One-dimensional scattering mediated by non-Hermitian Hamiltonians is studied. A schematic set of models is used which simulate two point interactions at a variable strength and distance. The feasibility of the exact construction of the amplitudes is achieved via the discretization of the coordinate. By direct construction it is shown that in all our models the probability is conserved. This feature is tentatively attributed to the space- and time-reflection symmetry (a.k.a. PT-symmetry) of our specific Hamiltonians.
Group symmetric LOCC measurement for detecting maximally entangled state is considered. Usually, this type measurement has continuous-valued outcomes. However, any realizable measurement has finite-valued outcomes. This paper proposes discrete realizations of such a group symmetric LOCC measurement.
We investigate the properties of two anti-parity-time (anti-PT )-symmetric four-site scattering centers. The anti-PT -symmetric scattering center may have imaginary couplings, real couplings, and real on-site potentials. The only difference between the two scattering centers is the coupling between two central sites of the scattering center, which plays a crucial role in determining the parity of anti-PT symmetry and significantly affects the scattering properties. For the imaginary coupling, the even-parity anti-PT -symmetric scattering center possesses nonunitary scattering and the differenc…
Discrete PT-symmetric square wells are studied. Their wave functions are found proportional to classical Tshebyshev polynomials of complex argument. The compact secular equations for energies are derived giving the real spectra in certain intervals of non-Hermiticity strengths Z. It is amusing to notice that although the known square well re-emerges in the usual continuum limit, a twice as rich, upside-down symmetric spectrum is exhibited by all its present discretized predecessors.
The general Weyl -- Wigner formalism in finite dimensional phase spaces is investigated. Then this formalism is specified to the case of symmetric ordering of operators in an odd -- dimensional Hilbert space. A respective Wigner function on the discrete phase space is found and the limit, when the dimension of Hilbert space tends to infinity, is considered. It is shown that this limit gives the number -- phase Wigner function in quantum optics. Analogous results for the `almost' symmetric ordering in an even -- dimensional Hilbert space are obtained. Relations between the discrete Wigner funct…
The problem of the Hamiltonian matrix in the oscillator and Laguerre basis construction from the S-matrix is treated in the context of the algebraic analogue of the Marchenko method.
This research focuses on the possibility of the surjective relation between symmetric potential function and its scattering matrix in 1-dimension. The theory bases on the property of wave function symmetry and boundary conditions. This research shows the surjective relation in some particular cases, delta function potential, and finite square wall potential, and disproves the injective relation of the arbitrary potential function and its S-matrix.
A general formalism is worked out for the description of one-dimensional scattering by non-local separable potentials and constraints on transmission and reflection coefficients are derived in the cases of P, T, or PT invariance of the Hamiltonian. The case of a solvable Yamaguchi potential is discussed in detail.
Exact solvability of the discretized N-point version of the PT-symmetric square-well model is pointed out. Its wave functions are found proportional to the classical Tshebyshev polynomials of a complex argument. At all N a compact secular equation is derived giving the real spectrum of energies at any non-Hermiticity strength Z below its finite and weakly N-dependent critical value. In the limit of vanishing Z the model degenerates to a Hermitian Hueckel Hamiltonian.
In this letter, we study the scattering of spin-$\frac{1}{2}$ particles from a spin-independent parity time ($\cal PT$)-symmetric complex potential, and for the first time, theoretically demonstrate the coexistence of $\cal PT$-symmetric and $\cal PT$-broken phases for broadband energy spectra in this system. We also show the existence of anisotropic transmission resonances, accessible through the tuning of energy. Our results are promising for applications in spintronics, semiconductor-based devices, and a better understanding of the topological surface states.
We investigate the approximate bound state solutions of the Schrödinger equation for the PT-/non-PT-symmetric and non Hermitian Hellmann potential. Exact energy eigenvalues and corresponding normalized wave functions are obtained. Numerical values of energy eigenvalues for the bound states are compared with the ones obtained before. Scattering state solutions are also studied. Phase shifts of the potential are written in terms of the angular momentum quantum number $\ell$.
We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q-function. Using these coherent states we establish a partial order in the discrete phase space, which allows us to picture some n-qubit states as apparent distributions. We also analyze correlations in terms of sums of squared Q-functions.
Scattering on the ${\cal PT}$-symmetric Coulomb potential is studied along a U-shaped trajectory circumventing the origin in the complex $x$ plane from below. This trajectory reflects ${\cal PT}$ symmetry, sets the appropriate boundary conditions for bound states and also allows the restoration of the correct sign of the energy eigenvalues. Scattering states are composed from the two linearly independent solutions valid for non-integer values of the 2L parameter, which would correspond to the angular momentum in the usual Hermitian setting. Transmission and reflection coefficients are written …
A general formalism is worked out for the description of one-dimensional scattering in non-hermitian quantum mechanics and constraints on transmission and reflection coefficients are derived in the cases of P, T, or PT invariance of the Hamiltonian. Applications to some solvable PT-symmetric potentials are shown in detail. Our main original results concern the association of reflectionless potentials with asymptotic exact PT symmetry and the peculiarities of separable kernels of non-local potentials in connection with hermiticity, T invariance and PT invariance.
The exact analytical solutions of the Schrödinger equation for the generalized symmetrical Woods-Saxon potential are examined for the scattering, bound and quasi-bound states in one dimension. The reflection and transmission coefficients are analytically obtained. Then, the correlations between the potential parameters and the reflection-transmission coefficients are investigated, and a transmission resonance condition is derived. Occurrence of the transmission resonance has been shown when incident energy of the particle is equal to one of the resonance energies of the quasi-bound states.
We complexify a 1-d potential which exhibits bound, reflecting and free states to study various properties of a non-Hermitian system. This potential turns out a PT-symmetric non-Hermitian potential when one of the parameters becomes imaginary. For one PT-symmetric case we have entire real bound state spectrum. Explicit scattering states are constructed to show reciprocity at certain discrete values of energy even though the potential is not parity symmetric. Coexistence of deep energy minima of transmissivity with the multiple spectral singularities (MSS) is observed. We further show that this…
We present a systematic treatment of scattering processes for quantum systems whose time evolution is discrete. We define and show some general properties of the scattering operator, in particular the conservation of quasi-energy which is defined only modulo $2 π$. Then we develop two perturbative techniques for the power series expansion of the scattering operator, the first one analogous to the iterative solution of the Lippmann-Schwinger equation, the second one to the Dyson series of perturbative Quantum Field Theory. We use this formalism to compare the scattering amplitudes of a continuo…
The theory of random walks on finite graphs is well developed with numerous applications. In quantum walks, the propagation is governed by quantum mechanical rules; generalizing random walks to the quantum setting. They have been successfully applied in the development of quantum algorithms. In particular, to solve problems that can be mapped to searching or property testing on some specific graph. In this paper we investigate the discrete time coined quantum walk (DTCQW) model using tools from non-commutative Fourier analysis. Specifically, we are interested in characterizing the DTCQW on Cay…
Perfect state transfer between two marked vertices of a graph by means of discrete-time quantum walk is analyzed. We consider the quantum walk search algorithm with two marked vertices, sender and receiver. It is shown by explicit calculation that for the coined quantum walks on star graph and complete graph with self-loops perfect state transfer between the sender and receiver vertex is achieved for arbitrary number of vertices $N$ in $O(\sqrt{N})$ steps of the walk. Finally, we show that Szegedy's walk with queries on complete graph allows for state transfer with unit fidelity in the limit o…
We study the effect of discrete symmetry breaking in inhomogeneous scattering media within the framework of generic wave propagation. Our focus is on one-dimensional scattering potentials exhibiting local symmetries. We find a class of spatially invariant nonlocal currents, emerging when the corresponding generalized potential exhibits symmetries in arbitrary spatial domains. These invariants characterize the wave propagation and provide a spatial mapping of the wave function between any symmetry related domains. This generalizes the Bloch and parity theorems for broken reflection and translat…
A simple formalism for exploring quantum scattering and possible bound states in an arbitrary symmetric and localized potential in a unified way is presented. The symmetric square barrier and well potentials are used for illustrating the method.
We investigate the scattering properties of an anti-parity-symmetric non-Hermitian system. The anti-parity-symmetric scattering center possesses imaginary nearest-neighbor hoppings and real onsite potentials, it has been experimentally realized through dissipative coupling and frequency detuning between atomic spin waves. We find that such anti-parity-symmetric system displays three salient features: Firstly, the reflection and transmission are both reciprocal. Secondly, the reflection and transmission probabilities satisfy R+T=1 or R-T=1, which depends on the parity of the scattering center s…
A physical realization of scattering by ${\cal{PT}}$-symmetric potentials is provided: we show that the Maxwell equations for an electromagnetic wave travelling along a planar slab waveguide filled with gain and absorbing media in contiguous regions, can be approximated in a parameter range by a Schrödinger equation with a ${\cal{PT}}$-symmetric scattering potential.
Generalized PT-symmetric operators acting an a Hilbert space $\mathfrak{H}$ are defined and investigated. The case of PT-symmetric extensions of a symmetric operator $S$ is investigated in detail. The possible application of the Lax-Phillips scattering methods to the investigation of PT-symmetric operators is illustrated by considering the case of 0-perturbed operators.
The S-matrices corresponding to PT-symmetric ρ-perturbed operators are defined and calculated by means of an approach based on an operator-theoretical interpretation of the Lax-Phillips scattering theory.
A new version of an elementary PT-symmetric square well quantum model is proposed in which a certain Hermiticity-violating end-point interaction leaves the spectrum real in a large domain of couplings $λ\in (-1,1)$. Within this interval we employ the usual coupling-independent operator P of parity and construct, in a systematic Runge-Kutta discrete approximation, a coupling-dependent operator of charge C which enables us to classify our P-asymmetric model as CPT-symmetric or, equivalently, hiddenly Hermitian alias cryptohermitian.
Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential, $V(x)$, where the spectrum is again discrete and real. Showing that the Hamiltonian, $p^2/2m+V(x)$, in the new cases is pseudo-Hermitian could be challenging, if possible.
The scattering of a Bose-Einstein Condensate (BEC) from a Gaussian well and Gaussian barrier is investigated over a wide range of depths and heights, respectively. We compare analytical and numerical results for a BEC scattering from Gaussian Obstacles, both in the presence and in the absence of PT-symmetric potential. And we find out that the Complex Ginzburg-Landau Equation (CGLE) method has limitations due to the limited number of variational parameters of the ansatz. We also find that the presence of the PT-symmetric potential controls the reflection and the transmission flux of the BEC th…
Recently, it has been shown that the generalized symmetric Woods-Saxon potential energy, in which surface interaction terms are taken into account, describes the physical processes better than the standard form. Therefore in this study, we investigate the scattering of Klein-Gordon particles in the presence of both generalized symmetric Woods-Saxon vector and scalar potential. In one spatial dimension we obtain the solutions in terms of hypergeometric functions for spin symmetric or pseudo-spin symmetric cases. Finally, we plot transmission and reflection probabilities for incident particles w…
One-dimensional scattering problem admitting a complex, PT-symmetric short-range potential V(x) is considered. Using a Runge-Kutta-discretized version of Schroedinger equation we derive the formulae for the reflection and transmission coefficients and emphasize that the only innovation emerges in fact via a complexification of one of the potential-characterizing parameters.
The upside-down $-x^4$, $-x^6$, and $-x^8$ potentials with appropriate PT-symmetric boundary conditions have real, positive, and discrete quantum-mechanical spectra. This paper proposes a straightforward macroscopic quantum-mechanical scattering experiment in which one can observe and measure these bound-state energies directly.
In this letter, we investigate the coherent tunneling process of photons between a defected circular resonator and a waveguide based on the recently developed discrete coordinate scattering methods (L. Zhou et al., Phys. Rev. Lett. 101, 100501 (2008)). We show the detailed microscopic mechanism of the tunneling and present a simple model for defect coupling in the resonator. The Finite-Difference Time-Domain(FDTD) numerical results is explored to illustrate the analysis results.
One-dimensional discrete-time quantum walks show a rich spectrum of topological phases that have so far been exclusively analysed in momentum space. In this work we introduce an alternative approach to topology which is based on the scattering matrix of a quantum walk, adapting concepts from time-independent systems. For gapped quantum walks, topological invariants at quasienergies 0 and π probe directly the existence of protected boundary states, while quantum walks with a non-trivial quasienergy winding have a discrete number of perfectly transmistting unidirectional modes. Our classificatio…
We establish that a perfect-transmission scattering problem can be described by a class of parity and time reversal symmetric operators and hereby we provide a scenario for understanding and implementing the corresponding quasi-Hermitian quantum mechanical framework from the physical viewpoint. One of the most interesting features of the analysis is that the complex eigenvalues of the underlying non-Hermitian problem, associated with a reflectionless scattering system, lead to the loss of perfect-transmission energies as the parameters characterizing the scattering potential are varied. On the…
I review how methods from mesoscopic physics can be applied to describe the multiple wave scattering and complex wave dynamics in non-hermitian PT-symmetric resonators, where an absorbing region is coupled symmetrically to an amplifying region. Scattering theory serves as a convenient tool to classify the symmetries beyond the single-channel case and leads to effective descriptions which can be formulated in the energy domain (via Hamiltonians) and in the time domain (via time evolution operators). These models can then be used to identify the mesoscopic time and energy scales which govern the…
Linear theory for phonon scattering by discrete breathers in the discrete nonlinear Schroedinger equation using the transfer matrix approach is presented. Transmission and reflection coefficients are obtained as a function of the wave vector of the input phonon. The occurrence of a nonzero transmission, which in fact becomes perfect for a symmetric breather, is shown to be connected with localized eigenmodes thresholds. In the weak-coupling limit, perfect reflection are shown to exist, which requires two scattering channels. A necessary condition for a system to have a perfect reflection is al…
N-site-lattice Hamiltonians H are introduced and perceived as a set of systematic discrete approximants of a certain PT-symmetric square-well-potential model with the real spectrum and with a non-Hermiticity which is localized near the boundaries of the interval. Its strength is controlled by one, two or three parameters. The problem of the explicit construction of a nontrivial metric which makes the theory unitary is then addressed. It is proposed and demonstrated that due to the not too complicated tridiagonal-matrix form of our input Hamiltonians the computation of the metric is straightfor…
In quantum scattering, Hermiticity is necessary for both reciprocity and unitarity. Reciprocity means that both reflectivity (R) and transmitivity (T) are insensitive to the direction of incidence of a wave (particle) at a scatterer from left/right. Unitarity means that R+T=1. In scattering from non-Hermitian PT-symmetric structures the (left/right) handedness (non-reciprocity) of reflectivity is known to be essential and unitarity remains elusive so far. Here we present a surprising occurrence of both reciprocity and unitarity in scattering from a complex PT-symmetric potential. In special ca…
A non-unitary version of quantum scattering is studied via an exactly solvable toy model. The model is merely asymptotically local since the smooth path of the coordinate is admitted complex in the non-asymptotic domain. At any real angular-momentum-like parameter the reflection R and transmission T are shown to change with the winding number (i.e., topology) of the path. The points of unitarity appear related to the points of existence of quantum-knot bound states.
For solving the $2\to 2,3$ three-body Coulomb scattering problem the Faddeev-Merkuriev integral equations in discrete Hilbert-space basis representation are considered. It is shown that as far as scattering amplitudes are considered the error caused by truncating the basis can be made arbitrarily small. By this truncation also the Coulomb Green's operator is confined onto the two-body sector of the three-body configuration space and in leading order can be constructed with the help of convolution integrals of two-body Green's operators. For performing the convolution integral an integration co…
We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, $R^{l/r}(k)$ and $T(k)$, of general ${\cal PT}$-symmetric scattering potentials. We use these identities to give a general proof of the relations, $|T(-k)|=|T(k)|$ and $|R^r(-k)|=|R^l(k)|$, conjectured in [Z. Ahmed, J. Phys. A 45 (2012) 032004], establish the generalized unitarity relation: $R^{l/r}(k)R^{l/r}(-k)+|T(k)|^2=1$, and show that it is a common property of both real and complex ${\cal PT}$-symmetric potentials. The same holds for $T(-k)=T(k)^*$ and $|R^r(-k)|=|R^l(k)|$.
Conformal symmetry is taken as an attribute of theories of massless fields in manifolds with specific dimensionalities. This paper shows that this is not an absolute truth; it is a consequence of the mathematical representation used for the physical interactions. It introduces a new kind of representation where the propagation of massive (invariant mass) and massless interactions are unifiedly described by a single conformally symmetric Green's function. Sources and fields are treated at a same footing, symmetrically, as discrete fields - the fields in this new representation - fields defined …
Boundary conditions in quantum graph vertices are generally given in terms of a unitary matrix $U$. Observing that if $U$ has at most two eigenvalues, then the scattering matrix $\mathcal{S}(k)$ of the vertex is a linear combination of the identity matrix and a fixed Hermitian unitary matrix, we construct vertex couplings with this property: For all momenta $k$, the transmission probability from the $j$-th edge to $\ell$-th edge is independent of $(j,\ell)$, and all the reflection probabilities are equal. We classify these couplings according to their scattering properties, which leads to the …
