Quantum Physics: Making Hidden Dynamics Visible with Complex Time

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Modern experiments increasingly exploit the interplay between quantum systems and their environment to create and control exotic many-body states. Yet the microscopic mechanisms underlying these phenomena remain largely hidden in strongly interacting systems. The reason is surprisingly fundamental: for open quantum systems, it is not sufficient to consider their energy levels alone. What matters is also how the system evolves together with its environment. This dynamics is described by a so-called Lindbladian. Its eigenmodes determine, among other things, how quickly different collective excitations of the system decay, making them particularly relevant for understanding its dynamical properties.

Existing methods for studying this interplay between a system and its environment beyond just a few quantum degrees of freedom have so far been largely limited to the stationary state and, in some cases, the dissipative gap. The latter describes how quickly an open quantum system returns to its stationary state after a perturbation. The physically interesting dynamical (excited) states, by contrast, have remained inaccessible because they are exponentially suppressed in existing methods.

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The team led by LMU physicist Dr. Sebastian Paeckel has developed an approach that makes these hidden modes accessible and was recently published in Physical Review X. The crucial step is to extend the time evolution in a computer simulation into the realm of complex time, building on a long-established connection in physics between imaginary time and thermal properties. This makes it possible to selectively control which dynamical modes of the system contribute most strongly to the simulation.

Using this approach, the dynamics of a system can be manipulated within a simulation such that the relevant, short-lived states of the system and its environment are exponentially enhanced and thus become visible in the simulation before they decay away again. „In this way, efficiently tractable simulations of short-time dynamics become a window into the physical behavior at long times,” says first author Philipp Westhoff, who worked on the project as part of his master’s thesis and received a research award from the MCQST Cluster of Excellence for his work.

The researchers applied their method to a driven Bose-Hubbard model. This model describes many interacting bosons, such as ultracold atoms, with the system additionally being driven externally in a time-dependent manner. It is relevant, for example, to experiments with ultracold quantum gases. Using the new method, the team observed signatures of an unusual form of relaxation in which disturbances propagate through the system differently from ordinary diffusion.

Another surprising observation was the so-called Mpemba effect. Knowledge of the collective modes makes it possible to deliberately choose the initial state such that the system relaxes significantly faster into the desired state—in this case, into a Bose-Einstein condensate. “The results show how the new spectral information can be used not only to understand quantum dynamics, but also to control it,” says Paeckel. The findings open up new possibilities for systematic investigations of complex open quantum systems.

Philipp Westhoff, Mattia Moroder, Ulrich Schollwöck and Sebastian Paeckel: Tensor-Network Framework for Lindbladian Spectra and Steady States. Physical Review X 2026