Phonon Decoherence of Quantum Dots in Photonic Structures: Broadening of the Zero-Phonon Line and the Role of Dimensionality

We develop a general microscopic theory describing the phonon decoherence of quantum dots and indistinguishability of the emitted photons in photonic structures. The coherence is found to depend fundamentally on the dimensionality of the structure resulting in vastly different performance for quantum dots embedded in a nanocavity (0D), waveguide (1D), slab (2D), or bulk medium (3D). In bulk, we find a striking temperature dependence of the dephasing rate scaling as $T^{11}$ implying that phonons are effectively “frozen out” for $T\lesssim 4\mathrm{K}$. The phonon density of states is strongly modified in 1D and 2D structures leading to a linear temperature scaling for the dephasing strength. The resulting impact on the photon indistinguishability can be important even at sub-Kelvin temperatures. Our findings provide a comprehensive understanding of the fundamental limits to photon indistinguishability in photonic structures.

Figure 1

Phonon dephasing of spontaneous emission from QDs. (a) The emission spectrum consists of a ZPL and broad sidebands. (b) The quadratic coupling represents scattering of phonons through virtual excitations to a higher state and leads to ZPL broadening. (c) The linear coupling is associated with the emission or absorption of phonons by the QD. In bulk this leads to phonon sidebands in the emission. In nanostructures, an additional mechanism broadens the ZPL through long-wavelength deformations. (d) Error in two-photon interference versus temperature for QDs embedded in structures with different dimensionality. 0D corresponds to a QD in the center of a sphere with radius $R=80\mathrm{nm}$, 1D to a cylindrical waveguide with radius $\rho =80\mathrm{nm}$ and the QD placed in the cross-sectional center (1D) or halfway off center ($1{\mathrm{D}}^{\prime }$), 2D to a QD in the center of a freestanding membrane with height $2h=160\mathrm{nm}$, and 3D to a bulk medium. Each structure is represented by two curves that correspond to a large ($L=4.5\mathrm{nm}$) and small ($L=1.5\mathrm{nm}$) wave function denoted with large and small triangles, respectively.

Figure 2

Phonon dephasing in a bulk medium. (a) The linear (quadratic) exciton-phonon coupling affects the short-time (long-time) decay of coherence. Parameters: $T=10\mathrm{K}$, $L=3\mathrm{nm}$. (b) Phonon dephasing rate versus temperature for $L=1.5\mathrm{nm}$ (dashed line), $L=3\mathrm{nm}$ (solid line) and $L=4.5\mathrm{nm}$ (dotted line). The natural linewidth in a bulk medium is indicated by the dash-dotted line.

Figure 3

Phonon dephasing in photonic structures and the role of dimensionality. (a) Decay of coherence for the same structures as in Fig. 1 except the 0D structure ($R=20\mathrm{nm}$). Inset: initial decay of coherence (0–20 ps) for bulk (dotted line), sphere (solid line), and the $t^2$ approximation for the sphere. (b) Corresponding emission spectrum. (c) Two-photon indistinguishability versus size of the structure. All plots are for $T=5\mathrm{K}$, $L=3\mathrm{nm}$.

Figure 4

Suppressing phonon dephasing by clamping the photonic structures. The plot shows the photon infidelity versus thickness of ${\mathrm{SiO}}_2$ (gray) surrounding a GaAs waveguide (dark gray) of radius 80 nm with a QD (small black triangle) in the center (solid line) and halfway off center (dashed line). Parameters: $T=5\mathrm{K}$, $L=3\mathrm{nm}$, $v_{s,\mathrm{SiO}2}=5848{\mathrm{ms}}^{-1}$, ${\rho }_{m,\mathrm{SiO}2}=2.2\mathrm{g}{\mathrm{cm}}^{-3}$, ${\nu }_{\mathrm{SiO}2}=0.17$.