Random singlet phase of cold atoms coupled to a photonic crystal waveguide

David Z. Li, Marco T. Manzoni, and Darrick E. Chang

Phys. Rev. A 104, 013523 – Published 26 July 2021

Abstract

Systems consisting of cold atoms trapped near photonic crystal waveguides have recently emerged as an exciting platform for quantum atom-light interfaces. Such a system enables realization of tunable long-range interactions between internal states of atoms (spins), mediated by guided photons. Currently, experimental platforms are still limited by low filling fractions, where the atom number is much smaller than the number of sites at which atoms can potentially be trapped. Here, we show that this regime in fact enables interesting many-body quantum phenomena, which are typically associated with short-range disordered systems. As an example, we show how the system can realize the so-called “random singlet phase” (RSP), in which all atoms pair into entangled singlets, but the pairing occurs over a distribution of ranges as opposed to nearest neighbors. We use a renormalization group method to obtain the distribution of spin entanglement in the RSP, and show how this state can be approximately reached via adiabatic evolution from the ground state of a noninteracting Hamiltonian. We also discuss how experimentally this RSP can be observed. We anticipate that this work will accelerate the route toward the exploration of strongly correlated matter in atom-nanophotonics interfaces, by avoiding the requirement of perfectly filled lattices.

Received 5 May 2020
Revised 13 May 2021
Accepted 29 June 2021

DOI:https://doi.org/10.1103/PhysRevA.104.013523

Physics Subject Headings (PhySH)

Cold gases in optical lattices Entanglement in quantum gases Nanophotonics Photonic crystals

Atomic, Molecular & Optical

Authors & Affiliations

David Z. Li ¹, Marco T. Manzoni¹, and Darrick E. Chang^1,2

¹ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, E-08860 Castelldefels (Barcelona), Spain
²ICREA-Institució Catalana de Recerca i Estudis Avançats, E-08015 Barcleona, Spain

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Vol. 104, Iss. 1 — July 2021

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Images

Figure 1
(a) Schematic illustration of setup, consisting of a sparse and random filling of cold atoms [shaded (green) balls] tightly trapped in a lattice potential [(blue) periodic curve] near a 1D PCW. The atoms have a $Λ$ -level scheme, with the ground ( $g$ ) to excited ( $e$ ) state transition frequency being $ω_{e g}$ , and the excited and a metastable state ( $s$ ) coupled by a Raman laser with Rabi frequency $Ω$ and detuning $δ_{L}$ . If $ω_{e g}$ lies in the bandgap of the PhC, a photon bound state can form around the atom, illustrated as the (red) decaying envelope. (b) Typical band structure of a 1D PhC with bandgaps, with guided mode frequency $ω_{k}$ as a function of Bloch wave vector $k$ . The zoom-in rectangle shows the atomic transition frequency $ω_{e g}$ situated in a bandgap and close to a band edge.
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Figure 2
(a) Illustration of renormalization process and interactions mediated by singlet pairs for the case of four atoms. In the top line, atoms 2 and 3 experience the strongest interaction (thick black line) due to their proximity, so (in the second line) they form a singlet pair [indicated by the (red) line above the atoms] and can be “frozen out” of the 1D chain (indicated by transparent red). Below, we indicate half of the singlet state, with atom 2 initially in state $| ↑ 〉$ and atom 3 in state $| ↓ 〉$ . (First line) If atoms 1 and 4 are in states $| ↓ 〉$ and $| ↑ 〉$ , respectively, atoms 1 and 2 can virtually exchange their spins [dashed (red) circle] at a high energy cost. Atoms 2 and 3 can return to the singlet state (third line) if 3 and 4 virtually exchange their spins as well [second line, dashed (red) circle]. The entire process overall results in an effective interaction between atoms 1 and 4 mediated by the singlet pair of atoms 2 and 3. The effective distance $l$ between atoms 1 and 4 is therefore “renormalized” and shrunk by an amount of $d_{eff}$ (given in the text). This procedure generates pair nesting configurations where singlet pairs sit inside longer ones. (b) A representative ground state of the RSP for 10 atoms.
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Figure 3
(Left) Fraction of atoms $N (l_{m}) / N$ left unpaired, after pairs of atoms with an effective interaction distance up to $l_{m}$ have been renormalized into singlets. The solid (red) and dash-dotted (green) curves denote the results predicted by the RG flow equations and by numerical MPS simulations, respectively. For comparison, the dashed (black) curve denotes the unpaired atoms without RG, i.e., without allowing nested pairs to occur. (Right) Among the paired atoms, we plot the fractions of nested pairs at $l_{m}$ . The solid and dashed lines are for RG and MPS simulated results, respectively. The black, dark gray (red), and light gray (green) correspond to nesting orders $n_{l_{m}} = 0, 1, 2$ , respectively. Plotting parameters are $L = 5 a$ and 30% filling.
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Figure 4
Slew rate at pair breaking vs binding energy for 1000 random distributions of 12 atoms on 100 trapping sites. For comparison, the result for two atoms is also shown. The relevant parameters are $ε_{0} = J_{0}, L = 5 a$ , and $ϕ_{0} = π / 6$ . The black line is a guide to the eye, showing a $ω \propto {\tilde{J}}_{i j}$ scaling.
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Figure 5
Illustration of various terms in Eq. (C1). The atoms involved in the RG are denoted by circles. The strongest interacting pair, separated by effective distance $l_{m}$ (red online), is integrated out, leading to new effective distances between the remaining pair of atoms.
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Figure 6
Fractions of un-nested ( $n = 0$ ) and nested ( $n = 1, 2$ ) pairs at pair length $l_{m}$ for 12% filling fraction.
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