Coherent MagnetoOptical Polarisation Dynamics in a Single Chiral

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Coherent Magneto-Optical Polarisation Dynamics in a Single Chiral Carbon Nanotube Gaby Slavcheva 1 and

Coherent Magneto-Optical Polarisation Dynamics in a Single Chiral Carbon Nanotube Gaby Slavcheva 1 and Philippe Roussignol 2 1 The Blackett Laboratory, Imperial College London, United Kingdom 2 Laboratoire Pierre Aigrain, Ecole Normale Supérieure, Paris, France PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Motivation ● Fundamental point of view ØFormulation of a theory and model of the

Motivation ● Fundamental point of view ØFormulation of a theory and model of the magneto-optical activity in chiral molecules (SWCNTs) in the nonlinear coherent regime How the chirality affects the ultrafast nonlinear optical and magnetooptical response? ● Novel class of ultrafast polarisation-sensitive integrated optoelectronic devices, based on SWCNTs ● Time-resolved magnetic circular dichroism (MCD) and magnetooptical rotatory dispersion (MORD) techniques provide spectroscopic information, different or impossible to obtain by other means: e. g. TRFaraday rotation for spin dynamics ● Chiral materials exhibit negative refractive index: Ø artificial chiral negative refractive index metamaterials: exhibit giant gyrotropy ØCNTs: promising candidates in the visible range PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Outline ●Relationship between chiral symmetry and optical activity ●Theoretical framework for description of the

Outline ●Relationship between chiral symmetry and optical activity ●Theoretical framework for description of the natural optical activity in a chiral SWCNT in the nonlinear coherent regime ●Simulation results for the ultrafast nonlinear dynamics of the natural optical activity in chiral SWCNTs Ø time-resolved circular dichroism Ø time-resolved circular birefringence and rotatory power ● Model of the Faraday effect in SWCNTs in an axial B Ø Zeeman splitting Ø Aharonov-Bohm flux ● Simulation results for nonlinear Faraday rotation ● Summary and conclusions PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

SWCNT with chiral symmetry Primary classification of nanotubes: achiral (superimposable mirror image): zig-zag and

SWCNT with chiral symmetry Primary classification of nanotubes: achiral (superimposable mirror image): zig-zag and armchair ● chiral (non-superimposable) Chiral vector: Chiral angle: AL-handed or AR-handed SWCNT: depending on the rotation of 2 of the 3 armchair (A) chains of C-atoms to the L or R when looking against z: AL (5, 4) AR (4, 5) PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Electronic band structure of SWCNT L=|Ch| - tube circumference - quasiangular momentum quantum number

Electronic band structure of SWCNT L=|Ch| - tube circumference - quasiangular momentum quantum number Graphene dispersion -2 -1 PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Optical dipole transitions for circularly polarised light Geometry of optical experiments on isolated SWCNTs

Optical dipole transitions for circularly polarised light Geometry of optical experiments on isolated SWCNTs Linear Depolarisation: Ex polarisation suppressed E||=Ez 1 D electronic density of states at the K-point > 0 E =Ex Dipole selection rules: m=0 m=± 1 both -1 and +1 symmetry allowed m=± 1 only one Circular polarisation: transition -1 or +1 PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Energy-level structure at the point K (K′) of the lowest subbands AR-handed SWCNT AL-handed

Energy-level structure at the point K (K′) of the lowest subbands AR-handed SWCNT AL-handed SWCNT Non-superimposable energy-level diagrams Absorption of σ + light induces -1 ( +1 ) transition in AL-handed SWCNT Absorption of σ + light induces -1 transition in AR-handed SWCNT Difference in dipole selection rules for L and R circularly polarised light gives rise to optical activity Samsonidze et al. , Phys. Rev. B 69, 205402 (2004) PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Energy dispersion and 1 D DOS of a AL-(5, 4) SWCNT Nanotube diameter 0.

Energy dispersion and 1 D DOS of a AL-(5, 4) SWCNT 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 (kzmax) (m-1) 9. 4422 e+08 Bandgap Magnitude E , (e. V) 1. 321, =939 nm E , ± 1 =1. 982 e. V, =626. 5 nm pulse duration = 60 fs, excitation fluence S=20 m. J/m 2 J. -S. Lauret, C. Voisin, G. Cassabois, C. Delalande, Ph. Roussignol, O. Jost, and L. Capes, Phys. Rev. Lett. 90, 057404 (2003) PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Dielectric response function and optical dipole matrix element ● Effective medium theory: Dipolarisability of

Dielectric response function and optical dipole matrix element ● Effective medium theory: Dipolarisability of a SWCNT m per unit length in a quasistatic approximation (m=1) Ø Introduce equivalent isotropic dielectric function of a solid cylinder with radius R y Lü et al. , Phys. Rev. B 63, 033401 (2000), Henrard and Lambin. J. Phys. B 29, 5127 (1996) x 2 r y x 2 R 2 R z z Ø ordinary and extraordinary ray in graphite no=2. 64, ne=2. 03, n. SWCNT= =2. 3 · Estimate of the dipole matrix element for optical transitions excited by circularly polarised light Ø Upper and lower bounds from the extension of the effective mass method applied to chirality effects in CNTs(coupling between the orbital momentum k ( ) and kz ) ~10 -31 - 10 -29 Cm ØEstimate from radiative lifetime of an e-h pair spont ~10 ns *: ~ 3. 579× 10 -29 Cm Ivchenko and Spivak, Phys. Rev. B 66, 155404 (2002) *Wang et al. , Phys. Rev. Lett. 92, 177401 (2000) PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Theoretical formalism Dynamical evolution of an N-level quantum system Pseudospin equation for the real

Theoretical formalism Dynamical evolution of an N-level quantum system Pseudospin equation for the real state coherence vector S =(S 1, S 2, . . . , S ) (Heisenberg picture)*: Liouville equation (Schrödinger picture): N N-1 i N N-1 i 3 2 3 2 Using Gell-Mann’s -generators of the SU(N) Lie algebra: 1 1 (E=0) torque vector *Hioe and Eberly PRL, 47, 838, 1981 PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Optical excitation of ± 1 transition by +(-)-polarised pulse N=4, SU(4) Lie group =60

Optical excitation of ± 1 transition by +(-)-polarised pulse N=4, SU(4) Lie group =60 fs -pulse System Hamiltonian Rabi frequencies torque vector Relaxation times estimated from spontaneous emission rate 1= 2. 91 ns-1, 2= 9. 81 ns-1, 3= 1. 23 ns-1, = 130 fs-1, = 0. 8 ps-1, -1= 1. 6 ps-1 Gaussian -pulse with =60 fs: E 0=6. 098 108 Vm-1 ; Resonant wavelength: 0=626. 5 nm, E , ± 1=1. 9815 e. V Density of resonant absorbers Na=6. 811 1024 m-3 PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Master equation for resonant excitation by + (-) pulse Maxwell curl equations Medium polarisation

Master equation for resonant excitation by + (-) pulse Maxwell curl equations Medium polarisation Source optical field Pseudospin equations Finite-Difference Time-Domain (FDTD) solution: timestepping algorithm with predictor-corrector iterative scheme Slavcheva, Phys. Rev. B 77, 115347 (2008) PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Spatially resolved temporal dynamics for - and + excitations d z=0 - (a) z=z

Spatially resolved temporal dynamics for - and + excitations d z=0 - (a) z=z 1 + (a) - 50 nm z 1 500 nm z 2 z 3 z 4 50 nm z=L - (b) - (c) - (d) z=z 2 z=z 3 z=z 4 + (b) + + (c) z + (d) Chirality determination from the ultrafast nonlinear response using ultrashort pulses with both helicities PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Time evolution of a linearly polarised pulse Rotation of the polarisation plane during the

Time evolution of a linearly polarised pulse Rotation of the polarisation plane during the pulse propagation Source optical field z=z 1 Transmission spectra of Ex, Ey at the output facet vs z=z 4 PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Spatially resolved gain coefficient spectra for - and +- pulse Natural circular dichroism Theor.

Spatially resolved gain coefficient spectra for - and +- pulse Natural circular dichroism Theor. Value* ~ 1. 03 m-1 ; Experiment (artificial helicoidal bilayer)**: 1. 15 -2. 07 m-1 *Ivchenko and Spivak, Phys. Rev. B 66, 155404 (2002); **Rogacheva et al. , Phys Rev. Lett. 97, 177401 (2006) PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Spatially resolved phase shift spectra for - and +- pulse Comparison with rotatory power

Spatially resolved phase shift spectra for - and +- pulse Comparison with rotatory power of birefringent materials: Na. Br. O 3 = 2. 24 /mm quartz 21. 7 /mm, |n. L-n. R|=7. 1 10 -5 Cinnabar (Hg. S) 32. 5 /mm Ag. Ga. S 2 522 /mm Liquid substances: Turpentine -0. 37 /mm Corn syrup 1. 18 /mm Cholesteric liquid crystals ~1000 /mm, Artificial photonic metamaterials: ~ 2500 /mm Sculptured thin films ~ 6000 /mm Specific rotatory power : Circular birefringence: PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Single chiral CNT in an axial magnetic field Magnetic energy bands H. Ajiki and

Single chiral CNT in an axial magnetic field Magnetic energy bands H. Ajiki and T. Ando, J. Phys. Soc. Jap. 62, 1255 (1993) Jiang et al. , PRB 62, 13209 (2000); Minot et al. , Nature 428, 536 (2004) PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Single chiral CNT in an axial magnetic field Energy-level structure significantly modified: ● Zeeman

Single chiral CNT in an axial magnetic field Energy-level structure significantly modified: ● Zeeman splitting Type I tube: n-m=3 q, q integer B=8 T: E Z~ 0. 46 me. V, z~7. 03 1011 rad/s ● Orbital effects - Aharonov-Bohm phase due to the flux through the tube uniform shift in the energy levels ● Energy band gap oscillates with B (or magnetic flux ), / 0=0. 00057 Type II tube: n-m=3 q± 1, q integer Energy gap shift (band gap reduction): B= 8 T, EAB ~ 3. 37 me. V, AB~ 5. 12 1012 rad/s Band gap renormalisation (K-point) : 0 0 - AB H. Ajiki and T. Ando, J. Phys. Soc. Jap. 62, 1255 (1993) Jiang et al. , PRB 62, 13209 (2000); PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Original (B=0) and reduced energy-level systems in an axial magnetic field Jz=+1/2 4 Jz=+3/2

Original (B=0) and reduced energy-level systems in an axial magnetic field Jz=+1/2 4 Jz=+3/2 ” B Jz= -1/2 B 1 1 B 3 E=0 1 1’ 3” 4’ Jz= -1/2 3’ B B Jz=+3/2 l=1 Jz=+1/2 3 B 1 B 2 B 0 1 3 B 2 Eg 0 1” Jz=+1/2 2’ l=0 l=1 4” z Jz=+1/2 PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010 l=0 3

Theoretical Formalism 2’ Jz=+1/2 2 B Jz=+1/2 3 B 1 B Jz= -1/2 4’

Theoretical Formalism 2’ Jz=+1/2 2 B Jz=+1/2 3 B 1 B Jz= -1/2 4’ 3’ 2” 1” Jz= -1/2 Jz=+1/2 3 B 2 B Jz=+3/2 1’ 4” 3 -level -system System Hamiltonian Jz=+1/2 1 B 3” Torque vector Polarisation vector components 1= 2. 91 ns-1, 2= 9. 79 ns-1, 3= 9. 77 ns; = 130 fs-1, = 0. 8 ps-1, -1= 1. 6 ps-1 E 0=6. 098 108 Vm-1, Eres= ( 0 - AB-2 z) , B= 8 T: *coup=3. 62054 10 -29 Cm , Na=6. 811 1024 m-3 Ivchenko and Spivak, Phys. Rev. B 66, 155404 (2002) PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Simulation results for Faraday rotation Spatially resolved absorption/gain coefficient spectra for - and +-

Simulation results for Faraday rotation Spatially resolved absorption/gain coefficient spectra for - and +- pulse at B=8 T Absorption dip at resonance for + excitation Magnetic circular dichroism PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Spatially resolved phase shift spectra for - and +- pulse at B=8 T Double-peaked

Spatially resolved phase shift spectra for - and +- pulse at B=8 T Double-peaked phase shift curve at resonance for + excitation Specific rotatory power : Magneto-chiral effect PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Summary · Dynamical model proposed of the optical activity and the Faraday effect of

Summary · Dynamical model proposed of the optical activity and the Faraday effect of a SWCNT in the nonlinear coherent regime Provided an estimate for the dielectric response function and dipole matrix element for circularly polarised light in a single CNT SWCNT handedness determined by optical spectroscopy using circularly and linearly polarised light Giant natural gyrotropy demonstrated (~ 3000 /mm) in a (5, 4) SWCNT Model of nonlinear Faraday rotation in a single chiral CNT Enhancement of magneto-chiral circular dichroism and rotatory power in an external B Method valid for an arbitrary nanotube chirality and pulse polarisation; Valid for ultrashort optical pulses and arbitrary pulse shape (including cw) Outlook: study of the rotation angle dependence on chirality with possibility of engineering rotatory power; study of the B-field dependence of the specific rotation angle · · · · PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

Acknowledgements G. Bastard R. Ferreira C. Flytzanis C. Voisin, LPA, ENS, Paris Thank you

Acknowledgements G. Bastard R. Ferreira C. Flytzanis C. Voisin, LPA, ENS, Paris Thank you for your attention! PLMCN 10, Cuernavaca (Mexico), 12 -16 April 2010

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