Intertwined and vestigial order with ultracold atoms in multiple cavity modes

Sarang Gopalakrishnan, Yulia E. Shchadilova, and Eugene Demler

Phys. Rev. A 96, 063828 – Published 19 December 2017

Abstract

Atoms in transversely pumped optical cavities “self-organize” by forming a density wave and emitting superradiantly into the cavity mode(s). For a single-mode cavity, the properties of this self-organization transition are well characterized both theoretically and experimentally. Here, we explore the self-organization of a Bose-Einstein condensate in the presence of two cavity modes—a system that recently was realized experimentally [Léonard et al., Nature (London) 543, 87 (2017)]. We argue that this system can exhibit a “vestigially ordered” phase in which neither cavity mode exhibits superradiance but the cavity modes are mutually phase locked by the atoms. We argue that this vestigially ordered phase should generically be present in multimode cavity geometries.

Received 22 July 2017
Corrected 16 August 2018

DOI:https://doi.org/10.1103/PhysRevA.96.063828

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Quantum simulation Superradiance & subradiance

Bose-Einstein condensates

Atomic, Molecular & Optical

Corrections

16 August 2018

Erratum

Publisher's Note: Intertwined and vestigial order with ultracold atoms in multiple cavity modes [Phys. Rev. A 96, 063828 (2017)]

Sarang Gopalakrishnan, Yulia E. Shchadilova, and Eugene Demler

Phys. Rev. A 98, 029901 (2018)

Authors & Affiliations

Sarang Gopalakrishnan¹, Yulia E. Shchadilova², and Eugene Demler²

¹Department of Engineering Science and Physics, CUNY College of Staten Island, Staten Island, New York 10314, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

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Vol. 96, Iss. 6 — December 2017

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Images

Figure 1
Sketch of the geometry considered in this paper. Panel (a) shows the assumed experimental setup with a retroreflected pump laser and two cavities at an angle $θ$ relative to one another. The atoms are Bose condensed and lie at the intersection of the two cavity axes and the pump axis. The relevant low-energy atomic momentum modes that are coupled either directly or indirectly to light are shown in panel (b).
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Figure 2
Nature of mode softening in the two-cavity setup. The thin black lines indicate the atomic and photonic polaritons that are familiar from the single-mode problem. These are at energies $E_{R}$ and $Δ_{C}$ in the uncoupled problem, and the former softens giving the usual Dicke transition in a single-mode cavity. Note that there are additional atomic modes (the thin gray lines), which are weakly coupled to the photons and are insensitive to the self-organization transition. The $χ$ mode, which softens to give vestigial order, is depicted by a thick black line. Photon-mediated interactions mix atomic density waves with photons and mix the $χ$ mode with the two-polariton branch (the dashed gray line).
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Figure 3
(a) Phase diagram of the low-energy Hamiltonian (11) and order parameters (b) $〈{\hat{x}}_{χ}〉$ , (c) $〈{\hat{x}}_{1}〉$ , and (d) $〈{\hat{x}}_{2}〉$ as a function of the detunings of the two cavities $λ_{1}$ and $λ_{2}$ . The diagram is calculated for $E_{χ} = 0.01, ξ = 1$ , and $γ = γ^{'} = 0.1$ . The different colors in panel (a) correspond to different phases (see the text for discussion). All phase transitions are of the second order.
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Figure 4
Three-mode mean-field (a) phase diagram and order parameters (b) $〈{\hat{x}}_{χ}〉$ , (c) $〈{\hat{x}}_{1}〉$ , and (d) $〈{\hat{x}}_{2}〉$ as a function of the detunings of the two cavities $λ_{1}$ and $λ_{2}$ . The situation shown here is for the case when $E_{χ}$ is fixed at 0.01 in units where the strength of the cubic anisotropy is set to $ξ = 1$ and the quartic anisotropy is $γ = 0.1$ .
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Figure 5
(a) Phase diagram of the low-energy Hamiltonian (11) and order parameters (b) $〈{\hat{x}}_{χ}〉$ , (c) $〈{\hat{x}}_{1}〉$ , and (d) $〈{\hat{x}}_{2}〉$ as a function of the detunings of the two cavities $λ_{1}$ and $λ_{2}$ . The diagram is calculated for $E_{χ} = 0.01, ξ = 1$ , and $γ = γ^{'} = 0.2$ . Different colors in panel (a) correspond to different phases (see the text for discussions). All phase transitions are of the second order.
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Figure 6
Energy difference between the VO ground state and the trivial phase of the low-energy Hamiltonian (11) as a function of the detunings of the two cavities $λ_{1}$ and $λ_{2}$ . The energy gap is shown for $E_{χ} = 0.01, ξ = 1$ , and $γ = γ^{'} = 0.1 s$ .
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