Probing Quantum Speed Limits with Ultracold Gases

Quantum speed limits (QSLs) rule the minimum time for a quantum state to evolve into a distinguishable state in an arbitrary physical process. These fundamental results constrain a notion of distance traveled by the quantum state, known as the Bures angle, in terms of the speed of evolution set by nonadiabatic energy fluctuations. I theoretically propose how to measure QSLs in an ultracold quantum gas confined in a time-dependent harmonic trap. In this highly-dimensional system of continuous variables, quantum tomography is prohibited. Yet, QSLs can be probed whenever the dynamics is self-similar by measuring as a function of time the cloud size of the ultracold gas. This makes it possible to determine the Bures angle and energy fluctuations, as I discuss for various ultracold atomic systems.

Figure 1

QSL for an expansion induced by a linear frequency ramp and a shortcut to adiabaticity. (a) Scaling factor and (b) logarithmic fidelity as a function of time for a fourfold expansion with $\tau =10/{\omega }_0$ for a linear ramp (blue line) and a STA (red line). Orthogonality catastrophe is encoded in the constant ${\sigma }^2$, which captures the dependence on the system size. (c) Path length $\gamma (\tau )$ (solid line) in Hilbert space lower bounded by the geodesic $\mathcal{L}(\tau )$ (dashed line) with $N=1$. (d) While the excess Bures angle $\delta \mathcal{L}$ increases for a linear ramp as the adiabatic limit is approached, the converse is true for the STA.

Figure 2

Excess Bures angle under adiabatic evolution and TQD. $\delta \mathcal{L}(\tau )$ is shown for different values of ${\sigma }^2$, increasing from bottom to top. The Mandelstam-Tamm QSL is saturated only when the ratio $x=\omega (\tau )/{\omega }_0$ approaches unity, as $\delta \mathcal{L}(\tau )$ vanishes. Many particle effects increase $\delta \mathcal{L}(\tau )$, hindering the driving at the QSL.