| Summary: | $^{87}{\rm Rb}$ atoms are known to have long-lived Rydberg
excited states with controllable excitation amplitude (detuning) and
strong repulsive van der Waals interaction $V_{{\bf r} {\bf r'}}$
between excited atoms at sites ${\bf r}$ and ${\bf r'}$. Here we
study such atoms in a two-leg ladder geometry in the presence of
both staggered and uniform detuning with amplitudes $\Delta$ and
$\lambda$ respectively. We show that when $V_{{\bf r r'}} \gg(\ll)
\Delta, \lambda$ for $|{\bf r}-{\bf r'}|=1(>1)$, these ladders host
a plateau for a wide range of $\lambda/\Delta$ where the ground
states are selected by a quantum order-by-disorder mechanism from a
macroscopically degenerate manifold of Fock states with fixed
Rydberg excitation density $1/4$. Our study further unravels the
presence of an emergent Ising transition stabilized via the
order-by-disorder mechanism inside the plateau. We identify the
competing terms responsible for the transition and estimate a
critical detuning $\lambda_c/\Delta=1/3$ which agrees well with
exact-diagonalization based numerical studies. We also study the
fate of this transition for a realistic interaction potential
$V_{{\bf r} {\bf r'}} = V_0 /|{\bf r}-{\bf r'}|^6$, demonstrate that
it survives for a wide range of $V_0$, and provide analytic estimate
of $\lambda_c$ as a function of $V_0$. This allows for the
possibility of a direct verification of this transition in standard
experiments which we discuss.
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