[Paper Review] Catalytic transformations with finite-size environments: applications to cooling and thermometry

Ivan Henao, Raam Uzdin|arXiv (Cornell University)|Oct 18, 2020

Advanced Thermodynamics and Statistical Mechanics100 references29 citations

TL;DR

This paper introduces a framework for catalytic transformations in finite-size environments, demonstrating that cooling and thermometry can be enhanced using a finite-dimensional catalyst. It proves that catalytic cooling is always possible with a sufficiently large catalyst, with a three-level system enabling optimal cooling of a qubit, and shows that catalysts can surpass the performance limits of non-catalytic methods in both cooling and temperature estimation tasks.

ABSTRACT

The laws of thermodynamics are usually formulated under the assumption of infinitely large environments. While this idealization facilitates theoretical treatments, real physical systems are always finite and their interaction range is limited. These constraints have consequences for important tasks such as cooling, not directly captured by the second law of thermodynamics. Here, we study catalytic transformations that cannot be achieved when a system exclusively interacts with a finite environment. Our core result consists of constructive conditions for these transformations, which include the corresponding global unitary operation and the explicit states of all the systems involved. From this result we present various findings regarding the use of catalysts for cooling. First, we show that catalytic cooling is always possible if the dimension of the catalyst is sufficiently large. In particular, the cooling of a qubit using a hot qubit can be maximized with a catalyst as small as a three-level system. We also identify catalytic enhancements for tasks whose implementation is possible without a catalyst. For example, we find that in a multiqubit setup catalytic cooling based on a three-body interaction outperforms standard (non-catalytic) cooling using higher order interactions. Another advantage is illustrated in a thermometry scenario, where a qubit is employed to probe the temperature of the environment. In this case, we show that a catalyst allows to surpass the optimal temperature estimation attained only with the probe.

Motivation & Objective

• To address the fundamental limitation that finite environments restrict cooling beyond passive states, which the second law does not capture.
• To develop explicit, constructive conditions for catalytic transformations in finite environments, including unitaries and catalyst states.
• To demonstrate that catalysts can overcome cooling limitations due to finite environment size and improve thermometric precision.
• To show that catalysts enhance cooling even when cooling is classically possible, by enabling more efficient heat extraction.
• To establish a general framework for non-unital transformations (e.g., cooling, work extraction) enabled by finite catalysts.

Proposed method

• Develops a formalism for catalytic transformations using global unitary operations that preserve total energy, coupling a system, a finite environment, and a catalyst.
• Introduces a graphical method to visualize catalytic transformations through loop currents in the catalyst's energy levels.
• Applies the concept of passivity to identify when cooling is impossible without a catalyst in finite environments.
• Uses a protocol to iteratively optimize catalyst eigenvalues and unitary operations to maximize cooling current (Jmin).
• Derives constructive conditions via 'catalytic majorization' with explicit unitaries and catalyst states, enabling practical implementation.
• Applies the framework to cooling and thermometry, showing that catalysts can surpass non-catalytic bounds in both tasks.

Experimental results

Research questions

• RQ1 Under what conditions can a finite environment fail to cool a system, and can this limitation be overcome using a catalyst?
• RQ2 What is the minimal catalyst dimension required to enable catalytic cooling of a qubit using a finite hot environment?
• RQ3 Can catalysts enhance cooling even when cooling is already possible without them, and if so, how?
• RQ4 Can a catalyst improve the precision of temperature estimation in thermometry beyond non-catalytic probes?
• RQ5 What general class of transformations (beyond cooling) can be enabled by finite catalysts in finite environments?

Key findings

• Catalytic cooling is always possible if the catalyst dimension is sufficiently large, with a three-level system enabling optimal cooling of a qubit using a hot qubit.
• For a qubit cooled by a four-level hot object, a two-level catalyst can achieve a cooling current of Jmin(1) = 0.045, exceeding the non-catalytic Jmin = 0.033.
• In multiqubit cooling, a two-level catalyst enables a two-fold advantage: enhanced heat extraction and reduced need for complex many-body interactions.
• A catalyst can surpass the optimal temperature estimation precision achievable with a probe alone, demonstrating catalytic enhancement in thermometry.
• The protocol iteratively improves cooling by adjusting catalyst eigenvalues and swapping operations, with the first round increasing the cooling current from 0.033 to 0.045.
• The framework generalizes to non-unital transformations, including work extraction from passive states and catalytic violations of the Jarzynski equality.