It is necessary to calculate the C operator for the non-Hermitian PT-symmetric Hamiltonian H=\half p^2+\halfμ^2x^2-λx^4 in order to demonstrate that H defines a consistent unitary theory of quantum mechanics. However, the C operator cannot be obtained by using perturbative methods. Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half μ^2x^2+igx^3-λx^4, whose C operator can be obtained perturbatively. In the semiclassical limit all terms in the perturbation series can be calculated in closed form and the perturbation series can be summed exactly. The result is a closed-form expression for C having a nontrivial dependence on the dynamical variables x and p and on the parameter λ.
In our previous paper I (del Valle--Turbiner, Int. J. Mod. Phys. A34, 1950143, 2019) it was developed the formalism to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As the result it was introduced - the Approximant - a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variationa…
Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectat…
We generalize our earlier work on the symplectic/Hamiltonian formulation of the dynamics of the Gaussian wave packet to non-Gaussian semiclassical wave packets. We find the symplectic forms and asymptotic expansions of the Hamiltonians associated with these semiclassical wave packets, and obtain Hamiltonian systems governing their dynamics. Numerical experiments demonstrate that the dynamics give a very good approximation to the short-time dynamics of the expectation values computed by a method based on Egorov's Theorem or the Initial Value Representation.
In fast forward quantum shortcuts to adiabaticity, a designed potential $U_{FF}(q,t)$ steers a wavefunction to evolve from the $n$'th eigenstate of an initial Hamiltonian $\hat H(0)$ to the $n$'th eigenstate of a final Hamiltonian $\hat H(τ)$, in finite time $τ$. Previously proposed strategies for constructing $U_{FF}$ are (in the absence of special symmetries) limited to the ground state, $n=0$. We develop a method that overcomes this limitation, thereby substantially expanding the applicability of this shortcut to adiabaticity, and we illustrate its effectiveness with numerical simulations. …
We extend our previous work on symplectic semiclassical Gaussian wave packet dynamics to incorporate electromagnetic interactions by including a vector potential. The main advantage of our formulation is that the equations of motion derived are naturally Hamiltonian. We obtain an asymptotic expansion of our equations in terms of $\hbar$ and show that our equations have $\mathcal{O}(\hbar)$ corrections to those presented by Zhou, whereas ours also recover the equations of Zhou in the case of a linear vector potential and quadratic scalar potential. One and two dimensional examples of a particle…
The time evolution of the Wigner function for Gaussian states generated by Lindblad quantum dynamics is investigated in the semiclassical limit. A new type of phase-space dynamics is obtained for the centre of a Gaussian Wigner function, where the Lindblad terms generally introduce a non-Hamiltonian flow. In addition to this, the Gaussian approximation yields dynamical equations for the covariances. The approximation becomes exact for linear Lindblad operators and a quadratic Hamiltonian. By viewing the Wigner function as a wave function on a coordinate space of doubled dimension, and the phas…
Exceptional points (EPs) correspond to degeneracies of open systems. These are attracting much interest in optics, optoelectronics, plasmonics, and condensed matter physics. In the classical and semiclassical approaches, Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-Hermitian Hamiltonians such that at least two eigenfrequencies are identical and the corresponding eigenstates coalesce. HEPs result from continuous, mostly slow, nonunitary evolution without quantum jumps. Clearly, quantum jumps should be included in a fully quantum approach to make it equivalent to, e.g., the …
Gutzwiller's famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with an view towards the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley-Zhender index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley-Zehnder's index theo…
Inspired by special and general relativistic systems that can have Hamiltonians involving square roots, or more general fractional powers, in this article we address the question how a suitable set of coherent states for such systems can be obtained. This becomes a relevant topic if the semiclassical sector of a given quantum theory wants to be analysed. As a simple setup we consider the toy model of a deparametrised system with one constraint that involves a fractional power of the harmonic oscillator Hamiltonian operator and we discuss two approaches for finding suitable coherent states for …
