Certifying quantum enhancements in thermal machines beyond the Thermodynamic Uncertainty Relation

TL;DR

The paper develops a framework to certify genuine quantum thermodynamic advantages in steady-state thermal machines by comparing quantum machines with carefully constructed classical equivalents that use the same incoherent resources. It separates coherence into Hamiltonian-induced and noise-induced categories, deriving classical equivalents and applying full counting statistics to compare current fluctuations. A universal quantum advantage emerges for Hamiltonian-induced coherence: quantum machines outperform their classical equivalents in nonequilibrium, with $\mathcal{R}>0$ (and detuning constraints) under weak coupling and driving. For noise-induced coherence, the advantage is not universal; $\mathcal{R}$ can be positive or negative depending on the model and parameters, and TUR violations are not a necessary indicator of quantum advantages. The authors validate their results through three prototypical models (three-level amplifier, three-qubit refrigerator, NIC machine) and provide explicit expressions for the classical equivalents and fluctuation bounds, highlighting the conditions under which genuine quantum thermodynamic advantages arise and how to certify them in practice.

Abstract

Quantum coherence has been shown to impact the operational capabilities of quantum systems performing thermodynamic tasks in a significant way, and yet the possibility and conditions for genuine coherence-enhanced thermodynamic operation remain unclear. Introducing a comparison with classical machines using the same set of thermodynamic resources, we show that for steady-state quantum thermal machines -- both autonomous and externally driven -- that interact weakly with thermal reservoirs and work sources, the presence of coherence induced by perturbations in the machine Hamiltonian guarantees a genuine thermodynamic advantage under mild conditions. This advantage applies to both cases where the induced coherence is between levels with different energies or between degenerate levels. On the other hand, we show that engines subjected to noise-induced coherence can be outperformed by classical stochastic engines using exactly the same set of (incoherent) resources. We illustrate our results with three prototypical models of heat engines and refrigerators: the three-level amplifier, the three-qubit autonomous refrigerator, and a noise-induced-coherence machine.

Figures (7)

• Figure 1: Schematic representation of three quantum thermal machine models. (a) The coherent three-level amplifier with couplings to baths at hot $\beta_\mathrm{h}$(red) and cold $\beta_\mathrm{c}$(blue) temperatures, as well as coherent external driving (green thunderbolt), (b) the three-qubit autonomous (absorption) refrigerator where each qubit is locally coupled to baths at hot (red), cold (blue), and intermediate (yellow) temperatures $\beta_\mathrm{m}$, and (c) the noise-induced-coherence machine showing collective jumps induced by the baths at hot (red) and cold (blue) temperatures, together with a classical work source given by an infinite-temperature bath (green). Plain double arrows represent jumps between the machine energy levels induced by the baths while wavy arrows represent coherent interactions. Collective jumps in (c) are represented by triple arrows.
• Figure 2: Schematic representation of a generic $N$ level thermal machine with Hamiltonian-induced coherence in a subspace composed by two levels (green levels). The transitions between all other levels are produced by thermal baths at possibly different temperatures (simple arrows), and one of its transitions is being monitored (black detector). A reduced version of the machine model is obtained by introducing the coarse-grained state $S$ (purple shaded box) including $N-3$ levels. In the unicycle case the rates of the transitions $\ket{\rm v}\leftrightarrow \ket{S}$ and $\ket{\rm u} \leftrightarrow \ket{\rm m}$ are set to zero, as well as the transitions between levels within the coarse-grained state $\ket{S}$ leading to multiple cycles.
• Figure 3: Testing fluctuations in the three-level amplifier. a) Histogram of sampled values of the ratio between the fluctuations of the system and its classical analog $\mathcal{R}$ for different ratios of the bath temperatures (see legend). The values corresponds to an exploration of the following region in the parameters space of the system: $\beta_{\mathrm{c}} = 1$, $\epsilon_{2} = 5$, $\omega_{\mathrm{d}} \in [0.1,4.9]$, $\gamma_{\mathrm{h}/\mathrm{c}} \in [10^{-5}, 10^{-2}]$ and $g \in [10^{-5},10^{-2}]$. b) Colour maps of the fluctuation ratio $R$ and c) the TUR ratio $\mathcal{Q}_\mathrm{he}$ as a function of the bath interaction strengths and the driving force. Solid black lines in both plots correspond to the TUR ratio saturation $\mathcal{Q}_{\rm he} = 2$. The other systems parameters are: $\beta_c = 1$, $\beta_{\mathrm{h}} = 0.1/\beta_c$, $\epsilon_2 = 5$, $\omega_{\mathrm{d}} = 2.5$ and $\gamma_{\mathrm{c}} = 10^{-3}$ (energies are given in $k_{\rm B} T_{\rm c} = 1$ units).
• Figure 4: Histograms for the three-qubit refrigerator of sampled values of the fluctuation ratio $\mathcal{R}$ and TUR ratio $\mathcal{Q}_{\rm abs}$ (inset plot). The positive values of $\mathcal{R}$ indicate reduced fluctuations in the quantum model with respect to the classical equivalent that is not related with a violation of the TUR. The values corresponds to an exploration of the following region in the parameters space of the system: $\beta_{\mathrm{m}}/\beta_{\mathrm{c}} \in [0,1]$, $\beta_{\mathrm{h}}/\beta_{\mathrm{m}} \in [0,1]$, $\epsilon_{1} \in [0.1,4.9]$, $\gamma_{\mathrm{c}/\mathrm{m}/\mathrm{h}} \in [10^{-5}, 10^{-2}]$ and $g \in [10^{-4},10^{-2}]$. Fixed parameter are $\beta_{\mathrm{c}} = 1$, $\epsilon_{2} = 5$ (energies are given in $k_{\rm B} T_{\rm c} = 1$ units).
• Figure 5: (a) Schematic representation of the level transitions in the NIC machine after the change of variables. Here one of the double coherent transitions are replaced by a simple jump ($\ket{0} \leftrightarrow \ket{\alpha}$). (b) Graphical representation of the inequality (\ref{['eq: validity of the equivalent']}) for $\gamma^{\rm a}_{\rm c} = \gamma^{\rm b}_{\rm c} =: \gamma_{\rm c}$, separating the regimes where we can define the classical equivalent for noise-induced coherence (blue zone) and where we cannot (grey zone). In the symmetric case (blue diagonal) the classical equivalent can always be defined.