  A method for measuring small energy level shifts in a qubit by coherent
amplification of their effect is proposed. It is based on the repeated
application of the same interaction pulse in two manners: with the same phase
of each subsequent pulse, and with an alternating phase shift of $\pi$ (i.e. a
minus sign) from pulse to pulse. Two specific types of pulses are considered: a
resonant $\pi$ pulse and an adiabatic chirped pulse, both of which produce
complete population inversion with high fidelity. In the presence of a weak
ambient external electric or magnetic field, the ensuing Stark or Zeeman shift
leads to an energy level shift and hence a static detuning. In both the
resonant and adiabatic approaches, a small level shift does not alter the
transition probability very much; however, it can significantly change the
dynamical phases in the propagator. The repeated application of the same pulse
greatly amplifies the changes in the dynamical phases and maps them onto the
populations. Hence the effect of the level shift can be measured with good
accuracy. It is found that sequences of pulses with alternating phases deliver
much greater error amplification and much steeper excitation profiles around
resonance, thereby providing much higher sensitivity to small energy level
shifts. Explicit analytic estimates of the sensitivity are derived using the
well-known non-crossing Rosen-Zener and Rabi models and the level-crossing
Demkov-Kunike model. This recipe provides a simple tool for rapid and accurate
sensing of weak electric and magnetic fields by using the same pulse generating
an inversion quantum gate, without sophisticated tomography or entangling
operations.
