scholarly journals Bulk–edge correspondence and stability of multiple edge states of a $\mathcal{PT}$-symmetric non-Hermitian system by using non-unitary quantum walks

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Makio Kawasaki ◽  
Ken Mochizuki ◽  
Norio Kawakami ◽  
Hideaki Obuse
Keyword(s):  
Symmetry Breaking ◽  
Open Quantum Systems ◽  
Quantum Walk ◽  
Quantum Systems ◽  
Quantum Walks ◽  
Multiple Edge ◽  
Exceptional Point ◽  
Topological Phase ◽  
Edge States ◽  
The Stability

Abstract Topological phases and the associated multiple edge states are studied for parity and time-reversal ($\mathcal{PT}$)-symmetric non-Hermitian open quantum systems by constructing a non-unitary three-step quantum walk retaining $\mathcal{PT}$ symmetry in one dimension. We show that the non-unitary quantum walk has large topological numbers of the $\mathbb{Z}$ topological phase and numerically confirm that multiple edge states appear as expected from the bulk–edge correspondence. Therefore, the bulk–edge correspondence is valid in this case. Moreover, we study the stability of the multiple edge states against a symmetry-breaking perturbation so that the topological phase is reduced to $\mathbb{Z}_2$ from $\mathbb{Z}$. In this case, we find that the number of edge states does not become one unless a pair of edge states coalesce at an exceptional point. Thereby, this is a new kind of breakdown of the bulk–edge correspondence in non-Hermitian systems. The mechanism of the prolongation of edge states against the symmetry-breaking perturbation is unique to non-Hermitian systems with multiple edge states and anti-linear symmetry. Toward experimental verifications, we propose a procedure to determine the number of multiple edge states from the time evolution of the probability distribution.

scholarly journals Shannon Entropy and Diffusion Coefficient in Parity-Time Symmetric Quantum Walks

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1145
Author(s):  
Zhiyu Tian ◽  
Yang Liu ◽  
Le Luo
Keyword(s):  
Diffusion Coefficient ◽  
Shannon Entropy ◽  
Abrupt Change ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Bulk Property ◽  
Topological Phase ◽  
Edge States ◽  
Peak Structure ◽  
Topological State

Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk–boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient was found to show unique features with the topological phase transitions driven by parity–time (PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite the fact that artificial boundaries are not constructed by an inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated with the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented herein may open up a new avenue for studying the topological state in non-Hermitian quantum walk systems.

scholarly journals Emergence of PT-symmetry breaking in open quantum systems

2020 ◽  
Vol 9 (4) ◽  
Author(s):  
Julian Huber ◽  
Peter Kirton ◽  
Stefan Rotter ◽  
Peter Rabl
Keyword(s):  
Symmetry Breaking ◽  
Open Quantum Systems ◽  
Symmetry Transformation ◽  
Quantum Systems ◽  
Coupled Systems ◽  
Physical Systems ◽  
Open Quantum ◽  
Parity Symmetry ◽  
Classical Correspondence ◽  
Liouville Operators

The effect of \mathcal{PT}𝒫𝒯-symmetry breaking in coupled systems with balanced gain and loss has recently attracted considerable attention and has been demonstrated in various photonic, electrical and mechanical systems in the classical regime. However, it is still an unsolved problem how to generalize the concept of \mathcal{PT}𝒫𝒯 symmetry to the quantum domain, where the conventional definition in terms of non-Hermitian Hamiltonians is not applicable. Here we introduce a symmetry relation for Liouville operators that describe the dissipative evolution of arbitrary open quantum systems. Specifically, we show that the invariance of the Liouvillian under this symmetry transformation implies the existence of stationary states with preserved and broken parity symmetry. As the dimension of the Hilbert space grows, the transition between these two limiting phases becomes increasingly sharp and the classically expected \mathcal{PT}𝒫𝒯-symmetry breaking transition is recovered. This quantum-to-classical correspondence allows us to establish a common theoretical framework to identify and accurately describe \mathcal{PT}𝒫𝒯-symmetry breaking effects in a large variety of physical systems, operated both in the classical and quantum regimes.

scholarly journals Symmetry Breaking and Error Correction in Open Quantum Systems

2020 ◽  
Vol 125 (24) ◽  
Author(s):  
Simon Lieu ◽  
Ron Belyansky ◽  
Jeremy T. Young ◽  
Rex Lundgren ◽  
Victor V. Albert ◽  
...  
Keyword(s):  
Symmetry Breaking ◽  
Error Correction ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Open Quantum

scholarly journals General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

2020 ◽  
Vol 19 (10) ◽  
Author(s):  
Michael Manighalam ◽  
Mark Kon
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Systems ◽  
Quantum Walks ◽  
Time Limit ◽  
Continuous Limit ◽  
Continuous Space ◽  
Time Quantum ◽  
Space Limit

Abstract Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.

scholarly journals Persistence of topological phases in non-Hermitian quantum walks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Vikash Mittal ◽  
Aswathy Raj ◽  
Sanjib Dey ◽  
Sandeep K. Goyal
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Topological Phases ◽  
Topological Order ◽  
Topological Phase ◽  
Two Dimensional ◽  
One Dimensional ◽  
Topological Phase Transition ◽  
Topological States ◽  
Time Quantum

AbstractDiscrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a lossy environment may destroy these phases. We investigate the behaviour of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian respects exact $${{\mathcal {P}}}{{\mathcal {T}}}$$ P T -symmetry. Although the topological nature persists in two-dimensional quantum walks as well, the $${{\mathcal {P}}}{{\mathcal {T}}}$$ P T -symmetry has no role to play there. Furthermore, we observe topological phase transition in two-dimensional quantum walks that is induced by losses in the system.

scholarly journals Impact of Lattice Vibrations on the Dynamics of a Spinor Atom-Optics Kicked Rotor

2019 ◽  
Vol 4 (1) ◽  
pp. 10 ◽  
Author(s):  
Caspar Groiseau ◽  
Alexander Wagner ◽  
Gil Summy ◽  
Sandro Wimberger
Keyword(s):  
Quantum Walk ◽  
Quantum Walks ◽  
Atom Optics ◽  
Lattice Vibrations ◽  
Noise Sources ◽  
The Road ◽  
Spin Degree ◽  
Kicked Rotor ◽  
Time Quantum ◽  
The Stability

We investigate the effect of amplitude and phase noise on the dynamics of a discrete-time quantum walk and its related evolution. Our findings underline the robustness of the motion with respect to these noise sources, and can explain the stability of quantum walks that has recently been observed experimentally. This opens the road to measure topological properties of an atom-optics double kicked rotor with an additional internal spin degree of freedom.

scholarly journals Gravitationally induced entanglement dynamics between two quantum walkers

2021 ◽  
Vol 81 (5) ◽  
Author(s):  
Himanshu Badhani ◽  
C. M. Chandrashekar
Keyword(s):  
Potential Role ◽  
Gravitational Interaction ◽  
Quantum Walk ◽  
Quantum Systems ◽  
Quantum Walks ◽  
Quantum Mechanical ◽  
Entanglement Dynamics ◽  
Position Space ◽  
Entanglement Generation

AbstractQuantum walk is a synonym for multi-path interference and faster spread of a particle in a superposition of position space. We study the effects of a quantum mechanical interaction modeled to mimic quantum mechanical gravitational interaction between the two states of the walkers. The study has been carried out to investigate the entanglement generation between the two quantum walkers that do not otherwise interact. We see that the states do in fact get entangled more and more as the quantum walks unfold, and there is an interesting dependence of entanglement generation on the mass of the two particles performing the walks. With the introduction of noise into the dynamics, we also show the sensitivity of entanglement between the two walkers on the noise introduced in one of the walks. The signature of quantum effects due to gravitational interactions highlights the potential role of quantum systems in probing the nature of gravity.

scholarly journals Continuous Dissipative Phase Transitions with or without Symmetry Breaking

2021 ◽  
Author(s):  
Fabrizio Minganti ◽  
Ievgen Arkhipov ◽  
Adam Miranowicz ◽  
Franco Nori
Keyword(s):  
Phase Transitions ◽  
Symmetry Breaking ◽  
Open Quantum Systems ◽  
Quantum Systems ◽  
Second Order ◽  
Necessary Condition ◽  
Symmetric Model ◽  
Open Quantum ◽  
Modern Physics ◽  
Laser Model

Abstract The paradigm of second-order phase transitions (PTs) induced by spontaneous symmetry breaking (SSB) in thermal and quantum systems is a pillar of modern physics that has been fruitfully applied to out-of-equilibrium open quantum systems. Dissipative phase transitions (DPTs) of second order are often connected with SSB, in close analogy with well-known thermal second-order PTs in closed quantum and classical systems. That is, a second-order DPT should disappear by preventing the occurrence of SSB. Here, we prove this statement to be wrong, showing that, surprisingly, SSB is not a necessary condition for the occurrence of second-order DPTs in \textit{out-of-equilibrium open quantum systems}. We analytically prove this result using the Liouvillian theory of dissipative phase transitions, and demonstrate this anomalous transition in two exemplary models: a paradigmatic laser model, where we can arbitrarily remove SSB while retaining criticality, and a $Z_2$-symmetric model of a two-photon Kerr resonator.

scholarly journals A 2D Quantum Walk Simulation of Two-Particle Dynamics

Science ◽  
2012 ◽  
Vol 336 (6077) ◽  
pp. 55-58 ◽  
Author(s):  
Andreas Schreiber ◽  
Aurél Gábris ◽  
Peter P. Rohde ◽  
Kaisa Laiho ◽  
Martin Štefaňák ◽  
...  
Keyword(s):  
Topological Structure ◽  
Dynamic Control ◽  
Quantum Walk ◽  
Particle Dynamics ◽  
Quantum Systems ◽  
Quantum Walks ◽  
Transport Systems ◽  
Graph Structure ◽  
Fiber Network ◽  
12 Steps

Multidimensional quantum walks can exhibit highly nontrivial topological structure, providing a powerful tool for simulating quantum information and transport systems. We present a flexible implementation of a two-dimensional (2D) optical quantum walk on a lattice, demonstrating a scalable quantum walk on a nontrivial graph structure. We realized a coherent quantum walk over 12 steps and 169 positions by using an optical fiber network. With our broad spectrum of quantum coins, we were able to simulate the creation of entanglement in bipartite systems with conditioned interactions. Introducing dynamic control allowed for the investigation of effects such as strong nonlinearities or two-particle scattering. Our results illustrate the potential of quantum walks as a route for simulating and understanding complex quantum systems.

scholarly journals Effects of symmetry breaking of the structurally-disordered Hamiltonian ensembles on the anisotropic decoherence

2021 ◽  
Author(s):  
Hong-Bin Chen
Keyword(s):  
Probability Distribution ◽  
Symmetry Breaking ◽  
Canonical Form ◽  
Open Quantum Systems ◽  
Dynamical Behavior ◽  
Structural Disorder ◽  
Quantum Systems ◽  
New Approach ◽  
Pure Dephasing ◽  
Different Types

Abstract It is commonly known that the dephasing in open quantum systems is due to the establishment of bipartite correlations with an environment. Recently, a new approach of average over disordered Hamiltonian ensemble is developed and shown to capable of describing both the incoherent dynamical behavior and the nonclassicality of dynamical processes. Here we further extend the approach of Hamiltonian ensemble in the canonical form to the realm of structural disorder. Under the separation of the probability distribution within the Hamiltonian ensemble, the geometrical structural is easily visualized and can be characterized according to the degree of symmetry. We demonstrate four degrees and investigate the effects of different types of symmetry breaking on the incoherent dynamics. With these effects, we obtain rather general master equations, going beyond the previous frameworks of pure dephasing or isotropic depolarization. The practicality of the Hamiltonian ensemble and the theory of process nonclassicality is significantly enhanced.

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