Brazilian researchers incorporate quantum fluctuation into the concept of entropy

Theoretical study published in Physical Review Letters could have future applications in nanodevices and computation, and in quantum encryption and communication.

By José Tadeu Arantes  |  Agência FAPESP – Classical thermodynamics was born in the first half of the nineteenth century as a response to the industrial revolution’s need for optimized machines, engines and motors. It focused on calculating such quantities as useful work, dissipated energy and efficiency. According to the second law of thermodynamics, mechanical energy can be completely converted into thermal energy but thermal energy cannot be completely converted into mechanical energy. This asymmetry imposes a direction on material processes and hence time’s arrow, which moves toward increasingly disorganized energy configurations. It inspired German physicist Rudolf Clausius (1822-1888) to define a new property called “entropy”, which refers to the proportion of thermal energy that can no longer be converted into useful work, and and hence adds to the system’s degree of irreversibility.

Is it possible to understand the macroscopic concepts of thermodynamics at the atomic or subatomic scale? What would change if a motor with a single atom were built? How are the laws of thermodynamics affected by quantum mechanics? These questions underpin an article on the “Wigner entropy production rate” by Brazilian researchers Jader Pereira dos Santos (Federal University of the ABC), Gabriel Teixeira Landi (University of São Paulo) and Mauro Paternostro (Queen’s University Belfast, UK), published as an “Editors' Suggestion” in the journal Physical Review Letters.

The study was supported by a grant from FAPESP for the project “Stochastic modeling of non-equilibrium quantum systems” conducted by Landi, who supervises Jader Pereira dos Santos’s postdoctoral research.

“The approximation between thermodynamics and quantum mechanics is fairly recent. It’s been happening for only a few decades, since it became possible to exert very fine control in manipulating atoms and literally build atomic-scale motors. Although our study deals with pure knowledge issues in fundamental physics, we can envisage various applications in microscopic systems, such as computational nanodevices, or quantum encryption and communication,” Landi told Agência FAPESP.

The researchers specifically addressed the production of entropy as a measure of irreversibility in quantum contexts for which there was previously no well-established theory. “There were very good theories about measuring irreversibility in the classical context, i.e. at the macroscopic scale, but there were no good theories about how to measure the degree to which a quantum process is irreversible,” Landi said. “Previous proposals were incomplete and had several gaps. This was basically because they were designed for classical systems and not for quantum systems. Our contribution addresses this problem. We propose a new theory to describe this kind of phenomenon.”

We know that the energy of a closed system is conserved (first law of thermodynamics), but entropy always tends to increase (second law of thermodynamics) because energy reconfigures in a less organized form with each transformation owing to irreversibility. It is possible to speak of energy degradation and define entropy as the measure of this spontaneous increase in disorder. The researchers’ goal in their purely theoretical study was to include the contributions of quantum systems in the process.

“The idea is that every system displays two types of fluctuation simultaneously: heat fluctuation deriving from external particle agitation, and quantum fluctuation, which is an intrinsic phenomenon,” Landi explained. “At high energy levels such as those obtained in the laboratory using a particle collider, quantum fluctuation is responsible for the creation and annihilation of pairs of particles and antiparticles, but quantum fluctuation also occurs at low energy levels, and ideally even at absolute zero. In macroscopic processes, heat fluctuation is generally more important, but there are situations in which quantum fluctuation predominates and contributes more significantly to entropy.”

Classical thermodynamics worked only with heat fluctuation, but at the atomic and subatomic scale, where quantum physics becomes necessary to describe entities and phenomena, the disorder due to quantum fluctuation must be considered and computed. According to quantum mechanics, even if a system is in an ideal state in which there is no thermal agitation (defined as absolute zero or zero kelvin), it nevertheless displays an implicit tendency to become disorganized owing to quantum fluctuation, associated with the uncertainty principle formulated by Werner Heisenberg (1901-1976).

According to the uncertainty principle, complementary variables such as position and linear momentum (mass times velocity), for example, cannot be accurately measured at the same time. Uncertainty is manifested, for example, in the particle-wave duality. An object cannot be perfectly located in space owing to its behavior as a wave, and presents itself to an observer spread out, as it were, or fluctuating between various possible positions.

“Eugene Wigner [Budapest, Hungary, 1902-Princeton, USA, 1995], winner of the 1963 Nobel Prize for Physics, formulated a probabilistic interpretation of quantum mechanics,” Landi said. “The Wigner function takes into account both heat fluctuation and quantum fluctuation. Working with the Wigner function, we succeeded in reformulating the theory of irreversibility so as to incorporate quantum fluctuation into the concept of entropy. This is what we call Wigner entropy in our article. We defined entropy as the disorder associated with the statistical distribution described by the Wigner function. Construction of the new theory and its application to quantum systems followed naturally.”

The main novelty, according to Landi, is that the results obtained can be applied even to systems at absolute zero. Prior to this study there was nothing in the theoretical repertoire to explain the effect of quantum fluctuation on the increase in entropy at absolute zero. “Although absolute zero is never reached in practice, there may be situations, including in the lab, where temperatures are low enough – only a few kelvins – to make quantum fluctuation more important than heat fluctuation. In quantum optics systems involving lasers, quantum fluctuation may be dominant even at room temperature,” he said.

Another aspect explored by the researchers is the system’s interaction with its heat reservoir. “It’s possible to construct reservoirs with special properties that are different from the properties of a classical reservoir,” Landi said. “We call these reservoirs ‘structured’. Conventional reservoirs input symmetrical fluctuation into the system, whereas structured reservoirs can input asymmetrical fluctuation.”

This asymmetry could possibly be used to encode information in the context of quantum computing. This would be a more distant objective, however. In the shorter term, for future research the three scientists are considering possible applications in communications using light. “The idea is to use the concept of irreversibility to quantify losses in fiber optic communications. Besides energy loss, there’s also a loss of light coherence. Our formalism is capable of accounting for all these kinds of loss,” Landi said.

Another focus of interest is entanglement, which occurs when pairs or groups of particles are generated or interact in such a way that the quantum state of each particle cannot be described independently, since it depends on the system as a whole. Maintenance of entanglement is essential in quantum computing. However, the system’s interaction with the environment produces a loss of entanglement. “The idea is to use our formalism to estimate this loss and develop strategies to minimize it,” Landi said.