  We describe the quantum phase transition in the $N$-state chiral clock model
in spatial dimension $d=1$. With couplings chosen to preserve time-reversal and
spatial inversion symmetries, such a model is in the universality class of
recent experimental studies of the ordering of pumped Rydberg states in a
one-dimensional chain of trapped ultracold alkali atoms. For such couplings and
$N=3$, the clock model is expected to have a direct phase transition from a
gapped phase with a broken global $\mathbb{Z}_N$ symmetry, to a gapped phase
with the $\mathbb{Z}_N$ symmetry restored. The transition has dynamical
critical exponent $z \neq 1$, and so cannot be described by a relativistic
quantum field theory. We use a lattice duality transformation to map the
transition onto that of a Bose gas in $d=1$, involving the onset of a single
boson condensate in the background of a higher-dimensional $N$-boson
condensate. We present a renormalization group analysis of the strongly coupled
field theory for the Bose gas transition in an expansion in $2-d$, with $4-N$
chosen to be of order $2-d$. At two-loop order, we find a regime of parameters
with a renormalization group fixed point which can describe a direct phase
transition. We also present numerical density-matrix renormalization group
studies of lattice chiral clock and Bose gas models for $N=3$, finding good
evidence for a direct phase transition, and obtain estimates for $z$ and the
correlation length exponent $\nu$.
