How coherence measurements of a qubit steer its quantum environment

Repetitive Ramsey interferometry measurements (RIMs) are often used to measure qubit coherence, assuming that the environment remains unaffected after each measurement and the outcomes of all measurements are independent and identically distributed (i.i.d.). While this assumption is often valid for a classical environment, it may not hold for a quantum environment due to non-negligible backaction. Here we present a general theoretical framework to account for the measurement backaction in sequential RIMs. We show that an RIM induces a quantum channel on the quantum environment, and sequential RIMs gradually steer the quantum environment to the fixed points of the channel. For the first time, we reveal three distinct environment steering effects – polarization, depolarization and metastable polarization, depending on the commutativity of the noise operator B and the environment Hamiltonian H_e: (1) When [B,H_e]=0, the quantum environment is gradually polarized to different eigenstates of B as the number m of repetitive RIMs increases; (2) When [B,H_e]≠ 0, the quantum environment is gradually depolarized to a maximally mixed state of its whole Hilbert space or a Hilbert subspace; (3) When [B,H_e]≠ 0 but one of H_e and B is a small perturbation on the other, metastable polarization can happen, such that the quantum environment is first polarized for a finite range of m but becomes gradually depolarized as m increases further. The environment steering also makes the measurement statistics of sequential RIMs develop non-i.i.d. features. Realistic examples of central spin models are also presented. Our work not only elucidates the measurement backaction and statistics of repetitive qubit coherence measurements, but is also useful for designing protocols to engineer the state or dynamics of a quantum environment with a qubit ancilla.