We investigate a measurement-feedback process of repeated operations with
time delay. During a finite-time interval, measurement on the system is
performed and the feedback protocol derived from the measurement outcome is
applied with time delay. This protocol is maintained into the next interval
until a new protocol from the next measurement is applied. Unlike a feedback
process without delay, both memories associated with previous and present
measurement outcomes are involved in the system dynamics, which naturally
brings forth a joint system described by a system state and two memory states.
The thermodynamic second law provides a lower bound for heat flow into a
thermal reservoir by the (3-state) Shannon entropy change of the joint system.
However, as the feedback protocol depends on memory states sequentially, we can
deduce a tighter bound for heat flow by integrating out irrelevant memory
states during dynamics. As a simple example, we consider the so-called cold
damping feedback process where the velocity of a particle is measured and a
dissipative feedback protocol is applied to decelerate the particle. We confirm
that the heat flow is well above the tightest bound. We also examine the
long-time limit of this feedback process, which turns out to exhibit an
interesting instability transition as well as heating by controlling parameters
such as measurement errors, time interval, protocol strength, and time delay
length. We discuss the underlying mechanism for instability and heating, which
might be unavoidable in reality.Comment: 5 pages, 4 figure
