Nonlinear and quantum optics of twodimensional

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1 Nonlinear and quantum optics of twodimensional systems Alexey Belyanin Department of Physics and Astronomy Texas A&M University Xianghan Yao, Ryan Kutayah, and Yongrui Wang Texas A&M University Collaborations: Mikhail Tokman, Russian Academy of Sciences Dmitry Revin University of Sheffield Tony Heinz, Stanford University Manfred Helm, Dresden

2 Work on the project in 2016 Rigorous quantum mechanical theory of the secondorder optical nonlinearities of massless Dirac fermions Second-order nonlinear optical response of graphene, Phys. Rev. B submitted; Collaboration with Tony Heinz: Nonlocal second-order nonlinear response of massless Dirac fermions in graphene, Phys. Rev. Lett. to be submitted Laser-driven parametric instability and generation of entangled photon-plasmon states in graphene and topological insulators Phys. Rev B. 93, (2016). Nonlinear optics of graphene in a magnetic field Collaboration with Manfred Helm: Four-wave mixing in Landau-quantized graphene, Phys. Rev. Lett. to be submitted

3 Work on the project in 2016 part 2 Optical nonlinearities and many-body physics of 2D electron gas in quantum wells Collaboration with Jun Kono: Terahertz Magnetospectroscopy of High-Density 2D Electron- Hole Pairs: Absence of Mott Transition in High Magnetic Fields, Phys. Rev. Lett. submitted D.G. Revin, M. Hemingway, Y. Wang, J.W. Cockburn, A. Belyanin, Active mode locking of quantum cascade lasers operating in external ring cavity, Nature Comm. 7, (2016).

4 Work on the project in 2016 Rigorous quantum mechanical theory of the secondorder optical nonlinearities of massless Dirac fermions Second-order nonlinear optical response of graphene, Phys. Rev. B submitted; Laser-driven parametric instability and generation of entangled photon-plasmon states in graphene and topological insulators Phys. Rev B. 93, (2016). Collaboration with Tony Heinz: Nonlocal second-order nonlinear response of massless Dirac fermions in graphene, Phys. Rev. Lett. to be submitted Nonlinear optics of graphene in a magnetic field Collaboration with Manfred Helm: Four-wave mixing in Landau-quantized graphene, Phys. Rev. Lett. to be submitted

5 Layered Van der Waals materials Strong in-plane bonding, weak van der Waals bonding between layers Easy to peel off and possible to make a single layer Properties at a single layer are very different from the bulk Easy to stack different materials to form a heterostructure semimetal insulator semiconductor Graphite lpmmc.grenoble.cnrs.fr Hexagonal Boron Nitride MoS 2 layered metal dichalcogenide Radislavljevic et al., Nat. Nano 2011

6 Amazing structural diversity of carbon 4 valence electrons Ground-state atomic configuration: 2s 2 2p 2 Key to the structural diversity: formation of hybridized sp electron states Diamond sp 3 bonding: 4 electrons Covalent bonds Cubic lattice, Hardest natural mineral Large-gap insulator E g = 5.5 ev Graphite: sp 2 bonding: 3 electrons form inplane covalent bonds One p-electron is delocalized (π-bond) Weakly connected single-atom sheets Conductor Since 1564

7 Graphite pencil Discovery of pure graphite deposits in Grey Knotts (Lake District, England) around 1564 England remained a monopoly until a process of mixing powdered graphite with clay was discovered (Joseph Hardtmuth & Koh-I-Noor, 1790).

8 Novoselov s scotch tape at the Nobel museum in Stockholm

9 Band structure of graphene Linear dependence and conical selfcrossing for E(p) at E = 0 in K,K points Fermi energy is at E = 0 in undoped graphene Low-energy excitations are massless chiral Dirac fermions! Geim, Phys. Today 2007

10 Band structure near K- or K -point One can assign pseudospin ±½ and write Hamiltonian in terms of Pauli matrices. It has nothing to do with real spin. Energy velocity Eigenstates: æ s x = 0 1 ö æ ç,s è 1 0 y = 0 -i ö ç ø è i 0 ø Chirality: definite pseudospin direction (parallel or antiparallel to p)

11 Band insulator in the bulk, 2D gapless metallic states on the surface 3D topological insulators: Bi 2 Se 3, Sb 2 Te 3 : large bulk band gap ~ 0.3 ev jarilloherrero.mit.edu wikipedia Xia et al. 2009

12 Graphene vs. topological insulators Graphene: pseudospin ±½ related to A,B sublattices; Spin and valley degeneracy Topological insulator: real spin; no degeneracy u F»10 8 cm/s; u' F» cm/s No valley and spin degeneracy One spin direction per surface state Spin locked to the direction of momentum k and rotates with k. No two states at same energy and spin Momentum scattering is suppressed

13 Coupling to the electromagnetic field Intraband (Drude-like) Interaction Hamiltonian: E F interband -Linear in A - No momentum dependence Problems: divergence in A-description; need to renormalize the linear response ; Falkovsky 2007

14 Magnitude of linear optical response Dipole matrix element of the optical transition: m e ~ u F w ~ u F c l 2p ~ l 1800 Note fast growth with λ! Same scaling for intraband transitions Promising for nonlinear optics at long wavelengths Interband absorbance: Similar to interband absorbance in quantum wells, but coming from one monolayer!

15 Even one monolayer of atoms is visible! Monolayer graphene Graphene flakes of different thickness MoS 2 bulk Observing the fine structure constant with the naked eye MoS 2 monolayer Novoselov, RMP 2011 Benameur et al. Nanotech. 2011

16 Oblique incidence µe iqx-iwt Both S- and P-polarizations E,H q z y x Nonzero q : Couples to surface plasmon-polariton modes Breaks inversion symmetry; enables nonzero χ (2)

17 Optical nonlinearities of massless Dirac electrons P = 1 V å ex k = c (1) E + c (2) E 1 E 2 + c (3) E 1 E 2 E Normal incidence: symmetry in the (x,y) plane No second-order nonlinearity: χ (2) = 0 Only third-order nonlinear processes are possible, such as four-wave mixing Oblique incidence: symmetry is broken by q- vector direction Spatial dispersion! χ (2) (ω,q) Three-wave χ (2) processes become possible: - Second-harmonic generation, - Difference frequency generation - Parametric decay

18 Note: spatial dispersion is equivalent to nonlocal response beyond electric dipole approximation Next-order terms ~ in-plane photon wave number q and contain extra (v F /c) ~ 1/300 Not too bad since electron velocity v F is very high in graphene and Bi 2 Se 3 Resembles spatial dispersion due to hot electron motion in plasma For a two-color field: Painful

19 SHG at high frequencies ω >> v F k F H(ω) E(ω) E(2ω) Quadratic in k F H(2ω) Main contribution: S-polarized pump to P-polarized SH Agrees with recent experiments by Tony Heinz for graphene

20 How to make 3-wave mixing efficient Go to longer wavelengths E F Have one of the frequencies close to Leads to enhanced c (2) c (2) / L z ~10-3 m/v Compare with c (2) 3D ~10-10 m/v Have one of the waves close to surface plasmon resonance Leads to field enhancement ~ w g

21 Nonlinear conductivity for SHG ~ 1 w 2 E F = 200 mev

22 Much larger nonlinear conductivity for DFG

23 frequency, THz Difference-frequency generation: - going to low frequencies; - Field enhancement at phase-matched plasmon resonance! w b w a q s w s Re[D(w,q)] = 0 w b -w a = w s (q s ), q bx - q ax = q b + q a = q s w s (q s ) : from Re[D(w, q)] = 0 Yao, Tokman, and Belyanin PRL 112, (2014) 0.5 q s Wavenumber, cm -1 E pl (w,q) = 4pP NL D(w,q) 4pP NL Im[D(w,q)]

24 Experimental realization Hendry group, Nature Phys Observed enhancement in the pump reflection. DFG efficiency ~ 10-5 Theoretically, efficiency ~ 0.1/W is possible for mid-ir to THz down-conversion

25 Parametric instability of pump decay into plasmon-photon pairs w p = w i +w s, q p = q i + q s Mid-IR pump q p q s detector Mid-IR idler q i THz surface plasmon signal Process occurs with exponential gain for high enough pump intensity and nonlinearity Tokman, Wang, Oladyshkin, Belyanin, PRB 93, (2016)

26 Heisenberg-Langevin approach for quantization of fields with dissipation and dispersion Idler noise Quantum and thermal noise Nonlinear polarization

27 Condition for parametric instability: Pump intensity For γ s ~ s -1 gain exceeds losses when I p ~ 10 8 W/cm 2 at 10 μm wavelength Exponential growth of signal and idler fields: Group velocity of surface plasmons << c Gain enhancement!

28 Parametric gain and plasmon frequency q q p = q i p q s q p q s Idler emission angle Tokman, Wang, Oladyshkin, Belyanin, PRB 93, (2016)

29 Entanglement of plasmons and idler photons Using idler field for surface plasmon diagnostics and control Idler flux on the detector: ω p ω i Incident noise Entanglement: Corresponds to one of the Bell states: ω s L y L x

30 Massless Dirac fermions in a magnetic field N = A 2pl c 2 n = 2 n = 1 n = 0 n = -1 Leggett lectures B B

31 GaAs quantum well Selection rules: m/e ~ l c µ l æ e n µ n + 1 ö ç B è 2ø e n µ Massless Dirac fermions Selection rules: m 01 e = u F 2w c = l c 2 n B» 13 nm B(Tesla) µ l n = 3 n = 2 n = 1 B n = 0 B PRL 108, (2012), PRL 110, (2013) PRL 112, (2014) LHC: n f = n i + 1 RHC: n f = n i - 1 n = - 1 n = - 2 n = - 3

32 Enhanced optical nonlinearity No magnetic field Magnetic field Inter-Landau Level transition 4-wave-mixing from interband transitions c (3) ~ 10-7 esu In the near-ir Hendry et al. PRL 105,097401, 2010 Enhanced density of states ~ w c g Single- and multi-photon resonance Much stronger optical nonlinearity Yao & Belyanin, PRL 108, (2012) PRL 110, (2013)

33 Resonant 3 rd order nonlinearity P (3) (w 3 ) = c (3) (w 3 )E 1 (w 1 )( E * 2 (w 2 )) 2 w 3 = w 1-2w 2 w 1 w 2 w 2 Assuming the same scattering rates γ Divided by thickness Bulk susceptibility For pump below saturation, f(ρ) ~ 1 and χ (3) 0.4 esu at B = 1 T. Scales as 1/B Compare with χ (3) 10-7 esu for asymmetric quantum wells

34 4-wave mixing in Landau-quantized graphene Experiment: Manfred Helm, Dresden

35 Generation of polarizationentangled photons by massless Dirac fermions The scheme utilizes unique transitions involving n = 0 quantum state: n = +1Þ 0 Þ -1 Laser Pump ωlf ωhf B LHS RHS E F Tokman, Yao, & Belyanin, PRL 2013

36 Geim Nature 2013 Van der Waals heterostructures graphene h-bn MoS 2 graphene

37 3D massless Dirac fermions! Discovery of 3D Dirac/Weyl semimetal Na 3 Bi: Liu et al., Science 2014 Dirac Hamiltonian (4X4): Weyl Hamiltonian (2X2): H = c(a x p x +a y p y +a z p z )+ bmc 2 H =u F (s x p x +s y p y +s z p z )

38 Outlook Exciting physics: giant nonlinearities, quantum control of electron, photon, and plasmon states; marriage of optics with spin- and valleytronics Van der Waals heterostructures Ultra-compact optical devices, unique functionalities and characteristics: ultrabroadband, ultrafast New materials continue to appear