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Register a new userHidden symmetries of two-electron quantum dots in a magnetic field
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Using a classical and quantum mechanical analysis, we show that the magnetic field gives rise to dynamical symmetries of a three-dimensional axially symmetric two-electron quantum dot with a parabolic confinement. These symmetries manifest themselves as near-degeneracies in the quantum spectrum at specific values of the magnetic field and are robust at any strength of the electron-electron interaction.
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We use the entanglement measure to study the evolution of quantum correlations in two-electron axially-symmetric parabolic quantum dots under a perpendicular magnetic field. We found that the entanglement indicates on the shape transition in the density distribution of two electrons in the lowest state with zero angular momentum projection at the specific value of the applied magnetic field.
We present a thorough analysis of the electron density distribution (shape) of two electrons, confined in the three-dimensional harmonic oscillator potential, as a function of the perpendicular magnetic field.Explicit algebraic expressions are derived in terms of the systems parameters and the magnetic field strength to trace the shape transformations in the ground and low-lying excited states. We found that the interplay of the classical and quantum properties lead to a quantum shape transition from a lateral to a vertical localization of electrons in low-lying excited states at relatively strong Coulomb interaction with alteration of the magnetic field. In contrast, in that regime in the ground states the electrons form always a ring type distribution in the lateral plane. The analytical results demonstrate a good agreement with quantum numerical results near the transition point and at high magnetic field.
A cornerstone of quantum mechanics is the characterisation of symmetries provided by Wigners theorem. Wigners theorem establishes that every symmetry of the quantum state space must be either a unitary transformation, or an antiunitary transformation. Here we extend Wigners theorem from quantum states to quantum evolutions, including both the deterministic evolution associated to the dynamics of closed systems, and the stochastic evolutions associated to the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time symmetric formulation of quantum theory must either restrict the set of the allowed evolutions, or modify the operational interpretation of quantum states and processes. Here we propose a time symmetric formulation of quantum theory where the allowed quantum evolutions are restricted to a suitable set, which includes both unitary evolution and projective measurements, but excludes the deterministic preparation of pure states. The standard operational formulation of quantum theory can be retrieved from this time symmetric version by introducing an operation of conditioning on the outcomes of past experiments.
For every ADE Dynkin diagram, we give a realization, in terms of usual fusion algebras (graph algebras), of the algebra of quantum symmetries described by the associated Ocneanu graph. We give explicitly, in each case, the list of the corresponding twisted partition functions
We extend the class of QM problems which permit for quasi-exact solutions. Specifically, we consider planar motion of two interacting charges in a constant uniform magnetic field. While Turbiner and Escobar-Ruiz (2013) addressed the case of the Coulomb interaction between the particles, we explore three other potentials. We do this by reducing the appropriate Hamiltonians to the second-order polynomials in the generators of the representation of $SL(2,C)$ group in the differential form. This allows us to perform partial diagonalisation of the Hamiltonian, and to reduce the search for the first few energies and the corresponding wave functions to an algebraic procedure.
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