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 arXiv: 0807.4276v1 [math-ph]
Dec 18, 2015
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