Multi-dimensional photonic states from a quantum dot

We perform our experiments using a single-hole charged quantum dot in a micropillar cavity (figure 1(b)), a system which has been shown to be a good source of indistinguishable photons [34]. The device we use is nominally undoped, emission from the positively charged state can be activated by weak non-resonant light at a wavelength of 850 nm [34]. Applying a Voigt geometry magnetic field results in a double-Λ system. The magnetic field and temperature are tuned so that a single vertical transition is cavity enhanced (figure 1(c)). The states of the system are aligned parallel/anti-parallel to the magnetic field. We label the hole spin states, $| h\rangle /| \bar{h}\rangle =\tfrac{| \Uparrow \rangle \,\pm \,| \Downarrow \rangle }{\sqrt{2}}$ and the trion states as $| T\rangle /| \bar{T}\rangle =\tfrac{| \Uparrow \Downarrow \uparrow \rangle \pm | \Uparrow \Downarrow \uparrow \rangle }{\sqrt{2}}$.

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In order to maximise the number of operations that can be performed on a spin per unit time, it is desirable to perform spin state preparation as rapidly as possible. In the Voigt configuration the spin preparation typically takes several nanoseconds [35].

In our experiments we use a micropillar cavity to decrease the spin preparation time. In the Raman process a photon from the laser is scattered resulting in a photon that we detect at the energy of the $| \bar{T}\rangle \to | h\rangle$ transition. Driving the $| \bar{h}\rangle \to | \bar{T}\rangle$ transition we observe cavity-stimulated Raman emission [36]. This increases the speed of spin preparation because the system is more likely to decay via the enhanced vertical transition than the non-enhanced diagonal transition. As a result we can perform spin preparation in 270 ± 6 ps—over an order of magnitude faster than is typically seen for similar non-cavity-enhanced systems (figure 2). We note that there is a continued background emission in figure 2(b) once spin preparation has occurred—we attribute the to the similarity in energy of the diagonal transitions in this quantum dot. The resonant laser may weakly drive the $| h\rangle \to | T\rangle$ transition as well as the intended transition, lowering the spin preparation fidelity in this case. Higher magnetic fields or a quantum dot with a different combination of electron and whole g-factors could remove this effect.

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We also note that the selective enhancement of one transition of the lambda system means that it could function as a cycling transition to be used for spin readout [37, 38]. The ability to perform rapid spin preparation and reliable spin readout in the Voigt geometry will help make trapped spins in quantum dots appealing candidates for stationary qubits and light matter interfaces. Our pillar microcavity has negligible polarisation splitting and although it enhances transitions resonant with the mode, it does not suppress detuned transitions. Non-resonant excitation of the 4 level system in figure 1(d) results in equal populations in states $| \bar{T}\rangle$ and $| T\rangle$, which decay via 4 unequal decay channels, which we collect with unequal efficiencies. The ratio of the area of the longest and shortest wavelength peaks in the spectrum of the X⁻ transitions at 9 T (figure 1(d)) suggest the cavity-enhanced decay is a factor of ∼4.1 faster than the non-enhanced case. Similarly, comparing the spin preparation time recorded by driving the $| \bar{h}\rangle \to | \bar{T}\rangle$ and measuring light from the $| \bar{T}\rangle \to | h\rangle$ (cavity-enhanced preparation) and the spin preparation time recorded by driving the $| h\rangle \to | \bar{T}\rangle$ and measuring light from the $| \bar{T}\rangle \to | \bar{h}\rangle$ gives us a ratio of ∼4.8.

The coherence time of the Raman scattered photons is determined by the coherence of the long lived ground state hole spin and the laser rather than the shorter coherence time of the exciton [39, 40]. This means that the coherence of the photon state is in part inherited from the highly coherent laser light. A further useful property of Raman emission is that the wavelength of the emitted photons can be tuned by tuning the wavelength of the excitation laser [36, 41]. In our system we observe a tuning range of over 65 μeV. This would allow for the generation of indistinguishable photons from different sources—a key requirement for several quantum photonic technologies [16, 42].

Here we implement a scheme for the generation of a single-photon W-state. W-states are of particular interest as they represent one of two types of maximally entangled tripartite states and maintain their entanglement in the presence of dissipation [43]. The concept can be generalized to include W-states with more than three qubits.

We use a series of weak resonant pulses (shown in figure 3(a)) to drive the $| \bar{h}\rangle \to | \bar{T}\rangle$ transition (shown as the red arrow in figure 1(c)). The power of these pulses is tuned so that they each have the same probability of driving the transition and generating a photon. When the $| \bar{h}\rangle \to | \bar{T}\rangle$ transition is driven, the system preferentially decays vertically (shown as the blue transition in figure 1(c)). The emitted photon has an equal probability of being measured in each time bin (figure 3(b)). The light produced by this mechanism is emitted as single quanta because the detection of a photon at the wavelength of the enhanced transition heralds the preparation of the hole spin in the $| h\rangle$ state. Once in the $| h\rangle$ state the system cannot be re-excited by the laser driving the $| \bar{h}\rangle \to | \bar{T}\rangle$ transition until a spin-flip occurs. Spin-flips typically take three orders of magnitude longer than the W-state generation process and so are not of concern here [44]. In contrast, the single-photon emission that is usually observed from quantum dots relies on the spontaneous decay time of a transition being greater than the applied pulse length [45].

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The setup for our scheme is shown in figure 1(a). A non-resonant pulse ensures that there is a non-zero population in the $| \bar{h}\rangle$ state. Then resonant pulses produced using a continuous-wave laser and an electro-optic modulator (EOM) are used to drive the Raman transition.

First, we use a single resonant pulse and second-order correlation function measurement to demonstrate that the system can act as a single-photon source. figure 4(a) shows a time resolved measurement of the output photons and figure 4(e) shows the second-order correlation function measurement. The low ${g}^{(2)}(0)\approx 0.02$ indicates that the output light is primarily composed of single photons.

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We then use a two pulse sequence to create a photon superposed across two time bins (figure 4(b)). Extending this to a three pulse sequence we generate a single-photon three-part W-state (figure 4(c)). Finally, we use a four pulse sequence to create a four part W-state and demonstrate the scalability of our scheme (figure 4(d)). The output light for all of the pulse sequences have a greatly reduced peak in the second-order correlation function at τ = 0, indicating that the output is dominated by single-photon emission (figures 4(e)–(h)). We work in the low efficiency regime using a resonant laser power well below saturation. This means that the $| \bar{h}\rangle$ population is not completely depleted by the final pulse of a pulse sequence, allowing us to add additional pulses to the end of the sequence to increase the dimensionality of the photonic state without requiring the recalibration of each pulse power.

We also observe anti-bunching over tens of nanoseconds; the peaks at τ = ±12.5 ns are lower than the other peaks away from τ = 0. This indicates that the non-resonant pulse does not completely randomise the hole spin state meaning that a photon is less likely to be generated if a photon has been produced during the preceding excitation sequence. This effect could be removed by deterministic preparation of the spin in the $| \bar{h}\rangle$ state prior to the incidence of the resonant laser pulses.

Finally, we probe the coherence between neighbouring time bins of the three-part W-state by time resolving the output of an unbalanced Michelson interferometer. This measurement shows that the photons are in a coherent superposition between different time bins and not simply emitted into one time bin or another probabilistically. The long arm of the interferometer delays the light by one time-bin relative to the short arm (figures 4(i) and (j)). This means that we expect to see interference between time-bin 1 and time-bin 2, and between time-bin 2 and time-bin 3 of a single 3-time-bin photon [46].

By varying the phase difference between the two arms we observe interference between the overlapping time bins (figure 4(k)). The interference measurements were also performed for the 2 and 4 part W-states; the visibilities obtained by fitting a sinusoidal function to the measured intensities of the overlapping time bins as a function of phase are shown in table 1. The phase of the photon is determined by the phase of the laser and the phase of the hole spin. We attribute the deviation from perfect interference visibility to the non-zero probability of a decay along the $| \bar{T}\rangle \to | \bar{h}\rangle$ transition and the decoherence of the hole spin.

Table 1.  Coherent single-photon states.

# Time bins	${g}^{(2)}(0)$	Visibility
1	0.0233 ± 0.0009	—
2	0.0463 ± 0.0012	65.6 ± 4.3%
3	0.0309 ± 0.0004	69.6 ± 5.4%
4	0.0577 ± 0.0004	67.7 ± 5.7%

As quantum dots in high Q cavities can act as deterministic photon sources [47], we propose an extension to our scheme to generate generate W-States deterministically. We assume that a future experiment will be performed with a quantum dot deterministically charged with a single charge, either an electron or a hole, which could be introduced by tunneling in an appropriately biased diode structure [48] or a deliberately doped dot [49]. Our proposal is as follows:

• 1.  
  Ensure that the system in the state $| \bar{h}\rangle$ by first preparing the $| h\rangle$ state as demonstrated earlier, then using an off-resonant pulse to perform a π-rotation of the hole spin [25, 37, 50].
• 2.  
  Apply a $\tfrac{\pi }{3}$ pulse to the diagonal transition indicated in figure 2(a). The resulting Raman emission leaves us with the state $\sqrt{\tfrac{2}{3}}| \bar{h}\rangle | {0}_{\tau =1}\rangle +\tfrac{1}{\sqrt{3}}| h\rangle | {1}_{\tau =1}\rangle$, where $| {0}_{\tau =1}\rangle$ $(| {1}_{\tau =1}\rangle )$ indicates the absence (presence) of a photon in time-bin 1.
• 3.  
  A $\tfrac{\pi }{2}$ pulse leaves us with the state $\tfrac{1}{\sqrt{3}}| \bar{h}\rangle | {0}_{\tau =2}{0}_{\tau =1}\rangle +\tfrac{1}{\sqrt{3}}| h\rangle | {1}_{\tau =2}{0}_{\tau =1}\rangle +\tfrac{1}{\sqrt{3}}| h\rangle | {0}_{\tau =2}{1}_{\tau =1}\rangle$.
• 4.  
  A final π pulse leaves us with the state $\tfrac{1}{\sqrt{3}}| h\rangle | {1}_{\tau =3}{0}_{\tau =2}{0}_{\tau =1}\rangle +\tfrac{1}{\sqrt{3}}| h\rangle | {0}_{\tau =3}{1}_{\tau =2}{0}_{\tau =1}\rangle +\tfrac{1}{\sqrt{3}}| h\rangle | {0}_{\tau =3}{0}_{\tau =2}{1}_{\tau =1}\rangle$.

Ignoring the hole state and considering the state of the photon in isolation it is clear that this gives us with a photon in the state $\tfrac{1}{\sqrt{3}}(| 001\rangle +| 010\rangle +| 100\rangle )$ — a single-photon W-state. For the success of this scheme, future work must investigate whether the use of a sequence of Raman pulses that result in a deterministic photon generation are going to reduce the coherence of the single, multi-time-bin Raman photon or whether the improved coherence properties of the emitted light due to the Purcell enhancement (discussed in [40] and [34]) will be enough to counter any detrimental effects of a higher laser power.

The experiments were performed using an InAs quantum dot in a GaAs/AlGaAs micropillar cavity with a Q-factor of ∼7500. The dot-cavity system was cooled to 5 K and placed in a 9 T Voigt geometry magnetic field. The resonant pulses are generated using a continuous-wave diode laser at ∼940 nm and electro-optic modulator. The resonant pulses are separated from the emitted light by both polarisation and spectral filtering. To record the HBT measurements we time-tag each photon, this allows us to remove the effect of photons generated from the non-resonant pulses on the correlation measurements through temporal filtering. The unbalanced Michelson interferometer used was kept at a constant temperature in order to minimise drift.