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

High-frequency analysis of periodically driven quantum system with slowly varying amplitude Viktor Novi č enko 2017 Zakopane, Poland

Main message        φ  ω φ  i t H t , t t  t     ω π ω   periodic dependence on the first argument: H t 2 , t H t , t ω  any other characteristic energies of the system         φ φ   i t H t t  eff t V. Novičenko, E. Anisimovas, G. Juzeliūnas, Phys. Rev. A 95 , 023615 (2017) Motivation shaken optical lattice R. Desbuquois, M. Messer, F. Görg, K. Sandholzer, G. Jotzu, T. Esslinger, arXiv:1703.07767 (2017)

Extension of the space Let us study whole family of the solutions:          θ  π φ  ω  θ φ initial conditions:  0 , 2 i t H t , t t θ θ  t     φ  φ t t θ  π θ 2 init init   Hamiltonian acts on a Hilbert space H ω  θ H t , t Introduce the space T of θ - periodic functions Construct the space L=H T      ω t Apply unitary transformation U exp    θ      Orthonormal basis of the space T      ω  θ † †   K U HU i U U i H , t  θ e in  θ n          ω   m n  K n n 1 n m H t n     n m , n where the Fourier expansion of the Hamiltonian         θ ,  θ l il H t H t e   l

“Kamiltonian” matrix                       ω     H 0 1 2 3  1 H t t H t H t                    H 1 H 0 1 2  t t H t H t    K t                  H 2 H 1 1  ω t t H 0 H t   t 1                   ω H 3  H 2 H 1 0   t t t H t 2 1    

Floquet band structure of the “Kamiltonian” operator

Block diagonalization of the “Kamiltonian”                        ω † †   K t D t K t D t i D t D t n H t n n D eff   n         ω   H eff t 1 0 0 0    H eff t  0 0 0    K D t    ω 0 0 0   H eff t 1     ω   H t 2 1  0 0 0 eff    

High-frequency expansion                                    ω      ω 3 3 D t n 1 n D t D t O H t H t H t H t O 1 2 eff eff 0 eff 1 eff 2   n                     ω † †   D t K t D t i D t D t n H t n n eff   n    0 H H   eff 0    1      m H   m H H ,   ω eff 1   m 1                                   m 0 m  m m m m n n 1 H , H , H i H , H H , H , H       H     eff 2  ω 2 2   2 m 3 mn     m 0 n 0 , m        φ  ω φ  Our original problem: i t H t , t t  t           φ  ω ω φ † t U t , t U t , t U t , t t fin Micro fin fin eff fin init Micro init init init        χ  χ  i t H t t  eff t

High-frequency expansion                                    ω      ω 3 3 D t n 1 n D t D t O H t H t H t H t O 1 2 eff eff 0 eff 1 eff 2   n                     ω † †   D t K t D t i D t D t n H t n n eff   n     0 0 H H   eff 0     1 0      m H   m H H ,   ω eff 1   m 1                                   m 0 m m m m m n n  1  H , H , H i H , H  H , H , H     H      ω eff 2 2 2  2 m 3 mn      m 0 n 0 , m        φ  ω φ  Our original problem: i t H t , t t  t           φ  ω ω φ † t U t , t U t , t U t , t t fin Micro fin fin eff fin init Micro init init init        χ  χ  i t H t t  eff t

Spin in an oscillating magnetic field The system Hamiltonian:       ω  F  ω H t , t g B t cos t F The non-zero Fourier components:        g       F  1 1 F H t H t B t 2 The effective Hamiltonian is non-zero only due to “slow” time derivative:           i      A      1 1 H t H t H , H B   ω eff eff 2 2      2 2 ω  2   where we introduce the geometric matrix valued non-Abelian vector potential A g F B F The effective evolution:   t   fin   i T       U t , init t exp A d B t eff fin      t init    B  ϕ If and performs rotation in a plane by an angle B t const     2 2   i g B B B ϕ   γ  γ  ϕ  F U n , exp F n , where and n   ϕ ϕ ω  eff    2  4 B B

Numerical demonstration for a spin ½             Magnetic field amplitude: B t B e cos t e sin t z y        φ    Wave function: t c t c t         c t 1 , c t 0   init init  l 10 rotations

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