scholarly journals BLOCH THEOREM FOR THE EVOLUTION OF STATES IN THE CYCLIC QUANTUM SYSTEMS AND THE UNIFICATION OF RESONANT GEOMETRIC PHASES

1997 ◽  
Vol 46 (2) ◽  
pp. 227
Author(s):  
LI BO-ZANG ◽  
ZHANG DE-GANG ◽  
WU JIAN-HUA ◽  
YAN FENG-LI
Keyword(s):  
Quantum Systems ◽  
Geometric Phases ◽  
Bloch Theorem

scholarly journals Photonic implementation of Majorana-based Berry phases

2018 ◽  
Vol 4 (10) ◽  
pp. eaat6533 ◽  
Author(s):  
Jin-Shi Xu ◽  
Kai Sun ◽  
Jiannis K. Pachos ◽  
Yong-Jian Han ◽  
Chuan-Feng Li ◽  
...  
Keyword(s):  
Quantum Systems ◽  
Majorana Fermions ◽  
Topological Characteristic ◽  
Fundamental Physics ◽  
Geometric Phases ◽  
Physical Systems ◽  
All Optical ◽  
Topological Version ◽  
Berry Phases ◽  
Statistical Evolution

Geometric phases, generated by cyclic evolutions of quantum systems, offer an inspiring playground for advancing fundamental physics and technologies alike. The exotic statistics of anyons realized in physical systems can be interpreted as a topological version of geometric phases. However, non-Abelian statistics has not yet been demonstrated in the laboratory. Here, we use an all-optical quantum system that simulates the statistical evolution of Majorana fermions. As a result, we experimentally realize non-Abelian Berry phases with the topological characteristic that they are invariant under continuous deformations of their control parameters. We implement a universal set of Majorana-inspired gates by performing topological and nontopological evolutions and investigate their resilience against perturbative errors. Our photonic experiment, though not scalable, suggests the intriguing possibility of experimentally simulating Majorana statistics with scalable technologies.

scholarly journals A Proof of the Bloch Theorem for Lattice Models

2019 ◽  
Vol 177 (4) ◽  
pp. 717-726 ◽  
Author(s):  
Haruki Watanabe
Keyword(s):  
Boundary Condition ◽  
Tight Binding ◽  
Lattice Models ◽  
Quantum Systems ◽  
Open Boundary ◽  
Current Operator ◽  
Open Boundary Condition ◽  
Expectation Value ◽  
Bloch Theorem ◽  
Large Quantum

Abstract The Bloch theorem is a powerful theorem stating that the expectation value of the U(1) current operator averaged over the entire space vanishes in large quantum systems. The theorem applies to the ground state and to the thermal equilibrium at a finite temperature, irrespective of the details of the Hamiltonian as far as all terms in the Hamiltonian are finite ranged. In this work we present a simple yet rigorous proof for general lattice models. For large but finite systems, we find that both the discussion and the conclusion are sensitive to the boundary condition one assumes: under the periodic boundary condition, one can only prove that the current expectation value is inversely proportional to the linear dimension of the system, while the current expectation value completely vanishes before taking the thermodynamic limit when the open boundary condition is imposed. We also provide simple tight-binding models that clarify the limitation of the theorem in dimensions higher than one.

Geometric Phases forSU(3) Representations and Three Level Quantum Systems

1997 ◽  
Vol 253 (1) ◽  
pp. 55-82 ◽  
Author(s):  
G. Khanna ◽  
S. Mukhopadhyay ◽  
R. Simon ◽  
N. Mukunda
Keyword(s):  
Quantum Systems ◽  
Geometric Phases

scholarly journals Ray space ‘Riccati’ evolution and geometric phases for N-level quantum systems

Pramana ◽  
2007 ◽  
Vol 69 (3) ◽  
pp. 317-327 ◽  
Author(s):  
S. Chaturvedi ◽  
E. Ercolessi ◽  
G. Marmo ◽  
G. Morandi ◽  
N. Mukunda ◽  
...  
Keyword(s):  
Quantum Systems ◽  
Geometric Phases ◽  
N Level

Experimental Detection of Geometric Phases I: Quantum Systems in Classical Environments

2003 ◽  
pp. 225-253 ◽  
Author(s):  
Arno Bohm ◽  
Ali Mostafazadeh ◽  
Hiroyasu Koizumi ◽  
Qian Niu ◽  
Joseph Zwanziger
Keyword(s):  
Quantum Systems ◽  
Geometric Phases ◽  
Experimental Detection
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