We study the squeezed states of the phonon modes and the associated
supercurrent in superconducting moleclular transistors with strong
electron-phonon interaction. It is suggested that the phonon squeezing can
be manipulated by means of continuous quantum measurement of the charge on
the molecule. We use both a numerical method (to solve the master equation)
and the nonequilibrium Keldysh formalism to show that the continuous
measurement gives rise to the quantum Zeno effect and hence the control of
the phonon squeezing.
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