A phase diagram for quantum states with adjustable energy

Depending on external control parameters, the physically realized states of a given system can be grouped into phases that are defined by a measurable order parameter. For ultracold systems, where quantum fluctuations dominate thermal fluctuations, quantum phases arise, which are separated by quantum phase transitions (QPTs) with a vanishing energy gap between the ground state and the first excited state. Today, ultracold quantum many-body systems can also be prepared at non-zero energy and protected from the environment to suppress thermalization. For such systems, it is possible to define excited-state quantum phase transitions (ESQPTs) by an analogous divergence of the density of states. While signatures of such singularities were detected in molecular spectra [1-3], an experimental classification of excited-state quantum phases has not yet been demonstrated.

Here we present the experimental determination of a quantum phase diagram, where the energy of the system is one of the control parameters. The quantum phases are detected by the measurement of an interferometric order parameter [4] that abruptly changes at the ESQPTs. The concept is demonstrated with an atomic Bose-Einstein condensate with a spin degree of freedom. We show that the three ground-state phases that appear as a function of the effective magnetic field, are zero-energy boundaries of equivalent excited-state quantum phases. We pinpoint the excited-state phase transitions by varying three order parameters: the energy, the effective magnetic field, and an adjustable phase between the spin components. Our work presents an example how the extensive Hilbert state of quantum many-body systems can be structured by the definition and measurement of well-defined order parameters.

[1] B. P. Winnewisser, et al., Phys. Rev. Lett. 95, 243002 (2005).

[2] N. F. Zobov, et al., Chem. Phys. Lett. 414, 193 (2005).

[3] D. Larese, F. Pérez-Bernal, F. Iachello, J. Mol. Struct. 1051, 310 (2013).

[4] P. Feldmann, et al., Phys. Rev. Lett. 126, (2021)