[PPT] - High-frequency analysis of periodically driven quantum system with PowerPoint Presentation

SLIDE 1

High-frequency analysis of periodically driven quantum system with slowly varying amplitude

Viktor Novičenko 2017 Zakopane, Poland

SLIDE 2

Main message

     

t t t H t t i φ ω φ ,    

   

t t H t t H , , 2 ω π ω   periodic dependence on the first argument:  ω  any other characteristic energies of the system

     

t t H t t i φ φ

eff

   

V. Novičenko, E. Anisimovas, G. Juzeliūnas, Phys. Rev. A 95, 023615 (2017)

Motivation

R. Desbuquois, M. Messer, F. Görg, K. Sandholzer, G. Jotzu, T. Esslinger, arXiv:1703.07767 (2017)

shaken optical lattice

SLIDE 3

Extension of the space

     

t t t H t t i

θ θ

φ θ ω φ ,      Let us study whole family of the solutions:

 

π θ 2 ,  initial conditions:

   

init init 2

t t

θ π θ

φ φ 



 

  



  



l il l

e t H t H

θ

θ, Hamiltonian acts on a Hilbert space H

 

t t H , θ ω  Introduce the space T of

periodic functions

θ Construct the space L=H T           θ ωt U exp Apply unitary transformation

 

t H i U U i HU U K ,

† †

θ θ ω           Orthonormal basis of the space T n ein 

θ

  

 

      

 

n m n m n

n t H m n n n K

,

1 ω  where the Fourier expansion of the Hamiltonian

SLIDE 4

“Kamiltonian” matrix

  

t H 0

  

1 ω   t H 0

  

1 ω   t H 0

  

t H

1 

  

t H

1 

  

t H 1

  

t H 1

  

t H 2

  

t H

2 

  

1 ω  2  t H

  

t H

1 

  

t H 1

  

t H 2

  

t H

2 

  

t H

3 

  

t H 3

 

t K                

SLIDE 5

Floquet band structure of the “Kamiltonian” operator

SLIDE 6

Block diagonalization of the “Kamiltonian”

 

t Heff

 

1 ω   t Heff

 

1 ω   t Heff

 

1 ω  2

eff

 t H

 

t KD                

               



  

   

n D

n n t H n t D t D i t D t K t D t K ω   

eff † †

SLIDE 7

High-frequency expansion

 

        

 

3 2 eff 1 eff eff eff 

    ω O t H t H t H t H

 

     

 

3 2 1    

     ω O t D t D n n t D

n

1

     

t t t H t t i φ ω φ ,    

         

init init init † Micro init fin eff fin fin Micro fin

, , , t t t U t t U t t U t φ ω ω φ 

     

t t H t t i χ χ

eff

   

   

eff

H H 

     

 



  



1 1 eff

, 1

m m m H

H H ω 

 

     

   

   

 

     

   

 

 

     

        

, 2 2 2 eff

3 , , 2 , , , 1

m m n n n m m m m m m

mn H H H m H H i H H H H   ω Our original problem:

             



  

  

n

n n t H n t D t D i t D t K t D ω   

eff † †

SLIDE 8

High-frequency expansion

 

        

 

3 2 eff 1 eff eff eff 

    ω O t H t H t H t H

 

     

 

3 2 1    

     ω O t D t D n n t D

n

1

     

t t t H t t i φ ω φ ,    

         

init init init † Micro init fin eff fin fin Micro fin

, , , t t t U t t U t t U t φ ω ω φ 

     

t t H t t i χ χ

eff

   

   

eff

H H 

     

 



  



1 1 eff

, 1

m m m H

H H ω 

 

     

   

   

 

     

   

 

 

     

        

, 2 2 2 eff

3 , , 2 , , , 1

m m n n n m m m m m m

mn H H H m H H i H H H H   ω Our original problem:

             



  

  

n

n n t H n t D t D i t D t K t D ω   

eff † †

 

SLIDE 9

Spin in an oscillating magnetic field

     

t t g t t H

F

ω ω cos , B F  The system Hamiltonian: The non-zero Fourier components:

     

 

t g t H t H

F

B F  



2

1 1

The effective Hamiltonian is non-zero only due to “slow” time derivative:

 

  

 

   

 

B A         

1 1 2 2 eff eff

, H H i t H t H ω where we introduce the geometric matrix valued non-Abelian vector potential

   

B F A  

2 2 2ω F

g The effective evolution:

   

          



fin init

exp , init

fin eff t t

t d i t t U B A  T If and performs rotation in a plane by an angle

 

const   B t B ϕ

 

B B B B n n F n                 and 4 where , exp ,

2 2 2 eff

ω ϕ γ γ ϕ

ϕ ϕ

B g i U

F

SLIDE 10

Numerical demonstration for a spin ½

Magnetic field amplitude:

     

 

t t B t

y z

    sin cos e e B Wave function:

      

  

 

t c t c t φ rotations 10  l

   

, 1

init init

 

 

t c t c

SLIDE 11

The end