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Physical Review A

Ratchet current in a PT -symmetric Floquet quantum system with symmetric sinusoidal driving

Zhiqiang Li, Xiaoxiao Hu, Jinpeng Xiao, Yajiang Chen, Xiaobing Luo

Published Nov 9, 2023• 6 citations
Publisher

Abstract

We consider the ratchet dynamics in a $\mathcal{PT}$-symmetric Floquet quantum system with symmetric temporal (harmonic) driving. In the exact $\mathcal{PT}$-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-$\mathcal{PT}$-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.

DOI
10.1103/physreva.108.052211

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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.
We investigate the directed momentum current in the quantum kicked rotor model with PT-symmetric deriving potential. For the quantum nonresonance case, the values of quasienergy become complex when the strength of the imaginary part of the kicking potential exceeds a threshold value, which demonstrates the appearance of the spontaneous PT symmetry breaking. In the vicinity of the transition point, the momentum current exhibits a staircase growth with time. Each platform of the momentum current corresponds to the mean momentum of some eigenstates of the Floquet operator whose imaginary parts of the quasienergy are significantly large. Above the transition point, the momentum current increases linearly with time. Interestingly, its acceleration rate exhibits a kind of "quantized" increment with the kicking strength. We propose a modified classical acceleration mode of the kicked rotor model to explain such an intriguing phenomenon. Our theoretical prediction is in good agreement with numerical results.
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.
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.
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.
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.
Abstract We numerically study the quantum dynamics of a P T -symmetric kicked harmonic oscillator. The P T symmetry breaks spontaneously when the non-Hermitian driving strength surpasses the exceptional point. In the P T -broken phase, we observe that directed current of momentum and ballistic diffusion of energy coexist under the non-resonant conditions, whereas both the momentum and energy oscillate as damped cosine functions with identical frequencies under the resonant conditions. The research shows that the directed current of momentum and ballistic diffusion of energy arise from nearest-neighbor hopping between momentum eigenstates with the non-Hermitian driving, while the damped oscillations of momentum and energy originate from resonant coupling between the non-Hermitian driving and the harmonic oscillator. Our findings indicate that the non-Hermiticity and the frequency characteristic of this system collectively result in these distinctive dynamical behaviors.