Quasiclassical analysis of Bloch oscillations in non-hermitian tightbinding

Transcription

1 Quasiclassical analysis of Bloch oscillations in non-hermitian tightbinding 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

2 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

3 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

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

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

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

7 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

8 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

9 Algebraic formulation Ĥ = Fd ˆN J 2 ˆK + ˆK «With the shift algebra ˆK = X n X nihn +1, n +1ihn, and ˆN = n nihn ˆK = X n [ ˆK, ˆN] = ˆK, [ ˆK, ˆN] = ˆK, [ ˆK, ˆK ]=0 n

10 Algebraic formulation «With the shift algebra ˆK = X n Ĥ = Fd ˆN J ˆK + ˆK 2 X nihn +1, n +1ihn, and ˆN = n nihn ˆK = X n [ ˆK, ˆN] = ˆK, [ ˆK, ˆN] = ˆK, [ ˆK, ˆK ]=0 «Define quasimomentum operator [ ˆN,ˆapple] =i ˆapple via ˆK =e iˆapple «Conjugate of the discrete position operator: n

11 Bloch oscillations quasiclassical explanation Ĥ = E(ˆapple)+Fd ˆN, with E(ˆapple) = J 2 cos(ˆapple) «Heisenberg equations of motion d dt hˆapplei = Fd and d dt h «Acceleration theorem: hˆapplei(t) = «Ehrenfest theorem: N(t) N 0 + E(apple 0) E(apple(t)) Fd Fdt+ hˆapplei(0)

12 EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) Ĥ = Non-Hermitian tight-binding lattice +1X n= 1 g 1 nihn +1 + g 2 n +1ihn +2Fn nihn g 1,2 2 C, F 2 R «Unbroken PT-symmetry apple! apple, i! i, N! N

13 EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) Ĥ = Non-Hermitian tight-binding lattice +1X n= 1 g 1 nihn +1 + g 2 n +1ihn +2Fn nihn g 1,2 2 C, F 2 R «Unbroken PT-symmetry apple! apple, i! i, N! N «Quasiclassical dynamics?

14 Ĥ = Non-Hermitian tight-binding lattice +1X n= 1 g 1 nihn +1 + g 2 n +1ihn +2Fn nihn g 1,2 2 C, F 2 R «Unbroken PT-symmetry apple! apple, i! i, N! N «Quasiclassical dynamics? «Modified Heisenberg equations of motion i~ d dt hâi = h[â, ĤR]i i h[â, ĤI] + i 2hÂihĤIi H = H R ih I not directly useful EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.)

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

16 EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) Semiclassical limit for non-hermitian systems H = H R ih I ṗ pp q R pq «With covariance matrix

17 EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) Semiclassical limit for non-hermitian systems H = H R ih I ṗ pp q R pq «With covariance matrix pp = 2 ~ ( p)2, qq = 2 ~ ( q)2, pq = qp = 2 ~ pq

18 EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) Semiclassical limit for non-hermitian systems H = H R ih I ṗ pp q R pq «With covariance matrix pp = 2 ~ ( p)2, qq = 2 ~ ( q)2, pq = qp = 2 ~ pq = H 00 R H 00 R H 00 I H 00 I

19 EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) Non-Hermitian semiclassical dynamics «Dynamics of position and momentum ṗ pp q R pq «Coupled to covariance dynamics = H 00 R H 00 R H 00 I H 00 I «Resulting dynamics of squared norm/total power: P = 2H I 1 2 Tr( H00 I 1 ) P

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

21 EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) Limit of narrow momentum packets «Can be analytically solved to yield: «Acceleration theorem: p(t) =p 0 2Ft «Dynamics of centre: q Fieldfree dispersion relation: E(p) =g 1 e ip + g 2 e ip pq Constant covariance

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

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

24 EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) Example: Hatano-Nelson lattice Ĥ = +1X n= 1 ge +µ nihn +1 + ge µ n +1ihn +2Fn nihn «Simple mapping to Hermitian Hamiltonian and analytical solution «Experimental realisation in optical resonator structures proposed by Longhi «Classical Hamiltonian: H =2g cosh µ cos p + 2ig sinh µ sin p +2Fq

25 EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) Example: Hatano-Nelson lattice Propagation of (wide) Gaussian beam P t/t B 1 t/t B n n t/t B 2 3 F =0.1, g =1, µ =0.2

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

27 Breathing modes in a Hatano-Nelson lattice 3 2 t/t B n 50

28 Quasiclassical breathing mode «Interpret Fourier transform of initially localised state as (incoherent) ensemble of infinitely Z narrow momentum wavepackets! 2 e ipn dp n = Quantum propagation: 3 Classical ensemble: 2 2 t/t B t/t B n EMG, H.J. Korsch, and A. Rush, arxiv: (to appear in New J. Phys.) n

29 Happy Birthday, Miloslav!