  Semi-device independent (Semi-DI) quantum random number generators (QRNG)
gained attention for security applications, offering an excellent trade-off
between security and generation rate. This paper presents a proof-of-principle
time-bin encoding semi-DI QRNG experiments based on a prepare-and-measure
scheme. The protocol requires two simple assumptions and a measurable
condition: an upper-bound on the prepared pulses' energy. We lower-bound the
conditional min-entropy from the energy-bound and the input-output correlation,
determining the amount of genuine randomness that can be certified. Moreover,
we present a generalized optimization problem for bounding the min-entropy in
the case of multiple-input and outcomes in the form of a semidefinite program
(SDP). The protocol is tested with a simple experimental setup, capable of
realizing two configurations for the ternary time-bin encoding scheme. The
experimental setup is easy-to-implement and comprises commercially available
off-the-shelf (COTS) components at the telecom wavelength, granting a secure
and certifiable entropy source. The combination of ease-of-implementation,
scalability, high-security level, and output-entropy make our system a
promising candidate for commercial QRNGs.
