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Physical Review A

Sudden transition from a stable to an unstable harmonic trap as the adiabatic potential parameter is varied in a time-periodic harmonic trap

Nimrod Moiseyev

Published Sep 30, 2013• 4 citations
Publisher

Abstract

It has been shown already that when the harmonic trap is opened (or closed) as a function of time while keeping the adiabatic parameter $\ensuremath{\mu}=[d\ensuremath{\omega}(t)/dt]/{\ensuremath{\omega}}^{2}(t)$ fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at $\ensuremath{\mu}=2$ [Uzdin, Dalla Torre, Kosloff, and Moiseyev, Phys. Rev. A 88, 022505 (2013)]. Here we show that by using time-dependent linear coordinate transformation the time-dependent Schr\"odinger equation with Hermitian time-dependent Hamiltonian is transformed into a time-dependent Schr\"odinger equation with a time-independent harmonic oscillator with the dimensionless frequency of $\sqrt{1\ensuremath{-}{(\ensuremath{\mu}/2)}^{2}}$. At $\ensuremath{\mu}=2$ a transition to a non-Hermitian Hamiltonian is obtained as the potential well is transformed to a parabolic potential barrier. While in a harmonic trap noninteracting particles have classical periodic motions, they are pushed apart exponentially in time as the potential well is suddenly transformed into a parabolic potential barrier in the new variable representation.

DOI
10.1103/physreva.88.034502

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It has been shown already that when the harmonic trap is opened (or closed) as a function of time while keeping the adiabatic parameter \ensuremathμ=[d\ensuremathω(t)/dt]/\ensuremathω2(t)\ensuremath{\mu}=[d\ensuremath{\omega}(t)/dt]/{\ensuremath{\omega}}^{2}(t)\ensuremathμ=[d\ensuremathω(t)/dt]/\ensuremathω2(t) fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at \ensuremathμ=2\ensuremath{\mu}=2\ensuremathμ=2 [Uzdin, Dalla Torre, Kosloff, and Moiseyev, Phys. Rev. A 88, 022505 (2013)]. Here we show that by using time-dependent linear coordinate transformation the time-dependent Schr"odinger equation with Hermitian time-dependent Hamiltonian is transformed into a time-dependent Schr"odinger equation with a time-independent harmonic oscillator with the dimensionless frequency of 1\ensuremath−(\ensuremathμ/2)2\sqrt{1\ensuremath{-}{(\ensuremath{\mu}/2)}^{2}}1\ensuremath−(\ensuremathμ/2)2​. At \ensuremathμ=2\ensuremath{\mu}=2\ensuremathμ=2 a transition to a non-Hermitian Hamiltonian is obtained as the potential well is transformed to a parabolic potential barrier. While in a harmonic trap noninteracting particles have classical periodic motions, they are pushed apart exponentially in time as the potential well is suddenly transformed into a parabolic potential barrier in the new variable representation.
The time evolution of a single particle in a harmonic trap with time-dependent frequency $\ensuremath{\omega}(t)$ has been well studied. Nevertheless, here we show that when the harmonic trap is opened (or closed) as a function of time while keeping the adiabatic parameter $\ensuremath{\mu}=[d\ensuremath{\omega}(t)/dt]/{\ensuremath{\omega}}^{2}(t)$ fixed, a sharp transition from an oscillatory to a monotonic exponential dynamics occurs at $\ensuremath{\mu}=2$. At this transition point, the time evolution has an exceptional point (EP) at all instants. This situation, where an EP of a time-dependent Hermitian Hamiltonian is obtained at any given time, is very different from other known cases. In the present case, we show that the order of the EP depends on the set of observables used to describe the dynamics. Our finding is relevant to the dynamics of a single ion in a magnetic, optical, or rf trap, and of diluted gases of ultracold atoms in optical traps.
In quantum physics the free particle and the harmonically trapped particle are arguably the most important systems a physicist needs to know about. It is little known that, mathematically, they are one and the same. This knowledge helps us to understand either from the viewpoint of the other. Here we show that all general time-dependent solutions of the free-particle Schrödinger equation can be mapped to solutions of the Schrödinger equation for harmonic potentials, both the trapping oscillator and the inverted “oscillator”. This map is fully invertible and therefore induces an isomorphism between both types of system, they are equivalent. A composition of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant and different center position. The map is independent of the state of the system, consisting only of a coordinate transformation and multiplication by a form factor, and can be chosen such that the state is identical in both systems at one point in time. This transition point in time can be chosen freely, the wave function of the particle evolving in time in one system before the transition point can therefore be linked up smoothly with the wave function for the other system and its future evolution after the transition point. Such a cut-and-paste procedure allows us to describe the instantaneous changes of the environment a particle finds itself in. Transitions from free to trapped systems, between harmonic traps of different spring constants or center positions, or, from harmonic binding to repulsive harmonic potentials are straightforwardly modelled. This includes some time-dependent harmonic potentials. The mappings introduced here are computationally more efficient than either state-projection or harmonic oscillator propagator techniques conventionally employed when describing instantaneous (non-adiabatic) changes of a quantum particle’s environment.
We theoretically investigate the stability and linear oscillatory behavior of a naturally unstable particle whose potential energy is harmonically modulated. We find this fundamental dynamical system is analogous in time to a quantum harmonic oscillator. In a certain modulation limit, a.k.a. the Kapitza regime, the modulated oscillator can behave like an effective classic harmonic oscillator. But in the overlooked opposite limit, the stable modes of vibrations are quantized in the modulation parameter space. By analogy with the statistical interpretation of quantum physics, those modes can be characterized by the time-energy uncertainty relation of a quantum harmonic oscillator. Reducing the almost-periodic vibrational modes of the particle to their periodic eigenfunctions, one can transform the original equation of motion to a dimensionless Schrodinger stationary wave equation with a harmonic potential. This reduction process introduces two features reminiscent of the quantum realm: a wave-particle duality and a loss of causality that could legitimate a statistical interpretation of the computed eigenfunctions. These results shed new light on periodically time-varying linear dynamical systems and open an original path in the recently revived field of quantum mechanical analogs.
The temporal evolution of the state vector relative to a harmonic oscillator with time-dependent frequency is examined. The Schrödinger equation is solved by choosing the instantaneous eigenstates of the Hamiltonian as the basis, thus getting an infinite set of coupled linear differential equations. This formulation is particularly suitable for studying the cases in which the Hamiltonian undergoes a very slow or a sudden variation from an initial constant value into a final one. A rigorous proof of adiabatic invariance to all orders in the slowness parameter is given for the transition probabilities. An application to the evolution of an initially coherent state is made.
We solve the time-dependent Schrödinger equation describing a generalized harmonic oscillator in the coordinate representation with constant mass but time-dependent stiffness and correlation parameters. Exact solutions are obtained in terms of confluent hypergeometric functions or parabolic cylinder functions. The dynamics of mean energy is studied in several different regimes, including the special case of imaginary correlation parameter, corresponding to the non-Hermitian Ahmed–Swanson model. In this case, real time-dependent mean values of the Hamiltonian are obtained by using dual wave functions, evolving according to the time-dependent Schrödinger equations with the given Hamiltonian and its Hermitially conjugated partner. The adiabatic evolution is considered as one of examples.
It is shown that general solutions of the free-particle Schroedinger equation can be mapped onto solutions of the Schroedinger equation for the harmonic oscillator. This is done in such a way that the time evolution of a free particle subjected to a sudden transition to a harmonic potential can be described by a simple coordinate transformation applied at the transition time. This procedure is computationally more efficient than either state-projection or propagator techniques. A concatenation of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant.