Spin–cavity interactions between a quantum dot molecule and a photonic crystal cavity

The integration of InAs/GaAs quantum dots into nanophotonic cavities has led to impressive demonstrations of cavity quantum electrodynamics. However, these demonstrations are primarily based on two-level excitonic systems. Efforts to couple long-lived quantum dot electron spin states with a cavity are only now succeeding. Here we report a two-spin–cavity system, achieved by embedding an InAs quantum dot molecule within a photonic crystal cavity. With this system we obtain a spin singlet–triplet Λ-system where the ground-state spin splitting exceeds the cavity linewidth by an order of magnitude. This allows us to observe cavity-stimulated Raman emission that is highly spin-selective. Moreover, we demonstrate the first cases of cavity-enhanced optical nonlinearities in a solid-state Λ-system. This provides an all-optical, local method to control the spin exchange splitting. Incorporation of a highly engineerable quantum dot molecule into the photonic crystal architecture advances prospects for a quantum network.


Supplementary Note 1. QDM energy level structure
Charged QDMs exhibit bias-dependent photoluminescence spectra that are considerably more complicated than single QDs, but are well understood. 1 This is apparent in Supplementary Fig.  1a, where we present a PL bias map obtained under nonresonant excitation with a Ti-sapphire laser at 860 nm. In this particular set of data we have red detuned the cavity by gas adsorption in order to prevent the bright cavity emission from obscuring spectral features. Charging steps are observed which we assign to the 0e, 1e, 2e, and 3e charge states. Direct calculations of the doubly charged QDM transition spectrum following the methods of Refs. 1 and 2 using a tunnel coupling t ~ 1.75 meV agree with the experiment over the range where the two electron state is stable, reproducing the singlet, triplet, and other weak transitions in the data ( Supplementary Fig.  1b). Additional features such as the "x-patterns" common in QDM spectra appear in the calculation but are absent from experiment due to the fact that they lie outside the bias region where the two electron state is stable. In Supplementary Fig. 1c we plot the energies of the singlet, triplet, and the excited states in a doubly charged QDM from which the spectrum in Supplementary Fig. 1b can be obtained by taking the difference between the excited and ground states to obtain the transition energies and then using the state vectors to determine the intensity.

Supplementary Note 2. Cavity-QDM coupling
We extract the cavity-QDM coupling constant, , by sweeping the cavity through the triplet transition via a gas adsorption technique. The resulting time-photoluminescence map in Supplementary Fig. 2 exhibits a clear anticrossing. The minimum separation between the peaks ( ) at the anticrossing is ~80 μeV and is related to by 3

√
(1) where = 190 μeV is the cavity linewidth and = 28 μeV is the triplet linewidth estimated from Supplementary Fig. 3b. Using this equation we find that ≈ 57 μeV and therefore our cavity-QDM system is just in the strong coupling regime ( > κ/4 ≈ 47.5 μeV). In addition, note the enhancement of the anti-Stokes Raman as the cavity is swept through it.

Supplementary Note 3. Bias-dependent Raman linewidth
A single photon source based on a Raman process has many advantages over other schemes, one of which is that the linewidth of the photon is determined by the spin-dephasing time, not the spontaneous emission lifetime. However, utilizing a scanning Fabry-Perot interferometer we observe relatively broad Raman photons (~11 μeV) compared to earlier work with a single spin (3 μeV). 4 There are number of processes which could cause this effect, one of which is fast spinrelaxation due to the large exchange energy. 5 The Raman linewidth can also be increased by electric field fluctuations in either the voltage source or the local environment that cause jitter in the singlet, triplet, and excited state energy levels. Such spectral wandering effects can be significant in QDMs as the transition energies strongly depend on bias.
To test this, we extract the singlet, triplet, and anti-Stokes Raman full width at half maximum (FWHM) linewidths from the data in Fig. 3a as a function of bias. Example one Lorentzian (used for the singlet and triplet) and two Lorentzian (used for the cavity-stimulated Raman) fits to the data are shown in Supplementary Fig. 3a, clearly demonstrating that the Raman linewidth can vary with sample bias. The full dependence of the Raman linewidth on bias is shown in Supplementary Fig. 3b and the variation is significant, reaching a minimum value at the "sweet spot" ( ). The singlet and triplet transitions also exhibit variations, although they are less severe. We find that we can fit the bias-dependent Raman linewidth with the following model that accounts for voltage fluctuations of the exchange energy: is the experimentally measured anti-Stokes Raman linewidth, is the amplitude of the voltage fluctuations experienced by the sample, is the bias-dependent exchange energy, is the spectrometer resolution, and is the intrinsic Raman linewidth. The fit to the data is excellent (Supplementary Fig. 3b) when we use values of 4.5 mV (a value consistent with observations on past QDM samples), 15 μeV (the spectrometer resolution), and 8 μeV. The extracted value of agrees well with our Fabry Perot measurements suggesting that while electric field fluctuations do play a role, there are additional mechanisms broadening the Raman linewidth. It may be possible to reduce Raman linewidth by decreasing the exchange splitting. This will both extend the voltage range over which and also reduce the spin dephasing rate arising from cotunneling to the electron reservoir.

Supplementary Note 4. Raman power dependence
In Fig. 4b of the manuscript we plot the integrated intensity of the cavity-stimulated and resonant anti-Stokes Raman at a single laser power. In Supplementary Fig. 4 we present a complete characterization of the effect of laser power on the peak intensity, linewidth, and integrated intensity of the anti-Stokes Raman. These values are extracted by fitting a sum of three Lorentzians to the data. Supplementary Fig. 4a shows the Raman peak intensity for the resonant case (solid markers) and the cavity-stimulated case (open markers) at three detunings. Signs of saturation are evident in the resonant condition but less-so for the cavity-assisted process. We observe that at high powers the peak intensity of the cavity-assisted Raman can actually become comparable to resonant Raman, even at a cavity detuning of Δ CS = -400 μeV. Finally, we note that the Raman intensities in Fig. 3 and Supplementary Fig. 4 differ slightly. We attribute this to variations in the collection efficiency as these data were acquired on different days.
The Raman linewidth exhibits a surprising dependence on the laser power ( Supplementary Fig.  4b). This is particularly pronounced for the resonant cases where the linewidth increases as the laser power exceeds ~10 μW for all cavity detunings. In addition, Raman photons for the doubly resonant case (Δ CS = 0 μeV) appear to have a larger linewidth than the detuned cases. The linewidth of cavity-stimulated Raman photons increases only marginally.
Finally, while the peak intensity saturates for the resonant Raman, the integrated intensity does not ( Supplementary Fig. 4c). This occurs because of the substantial broadening of the resonant Raman photons. Why this broadening is not compensated for by a concomitant reduction in peak intensity remains an open question.

Supplementary Note 5. Expressions for the Autler-Townes and AC Stark effects
The Autler-Townes effect can most easily be described in a strongly-driven two-level system. If the drive field is made sufficiently strong, the levels become "dressed" by the laser field and form polariton-like states. As a result both the ground ( ) and excited ( ) states split, forming a pair levels separated by the generalized Rabi frequency, is the Rabi frequency of the laser, is the laser detuning from the transition, and corresponds to the unperturbed state energies. This splitting can be directly resolved in a threelevel system by observing emission or absorption between the dressed excited states and a spectator state. The spectral signature is a doublet spilt by on resonance (=0).
In the limit of large detuning ( ) the eigenstates are well described by the bare states, which are weakly dressed and experience small energy shifts. The magnitude of this shift can be found by expanding Eqs. (3,4) to second order in and keeping only the solution close to . This results in and is known as the AC Stark shift. Note that the amplitude of the shift scales linearly with the laser intensity, in contrast to the Autler-Townes splitting.
Since the cavity is strongly driven, we anticipate that a good approximation is to ignore the effect of the QDM on the cavity which is weak compared to the laser. The steady state of the cavity can then be modeled as an effective field driving the QDM. Following this process we find that the Rabi frequency appearing in Eqs. (3-6) is replaced by a cavity-enhanced Rabi frequency: Here, is the coupling of the laser to the cavity and is the detuning of the laser from the cavity (Fig. 2b). We find that these laser-cavity interactions cause remarkable modifications of the Autler-Townes and AC Stark effects as shown in Fig. 5 and Fig. 6 of the manuscript.

Supplementary Note 6. Cavity-QDM modeling
When the system is driven by vertically polarized light the cavity is driven very strongly. The interaction of the cavity with the external field is much stronger than the interaction of the cavity with the quantum dot, which to first approximation we can ignore. Our strategy is therefore to solve the equation of motion of the cavity annihilation/creation operator for the driven and lossy cavity in the absence of the quantum dot. This leads to a steady state solution for the cavity operators which can then be plugged into the equation of motion for the quantum dot degrees of freedom, allowing us to obtain an effective Hamiltonian for the QDM alone. This approximation is justified because there is an asymmetry in how strongly the QDM affects the cavity versus how much the cavity affects the QDM. It should be a valid approximation in the limit where the cavity is populated with a large number of photons, . Following this process we derive an effective cavity-induced driving of the QDM, effectively amplifying the driving field the quantum dots experience.