Quasiclassical analysis of Bloch oscillations in non-Hermitian - PowerPoint PPT Presentation

Quasiclassical analysis of Bloch oscillations in non-Hermitian tight- binding lattices Eva-Maria Graefe Department of Mathematics, Imperial College London, UK joint work with Hans Jürgen Korsch, and Alexander Rush Department of Physics, TU Kaiserslautern, Germany Department of Mathematics, Imperial College London, UK AAMP13, June 2016 Villa Lanna Prague

Single-band tight-binding Hamiltonian On-site energy Tunneling/hopping between sites T. Hartmann, F. Keck, S. Mossmann, and H.J. Korsch, New J. Phys. 6 (2004) 2

Single-band tight-binding Hamiltonian On-site energy Tunneling/hopping between sites Lattice site T. Hartmann, F. Keck, S. Mossmann, and H.J. Korsch, New J. Phys. 6 (2004) 2

Bloch oscillations - experimental observations « Original context: Electrons in periodic potential of nuclei in conductor with static electric field

Bloch oscillations - experimental observations « Original context: Electrons in periodic potential of nuclei in conductor with static electric field

Bloch oscillations - experimental observations « Original context: Electrons in periodic potential of nuclei in conductor with static electric field « Semiconductor superlattices

Bloch oscillations - experimental observations « Original context: Electrons in periodic potential of nuclei in conductor with static electric field « Semiconductor superlattices « Ultracold atoms in optical lattices

Bloch oscillations - experimental observations « Original context: Electrons in periodic potential of nuclei in conductor with static electric field « Semiconductor superlattices « Ultracold atoms in optical lattices « Optical waveguide structures

Algebraic formulation N − J ⇣ K † ⌘ H = Fd ˆ ˆ K + ˆ ˆ 2 « With the shift algebra K † = ˆ X ˆ X ˆ X | n ih n + 1 | , | n + 1 ih n | , n | n ih n | K = and N = n n n K † , [ ˆ K, ˆ N ] = ˆ [ ˆ K † , ˆ N ] = − ˆ [ ˆ K, ˆ K † ] = 0 K ,

Algebraic formulation N − J ⇣ K † ⌘ H = Fd ˆ ˆ K + ˆ ˆ 2 « With the shift algebra K † = ˆ X ˆ X ˆ X | n ih n + 1 | , | n + 1 ih n | , n | n ih n | K = and N = n n n K † , [ ˆ K, ˆ N ] = ˆ [ ˆ K † , ˆ N ] = − ˆ [ ˆ K, ˆ K † ] = 0 K , « Define quasimomentum operator via K = e iˆ ˆ ˆ κ κ « “Conjugate” of the discrete position operator: [ ˆ N, ˆ κ ] = i

Bloch oscillations – quasiclassical explanation κ ) = − J ˆ κ ) + Fd ˆ H = E (ˆ with E (ˆ 2 cos(ˆ κ ) N, « Heisenberg equations of motion ⌧ ∂ E (ˆ � d d κ ) d t h ˆ d t h ˆ κ i = � Fd and N i = ∂ ˆ κ « Acceleration theorem: h ˆ κ i ( t ) = � Fdt + h ˆ κ i (0) « Ehrenfest theorem: N ( t ) ≈ N 0 + E ( κ 0 ) − E ( κ ( t )) Fd

Non-Hermitian tight-binding lattice + ∞ X ˆ � � g 1 | n ih n + 1 | + g 2 | n + 1 ih n | + 2 Fn | n ih n | H = n = −∞ g 1 , 2 ∈ C , F ∈ R « Unbroken PT-symmetry κ → − κ , i → − i , N → N EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Non-Hermitian tight-binding lattice + ∞ X ˆ � � g 1 | n ih n + 1 | + g 2 | n + 1 ih n | + 2 Fn | n ih n | H = n = −∞ g 1 , 2 ∈ C , F ∈ R « Unbroken PT-symmetry κ → − κ , i → − i , N → N « Quasiclassical dynamics? EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Non-Hermitian tight-binding lattice + ∞ X ˆ � � g 1 | n ih n + 1 | + g 2 | n + 1 ih n | + 2 Fn | n ih n | H = n = −∞ g 1 , 2 ∈ C , F ∈ R « Unbroken PT-symmetry κ → − κ , i → − i , N → N « Quasiclassical dynamics? « Modified Heisenberg equations of motion i ~ d ⇣ ⌘ d t h ˆ A i = h [ ˆ A, ˆ h [ ˆ A, ˆ H I ] + i � 2 h ˆ A ih ˆ H R ] i � i H I i not directly useful… H = H R − i H I EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

The semiclassical limit with Gaussian states « Gaussian states stay Gaussian under evolution with quadratic Hamiltonian! « Ansatz for time evolved Wigner function: ( � π ) n e − 1 1 � ( z − Z ( t )) · G ( t )( z − Z ( t )) W ( t, z ) = « Quadratic Taylor expansion around the central trajectory Z(t) « Yields semiclassical evolution: ˙ = Ω ⇥ H ( Z ) Z ˙ H �� ( Z ) Ω G � G Ω H �� ( Z ) = G anharmonic oscillator Hepp, Heller, Littlejohn 1970‘s

Semiclassical limit for non-Hermitian systems H = H R − i H I p = − ∂ H R ∂ H I ∂ H I − Σ pp ∂ p − Σ pq ˙ ∂ q ∂ q q = ∂ H R ∂ H I ∂ H I ˙ − Σ pq ∂ p − Σ qq ∂ p ∂ q « With covariance matrix Σ EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Semiclassical limit for non-Hermitian systems H = H R − i H I p = − ∂ H R ∂ H I ∂ H I − Σ pp ∂ p − Σ pq ˙ ∂ q ∂ q q = ∂ H R ∂ H I ∂ H I ˙ − Σ pq ∂ p − Σ qq ∂ p ∂ q « With covariance matrix Σ Σ pp = 2 ~ ( ∆ p ) 2 , Σ qq = 2 ~ ( ∆ q ) 2 , Σ pq = Σ qp = 2 ~ ∆ pq EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Semiclassical limit for non-Hermitian systems H = H R − i H I p = − ∂ H R ∂ H I ∂ H I − Σ pp ∂ p − Σ pq ˙ ∂ q ∂ q q = ∂ H R ∂ H I ∂ H I ˙ − Σ pq ∂ p − Σ qq ∂ p ∂ q « With covariance matrix Σ Σ pp = 2 ~ ( ∆ p ) 2 , Σ qq = 2 ~ ( ∆ q ) 2 , Σ pq = Σ qp = 2 ~ ∆ pq ˙ Σ = Ω H 00 R Σ − Σ H 00 R Ω − Ω H 00 I Ω − Σ H 00 I Σ EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Non-Hermitian semiclassical dynamics « Dynamics of position and momentum p = − ∂ H R ∂ H I ∂ H I − Σ pp ∂ p − Σ pq ˙ ∂ q ∂ q q = ∂ H R ∂ H I ∂ H I − Σ pq ∂ p − Σ qq ˙ ∂ p ∂ q « Coupled to covariance dynamics ˙ Σ = Ω H 00 R Σ − Σ H 00 R Ω − Ω H 00 I Ω − Σ H 00 I Σ « Resulting dynamics of squared norm/total power: ˙ I ΩΣ � 1 ) � 2 H I − 1 � 2 Tr( Ω H 00 P = − P EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Limit of narrow momentum packets « Can be analytically solved to yield: « Acceleration theorem: p ( t ) = p 0 − 2 Ft EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Limit of narrow momentum packets « Can be analytically solved to yield: « Acceleration theorem: p ( t ) = p 0 − 2 Ft Constant « Dynamics of centre: covariance q = ∂ Re( E ( p )) + ∂ Im( E ( p )) ˙ Σ pq ∂ p ∂ p Fieldfree dispersion relation: E ( p ) = g 1 e i p + g 2 e − i p EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Limit of narrow momentum packets « Can be analytically solved to yield: « Acceleration theorem: p ( t ) = p 0 − 2 Ft Constant « Dynamics of centre: covariance q = ∂ Re( E ( p )) + ∂ Im( E ( p )) ˙ Σ pq ∂ p ∂ p Fieldfree dispersion relation: E ( p ) = g 1 e i p + g 2 e − i p ˙ « Evolution of total power: P = − 2Im( E ( p )) P EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Limit of narrow momentum packets « Can be analytically solved to yield: « Acceleration theorem: p ( t ) = p 0 − 2 Ft Constant « Dynamics of centre: covariance q = ∂ Re( E ( p )) + ∂ Im( E ( p )) ˙ Σ pq ∂ p ∂ p Fieldfree dispersion relation: E ( p ) = g 1 e i p + g 2 e − i p ˙ « Evolution of total power: P = − 2Im( E ( p )) P « Exact for vanishing momentum width! EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Example: Hatano-Nelson lattice + ∞ X ˆ � g e + µ | n ih n + 1 | + g e − µ | n + 1 ih n | + 2 Fn | n ih n | � H = n = −∞ « Simple mapping to Hermitian Hamiltonian and analytical solution « Experimental realisation in optical resonator structures proposed by Longhi « Classical Hamiltonian: H = 2 g cosh µ cos p + 2i g sinh µ sin p + 2 Fq EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Example: Hatano-Nelson lattice Propagation of (wide) Gaussian beam 3 3 1 P 2 2 t/T B 0.5 t/T B 1 1 0 0 0 t/T B -20 0 20 40 -20 0 20 40 0 1 2 3 n n F = 0 . 1 , g = 1 , µ = 0 . 2 EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Breathing modes Propagation of single site initial state in Hermitian case Quasiclassical dynamics not valid, but can be explained as classical ensemble T. Hartmann, F. Keck, S. Mossmann, and H.J. Korsch, New J. Phys. 6 (2004) 2

Breathing modes in a Hatano-Nelson lattice 3 2 t/T B 1 0 n -50 0 50

Quasiclassical breathing mode « Interpret Fourier transform of initially localised state as (incoherent) ensemble of infinitely narrow Z 2 π momentum wavepackets! e i pn d p 1 δ n = 2 π 0 Classical ensemble: Quantum propagation: 3 3 2 2 t/T B t/T B 1 1 0 0 n -50 0 50 -50 0 50 n EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

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