  Lasers with high spectral purity are indispensable for optical clocks and
coherent manipulation of atomic and molecular qubits for applications such as
quantum computing and quantum simulation. Stabilisation of the laser to a
reference can provide a narrow linewidth and high spectral purity. However,
widely-used diode lasers exhibit fast phase noise that prevents high fidelity
qubit manipulation. Here we demonstrate a self-injection locked diode laser
system utilizing a medium finesse cavity. The cavity not only provides a stable
resonance frequency, but at the same time acts as a low-pass filter for phase
noise beyond the cavity linewidth of around 100 kHz, resulting in low phase
noise from dc to the injection lock limit.
  We model the expected laser performance and benchmark it using a single
trapped $^{40}$Ca$^{+}$-ion as a spectrum analyser. We show that the fast phase
noise of the laser at relevant Fourier frequencies of 100 kHz to >2 MHz is
suppressed to a noise floor of between -110 dBc/Hz and -120 dBc/Hz, an
improvement of 20 to 30 dB over state-of-the-art Pound-Drever-Hall-stabilized
extended-cavity diode lasers. This strong suppression avoids incoherent
(spurious) spin flips during manipulation of optical qubits and improves
laser-driven gates in using diode lasers with applications in quantum logic
spectroscopy, quantum simulation and quantum computation.
