## Low-frequency anomalies in dynamic localization

Stefano Longhi

Quantum mechanical spreading of a particle hopping on tight binding lattices can be suppressed by the application of an external ac force, leading to periodic wave packet reconstruction. Such a phenomenon, referred to as dynamic localization (DL), occurs for certain magic values of the ratio $$\Gamma=F_0/\omega$$ between the amplitude F0 and frequency ω of the ac force. It is generally believed that in the low-frequency limit $$(\omega\to0)$$ DL can be achieved for an infinitesimally small value of the force F0, i.e. at finite values of $$\Gamma$$. Such a normal behavior is found in homogeneous lattices as well as in inhomogeneous lattices of Glauber-Fock type. Here we introduce a tight-binding lattice model with inhomogeneous hopping rates, referred to as pseudo Glauber-Fock lattice, which shows DL but fails to reproduce the normal low-frequency behavior of homogeneous and Glauber-Fock lattices. In pseudo Glauber-Fock lattices, DL can be exactly realized, however at the DL condition the force amplitude $$F_0$$ remains finite as $$\omega\to0$$. Such an anomalous behavior is explained in terms of a PT symmetry breaking transition of an associated two-level non-Hermitian Hamiltonian that effectively describes the dynamics of the Hermitian lattice model.

http://arxiv.org/abs/1405.2549

Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el)

## Exceptional points and Bloch oscillations in non-Hermitian lattices with unidirectional hopping

Stefano Longhi

The spectral and transport properties of a non-Hermitian tight-binding lattice with unidirectional hopping are theoretically investigated in three different geometrical settings. It is shown that, while for the infinitely-extended (open) and for the ring lattice geometries the spectrum is complex, lattice truncation makes the spectrum real. However, an exceptional point of order equal to the number of lattice sites emerges. When a homogeneous dc force is applied to the lattice, in all cases an equally-spaced real Wannier-Stark ladder spectrum is obtained, corresponding to periodic oscillatory dynamics in real space. Possible physical realizations of non-Hermitian lattices with unidirectional hopping are briefly discussed.

http://arxiv.org/abs/1404.3662
Quantum Physics (quant-ph)

## Spectral and transport properties of time-periodic PT-symmetric tight-binding lattices

Giuseppe Della Valle, Stefano Longhi

We investigate the spectral properties and dynamical features of a time-periodic PT-symmetric Hamiltonian on a one-dimensional tight-binding lattice. It is shown that a high-frequency modulation can drive the system under a transition between the broken-PT and the unbroken-PT phases. The time-periodic modulation in the unbroken-PT regime results in a significant broadening of the quasi-energy spectrum, leading to a hyper-ballistic transport regime. Also, near the PT-symmetry breaking the dispersion curve of the lattice band becomes linear, with a strong reduction of quantum wave packet spreading.

http://arxiv.org/abs/1306.1048
Quantum Physics (quant-ph)

Boyan T. Torosov, Giuseppe Della Valle, Stefano Longhi

A non-Hermitian shortcut to adiabaticity is introduced. By adding an imaginary term in the diagonal elements of the Hamiltonian of a two state quantum system, we show how one can cancel the nonadiabatic losses and perform an arbitrarily fast population transfer, without the need to increase the coupling. We apply this technique to two popular level-crossing models: the Landau-Zener model and the Allen-Eberly model.

http://arxiv.org/abs/1306.0698
Quantum Physics (quant-ph)

## Absence of Floquet scattering in oscillating non-Hermitian potential wells

S. Longhi, G. Della Valle

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\”{o}dinger equation. For one of the two families of scattering-free potentials, the oscillating potential turns out to be fully invisible.

http://arxiv.org/abs/1306.0675
Quantum Physics (quant-ph)

## Invisible defects in complex crystals

S. Longhi, G. Della Valle

We show that invisible localized defects, i.e. defects that can not be detected by an outside observer, can be realized in a crystal with an engineered imaginary potential at the defect site. The invisible defects are synthesized by means of supersymmetric (Darboux) transformations of an ordinary crystal using band-edge wave functions to construct the superpotential. The complex crystal has an entire real-valued energy spectrum and Bragg scattering is not influenced by the defects. An example of complex crystal synthesis is presented for the Mathieu potential.

http://arxiv.org/abs/1306.0667
Quantum Physics (quant-ph)