Nonequilibrium thermodynamics and power generation in open quantum optomechanical systems

Paulo J. Paulino, Igor Lesanovsky, and Federico Carollo

Phys. Rev. A 108, 023516 – Published 21 August 2023

Abstract

Cavity optomechanical systems are a paradigmatic setting for the conversion of electromagnetic energy into mechanical work. Experiments with atoms coupled to cavity modes are realized in nonequilibrium conditions, described by phenomenological models encoding nonthermal dissipative dynamics and falling outside the framework of weak system-bath couplings. This fact makes their interpretation as quantum engines, e.g., the derivation of a well-defined efficiency, quite challenging. Here, we present a consistent thermodynamic description of open quantum cavity-atom systems. Our approach takes advantage of their nonequilibrium nature and arrives at an energetic balance which is fully interpretable in terms of persistent dissipated heat currents. The interaction between atoms and cavity modes can further give rise to nonequilibrium phase transitions and emergent behavior and allows us to assess the impact of collective many-body phenomena on the engine operation. To enable this, we define two thermodynamic limits, one related to a weak optomechanical coupling and one related to a strong optomechanical coupling. We illustrate our ideas by focusing on a time-crystal engine and discuss power generation, energy-conversion efficiency, and the emergence of metastable behavior in these limits.

Received 16 January 2023
Revised 24 July 2023
Accepted 26 July 2023

DOI:https://doi.org/10.1103/PhysRevA.108.023516

Physics Subject Headings (PhySH)

Nonequilibrium & irreversible thermodynamics Nonequilibrium statistical mechanics Open quantum systems

Nonequilibrium systems

Lindblad equation Mean field theory

Statistical Physics & Thermodynamics

Authors & Affiliations

Paulo J. Paulino ^1,*, Igor Lesanovsky ^1,2, and Federico Carollo ¹

¹Institut für Theoretische Physik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14, 72076 Tübingen, Germany
²School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom

^*paulo.paulino.souza96@gmail.com

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Issue

Vol. 108, Iss. 2 — August 2023

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Images

Figure 1
Nonequilibrium cavity-atom optomechanical engine. (a) Atoms in a cavity are driven by a laser with Rabi frequency $Ω$ and detuning $Δ$ . One of the cavity mirrors can move, allowing for oscillations of the cavity length $ℓ$ from its equilibrium position $ℓ_{0}$ . (b) The laser provides energy which flows through the cavity to the mirror. Atom and photon losses at rates $ν$ and $κ$ , respectively, determine energy dissipation. The mechanical power delivered by the engine equals the power the mirror dissipates due to friction. (c) Two possible thermodynamic limits for the system. The first features a finite density of atoms and results in a weak optomechanical coupling. The second features an infinite density of atoms, giving rise to a strong optomechanical coupling.
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Figure 2
Mechanical power output of the time-crystal quantum engine. (a) Power as a function of $Ω$ and $g$ , given in units of $γ {(Λ ω)}^{2}$ , with $Λ = ℏ G N / (m ω^{2})$ . Note that, in our scaling limits, $Λ$ remains finite for all $N$ . The parameter $γ$ is instead proportional to $N$ in the infinite-density limit in which the optomechanical engine thus delivers extensive power output. The stationary phase (zero power) and the time-crystal phase (finite power) are separated by a critical line. (b) Three sections across (a) for different values of $g$ . The solid line refers to $g / ω = 1 / 3$ , the dashed line refers to $g / ω = 2 / 3$ , and the dot-dashed line is associated with $g / ω = 1$ . The atoms are all initialized in the ground state and the light field in the vacuum. The parameters used are $κ = γ / m = ω$ .
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Figure 3
Metastable time-crystal engine and efficiency in the infinite-density limit. (a) Mirror position $x$ in units of $Λ$ as a function of time for $Ω = g = ω$ . The plot shows an emergent metastable regime in which the mirror features oscillations (see inset), unveiling that the quantum system is in a time-crystal phase. The smaller $G Λ / ω$ (see arrows in the plot) is, the longer the metastable regime lasts before the system ends up in the stationary state. (b) Growth of the fluctuation $δ α$ , associated with the light-field expectation $α$ , due to the back-action of the dynamics of the mirror on the time evolution of the quantum system. The different lines refer to different values of $G Λ / ω$ , as indicated by the arrows in the plot. (c) Efficiency (in the metastable regime) in units of $γ Λ / (κ m ℓ_{0})$ , obtained from Eq. (20). (d)–(f) Normalized power, light-field dissipation, and efficiency as a function of $Ω$ for different values of $g$ . The unspecified parameters are $κ = γ / m = ω$ .
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