Interplay of micromotion and interactions in fractional Floquet - PowerPoint PPT Presentation

Interplay of micromotion and interactions in fractional Floquet Chern insulators Egidijus Anisimovas and André Eckardt Vilnius University and Max-Planck Institut Dresden Quantum Technologies VI Warsaw 2015-06-25

Outline 1. Optical lattices 2. Floquet engineering 3. FFCI in a driven hexagonal lattice Acknowledgements Nathan Goldman, Adolfo Grushin, This research was supported by Gediminas Juzeliūnas, Viktor Novičenko, the European Social Fund Mantas Račiūnas, Giedrius Žlabys under the Global Grant measure quantum technologies vi • 2015-06-25 2 / 23 E Anisimovas micromotion and interactions •

1. Optical lattices 2. Floquet engineering 3. FFCI in a driven hexagonal lattice quantum technologies vi • 2015-06-25 3 / 23 E Anisimovas micromotion and interactions •

Optical lattices tunable spatially periodic potentials for cold atoms 1D – two beams 2D hexagonal – three beams two counterpropagating beams E 1 = E 0 cos ( ω t − kx ) E 2 = E 0 cos ( ω t + kx ) intensity distribution 0 cos 2 kx ( E 1 + E 2 ) 2 � = 2 E 2 I ( x ) = � ψ = 60 ◦ translates to potential distribution quantum technologies vi • 2015-06-25 4 / 23 E Anisimovas micromotion and interactions •

Quantum mechanics on a lattice Lattice is a collection of sites ℓ … a † ˆ – creation operator ℓ ˆ – annihilation operator a ℓ a † n ℓ = ˆ ˆ ℓ ˆ a ℓ – particle number operator ... and links � ℓ ′ ℓ � � J ℓ ′ ℓ e i θ ℓ ′ ℓ ˆ a † − ℓ ′ ˆ – hopping in the presence of a gauge field a ℓ � ℓ ′ ℓ � Description of interactions � v ℓ ˆ – single-particle on-site energies n ℓ ℓ U � ˆ n ℓ (ˆ n ℓ − 1) – on-site (bosonic) particle interactions 2 ℓ quantum technologies vi • 2015-06-25 5 / 23 E Anisimovas micromotion and interactions •

Shaken lattice also: driven lattice or dynamic lattice Time-dependent relative phases E 1 = E 0 cos ( ω t + ϕ 12 ( t ) − kx ) E 2 = E 0 cos ( ω t + kx ) absorbed into a rigid translation x → x + δ ( t ) Shaking as quantum engineering ◮ neutral atoms – no direct coupling to natural gauge fields ◮ lattice shaking – powerful method of control ◮ no internal atomic structure involved (cf. control by light) ◮ bandstructure engineering and emulation of artificial gauge fields possible next: Floquet engineering by time-periodic driving quantum technologies vi • 2015-06-25 6 / 23 E Anisimovas micromotion and interactions •

1. Optical lattices 2. Floquet engineering 3. FFCI in a driven hexagonal lattice quantum technologies vi • 2015-06-25 7 / 23 E Anisimovas micromotion and interactions •

Effective long-term dynamics versus micromotion intuitive picture An example – motion of a “slow train” 16 start-stop cycles 16 start-stop cycles long-time trend micromotion separation of complete dynamics into a superposition of long-term dynamics and periodic micromotion quantum technologies vi • 2015-06-25 8 / 23 E Anisimovas micromotion and interactions •

Effective long-term dynamics versus micromotion general and consistent approach time-periodic driven Hamiltonian H ( t ) = ˆ ˆ H ( t + T ) corresponding quantum-mechanical evolution operator � t 2 � � − i ˆ dt ˆ U ( t 2 , t 1 ) = T exp H ( t ) � t 1 Factorization H F / � ˆ U F ( t 2 ) e − i ( t 2 − t 1 ) ˆ U ( t 2 , t 1 ) = ˆ ˆ U † F ( t 1 ) with ˆ H F – stationary effective Hamiltonian ˆ U F ( t ) – time-periodic unitary micromotion operator t 2 , t 1 – arbitrary time instances quantum technologies vi • 2015-06-25 9 / 23 E Anisimovas micromotion and interactions •

Effective long-term dynamics general and consistent approach time-periodic driven Hamiltonian as Fourier series ∞ � ˆ H s e is ω t ˆ H ( t ) = s = −∞ factorization of time evolution H F / � ˆ U F ( t 2 ) e − i ( t 2 − t 1 ) ˆ U ( t 2 , t 1 ) = ˆ ˆ U † F ( t 1 ) how does one compute ˆ H F ? High-frequency expansion H (1) ˆ = ˆ H 0 F ∞ [ ˆ H s , ˆ = 1 H − s ] � H (2) ˆ F � ω s s =1 ∞ [ ˆ H − s , [ ˆ H 0 , ˆ 1 H s ]] H (3) ˆ � = + · · · F 2( � ω ) 2 s 2 s =1 quantum technologies vi • 2015-06-25 10 / 23 E Anisimovas micromotion and interactions •

Effective Hamiltonian extended Floquet Hilbert space Block diagonalization � 0 − ℏ𝜕 � −1 � −2 � −3 � 𝐺 − ℏ𝜕 𝐼 𝐼 𝐼 𝐼 𝐼 0 0 0 � 1 � 0 � −1 � −2 � 𝐺 𝐼 𝐼 𝐼 𝐼 0 𝐼 0 0 � 2 � 1 � 0 + ℏ𝜕 � −1 � 𝐺 + ℏ𝜕 𝐼 𝐼 𝐼 𝐼 0 0 𝐼 0 � 3 � 2 � 1 � 0 + 2ℏ𝜕 � 𝐺 + 2ℏ𝜕 𝐼 𝐼 𝐼 𝐼 0 0 0 𝐼 Eckardt and Anisimovas, arXiv:1502.06477, to appear in NJP (2015) quantum technologies vi • 2015-06-25 11 / 23 E Anisimovas micromotion and interactions •

Effective long-term dynamics versus micromotion somewhat simple-minded stroboscopic approach time-periodic driven Hamiltonian H ( t ) = ˆ ˆ H ( t + T ) corresponding stroboscopic evolution operator � t 0 + T � � − i ˆ dt ˆ U ( t 0 + T , t 0 ) = T exp H ( t ) � t 0 Effective (Magnus-Floquet) Hamiltonian define − i � � ˆ ˆ H F U ( t 0 + T , t 0 ) = exp t 0 T � warning: the effective Hamiltonian ˆ H F t 0 will depend parametrically on t 0 quantum technologies vi • 2015-06-25 12 / 23 E Anisimovas micromotion and interactions •

Effective long-term dynamics stroboscopic approach use the definitions i � d U ( t , t ′ ) = ˆ ˆ H ( t ) ˆ U ( t , t ′ ) dt � − i � ˆ H F ˆ U ( t 0 + T , t 0 ) = exp t 0 T � and the Magnus expansion to obtain High-frequency (Magnus-Floquet) expansion H F (1) ˆ = ˆ H 0 t 0 ∞ [ ˆ H s , ˆ + e is ω t 0 [ ˆ H 0 , ˆ − e − is ω t 0 [ ˆ H 0 , ˆ � � = 1 H − s ] H s ] H − s ] � H F (2) ˆ t 0 � ω s s s s =1 terms in red – artifactual dependence on the driving phase t 0 quantum technologies vi • 2015-06-25 13 / 23 E Anisimovas micromotion and interactions •

Effective long-term dynamics versus micromotion summary of the consistent approach Partitioning the evolution H F / � ˆ U F ( t 2 ) e − i ( t 2 − t 1 ) ˆ U ( t 2 , t 1 ) = ˆ ˆ U † F ( t 1 ) Effective Hamiltonian H (1) ˆ = ˆ H 0 F ∞ [ ˆ H s , ˆ = 1 H − s ] � H (2) ˆ F � ω s s =1 ∞ [ ˆ H − s , [ ˆ H 0 , ˆ 1 H s ]] H (3) ˆ � = + · · · F 2( � ω ) 2 s 2 s =1 next: application to a simple paradigmatic model quantum technologies vi • 2015-06-25 14 / 23 E Anisimovas micromotion and interactions •

1. Optical lattices 2. Floquet engineering 3. FFCI in a driven hexagonal lattice quantum technologies vi • 2015-06-25 15 / 23 E Anisimovas micromotion and interactions •

Lattice and driving protocol original formulation driven lattice B ˆ � � a † H dr ( t ) = − J ˆ ℓ ′ ˆ a ℓ + v ℓ ( t )ˆ n ℓ � ℓ ′ ℓ � ℓ A effect of the driving force – on-site potentials v ℓ ( t ) = − r ℓ · F ( t ) circular driving nearest-neighbor hopping � ℓ ′ ℓ � F ( t ) = − F [cos ω t e x + sin ω t e y ] distance d , direction ϕ ℓ ′ ℓ Strong forcing regime The effect of the force cannot be treated perturbatively Fd � ω � 1 one must switch to the interaction representation quantum technologies vi • 2015-06-25 16 / 23 E Anisimovas micromotion and interactions •

Lattice and driving protocol interaction representation original driven Hamiltonian ˆ � � a † H dr ( t ) = − J ˆ ℓ ′ ˆ a ℓ + v ℓ ( t )ˆ n ℓ � ℓ ′ ℓ � ℓ unitary transformation to remove on-site terms � � � � ˆ χ ℓ ( t ) = − � − 1 dt ′ v ℓ ( t ′ ) U ( t ) = exp χ ℓ ( t )ˆ i n ℓ ℓ resulting translationally invariant Hamiltonian U † ( t ) � ˆ � ˆ � H ( t ) = ˆ ˆ J e i θ ℓ ′ ℓ ( t ) ˆ a † H dr ( t ) − i � d t U ( t ) = − ℓ ′ ˆ a ℓ � ℓ ′ ℓ � direction-dependent phases θ ℓ ′ ℓ ( t ) = Fd � ω sin( ω t − ϕ ℓ ′ ℓ ) quantum technologies vi • 2015-06-25 17 / 23 E Anisimovas micromotion and interactions •

High-frequency expansion how stuff works Starting point In the interaction representation, the driven Hamiltonian � ˆ J e i θ ℓ ′ ℓ ( t ) ˆ a † H ( t ) = − ℓ ′ ˆ a ℓ � ℓ ′ ℓ � has the Fourier components ˆ � J J s ( α )e − is ϕ ℓ ′ ℓ ˆ a † H s = − ℓ ′ ˆ a ℓ � ℓ ′ ℓ � ◮ physical nature – nearest-neighbour hopping � ℓ ′ ℓ � ◮ hopping amplitudes are renormalized by Bessel functions J s ( α ) ◮ hopping amplitudes incorporate direction-dependent phases ϕ ℓ ′ ℓ quantum technologies vi • 2015-06-25 18 / 23 E Anisimovas micromotion and interactions •

High-frequency expansion first order – time averaging High-frequency limit B H (1) ˆ = ˆ H 0 F zeroth Fourier component (time average) A � ˆ a † H 0 = − J J 0 ( α )ˆ ℓ ′ ˆ a ℓ � ℓ ′ ℓ � Physics nearest-neighbor hopping with renormalized amplitude J → J J 0 ( α ) quantum technologies vi • 2015-06-25 19 / 23 E Anisimovas micromotion and interactions •