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Bosons at play: evolutionary games of condensates in coupled birth-death processes

Johannes Knebel, Markus F. Weber, Torben Krüger, Erwin Frey

LVBEC_smallBosons are quantum particles that like to be together. When a dilute gas of bosons is cooled to a temperature close to absolute zero, a macroscopic fraction of the bosons will occupy the energetic ground state of the system; the bosons form a Bose-Einstein condensate. Only recently has it been proposed that bosons may also group into multiple condensates when they are driven by a time-periodic force and dissipate into the environment.
In our manuscript, we show that the condensation of bosons in such a nonequilibrium system corresponds to the selection of strategies in evolutionary game theory. We applied mathematical concepts of this theory to explain why and under which conditions bosons form multiple condensates. We found that the condensation dynamics follows a simple physical guiding principle: the vanishing of relative entropy production guides the selection of condensates.

Bosons like to cluster, fermions avoid each other – this categorization of particles lies at the heart of quantum physics and has dramatic consequences for their behavior. In thermodynamic equilibrium, a dilute gas of bosons forms the so-called Bose-Einstein condensate at very low temperatures. In this state of matter, a macroscopic fraction of the bosons occupies the system’s lowest energy state. The possibility to create such a condensate was first proposed theoretically by Bose and Einstein in 1924. During the 1990s, experimentalists eventually confirmed this long-standing prediction by studying ultracold atomic gases.

During recent years, scientists have begun to explore whether novel condensation phenomena may occur when a bosonic system is out of equilibrium. It has been proposed that non-interacting bosons may not only group into a single, but also into multiple condensates. In order for this to happen, the bosons need to be in an open system. A suitable system is, for example, a driven-dissipative system into which energy is periodically pumped from the outside and in which each boson may release its energy into the environment. Periodic energy input and dissipation may balance such that the bosons aggregate into multiple quantum states. Thus far, it has remained elusive, which of the quantum states become the condensates and how the condensation proceeds.

In our article, we applied concepts from evolutionary game theory to explain the formation of multiple condensates in such driven-dissipative bosonic systems. The strength of game theory lies in its ability to explain the behavior and interactions of agents in a collective. Each agent has its own strategy, whether it be a predator that is hunting for prey, or a participant in the children’s game “rock-paper-scissors” who chooses to play the “rock” strategy. We showed that non-interacting bosons that transition between different quantum states correspond to interacting agents who may change their strategies. In an evolutionary game, when many agents with different strategies compete against each other, only the successful strategies prevail. The other strategies vanish over time. Similarly, because energy is allowed to flow in and out of the system, the bosons eventually group into particular quantum states, whereas the other states become depleted. An evolutionary game in which multiple strategies survive corresponds to a bosonic system in which multiple condensates form. Hence, order emerges with time.

LVBEC2 On a mathematical level, the dynamics of both condensation of bosons and selection of strategies are captured by a coupled birth-death process. By identifying the strategies that prevail in an evolutionary game, we identified the quantum states that become the condensates in a driven-dissipative bosonic system. The formation of order is determined by a collective quantity: the vanishing of relative entropy production guides the selection of condensates. The condensation dynamics proceeds exponentially fast, but the system of bosons never comes to rest. Instead, the occupation numbers of condensates may oscillate and the bosons may engage in complex games. We show that the rules of these games can be tuned, for example to design a rock-paper-scissors game of condensates.

How such evolutionary games of condensates may be realized in an experiment poses an interesting question for future research. Experiments with ultracold atomic gases, such as those being conducted in the group led by NIM investigator Prof. Immanuel Bloch (LMU Munich and Max-Planck-Institute for Quantum Optics), offer promising candidates to study bosons out of equilibrium.