Doppler-broadened quantum electromagnetically-induced-transparency heat engines

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Doppler-broadened quantum electromagnetically-induced-transparency heat engines

Xiao-Jun Zhang, G. C. La Rocca, M. Artoni, Hai-Hua Wang, and Jin-Hui Wu

Phys. Rev. A 103, 062205 – Published 2 June 2021

Abstract

A nontraditional quantum heat engine based on electromagnetically induced transparency has recently been suggested and experimentally demonstrated in ultracold atoms. In more practical setups with warm atoms, thermal atomic motions might hamper this heat engine mechanism. We here show that a Doppler-broadened atomic sample can still behave like an engine. However, only photons emitted in the direction of the coupling laser have the same brightness as for the Doppler-free engine, while the larger the angular deviation of emission, the lower the brightness. Our results suggest that the lower brightness can be seen as an act of Doppler broadening as if the thermal occupation number of the reservoir serving as entropy sink is increased, and the quantum heat engines may be feasible in warm atomic interfaces.

Received 20 December 2020
Accepted 17 May 2021

DOI:https://doi.org/10.1103/PhysRevA.103.062205

Physics Subject Headings (PhySH)

Light-matter interaction Optical coherence Quantum optics Quantum thermodynamics

Atomic, Molecular & Optical

Authors & Affiliations

Xiao-Jun Zhang ¹, G. C. La Rocca^2,*, M. Artoni^3,4,†, Hai-Hua Wang⁵, and Jin-Hui Wu^1,‡

¹School of Physics, Northeast Normal University, Changchun 130024, China
²Scuola Normale Superiore and CNISM, 56126 Pisa, Italy
³European Laboratory for Nonlinear Spectroscopy, 50019 Firenze, Italy
⁴National Institute Optics (National Research Council), 50019 Firenze, Italy
⁵College of Physics, Jilin University, 130023 Changchun, China

^*giuseppe.larocca@sns.it
^†artoni@lens.unifi.it
^‡jhwu@nenu.edu.cn

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Vol. 103, Iss. 6 — June 2021

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Images

Figure 1
Doppler-broadened quantum EIT heat engine: (a) Three-level atoms generate the field $Ω_{p}$ (purple arrow) as output in the presence of thermal radiation from two black-bodies of different temperatures (red shapes) and of a coherent coupling field $Ω_{c}$ (orange arrow). (b) Owning to the thermal motion of the atoms, the brightness of the output is not isotropic—it depends on the angle $θ$ between the traveling direction of the output field and the $z$ axis along which the laser beam $Ω_{c}$ propagates. The atoms are contained in a two-dimensional cell the size of which is much larger than the characteristic length $ξ_{o}$ , represented by the white semicircular dashed line, over which the output field reaches its asymptotic value (see Sec. 2).
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Figure 2
The averaged (a) absorption and (b) emission coefficients: The different curves correspond to different angles $θ$ . The Gaussian Doppler broadening $f_{G}$ is employed with $k v_{T} = 5 γ_{31}$ , and $Ω_{c} = γ_{31}$ , $T_{13} = 4000$ K, $T_{23} = 5000$ K. The atomic parameters are appropriate for the ${}^{87}{Rb} D_{2}$ line: $Γ_{31} = Γ_{32} = 2 π \times 6$ MHz, $γ_{0} = 2 π \times 16$ kHz, $λ_{31} = 780$ nm, and $ω_{21} = 2 π \times 6.8$ GHz.
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Figure 3
Brightness $B (δ_{p}, θ, \infty)$ under different widths of the Doppler broadening ( $k v_{T}$ ) plotted as a function of $δ_{p} / γ_{31}$ and the angle $θ$ . Data in panel (a) are obtained under $k v_{T} = 10 γ_{31}$ , (b) $k v_{T} = 5 γ_{31}$ , and (c) $k v_{T} = γ_{31}$ . The other parameters are the same as in Fig. 2.
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Figure 4
(a) The $ξ − θ$ dependence of the relative brightness $B_{0} / {\bar{n}}_{13}$ . The Doppler broadening is set to $k v_{T} = 5 γ_{31}$ and $Ω_{c} = 1.3 γ_{31}$ , which is equivalent to $w = v_{T}$ [being the parameter $w$ given in Eq. (9)], and the Gaussian-broadening function $f_{G} (v_{i})$ is adopted in the calculation. (b) Relation between $w$ and $Ω_{c}$ (green solid line) with $k w = 5 γ_{31}$ marked by the pink horizontal line. (c) Brightness obtained from analytic result (10) presented as a function of $θ$ (red solid line). The numerical results with $f_{G} (v_{i})$ (gray solid line) and $f_{L} (v_{i})$ (black dotted line) modeling the Doppler broadening are plotted for comparison. $k v_{T} = 5 γ_{31}$ , and $w = v_{T}$ just as in panel (a). The other parameters are identical with that in Fig. 2.
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Figure 5
(a) $B_{0} (θ = 1^{\circ})$ from Eq. (10) (black solid line) and from Eq. (13) (orange circle marked line), the horizontal dash-dotted line corresponds to the ideal quantum heat engine. Here $k v_{T} = 5 γ_{31}$ . (b) Entropy of the output radiation under different Doppler broadenings $k v_{T} = γ_{31}$ (blue dotted line), $3 γ_{31}$ (purple dashed line), $5 γ_{31}$ (orange dash-dotted line), and $10 γ_{31}$ (red solid line). The horizontal dashed line and the horizontal dash-dotted line indicate, respectively, the limits due to the thermodynamics inequalities of Eq. (18) (blackbody reservoir only) and Eq. (19) (ideal quantum heat engine). (c) Emission rates of the output plotted for $k v_{T} = γ_{31}$ , $3 γ_{31}$ , $5 γ_{31}$ , and $10 γ_{31}$ are plotted with respect to $Ω_{c}$ . The same correspondence between line style and value of $k v_{T}$ as in panel (b) is used here, and other parameters are identical to those in Fig. 2.
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