Abstract
We consider the nonadiabatic energy fluctuations of a many-body system in a time-dependent harmonic trap. In the presence of scale-invariance, the dynamics becomes self-similar and the nondiabatic energy fluctuations can be found in terms of the initial expectation values of the second moments of the Hamiltonian, square position, and squeezing operators. Nonadiabatic features are expressed in terms of the scaling factor governing the size of the atomic cloud, which can be extracted from time-of-flight images. We apply this exact relation to a number of examples: the single-particle harmonic oscillator, the one-dimensional Calogero-Sutherland model, describing bosons with inverse-square interactions that includes the non-interacting Bose gas and the Tonks-Girdardeau gas as limiting cases, and the unitary Fermi gas. We illustrate these results for various expansion protocols involving sudden quenches of the trap frequency, linear ramps and shortcuts to adiabaticity. Our results pave the way to the experimental study of nonadiabatic energy fluctuations in driven quantum fluids.
Highlights
Nonequilibrium quantum phenomena in many-body systems are often hard to describe by numerical methods due to the growth of entanglement with the time of evolution
Scale invariance is an important symmetry that governs the dynamics of a variety of systems, including many harmonically-trapped ultracold atomic gases
It is of relevance to time-of-flight imaging techniques, but it plays an important role in the implementation of quantum control techniques and the characterization of finite-time thermodynamics
Summary
Nonequilibrium quantum phenomena in many-body systems are often hard to describe by numerical methods due to the growth of entanglement with the time of evolution. A prominent example is the presence of scale-invariance in the dynamics of a broad family of quantum fluids in time-dependent traps, which is exploited with routine in ultracold atom laboratories [1,2]. Beyond the use of time-of-flight imaging techniques, scale invariance has applications in a wide variety of contexts It allows for the engineering of fast control protocols known as shortcuts to adiabaticity [13,14,15,16,17], as demonstrated in the laboratory [18,19,20,21,22,23]. Energy fluctuations can be used to quantify the cost of thermodynamic processes [41,42], including shortcuts to adiabaticity, and their minimization paves the way to the finding of optimal protocols, e.g., the solution of the quantum brachistochrone problem [43,44]
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