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arXiv:2408.00635 · Physical review. E

Driving of an open quantum system at finite temperature across first- and second-order quantum phase transitions

Felipe Matus, Pavel Cejnar

Published Apr 3, 2025• 0 citations
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Abstract

An open fully connected system of qubits at nonzero temperature is driven within a finite time interval along various paths in the space of its control parameters. The driving leads across finite-size precursors of first- and second-order quantum phase transition from factorized to entangled ground-state phases, aiming at the preparation of the complex ground state of the system at the final parameter point with maximal fidelity. During the drive, the system is coupled to a heat bath at a constant temperature, the dynamics being determined in a nonperturbative way by the method of hierarchical equations of motion. It is shown that the presence of the heat bath in combination with specific patterns of avoided crossings affecting the ground and excited states in the parameter region around the quantum phase transition may considerably improve the fidelity of preparation of the target ground state.

DOI
10.1103/physreve.111.044104

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An open fully connected system of qubits at nonzero temperature is driven within a finite time interval along various paths in the space of its control parameters. The driving leads across finite-size precursors of first- and second-order quantum phase transition from factorized to entangled ground-state phases, aiming at the preparation of the complex ground state of the system at the final parameter point with maximal fidelity. During the drive, the system is coupled to a heat bath at a constant temperature, the dynamics being determined in a nonperturbative way by the method of hierarchical equations of motion. It is shown that the presence of the heat bath in combination with specific patterns of avoided crossings affecting the ground and excited states in the parameter region around the quantum phase transition may considerably improve the fidelity of preparation of the target ground state.
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.
Using the adiabatic perturbation theory of driven dynamics [Phys. Rev. A 78, 052508 (2008)] we design a hierarchy of quantum state preparation protocols that systematically increase the fidelity at very long driving times. We test these and other protocols, including those based on the geometric analysis of the parameter space, in a single-qubit system and in a fully connected multiqubit system showing in its infinite-size limit several quantum phase transitions. Our protocols excel in the asymptotic driving regime, above a crossover time from the Landau-Zener regime which increases with a decreasing minimal energy gap along the driving path (with the size of the system). In the medium-time domain, the performance of all tested protocols is indecisive.
We study the ground state entanglement properties of a finite size ensemble of interacting qubits driven by a quantized field. In the absence of driving, the system reduces to a Lipkin-Meshkov-Glick model that presents maximal entanglement at the first of a series of quantum phase transitions; i.e. a W-state of the qubit ensemble. We present exact and numerical solutions, within and without the rotating wave approximation, in that order, showing that a large finite probability of having the maximal entangled state in the ground state is preserved in the driven system.
We study the ground state of a finite-size ensemble of interacting qubits driven by a quantum field. We find a maximally entangled WWW state in the ensemble part of the system for a certain region of the coupling parameters. The area of this region decreases as the ensemble size increases and, in the classical limit, becomes the line in parameter space that defines the phase transition of the system. In the classical limit, we also study the dynamics of the system and its transition from order to disorder for initial energies close to the ground-state energy. We find that a critical energy providing this transition is related to the minimum of the projection of the total angular momentum of the quantum system in the zzz direction.
First order quantum phase transitions (1QPTs) are signalled, in the thermodynamic limit, by discontinuous changes in the ground state properties. These discontinuities affect expectation values of observables, including spatial correlations. When a 1QPT is crossed in the vicinity of a second order one, due to the correlation length divergence of the latter, the corresponding ground state is modified and it becomes increasingly difficult to determine the order of the transition when the size of the system is finite. Here we show that, in such situations, it is possible to apply finite size scaling (FSS) to entanglement measures, as it has recently been done for the order parameters and the energy gap, in order to recover the correct thermodynamic limit (Campostrini et al 2014 Phys. Rev. Lett. 113 070402). Such a FSS can unambiguously discriminate between first and second order phase transitions in the vicinity of multicritical points even when the singularities displayed by entanglement measures lead to controversial results.
Existing optimal control methods of open quantum systems rely on extensive numerical simulations of the dynamics in the presence of a bath, or alternatively ignore the exact bath dynamics. If the bath effects are to be treated properly on both Markovian and non-Markovian timescales using numerical simulations, the number of bath modes cannot be large. This may affect the ability to simulate realistic scenarios. Even if realistic, such simulations are hard to interpret physically. An alternative approach advocated here is to resort to a perturbative analysis provided the system–bath coupling is weak. This analysis would allow for the effects of any given bath (finite or infinite, Markovian or non-Markovian) and any control at our disposal. This poses the challenge of constructing a method for the optimization of various operations requiring proper manipulation of the system, based on a general perturbative treatment to second order in the system–bath coupling. This proposed treatment yields a universal tool for optimizing the fidelity of a given operation. It involves a fidelity-control matrix: a construct that allows us to prioritize the use of available control resources so as to maximize the operation fidelity in any given bath. As an analytically solvable example of this general method, we analyse quantum state-transfer optimization, from a ‘noisy’ (write-in) qubit to its ‘quiet’ counterpart (storage qubit). Intriguing interplay is revealed between our ability to avoid bath-induced errors that profoundly depend on the bath-memory time and the limitations imposed by leakage out of the operational subspace. Counterintuitively, under no circumstances is the fastest transfer optimal (for a given transfer energy).