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arXiv:2309.04419

Quantum work statistics of controlled evolutions

Steve Campbell

Published Jan 1, 2023• 0 citations
arXivHTMLPDF

Abstract

We use the quantum work statistics to characterize the controlled dynamics governed by a counterdiabatic driving field. Focusing on the Shannon entropy of the work probability distribution, $P(W)$, we demonstrate that the thermodynamics of a controlled evolution serves as an insightful tool for studying the non-equilibrium dynamics of complex quantum systems. In particular, we show that the entropy of $P(W)$ recovers the expected scaling according to the Kibble-Zurek mechanism for the Landau-Zener model. Furthermore, we propose that the entropy of the work distribution provides a useful summary statistic for characterizing the need and complexity of the control fields for many-body systems.

DOI
10.17169/refubium-42521

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We use the quantum work statistics to characterize the controlled dynamics governed by a counterdiabatic driving field. Focusing on the Shannon entropy of the work probability distribution, P(W)P(W)P(W), we demonstrate that the thermodynamics of a controlled evolution serves as an insightful tool for studying the non-equilibrium dynamics of complex quantum systems. In particular, we show that the entropy of P(W)P(W)P(W) recovers the expected scaling according to the Kibble-Zurek mechanism for the Landau-Zener model. Furthermore, we propose that the entropy of the work distribution provides a useful summary statistic for characterizing the need and complexity of the control fields for many-body systems.
Abstract We use the quantum work statistics to characterize the controlled dynamics governed by a counterdiabatic driving field. Focusing on the Shannon entropy of the work probability distribution, P ( W ), we demonstrate that the thermodynamics of a controlled evolution serves as an insightful tool for studying the non-equilibrium dynamics of complex quantum systems. In particular, we show that the entropy of P ( W ) recovers the expected scaling according to the Kibble-Zurek mechanism for the Landau-Zener model. Furthermore, we propose that the entropy of the work distribution provides a useful summary statistic for characterizing the need and complexity of the control fields for many-body systems.
Abstract Driving a quantum many-body system across the quantum phase transition (QPT) in the finite time has been concerned in different branches of physics to explore various fundamental questions. Here, we analyze how the underlying QPT affects the work distribution P (W ), when the control parameter of a ferromagnetic spinor Bose-Einstein condensates is tuned through the critical point in the finite time. We show that the work distribution undergoes a dramatic change with increasing the driving time τ. To capture the characteristics of the work distribution, we analyze the entropy of P (W ) and find three different regions in the evolution of entropy as a function of τ. Specifically, the entropy is insensitive to the driving time in the region of very short τ, while it exhibits a universal power-law decay in the region with intermediate value of τ. In particular, the power-law scaling of the entropy is according with the well-known Kibble-Zurek mechanism. For the region with large τ, the validity of the adiabatic perturbation theory leads to the entropy decay as τ-2 ln τ. Our results verify the usefulness of the entropy of the work distribution for understanding the critical dynamics and provide an alternative way to experimentally study nonequilibrium properties in quantum many-body systems.
Many years have passed since the conception of the quintessential method of shortcut to adiabaticity known as counterdiabatic driving (or transitionless quantum driving). Yet, this method appears to be energetically cost-free and thus continually challenges the task of quantifying the amount of energy it demands to be accomplished. This paper proposes that the energy cost of controlling a closed quantum system using the counterdiabatic method can also be assessed using the instantaneous excess work during the process and related quantities, as the time-averaged excess work. Starting from the Mandelstam-Tamm bound for driven dynamics, we have shown that the speed-up of counterdiabatic driving is linked with the spreading of energy between the eigenstates of the total Hamiltonian, which is necessarily accompanied by transitions between these eigenstates. Nonetheless, although excess work can be used to quantify energetically these transitions, it is well known that the excess work is zero throughout the entire process under counterdiabatic driving. To recover the excess work as an energetic cost quantifier for counterdiabatic driving, we will propose a different interpretation of the parameters of the counterdiabatic Hamiltonian, leading to an excess work different from zero. We have illustrated our findings with the Landau-Zener model.
Geometric quantum speed limits quantify the tradeoff between the rate at which quantum states can change and the resources that are expended during the evolution.Counterdiabatic driving is a unique tool from shortcuts to adiabaticity to speed up quantum dynamics while completely suppressing nonequilibrium excitations.We show that the quantum speed limit for counterdiabatically driven systems undergoing quantum phase transitions fully encodes the Kibble-Zurek mechanism by correctly predicting the transition from adiabatic to impulse regimes.Our findings are demonstrated for three scenarios, namely the transverse field Ising model, the Landau-Zener model, and the Lipkin-Meshkov-Glick model.
We develop a field-theoretic framework for work statistics in O(N)O(N)O(N) models driven through criticality. By analyzing the dynamic renormalization group flow of composite power operators, we find the Kibble-Zurek scaling laws as a natural consequence of the flow, and we derive the scaling of work cumulants relevant to Kibble-Zurek scaling of topological defects from first principles, bypassing heuristic freeze-out argument. This yields the universal scaling cn∼τQ−αnc_n \sim τ_Q^{-α_n}cn​∼τQ−αn​​ for the nnn-th work cumulant density: isolated quantum systems exhibit a scaling of αn=p(d+nz)ν/(1+pzν)α_n = p(d+nz)ν/(1+pzν)αn​=p(d+nz)ν/(1+pzν), whereas open quantum and classical systems undergo a dimensional collapse to αn=pdν/(1+pzν)α_n = pdν/(1+pzν)αn​=pdν/(1+pzν). Validated by exact Gaussian solutions and numerical simulations, our theory establishes a foundation for general work statistics far from equilibrium, thereby bridging stochastic thermodynamics and the renormalization group theory.
Studying the implications and characterizations of the excited-state quantum phase transitions (ESQPTs) would enable us to understand various phenomena observed in quantum many-body systems. In this work, we delve into the affects and characterizations of the ESQPTs in the anharmonic Lipkin-Meshkov-Glick (LMG) model by means of the entropy of the quantum work distribution. The entropy of the work distribution measures the complexity of the work distribution and behaves as a valuable tool for analyzing nonequilibrium work statistics. We show that the entropy of the work distribution captures salient signatures of the underlying ESQPTs in the model. In particular, a detailed analysis of the scaling behavior of the entropy verifies that it not only acts as a witness of the ESQPTs but also reveals the difference between different types of ESQPTs. We further demonstrate that the work distribution entropy also behaves as a powerful tool for understanding the features and differences of ESQPTs in the energy space. Our results provide further evidence of the usefulness of the entropy of the work distribution for investigating various phase transitions in quantum many-body systems and open up a promising way for experimentally exploring the signatures of ESQPTs.