A unidirectional invisible PT-symmetric complex crystal with arbitrary thickness

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)

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)

Bound states in the continuum in PT-symmetric optical lattices

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)

PT-symmetric optical superlattices

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)

Non-Hermitian quantum rings

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)

Convective and absolute PT symmetry breaking in tight-binding lattices

Stefano Longhi

We investigate the onset of parity-time (PT) symmetry breaking in non-Hermitian tight-binding lattices with spatially-extended loss/gain regions in presence of an advective term. Similarly to the instability properties of hydrodynamic open flows, it is shown that PT-symmetry breaking can be either absolute or convective. In the former case, an initially-localized wave packet shows a secular growth with time at any given spatial position, whereas in the latter case the growth is observed in a reference frame moving at some drift velocity while decay occurs at any fixed spatial position. In the convective unstable regime, PT-symmetry is restored when the spatial region of gain/loss in the lattice is limited (rather than extended). We consider specifically a non-Hermitian extension of the Rice-Mele tight binding lattice model, and show the existence of a transition from absolute to convective symmetry breaking when the advective term is large enough. An extension of the analysis to ac-dc-driven lattices is also presented, and an optical implementation of the non-Hermitian Rice-Mele model is suggested, which is based on light transport in an array of evanescently-coupled optical waveguides with a periodically-bent axis and alternating regions of optical gain and loss.

http://arxiv.org/abs/1310.5004
Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

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)

Invisibility in PT-symmetric complex crystals

Stefano Longhi

Bragg scattering in sinusoidal PT-symmetric complex crystals of finite thickness is theoretically investigated by the derivation of exact analytical expressions for reflection and transmission coefficients in terms of modified Bessel functions of first kind. The analytical results indicate that unidirectional invisibility, recently predicted for such crystals by coupled-mode theory [Z. Lin et al., Phys. Rev. Lett. 106, 213901 (2011)], breaks down for crystals containing a large number of unit cells. In particular, for a given modulation depth in a shallow sinusoidal potential, three regimes are encountered as the crystal thickness is increased. At short lengths the crystal is reflectionless and invisible when probed from one side (unidirectional invisibility), whereas at intermediate lengths the crystal remains reflectionless but not invisible; for longer crystals both unidirectional reflectionless and invisibility properties are broken.

http://arxiv.org/abs/1111.3448
Quantum Physics (quant-ph)