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Zafar Ahmed

Publications and source records attributed to Zafar Ahmed.

At least 55 records · Page 3Linked to original sources
arXiv2014

Coherent perfect absorption with and without lasing in complex potentials

We prove that coherent perfect absorption (CPA) without lasing is {\bf not} possible in the PT-symmetric domain as the s-matrix is such that $|\det S(\pm k)|=1$. We study coherent scattering from three complex potentials, one solved analytically and the other two numerically. We conjecture that in the domain of unbroken symmetry (when the potential has real discrete spectrum) neither spectral singularity nor CPA can occur. We show that Scarf II potential is a special model that can analytically and explicitly exhibit these as well as other novel phenomena and their subtleties. \pacs{03.65-w, 03.65.Nk, 11.30.Er, 42.25.Bs, 42.79.Gn

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arXiv2014

Non-catastrophic resonant states in one dimensional scattering from a rising exponential potential

Investigation of scattering from rising potentials has just begun, these unorthodox potentials have earlier gone unexplored. Here, we obtain reflection amplitude ($r(E)$) for scattering from a two-piece rising exponential potential: $V(x\le 0)=V_1[1-e^{-2x/a}], V(x > 0)=V_2[e^{2x/b}-1]$, where $V_{1,2}>0$. This potential is repulsive and rising for $x>0$; it is attractive and diverging (to $-\infty$) for $x<0$. The complex energy poles (${\cal E}_n= E_n-iΓ_n/2, Γ_n>0$) of $r(E)$ manifest as resonances. Wigner's reflection time-delay displays peaks at energies $E(\approx E_n$) but the eigenstates do not show spatial catastrophe for $E={\cal E}_n$.

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arXiv2014

One dimensional scattering from two-piece rising potentials: a new avenue of resonances

We study scattering from potentials that rise monotonically on one side; this is generally avoided. We report that resonant states are absent in such potentials when they are smooth and single-piece having less than three real turning points (like in the cases of Morse oscillator, exponential and linear potentials). But when these potentials are made two-piece, resonances can occur. We further show that rising potentials next to a well/step/barrier are rich models of multiple resonances (Gamow's decaying states) in one- dimension. We use linear, parabolic and exponential profiles as rising part and find complex-energy poles, ${\cal E}_n=E_n-iΓ_n/2$ $(Γ_n > 0)$, in the reflection amplitude (s-matrix). The appearance of peaks in Wigner's (reflection) time-delay at $E=ε_n$ (close to $E_n$) and spatial catastrophe in the eigenfunction confirm the existence of resonances and meta-stable states in these systems. PACS Nos.: 03.65.-w, 03.65.Nk

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arXiv2014

Anti-symmetric square well and barrier potential between two rigid walls

We study an anti-symmetric (square) well and barrier potential of depth/height $(V_0)$ placed between two rigid walls. Unlike the usual double-well, here the closely lying sub-barrier doublets need not be the lowest ones in the spectrum. When $V_0$ admits certain calculable values, $E=0$ or $E=V_0$ or both could become energy eigenvalues of the special eigenstate which emerge only if one seeks a linear solution of Schr{ö}dinger equation in the appropriate regions.

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arXiv2014

Position-momentum uncertainty products

We point out two interesting features of position-momentum uncertainty product: $U=Δx Δp$. We show that two special (non-differentiable) eigenstates of the Schr{ö}dinger operator with the Dirac Delta potential $[V(x)=-V_0 δ(x)],V_0>0$, also satisfy the Heisenberg's uncertainty principle by yielding $U> \frac{\hbar}{2}$. One of these eigenstates is a zero-energy and zero-curvature bound state.

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arXiv2014

Special rectangular (double-well and hole) potentials

We revisit a rectangular barrier as well as a rectangular well (pit) between two rigid walls. The former is the well known double-well potential and the latter is a hole potential. Let $|V_0|$ be the height (depth) of the barrier (well) then for a fixed geometry of the potential, we show that in the double-well, $E=V_0(>0)$, and in the hole potential ($V_0 <0$), $E=0$, can be energy eigenvalues provided $V_0$ admits some special discrete values. These states have been missed out earlier which emerge only when one seeks the special zero-energy solution of one-dimensional Schr{ö}dinger equation as $ψ(x)=Bx+C$.

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arXiv2013

Non-unitarity and non-reciprocity in scattering from real potentials in presence of confined non-linearity

Investigations of scattering in presence of non-linearity which have just begun require the confinement of both the potential, $V(x)$, and the non-linearity, $γf(|ψ|)$. There could be two options for the confinement. One is the finite support on $x \in [-L,L]$ and the other one is on $x \in [0,L]$. Here, we consider real Hermitian potentials and report a surprising disparate behaviour of these two types of confinements. We prove that in the first option the symmetric potential enjoys reciprocity of both reflectivity ($R$) and transmitivity ($T$) and their unitarity. More interestingly, the asymmetry in $V(x)$ causes non-unitarity ($ R+T\ne 1$) and the non-reciprocity (reciprocity) of $T (R)$. On the other hand, the second option of confinement gives rise to an essential non-unitarity even when $V(x)$ is symmetric about a point in $[0,L]$. In the absence of symmetry there occurs non-reciprocity of both $R$ and $T$.

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arXiv2013

Reciprocity and unitarity in scattering from a non-Hermitian complex PT-symmetric potential

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 cases, we show that this potential can even become invisible (R=0, T=1) from both left and right sides. We also find that this potential in a parametric regime enjoys a pseudo-unitarity of the type $T+\sqrt{R_{left}R_{right}}=1$.

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arXiv2012

Spectral singularity and deep multiple minima in the reflectivity in non-Hermitian (complex) Ginocchio potential

We bring out the existence of at most one spectral singularity (SS) and deep multiple minima in the reflectivity of the non-Hermitian (complex) Ginocchio potential. We find a parameter dependent single spectral singularity in this potential provided the imaginary part is emissive (not absorptive). The reflectionlessness of the real Hermitian Ginocchio's potential at discrete positive energies gives way to deep multiple minima in reflectivity when this potential is perturbed and made non-Hermitian (complex). A novel co-existence of a SS with deep minima in reflectivity is also revealed wherein the first reflectivity zero of the Hermitian case changes to become a SS for the non-Hermitian case. {PACS: 03.65.Nk,11.30.Er,42.25.Bs}

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arXiv2011

New features of scattering from a one-dimensional non-Hermitian (complex) potential

For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time-reversal: $R(-k)\ne R(k)$ and $T(-k) \ne T(k)$, unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that $R_{left}(-k)=R_{right}(k)$ and $T(-k)=T(k)$. So far, the spectral singularities (SS) of a one-dimensional non-Hermitian scattering potential are witnessed/conjectured to be at most one. We present a new non-Hermitian parametrization of Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies ($E_*=α^2,β^2$) either in $T(k)$ or in $T(-k)$, when $αβ>0$. Thirdly, when $αβ<0$ it possesses one SS in $T(k)$ and the other in $T(-k)$. Fourthly, when the potential becomes PT-symmetric $[(α+β)=0]$, we get $T(k)=T(-k)$, it possesses a unique SS at $E=α^2$ in both $T(-k)$ and $T(k)$. Lastly, for completeness, when $α=iγ$ and $β=iδ$, there are no SS, instead we get two negative energies $-γ^2$ and $-δ^2$ of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound state eigenvalues and no spectral singularity exists in this case. We find them as $E^{+}_{M}=-(γ-M)^2$ and $E^{-}_{N}=-(δ-N)^2$; $M(N)=0,1,2,...$ with $0 \le M (N)< γ(δ)$. {PACS: 03.65.Nk,11.30.Er,42.25.Bs}

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arXiv2011

Transmission through a non-overlapping well adjacent to a finite barrier

We point out that a non-overlapping well (at negative energies) adjacent to a finite barrier (at positive energies) is a simple potential which is generally missed out while discussing the one-dimensional potentials in the textbooks of quantum mechanics. We show that these systems present interesting situations wherein transmitivity $(T_b(E))$ of a finite barrier can be changed both quantitatively and qualitatively by varying the depth or width of the well or by changing the distance between the well and the barrier. Using delta (thin) well near a delta (thin) barrier we show that the well induces energy oscillations riding over $T_b(E)$ in the transmitivity $T(E)$ at both the energies below and above the barrier. More generally we show that a thick well separated from a thick barrier also gives rise to energy oscillations in $T(E)$. A well joining a barrier discontinuously (a finite jump) reduces $T(E)$ (as compared to $T_b(E))$ over all energies. When the well and barrier are joined continuously, $T(E)$ increases and then decreases at energies below the barrier. At energy above the the barrier the changes are inappreciable. In these two cases if we separate the well and the barrier by a distance, $T(E)$ again acquires oscillations. Paradoxically, it turns out that a distant well induces more energy oscillations in $T(E)$ than when it is near the barrier.

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arXiv2009

Zero width resonance (spectral singularity) in a complex PT-symmetric potential

We show that the complex PT-Symmetric potential, $V(x)=-V_1 {sech}^2x + iV_2 {sech}x ~\tanh x, $, entails a single zero-width resonance (spectral singularity) when $V_1+|V_2|=4n^2+4n+{3\over 4}(n=1,2,3.., |V_2|>|V_1|+ {{sgn}(V_1) \over 4})$ and the positive resonant energy is given as $E_*={1 \over 4}[|V_2|-(1/4+V_1)]$.

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arXiv2006

Eigenvalue problems for the complex PT-symmetric potential V(x)= igx

The spectrum of complex PT-symmetric potential, $V(x)=igx$, is known to be null. We enclose this potential in a hard-box: $V(|x| \ge 1) =\infty $ and in a soft-box: $V(|x|\ge 1)=0$. In the former case, we find real discrete spectrum and the exceptional points of the potential. The asymptotic eigenvalues behave as $E_n \sim n^2.$ The solvable purely imaginary PT-symmetric potentials vanishing asymptotically known so far do not have real discrete spectrum. Our solvable soft-box potential possesses two real negative discrete eigenvalues if $|g|<(1.22330447)$. The soft-box potential turns out to be a scattering potential not possessing reflectionless states.

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arXiv2005

Reflectionless Potentials and PT Symmetry

Large families of Hamiltonians that are non-Hermitian in the conventional sense have been found to have all eigenvalues real, a fact attributed to an unbroken PT symmetry. The corresponding quantum theories possess an unconventional scalar product. The eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane. For a special class of such systems, it is possible to impose the PT-symmetric boundary conditions on the real axis, which lies on the edges of the wedges. The PT-symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless.

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arXiv2004

A pseudo-unitary ensemble of random matrices, PT-symmetry and the Riemann Hypothesis

An ensemble of 2 x 2 pseudo-Hermitian random matrices is constructed that possesses real eigenvalues with level-spacing distribution exactly as for the Gaussian Unitary Ensemble found by Wigner. By a re-interpretation of Connes' spectral interpretation of the zeros of the Riemann zeta function, we propose to enlarge the scope of search of the Hamiltonian connected with the celebrated Riemann Hypothesis by suggesting that the Hamiltonian could also be PT-symmetric (or pseudo-Hermitian).

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arXiv2004

Gaussian-random Ensembles of Pseudo-Hermitian Matrices

Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or, well-known number sequences may display strong signatures that the Hamiltonian yielding them as eigenvalues is PT-symmetric (Pseudo-Hermitian). We find that the random matrix theory of pseudo-Hermitian Hamiltonians gives rise to new universalities of level-spacing distributions other than those of GOE, GUE and GSE of Wigner and Dyson. We call the new proposals as Gaussian Pseudo-Orthogonal Ensemble and Gaussian Pseudo-Unitary Ensemble. We are also led to speculate that the enigmatic Riemann-zeros (${1 \over 2}\pm i t_n)$ would rather correspond to some PT-symmetric (pseudo-Hermitian) Hamiltonian.

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arXiv2003

Handedness of complex PT-symmetric potential barriers

Generally, when imaginary part of an optical potential is non-symmetric the reflectivity, $R(E)$, shows left/right handedness, further if it is not negative-definite the reflection and transmission, $T(E)$, coefficients become anomalous in some energy intervals and absorption is indefinite ($\pm$). We find that the complex PT-symmetric potentials could be exceptional in this regard. They may act effectively like an absorptive potential for any incident energy provided the particle enters from the preferred (absorptive) side.

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