Physicists have derived a fundamental quantum limit on how fast any physical system can reach thermal equilibrium, a finding that constrains everything from boiling a kettle to cooling a beer. The result, published in Nature Physics, shows that the minimum thermalization time is approximately ℏ/(2 k_B T), where ℏ is Planck's constant, k_B is Boltzmann's constant, and T is the temperature. Away from absolute zero, this bound applies universally, independent of the specific machine or system involved.
Thermalization occurs when an object stops exchanging net heat with its surroundings, reaching a state of thermal equilibrium. While physicists had long suspected that this process could not happen arbitrarily quickly, the new work provides a rigorous mathematical framework. «There was suggestive evidence from experiments, numerical simulations and analytical calculations that a physical system can't be made to thermalize faster than some fundamental limit,» says Martí Perarnau-Llobet, a quantum scientist at the Autonomous University of Barcelona and one of the paper's authors.
To guarantee thermalization, the researchers—from institutions in Barcelona, Vienna, and Geneva—introduced the concept of a thermalization machine. Given a quantum system in some initial state, this machine evolves it to a thermal state at a well-defined temperature. A refrigerator cooling a beer or a kettle boiling water would be everyday examples. The team then imposed a crucial condition: the same machine must also thermalize systems with slightly different properties, such as a cup of coffee with varying amounts of sugar or milk. This rules out tricks where a machine is finely tuned to a specific system, which could otherwise thermalize it arbitrarily fast.
«Such examples where you can finely tune your algorithm to thermalize a specific system can be avoided by carefully defining thermalization,» explains team member Pavel Sekatski of the University of Geneva. The researchers used a mathematical object called a Hamiltonian, which describes the possible energies of a system, to formalize this requirement. If a quantum system has a Hamiltonian close to the original one, the thermalization machine must be able to handle it as well.
Based on these assumptions, the team showed that the fastest thermalization time is set by the fundamental constants ℏ and k_B, scaled by temperature. «These restrictions on these thermal machines are actually very weak, and the notion is very general,» notes Perarnau-Llobet. The bound applies away from absolute zero, where quantum effects become negligible.
The work also connects to problems in computer science. One task, Gibbs state sampling (GSS), involves computing a thermal state from a known Hamiltonian and can be performed arbitrarily fast. Another, Hamiltonian learning (HL), requires determining the Hamiltonian from a quantum system and is subject to a time bound. The researchers' task lies between the two: they have access to the quantum system but do not know its Hamiltonian, yet they want to produce the thermal state. «We don't know what the Hamiltonian of the quantum system is, but we do have access to the quantum system, and we want to produce the thermal state,» says Paolo Abiuso of the Institute for Quantum Optics and Quantum Information in Vienna. His Barcelona co-author John Calsamiglia adds that the task is «harder than GSS, it is still easier than HL».
Beyond computer science, the team believes their approach could shed light on fundamental questions in physics. «We are now investigating what this approach can tell us about the fundamental limits imposed by the second and third laws of thermodynamics,» says Alberto Rolandi of TU Wien. The second law states that entropy in an isolated system never decreases, while the third law concerns the impossibility of reaching absolute zero. The new bound may offer insights into how these laws constrain thermalization processes.
The research was a collaboration involving the Autonomous University of Barcelona, the University of Geneva, the Institute for Quantum Optics and Quantum Information in Vienna, and TU Wien. It was published in Nature Physics. The exact conditions under which thermalization is always guaranteed remain an open puzzle, but the team focused on systems that do have this feature, providing a general framework for understanding the speed limits of thermal processes.