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

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 2 / 23

• 1. Optical lattices
• 2. Floquet engineering
• 3. FFCI in a driven hexagonal lattice

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 3 / 23

Optical lattices

tunable spatially periodic potentials for cold atoms

1D – two beams

two counterpropagating beams E1 = E0 cos (ωt − kx) E2 = E0 cos (ωt + kx) intensity distribution I(x) = (E1 + E2)2 = 2E2

0 cos2 kx

translates to potential distribution

2D hexagonal – three beams

ψ = 60◦

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 4 / 23

Quantum mechanics

• n a lattice

Lattice is a collection of sites ℓ … ˆ a†

ℓ

– creation operator ˆ aℓ – annihilation operator ˆ nℓ = ˆ a†

ℓ ˆ

aℓ – particle number operator ... and links ℓ′ℓ −

• ℓ′ℓ

Jℓ′ℓ eiθℓ′ℓˆ a†

ℓ′ˆ

aℓ – hopping in the presence of a gauge field Description of interactions

• ℓ

vℓ ˆ nℓ – single-particle on-site energies U 2

• ℓ

ˆ nℓ(ˆ nℓ − 1) – on-site (bosonic) particle interactions

E Anisimovas

• micromotion and interactions

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Shaken lattice

also: driven lattice or dynamic lattice

Time-dependent relative phases E1 = E0 cos (ωt + ϕ12(t) − kx) E2 = E0 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 • 2015-06-25 6 / 23

• 1. Optical lattices
• 2. Floquet engineering
• 3. FFCI in a driven hexagonal lattice

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 7 / 23

Effective long-term dynamics versus micromotion

intuitive picture

An example – motion of a “slow train”

long-time trend micromotion 16 start-stop cycles 16 start-stop cycles

separation of complete dynamics into a superposition of long-term dynamics and periodic micromotion

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 8 / 23

Effective long-term dynamics versus micromotion

general and consistent approach

time-periodic driven Hamiltonian ˆ H(t) = ˆ H(t + T) corresponding quantum-mechanical evolution operator ˆ U(t2, t1) = T exp

• − i
• t2

t1

dt ˆ H(t)

• Factorization

ˆ U(t2, t1) = ˆ UF(t2) e−i(t2−t1) ˆ

HF / ˆ

U †

F(t1)

with ˆ HF – stationary effective Hamiltonian ˆ UF(t) – time-periodic unitary micromotion operator t2, t1 – arbitrary time instances

E Anisimovas

• micromotion and interactions

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Effective long-term dynamics

general and consistent approach

time-periodic driven Hamiltonian as Fourier series ˆ H(t) =

∞

• s=−∞

ˆ Hs eisωt factorization of time evolution ˆ U(t2, t1) = ˆ UF(t2) e−i(t2−t1) ˆ

HF / ˆ

U †

F(t1)

how does one compute ˆ HF?

High-frequency expansion

ˆ H (1)

F

= ˆ H0 ˆ H (2)

F

= 1 ω

∞

• s=1

[ ˆ Hs, ˆ H−s] s ˆ H (3)

F

= 1 2(ω)2

∞

• s=1

[ ˆ H−s, [ ˆ H0, ˆ Hs]] s2 + · · ·

E Anisimovas

• micromotion and interactions

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Effective Hamiltonian

extended Floquet Hilbert space

Block diagonalization

𝐼 0 − ℏ𝜕 𝐼 𝐼 0 + ℏ𝜕 𝐼 −1 𝐼 −1 𝐼 −1 𝐼 1 𝐼 1 𝐼 1 𝐼 2 𝐼 2 𝐼 −2 𝐼 3 𝐼 −2 𝐼 −3 𝐼 0 + 2ℏ𝜕 𝐼 𝐺 − ℏ𝜕 𝐼 𝐺 𝐼 𝐺 + ℏ𝜕 𝐼 𝐺 + 2ℏ𝜕

Eckardt and Anisimovas, arXiv:1502.06477, to appear in NJP (2015)

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 11 / 23

Effective long-term dynamics versus micromotion

somewhat simple-minded stroboscopic approach

time-periodic driven Hamiltonian ˆ H(t) = ˆ H(t + T) corresponding stroboscopic evolution operator ˆ U(t0 + T, t0) = T exp

• − i
• t0+T

t0

dt ˆ H(t)

• Effective (Magnus-Floquet) Hamiltonian

define ˆ U(t0 + T, t0) = exp

• − i
• ˆ

H F

t0T

• warning: the effective Hamiltonian ˆ

H F

t0 will depend parametrically on t0

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 12 / 23

Effective long-term dynamics

stroboscopic approach

use the definitions i d dt ˆ U(t, t′) = ˆ H(t) ˆ U(t, t′) ˆ U(t0 + T, t0) = exp

• − i
• ˆ

H F

t0T

• and the Magnus expansion to obtain

High-frequency (Magnus-Floquet) expansion

ˆ H F(1)

t0

= ˆ H0 ˆ H F(2)

t0

= 1 ω

∞

• s=1
• [ ˆ

Hs, ˆ H−s] s + eisωt0 [ ˆ H0, ˆ Hs] s − e−isωt0 [ ˆ H0, ˆ H−s] s

• terms in red – artifactual dependence on the driving phase t0

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 13 / 23

Effective long-term dynamics versus micromotion

summary of the consistent approach

Partitioning the evolution

ˆ U(t2, t1) = ˆ UF(t2) e−i(t2−t1) ˆ

HF / ˆ

U †

F(t1)

Effective Hamiltonian

ˆ H (1)

F

= ˆ H0 ˆ H (2)

F

= 1 ω

∞

• s=1

[ ˆ Hs, ˆ H−s] s ˆ H (3)

F

= 1 2(ω)2

∞

• s=1

[ ˆ H−s, [ ˆ H0, ˆ Hs]] s2 + · · ·

next: application to a simple paradigmatic model

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 14 / 23

• 1. Optical lattices
• 2. Floquet engineering
• 3. FFCI in a driven hexagonal lattice

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 15 / 23

Lattice and driving protocol

• riginal formulation

driven lattice ˆ Hdr(t) = −J

• ℓ′ℓ

ˆ a†

ℓ′ˆ

aℓ +

• ℓ

vℓ(t)ˆ nℓ effect of the driving force – on-site potentials vℓ(t) = −rℓ · F(t) circular driving F(t) = −F[cos ωt ex + sin ωt ey]

A B

nearest-neighbor hopping ℓ′ℓ distance d, direction ϕℓ′ℓ

Strong forcing regime

The effect of the force cannot be treated perturbatively Fd ω 1

• ne must switch to the interaction representation

E Anisimovas

• micromotion and interactions

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Lattice and driving protocol

interaction representation

• riginal driven Hamiltonian

ˆ Hdr(t) = −J

• ℓ′ℓ

ˆ a†

ℓ′ˆ

aℓ +

• ℓ

vℓ(t)ˆ nℓ unitary transformation to remove on-site terms ˆ U(t) = exp

• i
• ℓ

χℓ(t)ˆ nℓ

• χℓ(t) = −−1
• dt′vℓ(t′)

resulting translationally invariant Hamiltonian

ˆ H(t) = ˆ U †(t) ˆ Hdr(t) − idt

ˆ

U(t) = −

• ℓ′ℓ

Jeiθℓ′ℓ(t)ˆ a†

ℓ′ˆ

aℓ direction-dependent phases θℓ′ℓ(t) = Fd ω sin(ωt − ϕℓ′ℓ)

E Anisimovas

• micromotion and interactions

quantum technologies vi • 2015-06-25 17 / 23

High-frequency expansion

how stuff works

Starting point

In the interaction representation, the driven Hamiltonian ˆ H(t) = −

• ℓ′ℓ

Jeiθℓ′ℓ(t)ˆ a†

ℓ′ˆ

aℓ has the Fourier components ˆ Hs = −

• ℓ′ℓ

JJs(α)e−isϕℓ′ℓˆ a†

ℓ′ˆ

aℓ

◮ physical nature – nearest-neighbour hopping ℓ′ℓ ◮ hopping amplitudes are renormalized by Bessel functions Js(α) ◮ hopping amplitudes incorporate direction-dependent phases ϕℓ′ℓ

E Anisimovas

• micromotion and interactions

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High-frequency expansion

first order – time averaging

High-frequency limit

ˆ H (1)

F

= ˆ H0 zeroth Fourier component (time average) ˆ H0 = −

• ℓ′ℓ

JJ0(α)ˆ a†

ℓ′ˆ

aℓ

A B

Physics

nearest-neighbor hopping with renormalized amplitude J → JJ0(α)

E Anisimovas

• micromotion and interactions

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High-frequency expansion

second order – combining two hopping events

Second order correction

ˆ H (2)

F

= 1 ω

∞

• s=1

[ ˆ Hs, ˆ H−s] s combining to NN hopping events ˆ Hs = −

• ℓ′ℓ

JJs(α)e−isϕℓ′ℓˆ a†

ℓ′ˆ

aℓ into an NNN hopping process

A B

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

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Role of interactions

fractional Floquet Chern insulators

Flat band

Single-particle bands are topological and have flat segments

𝐵 𝐶

−𝐾

(a)

> > > > > > >

−𝐾(1)

(b)

− 𝐾(2) 𝑓𝑗𝑗

𝑏

repulsive interactions may stabilize fractional Chern insulating phases Numerics: driven Hubbard Hamiltonian (e. g., hardcore bosons) ˆ H(t) + ˆ Hint ˆ Hint = U 2

• ℓ

ˆ nℓ(ˆ nℓ − 1) interactions influence only the static m = 0 Fourier component

E Anisimovas

• micromotion and interactions

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High-frequency expansion

third order – role of interactions

Interaction effects show up also in third-order terms ˆ H (3)

F

= 1 2(ω)2

∞

• s=1

[ ˆ H−s, [ ˆ Hint, ˆ Hs]] s2

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

ˆ H (3)

F

= −2zη U

• i

ˆ ni (ˆ ni − 1) + 4η U

• ij

ˆ niˆ nj + 2η U

• ij

ˆ a†

i ˆ

a†

i ˆ

ajˆ aj − 1

2η U

• ijk

ˆ a†

i (4ˆ

nj − ˆ ni − ˆ nk)ˆ ak − 1

2η U

• ijk
• ˆ

a†

j ˆ

a†

j ˆ

aiˆ ak + h.c. .

EA et al., PRB 91, 245135 (2015)

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• micromotion and interactions

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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 ( ˆ

H (2)

F

)

◮ fractional phases possible in flat bands ◮ interplay of micromotions and interactions: in many cases detrimental

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• micromotion and interactions

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