Quantum order-by-disorder induced phase transition in Rydberg ladders with staggered detuning
Madhumita Sarkar, Mainak Pal, Arnab Sen, K. Sengupta
Abstract
^{87} Rb <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mi/> <mml:mn>87</mml:mn> </mml:msup> <mml:mi>R</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> atoms are known to have long-lived Rydberg excited states with controllable excitation amplitude (detuning) and strong repulsive van der Waals interaction V_{{r} {r'}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>′</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:math> between excited atoms at sites {r} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>r</mml:mi> </mml:math> and {r'} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>′</mml:mi> </mml:mrow> </mml:math> . Here we study such atoms in a two-leg ladder geometry in the presence of both staggered and uniform detuning with amplitudes \Delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Δ</mml:mi> </mml:math> and \lambda <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>λ</mml:mi> </mml:math> respectively. We show that when V_{{r r'}} \gg(\ll) \Delta, \lambda <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>r</mml:mi> <mml:mi>′</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≫</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mo>≪</mml:mo> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>λ</mml:mi> </mml:mrow> </mml:math> for |{r}-{r'}|=1(&gt;1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mi>r</mml:mi> <mml:mo>−</mml:mo> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>′</mml:mi> </mml:mrow> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> , these ladders host a plateau for a wide range of \lambda/\Delta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mi>/</mml:mi> <mml:mi>Δ</mml:mi> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mi>/</mml:mi> <mml:mi>Δ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> which agrees well with exact-diagonalization based numerical studies. We also study the fate of this transition for a realistic interaction potential V_{{r} {r'}} = V_0 /|{r}-{r'}|^6 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>′</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>0</mml:mn>