  We theoretically investigate resonant dipole-dipole interaction (RDDI)
between artificial atoms in a 1D geometry, implemented by N transmon qubits
coupled through a transmission line. Similarly to the atomic cases, RDDI comes
from exchange of virtual photons of the unexcited modes, and causes the
so-called collective Lamb shift (CLS). To probe the shift, we effectively set
one end of the transmission line as a mirror, and examine the reflection
spectrum of the probe field from the other end. Our calculation shows that when
a qubit is placed at the node of the standing wave formed by the incident and
reflected waves, even though it is considered to be decoupled from the field,
it results in large energy splitting in the spectral profile of a resonant
qubit located elsewhere. This directly signals the interplay of virtual photon
processes and explicitly demonstrates the CLS. We further derive a master
equation to describe the system, which can take into account mismatch of
participating qubits and dephasing effects. Our calculation also demonstrates
the superradiant and subradiant nature of the atomic states, and how the CLS
scales when more qubits are involved.
