Transform-limited single photons from a single quantum dot

Developing a quantum photonics network requires a source of very-high-fidelity single photons. An outstanding challenge is to produce a transform-limited single-photon emitter to guarantee that single photons emitted far apart in the time domain are truly indistinguishable. This is particularly difficult in the solid-state as the complex environment is the source of noise over a wide bandwidth. A quantum dot is a robust, fast, bright and narrow-linewidth emitter of single photons; layer-by-layer growth and subsequent nano-fabrication allow the electronic and photonic states to be engineered. This represents a set of features not shared by any other emitter but transform-limited linewidths have been elusive. Here, we report transform-limited linewidths measured on second timescales, primarily on the neutral exciton but also on the charged exciton close to saturation. The key feature is control of the nuclear spins, which dominate the exciton dephasing via the Overhauser field.

spin noise (b) for both X 0 (two lasers with frequency splitting equal to the fine structure) and X 1− (one laser       To compare the optical linewidth to the ideal case it is clearly necessary to know the transform-limit, equivalently the radiative lifetime τ r . We measure the radiative lifetime in three different ways, Supplementary Fig. 1. First, we measure the linewidth as a function of scanning frequency. The linewidth is a constant and then decreases between scanning frequencies of ∼ 5 kHz and ∼ 50 kHz to another constant value [1]. Given the known absence of upper level decoherence under these conditions (low temperature, high quality material, weak resonant excitation) this constant value at high scanning frequencies corresponds to the transform-limit. Secondly, we measure the radiative lifetime by recording a decay curve following excitation with a non-resonant pulse. Analysis of the decay, taking account the timing jitter of the detector, reveals the radiative lifetime. Finally, we record an intensity correlation function g (2) with a Hanbury Brown-Twiss setup, fitting the results to the behaviour of a resonantly-driven two-level atom convoluted with the jitter function of the detectors. We find that all three results give the same value for the radiative lifetime to within the measurement uncertainty of ∼ 5%.
From quantum dot to quantum dot, there is a systematic difference between X 0 and X 1− : the radiative decay rate is larger for X 0 than for X 1− (Fig. 2c of the main article). The difference however from quantum dot to quantum dot is small, just ∼ 5%.

Supplementary Note 2: Quantum dot noise spectrum
To determine the QD noise spectrum the arrival time of each photon is recorded over the entire measurement time T . Post measurement, a binning time t bin is selected, typically 1 µs. The number of counts in each time bin is S(t), the average number of counts per bin S(t) . The fast Fourier transform of the normalized RF signal S(t)/ S(t) is calculated to yield a spectrum of the noise power N RF (f ), specifically N RF (f ) has the same spectrum independent of the choice of t bin and T : smaller values of t bin allow N RF (f ) to be determined to higher values of frequency f ; larger values of T allow N RF (f ) to be determined with higher resolution. The high frequency limit of our experiment is only limited by the photon flux.
All Fourier transforms are normalized [2] such that the integral of the noise power N x (f ) over all positive frequencies equals the variance of the fluctuations δx, To record a noise spectrum of the experiment alone, the QD is detuned by > 100 linewidths relative to  Fig. 2 (b)) and not to S(t) 1/2 due to the normalization of S(t) by S(t) in the calculation of the spectrum. N shot is comparable to N QD (f ) at low frequencies (f ∼ 10 Hz), and exceeds N QD (f ) at higher frequencies, Supplementary   Fig. 2 (c).
The noise spectrum of the QD alone is then determined using

Supplementary Note 3: Noise spectra modelling
Our previous experiments [1] demonstrate that the spectrum of the noise in the RF is dominated by charge noise at low frequency, spin noise at high frequency. The noise sensor, the RF from a single quantum dot, has a trivial dependence on the fluctuating electric F (t) and magnetic fields B N (t) only for small fluctuations in the detunings around particular values of detuning δ. Monte Carlo simulations allow us to determine both the electric field and magnetic field noise accurately by describing the response of the sensor for all δ, treating charge noise and spin noise on an equal footing.
The basic approach is to calculate F (t) and B N (t), in each case from an ensemble of independent, but identical, 2-level fluctuators using a Monte Carlo method; to calculate the RF signal S(t) from F (t) and B N (t); and to compute the noise N (f ) from S(t) using exactly the same routine as for the experiments (but without the correction for extrinsic noise of course). Here, we discuss the spin noise modelling of the neutral exciton X 0 used to extract the root-mean-square (rms) values of the magnetic field B N,rms in Supplementary Fig. 4 (c) of the main article. The modelling of charge noise is explained in detail elsewhere [1].
For X 0 , the RF depends on the electric and magnetic fields according to where a is the dc Stark coefficient, g the electron g-factor and ∆ the fine structure splitting. For the blue Zeeman branch δ 0 (t) is positive, for the red one negative, respectively.
An ensemble of identical 2-level fluctuators fully describes spin noise, Fig. 4 (a) of the main article.
A 2-level fluctuator occupies either state 0 with lifetime τ 0 or state 1 with lifetime τ 1 . The probability p of being, at any time, in state 1 is τ 1 /(τ 0 + τ 1 ); the probability of being in state 0 is τ 0 /(τ 0 + τ 1 ). The configuration C(t) of a 2-level fluctuator, either 0 or 1, is determined by the probabilities of a 0 → 1 and a 1 → 0 transition, where δt denotes the time over which the system evolves. The power spectrum of a 2-level fluctuator S(ω) is Lorentzian [3], The calculation of the time trace of the magnetic field B N (t) is simplified, such that each nucleus is treated as a two-level fluctuator, with equal 0 → 1, 1 → 0 transition rates, 1/τ . At t = 0, each nucleus is initialized by a random number generator giving a configuration of nuclear spins C(0). At a later time, δt, C(δt) is calculated from C(0) again with a random number generator using the probabilities p 1→0 (δt) and p 0→1 (δt) from the theory of a two-level fluctuator. The nuclei are treated independently.
The nuclear magnetic field, the so-called Overhauser field B N , is given by [4] where v 0 is the atomic volume, A i the hyperfine interaction constant, r i is the position of the nuclei i with spin I i , and ψ(r) is the normalized electron envelope function. By using an average hyperfine constant [5] A = 90 µeV and approximating the electron envelope function ψ(r) by a top hat, Eq. (8) simplifies to N eff denotes the number of nuclear spins inside the top hat envelope function.
Regarding the dimensionality of B N , a 1D model for the nuclear spins is appropriate for X 0 . The isotropic part of the electron-hole exchange interaction "protects" the X 0 from the in-plane fluctuations of the nuclear magnetic field. Specifically, the z-component of the Overhauser field enters along the diagonals of the exchange/Zeeman Hamiltonian [6] in the |⇑↓ , |⇓↑ , |⇑↑ , |⇓↓ basis and results in the dispersion of Eq. 4. The in-plane components of the Overhauser field couple |⇑↓ ↔ |⇑↑ and |⇓↑ ↔ |⇓↓ but these states are split by the dark-bright splitting, 100s of µeV, determined by the isotropic part of the exchange interaction. As a result the dependence of the exciton energy on the in-plane fields is negligible.
We assume that each nuclear spin I can be represented by a spin-1 2 , a 2-level fluctuator. To account for an underestimate of the hyperfine interaction (the real spins are larger than 1 2 ) the Overhauser field is enhanced via a reduction in the total number of nuclei, N → N eff . Equivalently, we could work with a higher N eff and larger A. The model represents a phenomenological way to create B N (t) which mimics the experiment. B N (t) is unique, the route to B N (t) is not.
There are two independent parameters that control spin noise in the simulation: the correlation time τ and the rms field B N,rms . For the simulation shown in Fig. 4 (a) of the main article A = 90 µeV, N eff = 178, corresponding to B N,rms = 116 mT, and τ = 6.0 µs were used. The noise spectra at higher Rabi energies were fitted by decreasing N eff (increasing B N,rms ) and the same τ .
The sensitivity in the RF to charge noise and spin noise depends on the laser detuning δ, Supplementary   Fig. 3. For X 1− , only one laser is required to distinguish charge noise and spin noise yet two lasers with frequencies separated by the fine structure splitting are required for X 0 . Both charge noise ( Supplementary   Fig. 3 (a)) and spin noise ( Supplementary Fig. 3 (b)) exhibit the same detuning dependence for X 1− (one laser) and X 0 (two lasers). On detuning the laser/both lasers (X 1− /X 0 ) from δ = 0 to δ = Γ/2, the sensitivity to charge noise changes from second order to first order yet the sensitivity to spin noise decreases by a factor ∼2.

Supplementary Note 6: The dc Stark effect
The Stark shift is determined by recording the resonance position in V g for many laser frequencies, the laser frequency measured in each case with an ultra-precise wavemeter. The Stark shift is linear in ∆F for the small windows of V g used here, Supplementary Fig. 4 (a). The neutral exciton X 0 has a larger Stark shift (a = 0.0306 µeVcm/V) than the charged exciton X 1− (a = 0.0219 µeVcmV −1 ) and thus it is more sensitive to charge noise. X 1− has a larger linewidth (Γ = 1.48 µeV) compared to X 0 (Γ = 1.28 µeV) despite the smaller Stark shift. Also, experiments on several QDs reveal no dependence of the linewidth on the Stark shift coefficient, Supplementary Fig. 4 (b). The Stark shift varies from quantum dot to quantum dot by up to 50% without a correlated change in linewidth. Both these facts support the dominant influence of spin noise and not charge noise on the X 0 and X 1− linewidths.

Supplementary Note 7: Power broadening
The linewidth of the optical resonance increases with increasing resonant excitation power, Supplementary Fig. 5. The additional contribution to the linewidth is known as power broadening, described for an ideal 2-level system by [7] Γ(Ω) = Γ 2 0 + 2Ω 2 + γ, Γ 0 =h/τ R with Rabi energy Ω and radiative lifetime τ R . An inhomogeneous broadening is included with the term γ.
For X 1− , the 2-level model with constant γ describes the data very well, Supplementary Fig. 5 (a). The inhomogeneous broadening γ is constant at low power, decreasing at high power but only when power broadening dominates, such that a constant γ allows the experimental data to be described very well (see main article). By fitting the 2-level model to the data a resonant excitation power measured by a photo diode beneath the sample can be converted to a Rabi energy, Supplementary Fig. 5 (a).
Conversely for X 0 , the 2-level model with constant γ does not describe the data well. The inhomogeneous broadening is strongly power dependent: γ increases significantly with increasing resonant excitation power (see main article).
A phonon-induced dephasing process as observed at very high Rabi couplings [8] and in pulsed experiments [9] is negligible at these Rabi couplings.
In a two-laser experiment at zero magnetic field with a resonant pump laser the optical resonance of the charged exciton splits into two resonances. The splitting reflects a static electron Zeeman splitting in the single electron ground-state and not an Autler-Townes splitting [11]. We can rule out an Autler-Townes splitting as first, the splitting is not given by the Rabi energy Ω as is the case for an Autler-Townes splitting, and secondly, we do not observe an optically-induced splitting when the X 1− resonance is pulled apart in a small magnetic field, Supplementary Fig. 8.
The samples only differ in the gate thickness: they are fabricated from the same wafer.
The number of electrons confined to the QD can be precisely controlled by the gate voltage V g as illustrated in Supplementary Fig. 9 (b). A change of gate voltage yields a change of the QD's local potential φ by where λ = 18.3 denotes the sample's lever arm, defined as the ratio of back contact to gate distance d and tunnel barrier thickness. The exciton energy E is detuned with respect to the constant laser frequency by exploiting the dc Stark effect, with Stark shift coefficient a and electric field F .