Introduction

A key goal in quantum communication is to create highly indistinguishable photons that are separated in space by more than 100 km for device-independent quantum key distribution and for a quantum repeater1. This is potentially possible using a solid-state source, a semiconductor quantum dot. A single quantum dot mimics a two-level atom and single photons are generated either by spontaneous emission from the upper level2,3,4 or by coherent scattering of a resonant laser5,6,7. The radiative lifetime is typically τr=800 ps (ref. 8). There is evidence that on this timescale and at low temperature, there is negligible pure upper level decoherence5,6,7,9. Photons emitted subsequently are close to indistinguishable3,10. (At higher temperatures9,11,12,13, equivalently at low temperature but at high Rabi couplings14,15, phonons dephase the upper level.) A key remaining issue concerns the wandering of the centre frequency over times much longer than τr (refs 16, 17, 18). This wandering is highly problematic in any quantum photonics network: the quantum dot detunes from the common optical frequency and becomes dark; equivalently, the indistinguishability of quantum dot single photons generated far apart in the time domain is reduced. Active single-quantum-dot stabilization is possible but is presently limited to correcting for very slow drifts and in any case comes at the expense of complexity18,19. Eliminating the spectral wanderings would be highly advantageous.

The spectral wanderings can be conveniently probed simply by measuring the optical linewidth. Measured on millisecond or even second timescales, the quantum dot optical linewidth Γ is larger than the transform limit Γ0=ℏ/τr (refs 16, 17, 20, 21). In fact single-quantum-dot linewidths have remained stubbornly 50–100% above the transform limit even under the most favourable conditions (high-quality material, low temperature, charge control via Coulomb blockade and resonant excitation). We report here two regimes in which we observe transform-limited quantum dot optical linewidths even when measured on second timescales. One regime applies to the neutral exciton, X0, the other to the charged exception, X1−.

The X0 transition is split into two linearly polarized transitions by the electron–hole exchange, the so-called fine structure, corresponding to an admixture of the spin ±1 states (Fig. 1a). The splitting between the two transitions increases in an applied magnetic field, quadratically initially (Fig. 1c). The magnetic field is applied externally or it arises from a net polarization in the nuclear spins, which acts on the electron spin via the Overhauser field, BN. The X1− exhibits a single line at zero magnetic field (Fig. 1b) splitting linearly in magnetic field, again via an external field or Overhauser field (Fig. 1d). Both excitons exhibit large and similar d.c. Stark shifts (dependence of energy on electric field F), ∼25 μeV cm kV−1 (ref. 22). Charge noise leads to an inhomogeneous broadening of both X0 and X1− transitions via the d.c. Stark shift. This determines the inhomogeneous broadening for quantum dots in poor-quality material or quantum dots in high-quality material but with non-resonant excitation. In addition, both excitons are sensitive to spin noise, that is, fluctuations in the Overhauser field, but with different sensitivities. For X0, the sensitivity is second order as the hole ‘shields’ the electron from the spin noise; for X1− the sensitivity is first order on account of the unpaired electron in the X1− ground state. For instance, a typical Overhauser field of 20 mT (ref. 23) (arising from incomplete cancellation of the ∼105 nuclear spins24,25) leads to a linewidth contribution in the case of X1− of ∼0.5–1.0 μeV. Experimentally, there is strong evidence that in this cold, clean limit, spin noise and not charge noise is responsible for the X1− inhomogeneous broadening18,21. Despite the different sensitivity to spin noise the X0 and X1− linewidths are very similar16,17,21.

Figure 1: Neutral and charged excitons.
figure 1

(a) Energy levels of the neutral exciton X0 at zero magnetic field B=0, showing the fine structure splitting Δ. (b) Energy levels of the charged exciton X1− in an Overhauser field BN, introducing an electron Zeeman splitting Ze. (c,d) X0, X1− energy levels versus BN with Δ=11.5 μeV and electron g-factor g=−0.5.

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The approach here is to suppress the effects of charge noise by working in the ideal limit (high-quality material at low temperature, resonant excitation on a quantum dot in the Coulomb blockade regime), to compare X0 and X1− on the same quantum dot and to suppress the effects of spin noise by a search of the available parameter space. The improvement in the optical linewidth arises as a consequence of improved optical control of the nuclear spins associated with the quantum dot. It is noteworthy that the hyperfine interaction26,27 limits also the entanglement in the biexciton cascade28,29.

Results

X0 single-quantum-dot optical linewidth

A typical X0 resonance fluorescence (RF) spectrum is shown in Fig. 2a with Ω/Γ0=0.5 where Ω is the Rabi coupling. The linewidth is a factor of 1.4 larger than the transform limit (for this particular quantum dot, ). The transform limit Γ0 is measured by scanning the optical resonance very quickly such that the fluctuations are frozen during the measurement21 (Fig. 2c). The result is corroborated by measuring the radiative lifetime, either by recording a decay curve following pulsed excitation or by recording an intensity correlation g(2): the results agree to within the random errors of ∼5% (Supplementary Note 1 and Supplementary Fig. 1).

Figure 2: Linewidth versus scanning frequency.
figure 2

(a,b) X0, X1− resonance fluorescence versus detuning δ at 4.2 K, B=0.0 mT with 100 ms integration time per point. The solid lines are Lorentzian fits to the data. The linewidths are , ; the Rabi energies Ω/Γ0=0.5 (X0), 0.4 (X1−); and transform limits , . (c) RF linewidth against scanning frequency dδ/dt/Γ0. Γ approaches Γ0 for scanning frequencies above 50 kHz. For each scanning frequency, the error bar represents the s.d. of several hundred linewidth scans. Solid lines represent a Lorentzian fit of the data with linewidth 30±3 kHz.

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Figure 3a shows Γ versus Vg on the neutral exciton, X0, measured below but close to saturation, Ω/Γ0=0.5. At the edges of the Coulomb blockade plateau, Γ rises rapidly on account of fast electron spin dephasing via co-tunnelling with the Fermi sea30. This process slows down as Vg moves away from the plateau edges. The new feature is that a ‘sweet spot’ exists close to the negative Vg end of the plateau with minimum linewidth 1.19±0.13 μeV (Fig. 3a,b). Accounting for the small power broadening, the ideal limit is . Within the measurement uncertainties of 10%, the transform limit is therefore achieved. As Vg raised to the positive side of the sweet spot, Γ increases beyond the ideal limit (Fig. 3a).

Figure 3: X0 spin noise within the Coulomb blockade plateau.
figure 3

(a) X0 optical linewidth measured at Rabi energies Ω/Γ0=0.50 for different gate voltages by sweeping the laser frequency through the resonance and integrating 100 ms per point. Γ decreases from 1.66 to 1.19 μeV with decreasing gate voltage. (b) X0 spectrum with Γ=1.15 μeV at Vg=−1.54 V. (c) X0 noise spectra recorded at Rabi energies Ω/Γ0=0.65 for different voltages, indicated in a by solid lines. Maximum/minimum spin noise (black/blue) is correlated with the largest/smallest Γ. (d) NQD(f) on X0 recorded with two lasers of frequencies f1 and f2 and a frequency splitting f1−f2 equal to the fine structure splitting for 〈δ〉=0 (blue) and 〈δ〉=Γ/2 (red). Inset shows the laser frequency detuning relative to the optical resonance.

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It is instructive to investigate the sources of noise. A diagnostic is a noise spectrum NQD(f), a Fourier transform of the RF time-trace (Supplementary Note 2 and Supplementary Fig. 2). From the known relationships between RF signal, detuning δ, Rabi coupling Ω, electric field F and the Overhauser field BN, the variances Fr.m.s. and BN,r.m.s. can be determined from the noise spectrum21 (Supplementary Note 3). The increase in linewidth above the transform limit represents a sum over all noise sources from the scanning frequency, about 1 Hz, to Γ0/ℏ, about 1 GHz. The noise spectra at the low-bias end (the sweet spot), the centre of the plateau and the positive-bias end are shown in Fig. 3c. There is a Lorentzian feature with linewidth 30 Hz (noise correlation time 30 ms) and a second Lorentzian feature at higher frequencies with linewidth 200 kHz (correlation time 5 μs). The origin of the two features in the noise spectrum can be identified by exploiting the different X0 response to charge noise and spin noise: charge noise moves both X0 peaks rigidly together; spin noise moves them apart or closer together, a ‘breathing’ motion. A two-laser experiment enables us to distinguish between these two possibilities. Specifically, we record X0 noise spectra with two lasers with frequencies separated in frequency by the fine structure. On detuning both lasers from δ=0 to δ=Γ/2, the sensitivity to charge noise increases (changing from second order to first order) yet the sensitivity to spin noise decreases (remaining second order but with a reduced pre-factor) (Supplementary Note 4 and Supplementary Fig. 3). In the experiment, switching from 〈δ〉=0 to 〈δ〉=Γ/2 causes the noise power of the low-frequency component to increase markedly (Fig. 3d) identifying it as charge noise. However, the frequency sum over the charge noise gives a contribution to Γ smaller than 0.05 μeV (Supplementary Note 5), a negligible value. (We note that both the d.c. Stark coefficient and Γ vary from quantum dot to quantum dot yet there is no correlation between the two (Supplementary Note 6 and Supplementary Fig. 4), pointing also to the unimportance of charge noise in the optical linewidth.) Conversely, the noise power of the high-frequency component decreases on detuning both lasers from δ=0 to δ=Γ/2, identifying it as spin noise (Fig. 3d). Furthermore, noise spectra measured at 〈δ〉=0 but with a single laser tuned to one of the X0 transitions show that the low-frequency noise, the charge noise, is similar for all three biases yet the high-frequency noise, the spin noise, increases with increasing bias (Fig. 3c). This confirms that the high-frequency noise, the spin noise, is responsible for the inhomogeneous linewidth: the integrated spin noise is vanishingly small at the sweet spot, increasing at the centre of the plateau, and increasing further at the positive-bias edge.

In general, the X0 Γ versus Ω curve does not follow exactly the textbook result for a two-level system (Supplementary Note 7 and Supplementary Fig. 5). The Ω dependence of NQD(f) is highly revealing (Fig. 4a, Supplementary Note 8 and Supplementary Fig. 6). In the centre of the plateau, as Ω increases the X0 spin noise also increases (Fig. 4a). increases roughly linearly with Ω reaching at the highest couplings extremely high values, 300 mT (Fig. 4c). ( is determined by a Monte Carlo simulation of NQD(f) including an ensemble of fluctuating nuclei—this is robust as X0 is sensitive only to the vertical component of BN (Supplementary Note 3).) The large would appear to prohibit transform-limited linewidths on X0 at all but the very lowest optical couplings. However, at the sweet spot, this mechanism is clearly suppressed: reduces to <50 mT and approaches the value for a quantum dot in the ground state.

Figure 4: Spin noise power dependence
figure 4

(a) NQD(f) on X0 for a series of Rabi couplings Ω with Vg at the centre of the Coulomb blockade plateau at B=10.0 mT. The experimental data for Ω=0.46 μeV (black curve) is accompanied by the Monte Carlo fit (red dashed curve). (b) NQD(f) on X1− for a series of Rabi couplings Ω taken at B=10.0 mT to enhance the sensitivity to spin noise (Supplementary Note 8). (c,d) BN,r.m.s. versus Ω for X0, X1−. For each Rabi coupling, the error bar in c represents an uncertainty of 10% on the determination of BN,r.m.s. from the Monte Carlo fit; the error bar in d represents the s.d. of several scans. r.m.s, root mean squared.

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The existence of the X0 sweet spot is a robust phenomenon. It exists on all the quantum dots investigated in this particular sample, on quantum dots from other samples from the same wafer and from samples from other wafers of a similar but non-identical design (Supplementary Note 9 and Supplementary Fig. 7). A very striking example is the observation of the sweet spot on a p-type field-effect device (Supplementary Fig. 7(e)). Choosing the correct bias allows us to achieve X0 transform-limited lifetimes (to within the random error of 0.1 μeV) in each case.

X1− single-quantum-dot optical linewidth

A typical X1− resonance fluorescence spectrum is shown in Fig. 2b with Ω/Γ0=0.4 (same quantum dot as in Fig. 2a). The linewidth is a factor of 2.0 larger than the transform limit (for this particular quantum dot, ). For X1−, it is clear that the nuclear spins are a significant source of inhomogeneous broadening. As a function of bias, the X1− linewidth is smallest in the centre of the Coulomb blockade plateau, rising at the edges. This is consistent with a co-tunnelling dominated mechanism30 (Supplementary Note 9 and Supplementary Fig. 7(a)). We investigate the spin noise and in particular its Ω dependence via the noise spectra. Figure 4b shows that the X1− spin noise decreases as Ω increases, corresponding to a decrease in (Fig. 4d). (The distinction between charge noise and spin noise can be made in the case of X1− simply by changing the detuning from 〈δ〉=0 to 〈δ〉=Γ/2 in a one-laser experiment21. X1− responds to all three components of BN, a more complex problem than that for X0, and instead is determined (Fig. 4d) with lower systematic error from the two-laser experiment described below.)

We address whether the spin noise reduction in the case of X1− is sufficient to achieve transform-limited optical linewidths. The Ω dependence of can be described extremely well with the two-level result including an inhomogeneous broadening γ (Fig. 5b, Supplementary Note 7 and Supplementary Fig. 5). At low Ω, Γ is determined by Γ0 and γ; at higher Ω, Γ increases (power broadening) and γ becomes irrelevant. We can therefore determine the ideal limit (Γ versus Ω with γ=0) and below saturation, the inhomogeneous broadening is clearly significant (Fig. 5b). However, this relatively simple linewidth measurement is complex to interpret as the spin noise is a function of both Rabi energy and detuning. To simplify matters, we performed the experiment with two lasers. The concept is that the stronger, constant frequency pump laser (Ω2, δ2) determines the spin noise, and the weaker probe laser (Ω1, δ1) measures the optical linewidth. Figure 5a shows measured by sweeping δ1 versus δ2 for Ω1=0.23, Ω2=0.80 μeV. For large δ2, the pump laser has no effect on Γ; power broadening is irrelevant and Γ is far from the transform limit. For small δ2 however, Γ decreases, despite the power broadening induced by Ω2. Taking into account power broadening, Γ reduces to the ideal limit. Figure 5b shows the results as Ω2 increases: for Ω/Γ0 >0.75, transform-limited optical linewidths are achieved (to within the random error of 10%). The spin noise reduction on driving X1− with the pump laser is accompanied by a profound change in the probe spectrum: the optical resonance now splits into two resonances (Fig. 5c). The splitting reflects a static electron Zeeman splitting in the single-electron ground state, BN=58 mT in Fig. 5c, with BN increasing with Ω2 (Fig. 5d, Supplementary Note 10 and Supplementary Fig. 8). Equivalently, even without an applied magnetic field31, a nuclear spin polarization is created by the optical coupling. This demonstrates that the laser locks the nuclear spins into an eigenstate of the ΣIz operator. (We comment that significant nuclear spin polarizations can be achieved in an applied magnetic field26,27, for instance via ‘dragging’ with resonant excitation32, but we find that the optical linewidths increase in this regime.)

Figure 5: The two-laser experiment (B=0.0 mT, T=4.2 K) on X1−.
figure 5

(a) Optical linewidth measured with the probe laser (Ω1=0.23 μeV) versus detuning of the pump laser δ2 for Ω2=0.80 μeV. The dashed lines show the ideal case (zero inhomogeneous broadening) in two limits, Ω=Ω1 and Ω=Ω1+Ω2, appropriate for large δ2 and δ2=0.0 μeV, respectively, the difference arising from power broadening. (b) Optical linewidth in one-laser experiment (black points) versus Ω with fit to two-level model (γ=0.65 μeV (dashed line), 0 (solid line)). The optical linewidth in two-laser experiment (Ω1=0.23 μeV, δ2=0) versus Ω2 (red points). (c) Probe spectrum with Ω1=0.23 μeV, Ω2=0.80 μeV and δ2=0.0 μeV (points) with a two Lorentzian fit (solid line, energy separation 1.6 μeV, linewidths 0.8±0.3, 1.2±0.3 μeV). (e) Splitting from d versus Ω2. The error bars represent the s.d. of several scans.

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Discussion

The experiments reveal a remarkable dependence of the spin noise on charge. In the centre of the plateaus, resonant excitation of X0 enhances spin noise yet resonant optical excitation of X1− suppresses spin noise. Concomitant with the different BN,r.m.s. values are the associated BN-correlation times, much shorter for X0 (5 μs) than for X1− (100 μs)21,33. We note that the scanning frequency dependence (Fig. 2c) reveals a 100-μs noise correlation time for both X0 and X1−: at higher enough scanning frequencies, X0 is driven too briefly for any nuclear noise enhancement to be active. This points to the fact that the reduced correlation time and increased amplitude of the spin noise as measured on X0 is related to the constant optical driving. Fortunately, at a particular bias, the nuclear spin ‘shake-up’ on driving X0 can be turned off and transform-limited linewidths can be achieved: the charge noise is too small to matter and the electron–hole exchange shields the exciton from the remaining nuclear spin noise.

Once the charge noise has been suppressed by using clean devices and the nuclear spin effects have been bypassed, the fidelity of the photons is limited by the phonons. The zero phonon line (ZPL) accounts for 95% of the emission12,19, a very high ratio for a solid-state emitter. The 5% non-ZPL photons can be filtered out without too much trouble but at the cost of a slight increase in shot noise. The phonon-related broadening of the ZPL is however very small at low temperature9,12. Once the device engineering described here has been combined with photonic mode engineering to boost the extraction efficiency34, there are excellent prospects for creating a fast and efficient source of indistinguishable photons using a semiconductor.

The mechanisms by which the nuclear spin noise respond to resonant optical excitation are unknown. For X1−, the data are compatible with a ‘narrowing’ of the nuclear spin distribution, perhaps caused by continuous weak measurement via the narrowband laser35. The correlation time is compatible with the nuclear spin dipole–dipole interaction. For X0 it is unlikely that the standard electron spin–nuclear spin contact hyperfine interaction can offer an explanation; it is also unlikely that the bare dipole–dipole interaction can account for the short correlation time. One possibility is that the hole in the X0 is important. First, a hole has a complex hyperfine interaction, containing a term (I+Jz+I−Jz), exactly the structure required to shake-up the nuclear spins on creation of a hole (I is the nuclear spin and J the hole pseudo-spin)36. While the coefficient of this term is likely to be small, it can have significant consequences should the dark X0 state be occupied for times far exceeding the radiative lifetime36. A second possibility is that the hyperfine interaction renders the X0 sensitive to the nuclear spins by altering the phase effects that account for the electric field dependence of the fine structure splitting37. We hope that our results will stimulate a refinement in understanding of the exciton–nuclear spin interaction.

In conclusion, we report transform-limited optical linewidths from a single semiconductor quantum dot even when measured on second timescales on both X0 and X1−. Generally speaking, controlling spin noise is key to operating a quantum dot-based spin qubit24,25,38,39,40. The same factor turns out also to be a key feature in creating a quantum dot-based high-fidelity single-photon source.

Methods

The semiconductor quantum dot sample

The quantum dots are self-assembled using InGaAs in high-purity GaAs and are embedded between an n+ back contact (25-nm tunnel barrier) and a surface gate17,21 (Supplementary Methods and Supplementary Fig. 9). The gate voltage Vg determines the electron occupation via Coulomb blockade41.

Resonance fluorescence

The quantum dot optical resonance is driven with a linearly polarized, resonant, continuous-wave laser (1 MHz linewidth) focused on to the sample surface. Reflected or scattered laser light is rejected with a dark-field technique using crossed linear polarizations for excitation and detection42. The laser excitation polarization is rotated by an angle of π/4 with respect to the neutral exciton’s linear polarization axes.

Resonance fluorescence is detected with a silicon avalanche photodiode in photon-counting mode. The experiment is not shielded against the earth’s magnetic field, thus Bmin ∼50 μT. All the experiments were performed with the sample at 4.2 K. Γ is determined by sweeping the laser frequency through the resonance, integrating the counts, typically 100 ms per point.

Additional information

How to cite this article: Kuhlmann, A. V. et al. Transform-limited single photons from a single quantum dot. Nat. Commun. 6:8204 doi: 10.1038/ncomms9204 (2015).