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Journal of Physics A Mathematical and Theoretical

Exact gaussian states under quadratic non-hermitian deformations

Kapil Shrivastava

Published Oct 28, 2026• 0 citations
Publisher

Abstract

Abstract We develop an exact deformation calculus for single-mode Hermite--Gaussian states under the quadratic analytic map \[ T_\phi=\exp\{2\phi(\partial_x)\}, \qquad \phi(z)=c_1+c_2z+c_3z^2 . \] The construction is not proposed as a general replacement for non-Hermitian quantum mechanics; rather, it provides an exactly solvable sector within the standard metric/quasi-Hermitian framework. For the literal similarity realization \[ H_\phi=T_\phi^{-1}HT_\phi, \] the physical inner product is generated by \(M_\phi=T_\phi^\dagger T_\phi\), time evolution is governed by the Schr"odinger equation with \(H_\phi\), and physical observables are self-adjoint with respect to the metric inner product. Separately, the deformation-adapted Hermite--Gaussian realization leads to closed coordinate- and Fourier-space expressions for coherent, squeezed, and squeezed-coherent states. Using the Hermite generating function and Mehler kernel, we obtain explicit Gaussian formulae controlled by \(\kappa=e^{i\theta}\tanh r\), \(\Delta_\phi=1-\kappa+8c_3\kappa\), and \(\Gamma_\phi=1+\kappa+8c_3\kappa\), together with their Wigner profiles. In these dimensionless coefficients, \(c_3\) is understood in oscillator-length units, as specified in the main text. The pure quadratic sector gives an exactly solvable benchmark for non-unitary Gaussian filtering and effective resolution broadening in continuous-variable optical models.

DOI
10.1088/1751-8121/aebacb

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Abstract We develop an exact deformation calculus for single-mode Hermite--Gaussian states under the quadratic analytic map [ T_\phi=\exp{2\phi(\partial_x)}, \qquad \phi(z)=c_1+c_2z+c_3z^2 . ] The construction is not proposed as a general replacement for non-Hermitian quantum mechanics; rather, it provides an exactly solvable sector within the standard metric/quasi-Hermitian framework. For the literal similarity realization [ H_\phi=T_\phi^{-1}HT_\phi, ] the physical inner product is generated by (M_\phi=T_\phi^\dagger T_\phi), time evolution is governed by the Schr"odinger equation with (H_\phi), and physical observables are self-adjoint with respect to the metric inner product. Separately, the deformation-adapted Hermite--Gaussian realization leads to closed coordinate- and Fourier-space expressions for coherent, squeezed, and squeezed-coherent states. Using the Hermite generating function and Mehler kernel, we obtain explicit Gaussian formulae controlled by (\kappa=e^{i\theta}\tanh r), (\Delta_\phi=1-\kappa+8c_3\kappa), and (\Gamma_\phi=1+\kappa+8c_3\kappa), together with their Wigner profiles. In these dimensionless coefficients, (c_3) is understood in oscillator-length units, as specified in the main text. The pure quadratic sector gives an exactly solvable benchmark for non-unitary Gaussian filtering and effective resolution broadening in continuous-variable optical models.
The aim of this chapter is to construct the quantum unitary propagator for Hamiltonians which are quadratic in position and momentum with time-dependent coefficients. We show that the quantum evolution is exactly solvable in terms of the classical flow which is linear. This allows to construct the metaplectic transformations which are unitary operators in L2(Rn)L^2(\mathbb R^n)L2(Rn) corresponding to symplectic transformations. Simple examples of such metaplectic transformations are the Fourier transform, which corresponds to the symplectic matrix J defined in (3.4) and the propagator of the harmonic oscillator, corresponding to rotations in the phase space. The main results of this chapter are computations of the quantum evolution operators for quadratic Hamiltonians acting on coherent states. We show that the time evolved coherent states are still Gaussian states which are recognized to be squeezed states centered at the classical phase space point (see Chap. 8). From these computations we can deduce most of properties concerning quantum quadratic Hamiltonians. In particular we get the explicit form of the Weyl symbols of the metaplectic transformations. These formulae are generalizations of the Mehler formula for the harmonic oscillator. Quadratic Hamiltonians are very important in quantum mechanics because more general Hamiltonians can be considered as non trivial perturbations of time dependent quadratic ones as we shall see in Chap. 5.
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Abstract In the conventional Schrödinger’s formulation of quantum mechanics the unitary evolution of a state ψ is controlled, in Hilbert space <?CDATA L{\mathcal{L}}L?> L , by a Hamiltonian ɧ which must be self-adjoint. In the recent, “quasi-Hermitian” reformulation of the theory one replaces ɧ by its isospectral but non-Hermitian avatar H = Ω−1 𝖍Ω with Ω†Ω = Θ ≠ I. Although acting in another, manifestly unphysical Hilbert space <?CDATA H{\mathcal{H}}H?> H , the amended Hamiltonian H ≠ H † can be perceived as self-adjoint with respect to the amended inner-product metric Θ. In our paper motivated by a generic technical “user-unfriendliness” of the non-Hermiticity of H we introduce and describe a specific new family of Hamiltonians H for which the metrics Θ become available in closed form.
We develop a comprehensive, gauge-covariant geometric framework for non-Hermitian quantum systems in the quasi-Hermitian regime, that is, the region of parameter space where the non-Hermitian Hamiltonian admits a real spectrum and a positive-definite metric operator. We build this framework by elevating the Dyson map to a central geometric object. This map is the transformation that converts a non-Hermitian Hamiltonian into an equivalent Hermitian one. From it we construct the Dyson connection and decompose it into Hermitian and anti-Hermitian parts, identified respectively as {\it stretching } and {\it rotation } components. This decomposition cleanly separates the genuine physical metric deformations from the unitary gauge redundancies. Working with manifestly gauge-covariant states, we then derive the complex non-Hermitian Berry phase and the quantum geometric tensor (QGT), and show that the non-Hermitian geometric curvature originates from the non-commutativity of the stretching components at the operator level. We further analyse the geometric singularities near an exceptional point (EP) and uncover a distinct hierarchy of divergences. For a general two-level non-Hermitian model, the quantum metric tensor (QMT) exhibits a leading-order divergence ∼∣εμ∣−2\sim |ε_μ|^{-2}∼∣εμ​∣−2, while the Berry curvature shows a weaker, subleading divergence ∼∣εμ∣−3/2\sim |ε_μ|^{-3/2}∼∣εμ​∣−3/2, with εμε_μεμ​ denoting the parameter displacement from the EP along an individual parameter axis μμμ. Finally, we examine physical realizations of this model, including the non-Hermitian Su--Schrieffer--Heeger (SSH) and Hatano--Nelson (HN) models, where exact analytical results confirm the predicted critical scaling laws and illustrate the metric-deformation-driven non-Hermitian geometries.
Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.
Abstract In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Θ-quasi-Hermitian) Hamiltonians H such that <?CDATA H†Θ=Θ HH^\dagger \Theta = \Theta\,HH†Θ=ΘH?> H † Θ = Θ H , the exactly solvable M-level bound-state models with arbitrary <?CDATA M⩽∞M\leqslant \inftyM⩽∞?> M ⩽ ∞ are rare. A new class of such models is proposed here, therefore. Its exact algebraic solvability (involving not only the closed formulae for wave functions but also the explicit description of all of the eligible metrics Θ) was achieved due to an extremely sparse (viz., just <?CDATA (2M−1)(2M-1)(2M−1)?> ( 2 M − 1 ) -parametric) but still nontrivial ‘zig-zag-matrix’ choice of the form of H.