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

Chirality is one of the main symmetries of the carbon nanotube geometry which determines the electronic and optical properties of single-walled carbon nanotubes (SWCNTs). SWCNTs are uniquely determined by the chiral vector **C**_{h}, or equivalently by a pair of integer numbers* (n,m)* in the planar graphene hexagonal lattice unit vector basis (Fig. 30).

**Fig. 30.** Schematic representation of a graphene sheet, a_{1}_{ }and a_{2}* are the primitive lattice translation vectors. Carbon nanotubes can be visualised as a graphene sheet rolled up into a cylinder, whereby the one of the 2D Bravais vectors, called also a chiral vector (C*_{h}*=na*_{1}*+ma*_{2}*) is mapped onto the cylinder circumference, **q =**Ð**(a*_{1}*, C*_{h}*) is the chiral angle 0*^{o}*<|**q* |<30^{o}

A primary classification of carbon nanotubes is the presence or the absence of the chiral symmetry. Achiral nanotubes, whose mirror image is superimposable, are subdivided into two classes: zig-zag *(m=0) *and armchair *(m=n) *nanotubes. The rest of the nanotubes belong to the most general class of chiral nanotubes, whose mirror reflection is not superimposable. Chiral molecules exist in two forms that are mirror images of each other (enantiomers). Similarly, carbon nanotubes exist in two (AL) left- and (AR) right-handed helical forms depending on the rotation of two of the three armchair (A) chains of carbon atoms counterclockwise or clockwise when looking against the nanotube z-axis. The two helical forms are shown for illustration in Fig. 31 for a *(5,4) *(a) and *(4,5)* (b) SWCNTs, where *m>n* corresponds to a left-handed (AL) nanotube, whereas* m<n* corresponds to a right-handed (AR) nanotube.

**Fig. 31. ** (5,4) AL-handed (a) and (4,5) AR-handed (b) depending on the rotation of the 2 of the 3 armchair (A) chains of C-atoms to the left (L), or to the right (R) when looking along z-axis.

The electronic band structure of a SWCNT can be described by the quantization of the wave vector around the graphene cylinder. Graphene dispersion is conical around the two points in k-space K_{1} and K_{2}, where the conduction and valence bands meet (Fig. 32).

**Fig. 32. **Graphene dispersion near the K-points in k-space. Horizontal lines on the left panel show the quantized values of **k**_{^}. Lines of allowed **k**_{^} intersect the two cones (red and blue curves), forming sub-bands in the conduction (upper branch) and valence (lower branch) band. The conduction states near K_{1}_{ }are in a minimum of the conduction band and move anticlockwise around the CNT, whereas the valence band states in a valence band maximum move clockwise.

When graphene is wrapped into a cylinder, the electron wave vector, perpendicular to the tube axis is quantized, satisfying the periodic boundary conditions:*k*_{^}*=2 *p m / *L, *m=0, *±1, **±2,...*where *L* is the tube circumference and m is an integer ( m>0 at K_{1}_{ }and m<0 at K_{2}), representing a quasi-angular momentum quantum number. The resulting allowed k-vector values correspond to the horizontal lines in Fig. 32 (left panel). The conic sections of the graphene dispersion cones by allowed k-vector values determine the CNT band structure near the Fermi level, the upper (lower) branches correspond to conduction (valence) states, forming sub-bands, labelled by the quasi-angular momentum quantum number m (Fig. 33)

**Fig. 33. **Energy dispersion vs transverse (along the tube axis) wavevector k_{t}, normalised with respect to the wavevector at the boundary of the Brillouin zone k_{tmax} of a (5,4) SWCNT. The subbands are labelled by the quasi-angular momentum quantum number*m*

The interaction of polarised light with chiral materials in the absence of magnetic fields gives rise to the phenomenon of natural optical (rotation) activity, whereby the polarisation plane is rotated continuously during the light propagation across the medium. Although the phenomenon of optical activity can be treated using classical electromagnetic theory, its in-depth understanding and full description requires a quantum-mechanical treatment at a microscopic level.

Formulation of a theory and model of the optical activity in chiral molecules, such as individual single-wall carbon nanotubes (SWCNTs), in the high-intensity nonlinear coherent regime and under an axial magnetic field, is of special interest from a fundamental point of view. To our knowledge, no such theory has been proposed and very little is known about the ultrafast nonlinear optical and magneto-optical response of a single carbon nanotube. A detailed understanding of the mechanisms underlying the optical and magneto-optical birefringence, circular dichroism and rotation in the nonlinear coherent regime would open up pathways for the development of a novel class of ultrafast polarisation-sensitive optoelectronic devices, based on single carbon nanotubes as the basic functional components of an integrated optoelectronic device.

We have formulated a theory and developed a model, of the optical and magneto-optical activity in chiral molecules, such as individual SWCNTs, in the high-intensity nonlinear coherent regime and under an axial magnetic field.

The majority of the optical experiments on isolated SWCNTs are performed with linearly polarised light in a geometry shown in Fig. 34 where the nanotube lies on a substrate (*x-z*) and the laser beam propagates normal to the substrate *(y)*, and thus normal to the tube axis *(z),* changing the direction of light E-vector from parallel (along *z*) to perpendicular (along *x*). In this geometry, the *E*_{x}* *component is suppressed due to the depolarisation effect (the tube polarisability along the tube axis is much larger than in a perpendicular direction). Therefore the dipole selection rules for light propagating along the tube axis Dm=0 only allow transitions between same sub-band index states (m®m).

**Fig. 34. **Geometry of an optical experiment on isolated SWCNT with linearly polarised laser beam perpendicular to the tibe axis; d is the tube diameter.

It is possible to design the geometry of an experiment, such that the laser beam propagates along the SWCNT axis (*z*). This is particularly relevant e.g. for aggregates or bundles of aligned SWCNTs grown by chemical vapour deposition in an electric field, or aligned in a polymer matrix. When a circularly polarised (in the *x-y* plane) laser pulse propagates along the *z*-axis of an AL or AR SWCNT (Fig. 35 (a)), only one of the two allowed transitions for a linearly polarised light (along *x* or *y*), between the quasi-angular momentum sub-band states m®m-1 and m®m+1, can be excited Fig. 35 (b)). Here we adopt the following convention: the left (s^{-}=x-iy) and right (s^{+}=x+iy) helicity of light corresponds to counter-clockwise and clockwise rotation of the electric field polarisation vector when looking towards the light source (against *z*-axis direction).

**Fig. 35. **(a) Geometry of an experiment with an optical excitation by circularly polarised pulse propagating along the SWCNT z-axis, E_{x}_{ }and E_{y} are the Cartesian components of the E-field vector; (b) 1D electronic density of states (DOS) vs energy at the K-point of the Brillouin zone (*m*>0) of a AL-handed (20,10) chiral SWCNT. The allowed dipole optical transitions for circularly polarised light are designated by arrows for left- (*s* ^{-}) and right- (*s* ^{+}) pulse helicity.

We model the lowest sub-bands of a single chiral nanotube band-edge structure at the K point of the Brillouin zone by an ensemble of identical four-level systems (non-superimposable for the two handednesses), corresponding to the dipole optically allowed transitions for AL and AR nanotube enantiomers. Chirality is modeled by a system Hamiltonian in a four-level basis corresponding to energy-level configurations, specific for each handedness, that are mirror reflections of each other (Fig. 36).

**Fig. 36.** Energy-level structure and allowed dipole optical transitions at the K (K'*) point of the lowest sub-bands labelled by the sub-band index **m** for an (a) AL-handed; (b) AR-handed SWCNT. The fundamental energy gap is shaded. Valence band states below the band gap are populated;**w*_{0}* is the resonant transition frequency and **D** is the energy separation between the lowest valence (conduction)sub-band and the second lowest sub-band near the band gap edge.*

We argue that the difference in the dipole selection rules and the relaxation times involved in the optical transitions for left and right circularly polarised light gives rise to the phenomenon optical activity.

Without loss of generality, we shall consider the special case of a AL-(5,4) SWCNT with the following parameters, calculated by tight-binding method (P_{z}-orbitals):

*Nanotube diameter: 0.611 nm*

*Chiral angle: 26.330*^{°}* *

*Length of unit cell: T = 3.3272 nm*

*Number of hexagons (unit cell) 122*

*Boundary of Brillouin Zone (k*_{zmax}) (m^{-1}) 9.4422*´*10^{8}

*Bandgap Magnitude *

*E*_{m,}_{m} (eV) 1.321, *l*=939 nm

*E*_{m,}_{m±1}* =1.982 eV, **l*=626.5 nm

The simulated structure and the 1D density of states of a *(5,4)* CNT and the optical transitions excited by circularly polarised light are shown in Fig. 37 and Fig. 38.

**F****ig. 37. **Simulation domain: The isolated SWCNT with length L_{n}=500 nm is placed between two free space regions, each 50 nm -long. The source pulse is launched from the left boundary z=0. The pulse duration is *t**=60 fs at an excitation fluence S=20 mJ/m*^{2}.

**Fig. 38. **1D-DOS vs energy, showing the lowest energy sub-bands near the band edge involved in dipole optical transitions induced by circularly polarised light. Only one transition can be excited at any one time by each helicity (*s** *^{- }*and **s*^{+}*)*

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The relaxation dynamics following a resonant optical excitation of m®m±1 transition by s^{+} ( s ^{-}) polarised pulse is shown in Fig. *39. The initial population of the lowest valence states below the band gap,**r*_{11i}= *r*_{33i}*=1/2. *All allowed longitudinal relaxation processes between the levels, associated with population transfer, are designated by wavy lines. G_{1 }is the spontaneous emission (radiative decay) rate of the interband transition |2ñ®|1ñ, which owing to the symmetry is assumed equal to the spontaneous emission rate of |4ñ®|3ñ;G_{2 }is the spontaneous emission rate for the m®m transition allowed for linearly polarised light; G_{3 }is the decay rate of the m-1®m-1 linearly polarised transition and g is the intraband relaxation rate. The transverse relaxation (dephasing) rates G_{m }and G_{m-1}_{ }are designated by arrows. The relaxation times are estimated from the spontaneous emission rate of a dipole transition, the intraband relaxation rate g=130 fs is taken from ultrafast experiments on individual SWCNTs. The effective dielectric constant, and hence the effective refractive index, are calculated using effective medium theory and accounting for the anisotropy, yielding a value for n=2.3. The dipole matrix element for optical transitions excited by circularly polarised light in SWCNT is estimated from the radiative lifetime of an e-h pair ~10 ns, giving a valueÃ~3.62´10^{-29 }Cm.

Fig, 39. Schematic representation of the asymmetry in the relaxation dynamics following (a) s ^{-} -pulse excitation and (b) s^{+ }-pulse excitation^{ }

The spatial dynamics of the electric field vector components, Ex and Ey of the circularly polarised s ^{-} - pulse (Fig. 39 (a)) and the induced level population dynamics along the nanotube length, following a resonant pi-pulse excitation with pulse duration t=60 fs, corresponding to initial electric field vector amplitude *E*_{0}=6.098*´*10^{8}* Vm*^{-1}** **is shown in the following movie (**click on the figures below)**:

The corresponding temporal dynamics is shown in the next movie:

The time evolution of the electric field vector components and the population of all four levels, corresponding to Fig. 39 (a), are shown at four locations along the tube length in Fig. 40.

**Fig. 40. **Time evolution of the electric field vector components E_{x}, E_{y} and the populations of all four levels in Fig.39 (a) under left circularly (s ^{-}) *polarised resonant Gaussian optical pulse excitation at four different locations along the nanotube axis (a) z=175nm; (b) z=300 nm; (c) z=425nm; (d) z=550 nm, measured from the left boundary z=0. The relaxation times are as follows: **G*_{1}*= 2.91 ns*^{-1}, *G*_{2}*= 9.81 ns*^{-1},*G*_{3}*= 1.23ns*^{-1}*, **g * = 130 fs^{-1}, *G*_{m}= 0.8 ps^{-1}, *G*_{m-1}*= 1.6 ps*^{-1}; pulse temporal width *t**=60 fs, initial electric field amplitude E*_{0}=6.098*´*10^{8}* Vm*^{-1} ;resonant wavelength:*l*_{0}*=626.5 nm, E*_{m,}_{m}_{±1}*=1.9815 eV ; density of resonant absorbers N*_{a}=6.811*´*10^{24 }m^{-3}, normalised to give single CNT within the simulation volume.

The spatial dynamics of the electric field vector components, *E*_{x} and *E*_{y}* *of the *s* ^{+ }circularly polarised pulse (Fig. 39 (b)) and the induced level population dynamics along the nanotube length, following a resonant p-pulse excitation is shown in the following movie:

The corresponding spatial and temporal dynamics following a *s* ^{+ }circularly polarised pulse is shown in the next movies **(click on the figures below)**:

Snapshots of the time evolution of the electric field vector components and the population of all four levels, corresponding to Fig. 39 (b), are shown at four locations along the tube length in Fig. 41.

Fig. 41. Time evolution of the electric field vector components Ex, Ey and the populations of all four levels in Fig.39 (a) under left circularly

From experimental point of view, it would be of particular interest to be able to distinguish between the spectra of the transmitted pulses at the output facet of the simulated domain for each helicity of the ultrashort excitation pulse.

We demonstrate that nanotube chirality can be determined from the transmission spectra of ultrafast pulses of both helicities, shown in Fig. 42.

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**Fig. 42. **Transmission (Fourier) spectra vs wavelength of E_{x} component of (a) input (blue curve) and output (red curve)

A comparison between Fig. 42 (a) and (b) reveals nearly an order of magnitude higher intensity of the peak, corresponding to the s ^{+ }excitation* *. The difference in the ultrafast transient response can be exploited in an experiment aiming to unambiguously determine the chirality of a single carbon nanotube by using ultrafast circularly polarised pulses of both helicities.

Rotation of the polarisation plane during the pulse propagation along the nanotube axis is numerically demonstrated by linearly polarised pulse excitation in Fig. 43.

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***Fig. 43. **(A – top panel) Time evolution of the electric field vector components E_{x} and E_{y} and the populations of all four levels under linearly polarised optical pulse excitation at four different locations along the nanotube axis (a) z=175nm; (b) z=300 nm; (c) z=425 nm; (d) z=550 nm ; measured from the left boundary z=0. The initial population is assumed equally distributed between the lower-lying levels. (B-bottom panel) *Expanded view of (A), clearly showing the build-up with time of the electric field vector E*_{y} component, equivalent to rotation of the electric field vector.

To quantify the optically-induced circular dichroism, we calculate the spacially resolved along the nanotube axis absorption/gain coefficient, shown in Fig. 44, leading to an average value of 0.083 mm^{-1}.

**Fig. 44. **Spatially resolved calculated gain coefficient per micron vs wavelength for E

The phase shift difference introduced by the nonsymmetric system response under *s* ^{+} (transition *|1ñ®|2ñ*) and *s* ^{-} (transition *|3ñ®|4ñ*)) circularly polarised pulse excitation represents a measure for the rotation angle. The spatially resolved calculated phase shift is shown in Fig. 45.

**Fig. 45.** Spatially resolved calculated phase shift vs wavelength for E_{x} (red) and E_{y} (green dot) components under *s** *^{+}* excitation and for E*_{x} (magenta line) and E_{y} (cyan dot) under *s** *^{-}*circularly polarised ultrashort optical excitation and the theoretical phase shift of a homogeneously broadened resonant two-level system (blue line).*

The average specific rotatory power *∼** *2962*.*24°*/*mm of this particular (5,4) nanotube exceeds the giant gyrotropy reported in the artificial photonic metamaterials of up to 2500°*/*mm. We should note, however, that the complexity of the carbon nanotube molecular structure allows for engineering the optical activity in a wide range. By comparison, the optical activity of e.g. quartz illuminated by the D line of sodium light l=589.3 nm, is 21.7° /mm, implying circular birefringence, or refractive indices difference ~7.1´10^{-5}; the specific rotatory power of cinnabar (HgS) is 32.5 ° /mm. A comparison of the specific rotatory power for a group of crystals shows a wide range of variation from 2.24 ° /mm for NaBrO_{3} to 522° /mm for AgGaS_{2}. Liquid substances exhibit much lower values of specific rotatorypower, e.g. -0.37° /mm for turpentine (T=10°, l=589.3 nm);1.18 ° /mm for corn syrup, etc. Cholesteric liquid crystals and sculptured thin films exhibit large rotatory power in the visible spectrum ~1000° /mm, and~6000° /mm, respectively.