Floquet-Bloch theory of high-harmonic generation in periodic structures

Faisal, Farhard; Kaminski, J. Z.

doi:10.1103/physreva.56.748

Cited by 169 publications

(118 citation statements)

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“…An important example is provided by the Floquet-Bloch states 11 , which emerge in solids owing to a coherent interaction between Bloch states inside the solid and a periodic driving potential. This is a consequence of the Floquet theorem 12 , which states that a Hamiltonian periodic in time with period T has eigenstates that are evenly spaced by the drive energy (2π/T ).…”

mentioning

confidence: 99%

Selective scattering between Floquet–Bloch and Volkov states in a topological insulator

Mahmood¹,

Chan²,

Alpichshev³

et al. 2016

Nature Phys

337

273

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The coherent optical manipulation of solids is emerging as a promising way to engineer novel quantum states of matter 1-5 . The strong time-periodic potential of intense laser light can be used to generate hybrid photon-electron states. Interaction of light with Bloch states leads to Floquet-Bloch states, which are essential in realizing new photo-induced quantum phases [6][7][8] . Similarly, dressing of free-electron states near the surface of a solid generates Volkov states, which are used to study nonlinear optics in atoms and semiconductors 9 . The interaction of these two dynamic states with each other remains an open experimental problem. Here we use time-and angle-resolved photoemission spectroscopy (Tr-ARPES) to selectively study the transition between these two states on the surface of the topological insulator Bi 2 Se 3 . We find that the coupling between the two strongly depends on the electron momentum, providing a route to enhance or inhibit it. Moreover, by controlling the light polarization we can negate Volkov states to generate pure Floquet-Bloch states. This work establishes a systematic path for the coherent manipulation of solids via light-matter interaction.The manipulation of solids using ultrafast optical pulses has opened up a new paradigm in condensed matter physics by allowing the study of emergent physical properties that are otherwise inaccessible in equilibrium 1,2,10 . An important example is provided by the Floquet-Bloch states 11 , which emerge in solids owing to a coherent interaction between Bloch states inside the solid and a periodic driving potential. This is a consequence of the Floquet theorem 12 , which states that a Hamiltonian periodic in time with period T has eigenstates that are evenly spaced by the drive energy (2π/T ). Floquet-Bloch states have generated a lot of interest recently both for realizing exotic states of matter such as a Floquet Chern insulator 7 , as well as understanding non-equilibrium periodic thermodynamics 13,14 . Experimental observation of these states requires the measurement of the transient electronic band structure of a crystal as it is perturbed by light. As has recently been demonstrated 15 in the topological insulator Bi 2 Se 3 , time-and angleresolved photoemission spectroscopy (Tr-ARPES) is a key tool that can achieve this. Characteristic signatures of Floquet-Bloch states in the Tr-ARPES spectra include replicas of the original band structure that are separated by the driving photon energy 15 .In addition to Floquet-Bloch states, light can also generate other coherent phenomena in solids [16][17][18] . In particular, it can dress freeelectron states near the surface of a solid (Fig. 1a), as the surface can provide the momentum conservation necessary for a photon to interact with a free electron. This dressing was first observed in time-resolved photoemission experiments 17 and has subsequently been referred to as laser-assisted photoemission (LAPE). LAPE is typically understood [19][20][21] by invoking the Volkov solution, which is an e...

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mentioning

confidence: 99%

Selective scattering between Floquet–Bloch and Volkov states in a topological insulator

Mahmood¹,

Chan²,

Alpichshev³

et al. 2016

Nature Phys

337

273

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“…Next, we allow for an oscillating electric field, at frequency ω, either due to an external electromagnetic radiation, or induced by periodically driving a voltage difference between a back and top gate enclosing the wire array. Driving at resonance across the band gap, that is with ℏω ¼ Δ g , opens a dynamical gap [87,88], essential for inducing nontrivial topology [52,57,89]. Using the Floquet representation [90][91][92], this gap arises as a splitting of degeneracies in the quasienergy spectrum of the Floquet operator.…”

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confidence: 99%

Topological Floquet Phases in Driven Coupled Rashba Nanowires

2016

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We consider periodically driven arrays of weakly coupled wires with conduction and valence bands of Rashba type and study the resulting Floquet states. This nonequilibrium system can be tuned into nontrivial phases such as topological insulators, Weyl semimetals, and dispersionless zero-energy edge mode regimes. In the presence of strong electron-electron interactions, we generalize these regimes to the fractional case, where elementary excitations have fractional charges e=m with m being an odd integer. DOI: 10.1103/PhysRevLett.116.176401 Introduction.-Topological effects in condensed matter systems have attracted attention for many years. From the quantum Hall effect over topological insulators (TIs) [1][2][3][4][5][6][7][8][9][10][11][12] and Weyl semimetals [13][14][15][16][17], to Majorana fermions and parafermions [39][40][41][42][43][44][45][46][47][48][49], the interest is driven both by fundamental physics and the promise for topological quantum computation. Despite the many proposals for topological systems, the search for the most optimal material still continues unabated.While most studies were focused on static structures, it has recently been proposed to extend topological phases to nonequilibrium systems, described by Floquet states . Remarkably, this approach no longer relies on given material properties, such as strong spin orbit interactions (SOIs), typically necessary for reaching topological regimes, but instead allows one to turn initially nontopological materials such as graphene [50] and non-bandinverted semiconducting wells into TIs [52] by applying an external driving field.An even bigger challenge is to describe topological effects that involve fractional excitations. This requires the presence of strong electron-electron interactions. However, given the difficulties in the search for conventional TIs, it would be even more surprising to expect such phases to occur naturally. Moreover, even if they existed, twodimensional (2D) systems with electron-electron interactions are difficult to describe analytically and often progress can come only from numerics [61].Here, we circumvent this difficulty by considering strongly anisotropic 2D systems [71][72][73][74] formed by weakly coupled Rashba wires (see Fig. 1), where each of them can be treated as a one-dimensional Luttinger liquid by bosonization [75][76][77][78][79][80][81][82][83][84][85][86]. This will allow us to introduce the Floquet version not only of TIs but also of Weyl semimetals in driven 2D systems. Importantly, in this way we can also address fractional regimes and are able to obtain the Floquet version of fractional TIs and Weyl semimetals.

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“…[10][11][12] Combining the effects of a strong, time-periodic driving field, with the nonlinearity of the Bloch oscillations leads to higher harmonic generation of the driving frequency. [13][14][15][16] This effect has been observed in semiconductor superlattices driven at terahertz frequencies with a free-electron laser, 17 and more recently in bulk ZnO crystals strongly driven by a pulsed infrared laser, 18 and in bulk GaSe crystals driven by short 30 THz pulses. 19 The application of the driving field in short, few-cycle pulses was necessary to ensure that the absorbed energy could be transferred to the lattice and dissipated.…”

Section: Introductionmentioning

confidence: 87%

Continuous third harmonic generation in a terahertz driven modulated nanowire

Hamilton

Kovalev

et al. 2015

Journal of Applied Physics

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We consider the possibility of observing continuous third-harmonic generation using a strongly driven, single-band one-dimensional metal. In the absence of scattering, the quantum efficiency of frequency tripling for such a system can be as high as 93%. Combining the Floquet quasienergy spectrum with the Keldysh Green's function technique, we derive a semiclassical master equation for a one-dimensional band of strongly and rapidly driven electrons in the presence of weak scattering by phonons. The power absorbed from the driving field is continuously dissipated by phonon modes, leading to a quasi-equilibrium in the electron distribution. We use the Kronig-Penney model with varying effective mass to establish the growth parameters of an InAs/ InP nanowire near optimal for third harmonic generation at terahertz frequency range. V C 2015 AIP Publishing LLC. [http://dx

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Floquet-Bloch theory of high-harmonic generation in periodic structures

Cited by 169 publications

References 35 publications

Selective scattering between Floquet–Bloch and Volkov states in a topological insulator

Selective scattering between Floquet–Bloch and Volkov states in a topological insulator

Topological Floquet Phases in Driven Coupled Rashba Nanowires

Continuous third harmonic generation in a terahertz driven modulated nanowire

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