Spectral relationships between kicked Harper

and on-resonance double kicked rotor

operators

Collaborators: Jiao Wang and Jiangbin Gong

Speakers: Anders S. Mouritzen (that’s me)

Wayne Lawton (coming soon)

arXiv: 0807.4276v1 [math-ph]

What this is about:We have: 1) Kicked Harper Model

An important theoretical model. No existing experimental realizations.

2) Kicked Rotor ModelAnother important theoretical model.Experimentally realizable.

Spectral relationships between kicked Harper and on-resonance double kicked rotor operators

What this is about:We have: 1) Kicked Harper Model

An important theoretical model. No existing experimental realizations.

2) Kicked Rotor ModelAnother important theoretical model.Experimentally realizable.

We wish to: Experimentally realize 1) using 2).

Spectral relationships between kicked Harper and on-resonance double kicked rotor operators

What this is about:We have: 1) Kicked Harper Model

An important theoretical model. No existing experimental realizations.

2) Kicked Rotor ModelAnother important theoretical model.Experimentally realizable.

We wish to: Experimentally realize 1) using 2).

What we get: An experimentally realizable operator with the same spectrum as 1), but

with different dynamics.

Spectral relationships between kicked Harper and on-resonance double kicked rotor operators

A closer look at the models:Kicked Harper operator:

On-resonance double kicked rotor operator:

Spectral relationships between kicked Harper and on-resonance double kicked rotor operators

πθdxdiαλκiπxiκkHU 2cos2exp2cos2exp,,,

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

1,0x

Use periodic boundary conditions.

Basis:

Ζnxnixn :2exp)(

Suggested experimental realization:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

Cold atomic gas or Bose-Einstein condensate

Suggested experimental realization:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

Laser Laser

Cold atomic gas or Bose-Einstein condensate

Suggested experimental realization:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

Laser Laser

A pulsed, standing wave sets up a potential energy: xxVxxV 2cos)(or 2cos)(

Suggested experimental realization:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

Laser Laser

A pulsed, standing wave sets up a potential energy: xxVxxV 2cos)(or 2cos)(

Suggested experimental realization:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

Laser Laser

The operator Uordkr is periodic in x with period 1.

Assumption: The state also has this periodicity.We only have to look at one such period in 1,0x

Suggested experimental realization:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

tj tj + α TR / 2

Laser intensity

Time

(1-α / 2)TR

TR

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

tj+1 = tj + TR

Suggested experimental realization:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

Free propagator:

2

2

2exp)(

dx

d

RTTiTU

Laser intensity

Timetj tj + α TR / 2 tj+1 = tj + TR

Note, that:

RTURTU

221

Suggested experimental realization:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

Laser potential:)2cos( xH

)(2cos xH

Neglect kinetic energy.

{

Laser intensity

Timetj tj + α TR / 2 tj+1 = tj + TR

Suggested experimental realization:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

2

2

4exp2cos2exp

2

2

4exp2cos2exp,,,

dx

dixi

dx

dixiordkrU

Estimates of obtainable experimental parameters:

1001.0,1001.0,2005.0

Let’s look at the spectrums, σ:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

,,,ordkrU

,,,κkHU

with grows Radius.8,4,2,1,5.0,25.0,0,1,13/8

Why are the spectrums different?

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

,,,ordkrU ,,,κkHU

0,138

The spectrum is:

1) Discontinuous at rational α for fixed θ (as in the figure below)

2) Continuous and independent of θ at irrational α.

Why are the spectrums different?

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

,,,ordkrU ,,,κkHU

0,138

The spectrum is:

1) Discontinuous at rational α for fixed θ (as in the figure below)

2) Continuous and independent of θ at irrational α.

A way out: (due to Hofstadter)

Something’s wrong with the physics! Take the union of the spectrums over θ – this is continuous in α.

Let’s look at the spectral unions:

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

,,,ordkrU

,,,κkHU

with grows Radius.8,4,2,1,5.0,25.0,1,13/8

Thanks for now ☺

Spectral relationships between kicked Harper and on resonance double kicked rotor operators

Please welcome Wayne.