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arXiv:2309.15305

Non-standard quantum algebras and finite dimensional $\mathcal{PT}$-symmetric systems

Ángel Ballesteros, Romina Ramírez, M. Reboiro

Published Sep 26, 2023• 0 citations
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Abstract

In this work, $\mathcal{PT}$-symmetric Hamiltonians defined on quantum $sl(2, \mathbb R)$ algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard $U_{z}(sl(2, \mathbb R))$ Hopf algebra deformation of $sl(2, \mathbb R)$. By making use of a particular boson representation of the generators of $U_{z}(sl(2, \mathbb R))$, both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the $\mathcal{PT}$-transformation. In terms of these operators, we construct several finite dimensional $\mathcal{PT}$-symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact $\mathcal{PT}$-symmetry and the broken $\mathcal{PT}$-symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter $z$ has to be identified with the detuning parameter of the system.

DOI
10.48550/arxiv.2309.15305

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Non-standard quantum algebras and finite dimensional PT\mathcal{PT}PT-symmetric systems
In this work, PT\mathcal{PT}PT-symmetric Hamiltonians defined on quantum sl(2,R)sl(2, \mathbb R)sl(2,R) algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard Uz(sl(2,R))U_{z}(sl(2, \mathbb R))Uz​(sl(2,R)) Hopf algebra deformation of sl(2,R)sl(2, \mathbb R)sl(2,R). By making use of a particular boson representation of the generators of Uz(sl(2,R))U_{z}(sl(2, \mathbb R))Uz​(sl(2,R)), both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the PT\mathcal{PT}PT-transformation. In terms of these operators, we construct several finite dimensional PT\mathcal{PT}PT-symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact PT\mathcal{PT}PT-symmetry and the broken PT\mathcal{PT}PT-symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter zzz has to be identified with the detuning parameter of the system.
Abstract In this work, P T -symmetric Hamiltonians defined on quantum s l ( 2 , R ) algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard U z ( s l ( 2 , R ) ) Hopf algebra deformation of s l ( 2 , R ) . By making use of a particular boson representation of the generators of U z ( s l ( 2 , R ) ) , both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the P T -transformation. In terms of these operators, we construct several finite dimensional P T -symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact P T -symmetry and the broken P T -symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter z has to be identified with the detuning parameter of the system.
Abstract In this work, we introduce a PT -symmetric infinite-dimensional representation of the U z ( s l ( 2 , R ) ) Hopf algebra, and we analyse a multiparametric family of Hamiltonians constructed from such representation of the generators of this non-standard quantum algebra. It is shown that all these Hamiltonians can be mapped to equivalent systems endowed with a position-dependent mass. From the latter presentation, it is shown how appropriate point canonical transformations can be further defined in order to transform them into Hamiltonians with constant-mass over suitable domains. By following this approach, the bound-state spectrum and the corresponding eigenfunctions of the initial PT -symmetric Hamiltonians can be determined. It is worth stressing that a relevant feature of some of the new U z ( s l ( 2 , R ) ) systems here presented is found to be their connection with double-well and Pöschl-Teller potentials. In fact, as an application we present a particular Hamiltonian that can be expressed as an effective double-well trigonometric potential, which is commonly used to model several relevant systems in molecular physics.
Combining non-Hermiticity and interactions yields unique effects in open quantum many-body systems. Here, we develop the generalized Schrieffer-Wolff transformation, discuss its symmetry properties, and derive the effective Hamiltonian suitable for various quasidegenerate non-Hermitian systems. We apply our results to an exemplary PT\mathcal{P}\mathcal{T}PT-symmetric circuit QED composed of two non-Hermitian qubits embedded in a lossless resonator. We consider a resonant quantum circuit as $|{\ensuremath{\omega}}_{r}\ensuremath{-}\mathrm{\ensuremath{\Omega}}|\ensuremath{\ll}{\ensuremath{\omega}}_{r}$, where $\mathrm{\ensuremath{\Omega}}$ and ${\ensuremath{\omega}}_{r}$ are qubits and resonator frequencies, respectively, providing well-defined groups of quasidegenerate resonant states. For such a system, using direct numerical diagonalization we obtain the dependence of the low-lying eigenspectrum on the interaction strength between a single qubit and the resonator ggg and the gain (loss) parameter $\ensuremath{\gamma}$, and compare that with the eigenvalues obtained analytically using the effective Hamiltonian of resonant states. We identify PT\mathcal{P}\mathcal{T}PT-symmetry broken and unbroken phases, trace the formation of exceptional points of the second and the third order, and provide a complete phase diagram $g\ensuremath{-}\ensuremath{\gamma}$ of low-lying resonant states. We relate the formation of exceptional points to the additional P\mathcal{P}P-pseudo-Hermitian symmetry of the system and show that non-Hermiticity mixes the ``dark'' and the ``bright'' states, which has a direct experimental consequence.
Abstract A γ -deformed version of s u ( 2 ) algebra with non-Hermitian generators has been obtained from a bi-orthogonal system of vectors in C 2 . The related Jordan–Schwinger map is combined with boson algebras to obtain a hierarchy of fusion polynomial algebras. This makes possible the construction of Higgs algebra of cubic polynomial type which eventually leads to several multi-boson non-Hermitian Hamiltonians. Finally, the notion of global and partial PT -symmetry have been introduced in a typical Fock space setting. The possibility of PT -symmetry breaking is also discussed. The deformation parameter γ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration. The symmetry behavior has been conceptualized in terms of a couple of toy models.
This thesis is centred around the role of non-Hermitian Hamiltonians in Physics both at the quantum and classical levels. In our investigations of two-level models we demonstrate [1] the phenomenon of fast transitions developed in the PT -symmetric quantum brachistochrone problem may in fact be attributed to the non-Hermiticity of evolution operator used, rather than to its invariance under PT operation. Transition probabilities are calculated for Hamiltonians which explicitly violate PT -symmetry. When it comes to Hilbert spaces of infinite dimension, starting with non-Hermitian Hamiltonians expressed as linear and quadratic combinations of the generators of the su(1; 1) Lie algebra, we construct [2] Hermitian partners in the same similarity class. Alongside, metrics with respect to which the original Hamiltonians are Hermitian are also constructed, allowing to assign meaning to a large class of non-Hermitian Hamiltonians possessing real spectra. The finding of exact results to establish the physical acceptability of other non-Hermitian models may be pursued by other means, especially if the system of interest cannot be expressed in terms of Lie algebraic elements. We also employ [3] a representation of the canonical commutation relations for position and momentum operators in terms of real-valued functions and a noncommutative product rule of differential form. Besides exact solutions, we also compute in a perturbative fashion metrics and isospectral partners for systems of physical interest. Classically, our efforts were concentrated on integrable models presenting PT - symmetry. Because the latter can also establish the reality of energies in classical systems described by Hamiltonian functions, we search for new families of nonlinear differential equations for which the presence of hidden symmetries allows one to assemble exact solutions. We use [4] the Painleve test to check whether deformations of integrable systems preserve integrability. Moreover we compare [5] integrable deformed models, which are thus likely to possess soliton solutions, to a broader class of systems presenting compacton solutions. Finally we study [6] the pole structure of certain real valued nonlinear integrable systems and establish that they behave as interacting particles whose motion can be extended to the complex plane in a PT -symmetric way.
Rotation-time symmetry in bosonic systems and the existence of exceptional points in the absence of PT{\mathscr{PT}}PT symmetry
We study symmetries of open bosonic systems in the presence of laser pumping. Non-Hermitian Hamiltonians describing these systems can be parity-time ([Formula: see text]) symmetric in special cases only. Systems exhibiting this symmetry are characterised by real-valued energy spectra and can display exceptional points, where a symmetry-breaking transition occurs. We demonstrate that there is a more general type of symmetry, i.e., rotation-time ([Formula: see text]) symmetry. We observe that [Formula: see text]-symmetric non-Hermitian Hamiltonians exhibit real-valued energy spectra which can be made singular by symmetry breaking. To calculate the spectra of the studied bosonic non-diagonalisable Hamiltonians we apply diagonalisation methods based on bosonic algebra. Finally, we list a versatile set rules allowing to immediately identifying or constructing [Formula: see text]-symmetric Hamiltonians. We believe that our results on the [Formula: see text]-symmetric class of bosonic systems and their spectral singularities can lead to new applications inspired by those of the [Formula: see text]-symmetric systems.