Virtual Temperatures Achieve Upper Bound for Otto Efficiency in Quantum Thermodynamics

Understanding how energy flows in quantum systems is a fundamental challenge in thermodynamics, and new research sheds light on the crucial role of ‘virtual temperatures’ in defining the behaviour of these systems. Sachin Sonkar and Ramandeep S. Johal, both from the Indian Institute of Science Education and Research Mohali, investigate these virtual temperatures as a way to quantify passive states during quantum thermodynamic processes. Their work utilises majorization theory to compare states resulting from different transformations, revealing how the minimum and maximum virtual temperatures dictate the direction of heat flow. By applying this approach to a model Otto engine, the researchers demonstrate that virtual temperature provides a quantifiable link between passivity, optimal performance, and classical thermodynamic principles, offering a novel tool for analysing quantum heat engines.

Their work utilises majorization theory to compare states resulting from different transformations, revealing how the minimum and maximum virtual temperatures dictate the direction of heat flow. By applying this approach to a model Otto engine, the researchers demonstrate that virtual temperature provides a quantifiable link between passivity, optimal performance, and classical thermodynamic principles, offering a novel tool for analysing quantum heat engines.

Virtual Temperature and Passive Quantum State Analysis

The study investigates passive quantum states through a novel application of majorization theory, focusing on the concept of virtual temperatures to characterise energy transfer. Researchers defined a mean virtual temperature, calculated as the weighted average of virtual temperatures between adjacent energy levels, to facilitate comparison of states resulting from both isoenergetic and isoentropic transformations. This mean temperature mirrors the role of final temperature in classical thermodynamics, enabling analysis of state evolution following these processes. To establish this, the team engineered a rigorous mathematical framework, defining virtual temperature between energy levels i and j as (Ej − Ei) / ln(pi/pj), where E represents energy and p denotes probability.

Crucially, the research highlights the significance of the minimum and maximum virtual temperatures in determining the direction of heat flow between a system and its environment. A passive state is predicted to lose energy to a heat reservoir at temperatures equal to or below its minimum virtual temperature, conversely gaining energy from environments exceeding its maximum virtual temperature. Experiments employed a coupled-spin system as a concrete example, allowing scientists to explicitly demonstrate the relationship between min-max virtual temperatures and the upper bound of Otto cycle efficiency. The study pioneered a method for deriving this upper bound directly from the min-max virtual temperatures of the working medium, revealing a fundamental link between passivity and optimal thermal machine performance.

This approach enables a deeper understanding of how non-equilibrium quantum systems behave when interacting with thermal reservoirs. Furthermore, the work demonstrates the power of majorization, a mathematical tool for comparing probability distributions, to connect these findings to broader theoretical developments in quantum information and thermodynamics. The research establishes that a probability distribution P is majorized by Q if cumulative differences between corresponding probabilities are non-negative, signifying that P is more spread out than Q. This majorization relation underpins the analysis of state transformations and resource theories, solidifying virtual temperature as a key operational quantity linking passivity, majorization, and the performance of thermal machines.

Virtual Temperatures Define Quantum Heat Flow Limits Researchers

Scientists have established a novel method for characterizing passive quantum states through the introduction of ‘virtual temperatures’, positive values assigned to energy level differences within a system. The research defines a mean virtual temperature to facilitate comparison of passive states resulting from both isoenergetic and isoentropic transformations, drawing parallels with classical thermodynamic processes. Experiments revealed that the minimum and maximum virtual temperatures play a crucial role in determining the direction of heat flow between a quantum system and its environment, governed by majorization relations. The team measured the intermediate passive states within an Otto engine, deriving an upper bound for its efficiency expressed in terms of the min-max virtual temperatures of the working medium.

Specifically, the upper bound on Otto efficiency is defined as ηub = 1 − min i ω′ i ωi, where ωi and ω′ i represent energy level differences, and can also be expressed as ηub = 1 − Tmin Th, with Tmin representing the minimum virtual temperature and Th denoting the temperature of the hot reservoir. Tests confirm that for heat to flow from the system to a cold reservoir, the minimum virtual temperature must exceed the reservoir’s temperature, ensuring the upper bound respects the Carnot limit. Researchers explicitly calculated this bound using a coupled-spins system as a working medium, defined by the Hamiltonian H = B(σ(1) z ⊗I + I ⊗σ(2) z ) + J[(1 + γ)σ(1) x ⊗σ(2) x + (1 −γ)σ(1) y ⊗σ(2) y ]. For this system, with coupling strength J = 0.5, the quantum Otto efficiency and its upper bound were compared across varying anisotropy parameters γ. Measurements of virtual temperatures during the first adiabatic stroke yielded T1,3 = Th K2 −J K1 −J and T2 = Th, establishing the order T1 = T3 T ′ 2, again yielding the same efficiency bound. This work establishes a clear link between passivity, majorization theory, and the optimal performance of quantum thermal machines, offering insights into the fundamental limits of quantum thermodynamics.

Virtual Temperatures and Quantum Engine Efficiency

Researchers have established a method for characterizing passive quantum states through a newly defined set of virtual temperatures associated with adjacent energy levels within a system. By defining a mean virtual temperature, they have created a means of comparing passive states resulting from different transformations, drawing parallels with classical thermodynamic processes. Utilising majorization relations, the study demonstrates that the mean energy of a passive state is bounded by energies corresponding to the minimum and maximum virtual temperatures, effectively dictating the direction of heat flow. This work culminates in the derivation of an upper bound for the efficiency of a quantum Otto engine, expressed in terms of these min-max virtual temperatures, and validated through application to a coupled-spin system.

The findings highlight virtual temperature as a crucial operational quantity, linking passivity, majorization theory, and the optimal performance of quantum thermal machines. The authors acknowledge that their approach was demonstrated on a fifteen-qubit system and suggest extending the methodology to other working media and heat cycles, including refrigerators and heat pumps. Furthermore, they propose investigating the adaptation of this approach to finite-time cycles as a promising avenue for future research.