  Twin field quantum key distribution promises high key rates at long distance
to beat the rate distance limit. Here, applying the sending or not sending TF
QKD protocol, we experimentally demonstrate a secure key distribution breaking
the absolute key rate limit of repeaterless QKD over 509 km, 408 km ultra-low
loss optical fibre and 350 km standard optical fibre. Two independent lasers
are used as the source with remote frequency locking technique over 500 km
fiber distance; Practical optical fibers are used as the optical path with
appropriate noise filtering; And finite key effects are considered in the key
rate analysis. The secure key rates obtained at different distances are more
than 5 times higher than the conditional limit of repeaterless QKD, a bound
value assuming the same detection loss in the comparison. The achieved secure
key rate is also higher than that a traditional QKD protocol running with a
perfect repeaterless QKD device and even if an infinite number of sent pulses.
Our result shows that the protocol and technologies applied in this experiment
enable TF QKD to achieve high secure key rate at long distribution distance,
and hence practically useful for field implementation of intercity QKD.
