Heat-machine control by quantum-state preparation: From quantum engines to refrigerators

D. Gelbwaser-Klimovsky and G. Kurizki

Phys. Rev. E 90, 022102 – Published 4 August 2014

Abstract

We explore the dependence of the performance bounds of heat engines and refrigerators on the initial quantum state and the subsequent evolution of their piston, modeled by a quantized harmonic oscillator. Our goal is to provide a fully quantized treatment of self-contained (autonomous) heat machines, as opposed to their prevailing semiclassical description that consists of a quantum system alternately coupled to a hot or a cold heat bath and parametrically driven by a classical time-dependent piston or field. Here, by contrast, there is no external time-dependent driving. Instead, the evolution is caused by the stationary simultaneous interaction of two heat baths (having distinct spectra and temperatures) with a single two-level system that is in turn coupled to the quantum piston. The fully quantized treatment we put forward allows us to investigate work extraction and refrigeration by the tools of quantum-optical amplifier and dissipation theory, particularly, by the analysis of amplified or dissipated phase-plane quasiprobability distributions. Our main insight is that quantum states may be thermodynamic resources and can provide a powerful handle, or control, on the efficiency of the heat machine. In particular, a piston initialized in a coherent state can cause the engine to produce work at an efficiency above the Carnot bound in the linear amplification regime. In the refrigeration regime, the coefficient of performance can transgress the Carnot bound if the piston is initialized in a Fock state. The piston may be realized by a vibrational mode, as in nanomechanical setups, or an electromagnetic field mode, as in cavity-based scenarios.

Received 17 April 2014

DOI:https://doi.org/10.1103/PhysRevE.90.022102

Authors & Affiliations

D. Gelbwaser-Klimovsky and G. Kurizki

Weizmann Institute of Science, 76100 Rehovot, Israel

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Vol. 90, Iss. 2 — August 2014

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Figure 1
Schematic description of the autonomous quantized heat machine. (a) Scheme of energy and heat exchange among $P, C, H$ , and $S$ to elucidate the heat-pumping gain mechanism of an autonomous quantized heat engine. Reversal of all arrows describes refrigeration via energy investment by $P$ . (b) Spectrally separated response of the $C$ (thin lines) and $H$ (thick lines) baths is engineered using the spectral filtering procedure described in Sec. 5. The pointed-line (green) curves are the $q = - 1, 0 (ω_{-}, ω_{0})$ harmonics of their response. The $q = 1 (ω_{+})$ harmonic is missing because it does not overlap with either of the (solid) bath spectra: namely we assume that after the filtering $G_{H} (ω_{0}) ≫ G_{C} (ω_{0})$ and $G_{C} (ω_{-}) ≫ G_{C} (ω_{+}), G_{H} (ω_{\pm})$ . (c) Schematic drawing of work capacity: “charging” the piston in a coherent state by work that is subsequently extracted on coupling the piston to an external device.
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Figure 2
Work capacity evolution. (a) Comparison of work-capacity change (normalized by the initial work capacity) as a function of $| Γ | t$ for $E_{P} (t) / E_{P} (0)$ (solid thick curve), coherent, quadrature-squeezed, and Fock states of $P$ with the same initial mean energy $E_{P} (0) = ν 〈 n_{P} (0) 〉$ . Note their same mean-energy gain. (b) The efficiency bound of an initial coherent state may surprass the Carnot limit and remain above it well after the system has reached steady state (here after $10^{4}$ cycles). (c) The respective evolution of the phase-plane distribution shows that the high resilience of the initial coherent state against thermalization, as opposed to the lower resilience of quadrature-squeezed states and the fragility of a Fock state, is the key to their different work capacity evolution. The spectral separation conditions and the response harmonics are as in Fig. 1 to ensure gain in Eq. (30).
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Figure 3
Refrigeration. (a) COP of an autonomous QR as a function of time (in units of $Γ^{- 1}$ ) for different initial piston states: coherent (continuous), Fock (dot-dashed), and thermal (dashed). The Carnot COP bound (dotted, thin) is slightly below the coherent-state COP. Both thermal state and Fock state COP asymptotically coincide to the the absorption bound at $T_{crit}$ . While at short times the Fock state has the largest COP, the coherent state outperforms it at long times, with a COP above the Carnot efficiency. (b) Entropy (in units of $k_{B}$ ) of the same states as a function of time. The initially rapid entropy increase of the Fock state explains its high COP at short times, while for the coherent state the entropy increase is slow but steady, resulting in the highest COP at large times. (c) Phase-plane plots of the distribution evolution illustrate that the high resilience of a coherent state against thermalization, as opposed to the low resilience (fragility) of Fock and Schroedinger-cat states, explains the COP evolution. The spectral separation conditions are as in Fig. 1, but the piston frequency is chosen to ensure cooling $(Γ > 0)$ . The initial mean energy complies with Eq. (38).
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Figure 4
Realizations. (a) Realization of the model in Fig. 1, Eq. (17a), for a flux qubit in a coplanar resonator. The change in the resonator field $(P)$ affects the magnetic flux threading the qubit. (b) Realization of the same model for NV center defects in diamond mounted on a nanomechanical resonator. The strain-field $P$ affects $S$ that is subject to magnetic-field gradient. (c) Filtering two bath spectra (top) according to Eqs. (43) and (44) in order to allow work extraction (bottom right) or refrigeration (bottom, right) by adjusting the filtered bath-spectra overlap with $ω_{0}$ and its sidebands in the coupled $S − P$ system.
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