Nonlinear coherent magneto-optical response of a single chiral carbon nanotube

Slavcheva, G; Roussignol, Philippe

doi:10.1088/1367-2630/12/10/103004

Cited by 6 publications

(10 citation statements)

References 63 publications

(87 reference statements)

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“…Furthermore, considering the trace condition

Tr {\hat{ρ}} = 1

, the density matrix can be represented by

N^{2} - 1

nonredundant, real‐valued elements, which are conveniently written as a vector

boldS

. This nonredundant representation is, for example, achieved by the coherence vector (or pseudospin) representation, which has also been found useful for numerically efficient implementations of the MB equations . For this purpose, the density matrix operator

\overset{ρ}{true}

is composed as

\hat{ρ} = \frac{1}{N} \hat{I} + \frac{1}{2} \sum_{j = 1}^{N^{2} - 1} S_{j} {\overset{s}{true}}_{j}

Here,

{\hat{s}}_{j}

are generators of the Lie algebra of SU( N ) which are traceless Hermitian operators fulfilling the condition

Tr false{{\hat{s}}_{j} {\hat{s}}_{k} false} = 2 δ_{j k}

, and

\overset{I}{true}

is the identity operator.…”

Section: Optical Bloch Equationsmentioning

confidence: 88%

“…Furthermore considering the trace condition Tr {ρ} = 1, the density matrix can be represented by N 2 − 1 non-redundant, real-valued elements, which are conveniently written as a vector S. This non-redundant representation is for example achieved by the coherence vector (or pseudospin) representation, 150 which has also been found useful for numerically efficient implementations of the MB equations. 53,[151][152][153] For this purpose, the density matrix operator ρ is composed as…”

Section: Non-redundant Density Matrix Representationmentioning

confidence: 99%

“…302 The Gaussian distribution, Eq. (153), is also frequently used as a generic model if the distribution of resonance frequencies is not exactly known, e.g., to describe above mentioned inhomogeneous broadening in ensembles of quantum dots due to size fluctuations. 303 If the individual quantum systems contain more than one relevant optical transition with distributed resonance frequencies, in principle joint distribution functions have to be used.…”

Section: B Inhomogeneous Broadeningmentioning

confidence: 99%

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Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations

Jirauschek

Riesch

Tzenov

2019

Advcd Theory and Sims

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Due to their intuitiveness, flexibility, and relative numerical efficiency, the macroscopic Maxwell-Bloch (MB) equations are a widely used semiclassical and semi-phenomenological model to describe optical propagation and coherent light-matter interaction in media consisting of discrete-level quantum systems. This review focuses on the application of this model to advanced optoelectronic devices, such as quantum cascade and quantum dot lasers. The Bloch equations are here treated as a density matrix model for driven quantum systems with two or multiple discrete energy levels, where dissipation is included by Lindblad terms. Furthermore, the 1D MB equations for semiconductor waveguide structures and optical fibers are rigorously derived. Special analytical solutions and suitable numerical methods are presented. Due to the importance of the MB equations in computational electrodynamics, an emphasis is placed on the comparison of different numerical schemes, both with and without the rotating wave approximation. The implementation of additional effects which can become relevant in semiconductor structures, such as spatial hole burning, inhomogeneous broadening, and local-field corrections, is discussed. Finally, links to microscopic models and suitable extensions of the Lindblad formalism are briefly addressed. of a lower bandgap material than the adjacent layers, which restricts the free electron motion in that layer to the in-plane directions and gives rise to quantized energy states in growth direction. As a consequence of the further restriction of the energy spectrum and the even stronger carrier localization, additional improvement can be expected from 2D or 3D confinement, resulting in quantum wire/dash and quantum dot (QD) structures, respectively. Indeed, QD [1][2][3] and quantum dash [4] lasers and laser amplifiers have been shown to exhibit excellent characteristics. In Figure 1, the formation of quantized states in quantum wells, wires, and dots is schematically illustrated. The term quantum dash refers to an elongated nanostructure, that is, some kind of short quantum wire. By contrast, the term nanowire does not necessarily indicate strong quantum confinement. For example, in nanowire lasers, the nanowire geometry typically serves as a single-mode optical waveguide resonator, while the active region is based on a heterostructure or quantum well, as in a conventional laser diode. [5,6] Semiconductor optoelectronic devices usually rely on electron-hole recombination, that is, optical transitions between conduction and valence band states. The associated resonance wavelength is largely determined by the semiconductor bandgap, which establishes a lower bound on the transition energy. Thus, the coverage of a certain spectral region depends on the existence of suitable semiconductor materials, which for example restricts the availability of practical optoelectronic sources and detectors in the mid-infrared and terahertz regions. An alternative concept is based on intersubband devices, which employ so-called i...

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“…Furthermore, considering the trace condition

Tr {\hat{ρ}} = 1

, the density matrix can be represented by

N^{2} - 1

nonredundant, real‐valued elements, which are conveniently written as a vector

boldS

\overset{ρ}{true}

is composed as

\hat{ρ} = \frac{1}{N} \hat{I} + \frac{1}{2} \sum_{j = 1}^{N^{2} - 1} S_{j} {\overset{s}{true}}_{j}

Here,

{\hat{s}}_{j}

are generators of the Lie algebra of SU( N ) which are traceless Hermitian operators fulfilling the condition

Tr false{{\hat{s}}_{j} {\hat{s}}_{k} false} = 2 δ_{j k}

, and

\overset{I}{true}

is the identity operator.…”

Section: Optical Bloch Equationsmentioning

confidence: 88%

Section: Non-redundant Density Matrix Representationmentioning

confidence: 99%

Section: B Inhomogeneous Broadeningmentioning

confidence: 99%

See 1 more Smart Citation

Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations

Jirauschek

Riesch

Tzenov

2019

Advcd Theory and Sims

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“…Currently, only non-optical techniques, such as scanning tunnelling microscopy and transmission electron microscopy, can 4 / 20 provide high-resolution images for handedness identification of individual carbon nanotubes 43,44 , whereas they have proven to be unsuitable for scalable and routine characterization due to their limited accessibility and low efficiency. Till now, the handedness of single nanotubes remains the only structural degree of freedom to be determined by optical means, and the development of sophisticated handedness-related applications of single nanotubes has been greatly hindered [45][46][47] .…”

Section: / 20mentioning

confidence: 99%

Complete structural characterization of single carbon nanotubes by Rayleigh scattering circular dichroism

Yao

Liu

et al. 2021

Nat. Nanotechnol.

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The accurate characterization of the chiral indices (n,m) of carbon nanotubes has greatly facilitated fundamental investigations and practical applications ranging from electronic circuits and quantum emission to biological detection 1-4 . To meet the growing miniaturizationdriven demand, handedness, the key structural quantity associated with mirror symmetry breaking, needs to be identified both accurately and efficiently 5-7 . To date, optical spectroscopic techniques with unprecedented high-throughput and noninvasive characteristics have achieved great success in identifying chiral indices even at the single-tube level 8-13 . However, none of these optical methods are capable of handedness characterization for single nanotubes due to their extremely weak chiroptical signals (~10 -7 ) compared to the excitation light 14,15 . Here, we demonstrate the complete structure identification of single nanotubes in terms of both chiral indices and handedness by Rayleigh scattering circular dichroism. The success originates from the background-free feature of Rayleigh scattering collected at an oblique angle, which enhances the chiroptical signal from nanotubes by three to four orders of magnitude compared to that using conventional absorption circular dichroism. We measured a total of 30 single-walled carbon nanotubes (including both semiconducting and metallic nanotubes) and found that their absolute chiroptical signals show an obvious structure dependence, which can be qualitatively understood through tightbinding calculations. Our strategy not only opens up exciting opportunities for unlocking the sophisticated functionality of nanotubes but also provides a new platform for chiral discrimination and chiral device exploration at the level of individual nanomaterials.

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“…In the following analysis we concentrate on pairs of subbands with m = 4 separated by the energy gap E g / = 5.12 fs −1 , see Table 1. We assume that the CNT is illuminated by two consecutive laser pulses producing electric fields oscillating in the x [29,30] (first box), laser pulses parameters after [31] (second box), frequencies corresponding to the gap energies obtained from the tight-binding theory (third box), electron relaxation time (model parameter, see [32]) and empirical decoherence time after [33].…”

Section: Zitterbewegung In Carbon Nanotubesmentioning

confidence: 99%

Two-photon echo method for observing electron zitterbewegung in carbon nanotubes

Rusin¹,

Zawadzki²

2014

Semicond. Sci. Technol.

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The phenomenon of Zitterbewegung (ZB, trembling motion) of electrons is described in zigzag carbon nanotubes (CNT) excited by laser pulses. The tight binding approach is used for the band structure of CNT and the effect of light is introduced by the vector potential. Contrary to the common theoretical practice, no a priori assumptions are made concerning electron wave packet, the latter is determined as a result of illumination. In order to overcome the problem of various electron phases in ZB, a method of two-photon echo is considered and described using the density function formalism. The medium polarization of CNT is calculated by computing exact solutions of the time-dependent electron Hamiltonian. The signal of two-photon echo is extracted and it is shown that, using existing parameters of CNT and laser pulses, one should be able to observe the electron trembling motion. Effects of electron decoherence and relaxation are discussed.

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Nonlinear coherent magneto-optical response of a single chiral carbon nanotube

Cited by 6 publications

References 63 publications

Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations

Optoelectronic Device Simulations Based on Macroscopic Maxwell–Bloch Equations

Complete structural characterization of single carbon nanotubes by Rayleigh scattering circular dichroism

Two-photon echo method for observing electron zitterbewegung in carbon nanotubes

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