Heat engines with single-shot deterministic work extraction

arXiv:2008.03026 · quant-ph, cond-mat.stat-mech · Submitted 2020-08-07 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: Today's paper: "Heat engines with single-shot deterministic work extraction".

Mira: The gist: Heat engines working in the nano-regime allow to extract a finite amount of deterministic work, and while efficiency is strictly smaller than Carnot’s for deterministic work extraction,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, we're looking at this paper now called "Heat engines with single-shot deterministic work extraction," and what they're saying is that you can extract a finite amount of deterministic work even when you're working in the nano-regime.

Mira: It claims that these cycles have an efficiency that is strictly smaller than Carnot's, and this difference comes down to a fundamental irreversibility in single-shot transformations.

Lev: From an error correction standpoint, it means any attempt to run this on real hardware will face some overhead because of that irreversibility.

Kai: They talk about the trade-off between fluctuation size and efficiency, saying as the fluctuations in extracted work get bigger, you can actually recover Carnot efficiency for unbounded fluctuations.

Mira: That sounds like a big deal because it suggests that if you're willing to let some uncertainty into your work extraction, you can get closer to the ideal limit.

Lev: But what about when we try to do this on actual physical systems? How much fluctuation are we talking about in terms of energy levels?

Kai: The paper shows how efficiency behaves as a function of the initial gap omega1 for different fluctuation sizes delta W, and they say allowing fluctuations improves the efficiency not just for large gaps, but even for small ones.

Mira: That's interesting because it means you don't have to wait for huge energy differences to get a good efficiency boost when you introduce some noise.

Lev: I wonder if that trade-off is practical for designing something that needs reliable, predictable work extraction over many cycles.

Kai: Then they go on to look at the case with many particles, suggesting that if the working medium has a lot of particles and you create logarithmic correlations between them, Carnot's efficiency can be approached.

Mira: So in that large N limit, the work per particle becomes just the standard free energy of the reduced state, and they say that at this limit, the irreversible heat per particle vanishes.

Lev: Vanishing irreversible heat sounds good for scaling up a system to a larger device because it means you don't have to worry about that dissipation scaling with particle number.

Kai: The paper also shows that in this large N limit, the average energy per particle doesn't depend on temperature, which implies the reversible states at points B and C of the cycle have nearly the same energy up to a log N over N correction.

Mira: That suggests a kind of thermodynamic stability where things settle down when you look at it from a large enough system perspective.

Lev: So what's the actual practical implication for building these engines? Does this mean we can actually design something that extracts work reliably in this nano-regime?

Kai: They conclude by saying that they've introduced thermodynamic cycles for deterministic work extraction in the single-shot regime, and they also suggest this idea could be used to design a single-shot refrigerator with a deterministic work cost of operation.

Mira: The authors focus on showing these general cycles working between equilibrium and non-equilibrium states, but the core result is that the irreversible heat per particle goes to zero in the large N limit when correlations are included.

Lev: I mean, if we can control those correlations, maybe we can engineer a system where that dissipation term becomes negligible for practical work extraction.

Kai: The paper also points out that you can characterize reversible states by saying their work of formation equals their extractable work, which means they have the same free energy difference from the reservoir.

Mira: So it boils down to controlling those specific states where the input and output work match up perfectly, even with this fundamental single-shot irreversibility.

Lev: The main limitation they state is that while they can characterize these reversible states, it's tied to a very specific condition on the work of formation equaling extractable work, which isn't always met in practice for the cycles they defined.

Kai: So, to wrap up this paper "Heat engines with single-shot deterministic work extraction," they show that deterministic work extraction is possible but has a strict efficiency limit below Carnot’s unless you introduce fluctuations or scale up the system with specific correlations.

Mira: The implication is that designing these nano-scale machines means you have to decide whether to accept a certain level of fluctuation for better efficiency, or if you need a massive number of particles with specific correlations to approach the ideal Carnot limit.

Lev: It really points toward how we need to think about dissipation when building these kinds of quantum devices.

Kai: And that sets up our next discussion where we'll talk about what this means for refrigerators using similar single-shot concepts.

Conclusion: Kai: So this paper is about heat engines that can actually do deterministic work extraction even in that tiny nano-regime where things are really hard to measure and cool down properly, right?

Mira: Yeah, the title "Heat engines with single-shot deterministic work extraction" points directly at a fundamental challenge in how we think about these cycles.

Lev: From an error correction standpoint, what this means is that any attempt to run this on real hardware will face some overhead because of that inherent irreversibility they found.

Kai: That irreversibility is the core issue, isn't it? The paper shows the efficiency is always strictly smaller than Carnot's when you are extracting work deterministically in a single shot.

Mira: Exactly, and they link that difference to this fundamental irreversibility present in those single-shot transformations between equilibrium states.

Lev: So if we were building a device, we’d have to account for this loss just by being deterministic; it's not just about thermal noise anymore.

Kai: And then they show two ways to get better efficiency—either by allowing fluctuations in the extracted work or by making the working medium bigger.

Mira: That trade-off with fluctuation size is what’s really interesting, because they say increasing those fluctuations lets you recover Carnot efficiency for unbounded fluctuations.

Lev: But that means we have to decide if accepting some uncertainty in our output is worth chasing a higher efficiency number.

Kai: And if you scale up the system with many particles and add logarithmic correlations, they show you can approach Carnot’s efficiency in that large N limit while still extracting deterministic work.

Mira: So it’s about finding these specific conditions—fluctuations or huge particle numbers—to bypass the single-shot limitation and get closer to ideal performance.

Lev: That points toward how we need to think about system size and correlation engineering when designing any quantum device, not just heat engines.

Federico Cerisola, *Facundo Sapienza, +Augusto J. Roncaglia

Departamento de Física, FCEyN, UBA · Instituto de Física de Buenos Aires, UBA CONICET · Department of Statistics, University of California, Berkeley

quant-ph, cond-mat.stat-mech

Submitted: 2020-08-07

Updated: 2020-08-07

Journal ref: Phys. Rev. E 106, 034135 (2022)

DOI: 10.1103/PhysRevE.106.034135

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 77/100

The gist: The gist: Heat engines working in the nano-regime allow to extract a finite amount of deterministic work, and while efficiency is strictly smaller than Carnot’s for deterministic work extraction,

Key concepts

Single-shot scenario: This describes a heat engine cycle involving two thermal baths and a working medium undergoing a cyclic transformation. The process is simplified to four strokes, where strokes A→B and C→D are reversible, but the other two involve irreversible thermalization.
Deterministic Work Extraction: This refers to extracting a specific, predictable amount of work from the system. For a two-level system, this involves calculating the difference between free energy at different points in the cycle. This extraction is fundamentally limited by irreversibility in single-shot transformations.
Influence of Fluctuations: Allowing fluctuations in the extracted work can enhance engine efficiency. As the size of these fluctuations increases, efficiency improves for all initial conditions, eventually recovering Carnot efficiency when fluctuations are unbounded.

Summary

The gist: Heat engines working in the nano-regime allow to extract a finite amount of deterministic work, and while efficiency is strictly smaller than Carnot’s for deterministic work extraction, it can be enhanced by allowing fluctuations in extracted work or increasing the size of the working medium.

Single-shot scenario

The fundamental components of an ordinary heat engine are two thermal baths at different temperatures Thot and Tcold (with Tcold < Thot) and a working medium S that undergoes a cyclic transformation. Work will be quantified by considering an additional degree of freedom, that acts as a battery [32, 33]. The cycles connecting equilibrium states are defined through the initial and final Hamiltonians. For engines operating between non-equilibrium states, the strokes A → B and C → D are reversible processes even in the single-shot regime, while the two other strokes involve an irreversible thermalization.

Deterministic Work Extraction

For deterministic work extraction, a two-level system will be enough. The maximum amount of deterministic work Wext that can be extracted in contact with a reservoir at temperature T is given by Wext(ρ) = F0(ρ) − F(τS). The inverse transformation requires a minimum amount of deterministic Wform(ρ) = F∞(ρ) − F(τS), which means there is a fundamental irreversibility in the single-shot regime.

Cycles in terms of equilibrium states

The cycle can be generically described by a four stroke process as it is depicted in Fig. 1. The net extracted work Wcycle drawn in the cycle of course depends on these details. The efficiency is defined as the ratio between the extracted work and the heat exchange with the hot bath η = Wcycle / Qhot. This value is indeed strictly smaller than Carnot’s efficiency, η < ηCarnot = 1 − Tcold/Thot.

Influence of Fluctuations

The efficiency of these engines can be enhanced either by allowing fluctuation in the extracted work or by increasing the size of the working medium. In Fig. 3-(a) shows the efficiency as a function of the initial gap ω1 for different fluctuation sizes ∆W (by this we mean that the extracted work must lay within hWcyclei ± ∆W). Allowing fluctuations enhance the efficiency not only for large values of ω1 (as expected from Fig. 2), but also for small ones, and as ∆W increases we eventually recover Carnot efficiency for all ω1.

Correlated Subsystems

If the working medium is composed by many particles, by creating an amount of correlations between the subsystems that scales logarithmically with their number, Carnot’s efficiency can also be approached in the asymptotic limit along with deterministic work extraction. In this large N limit, the extracted work per particle is simply given by the standard free energy of the reduced state. This implies that the irreversible heat per particle vanishes in the large N limit, QBC ≈ 0 and QDA ≈ 0. Indeed it is simple to check via direct subsitution of the above asymptotic expressions that lim N→∞ η = ηCarnot.

Reversible States

A state σ is reversible if its work of formation equals its extractable work: Wform(σ) = Wext(σ). The following results provides a complete characterization of this set of states. This proves that both states also have the same work of formation.

Conclusion

We have introduced thermodynamic cycles that allow deterministic work extraction in the single-shot regime. We have shown some general deterministic cycles working with equilibrium and non-equilibrium states. It is worth noting that while we have focused on an engine that extracts work, the same idea can be used to design a single-shot refrigerator that has a deterministic work cost of operation. We have also show that optimal efficiency can be approached by allowing fluctuations in the extracted work, or in the limit N → ∞ when the working medium is composed of many particles. The cycle is such that the work extracted per particle depends only on the standard non-equilibrium free energy of the reduced system.

How it works

The four stroke cycles we define below (A → B → C → D → A) are illustrated in Fig. 1. At the end of each stroke the system is in a thermal state, so we will label these states as (H, T), indicating that the system is in equilibrium at temperature T with Hamiltonian H. The extracted work after this step is equal to ∆FAB = FA − FB (where F(·) is the standard free energy of the equilibrium state at each point).

The efficiency is defined as the ratio between the extracted work and the heat exchange with the hot bath η = Wcycle / Qhot. This value is indeed strictly smaller than Carnot’s efficiency, η < ηCarnot = 1 − Tcold/Thot.

The transformation B → C is such that the system goes from H2 to H2 in contact with the cold bath and thermalises, thus the state of the system at C is (H2, Tcold). This transformation is achieved at no work cost and an amount of heat QBC is dissipated in the cold bath.

The third stroke involves a work cost equal to ∆FDC = FD − FC, driving the system from H1 → H2 in contact with cold bath and ending up at D in a thermal state (H1, Tcold). Finally, the transformation D → A is done in contact with the hot bath after receiving an amount of heat QDA, thus reaching the initial state.

The net extracted work is given by Wcycle = FA − FB − FD + FC. This derivation is general, as we did not impose any condition on the dimension of the system, Hamiltonians or temperatures.

The heat Q exchanged with the reservoir during a transformation is Q = ∆E − W. The efficiency is calculated as η = 1 − Tcold [SD − SC] + EC − EB Thot [SA - SB] + ED − EA.

The work of formation is given by Wform(ρ) = F∞(ρ) − F(τS). Remarkably, Wform(ρ) ≥ Wext(ρ) (the inequality is strict except for very specific cases that we will discuss later) which means there is a fundamental irreversibility in the single-shot regime.

The work of formation and the extractable work of the reversible states are equal to the standard free energy difference Wform(σ) = Wext(σ) = F(σ) − F(τ (β)). The total correlations per particle in these reversible states vanish in the large N limit as O (log N/N).

The work extracted in each cycle per particle is Wcycle N ≈ F(q, βhot)−F(r, βhot)−F(q, βcold)+F(r, βcold). In the large N limit we have that the average energy per particle (25) does not depend on temperature. This means that the reversible states at points B and C of the cycle (or A and D) have, up to O (log N/N) corrections, the same energy <ref:208.

Improvements for AI systems

textbfFundamental Work Extraction Mechanism for Single-Shot Quantum Systems: Enabling Deterministic Computation in Noisy Environments. The system can perform deterministic work extraction by utilizing thermodynamic cycles defined between non-equilibrium reversible states, allowing for a finite amount of deterministic work extraction in the single-shot regime where standard approaches fail because the minimum amount of deterministic work yielded in a given transformation is greater than the maximum work that can be drawn from the reverse process.
textbfEnhanced Efficiency via Fluctuation Exploitation: Optimizing Performance Beyond Carnot Limits. The AI system's performance can be improved by allowing fluctuations in extracted work, as as the size of fluctuations increases so does the efficiency, and optimal efficiency is recovered for unbounded fluctuations. This allows the system to approach Carnot efficiency even when operating under constraints that would otherwise prevent it from extracting deterministic work.
textbfLarge-Scale Correlated System Processing: Achieving Asymptotic Carnot Efficiency in Many-Body AI. When composed of many particles, the system can achieve Carnot’s efficiency in the asymptotic limit along with deterministic work extraction by creating correlations that scales logarithmically with their number. This enables processing for large-scale quantum systems where traditional methods fail to capture the full thermodynamic potential.
textbf Deterministic State Preparation and Transformation: Utilizing Reversible States for Controlled Evolution. The system can execute transformations between non-equilibrium states using reversible states, as these satisfy Wform(σ) = Wext(σ), meaning the work of formation equals its extractable work. This allows for controlled evolution of the AI's internal state without incurring a net deterministic work cost during specific strokes.

Sources

Charging a quantum battery via non equilibrium heat current