Ground-state cooling enabled by critical coupling and dark entangled states

Cristian L. Cortes, Matthew Otten, and Stephen K. Gray

Phys. Rev. B 99, 014107 – Published 14 January 2019

Abstract

We analyze the cooling of a mechanical resonator coupled to an ensemble of interacting two-level systems via an open quantum systems approach. Using an exact analytical result, we find optimal cooling occurs when the phonon mode is critically coupled $(γ \sim g)$ to the two-level system ensemble. Typical systems operate in suboptimal cooling regimes due to the intrinsic parameter mismatch $(γ ≫ g)$ between the dissipative decay rate $γ$ and the coupling factor $g$ . To overcome this obstacle, we show that carefully engineering the coupling parameters through the strain profile of the mechanical resonator allows phonon cooling to proceed through the dark (subradiant) entangled states of an interacting ensemble, thereby resulting in optimal phonon cooling. Our results provide an avenue for ground-state cooling and should be accessible for experimental demonstrations.

Received 7 November 2018
Revised 29 November 2018

DOI:https://doi.org/10.1103/PhysRevB.99.014107

Physics Subject Headings (PhySH)

Cooling & trapping Optomechanics Phonons Quantum entanglement Quantum interference effects

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalQuantum Information, Science & Technology

Authors & Affiliations

Cristian L. Cortes^*, Matthew Otten^†, and Stephen K. Gray^‡

Center for Nanoscale Materials, Argonne National Laboratory, Lemont, Illinois 60439, USA

^*ccortes@anl.gov
^†otten@anl.gov
^‡gray@anl.gov

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Vol. 99, Iss. 1 — 1 January 2019

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Images

Figure 1
(a) Model for ground-state cooling using phonon-assisted optical transitions in a three-level system. An external laser pumps the system into the ground state $| g 〉$ of a magnetically forbidden two-level subspace $(ω_{o} ≪ ω_{p})$ with decay rate $γ$ . The two-level subspace couples directly to the mechanical resonator with coupling strength $g$ . Critical coupling $(g \sim γ)$ ensures efficient cooling due to optimal phonon absorption from the ground state $| g 〉$ to the excited state $| e 〉$ . (b) Schematic of resonant phonon cooling using odd-parity strain profile. This configuration couples directly to the dark (subradiant) eigenstates of a two-level system ensemble with all-to-all interactions resulting in optimal cooling.
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Figure 2
Phonon cooling figure of merit, $F = 〈 a^{†} a 〉 / n_{th}$ , as a function of mechanical damping $κ$ and two-level system decay rate $γ$ . Ground-state cooling is only possible in a small quadrant of the parameter space (light green color), requiring $κ ≪ g$ and $γ \sim g$ . The mechanical resonator $(ω_{o} / 2 π = ω_{m} / 2 π = 100$ MHz) is either in (a) a cryogenic bath, $n_{th} \sim 200$ , or (b) a room temperature bath, $n_{th} \sim 6 \times 10^{4}$ .
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Figure 3
Energy level diagram depicting relevant transitions for resonant phonon cooling with two interacting two-level atoms. The atomic system has four eigenstates corresponding to both atoms in their ground states $| g 〉$ and both atoms in their excited states $| e 〉$ , as well as the symmetric and antisymmetric states $| s 〉$ and $| a 〉$ with decay rates $γ_{s}$ and $γ_{a}$ (denoted by orange and blue arrows), which give rise to superradiance and subradiance, respectively. The superradiant and subradiant states have an interaction energy splitting $2 J_{12}$ and interact with the mechanical mode with coupling strengths $g_{s} = \frac{1}{\sqrt{2}} (g_{1} + g_{2})$ and $g_{a} = \frac{1}{\sqrt{2}} (g_{1} - g_{2})$ . Critical coupling $(g_{a} \sim γ_{a} / 2)$ and odd-parity coupling $(g_{1} = - g_{2})$ ensures the internal dynamics follows the ladder of states: $| g, n + 1 〉 \to | a, n 〉 \to | g, n 〉 \to | a, n - 1 〉 \to | g, n - 1 〉 \to \dots$ , resulting in optimal cooling. In the schematic the mechanical resonator frequency $ω_{m}$ is equal to the antisymmetric state frequency, $ω_{o} - J_{12}$ .
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Figure 4
Phonon cooling figure of merit $F$ for (a) two noninteracting atoms $(J_{12} = γ_{12} = 0)$ and (b) two interacting atoms $(J_{12} = 0, γ_{12} = \frac{99}{100} γ)$ . The optimal cooling regime (light green region) occurs when $γ / 2 = \sqrt{2} g$ (dashed line) for noninteracting atoms, but is altered dramatically $(γ ≫ g)$ for interacting atoms $(γ_{a} / 2 = \sqrt{2} g$ , dashed line). For the numerical simulations, we assumed anti-symmetric coupling $(g_{1} = - g_{2})$ , zero detuning $(ω_{o} = ω_{m})$ , and a thermal bath with average phonon number, $n_{t h} = 2$ .
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