Orthogonality catastrophe and the speed of quantum evolution in a qubit-spin-bath system

Qing Wang,Zheng-Rong Zhu,Jian Zou and Bin Shao,*

1 School of Physics,Beijing Institute of Technology,Beijing 100081,China

2 Yangtze Delta Region Academy of Beijing Institute of Technology,Jiaxing 314019,China

Abstract The orthogonality catastrophe(OC)of quantum many-body systems is an important phenomenon in condensed matter physics.Recently,an interesting relationship between the OC and the quantum speed limit(QSL)was shown(Fogarty 2020 Phys.Rev.Lett.124 110601).Inspired by the remarkable feature,we provide a quantitative version of the quantum average speed as another different method to investigate the measure of how it is close to the OC dynamics.We analyze the properties of an impurity qubit embedded into an isotropic Lipkin-Meshkov-Glick spin model,and show that the OC dynamics can also be characterized by the average speed of the evolution state.Furthermore,a similar behavior of the actual speed of quantum evolution and the theoretical maximal rate is shown which can provide an alternative speed-up protocol allowing us to understand some universal properties characterized by the QSL.

Keywords:orthogonality catastrophe,speed of quantum evolution,LMG model

1.Introduction

The dynamical behaviors of many-body systems after switching of even a single,weak interacting impurity qubit are interesting.The phenomenon of Anderson’s orthogonality catastrophe(OC)[1]is then witnessed,when the initial manybody wave function loses its essential overlap with the perturbed one.It captures the main feature of many-body systems and reveals the sensitivity of them to local perturbations.The study of OC has aroused considerable interest within various areas of physics,such as quantum spin systems[2–6],quantum phase transitions(QPTs)[7–9]and nonequilibrium dynamics[10,11].Recently,the behavior of OC has been linked to the quantum speed limit(QSL),which originates from the uncertainty theory[12].It demonstrates that the OC follows as a consequence of the vanishing QSL time in manybody systems[13].In addition,the OC is also related to the decoherence of the impurity,characterizing the dynamical evolution of the impurity[14].Thus,it is natural to relate the evolution speed of the impurity qubit to the OC since the two quantities are associated with the fidelity.

In this paper,we consider a qubit-spin-bath system consisting of a central impurity qubit coupled to an isotropic Lipkin-Meshkov-Glick(LMG)model[15].The system was first proposed to describe how the QPT of the bath influences the quantum coherence of the central qubit[16].Tian et al analyzed the influences of the environment parameters to the non-Markovian effect of the system[17].The non-Markovian feature is the intrinsic mechanism for the quantum acceleration phenomenon of the central qubit in LMG bath[18].The LMG model,as a clean system,has also been used to study the time crystal effect.It shows that the model being infinite range and having an extensive amount of symmetry-breaking eigenstates is crucial to exhibit the time-crystal behavior[19,20].Within the considered system,we propose an alternative method of characterizing the OC dynamics through the speed of quantum evolution,which relies on a determined metric.Thus,we exploit the actual speed of quantum evolution in terms of the quantum Fisher information metric[21]to investigate its relationships to the OC and the QSL,respectively.

The rest of this paper is organized as follows.In section 2,we briefly review the qubit-spin-bath system and present the relationship between the OC dynamics and the quantum average speed.In section 3,we measure the QSL in an explicit form,and show the numerical comparison and analysis between the QSL and the actual speed quantum evolution speed.Section 4 is the conclusion.

2.OC and quantum average speed

The LMG model was first introduced to describe a set of N spin-1/2 mutually interacting in atomic nuclei,and this model is currently studied in various fields of physics ranging from magnetic molecules[22]to Bose–Einstein condensates[23,24],etc.For its unique phase transition feature,the LMG model has been widely used to investigate quantum critical phenomena[25–27].The total Hamiltonian of a central qubit interacting with an isotropic LMG spin model is written as

The purpose of this research is to investigate a connection between the OC and the actual speed of quantum evolution.Hence,we start by considering the overlap between the perturbed and unperturbed many-body states,which describes the OC.The original work of Anderson’s OC effect focused on stationary states[1],however,the many-body state after a local perturbation is generally time-dependent.The dynamical OC is given by a time-dependent overlap,read as

which measures the susceptibility of dynamics to a local perturbation.The decay dynamics of the fidelity are widely used in quantum information theory.

The speed of quantum evolution can be quantified by various measures[28–30].In this work,we look to the actual speed of quantum evolution derived by Cianciaruso et al due to the quantity relative to the time-dependent fidelity[21].Theoretical studies have proposed using the non-Markovian effect,and the associated information backflow from the environment,to enhance the actual evolution speed[31].Recent effort has observed that as much as the classical correlation of the channel increases,the speed of quantum evolution decreases[32].Based on the unified form of the Riemannian metric g,the squared infinitesimal distance between the neighboring states ρ and ρ+dρ is (ds)2=The speed of quantum evolutionat time t is given as follows(b) are functions of the coupling strength λ between the spins in the bath for different sizes of the bath,N=200(dotted–dashed,black line),N=500(dashed,blue line)and N=1000 (solid, red line).

3.QSL and actual speed of quantum evolution

The above results show that the dynamical occurrence of OC is attributed to the actual speed of quantum evolution,more specifically,the evolution speed scales with the environment parameters exhibits the OC effect.Previous research concluded that the OC follows as a consequence of the vanishing QSL time in many-body systems[13].It promotes us to further investigate a relationship between the quantum average speed and the QSL in the open quantum system.

4.Conclusion

We have applied the concept of the OC to the qubit-spin-bath system composed of a central qubit interacting with an isotropic LMG spin model to investigate the relationship between the OC and the actual speed of quantum evolution.Within the considered system,the numerical results of the fidelity exhibit how the OC dynamics are affected by the initial environment phase and the OC can only be observed in the symmetric phase of the LMG bath.In particular,we propose an alternative method of characterizing the dynamical occurrence of the OC by the quantum average speed since both are associated with fidelity.We demonstrate that the OC effect manifests itself followed by the vanishing fidelity and the sufficiently large but finite average speed.The actual speed of quantum evolution is as vital as the QSL time in[13]to the OC dynamics in many-body systems.Additionally,we reveal a similar behavior between the actual speed of quantum evolution and the QSL of the central qubit.The two quantities experience a dramatic acceleration around the second-order QPT point,which can be used to characterize the QPT in the LMG model.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos.11875086 and 11775019.