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 ${\mathscr{P}}{\mathscr{T}}$?> P T phase diagram are captured exactly. Two kinds of <?CDATA ${\mathscr{P}}{\mathscr{T}}$?> 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 ${\mathscr{P}}{\mathscr{T}}$?> 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 ${\mathscr{P}}{\mathscr{T}}$?> 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 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 ${\mathscr{P}}{\mathscr{T}}$?> P T phase diagram are captured exactly. Two kinds of <?CDATA ${\mathscr{P}}{\mathscr{T}}$?> 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 ${\mathscr{P}}{\mathscr{T}}$?> 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 ${\mathscr{P}}{\mathscr{T}}$?> 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 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 symmetry, which can be broken when the drive is sufficiently strong. We derive the boundaries of the 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 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.
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 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 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 report a novel mechanism of boundary-sensitive PT symmetry breaking in one-dimensional Floquet systems. By designing a time-periodic driving protocol, we realize a Floquet Hamiltonian that is Hermitian under periodic boundary conditions yet acquires non-Hermitian boundary terms under open boundary conditions due to the non-commutativity of driving Hamiltonians. We establish that a PT symmetry breaking transition occurs when the quasienergy bandwidth expands to cover the entire frequency Brillouin zone. This condition highlights a crucial difference from static non-Hermitian systems, where such transitions typically require band touching. Furthermore, we demonstrate that in the PT-broken phase, the eigenstates exhibit scale-free localization, a phenomenon arising from the specific system-size scaling of non-Hermitian terms. Finally, we provide a general framework for constructing multi-band models that exhibit this boundary-induced phase transition.
In recent years, there have been intensive studies of non-Hermitian physics and parity–time (PT) symmetry due to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric system is phase transition (spontaneous PT-symmetry breaking), i.e. an all-real energy spectrum changes into an all-complex one when the non-Hermitian parameter exceeds a certain threshold. However, the conditions for PT-symmetric system with real energy spectrum to occur are rather restrictive. The generalization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-real energy spectra is a currently important endeavor. A simple PT-symmetric two-level Floquet quantum system is now being actively explored, because it holds potential for the realization of non-unitary single-qubit quantum gate. However, studies of the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquet quantum system are still relatively rare. In this paper, we investigate the non-Hermitian physics of a periodically driven non-PT-symmetric two-level quantum system. By phase-space analysis, we find that there exist so-called pseudo fixed points in phase space representing the Floquet solutions with fixed population difference and a time-dependent relative phase between the two levels. According to these pseudo fixed points, we analytically construct a non-unitary evolution operator and then explore the dynamic behaviors of the non-PT-symmetric two-level quantum system in different parameter regions. We confirm both analytically and numerically that the two-level non-Hermitian Floquet quantum system, although it is non-parity-time-symmetric, still features a phase transition with the quasienergy spectrum changing from all-real to all-complex energy spectrum, just like the PT symmetric system. Furthermore, we reveal that a novel phenomenon called quasi-PT symmetric dynamics occurs in the time evolution process. The quasi-PT symmetric dynamics is so named in our paper, in the sense that the time-evolution of population probabilities in the non-PT-symmetric two-level system satisfies fully the time-space symmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not meet such a PT symmetry, due to the fact that time-evolution of the phases of the probability amplitudes on the two levels violates the requirement for the PT symmetry.
Abstract Open physical systems with balanced loss and gain, described by non-Hermitian parity-time (PT) P T reflection symmetric Hamiltonians, exhibit a transition which could engender modes that exponentially decay or grow with time, and thus spontaneously breaks the PT P T -symmetry. Such PT P T -symmetry-breaking transitions have attracted many interests because of their extraordinary behaviors and functionalities absent in closed systems. Here we report on the observation of PT P T -symmetry-breaking transitions by engineering time-periodic dissipation and coupling, which are realized through state-dependent atom loss in an optical dipole trap of ultracold 6Li atoms. Comparing with a single transition appearing for static dissipation, the time-periodic counterpart undergoes PT P T -symmetry breaking and restoring transitions at vanishingly small dissipation strength in both single and multiphoton transition domains, revealing rich phase structures associated to a Floquet open system. The results enable ultracold atoms to be a versatile tool for studying PT P T -symmetric quantum systems.
In recent years, there have been intensive studies on non-Hermitian physics and parity-time (PT) symmetry, due to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric systems is phase transition (spontaneous PT-symmetry breaking), where the energy spectrum changes from all real to complex when the non-Hermitian parameter exceeds a certain threshold. However, the conditions for PT-symmetric system with real energy spectrum to occur are rather restrictive. Generalization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-real spectra is a currently important endeavor. The simple PT-symmetric two-level Floquet quantum system is now being actively explored, because it holds potential for realization of non-unitary single-qubit quantum gate. However, studies on the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquet quantum system still remain relatively rare. In this paper, we investigate the non-Hermitian physics of a periodically driven non-PT-symmetric two-level quantum system. By phase-space analysis, we find that there exist so-called pseudo fixed points in phase space representing the Floquet solutions with fixed population difference and a time-dependent relative phase between the two levels. Based on these pseudo fixed points, we analytically construct the non-unitary evolution operator and then explore the dynamics of the non-PT-symmetric two-level quantum system in different parameter regions. We confirm both analytically and numerically that the two-level non-Hermitian Floquet quantum system, although being non-parity-time-symmetric, still features a phase transition with the quasienergy spectrum changing from all real to complex, just as for PT symmetric systems. Furthermore, we reveal that a novel phenomenon called quasi-PT symmetric dynamics occurs in the time evolution process. The quasi-PT symmetric dynamics is so named in our paper, in the sense that the time-evolution of population probabilities in the non-PT-symmetric two-level system respects fully the time-space symmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not, due to the fact that time-evolution of the phases of the probability amplitudes on the two levels violates the PT symmetry requirement.