Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries includes the Weyl-Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl-Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves invariant the Heisenberg commutation relations are essentially projective representations of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of Hamilton's equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.
The quantum phase leads to projective representations of symmetry groups in quantum mechanics. The projective representations are equivalent to the unitary representations of the central extension of the group. A celebrated example is Wigner's formulation of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group. However, Wigner's formulation makes no mention of the Weyl-Heisenberg group and the hermitian representation of its algebra that are the Heisenberg commutation relations fundamental to quantum physics. We put aside the relativistic …
From the principle that there is no absolute description of a physical state, we advance the approach according to which one should be able to describe the physics from the perspective of a quantum particle. The kinematics seen from this frame of reference is shown to be rather unconventional. In particular, we discuss several subtleties emerging in the relative formulation of central notions such as vector states, the classical limit, entanglement, uncertainty relations, and the complementary principle. A Hamiltonian formulation is derived as well which correctly encapsulates effects of ficti…
A new method of diagonalisation of the XY-Hamiltonian of inhomogeneous open linear chains with periodic (in space) changing Larmor frequencies and coupling constants is developed. As an example of application, multiple quantum dynamics of an inhomogeneous chain consisting of 1001 spins is investigated. Intensities of multiple quantum coherences are calculated for arbitrary inhomogeneous chains in the approximation of the next nearest interactions. {\it Key words:} linear spin chain, nearest--neighbour approximation, three--diagonal matrices, diagonalisation, fermions, multiple--quantum NMR, mu…
We demonstrate the fundamental links existing between two different descriptions of quantum electrodynamics in inhomogeneous, lossy and dispersive dielectric media which are based either on the Huttner-Barnett formalism for polaritons [B. Huttner and S. M. Barnett, Phys.Rev. A 46, 4306 (1992)] or the Langevin noise approach using fluctuating currents [T. Gruner and D.-G. Welsch, Phys.Rev.A 53, 1818 (1996)]. In this work we demonstrate the practical equivalence of the two descriptions by introducing the concept of effective photon state associated with some specific noise current distribution. …
The simplest Tree-Tensor-States (TTS) respecting the Parity and the Time-Reversal symmetries are studied in order to describe the ground states of Long-Ranged Quantum Spin Chains with or without disorder. Explicit formulas are given for the one-point and two-point reduced density matrices that allow to compute any one-spin and two-spin observable. For Hamiltonians containing only one-body and two-body contributions, the energy of the TTS can be then evaluated and minimized in order to obtain the optimal parameters of the TTS. This variational optimization of the TTS parameters is compared with…
In this work, we use the Hamiltonian of a modified Dzyaloshinskii-Moriya model and investigate the perfect transfer of the quantum state on the spin networks. In this paper, we calculate fidelity in which fidelity depends on magnetic field and another parameters. Then, by using the numerical analysis we show that the fidelity of the transferred state is determined by magnetic field $B$, exchange coupling $J$ and the Dzyaloshinskii- Moriya interaction $D$. We also found that the perfect transfer of the quantum state is possible with condition $B \gg Γ^2ω^{N/2}$ where $Γ=((J+iD)/2)$ and $ω=Γ^*/ …
Learning Inhomogeneous Heisenberg Hamiltonians in Nanographene Spin Chains
Aalto University（阿尔托大学）；Empa—Swiss Federal Laboratories for Materials Science and Technology（瑞士联邦材料科技与检测局）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）；Max Planck Institute of Microstructure Physics（马克斯·普朗克微结构物理研究所）；Technische Universität Dresden（德累斯顿工业大学）
We consider the time-inhomogeneous Markovian jump process introduced by John S. Bell [Phys.Rep. 137, 49] for a lattice quantum field theory, which runs on the associated configuration space. Its jump rates, tailored to give the process the quantum distribution $|Ψ_t|^2$ at all times $t$, typically exhibit singularities. We establish the existence of a unique such process for all times, under suitable assumptions on the Hamiltonian or the initial state vector $Ψ_0$. The proof of non-explosion takes advantage of the special role of the $|Ψ_t|^2$ distribution.
We study a three-parameter family of PT-symmetric Hamiltonians, related via the ODE/IM correspondence to the Perk-Schultz models. We show that real eigenvalues merge and become complex at quadratic and cubic exceptional points, and explore the corresponding Jordon block structures by exploiting the quasi-exact solvability of a subset of the models. The mapping of the phase diagram is completed using a combination of numerical, analytical and perturbative approaches. Among other things this reveals some novel properties of the Bender-Dunne polynomials, and gives a new insight into a phase trans…
Let group generators having finite-dimensional representation be realized as Hermitian linear differential operators without nhomogeneous terms as takes place, for example, for the SO(n) group. Then orresponding group Hamiltonians containing terms linear in generators (along with quadratic ones) give rise to quasi-exactly solvable models with a magnetic field in a curved space. In particular, in the two-dimensional case such models are generated by quantum tops. In the three-dimensional one for the SO(4) Hamiltonian with an isotropic quadratic part the manifold within which a quantum particle …
We investigate the ground-state Riemannian metric and the cyclic quantum distance of an inhomogeneous quantum Ising spin-1/2 chain in a transverse field. This model can be diagonalized by using a general canonical transformation to the fermionic Hamiltonian mapped from the spin system. The ground-state Riemannian metric is derived exactly on a parameter manifold ring $S^1$, which is introduced by performing a gauge transformation to the spin Hamiltonian through a twist operator. The ground-state cyclic quantum distance and the second derivative of the ground-state energy are studied in differe…
A quantum damped-polariton model is constructed for an inhomogeneous anisotropic linear dielectric with arbitrary dispersion in space and time. The model Hamiltonian is completely diagonalized by determining the creation and annihilation operators for the fundamental polariton modes as specific linear combinations of the basic dynamical variables. Explicit expressions are derived for the time-dependent operators describing the electromagnetic field, the dielectric polarization and the noise term in the latter. It is shown how to identify bath variables that generate the dissipative dynamics of…
We show, for quantum annealing, that a certain type of inhomogeneous driving of the transverse field erases first-order quantum phase transitions in the p-body interacting mean-field-type model with and without longitudinal random field. Since a first-order phase transition poses a serious difficulty for quantum annealing (adiabatic quantum computing) due to the exponentially small energy gap, the removal of first-order transitions means an exponential speedup of the annealing process. The present method may serve as a simple protocol for the performance enhancement of quantum annealing, compl…
As a nuclear spin model of scalable quantum register, the one-dimensional chain of the magnetic atoms with nuclear spins 1/2 substituting the basic atoms in the plate of nuclear spin free easy-axis 3D antiferromagnet is considered. It is formulated the generalized antiferromagnet Hamiltonian in spin-wave approximation (low temperatures) considering the inhomogeneous external magnetic field, which is directed along the easy axis normally to plane of the plate and has a constant gradient along the nuclear spin chain. Assuming a weak gradient, the asymptotic expression for coefficients of unitary…
The motion of quantum particles homogeneously constrained to a curved surface is affected by a curvature induced geometric potential. Here, we consider the case of inhomogeneous confinement and derive the effective Hamiltonian by extending thin-layer procedure, where an extra effective potential appears. This effective potential is relevant to the ground state energy perpendicular to the surface and the morphology of the confining potential. Tiny changes in the thickness are envisioned to induce considerable magnitude of the effective potential. To demonstrate the impact of the inhomogeneity, …
Quantum simulation of wave optics in weakly inhomogeneous media using block-encoding
Friedrich Schiller University Jena（耶拿弗里德里希·席勒大学）；Max Planck School of Photonics（马克斯·普朗克光子科学学院）；Fraunhofer Institute for Applied Optics and Precision Engineering IOF（弗劳恩霍夫应用光学和精密工程研究所）
The notion of geometric phase has been recently introduced to analyze the quantum phase transitions of many-body systems from the geometrical perspective. In this work, we study the geometric phase of the ground state for an inhomogeneous period-two anisotropic XY model in a transverse field. This model encompasses a group of familiar spin models as its special cases and shows a richer critical behavior. The exact solution is obtained by mapping on a fermionic system through the Jordan-Wigner transformation and constructing the relevant canonical transformation to realize the diagonalization o…
We give a pedagogical introduction to the Generalized Hydrodynamic approach to inhomogeneous quenches in integrable many-body quantum systems. We review recent applications of the theory, focusing in particular on two classes of problems: bipartitioning protocols and trap quenches, which represent two prototypical examples of broken translational symmetry in either the system initial state or post-quench Hamiltonian. We report on exact results that have been obtained for generic time-dependent correlation functions and entanglement evolution, and discuss in detail the range of applicability of…
Entanglement has become central for the characterization of quantum matter both in and out of equilibrium. In a dynamical context entanglement exhibits universal linear temporal growth in generic systems, which stems from the underlying linear light cones as they occur in planar geometries. Inhomogeneous spacetimes can lead, however, to strongly bent trajectories. While such bent trajectories crucially impact correlation spreading and therefore the light-cone structure, it has remained elusive how this influences the entanglement dynamics. In this work we investigate the real-time evolution of…
