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arXiv:2309.09829 · Physical review. B./Physical review. B

Schrieffer-Wolff transformation for non-Hermitian systems: Application for PT -symmetric circuit QED

Grigory A. Starkov, M. V. Fistul, Ilya Eremin

Published Dec 12, 2023• 3 citations
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Abstract

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 $\mathcal{P}\mathcal{T}$-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 $g$ and the gain (loss) parameter $\ensuremath{\gamma}$, and compare that with the eigenvalues obtained analytically using the effective Hamiltonian of resonant states. We identify $\mathcal{P}\mathcal{T}$-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 $\mathcal{P}$-pseudo-Hermitian symmetry of the system and show that non-Hermiticity mixes the ``dark'' and the ``bright'' states, which has a direct experimental consequence.

DOI
10.1103/physrevb.108.235417

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The non-Hermitian Hamiltonian describes the effective dynamics of a system coupled to a continuously measured bath, and can exhibit anti-unitary symmetries that give rise to exceptional points and broken phases with complex eigenvalues, features unique to non-Hermitian systems. Going beyond conventional Hermitian physics, we analyze the impact of non-Hermiticity in the quantum state transmission by employing a non-Hermitian spin chain that functions as a quantum data bus. By deriving a general expression for the fidelity of quantum state transfer for a U(1)-symmetric non-Hermitian Hamiltonian, we analyze PT-symmetric XX and SSH models, complemented by a numerical study of the RT-symmetric iXY model. We demonstrate that, in several parameter regimes, the transfer fidelity in the non-Hermitian setting exceeds the classical threshold and can even exceed the performance of the corresponding Hermitian models. In particular, for the SSH model with dominant inter-cell coupling, the broken phase supports near-unit-fidelity quantum state transfer, a level of performance that the corresponding Hermitian model fails to attain. Moreover, we establish a correspondence between the non-Hermitian and Hermitian descriptions by identifying related parameter regions in which the fidelity fails to surpass the classical bound.
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The PT -symmetric semiclassical Rabi model explores the fundamental interaction between a two-level atom and a classical field, revealing novel phenomena in open systems through the inclusion of non-Hermitian terms. We propose a single similarity transformation that yields an effective Hamiltonian in rotating-wave approximation, enabling an analytical solution. The phase boundary of the PT -broken phase, derived from the analytical eigenvalues, closely matches the numerically exact one over a wide range of atomic frequencies, demonstrating the effectiveness of the analytical approach, especially at the main resonance. The Floquet parity operator is also introduced, providing a deeper physical understanding of the emergence of the PT -broken phase. Furthermore, by analyzing the dynamics of excited-state population, we observe several stable oscillations in the Fourier spectrum, demonstrating the applicability of the analytical method beyond the single-photon resonance region. The Bloch-Siegert shift is also discussed, and surprisingly, resembles its Hermitian counterpart, except for the higher-order terms in the coupling strength. The present analytical treatment provides a concise and accurate description of the main physics of this non-Hermitian atom-field interaction system.
Pseudo-Hermitian operators generalize the concept of Hermiticity. This class of operators includes the quasi-Hermitian operators, which reformulate quantum theory while retaining real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices. In addition to the presented original research, scholars will appreciate the lengthy introduction to non-Hermitian physics. Local quasi-Hermitian observable algebras are examined. Expectation values of local quasi-Hermitian observables equal expectation values of local Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory. Exceptional points, which are branch points in the spectrum, are a perturbative feature unique to non-Hermitian operators. Cusp singularities of algebraic curves are related to higher-order exceptional points. To exemplify novelties of non-Hermiticity, one-dimensional lattice models with a pair of non-Hermitian defect potentials with balanced loss and gain, Δ±iγΔ\pm i γΔ±iγ, are explored. When the defects are nearest neighbour, the entire spectrum becomes complex when γγγ is tuned past a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the PT\mathcal{PT}PT-phase transition is dictated by the topological phase. Chiral symmetry and representation theory are used to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators.
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