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Floquet bound states in the continuum.

Longhi S, Della Valle G - Sci Rep (2013)

Bottom Line: Originally regarded as mathematical curiosities, BIC have found an increasing interest in recent years, particularly in quantum and classical transport of matter and optical waves in mesoscopic and photonic systems where the underlying potential can be judiciously tailored.Here we introduce a new kind of BIC, referred to as Floquet BIC, which corresponds to a normalizable Floquet state of a time-periodic Hamiltonian with a quasienergy embedded into the spectrum of Floquet scattered states.We discuss the appearance of Floquet BIC states in a tight-binding lattice model driven by an ac field in the proximity of the dynamic localization regime.

View Article: PubMed Central - PubMed

Affiliation: Dipartimento di Fisica- Politecnico di Milano and Istituto di Fotonica e Nanotecnologie - Consiglio Nazionale delle Ricerche Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy. stefano.longhi@polimi.it

ABSTRACT
Quantum mechanics predicts that certain stationary potentials can sustain bound states with an energy buried in the continuous spectrum of scattered states, the so-called bound states in the continuum (BIC). Originally regarded as mathematical curiosities, BIC have found an increasing interest in recent years, particularly in quantum and classical transport of matter and optical waves in mesoscopic and photonic systems where the underlying potential can be judiciously tailored. Most of our knowledge of BIC is so far restricted to static potentials. Here we introduce a new kind of BIC, referred to as Floquet BIC, which corresponds to a normalizable Floquet state of a time-periodic Hamiltonian with a quasienergy embedded into the spectrum of Floquet scattered states. We discuss the appearance of Floquet BIC states in a tight-binding lattice model driven by an ac field in the proximity of the dynamic localization regime.

No MeSH data available.


Related in: MedlinePlus

Numerically-computed quasi-energy spectrum versus the normalized forcing amplitude Γ = F0a/ω (upper panels) and corresponding maps of the participation ratio R(t) at time t = 0 of the N = 201 Floquet eigenstates (lower panels) in the ac-driven lattice of Fig. 1(a) for (a) κ/ω = 2, (b) κ/ω = 1, and (c) κ/ω = 0.3.The mode number on the vertical axis in the lower panels is ordered for increasing values of the quasi energy . In the upper panels, the dispersion curves that detach from the continuous band correspond to Floquet BOC modes. The circles in the lower panels indicate Floquet BIC states. The BIC modes are found at the forcing amplitudes Γ = 1.9940, 2.3121475, 2.35972, 2.49582, 2.6815 in (a), Γ = 2.2904, 2.356951, 2.5195 in (b), and Γ = 2.3800, 2.42875 in (c).
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f2: Numerically-computed quasi-energy spectrum versus the normalized forcing amplitude Γ = F0a/ω (upper panels) and corresponding maps of the participation ratio R(t) at time t = 0 of the N = 201 Floquet eigenstates (lower panels) in the ac-driven lattice of Fig. 1(a) for (a) κ/ω = 2, (b) κ/ω = 1, and (c) κ/ω = 0.3.The mode number on the vertical axis in the lower panels is ordered for increasing values of the quasi energy . In the upper panels, the dispersion curves that detach from the continuous band correspond to Floquet BOC modes. The circles in the lower panels indicate Floquet BIC states. The BIC modes are found at the forcing amplitudes Γ = 1.9940, 2.3121475, 2.35972, 2.49582, 2.6815 in (a), Γ = 2.2904, 2.356951, 2.5195 in (b), and Γ = 2.3800, 2.42875 in (c).

Mentions: To this aim, we considered the lattice model of Fig. 1(a) assuming ρ/κ = 0.7, and numerically computed the quasi-energy spectrum and associated Floquet eigenstates for a few values of the ratio κ/ω and for a normalized driving amplitue Γ spanning the interval (2, 2.8), near the DL point Γ = 2.405. The lattice comprises N = 201 sites. To avoid lattice truncation effects, periodic boundary conditions have been assumed in the numerical analysis (see Methods). The degree of localization of the Floquet states /u(t)〉 is described by the participation ratio R(t), which is defined as R(t) = 〈u(t)/u(t)〉2/〈u2(t)/u2(t)〉. Note that R(t) is periodic with period T, with R ~ 1 for localized modes while R ~ N for extended states. For the undriven lattice, all the modes are extended (with R ~ 134), and there are no signature of resonances; see Fig. 1(b). The scenario is fully modified when the driving field is switched on. Figure 2 shows the numerically-computed quasi-energy spectrum and participation ratio R(t) (in a log scale) at t = 0 of the corresponding N Floquet modes versus the normalized forcing Γ for the three values κ/ω = 2, κ/ω = 1 and κ/ω = 0.3. The eigenmode number of Floquet states is ordered for increasing values of the quasi energies. An inspection of the quasi energy diagrams shows that, in the neighborhood of the DL regime Γ = 2.405, Floquet BOC emerge in pairs, above and below the band of scattered states, which are clearly visible as isolated dispersion curves that detach from the continuous band of scattered states. In the participation ratio diagrams, the BOC modes correspond to the dark stripes at the upper and lower boundaries of the domain. The number of field-induced BOC states typically increases as the ratio κ/ω increases, according to previous studies35. As Γ is pushed far from the DL condition, the dispersion curves of the BOC states go inside the band of scattered states, and resonance states, i.e. unbounded states which retain a certain degree of localization, are clearly visible in the participation ratio diagrams as darker lines internal to the domain. Examples of BOC and resonance states are shown in Figs. 3(a) and (b), respectively.


Floquet bound states in the continuum.

Longhi S, Della Valle G - Sci Rep (2013)

Numerically-computed quasi-energy spectrum versus the normalized forcing amplitude Γ = F0a/ω (upper panels) and corresponding maps of the participation ratio R(t) at time t = 0 of the N = 201 Floquet eigenstates (lower panels) in the ac-driven lattice of Fig. 1(a) for (a) κ/ω = 2, (b) κ/ω = 1, and (c) κ/ω = 0.3.The mode number on the vertical axis in the lower panels is ordered for increasing values of the quasi energy . In the upper panels, the dispersion curves that detach from the continuous band correspond to Floquet BOC modes. The circles in the lower panels indicate Floquet BIC states. The BIC modes are found at the forcing amplitudes Γ = 1.9940, 2.3121475, 2.35972, 2.49582, 2.6815 in (a), Γ = 2.2904, 2.356951, 2.5195 in (b), and Γ = 2.3800, 2.42875 in (c).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3713529&req=5

f2: Numerically-computed quasi-energy spectrum versus the normalized forcing amplitude Γ = F0a/ω (upper panels) and corresponding maps of the participation ratio R(t) at time t = 0 of the N = 201 Floquet eigenstates (lower panels) in the ac-driven lattice of Fig. 1(a) for (a) κ/ω = 2, (b) κ/ω = 1, and (c) κ/ω = 0.3.The mode number on the vertical axis in the lower panels is ordered for increasing values of the quasi energy . In the upper panels, the dispersion curves that detach from the continuous band correspond to Floquet BOC modes. The circles in the lower panels indicate Floquet BIC states. The BIC modes are found at the forcing amplitudes Γ = 1.9940, 2.3121475, 2.35972, 2.49582, 2.6815 in (a), Γ = 2.2904, 2.356951, 2.5195 in (b), and Γ = 2.3800, 2.42875 in (c).
Mentions: To this aim, we considered the lattice model of Fig. 1(a) assuming ρ/κ = 0.7, and numerically computed the quasi-energy spectrum and associated Floquet eigenstates for a few values of the ratio κ/ω and for a normalized driving amplitue Γ spanning the interval (2, 2.8), near the DL point Γ = 2.405. The lattice comprises N = 201 sites. To avoid lattice truncation effects, periodic boundary conditions have been assumed in the numerical analysis (see Methods). The degree of localization of the Floquet states /u(t)〉 is described by the participation ratio R(t), which is defined as R(t) = 〈u(t)/u(t)〉2/〈u2(t)/u2(t)〉. Note that R(t) is periodic with period T, with R ~ 1 for localized modes while R ~ N for extended states. For the undriven lattice, all the modes are extended (with R ~ 134), and there are no signature of resonances; see Fig. 1(b). The scenario is fully modified when the driving field is switched on. Figure 2 shows the numerically-computed quasi-energy spectrum and participation ratio R(t) (in a log scale) at t = 0 of the corresponding N Floquet modes versus the normalized forcing Γ for the three values κ/ω = 2, κ/ω = 1 and κ/ω = 0.3. The eigenmode number of Floquet states is ordered for increasing values of the quasi energies. An inspection of the quasi energy diagrams shows that, in the neighborhood of the DL regime Γ = 2.405, Floquet BOC emerge in pairs, above and below the band of scattered states, which are clearly visible as isolated dispersion curves that detach from the continuous band of scattered states. In the participation ratio diagrams, the BOC modes correspond to the dark stripes at the upper and lower boundaries of the domain. The number of field-induced BOC states typically increases as the ratio κ/ω increases, according to previous studies35. As Γ is pushed far from the DL condition, the dispersion curves of the BOC states go inside the band of scattered states, and resonance states, i.e. unbounded states which retain a certain degree of localization, are clearly visible in the participation ratio diagrams as darker lines internal to the domain. Examples of BOC and resonance states are shown in Figs. 3(a) and (b), respectively.

Bottom Line: Originally regarded as mathematical curiosities, BIC have found an increasing interest in recent years, particularly in quantum and classical transport of matter and optical waves in mesoscopic and photonic systems where the underlying potential can be judiciously tailored.Here we introduce a new kind of BIC, referred to as Floquet BIC, which corresponds to a normalizable Floquet state of a time-periodic Hamiltonian with a quasienergy embedded into the spectrum of Floquet scattered states.We discuss the appearance of Floquet BIC states in a tight-binding lattice model driven by an ac field in the proximity of the dynamic localization regime.

View Article: PubMed Central - PubMed

Affiliation: Dipartimento di Fisica- Politecnico di Milano and Istituto di Fotonica e Nanotecnologie - Consiglio Nazionale delle Ricerche Piazza Leonardo da Vinci, 32, I-20133 Milano, Italy. stefano.longhi@polimi.it

ABSTRACT
Quantum mechanics predicts that certain stationary potentials can sustain bound states with an energy buried in the continuous spectrum of scattered states, the so-called bound states in the continuum (BIC). Originally regarded as mathematical curiosities, BIC have found an increasing interest in recent years, particularly in quantum and classical transport of matter and optical waves in mesoscopic and photonic systems where the underlying potential can be judiciously tailored. Most of our knowledge of BIC is so far restricted to static potentials. Here we introduce a new kind of BIC, referred to as Floquet BIC, which corresponds to a normalizable Floquet state of a time-periodic Hamiltonian with a quasienergy embedded into the spectrum of Floquet scattered states. We discuss the appearance of Floquet BIC states in a tight-binding lattice model driven by an ac field in the proximity of the dynamic localization regime.

No MeSH data available.


Related in: MedlinePlus