How coherence measurements of a qubit steer its quantum environment

A qubit suffers decoherence when coupled to a classical or quantum environment, and characterizing the qubit decoherence process is vital in quantum technologies. 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 valid for a classical environment, it may not hold for a quantum environment due to the non-negligible backaction from qubit to environment, especially when the environment has a memory time much longer than the duration of each RIM cycle. Here, we present a general theoretical framework to incorporate the measurement backaction from qubit to environment in sequential RIMs. We show that a RIM of a qubit induces a quantum channel on the quantum environment, and sequential RIMs gradually steer the quantum environment to the fixed points of the channel. We reveal three distinct environment steering effects---polarization, depolarization, and metastable polarization, depending on the commutativity of the noise operator $B$ and the free environment Hamiltonian ${H}_e$: (1) If $B$ commutes with ${H}_e$, i.e., $[B,{H}_e]=0$, the quantum environment is gradually polarized to different eigenstates of $B$ as the number $m$ of repetitive RIMs increases; (2) when $[B,{H}_e]\ensuremath{\ne}0$, the quantum environment is gradually depolarized to a maximally mixed state of its whole Hilbert space or a Hilbert subspace; (3) when $[B,{H}_e]\ensuremath{\ne}0$ 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, such that the measurement result distribution can display multiple peaks for a small quantum environment, corresponding to different fixed points of the quantum channel. Realistic examples of central spin models are presented to demonstrate the measurement statistics and environment steering effects. Our paper 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.