Fundamental Limits to Coherent Photon Generation with Solid-State Atomlike Transitions

Coherent generation of indistinguishable single photons is crucial for many quantum communication and processing protocols. Solid-state realizations of two-level atomic transitions or three-level spin-$\mathrm{\Lambda }$ systems offer significant advantages over their atomic counterparts for this purpose, albeit decoherence can arise due to environmental couplings. One popular approach to mitigate dephasing is to operate in the weak-excitation limit, where the excited-state population is minimal and coherently scattered photons dominate over incoherent emission. Here we probe the coherence of photons produced using two-level and spin-$\mathrm{\Lambda }$ solid-state systems. We observe that the coupling of the atomiclike transitions to the vibronic transitions of the crystal lattice is independent of the driving strength, even for detuned excitation using the spin-$\mathrm{\Lambda }$ configuration. We apply a polaron master equation to capture the non-Markovian dynamics of the vibrational manifolds. These results provide insight into the fundamental limitations to photon coherence from solid-state quantum emitters.

Figure 1

A quantum dot coupled to vibrational modes. (a) The equilibrium position of crystal lattice ions depends on the charge state of the quantum dot (schematic illustration) due to the deformation potential electron-phonon interaction. (b) Energy level schematic for the neutral exciton ($X^0$) in the weak-driving regime, showing elastic scattering from the excited state (green) and the inelastic Stokes (red) or anti-Stokes (blue) scattering due to the electronic relaxation from the excited-state to the ground-state vibrational manifold. (c) Spin-$\mathrm{\Lambda }$ energy level structure for the negatively charged exciton ($X^{1-}$) in the weak-driving regime, showing the inelastic zero-phonon (green) and stokes (red) and anti-stokes (blue) scatterings for the Raman spin-flip transition (yellow). The scattering from the spin-preserving transition is not shown. Single (double) arrows represent electron (heavy-hole) spin states. The energy separation between the Zeeman split ground (excited) states is given by ${\delta }_e$ (${\delta }_h$).

Figure 2

Resonance fluorescence spectra from a solid-state two-level transition. (a)–(c) Scattered photons at three different Rabi frequencies ($\mathrm{\Omega }\approx 0.1{\mathrm{\Omega }}_{\mathrm{sat}}$, ${\mathrm{\Omega }}_{\mathrm{sat}}$, $10{\mathrm{\Omega }}_{\mathrm{sat}}$) show a narrow ZPL accompanied by a broad PSB. The fit is produced by the non-Markovian model, using a super-Ohmic spectral density [68] with system-phonon coupling strength $\alpha =0.03{\mathrm{ps}}^2$ and frequency cutoff ${\omega }_c=2.2{\mathrm{ps}}^{-1}$. (d)–(f) Higher resolution ($\approx 0.1\mu \mathrm{eV}$) spectra, as a function of the detuning from the ZPL (1.287 eV) show the evolution from a coherent elastic peak at $\mathrm{\Omega }\ll {\mathrm{\Omega }}_{\mathrm{sat}}$ to a Mollow triplet at $\mathrm{\Omega }\gg {\mathrm{\Omega }}_{\mathrm{sat}}$, matching the behavior of a coherently driven two-level system. (g) Our theoretical curve (orange) matches the data and confirms that, in the weak-driving regime ($\mathrm{\Omega }\ll {\mathrm{\Omega }}_{\mathrm{sat}}$), the coherence of the transform-limited photons is limited by the branching ratio, given by the Debye-Waller factor, ${\alpha }_{\mathrm{DW}}\approx 0.91$. Blue data points are experimental data, obtained by computing the ratio of elastic peak [shaded region in (d)–(f)] to total spectrum. The red line shows the theoretical behavior for an atomic two-level system.

Figure 3

Two-photon interference between scattered photons from a neutral exciton $X^0$ as a function of driving strength. (a)–(c) Two-photon interference visibilities, $V_{\mathrm{HOM}}(\tau )$ at $\mathrm{\Omega }\approx 0.5{\mathrm{\Omega }}_{\mathrm{sat}}$, $\mathrm{\Omega }\approx {\mathrm{\Omega }}_{\mathrm{sat}}$, and $\mathrm{\Omega }\approx 3{\mathrm{\Omega }}_{\mathrm{sat}}$, respectively. (d) The coalescence time window normalized by the lifetime of the emitter ($T_1=0.625\mathrm{ns}$) deduced from the experimental (dots) and theoretical (solid line) CTW as a function of driving Rabi frequency $\mathrm{\Omega }$. The dashed line represents the theoretical CTW curve without the contribution from phonons. The shaded region represents the region inaccessible in experiment.

Figure 4

Spin-flip Raman photon spectra from a solid-state Spin-$\mathrm{\Lambda }$ system. (a) Scattered Raman photons from QD1 shows a narrow ZPL and a broad PSB at $\mathrm{\Omega }\approx 0.1{\mathrm{\Omega }}_{\mathrm{sat}}$. The fit is produced by the non-Markovian model using the same set of parameters as in Fig. 2. (b) The ZPL fraction is constant over 2 orders of magnitude in Rabi frequencies. It gives ${\alpha }_{\mathrm{DW}}=0.924(4)$ (QD1) and ${\alpha }_{\mathrm{DW}}=0.911(4)$ (QD2). (c)–(e) Emission spectra of the scattered Raman photons from QD2 at three different Rabi frequencies ($\mathrm{\Omega }\approx 0.1{\mathrm{\Omega }}_{\mathrm{sat}}$, ${\mathrm{\Omega }}_{\mathrm{sat}}$, $10{\mathrm{\Omega }}_{\mathrm{sat}}$) show an Autler-Townes splitting at $\mathrm{\Omega }⪆{\mathrm{\Omega }}_{\mathrm{sat}}$. Zero detuning corresponds to energy of ZPL at 1.277 eV. The broad linewidth ($\mathrm{\Gamma }\approx 2\mu \mathrm{eV}$) originates from the spin ground-state dephasing due to nuclear spin fluctuations.