Interplay of micromotion and interactions

Transcription

1 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

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

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

4 Optical lattices tunable spatially periodic potentials for cold atoms 1D two beams two counterpropagating beams 2D hexagonal three beams E 1 = E 0 cos (ωt kx) E 2 = E 0 cos (ωt + kx) intensity distribution I (x) = (E 1 + E 2) 2 = 2E 2 0 cos 2 kx ψ = 60 translates to potential distribution E Anisimovas micromotion and interactions quantum technologies vi / 23

5 Quantum mechanics on a lattice Lattice is a collection of sites l â l â l ˆn l = â l âl... and links l l J l l e iθ l lâ l â l l l creation operator annihilation operator particle number operator hopping in the presence of a gauge field Description of interactions v l ˆn l U 2 l ˆn l (ˆn l 1) l single-particle on-site energies on-site (bosonic) particle interactions E Anisimovas micromotion and interactions quantum technologies vi / 23

6 Shaken lattice also: driven lattice or dynamic lattice Time-dependent relative phases E 1 = E 0 cos (ωt + ϕ 12(t) kx) E 2 = E 0 cos (ωt + 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 E Anisimovas micromotion and interactions quantum technologies vi / 23

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

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

9 Effective long-term dynamics versus micromotion general and consistent approach time-periodic driven Hamiltonian Ĥ (t) = Ĥ (t + T) corresponding quantum-mechanical evolution operator { Û (t 2, t 1) = T exp i t2 t 1 dt Ĥ (t) } Factorization with Û (t 2, t 1) = ÛF(t 2) e i(t 2 t 1 )ĤF / Û F (t1) Ĥ F stationary effective Hamiltonian Û F (t) time-periodic unitary micromotion operator t 2, t 1 arbitrary time instances E Anisimovas micromotion and interactions quantum technologies vi / 23

10 Effective long-term dynamics general and consistent approach time-periodic driven Hamiltonian as Fourier series factorization of time evolution how does one compute ĤF? High-frequency expansion Ĥ (t) = s= Ĥ s e isωt Û (t 2, t 1) = ÛF(t 2) e i(t 2 t 1 )ĤF / Û F (t1) Ĥ (1) F = Ĥ0 Ĥ (2) F = 1 ω s=1 Ĥ (3) F = 1 2( ω) 2 [Ĥs, Ĥ s] s s=1 [Ĥ s, [Ĥ0, Ĥs]] + s 2 E Anisimovas micromotion and interactions quantum technologies vi / 23

11 Effective Hamiltonian extended Floquet Hilbert space Block diagonalization H 0 ħω H 1 H 2 H 3 H F ħω H 1 H 0 H 1 H 2 0 H F 0 0 H 2 H 1 H 0 + ħω H H F + ħω 0 H 3 H 2 H 1 H 0 + 2ħω H F + 2ħω Eckardt and Anisimovas, arxiv: , to appear in NJP (2015) E Anisimovas micromotion and interactions quantum technologies vi / 23

12 Effective long-term dynamics versus micromotion somewhat simple-minded stroboscopic approach time-periodic driven Hamiltonian Ĥ (t) = Ĥ (t + T) corresponding stroboscopic evolution operator { Û (t 0 + T, t 0) = T exp i t0 +T t 0 dt Ĥ (t) } Effective (Magnus-Floquet) Hamiltonian define { Û (t 0 + T, t 0) = exp i } Ĥ t F 0 T warning: the effective Hamiltonian Ĥ F t 0 will depend parametrically on t 0 E Anisimovas micromotion and interactions quantum technologies vi / 23

13 Effective long-term dynamics stroboscopic approach use the definitions and the Magnus expansion to obtain i d dt Û (t, t ) = Ĥ (t)û (t, t ) { Û (t 0 + T, t 0) = exp i } Ĥ t F 0 T High-frequency (Magnus-Floquet) expansion Ĥ F(1) t 0 = Ĥ0 Ĥ F(2) t 0 = 1 ω { [ Ĥ s, Ĥ s] s=1 s + e isωt [Ĥ0, Ĥs] 0 e isωt 0 s } [Ĥ0, Ĥ s] s terms in red artifactual dependence on the driving phase t 0 E Anisimovas micromotion and interactions quantum technologies vi / 23

14 Effective long-term dynamics versus micromotion summary of the consistent approach Partitioning the evolution Û (t 2, t 1) = ÛF(t 2) e i(t 2 t 1 )ĤF / Û F (t1) Effective Hamiltonian Ĥ (1) F = Ĥ0 Ĥ (2) F = 1 ω s=1 Ĥ (3) F = 1 2( ω) 2 [Ĥs, Ĥ s] s s=1 [Ĥ s, [Ĥ0, Ĥs]] + s 2 next: application to a simple paradigmatic model E Anisimovas micromotion and interactions quantum technologies vi / 23

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

16 Lattice and driving protocol original formulation driven lattice Ĥ dr (t) = J â l â l + v l (t)ˆn l l l l effect of the driving force on-site potentials B A v l (t) = r l F(t) circular driving F(t) = F[cos ωt e x + sin ωt e y] nearest-neighbor hopping l l 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 E Anisimovas micromotion and interactions quantum technologies vi / 23

17 Lattice and driving protocol interaction representation original driven Hamiltonian Ĥ dr (t) = J â l â l + v l (t)ˆn l l l l unitary transformation to remove on-site terms [ ] Û (t) = exp i χ l (t)ˆn l χ l (t) = 1 l dt v l (t ) resulting translationally invariant Hamiltonian Ĥ (t) = Û (t) [ ]Û Ĥ dr (t) i d t (t) = Je iθ l l (t) â l â l direction-dependent phases l l θ l l(t) = Fd ω sin(ωt ϕ l l) E Anisimovas micromotion and interactions quantum technologies vi / 23

18 High-frequency expansion how stuff works Starting point In the interaction representation, the driven Hamiltonian Ĥ (t) = Je iθ l l (t) â l â l l l has the Fourier components Ĥ s = JJ s(α)e isϕ l lâ l â 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 E Anisimovas micromotion and interactions quantum technologies vi / 23

19 High-frequency expansion first order time averaging High-frequency limit Ĥ (1) F = Ĥ0 zeroth Fourier component (time average) Ĥ 0 = JJ 0(α)â l â l l l B A Physics nearest-neighbor hopping with renormalized amplitude J JJ 0(α) E Anisimovas micromotion and interactions quantum technologies vi / 23

20 High-frequency expansion second order combining two hopping events Second order correction Ĥ (2) F = 1 ω s=1 [Ĥs, Ĥ s] s combining to NN hopping events Ĥ s = JJ s(α)e isϕ l lâ l â l l l B A into an NNN hopping process Physics next-nearest neighbor hopping involving phase ±π/2 emulation of gauge structure (artificial magnetic field) realization of the Haldane model featuring topological bandstructure E Anisimovas micromotion and interactions quantum technologies vi / 23

21 Role of interactions fractional Floquet Chern insulators Flat band Single-particle bands are topological and have flat segments (a) a J (b) J (1) > J (2) e ii A > > > B > > > repulsive interactions may stabilize fractional Chern insulating phases Numerics: driven Hubbard Hamiltonian (e. g., hardcore bosons) Ĥ (t) + Ĥint Ĥ int = U ˆn l (ˆn l 1) 2 interactions influence only the static m = 0 Fourier component E Anisimovas micromotion and interactions quantum technologies vi / 23 l

22 High-frequency expansion third order role of interactions Interaction effects show up also in third-order terms Ĥ (3) F = 1 2( ω) 2 s=1 [Ĥ s, [Ĥint, Ĥs]] s 2 Interplay of micromotions and interactions scale as U /( ω) 2 and may be significant, eg fractional Hall regime combination of interaction with two tunneling events Ĥ (3) F = 2zη U ˆn i (ˆn i 1) + 4η U ˆn i ˆn j i ij + 2η U â i â i â jâ j 1 η U â 2 i (4ˆn j ˆn i ˆn k )â k ij 1 2 η U ijk (â ijk j â j â iâ k + h.c. ). EA et al., PRB 91, (2015) E Anisimovas micromotion and interactions quantum technologies vi / 23

23 Summary Dynamic (shaken) lattices periodic driving means of control (Floquet engineering) theoretical analysis in terms of high-frequency expansion Fractional Floquet Chern insulators topological structure created by driving (Ĥ (2) F ) fractional phases possible in flat bands interplay of micromotions and interactions: in many cases detrimental E Anisimovas micromotion and interactions quantum technologies vi / 23