ 80 nm). The doped layer is located beneath QDs and is separated by a GaAs layer of thickness t_{B} = 37–52 nm, depending on the structure. Each sample is processed into a Schottky diode structure with an Au/(In–Ge) ohmic back contact^{45} annealed from the top surface, and a 5 nmthick semitransparent Ti Schottky top contact. External bias is applied to the top contract and controls the charge states of QDs. In order to form an electron spin qubit, the dot is charged deterministically with one electron (1e). This is achieved by tuning the energy of the 1e state to ≈10 meV below the Fermi energy E_{F} (Fig. 1a), while the twoelectron (2e) state remains depopulated, since its energy exceeds E_{F} by ≈10 meV, which is ≳20 times the thermal energy k_{B}θ at liquid helium temperature θ ≈ 4.2 K (Boltzmann constant k_{B} ≈ 86.17 μeV K^{−1}). The dot is then charged by an electron tunnelling from the Fermi reservoir.
The sample is mounted in a bath cryostat equipped with a superconducting coil producing magnetic field up to 8 T in Faraday geometry (field parallel to sample growth direction and optical axis z). An aspheric lens mounted near the sample is used for optical excitation of the QD and for light collection. Photoluminescence (PL) spectroscopy (see Supplementary Fig. 1) is used for initial QD characterisation. In nuclear spin dynamics experiments the dot is excited using diode lasers operating at 850 nm (resonant with InGaAs wetting layer). Nuclear spin polarisation (cooling) is achieved with a circularly polarised high power (≳100 times the power of ground state exciton saturation) laser, with typical pump pulse duration of T_{Pump} ≈ 8 s. A short (T_{Probe} ≈ 10 ms) low power (approximately corresponding to ground state exciton saturation) probe pulse is used to excite PL, which is then analysed on a double grating spectrometer to derive the hyperfine shifts E_{hf} in the splitting of a QD Zeeman doublet. The relaxation of the nuclear spin polarisation is derived by measuring E_{hf} in the probe as a function of delay T_{Dark} between the pump and the probe. The resulting E_{hf}(T_{Dark}) dependencies (e.g. Fig. 1f) are fitted with stretched or compressed exponentials \(\propto\! {{\rm{e}}}^{{({T}_{{\rm{Dark}}}/{T}_{1,{\rm{N}}})}^{\eta }}\), where η is the parameter describing stretching (η < 1) or compression (η > 1).
In ResFl experiments the dot is excited with a linearly polarised singlemode tunable diode laser. The scattered laser is rejected using crosspolarised detection^{46}, and the collected fluorescence is directed to an avalanche photodiode detector, whose photoncounting pulses are measured with a pulse counter and a digital oscilloscope. Typical linewidths measured in continuous excitation ResFl spectra on a negatively charged trion are ≈0.5 GHz at low power (nonsaturating excitation). Electron spin initialisation at finite magnetic field is witnessed through ResFl intensity I_{ResFl}, which is significantly reduced when electron is initialised into the \(\left\downarrow \right\rangle\) state, taking the dot out of resonance with optical driving of the \(\left\uparrow \right\rangle \leftrightarrow \left\uparrow \downarrow \Uparrow \right\rangle\) transition (compare squares and circles with triangles in Fig. 2a). Electron spin relaxation is accelerated and spin shelving is destroyed when the bias is tuned to the level where Fermi reservoir is resonantly tunnelcoupled with 1e (V_{S} ≈ 0.1 V in Fig. 2a) or 2e (V_{S} ≈ 0.33 V in Fig. 2a) quantum dot state, resulting in two peaks in the I_{ResFl}(V_{S}) dependence (squares and circles in Fig. 2a).
In electron spin dynamics measurements pulsing of the resonant laser is achieved with acoustooptical modulators providing on/off ratio better than 10^{7}. The power of the pump and probe pulses is close to ResFl saturation conditions and typical duration is T_{Pump} ≈ T_{Probe} ≈ 5–10 μs, which is significantly longer than the spin pumping time. As a result, timeresolved ResFl exhibits short pulses (Fig. 1d) with amplitudes I_{ResFl,Pump} and I_{ResFl,Probe}. At the start of each measurement cycle, and prior to optical pump pulse, the bias is adjusted for resonant electron tunnelling in order to counteract optical nuclear spin pumping and depolarise the electron (see further details in Supplementary Note 1). The rising edge of the fluorescence pulse corresponds to the rise time of the laser intensity in a pulse that pumps the initially depolarised electron. The exponentially decaying falling edge of the fluorescence pulse traces the gradual shelving (initialisation) of the dot into the \(\left\downarrow \right\rangle\) electron spin state. The loss of electron spin polarisation during the delay T_{Dark} results in partial recovery of the fluorescence intensity measured in the probe pulse. The residual polarisation P_{e} (i.e. electron spin polarisation at the start of the probe normalised by polarisation at the end of the pump pulse) is then derived as P_{e} = (I_{ResFl,Pump} − I_{ResFl,Probe})/I_{ResFl,Pump}. This way complete loss of electron polarisation (P_{e} = 0) is observed as I_{ResFl,Probe} = I_{ResFl,Pump}, while I_{ResFl,Probe} = 0 implies no loss (P_{e} = 1) or Auger recombination that empties the dot. By measuring P_{e} at different T_{Dark} the decay of electron spin polarisation is obtained as shown in Fig. 1e by the symbols.
Unless stated otherwise, all error estimates in the text and error bars in figures are 95% confidence intervals.
Modelling of the electron spin relaxation dynamics
We simulate the dynamics of the fourlevel system shown in Fig. 1b using a simplified noncoherent rate equation model. The relaxation rates of all possible transitions are shown in Fig. 1b, and when resonant optical pumping is present we add a transition \(\left\uparrow \right\rangle \to \left\uparrow \downarrow \Uparrow \right\rangle\) with rate P. We assume symmetric rates in electron spin flips \(\left\uparrow \right\rangle \leftrightarrow \left\downarrow \right\rangle\), which is justified when electron Zeeman energy is smaller than the thermal energy k_{B}θ. The system of firstorder differential rate equations is
In simulations we set γ_{R} = 10^{9} s^{−1}, which is typical for InAs/GaAs QDs^{47}. In order to simulate the pumpdelayprobe experiment we use initial population probabilities \(({p}_{\left0\right\rangle },{p}_{\left\uparrow \right\rangle },{p}_{\left\downarrow \right\rangle },{p}_{\left\uparrow \downarrow \Uparrow \right\rangle })=(0,1/2,1/2,0)\), and propagate the equations numerically over pump pulse (with P ≠ 0), dark delay (P = 0), and probe (P ≠ 0). The values of γ_{A}, P, β, r, ξ_{↑↓} are used as fitting parameters. The evolution of the trion state population \({p}_{\left\uparrow \downarrow \Uparrow \right\rangle }(t)\) reproduces the experimentally measured timeresolved ResFl (e.g. Fig. 1d). The simulated \({p}_{\left\uparrow \downarrow \Uparrow \right\rangle }(t)\) traces are integrated over pump and probe intervals and are used to calculate the residual electron polarisation P_{e} in the same way I_{ResFl,Pump} and I_{ResFl,Probe} are used to calculate P_{e} from experimental data (e.g. Fig. 1e). The fitting parameters are adjusted to achieve twoobjective optimisation: one objective is to minimise the root mean square difference between simulated and experimental P_{e}(T_{Dark}) traces, the other objective is to match the characteristic exponential time in the falling edge of the ResFl intensity produced by the pump pulse. An example of the Paretooptimal fitted P_{e}(T_{Dark}) is shown by the dashed line in Fig. 1e, in good agreement with experiment (squares). The calculated falling edge time (87 ns) is also in good agreement with the experimental value ≈91 ns.
From fitting we find r ≈ 1.26 × 10^{5} s^{−1} (95% confidence interval [0.47 × 10^{5} s^{−1}, 3.34 × 10^{5} s^{−1}]) and ξ_{↑↓} ≈ 1.65 s^{−1} [1.16 s^{−1}, 2.34 s^{−1}], which correspond to the characteristic timescales of the fast and slow components, respectively, in the twostage decay (squares in Fig. 1e). For Auger rate we obtain γ_{A} ≈ 1.09 × 10^{7} s^{−1} [0.66 × 10^{7} s^{−1}, 3.11 × 10^{7} s^{−1}]. This is approximately five times higher than γ_{A} ≈ 0.23 × 10^{7} s^{−1} reported previously from timeresolved ResFl experiments^{21}. The discrepancy could be due to the difference in QD structures, high optical pump power used in our experiments, limitations of a noncoherent rate equation model, and uncertainty in the fitted parameters. The uncertainty is increased by the interdependence of the fitting parameters γ_{A}, P, β, r, ξ_{↑↓}, which is inevitable since five parameters are used to fit essentially four degrees of freedom (fast and slow rates of the two stage P_{e}(T_{Dark}) decay, P_{e} following fast decay and the characteristic time of the falling edges in fluorescence pulses). This uncertainty can also be understood to arise from the limited information provided by the ResFl measurement, which does not distinguish between spin shelving into the \(\left\downarrow \right\rangle\) state and Auger recombination into the \(\left0\right\rangle\) state, restricting the ability to monitor the full dynamics of the fourlevel system. The remaining best fit values are P ≈ 2.5 × 10^{9} s^{−1} [0.34 × 10^{9} s^{−1}, 3.16 × 10^{9} s^{−1}] and β ≈ 5.2 × 10^{−3} [0.75 × 10^{−3}, 20.1 × 10^{−3}].
Modelling of the nuclear spin relaxation rate
We start by noting that at all magnetic fields used in this study (B_{z} = 0.15–8 T) nuclear spin relaxation in an empty dot (0e) is at least an order of magnitude slower than at the centre of the 1e charging plateau. (In 0e regime we measure Γ_{N} ≈ 6.0 × 10^{−4} s^{−1} at B_{z} = 0.15 T, which reduces at higher fields below the minimum measurable level of Γ_{N} < 10^{−4} s^{−1}.) This suggests that electron is the dominant mediator of nuclear spin relaxation in a Coulomb blockade regime (1e). Electron–nuclear coupling is given by the hyperfine Hamiltonian
where the summation goes over all nuclei, A_{hf,k} is the hyperfine constant of the kth nucleus, \(({\hat{s}}_{x},{\hat{s}}_{y},{\hat{s}}_{z})\) are the components of the electron spin1/2 operator and \(({\hat{I}}_{x,k},{\hat{I}}_{y,k},{\hat{I}}_{z,k})\) are the components of the spin operator of the kth nucleus. The flip–flop term \(\propto ({\hat{s}}_{x}{\hat{I}}_{x}+{\hat{s}}_{y}{\hat{I}}_{y})\) of this interaction permits spin exchange between nuclear spin I and electron spin s, but at sufficiently large magnetic field, where electron Zeeman energy significantly exceeds nuclear Zeeman and quadrupolar spin splitting, such flip–flops are strongly suppressed. For a depolarised nuclear spin ensemble in an InAs QD, this threshold field would be on the order of 0.03 T, corresponding to statistical fluctuation \(\propto {A}_{{\rm{hf}},k}\sqrt{N}\) of the nuclear hyperfine field acting on the electron. However, in selfassembled QDs the principal strain axis is generally misaligned^{36,37} from the growth axis z, resulting in nuclear eigenstates which are superpositions of \({\hat{I}}_{z}\) eigenstates. Under these conditions nuclear spin states are mixed by the nonflip–flop part of the hyperfine interaction \({A}_{{\rm{hf}},k}{\hat{I}}_{z,k}{\hat{s}}_{z}\). This noncollinear interaction enables transitions between nuclear spin states \(\lefti\right\rangle\) and \(\leftj\right\rangle\) without transfer of spin to the electron. Using firstorder perturbation and Weisskopf–Wigner approximation one arrives to Eq. (2) for nuclear spin relaxation rate between the pair of states \(\lefti\right\rangle\) and \(\leftj\right\rangle\), where we have assumed the same hyperfine constant A_{hf,k} = A_{hf}/N for all nuclei.
In order to calculate matrix element M_{ij} of \({\hat{I}}_{z}\) we consider spin I = 3/2 and assume that the principal component of electric field gradient is characterised by quadrupolar shift frequency ν_{Q}, and is tilted by angle α from the zaxis. Using firstorder perturbation approach we calculate M_{ij} in the opposite limits of small magnetic field (γ_{N}B_{z} ≪ ν_{Q}, where γ_{N} is the nuclear gyromagnetic ratio) and high magnetic field (γ_{N}B_{z} ≫ ν_{Q}). For each individual nucleus the pair (\(\lefti\right\rangle\), \(\leftj\right\rangle\)) with the largest offdiagonal matrix element is then selected leading to:
For intermediate fields we interpolate the matrix element with a monotonic function: \( M({B}_{z}){ }^{2}=1/( M{ }_{{B}_{z}\to 0}^{2}+ M{ }_{{B}_{z}\to \infty }^{2})\), and the nuclear spin transition frequency is taken to be \({\nu }_{ij}^{2}={({\gamma }_{{\rm{N}}}/2\pi )}^{2}({B}_{z}^{2}+{B}_{z,\min }^{2})\), where nonzero \({B}_{z,\min }\) reflects the fact that at B_{z} = 0 the spin states are split by nuclear quadrupolar effects. This model is a simplification since nuclear spin levels (anti)cross^{38} at B_{z} ≈ 2πν_{Q}/γ_{N}, where nucleus experiences a nearly zero effective magnetic field. The perturbative approach breaks down as M_{ij} is enhanced and ν_{ij} ≈ 0 at these anticrossing points (typically occurring at B_{z} ≈ 1 T)^{37}, hence the introduction of \({B}_{z,\min }\) which softens the singularities in Eq. (2). While spin relaxation would be enhanced in such resonant nuclei, it would also prevent their optical polarisation, thus we effectively neglect their contribution to the overall measured nuclear spin decay in our simplified model.
In order to calculate the nuclear spin decay rate according to Eq. (2), we take \({\tau }_{{\rm{e}}}={\varGamma }_{{\rm{e}}}^{1}\), equivalent to assuming that electron spin flips are the only source of noise acting on the nuclear spins. For Γ_{e} we use Eq. (1) taking best fit parameters for each QD sample. Based on NMR spectroscopy of similar QDs^{36,37}, we model strain inhomogeneity within the quantum dot by considering a uniformly distributed quadrupolar shift ν_{Q} ∈ [0, 16] MHz with principal axis uniformly distributed on a part of a sphere with α ∈ [0°, 76°]. The large values of α ≈ 76° account for As nuclei, whose quadrupolar shifts are dominated by atomic scale disorder, arising from Ga and In alloying^{36}. The gyromagnetic ratio is also varied uniformly γ_{N}/2π ∈ [7.4, 9.2]MHz T^{−1} to account for five different isotopes^{48} present in the dot (^{113}In, ^{115}In, ^{69}Ga, ^{71}Ga, ^{75}As). We take \({B}_{z,\min }=0.06\) T and use an average value A_{hf} = 50 μeV for all the isotopes^{49}. The number of nuclei is taken to be N = 4 × 10^{4}. The nuclear spin relaxation rate at a given magnetic field B_{z} is then calculated by averaging over all parameter distributions to take into account the contributions of the individual nuclei in a QD.
Despite the simplifications, the model is in good agreement with experimental dependence Γ_{N}(B_{z}) in the t_{B} = 37 nm sample (thin dashed line in Fig. 3a) and t_{B} = 42 nm sample (Supplementary Fig. 2). Qualitative comparison with Eq. (2) is possible in the high field limit where \({\varGamma }_{{\rm{e}}}\propto {B}_{z}^{4}\), \( {M}_{ij}{ }^{2}\propto {B}_{z}^{2}\) and \({\nu }_{ij}^{2}\propto {B}_{z}^{2}\) leading to \({\varGamma }_{{\rm{N}}}\propto {\rm{const}}\), which agrees with the weak field dependence observed for all samples at B_{z} ≳ 4 T. By contrast, at fields B_{z} ≲ 4 T, nuclear spin relaxation rate is determined by a combination of different factors prohibiting simple analytical description. Better description of the Γ_{N}(B_{z}) dependence, including the discrepancies with the thickbarrier (t_{B} = 52 nm) sample experiments, would require a more detailed model, which takes into account hyperfine fluctuations unrelated to electron spin flips, contributions of both noncollinear and direct hyperfine interaction, electron–nuclear spin feedback^{17}, quadrupolar anticrossings^{36,37} of the individual nuclear spin levels and electronmediated nuclear–nuclear interactions^{15}.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
I.M.G. processed samples into Schottky diodes with input and advice from S. Kumar, B. Royal, I. Farrer, N. Babazadeh, K. Kennedy, A.U., and B. Harrison. E.C. acknowledges P. Patil, I. Farrer and J. Heffernan. Royal Society provided funding support to E.A.C. through University Research Fellowship and grant RGF\EA\180117, and to G.G. and E.A.C. through grant RG150465. Experimental costs were partfunded through EPSRC grant EP/N031776/1.
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G.G., E.A.C. and A.U. developed timeresolved resonance fluorescence techniques. G.G., G.R., and E.A.C. developed nuclear spin lifetime measurement techniques. G.G. and E.A.C. conducted spin lifetime measurements. I.M.G., C.M. and E.A.C. conducted photoluminescence experiments. E.C. grew the samples. G.G. and E.A.C. analysed the data and wrote the manuscript. E.A.C. coordinated the project.
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Gillard, G., Griffiths, I.M., Ragunathan, G. et al. Fundamental limits of electron and nuclear spin qubit lifetimes in an isolated selfassembled quantum dot. npj Quantum Inf 7, 43 (2021). https://doi.org/10.1038/s41534021003782
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DOI: https://doi.org/10.1038/s41534021003782
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