Stability of $\mathcal{PT}$ and anti-$\mathcal{PT}$-symmetric Hamiltonians with multiple harmonics

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Stability of $PT$ and anti- $PT$ -symmetric Hamiltonians with multiple harmonics

Julia Cen, Yogesh N. Joglekar, and Avadh Saxena

Phys. Rev. Research 6, 013167 – Published 14 February 2024

Abstract

Hermitian Hamiltonians with time-periodic coefficients can be analyzed via Floquet theory, and have been extensively used for engineering Floquet Hamiltonians in standard quantum simulators. Generalized to non-Hermitian Hamiltonians, time periodicity offers avenues to engineer the landscape of Floquet quasienergies across the complex plane. We investigate two-level non-Hermitian $PT$ and anti- $PT$ -symmetric Hamiltonians with coefficients that have multiple harmonics using Floquet theory. By analytical and numerical calculations, we obtain their regions of stability, defined by real Floquet quasienergies, and contours of exceptional point (EP) degeneracies. We extend our analysis to study the phases that accompany these cyclic changes with the biorthogonality approach. Our results demonstrate that these time-periodic Hamiltonians generate a rich landscape of stable (real) and unstable (complex) regions.

Received 11 February 2023
Revised 13 December 2023
Accepted 10 January 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013167

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Floquet systems

PT-symmetry

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied PhysicsQuantum Information, Science & TechnologyInterdisciplinary Physics

Authors & Affiliations

Julia Cen ^1,*, Yogesh N. Joglekar ^2,†, and Avadh Saxena ^1,‡

¹Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
²Department of Physics, Indiana University-Purdue University Indianapolis, Indianapolis, Indiana 46202, USA

^*julia.cen@outlook.com
^†yojoglek@iupui.edu
^‡avadh@lanl.gov

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Vol. 6, Iss. 1 — February - April 2024

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Nonlinear Dynamics

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Figure 1
Stability of doubly driven non-Hermitian Hamiltonian with $β = 3$ . (a) Plot of $max Im ε_{α}$ in the parameter space of $ω$ and $γ$ for Hamiltonian (1) shows stable deep-blue regions punctuated by unstable regions at $γ ≪ 1$ at specific frequencies. The resonance at $ω = 2 J / β$ is clearly seen. (b) EP contours obtained from $G (T)$ for the piecewise constant Hamiltonian (2) also show qualitatively similar features, particularly at small $γ$ values. Note that stable regions at large gain-loss strengths $γ ≫ 1$ are more robust for the piecewise constant Hamiltonian, Eq. (2), than its continuous counterpart, Eq. (1).
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Figure 2
Stability of cosineX-cosineY $APT$ -symmetric Hamiltonian, Eq. (8), with $β = 3$ . (a) $max Im ε_{α} (γ, ω)$ shows triangular unstable regions near resonances $ω = 2, 2 / β$ and their subharmonics. (b) Corresponding EP contours from piecewise-constant Hamiltonian show same qualitative features, including the absence of stable regions at large gain-loss strengths. (c) Boundaries for the two main resonances from perturbation theory.
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Figure 3
Stability of cosineX-sineY Hamiltonian, Eq. (9), with $β = 3$ . (a) $max Im ε_{α} (γ, ω)$ shows a large triangular unstable region near $ω = 2$ and a small sliver near $ω = 2 / β$ , along with their subharmonics. (b) Corresponding EP contours from piecewise-constant Hamiltonian show same qualitative features. The clustering of EP contours at low frequencies $ω ≪ 1$ to the static threshold $γ_{E P} = 1$ is also clear.
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Figure 4
Real and imaginary parts of the complex Berry phase, Eq. (17), for the non-Hermitian $PT$ -symmetric system (18). Er denotes the real eigenvalues region, Ea is alternating real-imaginary eigenvalues region and HSA is the abbreviation for “half the solid angle.” (a)–(c) Top panel shows results for $β = 1$ , i.e., same frequency drives. Blue, red and orange, purple are the real and imaginary parts of the Berry phases from the two different initial eigenstates of the system. Black dash-dotted denote half the solid angle subtended by the closed curve evolved by the eigenstates. Here, $γ_{c} = \pm 1$ denotes the threshold above which the instantaneous eigenvalues become complex for some range of $s \sim T / 4, 3 T / 4$ . (d)–(f) Bottom panel shows corresponding results for $β = 3$ .
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Figure 5
Real and imaginary parts of the complex Berry phases $θ_{B α}$ for the $APT$ -symmetric Hamiltonian (19). Er, Ea, HSA are as in Fig. 4, and Ei denote the imaginary eigenvalues region. (a), (b) Top panel shows results for $β = 1$ , i.e., same frequency drives. Blue, red and orange, purple are the real and imaginary parts of the Berry phases from the two different initial eigenstates of the system. (c), (d) Bottom panel shows results for $β = 3$ . Here, $γ_{c}, γ_{c} 1, γ_{c} 2$ denote the transition of eigenvalue regions between real (blue region), pure imaginary (pink region), and alternating real and pure imaginary (purple region) cases.
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Physical Review Research

Stability of PT and anti-PT-symmetric Hamiltonians with multiple harmonics

Julia Cen, Yogesh N. Joglekar, and Avadh Saxena

Phys. Rev. Research 6, 013167 – Published 14 February 2024

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Stability of $PT$ and anti- $PT$ -symmetric Hamiltonians with multiple harmonics