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arXiv:1407.4535 · Physical Review A

PT spectroscopy of the Rabi problem

Yogesh N. Joglekar, Rahul Marathe, P. Durganandini, Rajeev K. Pathak

Published Oct 10, 2014• 66 citations
arXivHTMLPDF

Abstract

We investigate the effects of a time-periodic, non-Hermitian, $\mathcal{PT}\text{\ensuremath{-}}\mathrm{symmetric}$ perturbation on a system with two (or few) levels, and obtain its phase diagram as a function of the perturbation strength and frequency. We prove that when the perturbation frequency is close to one of the system resonances, even a vanishingly small gain-loss potential leads to $\mathcal{PT}$ symmetry breaking. We also find a restored $\mathcal{PT}\text{\ensuremath{-}}\mathrm{symmetric}$ phase at high frequencies, and at moderate perturbation strengths, we predict the existence of multiple frequency windows where $\mathcal{PT}$ symmetry is broken and restored. Our results imply that the $\mathcal{PT}\text{\ensuremath{-}}\mathrm{symmetric}$ Rabi problem shows exceptionally rich phenomena absent in its Hermitian or static counterparts.

DOI
10.1103/physreva.90.040101

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We investigate the effects of a time-periodic, non-Hermitian, PT\ensuremath-symmetric\mathcal{PT}\text{\ensuremath{-}}\mathrm{symmetric}PT\ensuremath-symmetric perturbation on a system with two (or few) levels, and obtain its phase diagram as a function of the perturbation strength and frequency. We prove that when the perturbation frequency is close to one of the system resonances, even a vanishingly small gain-loss potential leads to PT\mathcal{PT}PT symmetry breaking. We also find a restored PT\ensuremath-symmetric\mathcal{PT}\text{\ensuremath{-}}\mathrm{symmetric}PT\ensuremath-symmetric phase at high frequencies, and at moderate perturbation strengths, we predict the existence of multiple frequency windows where PT\mathcal{PT}PT symmetry is broken and restored. Our results imply that the PT\ensuremath-symmetric\mathcal{PT}\text{\ensuremath{-}}\mathrm{symmetric}PT\ensuremath-symmetric Rabi problem shows exceptionally rich phenomena absent in its Hermitian or static counterparts.
We study a non-Hermitian version of the Rabi model, where a two-level system is periodically driven with an imaginary-valued drive strength, leading to alternating gain and loss. In the Floquet picture, the model exhibits PT\mathcal{PT}PT symmetry, which can be broken when the drive is sufficiently strong. We derive the boundaries of the PT\mathcal{PT}PT phase diagram for the different resonances by doing perturbation theory beyond the rotating-wave approximation. For the main resonance, we show that the non-Hermitian analog of the Bloch-Siegert shift corresponds to maximal PT\mathcal{PT}PT breaking. For the higher-order resonances, we capture the boundaries to lowest order. We also solve the regime of high frequency by mapping to the Wannier-Stark ladder. Our model can be experimentally realized in waveguides with spatially modulated loss or in atoms with time-modulated spontaneous decay.
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.
We investigate the effect of a periodic modulation on parity-time symmetry in a non-Hermitian Rabi model with balanced gain or loss. It is shown that under the multiple-photon resonance conditions, this Rabi model supports exact Floquet solutions for certain special parameter conditions which allow us to find exact analytical results for the position of certain exceptional points (EPs). It is also found numerically that there exists a universal relation between the number of the EPs and the gain or loss strength.
Non-Hermitian, mathcalPT\\mathcal{PT}mathcalPT -symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, mathcalPT\\mathcal{PT}mathcalPT potentials for a non-relativistic particle confined in a circular geometry. We show that the mathcalPT\\mathcal{PT}mathcalPT -symmetry threshold can be tuned by introducing a second gain-loss potential or its Hermitian counterpart. Our results explicitly demonstrate that mathcalPT\\mathcal{PT}mathcalPT breaking in two dimensions has a rich phase diagram, with multiple re-entrant mathcalPT\\mathcal{PT}mathcalPT -symmetric phases.
We study a non-Hermitian two-level system with square-wave modulated dissipation and coupling. Based on the Floquet theory, we achieve an effective Hamiltonian from which the boundaries of the <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T phase diagram are captured exactly. Two kinds of <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T symmetry broken phases are found, whose effective Hamiltonians differ by a constant ω/2. For the time-periodic dissipation, a vanishingly small dissipation strength can lead to the <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T symmetry breaking in the (2k – 1)-photon resonance (Δ = (2k – 1) ω), with k = 1,2,3… It is worth noting that such a phenomenon can also happen in 2k-photon resonance (Δ = 2kΔ), as long as the dissipation strengths or the driving times are imbalanced, namely γ 0 ≠ – γ 1 or T 0 ≠ T 1. For the time-periodic coupling, the weak dissipation induced <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T symmetry breaking occurs at Δ eff = kω, where Δ eff = (Δ 0 T 0 + Δ 1 T 1)/T. In the high frequency limit, the phase boundary is given by a simple relation γ eff = ±Δ eff.
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].