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Quantum order-by-disorder induced phase transition in Rydberg ladders with staggered detuning

$^{87}{\rm Rb}$ 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 $Δ$ and $λ$ respectively. We show that when $V_{\bf r r'} \gg(\ll) Δ, λ$ for $|{\bf r}-{\bf r'}|=1(>1)$, these ladders host a plateau for a wide range of $λ/Δ$ 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 $λ_c/Δ=1/3$ 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 $λ_c$ as a function of $V_0$. This allows for the possibility of a direct verification of this transition in standard experiments which we discuss.