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Physical Review A

Non-Hermiticity-induced quantum control of localization in quantum walks

Rui Zhang, Yi-Fan Liu, Tian Chen

Published Aug 19, 2020• 10 citations
Publisher

Abstract

Non-Hermiticity-induced quantum control is an unique phenomenon, which cannot be observed in Hermitian quantum systems. Here we study the quantum control of localization in non-Hermitian QW systems. When a phase defect is introduced to a specific site of non-Hermitian QW, the walker will not only be localized at the site of phase defect but also at its neighboring sites. Moreover, we find that as the system transforms from the $\mathcal{PT}$-symmetry-unbroken regime to $\mathcal{PT}$-symmetry-broken regime, most of probability distributions can be found at the positions far away from the phase defect in a long period of time. Our discoveries about the control of walker in non-Hermitian QW can provide valuable references for the study of quantum search in dissipative quantum systems.

DOI
10.1103/physreva.102.022218

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Non-Hermiticity-induced quantum control is an unique phenomenon, which cannot be observed in Hermitian quantum systems. Here we study the quantum control of localization in non-Hermitian QW systems. When a phase defect is introduced to a specific site of non-Hermitian QW, the walker will not only be localized at the site of phase defect but also at its neighboring sites. Moreover, we find that as the system transforms from the PT\mathcal{PT}PT-symmetry-unbroken regime to PT\mathcal{PT}PT-symmetry-broken regime, most of probability distributions can be found at the positions far away from the phase defect in a long period of time. Our discoveries about the control of walker in non-Hermitian QW can provide valuable references for the study of quantum search in dissipative quantum systems.
Scale-free localization is a distinctive phenomenon in non-Hermitian systems, where specific eigenstates, known as scale-free localized states (LSs), possess localization lengths that scale linearly with the system size. Here, we show that a single non-Hermitian defect in one-dimensional quantum walks (QWs) induces scale-free localization. Driven by non-Hermiticity, the QW system undergoes a pseudo-Hermitian (PH) phase transition and the PH symmetry breaking is accompanied by the emergence of scale-free localization. As non-Hermiticity is further increased, the system undergoes a second phase transition, marked by the emergence of two pairs of bound states. Crucially, we identify the critical points of both transitions using localization properties. Furthermore, we extend our investigation to two-dimensional QWs, where a single non-Hermitian defect also induces scale-free localization, thereby demonstrating the universality of this phenomenon. Our work provides a valuable perspective for further exploring defect physics and a type of novel LS in non-Hermitian quantum systems.
We present a study on the long-term evolution of a non-Hermitian one-dimensional system with on-site dissipations in quantum-walk dynamics. By investigating various cases, including different evolution operations, lattice sizes, initial positions, and dissipations, we uncover remarkable and unconventional features. They include the emergence of distinct final states after a sufficiently long evolution time, the persistence of metastable states for extended periods, the survival of extended states only within the inner region of a domain-wall structure, and the occurrence of instantaneous ultrafar jumps during the evolution process. To provide insight into the underlying physical mechanisms behind these observations, we undertake a comprehensive investigation accompanied by detailed analysis, clarify the relation between time-evolved (meta)stable states and system eigenstates, and analyze the symmetry of final states. Moreover, our research reveals some phenomena that differ from those seen in previous experimental results, which can be attributed to the limited evolution time in the experiments. These theoretical findings underscore the significance of long-term evolution in non-Hermitian systems based on quantum walks.
Parity-time (PT\mathcal{P}\mathcal{T}PT) symmetry is extensively studied within non-Hermitian physical systems as it has the capability to ensure the presence of real eigenenergies of the systems characterized by non-Hermitian Hamiltonians. Previous studies have been mostly based on the Bloch band theory. Recent developments have unveiled intriguing phenomena which can only be explained by non-Bloch band theory. In this paper, we observe the PT\mathcal{P}\mathcal{T}PT phase transition in photonic quantum walks, which can only be explained by the non-Bloch band theory, which relies on the open boundary condition. Furthermore, we explore the impact of disorder on PT\mathcal{P}\mathcal{T}PT symmetry and demonstrate disorder-induced PT\mathcal{P}\mathcal{T}PT phase transitions. Our work provides a valuable perspective for further explorations on non-Hermitian physics.
Abstract A P T -symmetric, non-Hermitian Hamiltonian in the P T -unbroken regime can lead to unitary dynamics under the appropriate choice of the Hilbert space. The Hilbert space is determined by a Hamiltonian-compatible inner product map on the underlying vector space, facilitated by a ‘metric operator’. A more traditional method, however, involves treating the evolution as open system dynamics, and the state is constructed through normalization at each time step. In this work, we present a comparative study of the two methods of constructing the reduced dynamics of a system evolving under a P T -symmetric Hamiltonian. Our system is a one-dimensional quantum walk with the spin and position degrees of freedom forming its two subsystems. We compare the information flow between the subsystems under the two methods. We find that under the metric formalism, a power law decay of the information backflow to the subsystem gives a clear indication of the transition from P T -unbroken to the broken phase. This is unlike the information backflow under the normalized state method. We also note that even though non-Hermiticity models open system dynamics, pseudo-Hermiticity can increase entanglement between the subsystem in the metric Hilbert space, thus indicating that pseudo-Hermiticity cases can be seen as a resource in quantum mechanics.
Optical interferometers comprising birefringent-crystal beam displacers, wave plates, and phase shifters serve as stable devices for simulating quantum information processes such as heralded coined quantum walks. Quantum walks are important for quantum algorithms, universal quantum computing circuits, quantum transport in complex systems, and demonstrating intriguing nonlinear dynamical quantum phenomena. We introduce fully controllable polarization-independent phase shifters in optical pathes in order to realize site-dependent phase defects. The effectiveness of our interferometer is demonstrated through realizing single-photon quantum-walk dynamics in one dimension. By applying site-dependent phase defects, the translational symmetry of an ideal standard quantum walk is broken resulting in localization effect in a quantum walk architecture. The walk is realized for different site-dependent phase defects and coin settings, indicating the strength of localization signature depends on the level of phase due to site-dependent phase defects and coin settings and opening the way for the implementation of a quantum-walk-based algorithm.