  The length of time that a quantum system can exist in a superposition state
is determined by how strongly it interacts with its environment. This
interaction entangles the quantum state with the inherent fluctuations of the
environment. If these fluctuations are not measured, the environment can be
viewed as a source of noise, causing random evolution of the quantum system
from an initially pure state into a statistical mixture-a process known as
decoherence. However, by accurately measuring the environment in real time, the
quantum system can be maintained in a pure state and its time evolution
described by a quantum trajectory conditioned on the measurement outcome. We
employ weak measurements to monitor a microwave cavity embedding a
superconducting qubit and track the individual quantum trajectories of the
system. In this architecture, the environment is dominated by the fluctuations
of a single electromagnetic mode of the cavity. Using a near-quantum-limited
parametric amplifier, we selectively measure either the phase or amplitude of
the cavity field, and thereby confine trajectories to either the equator or a
meridian of the Bloch sphere. We perform quantum state tomography at discrete
times along the trajectory to verify that we have faithfully tracked the state
of the quantum system as it diffuses on the surface of the Bloch sphere. Our
results demonstrate that decoherence can be mitigated by environmental
monitoring and validate the foundations of quantum feedback approaches based on
Bayesian statistics. Moreover, our experiments suggest a new route for
implementing what Schrodinger termed "quantum steering"-harnessing action at a
distance to manipulate quantum states via measurement.
