Topology and Chiral Physics in Atomic, Molecular, and Optical systems

Topology and Chiral Physics in

Atomic, Molecular, and Optical systems

Tomoki Ozawa

RIKEN iTHEMS, Japan

@ Workshop on “Recent Developments in Chiral Matter and Topology”, Dec 8, 2018

Topology in various systems

• Nuclear physics / High energy physics

• Solid-state physics

• Atomic, molecular and optical (AMO) physics

✓ Physical Review C & Physical Review D Which Physical Review?

How about Physical Review E ? ・・・Topological soft matter, Topological origami In this conference, we cover:

✓ Physical Review B

✓ Physical Review A

/25

What is AMO?

3

AMO physics studies atoms, molecules, and light using laser

High controllability of system parameters allows one to realize various Hamiltonians

• ultracold atoms • exciton-polaritons

• photonics

b

Experiment

Outline

1. Topological physics in ultracold atomic gases 2. Topological physics in photonics

3. Synthetic dimensions and higher dimensional topological effects

/25

Quantum simulation - ultracold atoms

[Bloch group website @ MPQ, Munich]

[Bloch, Nat. Phys. 1, 23 (2005)]

Extreme controllability of the system:

Can choose bosons, fermions, or both

Can change interaction, underlying confinement (trap, lattice, box, etc…), spins, number of species, density , temperature, dimensionality, etc…

1995 : Realization of BEC - Colorado, Rice, MIT

1999 : Realization of degenerate Fermi gas - Colorado 1998, 2004 : Feshbach resonance - MIT, Colorado

2002 : Superfluid - Mott insulator transition - Max-Planck

Quantum simulation of topological models?

How can one simulate topologically nontrivial models?

How can one simulate quantum Hall effect?

How can one simulate an effect of a magnetic field at all?

p ² 2m

(p − eA) ²

2m ^?

p ²

2m − Ω · L = 1

2m (p − mΩ × r) ² − 1

2 m (Ω × r) ²

e A

corresponds to

Example: rotate the system

Equivalent to having an effective magnetic field B = A = 2m

e Ω

/25

Artificial magnetic field

Lin, Compton, Jiménez-García, Porto, and Spielman (NIST), Nature 462, 628 (2009)

F=1 hyperfine states of ⁸⁷ Rb

Consider when the internal degrees of freedom of an atom depends on position |χ(r)

The total state is where is the center-of-mass wavefunction ψ (r, t)|χ(r) ψ (r, t)

Assuming that the center-of-mass motion is adiabatic enough so that one stays in |χ(r) i ∂

∂t ψ (r, t) = 1

2m i i χ (r)| |χ((r)

2

ψ (r, t) + ˜ V (r)ψ(r, t)

≡ A(r) ：Berry connection Acts as an

artificial magnetic field

Artificial magnetic fields on lattice

In the presence of a periodic potential, when the lattice is sufficiently deep p ²

2m + V (r) ^{− J}

<i,j>

c ^†

j c _i + h.c. tight-binding model

In the presence of a magnetic field (p − A) ²

2m + V (r) − J

<i,j>

e ⁱ

R _rj

ri A·dr c ^†

j c _i + h.c.

Magnetic field appears as the Peierls phase in tight-binding models

Peierls phase

φ

E

Harper-Hofstadter model:

/25

Topological lattices in ultracold atoms

ω r2

a

x y

Φ Φ Φ

Φ Φ Φ

Φ Φ Φ

J _x J _y

∆ a

a

ω b2

ω r1

ω b1

Ketterle group @ MIT Bloch group @ Munich

Miyake et al., PRL 111, 185302 (2013)

Kennedy et al., Nature Physics 11, 859 (2015) Aidelsburger et al., PRL 111, 185301 (2013)

Aidelsburger et al., Nature Physics 11, 162 (2015)

• Harper-Hofstadter model • Haldane model

a

b

–π – π

2

0 0

6 3

AB /t ′

ν = –1 ν = +1

ν = 0 ν = 0

q _x q _y

E

Φ

x y Staggered

flux t _ij

A

A

A

B

B B

e ^{i ij} t′ _ij Tunnel couplings

√

–6 3 ^√

– + π

– 2 +π

Φ

Δ

Esslinger group @ ETH

Jotzu et al., Nature 515, 237 (2014)

Topological physics with ultracold gases

• Measurement of Chern number

Aidelsburger et al. (Munich), Nature Physics 11, 162 (2015).

• Measurement of Zak phase, Berry phase

Atala et al. (Munich), Nature Physics 9, 795 (2013); Duca et al. (Munich), Science 347, 288 (2015)

• Detection of chiral edge state

Mancini et al (Florence)., Science 349, 1510 (2015); Stuhl et al (Maryland)., Science 349, 1514 (2015).

• Measurement of Berry curvature

Li et al. (Munich), Science 352, 1094 (2016); Fläschner, et al. (Hamburg), Science 352, 1091 (2016).

• Realization of Su-Schrieffer-Heeger model

Meier et al. (Urbana), Nature communications 7, 13986 (2016).

• Topological charge pumping

Nakajima, et al. (Kyoto), Nature Physics 12, 296 (2016); Lohse et al. (Munich), Nature Physics 12, 350 (2016).

• Observation of quantized circular dichroism

Asteria et al. (Hamburg), arXiv:1805.11077.

/25

Outline

1. Topological physics in ultracold atomic gases 2. Topological physics in photonics

3. Synthetic dimensions and higher dimensional topological effects

Band structure can be classically realized

Tight-binding model can be realized classically. For example, consider a two-site model

x ₁ x

2

Consider two pendula coupled via a spring

m d ² x

1

dt ² = −mω ₁ ² x

1 + κ(x ₂ − x

1 ) m d ² x

2

dt ² = −mω ₂ ² x

2 + κ(x ₁ − x

2 )

1

d ² dt ²

✓x ₁ x ₂

◆

=

✓

− ω ²

1 − κ /m κ /m κ /m − ω ²

2 − κ /m

◆ ✓x ₁ x ₂

◆

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H = J ˆ ˆ c ^†

2 c ˆ

1 + J ˆ c ^†

1 c ˆ

2 + V ₁ c ˆ ^†

1 c ˆ

1 + V ₂ c ˆ ^†

2 c ˆ

2

= c ˆ ^†

1 c ˆ ^†

2

✓V 1 J J V

2

◆ ✓ˆ c

1

c ˆ

2

◆

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2

Eigen-energies are determined by the eigenvalues of this matrix

J

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V ₁

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V ₂

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Optical resonators and tight-binding model

Assume each resonator hosts localized mode E

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⁰ (r)

Align resonators in positions   to form a lattice The total electromagnetic field can be written as

E (r, t) = X

R _i

a i (t)E ⁰ (r − R ⁱ )

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R _i

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The coefficients a i (t) evolve in time with suitable coupling constants:

This is exactly the Heisenberg equation of motion of “quantum mechanical” tight-binding model

i ∂a _i (t)

∂t = − X

R _j

t _ij a _j (t)

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H = − ˆ X

i,j

t _ij a ˆ ^†

i a ˆ _j

Tight-binding models naturally appear in photonic

resonators

Harper-Hofstadter model with light

Hafezi, et al. (JQI), Nature Photonics 7, 907 (2011).

2πα

L ₁ + η L ₂

L ₂ L ₁

a

Por Por Link resonator

Site resonator

Probing waveguide

4

1 2

3

x ₁₂ x ₃₄ R

c a

b d

e

Input

Output

Intensity (normalized) Simulation

Simulation Experiment

Experiment

1.0 0.8 0.6 0.4 0.2 0.0 30 µm

Imaging topological edge states in silicon photonics

M. Hafezi * , S. Mittal, J. Fan, A. Migdall and J. M. Taylor

ARTICLES

PUBLISHED ONLINE: 20 OCTOBER 2013 | DOI: 10.1038/NPHOTON.2013.274

/25

Quantum Hall effect with drive and dissipation

Since photonic systems have dissipation, (Hall) current is usually not a good quantity to look at.

Instead, one can look at the steady-state reached as a result of drive and dissipation

Harper-Hofstadter model （φ = 1/4）

TO & Carusotto, PRL 112, 133902 (2014)

Brillouin zone area Loss

Chern number External force

hxi ⇡ 2πC ₁ F A _BZ γ

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/25

Topological laser

16

St-Jean, et al. (Marcoussis), Nature Photonics 11, 651 (2017)

(g)

QWs DBR

DBR

τ τ τ τ ^Energy

2 (b)

Number of state

5 10 15 20

-2 0

5 10 15 20

Number of state (e)

τ l

τ t

p _x

(c) τ t τ l

p _y (f)

Gap Gap

Energy

2

-2 0

Linewidth (μeV)

0 4

E - 1580 (meV)

Position (μm)

P-bands

(c) P = 0.1P _th (d) P = 1.5P _th

0 10

2

-2

Gap Gap

S-band D-bands

20 0 10 20

0 1

• Lasing in the topological edge state of SSH model in exciton-polariton micropillars

ARTICLES

DOI: 10.1038/s41566-017-0006-2

l l

/

Lasing in topological edge states of a one-dimensional lattice

P. St-Jean ¹ *, V. Goblot ¹ , E. Galopin ¹ , A. Lemaître    ¹ , T. Ozawa ² , L. Le Gratiet ¹ , I. Sagnes ¹ , J. Bloch ¹

and A. Amo ¹

/25

2D Topological laser

17

Bahari, et al. (UC San Diego), Science 358, 636 (2017)

REPORTS

Cite as: B. Bahari et al., Science 10.1126/science.aao4551 (2017).

Nonreciprocal lasing in topological cavities of arbitrary geometries

Babak Bahari, Abdoulaye Ndao, Felipe Vallini, Abdelkrim El Amili, Yeshaiahu Fainman, Boubacar Kanté*

Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093, USA.

ρ λ

ρ

Topological physics with photons

✓ Detection of chiral edge state

✓ Landau levels of photons

✓Measurement of Zak phase & Berry curvature

✓ Realization of anomalous Floquet topological insulators

✓ Realization of Su-Schrieffer-Heeger model

✓ Observation of three-dimensional Weyl dispersion

✓ Topological charge pumping

• Bosons instead of fermions

• More control on realizing various Hamiltonians

• Photons are lossy; sometimes one needs non-Hermitian Hamiltonians

/25

Outline

1. Topological physics in ultracold atomic gases 2. Topological physics in photonics

3. Synthetic dimensions and higher dimensional topological effects

Synthetic dimensions

Simulate higher-dimensions by regarding internal degrees of freedom as dimensions

• Choose degrees of freedom you want to use as synthetic dimensions Example ： Hyperfine degrees of freedom of ultracold atoms

Modes of photons in resonators

• Induce hopping (kinetic energy) along the synthetic dimension

0 1 2 3 4

0 1 2 3 4

/25

Experimental realization in ultracold gases

Florence (Fallani & Inguscio) - ¹⁷³ Yb (fermion)

Mancini et al., Science 349, 1510 (2015); Livi, et al., PRL 117, 220401 (2016)

Maryland (Spielman) - ⁸⁷ Rb (boson)

Stuhl et al., Science 349, 1514 (2015)

−2 −1 0 1 2

-1 0 1

m

x [sites]

+3/2

-5/2 -1/2

0 0.05 0.1 0.15 0.2 0.25

Florence

Maryland

• Use hyperfine degrees of freedom as a synthetic dimension

• Three sites along the synthetic direction

• Chiral propagation of edge states observed

Synthetic dimensions with photons

{ { {

{

• Use different angular momentum modes as a synthetic dimension

• Couple modes via external modulations of refractive index

[cf. Yuan, et al., Opt. Lett. 41, 741 (2016) ])

The resulting single-site effective Hamiltonian:

- 1D tight-binding Hamiltonian with hopping phases - Spatially aligning resonators, one can build up to 4D Hamiltonian

H = −

w

J e ^iθ b ^† _w

+1 b _w + h.c.

Synthetic dimensions with photons in a ring resonator

TO, Price, Goldman, Zilberberg, Carusotto, PRA 93, 043827 (2016)

/25

Harmonic potential eigenstates as synthetic dimensions

Price, TO, & Goldman, PRA 95, 023607 (2017)

cf. Lustig, et al., arXiv:1807.01983 for photonic realization

0 1 2 3 4

0 1 2 3 4

• Hopping among different states can be introduced by shaking the lattice

• In principle, one can simulate up to 6D (3 real dimensions + 3 harmonic potential directions)

H ⁰ = p ²

2 m + 1

2 mω ² x ² =

∞

λ=0

ωλ |λ λ|

H = H ⁰ + V (t)

λ

κ λ

8mω |λ 1 λ|e ^{i φ} + h.c.

−3 −2 −1

−3

−2

−1

40 80 120

0 9

1

Four-dimensional quantum Hall effect

2D: 4D:

Cold atom: Price, Zilberberg, TO, Carusotto, Goldman, PRL 115, 195303 (2015) ; PRB 93, 245113 (2016)

C _n

Quantum Hall effect occurs in any even dimensions (2, 4, 6, etc…), and characterized by the n-th Chern number:

4D quantum Hall effect can be explored with synthetic dimensions

j ^y = − e ²

h C

1 E _x j ^w = −

e ³

h ² C ₂ E _x B _yz

H = − J

x,y,z,w

c ^†

r+ˆ e _x c

r + c ^†

r+ˆ e _y c

r + e ^{iB xz x} c ^†

r+ˆ e _z c

r + e ^{iB yw y} c ^†

r+ˆ e _w c

r + h.c.

2 4 6 8 10 12 14

0.2 0.4 0.6 0.8 1

time 0

(b)

Extracted 2nd Chern number = —0.98

Simulated wavepacket dynamics in the above Hamiltonian to look for 4D quantum Hall effect

cf. Sugawa et al., Science 360, 1429 (2018)

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Summary

• Atomic, molecular, and optical systems provide powerful platforms to explore topological physics

• Ultracold gases are good for exploring many-particle and quantum properties

• Photons are good for exploring single-particle non-equilibrium properties

• They are both often bosons and often lossy

• Interaction effects?

• Non-Hermitian topological physics?

• Higher dimensional topology?

Review: Cooper, Dalibard, & Spielman, “Topological Bands for Ultracold Atoms,” arXiv:1803.00249 Review: Ozawa et al. “Topological Photonics,” arXiv:1802.04173

Both accepted for publication in Rev. Mod. Phys.