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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.

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(a) Schematic of an ac-driven tight-binding lattice with inhomogeneous hopping rate ρ < κ between lattice sites /0〉 and /±1〉. (b) Participation ratio R of the eigenstates of the undriven lattice comprising N = 201 sites for ρ/κ = 0.7. The mode number on the horizontal axis is ordered for increasing values of the energy , from  (lower axis limit) to  (upper axis limit). (c) Effective static lattice model in the high-frequency regime. The effective hopping rates κe, α and β are given in the text by Eq.(3).
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f1: (a) Schematic of an ac-driven tight-binding lattice with inhomogeneous hopping rate ρ < κ between lattice sites /0〉 and /±1〉. (b) Participation ratio R of the eigenstates of the undriven lattice comprising N = 201 sites for ρ/κ = 0.7. The mode number on the horizontal axis is ordered for increasing values of the energy , from (lower axis limit) to (upper axis limit). (c) Effective static lattice model in the high-frequency regime. The effective hopping rates κe, α and β are given in the text by Eq.(3).

Mentions: As a model system, we consider the coherent hopping dynamics of a quantum particle on a one-dimensional tight-binding lattice driven by an external sinusoidal field with inhomogeneous hopping rates, which is described by the Hamiltonian (with = 1) where /n〉 is the Wannier state localized at lattice site n (n = 0,±1,±2,…), κn is the hopping rate between sites n and (n + 1), a is the lattice period, and F(t) = F0 cos(ωt) is the external sinusoidal force of period T = 2π/ω. This model has been investigated in different physical contexts. It describes, for example, coherent transport of ultracold atoms in periodically-shaken optical lattices29, coherent electronic transport in irradiated semiconductor superlattices30, and light propagation in arrays of periodically-curved optical waveguides3132. We assume that the lattice is asymptotically homogeneous, i.e. that κn → κ as n → ±∞, and that in the absence of the driving force, i.e. for F(t) = 0, the static Hamiltonian has a purely continuous spectrum, i.e. that hopping inhomogeneities do not introduce bound states, neither outside nor inside the tight-binding energy band (−2κ, 2κ). For example, such a condition is satisfied by assuming κn = κ for n ≠ −1, 0 and κ−1 = κ0 = ρ < κ, see Fig. 1(a). In the presence of the periodic forcing, the eigenstates /ψ(t)〉 of the Hamiltonian are Floquet states of the form , where /u(t + T)〉 = /u(t)〉 and is the quasi-energy, which is assumed to vary in the interval (−ω/2, ω/2). A Floquet BIC state can be defined as a normalizable Floquet state of with a quasi-energy buried in the continuous quasi-energy spectrum of scattered states of the lattice. For a homogeneous lattice (κn = κ), the problem is integrable33 and the quasi-energy spectrum turns out to be purely continuous and defined by the dispersion relation30, where −π ≤ p < π, Γ = aF0/ω is the normalized forcing parameter, and J0 is the Bessel function of first kind and zero order. The two linearly-independent Floquet scattered states with quasi-energy in the ac-driven homogeneous lattice are given by the backward and forward propagating plane waves where and Φ(t) = Γ sin(ωt). As is well known, the quasi-energy spectrum shrinks and collapses as Γ → 2.405, at which J0(Γ) = 0 and dynamic localization (DL) is attained33. Hence, in a homogeneous lattice the role of the external force is to re-normalize the bandwidth of the undriven lattice band, however the spectrum remains purely continuous. Lattice defects or boundary effects make DL imperfect and bound states outside the continuum (BOC) can be induced by the ac field, as discussed in343536. Here we show that, besides BOC states, under certain driving conditions Floquet BIC states can appear as well.


Floquet bound states in the continuum.

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

(a) Schematic of an ac-driven tight-binding lattice with inhomogeneous hopping rate ρ < κ between lattice sites /0〉 and /±1〉. (b) Participation ratio R of the eigenstates of the undriven lattice comprising N = 201 sites for ρ/κ = 0.7. The mode number on the horizontal axis is ordered for increasing values of the energy , from  (lower axis limit) to  (upper axis limit). (c) Effective static lattice model in the high-frequency regime. The effective hopping rates κe, α and β are given in the text by Eq.(3).
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f1: (a) Schematic of an ac-driven tight-binding lattice with inhomogeneous hopping rate ρ < κ between lattice sites /0〉 and /±1〉. (b) Participation ratio R of the eigenstates of the undriven lattice comprising N = 201 sites for ρ/κ = 0.7. The mode number on the horizontal axis is ordered for increasing values of the energy , from (lower axis limit) to (upper axis limit). (c) Effective static lattice model in the high-frequency regime. The effective hopping rates κe, α and β are given in the text by Eq.(3).
Mentions: As a model system, we consider the coherent hopping dynamics of a quantum particle on a one-dimensional tight-binding lattice driven by an external sinusoidal field with inhomogeneous hopping rates, which is described by the Hamiltonian (with = 1) where /n〉 is the Wannier state localized at lattice site n (n = 0,±1,±2,…), κn is the hopping rate between sites n and (n + 1), a is the lattice period, and F(t) = F0 cos(ωt) is the external sinusoidal force of period T = 2π/ω. This model has been investigated in different physical contexts. It describes, for example, coherent transport of ultracold atoms in periodically-shaken optical lattices29, coherent electronic transport in irradiated semiconductor superlattices30, and light propagation in arrays of periodically-curved optical waveguides3132. We assume that the lattice is asymptotically homogeneous, i.e. that κn → κ as n → ±∞, and that in the absence of the driving force, i.e. for F(t) = 0, the static Hamiltonian has a purely continuous spectrum, i.e. that hopping inhomogeneities do not introduce bound states, neither outside nor inside the tight-binding energy band (−2κ, 2κ). For example, such a condition is satisfied by assuming κn = κ for n ≠ −1, 0 and κ−1 = κ0 = ρ < κ, see Fig. 1(a). In the presence of the periodic forcing, the eigenstates /ψ(t)〉 of the Hamiltonian are Floquet states of the form , where /u(t + T)〉 = /u(t)〉 and is the quasi-energy, which is assumed to vary in the interval (−ω/2, ω/2). A Floquet BIC state can be defined as a normalizable Floquet state of with a quasi-energy buried in the continuous quasi-energy spectrum of scattered states of the lattice. For a homogeneous lattice (κn = κ), the problem is integrable33 and the quasi-energy spectrum turns out to be purely continuous and defined by the dispersion relation30, where −π ≤ p < π, Γ = aF0/ω is the normalized forcing parameter, and J0 is the Bessel function of first kind and zero order. The two linearly-independent Floquet scattered states with quasi-energy in the ac-driven homogeneous lattice are given by the backward and forward propagating plane waves where and Φ(t) = Γ sin(ωt). As is well known, the quasi-energy spectrum shrinks and collapses as Γ → 2.405, at which J0(Γ) = 0 and dynamic localization (DL) is attained33. Hence, in a homogeneous lattice the role of the external force is to re-normalize the bandwidth of the undriven lattice band, however the spectrum remains purely continuous. Lattice defects or boundary effects make DL imperfect and bound states outside the continuum (BOC) can be induced by the ac field, as discussed in343536. Here we show that, besides BOC states, under certain driving conditions Floquet BIC states can appear as well.

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