How fast can a kettle boil water for your cup of tea, or a refrigerator cool down a bottle of beer? The answer depends on the details of the machines, but according to new work published in Nature Physics, the minimum wait time does not. Instead, it’s quantum mechanics that imposes a limit on how fast something can thermalize to a particular temperature.
Objects thermalize, or reach a state of thermal equilibrium, when they no longer exchange any net heat with their environment. Physicists already suspected that this process couldn’t happen arbitrarily quickly. “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 in the Quantum Information Group at the Autonomous University of Barcelona, Spain, and one of the paper’s authors.
Although the exact conditions under which thermalization is always guaranteed remain an unanswered puzzle, the researchers chose to focus on systems that do have this feature. To guarantee thermalization, Perarnau-Llobet and colleagues in Barcelona; Vienna, Austria; and Geneva, Switzerland came up with the mathematical notion of a thermalization machine. Given a quantum system in some initial state, this thermalization machine evolves it to a thermal state (that is, one that doesn’t spontaneously exchange net heat with its environment) at a particular well-defined temperature. In the previous example, the refrigerator and kettle would be considered thermalization machines, and your beer and water would be the systems.
No cheating
In some cases, one can engineer a machine that takes a particular system to its equilibrium state arbitrarily fast. For instance, a machine might quickly swap a hot cup of coffee for an iced one. But that would be cheating, says team member Pavel Sekatski, a researcher at the University of Geneva. “Such examples where you can finely tune your algorithm to thermalize a specific system can be avoided by carefully defining thermalization,” he tells Physics World.
The researchers did this using a mathematical object called a Hamiltonian, which describes the possible energies a system can have. If a quantum system has a Hamiltonian that is close to the original one, then they require that the thermalization machine must be able to thermalize that system, too. In other words, the same thermalization machine that cools your cup of coffee must also be able to cool down other cups of coffee that may have slightly different amounts of sugar or milk. This rules out the instant swap tricks as the machine can no longer be tailored to a specific cup of coffee.
Based on the assumptions that such a thermalization machine is consistent with laws of quantum mechanics and that it works for some quantum systems that are close to each other, the researchers show that, away from absolute zero, the fastest thermalization time is approximately ℏ/(2kBT), where ℏ is Planck’s constant and kB is Boltzmann’s constant. “These restrictions on these thermal machines are actually very weak, and the notion is very general,” notes Perarnau-Llobet.
Related problems in computer science
The researchers note that their work is closely related to interesting tasks in computer science. One such task, known as Gibbs state sampling (GSS), involves taking a known Hamiltonian and using it to work out the corresponding thermal state at a particular temperature. This task can be performed arbitrarily fast. In contrast, if you have access to the quantum system, but you don’t know its mathematical description (that is, its Hamiltonian) you must perform a task called Hamiltonian learning (HL). This task admits a bound on the time taken for it to be solved.
“The task that we considered was somewhere in between the two,“ explains team member Paolo Abiuso, a physicist at the Institute for Quantum Optics and Quantum Information in Vienna. “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.” His Barcelona co-author John Calsamiglia adds that while the task they considered is “harder than GSS, it is still easier than HL”.
Beyond these connections to computer science, the researchers say their strategy could lead to insights on broader 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.
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