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arXiv:1504.04747 · Journal of Physics A Mathematical and Theoretical

Enhancement of quantum speed limit time due to cooperative effects in multilevel systems

Pablo M. Poggi, Fernando C. Lombardo, Diego A. Wisniacki

Published Aug 11, 2015• 9 citations
arXivHTMLPDF

Abstract

Deriving minimum evolution times is of paramount importance in quantum mechanics. Bounds on the speed of evolution are given by the so called quantum speed limit (QSL). In this work we use quantum optimal control methods to study the QSL for driven many level systems which exhibit local two-level interactions in the form of avoided crossings (ACs). Remarkably, we find that optimal evolution times are proportionally smaller than those predicted by the well-known two-level case, even when the ACs are isolated. We show that the physical mechanism for such enhancement is due to non-trivial cooperative effects between the AC and other levels, which are dynamically induced by the shape of the optimized control field.

DOI
10.1088/1751-8113/48/35/35ft02

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Deriving minimum evolution times is of paramount importance in quantum mechanics. Bounds on the speed of evolution are given by the so called quantum speed limit (QSL). In this work we use quantum optimal control methods to study the QSL for driven many level systems which exhibit local two-level interactions in the form of avoided crossings (ACs). Remarkably, we find that optimal evolution times are proportionally smaller than those predicted by the well-known two-level case, even when the ACs are isolated. We show that the physical mechanism for such enhancement is due to non-trivial cooperative effects between the AC and other levels, which are dynamically induced by the shape of the optimized control field.
Quantum mechanics establishes a fundamental bound for the minimum evolution time between two states of a given system. Known as the quantum speed limit (QSL), it is a useful tool in the context of quantum control, where the speed of some control protocol is usually intended to be as large as possible. While QSL expressions for time-independent Hamiltonians have been well studied, the time-dependent regime has remained somewhat unexplored, albeit being usually the relevant problem to be compared with when studying systems controlled by external fields. In this paper we explore the relation between optimal times found in quantum control and the QSL bound, in the (relevant) time-dependent regime, by discussing the ubiquitous two-level Landau-Zener-type Hamiltonian.
We study time-optimal protocols for controlling quantum systems which show several avoided level crossings in their energy spectrum. The structure of the spectrum allows us to generate a robust guess which is time optimal at each crossing. We correct the field applying optimal control techniques in order to find the minimal evolution or quantum speed limit (QSL) time. We investigate its dependence as a function of the system parameters and show that the QSL time gets proportionally smaller to the well-known two-level case as the dimension of the system increases. Working at the QSL, we study the control fields derived from the optimization procedure and show that they present a very simple shape, which can be described by a few parameters. Based on this result, we propose a simple expression for the control field and show that the full time evolution of the control problem can be analytically solved.
Quantum mechanics establishes a fundamental bound for the minimum evolution time between two states of a given system. Known as the quantum speed limit (QSL), it is a useful tool in the context of quantum control, where the speed of some control protocol is usually intended to be as large as possible. While QSL expressions for time-independent Hamiltonians have been well studied, the time-dependent regime has remained somewhat unexplored, albeit being usually the relevant problem to be compared with when studying systems controlled by external fields. In this paper we explore the relation between optimal times found in quantum control and the QSL bound, in the (relevant) time-dependent regime, by discussing the ubiquitous two-level Landau-Zener–type Hamiltonian.
Limitations to the speed of evolution of quantum systems, typically referred to as quantum speed limits (QSLs), have important consequences for quantum control problems. However, in its standard formulation, is not straightforward to obtain meaningful QSL bounds for time-dependent Hamiltonians with unknown control parameters. In this paper we present a short introductory overview of quantum speed limit for unitary dynamics and its connection to quantum control. We then analyze potential methods for obtaining new bounds on control times inspired by the QSL. We finally extend the work in [Poggi, Lombardo and WisniackiEPL104 40005 (2013)] by studying the properties and limitations of these new bounds in the context of a driven two-level quantum system.
We apply the concept of quantum speed limit (QSL)---the minimal time needed to perform a driven evolution---to complex interacting many-body systems where the effects of interactions have to be taken into account. We introduce a general strategy to eliminate the detrimental effects of the interparticle repulsion and drive the system at the QSL by applying a compensating control pulse (CCP). To prove the principles we consider a prototypical many-body system, a bosonic Josephson junction, and investigate a transfer of atoms from the ground state of one well to the ground state of the neighboring well, at increasing levels of complexity---from a textbook two-level approximation to full many-body treatment. By tracing the efficiency of the CCP protocol we show that the driven dynamics does follow the geodetic pathway and, therefore, it is optimal. The CCP strategy, applicable for a general interacting quantum many-body system with strong driving, can be of a practical relevance for the experimental implementation of quantum technology protocols, as quantum simulations or matter-wave metrology.
In this work, we investigate the witnessing of the localization of quantum states through quantum speed limits (QSLs) in a two-level driven avoided-level crossing system. As the characteristic natures of the localized quantum states, the QSL presents the periodic oscillations and coherence. The coherence partition of QSL is much bigger than the population partition of QSL. Our study gives us the possibilities to manipulate dynamics of quantum states locally by employing the coherent destruction of tunneling, which is significant in quantum information process. In addition, we analyze the effects of the rotating-wave approximation and the generalized Van Vleck approach on QSL and show that they wipe out the quantum coherence.