  Broadband quantum-limited amplifiers would advance applications in quantum
information processing, metrology, and astronomy. However, conventional
traveling-wave parametric amplifiers (TWPAs) support broadband amplification at
the cost of increased added noise. In this work, we develop and apply a
multi-mode, quantum input-output theory to quantitatively identify the
sidebands as a primary noise mechanism in all conventional TWPAs. We then
propose an adiabatic Floquet mode scheme that effectively eliminates the
sideband-induced noise and subsequently overcomes the trade-off between quantum
efficiency (QE) and bandwidth. We then show that a Floquet mode Josephson
traveling-wave parametric amplifier implementation can simultaneously achieve
$>20\,$dB gain and a QE of $\eta/\eta_{\mathrm{ideal}}> 99.9\%$ of the quantum
limit over more than an octave of bandwidth. Crucially, Floquet mode TWPAs also
strongly suppress the nonlinear forward-backward wave coupling and are
therefore genuinely directional. Floquet mode TWPAs can thus be directly
integrated on-chip without isolators, making near-perfect measurement
efficiency possible. The proposed Floquet scheme is also widely applicable to
other platforms such as kinetic inductance traveling-wave amplifiers and
optical parametric amplifiers.
