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

Quasiclassical analysis of Bloch

• scillations in non-Hermitian tight-

binding lattices

Eva-Maria Graefe

Department of Mathematics, Imperial College London, UK AAMP13, June 2016 Villa Lanna Prague

joint work with Hans Jürgen Korsch, and Alexander Rush Department of Physics, TU Kaiserslautern, Germany Department of Mathematics, Imperial College London, UK

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

• T. Hartmann, F. Keck, S. Mossmann, and H.J. Korsch, New J. Phys. 6 (2004) 2

Lattice site

Bloch oscillations - experimental observations

« Original context: Electrons in periodic potential

• f nuclei in conductor with static electric field

Bloch oscillations - experimental observations

« Original context: Electrons in periodic potential

• f nuclei in conductor with static electric field

Bloch oscillations - experimental observations

« Original context: Electrons in periodic potential

• f nuclei in conductor with static electric field

« Semiconductor superlattices

Bloch oscillations - experimental observations

« Original context: Electrons in periodic potential

• f nuclei in conductor with static electric field

« Semiconductor superlattices « Ultracold atoms in optical lattices

Bloch oscillations - experimental observations

« Original context: Electrons in periodic potential

• f nuclei in conductor with static electric field

« Semiconductor superlattices « Ultracold atoms in optical lattices « Optical waveguide structures

Algebraic formulation

ˆ K = X

n

|nihn + 1| , ˆ K† = X

n

|n + 1ihn|, and ˆ N = X

n

n|nihn|

ˆ H = Fd ˆ N − J 2 ⇣ ˆ K + ˆ K†⌘

« With the shift algebra

[ ˆ K, ˆ N] = ˆ K , [ ˆ K†, ˆ N] = − ˆ K† , [ ˆ K, ˆ K†] = 0

Algebraic formulation

« Define quasimomentum operator via

ˆ H = Fd ˆ N − J 2 ⇣ ˆ K + ˆ K†⌘

« With the shift algebra

ˆ K = eiˆ

κ

[ ˆ N, ˆ κ] = i

ˆ κ

« “Conjugate” of the discrete position operator:

ˆ K = X

n

|nihn + 1| , ˆ K† = X

n

|n + 1ihn|, and ˆ N = X

n

n|nihn|

[ ˆ K, ˆ N] = ˆ K , [ ˆ K†, ˆ N] = − ˆ K† , [ ˆ K, ˆ K†] = 0

Bloch oscillations – quasiclassical explanation ˆ H = E(ˆ κ) + Fd ˆ N, with E(ˆ κ) = −J 2 cos(ˆ κ)

« Heisenberg equations of motion

d dthˆ κi = Fd and d dth ˆ Ni = ⌧∂E(ˆ κ) ∂ˆ κ

• « Acceleration theorem:

hˆ κi(t) = Fdt + hˆ κi(0)

« Ehrenfest theorem:

N(t) ≈ N0 + E(κ0) − E(κ(t)) Fd

Non-Hermitian tight-binding lattice

ˆ H =

+∞

X

n=−∞

• g1|nihn + 1| + g2|n + 1ihn| + 2Fn|nihn|
• g1,2 ∈ 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

ˆ H =

+∞

X

n=−∞

• g1|nihn + 1| + g2|n + 1ihn| + 2Fn|nihn|
• g1,2 ∈ 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

ˆ H =

+∞

X

n=−∞

• g1|nihn + 1| + g2|n + 1ihn| + 2Fn|nihn|
• g1,2 ∈ C, F ∈ R

« Unbroken PT-symmetry κ → −κ, i → −i, N → N « Quasiclassical dynamics? « Modified Heisenberg equations of motion i~ d dth ˆ Ai = h[ ˆ A, ˆ HR]i i ⇣ h[ ˆ A, ˆ HI]+i 2h ˆ Aih ˆ HIi ⌘ not directly useful…

H = HR − iHI

EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

The semiclassical limit with Gaussian states

« Ansatz for time evolved Wigner function:

W(t, z) =

1 (π)n e− 1

(z−Z(t))·G(t)(z−Z(t))

Hepp, Heller, Littlejohn 1970‘s

anharmonic oscillator

« Gaussian states stay Gaussian under evolution with quadratic Hamiltonian! « Quadratic Taylor expansion around the central trajectory Z(t) « Yields semiclassical evolution:

˙ Z = Ω⇥H(Z) ˙ G = H(Z)ΩG GΩH(Z)

Semiclassical limit for non-Hermitian systems H = HR − iHI ˙ p = −∂HR ∂q − Σpp ∂HI ∂p − Σpq ∂HI ∂q ˙ q = ∂HR ∂p − Σpq ∂HI ∂p − Σqq ∂HI ∂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 = HR − iHI ˙ p = −∂HR ∂q − Σpp ∂HI ∂p − Σpq ∂HI ∂q ˙ q = ∂HR ∂p − Σpq ∂HI ∂p − Σqq ∂HI ∂q

Σpp = 2

~(∆p)2,

Σqq = 2

~(∆q)2,

Σpq = Σqp = 2

~∆pq

« 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 = HR − iHI ˙ p = −∂HR ∂q − Σpp ∂HI ∂p − Σpq ∂HI ∂q ˙ q = ∂HR ∂p − Σpq ∂HI ∂p − Σqq ∂HI ∂q ˙ Σ = ΩH00

RΣ − ΣH00 RΩ − ΩH00 I Ω − ΣH00 I Σ

Σpp = 2

~(∆p)2,

Σqq = 2

~(∆q)2,

Σpq = Σqp = 2

~∆pq

« With covariance matrix

Σ

EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Non-Hermitian semiclassical dynamics

˙ p = −∂HR ∂q − Σpp ∂HI ∂p − Σpq ∂HI ∂q ˙ q = ∂HR ∂p − Σpq ∂HI ∂p − Σqq ∂HI ∂q

˙ Σ = ΩH00

RΣ − ΣH00 RΩ − ΩH00 I Ω − ΣH00 I Σ

« Dynamics of position and momentum « Coupled to covariance dynamics « Resulting dynamics of squared norm/total power: ˙ P = −

• 2HI − 1

2Tr(ΩH00 I ΩΣ1)

• 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:

p(t) = p0 − 2Ft

« Acceleration theorem:

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:

p(t) = p0 − 2Ft

« Acceleration theorem:

˙ q = ∂Re(E(p))

∂p

+ ∂Im(E(p))

∂p

Σpq

« Dynamics of centre:

Constant covariance

E(p) = g1eip + g2e−ip

Fieldfree dispersion relation:

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:

p(t) = p0 − 2Ft

« Acceleration theorem:

˙ q = ∂Re(E(p))

∂p

+ ∂Im(E(p))

∂p

Σpq

« Dynamics of centre:

Constant covariance

E(p) = g1eip + g2e−ip

Fieldfree dispersion relation:

˙ P = −2Im(E(p))P

« Evolution of total power:

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:

p(t) = p0 − 2Ft

« Acceleration theorem:

˙ q = ∂Re(E(p))

∂p

+ ∂Im(E(p))

∂p

Σpq

« Dynamics of centre:

Constant covariance

E(p) = g1eip + g2e−ip

Fieldfree dispersion relation:

˙ P = −2Im(E(p))P

« Evolution of total power:

« 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

ˆ H =

+∞

X

n=−∞

• ge+µ|nihn + 1| + ge−µ|n + 1ihn| + 2Fn|nihn|
• « Simple mapping to Hermitian Hamiltonian and

analytical solution « Classical Hamiltonian:

H = 2g cosh µ cos p + 2ig sinh µ sin p + 2Fq

« Experimental realisation in optical resonator structures proposed by Longhi

EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Example: Hatano-Nelson lattice

• 20

20 40

n

1 2 3

t/TB

• 20

20 40

n

1 2 3

t/TB

1 2 3

t/TB

0.5 1

P

Propagation of (wide) Gaussian beam

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

• 50

50

n

1 2 3

t/TB

Quasiclassical breathing mode

δn =

1 2π

Z 2π eipndp « Interpret Fourier transform of initially localised state as (incoherent) ensemble of infinitely narrow momentum wavepackets!

• 50

50

n

1 2 3

t/TB

• 50

50

n

1 2 3

t/TB

EMG, H.J. Korsch, and A. Rush, arXiv:1604.01885 (to appear in New J. Phys.)

Quantum propagation: Classical ensemble:

Happy Birthday, Miloslav!