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arXiv:1410.1555 · Physical Review Letters

Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model

Steve Campbell, Gabriele De Chiara, Mauro Paternostro, G. Massimo Palma, Rosario Fazio

Published May 1, 2015• 141 citations
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Abstract

We study transitionless quantum driving in an infinite-range many-body system described by the Lipkin-Meshkov-Glick model. Despite the correlation length being always infinite the closing of the gap at the critical point makes the driving Hamiltonian of increasing complexity also in this case. To this aim we develop a hybrid strategy combining a shortcut to adiabaticity and optimal control that allows us to achieve remarkably good performance in suppressing the defect production across the phase transition.

DOI
10.1103/physrevlett.114.177206

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We study transitionless quantum driving in an infinite-range many-body system described by the Lipkin-Meshkov-Glick model. Despite the correlation length being always infinite the closing of the gap at the critical point makes the driving Hamiltonian of increasing complexity also in this case. To this aim we develop a hybrid strategy combining a shortcut to adiabaticity and optimal control that allows us to achieve remarkably good performance in suppressing the defect production across the phase transition.
We apply the method of transitionless quantum driving for time-dependent quantum systems to spin systems. For a given Hamiltonian, the driving Hamiltonian is constructed so that the adiabatic states of the original system obey the Schrödinger equation. For several typical systems such as the XY spin chain and the Lipkin-Meshkov-Glick model, the driving Hamiltonian is constructed explicitly. We discuss possible interesting situations when the driving Hamiltonian becomes time independent and when the driving Hamiltonian is equivalent to the original one. For many-body systems, a crucial problem occurs at the quantum phase transition point where the energy gap between the ground and first excited states becomes zero. We discuss how the defect can be circumvented in the present method.
We explore quantum driving protocols that optimize ground-state fidelity in both closed and open quantum systems, using a model of interacting qubits governed by a Lipkin-Meshkov-Glick type of Hamiltonian. This model exhibits both ground-state and excited-state quantum phase transitions (QPTs and ESQPTs). For comparison, we also analyze a single-qubit system. The selected driving paths in parameter space traverse quantum critical regions, which have a large impact on the target ground-state fidelity. The influence of ESQPTs is recognized particularly in the case of an open system. We apply advanced theoretical techniques like quantum geometry, adiabatic perturbation theory and the method of hierarchical equations of motion.
One of the main difficulties in preparing many-body ground states is achieving the target state through simple counterdiabatic controls. For critical systems crossing a transition to a topological phase, this task becomes even more challenging due to the closing of the gaps in multiple symmetry sectors. This is the case of the Kitaev chain, whose transition between the trivial and topological phases involves states belonging to different symmetry sectors. In this work, we apply the recently introduced minimal action shortcut to adiabaticity (MA-STA) to a Kitaev chain and propose a multi-step strategy to obtain the optimal control protocol to drive the system across its different phases. Our results show that high fidelities can be achieved through the adapted MA-STA at time scales much shorter than those of linear ramp protocols. We also compare the performance of both controls in suppressing work fluctuations. These findings may guide the design of STA protocols in many-body systems where competing energy scales and symmetries shape the global dynamics.
We establish a set of nonequilibrium quantum phase transitions in the Lipkin-Meshkov-Glick model under monochromatic modulation of the interparticle interaction. We show that the external driving induces a rich phase diagram that characterizes the multistability in the system. Interestingly, the number of stable configurations can be tuned by increasing the amplitude of the driving field. Furthermore, by studying the quantum evolution, we demonstrate that the system exhibits a set of quantum phases that correspond to dynamically stabilized states.
The strategy of shortcuts to adiabaticity enables us to realize adiabatic dynamics in finite time. In the counter-diabatic driving approach, an auxiliary Hamiltonian which is called the counter-diabatic Hamiltonian is appended to an original Hamiltonian to cancel out diabatic transitions. The counter-diabatic Hamiltonian is constructed by using the eigenstates of the original Hamiltonian. Therefore, it is in general difficult to construct the counter-diabatic Hamiltonian for quantum many-body systems. Even if the counter-diabatic Hamiltonian for quantum many-body systems is obtained, it is generally non-local and even diverges at critical points. We construct an approximated counter-diabatic Hamiltonian for the infinite-range Ising model by making use of the mean-field approximation. An advantage of this method is that the mean-field counter-diabatic Hamiltonian is constructed by only local operators. We numerically demonstrate the effectiveness of this method through quantum annealing processes going the vicinity of the critical point. It is also confirmed that the mean-field counter-diabatic Hamiltonian is still well-defined in the limit to the critical point. The present method can take higher order contributions into account and is consistent with the variational approach for local counter-diabatic driving.
Adiabatic evolution is considered to be the ideal situation for most thermodynamic cycles, as it allows for the achievement of maximum efficiency. However, no power output is produced in light of the infinite amount of time required to perform a transformation. Such issue can be overcome through Shortcuts-To-Adiabaticity (STA) protocols, which allows a dynamics to mimic its adiabatic counterpart, but in a finite time. Transitionless quantum driving (TQD) is one form of STA. We develop and study the effects of TQD on a single-qubit system subjected to a changing magnetic field and a general two-qubit Heisenberg model with a time-dependent interaction strength. We establish a quantitative relation between the work produced across the dynamics and the entropy production resulting from the evolution at hand, focusing on the role played by the initial states of the work medium. We identify the states that extremize performance under the TQD protocol. The thermodynamic implications of STA are discussed in the end.