Stefano Longhi

We introduce a new class of \(\cal{PT}\)-symmetric complex crystals which are almost transparent and one-way reflectionless over a broad frequency range around the Bragg frequency, i.e. unidirectionally invisible, regardless of the thickness \(L\) of the crystal. The \(\cal{PT}\)-symmetric complex crystal is synthesized by a supersymmetric transformation of an Hermitian square well potential, and exact analytical expressions of transmission and reflection coefficients are given. As \(L\) is increased, the transmittance and reflectance from one side remain close to one and zero, respectively, whereas the reflectance from the other side secularly grows like ~\(L^2\) owing to unidirectional Bragg scattering. This is a distinctive feature as compared to the previously studied case of the complex sinusoidal \(\cal{PT}\)-symmetric potential \(V(x)=V_0\exp(−2ik_ox)\) at the symmetry breaking point, where transparency breaks down as \(L\to\infty\).

http://arxiv.org/abs/1410.5278

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

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)

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)

S. Longhi, G. Della Valle

The spectral, dynamical and topological properties of physical systems described by non-Hermitian (including PT-symmetric) Hamiltonians are deeply modified by the appearance of exceptional points and spectral singularities. Here we show that exceptional points in the continuum can arise in non-Hermitian (yet admitting and entirely real-valued energy spectrum) optical lattices with engineered defects. At an exceptional point, the lattice sustains a bound state with an energy embedded in the spectrum of scattered states, similar to the von-Neumann Wigner bound states in the continuum of Hermitian lattices. However, the dynamical and scattering properties of the bound state at an exceptional point are deeply different from those of ordinary von-Neumann Wigner bound states in an Hermitian system. In particular, the bound state in the continuum at an exceptional point is an unstable state that can secularly grow by an infinitesimal perturbation. Such properties are discussed in details for transport of discretized light in a PT-symmetric array of coupled optical waveguides, which could provide an experimentally accessible system to observe exceptional points in the continuum.

http://arxiv.org/abs/1402.3764

Quantum Physics (quant-ph); Optics (physics.optics)

Stefano Longhi

Bound states in the continuum (BIC), i.e. normalizable modes with an energy embedded in the continuous spectrum of scattered states, are shown to exist in certain optical waveguide lattices with PT-symmetric defects. Two distinct types of BIC modes are found: BIC states that exist in the broken PT phase, corresponding to exponentially-localized modes with either exponentially damped or amplified optical power; and BIC modes with sub-exponential spatial localization that can exist in the unbroken PT phase as well. The two types of BIC modes at the PT symmetry breaking point behave rather differently: while in the former case spatial localization is lost and the defect coherently radiates outgoing waves with an optical power that linearly increases with the propagation distance, in the latter case localization is maintained and the optical power increase is quadratic.

http://arxiv.org/abs/1402.3761

Quantum Physics (quant-ph); Optics (physics.optics)

Stefano Longhi

The spectral and localization properties of PT-symmetric optical superlattices, either infinitely extended or truncated at one side, are theoretically investigated, and the criteria that ensure the unbroken PT phase are derived. The analysis is applied to the case of superlattices describing a complex (PT-symmetric) extension of the Harper Hamiltonian in the rational case.

http://arxiv.org/abs/1402.3165

Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

Stefano Longhi

We investigate the spectral and dynamical properties of a quantum particle constrained on a ring threaded by a magnetic flux in presence of a complex (non-Hermitian) potential. For a static magnetic flux, the quantum states of the particle on the ring can be mapped into the Bloch states of a complex crystal, and magnetic flux tuning enables to probe the spectral features of the complex crystal, including the appearance of exceptional points. For a time-varying (linearly-ramped) magnetic flux, Zener tunneling among energy states is realized owing to the induced electromotive force. As compared to the Hermitian case, striking effects are observed in the non-Hermitian case, such as a highly asymmetric behavior of particle motion when reversing the direction of the magnetic flux and field-induced delayed transparency.

http://arxiv.org/abs/1312.4693

Quantum Physics (quant-ph)

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)

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)