Finite-size scaling of the photon-blockade breakdown dissipative quantum phase transition

A. Vukics1, A. Dombi1, J. M. Fink2, and P. Domokos1

1Wigner Research Centre for Physics, H-1525 Budapest, P.O. Box 49., Hungary
2Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria

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We prove that the observable telegraph signal accompanying the bistability in the photon-blockade-breakdown regime of the driven and lossy Jaynes–Cummings model is the finite-size precursor of what in the thermodynamic limit is a genuine first-order phase transition. We construct a finite-size scaling of the system parameters to a well-defined thermodynamic limit, in which the system remains the same microscopic system, but the telegraph signal becomes macroscopic both in its timescale and intensity. The existence of such a finite-size scaling completes and justifies the classification of the photon-blockade-breakdown effect as a first-order dissipative quantum phase transition.

First-order phase transitions characterized by the coexistence of phases are commonly observed in the surrounding world, e.g. in the freezing of water. Continuous – second-order – phase transitions also exist in classical physics, e.g. the transition between ferro- and paramagnetism at the Curie temperature. Whereas the latter class has seen straightforward generalizations to quantum systems for decades, the notion of a first-order quantum phase transition remains to be elucidated.

Bistability in certain small quantum systems has been identified as signature of first order quantum phase transitions, however, this identification is problematic: a randomly switching telegraph signal between two well-resolved attractors can also be observed in quantum dynamics distinct from phase transitions. For example, the famous electron-shelving scheme – used in atomic clocks or for qubit measurement in ion-trap quantum computers – produces a similar signal without any connection to phase transitions.

There is a missing element to support the interpretation of bistability as a first-order quantum phase transition: it must be shown that bistability is only a finite-size effect, and there exists an idealized thermodynamic limit, where temporal bistability is replaced by hysteresis. This idealized thermodynamic limit can be introduced such that the physical system remains a small quantum system with a few degrees of freedom, that is, the passage to the thermodynamic limit does not involve a quantum-to-classical transition. In this paper, we present a prototype of this procedure by constructing a finite-size scaling for the recently-observed photon-blockade-breakdown effect to justify its classification as a first-order dissipative quantum phase transition.

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[5] Jonathan B. Curtis, Igor Boettcher, Jeremy T. Young, Mohammad F. Maghrebi, Howard Carmichael, Alexey V. Gorshkov, and Michael Foss-Feig, "Critical theory for the breakdown of photon blockade", Physical Review Research 3 2, 023062 (2021).

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[7] I. Pietikäinen, J. Tuorila, D. S. Golubev, and G. S. Paraoanu, "Photon blockade and the quantum-to-classical transition in the driven-dissipative Josephson pendulum coupled to a resonator", Physical Review A 99 6, 063828 (2019).

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[9] Th. K. Mavrogordatos, "Strong-coupling limit of the driven dissipative light-matter interaction", Physical Review A 100 3, 033810 (2019).

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