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arXiv:1508.07001 · Physical Review A

PT -symmetric Rabi model: Perturbation theory

Tony E. Lee, Yogesh N. Joglekar

Published Oct 8, 2015• 66 citations
arXivHTMLPDF

Abstract

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 $\mathcal{PT}$ symmetry, which can be broken when the drive is sufficiently strong. We derive the boundaries of the $\mathcal{PT}$ 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 $\mathcal{PT}$ 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.

DOI
10.1103/physreva.92.042103

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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.
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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.
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 PT\mathcal{PT}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 PT\mathcal{PT}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.
We consider the ratchet dynamics in a PT\mathcal{PT}PT-symmetric Floquet quantum system with symmetric temporal (harmonic) driving. In the exact PT\mathcal{PT}PT-symmetry phase, for a finite number of resonant frequencies, we show that the long-lasting resonant currents can be generated with the symmetric time-continuous driving, which would otherwise forbid the generation of directed currents in the Hermitian limit. Such a non-Hermitian resonant current can be enhanced by increasing the non-Hermitian level and, in particular, the resonant current peaks (reaches the largest negative value) under the condition that the imaginary part of the potential depth is equal to the real part, at which the stable asymptotic current occurs owing to exceptional points mechanism. Moreover, the directed currents originating from the symmetry breaking are reported, which increase linearly with the driving frequency, the mechanism behind which is that the cutoff of the momentum eigenstates for the Floquet state with maximum imaginary quasienergy increases as the driving frequency is continuously increased. We also present a non-Hermitian three-level model that can account for the resonant currents and gives surprisingly good agreement with direct numerical results for weak driving, even in the broken-PT\mathcal{PT}PT-symmetry regime for the first-order resonance. Our results provide a means of realizing the non-Hermiticity-controlled ratchet current by means of a smooth continuous driving, previously used only to generate currents in Hermitian systems.
Monochromatically driven two-level systems (i.e., Rabi models) are ubiquitous in various fields of physics. Though they have been exactly solved, the physical pictures in these exact solutions are not clear. Recently, approximate analytical solutions with neat physics have been obtained by using the counter-rotating hybridized rotating wave (CHRW) method, which has been proven to be effective over a wider range of parameters than the previous analytical solutions. However, the CHRW depends on a parameter $\ensuremath{\xi}$, which has no solution in some regimes. Here, we combine the double-unitary-transformation approach with the generalized Van Vleck nearly degenerate perturbation theory, and present approximate analytical results with clear physics for almost all parameter regimes, which agree well with the numerical solutions and the previous experimental results. Moreover, the dynamic frequencies of the Rabi model are regular, and the frequency with the highest Fourier amplitude changes from the Rabi frequency to $2n\ensuremath{\omega}$ with driving frequency $\ensuremath{\omega}$ and integer nnn, as the driving intensity increases from weak to deep-strong. In addition, we further explore the Floquet dynamics of the dissipative open Rabi model. Remarkably, the dissipations are tunable in the rotating frame, and the approximate analytical results obtained by our method are in good agreement with the numerical results in the strong driving regime. These results pave the way to quantum control using strong and deep-strong driving with applications in quantum technologies.
So far the Bloch-Siegert shift of the Rabi model has only been calculated numerically or by means of perturbation theory valid in either the weak or strong driving regime only. Recently Yan, Lü, and Zheng [Phys.~Rev.~A {\bf 91}, 053834 (2015)] showed how to reduce the problem to solving a system of three nonlinear equations. Here, we pursue an alternative approach based on a perturbation expansion extrapolation technique. We are thus able to derive an explicit analytical approximation formula for the Bloch-Siegert shift of the Rabi model which is valid for all parameter regimes from weak to strong driving. Comparison with numerically exact results reveals an excellent agreement over the entire driving-strength range.