Scattering of a quantum particle from an oscillating barrier or well does not generally conserve the particle energy owing to energy exchange with the photon field, and an incoming particle-free state is scattered into a set of outgoing (transmitted and reflected) free states according to Floquet scattering theory. Here we introduce two families of oscillating non-Hermitian potential wells in which Floquet scattering is fully suppressed for any energy of the incident particle. The scattering-free oscillating potentials are synthesized by application of the Darboux transformation to the time-dependent Schr\"odinger equation. For one of the two families of scattering-free potentials, the oscillating potential turns out to be fully invisible.
Scattering of a quantum particle from an oscillating barrier or well does not generally conserve the particle energy owing to energy exchange with the photon field, and an incoming particle-free state is scattered into a set of outgoing (transmitted and reflected) free states according to Floquet scattering theory. Here we introduce two families of oscillating non-Hermitian potential wells in which Floquet scattering is fully suppressed for any energy of the incident particle. The scattering-free oscillating potentials are synthesized by application of the Darboux transformation to the time-dependent Schr"odinger equation. For one of the two families of scattering-free potentials, the oscillating potential turns out to be fully invisible.
In the framework of the ordinary non-relativistic quantum mechanics, it is known that a quantum particle in a rapidly oscillating bound potential with vanishing time average can be scattered off or even trapped owing to the phenomenon of dynamical (Kapitza) stabilization. A similar phenomenon occurs for scattering and trapping of optical waves. Such a remarkable result stems from the fact that, even though the particle is not able to follow the rapid external oscillations of the potential, these are still able to affect the average dynamics by means of an effective —albeit small— nonvanishing potential contribution. Here we consider the scattering and dynamical stabilization problem for matter or classical waves by a bound potential with oscillating ac amplitude f ( t ) in the framework of a non-Hermitian extension of the Schrödinger equation, and predict that for a wide class of imaginary amplitude modulations f ( t ) possessing a one-sided Fourier spectrum, the oscillating potential is effectively canceled, i.e. , it does not have any effect on the particle dynamics, contrary to what happens in the Hermitian case.
Scattering a particle on a one-dimensional δ-potential barrier: asymptotic superselection rule
As is stated, the modern mathematical (C∗-algebraic) scattering theory for the formal Hamiltonian with a one-dimensional short-range potential, developed on the basis of the operational definition of observables, describes unitary asymptotically free dynamics. But, as it s shown in this paper by the example of the δ-potential, this Hamiltonian can describe either unitary asymptotically not free dynamics (endless interaction process) or non-unitary asymptotically free dynamics (scattering process). In the case of scattering, the unboundedness of the position operator plays a key role. Each solution to the Schrodinger equation, describing the scattering process with a one-sided incidence of a particle on the barrier, is non-unique in the limit t→∞, when it is a superposition of unconnected states (left and right out-asymptotes) localized in the disjoint spatial regions located on opposite sides of the barrier --- it describes non-unitary quantum dynamics. Measurement of the coordinates of the transmitted and reflected particles using one experimental setup of finite dimensions, as is assumed in the operational approach, is impossible. We need two such setups --- one for the transmitted particles, and the other for the reflected ones. Thus, such a solution is a mixed vector state --- this process is governed by the asymptotic superselection rule.
The scattering of quantum particles by non-Hermitian (generally non-local) potentials in one dimension may result in asymmetric transmission and/or reflection from left and right incidence. After extending the concept of symmetry for non-Hermitian potentials, eight generalized symmetries based on the discrete Klein's four-group (formed by parity, time reversal, their product, and unity) are found. Together with generalized unitarity relations they determine selection rules for the possible and/or forbidden scattering asymmetries. Six basic device types are identified when the scattering coefficients (squared moduli of scattering amplitudes) adopt zero/one values, and transmission and/or reflection are asymmetric. They can pictorically be described as a one-way mirror, a one-way barrier (a Maxwell pressure demon), one-way (transmission or reflection) filters, a mirror with unidirectional transmission, and a transparent, one-way reflector. We design potentials for these devices and also demonstrate that the behavior of the scattering coefficients can be extended to a broad range of incident momenta.
The scattering of a particle by a target with periodic time dependence is usually formulated in Floquet extended space. This leads to time-independent transition probabilities which correspond to the scattering of wave packets that are close to being monoenergetic. Because of the wide space extension of such packets, many oscillations take place during the time particle and target interact. The purpose of this study consists in showing that the scattering amplitudes can also be used to describe the way the particle feels the instantaneous configuration of the target at the time of the collision. The application is made to an oscillating barrier in one dimension. Several tests of the formalism validate the analysis, by changing either the initial position of the wave packet, or the phase of the particle–target interaction. Two equivalent necessary conditions can be formulated to recover the results of standard scattering theory. One is that no energies differing by at least one quantum are present in the energy range effectively covered by the initial wave packet. The other is that there is appreciable self-overlapping of the wave packet after one period of oscillation.
In systems of ultracold atoms, pairwise interactions are resonantly enhanced by the application of an oscillating magnetic field that is parallel to the spin-quantization axis of the atoms. The resonance occurs when the frequency of the applied field is precisely tuned near the transition frequency between the scattering atoms and a diatomic molecule. The resulting cross section can be made more than two orders of magnitude larger than the cross section in the absence of the oscillating field. The low momentum resonance properties have a universal description that is independent of the atomic species. To arrive at these conclusions, we first develop a formal extension of Floquet theory to describe scattering of atoms with time-periodic, short-range interaction potentials. We then calculate the atomic scattering properties by modeling the atomic interactions with a square well potential with oscillating depth and then explicitly solving the time-dependent Schrodinger equation. We then apply the Floquet formalism to the case of atoms scattering with a contact interaction described by a time-periodic scattering length, obtaining analytic results that agree with those obtained by solving the time-dependent Schrodinger equation.
The non-Hermitian models, which are symmetric under parity ( P ) and time-reversal ( T ) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually demands precise control of nonlinear processes, limiting their application. In this paper, to bypass this obstacle, we introduce a class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet) that can describe a two-level system with temporally modulated on-site potential and couplings. We show that implementing an appropriate non-Unitary gauge transformation converts the original system to an effective one with a balanced gain and loss. This will allow us to derive the evolution of states analytically. Our proposed class of Hamiltonians can be employed in different platforms such as electronic circuits, acoustics, and photonics to design structures with hidden PT -symmetry potentially without imaginary onsite amplification and absorption mechanism to obtain an exceptional point.