# How to measure quantum speed limits experimentally

The common understanding of the uncertainty relation says that it is impossible to measure both the position and the momentum of a subatomic particle, in the same instant to unlimited accuracy. The more accurate is the measurement of the momentum, the less accurate is the measurement of the position in that instant, and vice versa.

Momentum is certainly related to time through velocity, but introducing time in the quantum realm turns everything a little bit blurry. The time-energy uncertainty relation is a fundamental result in quantum physics relating characteristic times to the inverse of energy fluctuations, Still, as Busch puts it ^{1}, different types of time energy uncertainty relation can indeed be deduced in specific contexts, but that there is no unique universal relation that could stand on equal footing with the position-momentum uncertainty relation.

The modern formulation of the time-energy uncertainty relation relies on quantum speed limits that bound the minimum time for a physical process to unfold in terms of energy fluctuations. As the state of a system in quantum mechanics is represented by a vector in a linear vector space that can have an infinite number of dimensions, called a Hilbert space, quantum speed limits represent quantum dynamics geometrically, where the quantum state of a system evolves in time by sweeping a distance in Hilbert space. Therefore, quantum speed limits involve the notions of speed and distance in Hilbert space.

Quantifying the distance between the initial and time-evolving quantum states requires estimating state overlaps, which is challenging, if not unfeasible, for many-particle systems with continuous variables.

In spite of the fundamental nature of quantum speed limits, there is currently a lack of experimental studies probing them. Now, Adolfo del Campo theoretically proposes ^{2} the experimental study of quantum speed limits with many-body systems of trapped ultracold atoms by measuring the mean atomic cloud size as a function of the evolution time.

The proposal relies on measuring the size of the atomic cloud in a given process, such an expansion or compression driven by a modulation of the trap frequency. The scaling factor can be determined by imaging the cloud size via different methods. From it, the distance traveled by the quantum state of the system in Hilbert space (Bures angle) during the evolution can be determined.

This approach circumvents the need for reconstructing a quantum state using measurements of the many-body quantum states of a continuous variable system. These results pave the way to the experimental study of the time-energy uncertainty relation and quantum speed limits in many-body quantum systems and their relation to the orthogonality catastrophe.

*Author: César Tomé López is a science writer and the editor of Mapping Ignorance*

*Disclaimer: Parts of this article may be copied verbatim or almost verbatim from the referenced research paper.*

## References

- P. Busch,
*The time-energy uncertainty relation*, in*Time in Quantum Mechanics*, edited by J. Muga, R. S. Mayato, and Í. Egusquiza (Springer, Berlin, 2008), pp. 73–105. ↩ - Adolfo del Campo (2021) Probing Quantum Speed Limits with Ultracold Gases
*Physical Review Letters*doi: 10.1103/PhysRevLett.126.180603 ↩