Optical decoherence studies of Tm$^{3+}$:Y$_3$Ga$_5$O$_{12}$

Decoherence of the 795 nm $^3$H$_6$ to $^3$H$_4$ transition in 1%Tm$^{3+}$:Y$_3$Ga$_5$O$_{12}$ (Tm:YGG) is studied at temperatures as low as 1.2 K. The temperature, magnetic field, frequency, and time-scale (spectral diffusion) dependence of the optical coherence lifetime is measured. Our results show that the coherence lifetime is impacted less by spectral diffusion than other known thulium-doped materials. Photon echo excitation and spectral hole burning methods reveal uniform decoherence properties and the possibility to produce full transparency for persistent spectral holes across the entire 56 GHz inhomogeneous bandwidth of the optical transition. Temperature-dependent decoherence is well described by elastic Raman scattering of phonons with an additional weaker component that may arise from a low density of glass-like dynamic disorder modes (two-level systems). Analysis of the observed behavior suggests that an optical coherence lifetime approaching one millisecond may be possible in this system at temperatures below 1 K for crystals grown with optimized properties. Overall, we find that Tm:YGG has superior decoherence properties compared to other Tm-doped crystals and is a promising candidate for applications that rely on long coherence lifetimes, such as optical quantum memories and photonic signal processing.

Tm$^{3+}$:Y$_3$Ga$_5$O$_{12}$ materials for spectrally multiplexed quantum memories

We investigate the relevant spectroscopic properties of the 795 nm $^3$H$_6$$leftrightarrow$$^3$H$_4$ transition in 1% Tm$^{3+}$:Y$_3$Ga$_5$O$_{12}$ at temperatures as low as 1.2 K for optical quantum memories based on persistent spectral tailoring of narrow absorption features. Our measurements reveal that this transition has uniform coherence properties over a 56 GHz bandwidth, and a simple hyperfine structure split by $pm$44 MHz/T with lifetimes of up to hours. Furthermore, we find a $^3$F$_4$ population lifetime of 64 ms -- one of the longest lifetimes observed for an electronic level in a solid --, and an exceptionally long coherence lifetime of 490 $mu$s -- the longest ever observed for optical transitions of Tm$^{3+}$ ions in a crystal. Our results suggest that this material allows realizing broadband quantum memories that enable spectrally multiplexed quantum repeaters.

Practical long-distance quantum communication using concatenated entanglement swapping

We construct a theory for long-distance quantum communication based on sharing entanglement through a linear chain of $N$ elementary swapping segments of length~$L=Nl$ where $l$ is the length of each elementary swap setup. Entanglement swapping is achieved by linear optics, photon counting and post-selection, and we include effects due to multi-photon sources, transmission loss and detector inefficiencies and dark counts. Specifically we calculate the resultant four-mode state shared by the two parties at the two ends of the chain, and we derive the two-photon coincidence rate expected for this state and thereby the visibility of this long-range entangled state. The expression is a nested sum with each sum extending from zero to infinite photons, and we solve the case $N=2$ exactly for the ideal case (zero dark counts, unit-efficiency detectors and no transmission loss) and numerically for $N=2$ in the non-ideal case with truncation at $n_text{max}=3$ photons in each mode. For the general case, we show that the computational complexity for the numerical solution is $n_text{max}^{12N}$.