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arXiv:1810.04137 · Journal of Physics A Mathematical and Theoretical

Taming Hamiltonian systems with balanced loss and gain via Lorentz interaction: general results and a case study with Landau Hamiltonian

Pijush K. Ghosh

Published Aug 30, 2019• 8 citations
arXivHTMLPDF

Abstract

Abstract The kinetic energy term of Hamiltonian systems with balanced loss and gain is not semi-positive-definite, leading to instabilities at the classical as well quantum level. It is shown that an additional Lorentz interaction in the Hamiltonian allows the kinetic energy term to be semi-positive-definite and thereby, improving the stability properties of the system. Further, a consistent quantum theory admitting bound states may be obtained on the real line instead of Stoke wedges on the complex plane. The Landau Hamiltonian in presence of balanced loss and gain is considered for elucidating the general result. The kinetic energy term is semi-positive-definite provided the magnitude of the applied external magnetic field is greater than the magnitude of the ‘analogous magnetic field’ due to the loss-gain terms. It is shown that the classical particle moves on an elliptical orbit with a cyclotron frequency that is less than its value in absence of the loss-gain terms. The quantum system share the properties of the standard Landau Hamiltonian, but, with the modified cyclotron frequency. It is shown that the Hall current has non-vanishing components along the direction of the external uniform electric field and to its transverse direction. The Pauli equation in presence of balanced loss and gain is shown to be supersymmetric.

DOI
10.1088/1751-8121/ab3f49

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Brazilian Journal of Physics•Sep 1, 1999•7 citations
Abstract The kinetic energy term of Hamiltonian systems with balanced loss and gain is not semi-positive-definite, leading to instabilities at the classical as well quantum level. It is shown that an additional Lorentz interaction in the Hamiltonian allows the kinetic energy term to be semi-positive-definite and thereby, improving the stability properties of the system. Further, a consistent quantum theory admitting bound states may be obtained on the real line instead of Stoke wedges on the complex plane. The Landau Hamiltonian in presence of balanced loss and gain is considered for elucidating the general result. The kinetic energy term is semi-positive-definite provided the magnitude of the applied external magnetic field is greater than the magnitude of the ‘analogous magnetic field’ due to the loss-gain terms. It is shown that the classical particle moves on an elliptical orbit with a cyclotron frequency that is less than its value in absence of the loss-gain terms. The quantum system share the properties of the standard Landau Hamiltonian, but, with the modified cyclotron frequency. It is shown that the Hall current has non-vanishing components along the direction of the external uniform electric field and to its transverse direction. The Pauli equation in presence of balanced loss and gain is shown to be supersymmetric.
Gain and loss to the wave function of quantum mechanics can in a convenient way be modelled by effective non-Hermitian Hamiltonians. Imaginary contributions to the potential introduce source and drain terms for the probability amplitude. A special class of non-Hermitian Hamiltonians are those which possess a parity-time symmetry. In spite of their non-Hermiticity these Hamiltonians allow for real energy eigenvalues, i.e. the existence of stationary states in the presence of balanced gain and loss. This effect has been identified theoretically in a large number of quantum systems. Its existence has also been proved experimentally in coupled optical wave guides. The wave guides are, however, only optical analogues of quantum systems. In the first part of this thesis it is shown from the theoretical side that Bose-Einstein condensates in a double-well setup are an ideal candidate for a first experimental realisation of a genuine quantum system with parity-time symmetry. When particles are removed from one well and coherently injected into the other the external potential is parity-time symmetric. To investigate the system the underlying time-independent and time-dependent Gross-Pitaevskii equations are solved numerically. It turns out that a subtle interplay between the nonlinearity of the Gross-Pitaevskii equation and the gain-loss effect leads to a complicated dynamics of the condensate wave function. However, the most important result is the existence of stationary states that are sufficiently stable to be observable in an experiment. Two suggestions for experimental realisations are presented. They are based on the idea of embedding the non-Hermitian parity-time-symmetric system into a larger structure described by a Hermitian Hamiltonian. A further effect of non-Hermitian Hamiltonians are so-called exceptional points, at which two resonances coalesce such that both their eigenvalues and wave functions become identical. It is shown that an exceptional point can unambiguously be identified by a characteristic non-exponential decay of the resonances. With numerically exact calculations for the hydrogen atom in crossed electric and magnetic fields this effect is verified in an experimentally accessible quantum system. The second part of the thesis is devoted to semiclassical Gaussian approximations to the Boltzmann operator, which have become an important tool for the investigation of thermodynamic properties of clusters of atoms at low temperatures. A numerically cheap frozen Gaussian approximation to the imaginary time propagator with a width matrix especially suited for the dynamics of clusters is developed. It is applied to the cases of Ar3 and Ar6. For these clusters classical-like transitions in one step from a bounded moiety to free particles are found for increasing temperatures. Additionally, the structure of the Ar6 cluster is studied in the bound configuration and during the dissociation. Quantum effects, i.e. differences with the purely classical case, manifest themselves in the low-temperature behaviour of the mean energy and specific heat as well as in a slight shift of the transition temperature. A first-order correction to the semiclassical propagator is used to improve the results of the calculation for Ar3, and it is shown how the correction can be used to objectively assess the validity of the frozen Gaussian approximation.
Abstract In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z -axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a + and a − . We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency ω c = ∣ q ∣ B m and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a + in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a − . The phase of the circular motion is the same as the phase of the complex eigen value of a + . So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z -axis. We also show that as the magnitude of the eigen value of a + becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.
Abstract We revisit the quantum dynamics of a charged particle in a time-dependent magnetic field, a fundamental problem exhibiting rich non-adiabatic behaviour, from the complementary perspective of the Madelung fluid formulation. We first analyse the system within standard quantum mechanics using perturbation theory around the Landau levels, and then address the same problem through the Madelung perspective. We show that the hydrodynamic formulation not only yields an intuitive derivation of the exact solution, it also provides a clear physical interpretation of non-adiabatic quantum evolution in terms of mechanical energy transfers. In this picture, the sloshing oscillations of the wave function arise from deviations from the force balance between the magnetic Lorentz force and the gradient of the Bohm potential within the Landau levels. More broadly, our study illustrates how the Madelung approach reveals unexpected analogies between quantum dynamics and phenomena familiar from geophysical fluid dynamics.
Probability backflow is investigated for charged spin-1/21/21/2 particles in a uniform magnetic field within the lowest-Landau-level approximation. While single-mode Landau states exhibit only weak and non-robust backflow, it is shown that two-mode interference gives rise to negative probability flux in both the Schrödinger and Pauli formulations. In the Pauli case, orbital and spin contributions to the probability current are explicitly separated, revealing an additional interference mechanism. The external magnetic field is found to act as a tunable control parameter, simultaneously controlling the spatial overlap of orbitals and the spin dynamics. In the two-mode regime, a resonance between orbital mismatch and spin precession leads to an enhancement of backflow. Finite spatial resolution is shown not to suppress the effect within realistic coarse-graining. These results establish magnetic-field-controlled quantum backflow in Landau systems as an accessible interference phenomenon.
We present a general construction of pseudo-hermitian matrices in an arbitrary large, but finite dimensional vector space. The positive-definite metric which ensures reality of the entire spectra of a pseudo-hermitian operator, and is used for defining a modified inner-product in the associated vector space is also presented. The construction for an N dimensional vector space is based on the generators of SU (N ) in the fundamental representation and the identity operator. We apply the results to construct a generic pseudo-hermitian lattice model of size N with balanced loss-gain. The system is amenable to periodic as well as open boundary conditions and by construction, admits entirely real spectra along with unitary time-evolution. The tight binding and Su-Schrieffer-Heeger(SSH) models with nearest neighbour(NN) and next-nearest neighbour(NNN) interaction with balanced loss-gain appear as limiting cases.
The motion of a particle with charge q and mass m in a magnetic field given by B = kB0 + B1 [icos(omegat) + jsin(omegat)] and an electric field which obeys Ñ ×E = - ¶ B/ ¶ t is analysed classically and quantum-mechanically. The use of a rotating coordinate system allows the analytical derivation of the particle classical trajectory and its laboratory wavefunction. The motion exhibits two resonances, one at omega = omegac = -qBo/m, the cyclotron frequency, and the other at omega = omegaL = -qBo/2m, the Larmor frequency. For omega at the first resonance frequency, the particle acquires a simple closed trajectory, and the effective hamiltonian can be interpreted as that of a particle in a static magnetic field. In the second case a term corresponding to an effective static electric field remains, and the particle orbit is an open line. The particle wave function and eigenenergies are calculated.