Exploring Hermitian and Non-Hermitian Topology inSynthetic Dimensions

Shanhui Fan
Department of Electrical Engineering, and Edward L. Ginzton Laboratory
Stanford University

Outline
Background

Hermitian Topology:
experiments

L0

Non-Hermitian Topology:
experiments

Non-Hermitian Topology:
theoryTopological physics is much richer in higher dimensions
Weyl-point physics in threedimensions

H = k xσ x + k yσ y + kzσ z + E0 I

The concept of synthetic dimension
Explore higher-dimensional physics in lower dimensional physical systems

3D physics
Add one syntheticdimension
2D representationTsomokos et al, PRA 82, 052311 (2010);
Boada et al, PRL 108, 133001 (2012);
Jukic and Buljan, PRA 87, 013814 (2013).

For a review on synthetic dimension inphotonics
L. Yuan et al, Optica 5, 1396 (2018).

A single ring resonator⋮⋮𝜔! + 2Ω𝜔! + Ω

In the absence of group velocity dispersion(GVD) in the ring waveguide, the ring supports a
set of resonances with equally spaced resonantfrequencies.

𝜔!
𝜔! − Ω
𝜔! − 2Ω⋮Ω = 2𝜋/𝑇⋮
where T is the round trip time

Gauge potential for lightΩ" = Ωa1a2

Modulation
frequency

𝑎# = 𝑎$ 𝑒 %&'%((*! +,-)⋮⋮

𝜔! + 2Ω𝜔!

/𝐻 = 𝑉 0 𝑎(/ 𝑎(,$ 𝑒 0%- + 𝑎(,$𝑎( 𝑒 %(••

𝜔! − Ω
𝜔! − 2Ω
⋮Modulation resonantly couples different
modes together

𝜙𝜔! + ΩModespacing

1D physics in 0D structure
𝜙 is the gauge potential which breaks
reciprocity⋮

K. Fang, Z. Yu and S. Fan, Nature Photonics 6, 782 (2012)

Tight-binding lattice in synthetic space
Array of coupled resonatork𝜙#𝜙$

𝜔! + 2Ω𝜔! + Ω𝜔!
𝜔! − Ω𝜔! − 2Ωk𝜙1kk𝜙k2𝜙

𝜙2k3𝜙4𝜙𝜙𝜙𝜙

A two-dimensional space, having a real-space axis and a frequency axis
with applied gauge field in the synthetic space.
L. Yuan, Y. Shi and S. Fan, Optics Letters 41, 741 (2016)
See also, Ozawa et al, PRA 93, 043827 (2016).

One-way edge mode in the synthetic space

Input ω 0

ω 0 + 3Ω

ωω01

ω 0 − 3ΩxSee the experiment by Segev’s group, Lustig et al, Nature 567, 356 (2019).

0Many non-trivial topological phenomena in synthetic space
Weyl semimetal

3D topological insulator

Higher-order
topological insulator

Lin et al, NatureCommunications 7,13731 (2016).

Lin et al, ScienceAdvances 4, eaat2774(2018).

Dutt et al, Light: Scienceand Applications, 9, 131(2020).

Band structure in synthetic space

𝐸!! = 2𝑉 cos 𝑘" Ω

𝑇#$% = 𝑒 &' ()*(,-./)/
𝐻 = 𝑉 0 𝑎(/ 𝑎(,$ + 𝑎(,$𝑎((Quasi-momentum in thesynthetic frequency dimension 𝑘3≡ time 𝑡Band structure from time-dependent transmission
𝑇 𝜔, 𝑘" =ring

𝛾1𝛾11[𝜔 − 𝐸 𝑘" ] +4=𝛾1𝛾11Δ𝜔 +4= 𝑇(Δ𝜔, 𝑡)

modulator
𝑉 𝑡 = 𝑉" cos Ω𝑡

Theoretical transmission (𝛾 = 𝑉/10)

Δ𝜔/𝑉

To determine the band structure,
one needs to measure thetransmission of the system as afunction of time (𝑡 or 𝑘3 ),for various detunings Δ𝜔.

−𝜋0𝑘" Ω (= 𝑡Ω)𝜋𝛾 = 𝑉/10

Experimental setup
Fiber loop length ~13m
Free spectral range: 15.1MHz

Narrow linewidth laser
Frequency sweep

LiNbO3 electro-optic modulator,
modulation frequency up to 2 GHz,
loss compensated with semiconductor optical amplifier

Modulation with a single frequency 𝛀 equal to free
spectral range

A. Dutt, M. Minkov, Q. Lin, L. Yuan, D. A. B. Miller, S. Fan, Nature Communications 10, 3122 (2019).

Creating boundary in frequency synthetic space

Modulator
𝜔2

𝜔2

𝜔2

𝜔2

Frequency

Frequency

A. Dutt, L. Yuan, K. Y. Yang, K. Wang, S. Buddhiraju, J. Vuckovic, and S. Fan,
Nature Communications 13, 3377 (2022).

Experimentally demonstrated boundary

Modulator
𝜔2

𝜔2
𝜔2

𝜔2

Fabry-Perot Oscillation Along Frequency Dimension

Modulator
𝜔2

𝜔2
𝜔2

𝜔2

Without boundaryWith boundary

Outline
Background

Hermitian Topology:
experiments

L0

Non-Hermitian Topology:
experiments

Non-Hermitian Topology:
theoryQuantum Hall systems
Bulk bands

Chiral edge state

Chiral edge state𝑦Magnetic field B𝑥

Chiral edgestates𝜓𝜓

Chiral two-legged Hall ladder…
Entire bulk sites removed → no bulk bands but chiral edge states preserved
Theory: D. Hugel and B. Paredes, Physical Review A 89, 023619 (2014).
Demonstrated in cold-atom systems with real-space lattices.

CW…

…

Two-legged synthetic Hall ladder
CCW≡
L0CCW
CW
pseudospin pseudospin

Δ𝜙 = 𝛽3 𝐿4 − 𝐿1CCW…

CW → CCW: L2
CCW → CW : L1CW

2𝜋𝑚=Δ𝐿𝐿2• Realize effective magnetic field in synthetic space
A. Dutt, Q. Lin, L. Yuan, M. Minkov, M. Xiao and S. Fan, Science 367, 59 (2020).

Spin-resolved band structure measurement

𝜔56

𝜔56

CW CCW

Measure clockwise (CW)transmission, yields bandstructure projected on CW
pseudospin by sweeping 𝜔56

Measure counter-clock-wise (CCW)transmission, yields band structure
projected on CCW pseudospin”Spin”: CW or CCW modesChiral bands and spin-momentum locking
CW

Theoreticalspin-projectedbands

Experimentallymeasuredbands

Floquetsimulations
𝐽/𝐾 = 2
𝜙# = 3𝜋/4

CCW𝑛!"𝑛!!"Chiral currents
Chiral current: 𝑗4 = ∑"5"$ 𝑃(𝑚) − ∑"6"$ 𝑃(𝑚)Δ𝜔 > 0Δ𝜔𝑚7𝑘"
Δ𝜔 < 0𝑚7
To demonstrate one-way transport, we measure theoutput frequency spectrum

Heterodyne measurement of the spectrum
Acousto-opticmodulator
𝑒 &9"#-

𝑒 & 9"#.:9 Detection
F 𝐴8 𝑒 &(9"#.8,)8

Electro-optic modulator
Aim of the measurement
1𝐼 𝑡 = F 𝐴8 𝑒 &(9"#.8,)- + 𝑒 & 9"#.:9 8Output of themodulatedcavity~ F 𝐴8 𝑒 & 8,;:9 - + 𝑐. 𝑐8

Small part of the
input signal,
frequency-shifted

Detected
intensity

Chiral current measurement results
B

Experiment
CW bandΔ𝜔

𝑃(𝑚)Simulation

A. Dutt, Q. Lin, L. Yuan, M. Minkov, M. Xiao and S. Fan, Science 367, 59 (2020).

Outline
Background

Hermitian Topology:
experiments

L0

Non-Hermitian Topology:
experiments

Non-Hermitian Topology:
theoryBand structure of Hermitian versus non-Hermitiansystems in one dimension
Hermitian

Non-Hermitian𝐸

Re(𝐸) Im(𝐸)

𝑘𝑘

Im(𝐸)

Im(𝐸)

Re(𝐸)

𝑘Re(𝐸)

Point-gap topology
Topology of a single non-Hermitian band with respect to a givenreference frequency 𝜖𝑘↑Winding numberIm(𝐸)#7 d𝑘𝑣≔F!

𝜖dln 𝐸 𝑘 − 𝜖2𝜋𝑖 d𝑘Re(𝐸)𝜖𝑣 = 1 if 𝜖 is in the loop
𝑣 = 0 if 𝜖 is outside the loop

K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys. Rev. X 9, 041015 (2019).
Z. Gong, et al., Phys. Rev. X 8, 031079 (2018).
N. Okuma, et al., Phys. Rev. Lett. 124, 086801 (2020).

Implication of point-gap topology: non-Hermitian skineffectInfinite (periodic) structure

Semi-infinite structure

…

…

…

Im(𝐸)

Im(𝐸)

𝜖Re(𝐸)

𝑣 = −1 at 𝜖Re(𝐸)There are exactly |𝑣| edge states
at every such 𝜖Theory: Z. Gong, et al., Phys. Rev. X 8, 031079 (2018); N. Okuma, et al., Phys. Rev. Lett. 124, 086801 (2020).
Experiment: S. Weidemann, et al., Science 368, 311 (2020); L. Xiao, et al., Nat. Phys. 16, 761 (2020);
T. Helbig, et al., Nat. Phys. 16, 747 (2020).

Hatano-Nelson model𝜅,…

…
𝜅0/O = 𝜅, 𝑎P(,$𝐻𝑎P( + 𝜅0 𝑎P(/ 𝑎P(,$𝜅, = 𝐶 + Δ

𝜅0 = 𝐶 − Δ

𝐶, Δ ∈ ℝ

Direction-dependent coupling amplitude
Im(𝐸)Complex-valued dispersion
𝐸 𝑘 = 2𝐶 cos 𝑘 + 2Δ𝑖 sin 𝑘

4Δ

Sign of CΔ determines thehandedness of the loop

Re(𝐸)

4𝐶
N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996).

Implementation of the Hatano-Nelson model in asynthetic dimension
𝜅,…

…
𝜅0/
O = 𝜅, 𝑎P(,$𝐻𝑎P( + 𝜅0 𝑎P(/ 𝑎P(,$

𝜅± = 𝐶 ± Δ
Hermitian part
𝑇<= = exp(−𝑖2𝐶 cos Ω𝑡)

Anti-Hermitian part
Amplitude modulation

𝜔# + 𝑛Ω…

Phase modulation

𝑇>3 = 1 + 2Δ sin Ω𝑡…
K. Wang*, A. Dutt*, K. Y. Yang, C. C. Wojcik, J. Vučković, and S. Fan, Science 371, 1240 (2021).

Complex band structure measurement in frequency
synthetic space
BS1CW laser input at 𝛿𝜔𝐼 𝑘, 𝛿𝜔 ∝𝜔# + 𝑛ΩInput
frequency

1Re 𝐸 − 𝛿𝜔 1 + Im 𝐸BS2

𝛿𝜔

Phase & amplitude
Time-resolved detectorEOMs𝐼 𝛿𝜔𝐼 𝛿𝜔𝛿𝜔/Ω𝐼 𝑘, 𝛿𝜔Re 𝐸𝑘/𝜋

Im 𝐸 = −0.02Ω𝑘=𝜋2

Im 𝐸 = −0.04Ω

𝑘=7𝜋4𝟐

Demonstrating the Hatano-Nelson modelExperiment

Theory𝑘/𝜋

𝑣=1K. Wang*, A. Dutt*, K. Y. Yang, C. C. Wojcik, J. Vučković, and S. Fan, Science 371, 1240 (2021).

Implementation of long-range coupling
𝐶 % 𝑒 &' + Δ′𝑒 &(%…

𝐶 + Δe&(𝐶 − Δe)&(…
𝐶 % 𝑒 )&' − Δ′𝑒 )&(%

Example: {𝑚} = {1,2}Phase modulation

𝑇>3 = 1 + 2Δ sin Ω𝑡 + 𝜙+ 𝟐𝚫@ 𝐬𝐢𝐧(𝟐𝛀𝒕 + 𝝓@ )

𝜔2 + 𝑛Ω…

𝑇<= = exp[−𝑖2𝐶 cos Ω𝑡 − 𝒊𝟐𝑪@ 𝐜𝐨𝐬 𝟐𝛀𝒕 + 𝜽 ]Amplitude modulation…
K. Wang*, A. Dutt*, K. Y. Yang, C. C. Wojcik, J. Vučković, and S. Fan, Science 371, 1240 (2021).

Experimental demonstration of complex winding of asingle non-Hermitian band
Longer-ranges of coupling can give rise to more nontrivial windings
𝐶 − ∆𝑒 !"#−∆𝑒 !"#

…

…

𝐶

…

−Δ$ 𝑒 !"#$…

Δ′𝑒 "#$∆𝑒 "#𝐶 + ∆𝑒 "#Experiment

Experiment𝑘/𝜋𝑘/𝜋𝑣 = 1 𝑣 = −1 𝑣 = 1𝑣=0𝑣=0𝑣 = −1 𝑣 = 1𝑣 = −1𝑣 = −1𝑣 = −2𝑣 = −1

K. Wang*, A. Dutt*, K. Y. Yang, C. C. Wojcik, J. Vučković, and S. Fan, Science 371, 1240 (2021).
35

Multiple energy bands of Hermitian versus
non-Hermitian systems in one dimension
Hermitian

Non-HermitianBraiding of non-Hermitian energy bands
𝐸 𝑘 = 0 = 𝐸(𝑘 = 2𝜋)
Non-Hermitian band structurein (Re(E), Im(E), k) – space

Closure of the braid forms a knot/ link

Braid diagram

Knot diagram

Examples of two-band braids and the correspondingknots/links

UnlinkUnknotHopf link

Trefoil𝑵 separable non-Hermitian bands in one dimension
can be classified by the braid group 𝔹l
𝜋$ 𝑈𝐶𝑜𝑛𝑓9 ℂ= 𝔹9An example with three energy bands (𝑁 = 3)

Figure-eight knot

C. C. Wojcik, X.-Q. Sun, T. Bzdušek, and S. Fan, Phys. Rev. B 101, 205417 (2020).

A model Hamiltonian for realizing two-band braids
44E (1) = G 𝑔 𝑎I34 𝑏K3 + 𝑔 𝑏K34 𝑎I3 − 𝑖𝛾 𝑎I34 𝑎I3 + 𝜅 𝑎I35"
𝐻𝑎I3 + 𝜅 𝑎I34 𝑎I35" + (𝐶 − Δ) 𝑎I351
𝑎I3 + 𝐶 + Δ 𝑎I34 𝑎I351
3𝑚=2𝑚=1… 𝑔𝐶−∆

𝐶−∆Sub-lattice a𝐶+∆

…

𝐶+∆

… 𝑔

Sub-lattice b

K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, Nature 598, 59 (2021)

𝜅Sub-lattice a…
Sub-lattice bExperimental realization of the model Hamiltonian

𝑚=2

Phase modulation𝐶−∆

𝑇<= = exp[−𝑖(2𝜅 cos Ω𝑡 + 2𝐶 cos 𝑚Ω𝑡)]
Amplitude modulation

𝐶+∆

… 𝑔

Sub-lattice a𝜅

…
Sub-lattice b

𝑇>3 = 1 + 2Δ sin 𝑚Ω𝑡Frequency modesInput

BS1

𝜔#Resonator aBS2Resonator bInput
frequency

OutputLoss𝜔#𝛿𝜔Experimental realization of a two-band complex-energy braid𝛿𝜔/Ω𝛿𝜔/ΩTransmission signal

K. Wang, A. Dutt, C. C. Wojcik, and S. Fan, Nature 598, 59 (2021)

𝑚=2

Demonstration of higher-order braids𝑚=3

𝑚=4𝑠Ω/2𝛾$#𝛿𝜔/Ω𝛿𝜔/Ω𝑠Ω/2𝛾$#

Experiment

ExperimentOutline
Background

Hermitian Topology:
experiments

L0

Non-Hermitian Topology:
experiments

Non-Hermitian Topology:
theoryEigenvalue topology in one dimension𝑘𝜋𝑎−𝜋𝑎𝜋𝑘=𝑎𝜋𝑘=−𝑎Eigenvalue topology in two dimensions
𝑘:𝜋𝑘A , 𝑘B = (0, )𝑎𝜋−𝑎𝜋𝑘A , 𝑘B = (0, − )𝑎𝜋𝑎𝜋Exceptional
points𝑎−𝜋𝑘A , 𝑘B = (− , 0)𝑎𝑘;

𝜋𝑎𝜋𝑘A , 𝑘B = ( , 0)𝑎Classification of N-band eigenvalue topology

Group homomorphism

Class of homotopy

𝑇 q − Δ, 𝑈𝐶𝑜𝑛𝑓l 𝐶

ddimensional
Brillouin zone

Exceptional
contour= 𝐻𝑜𝑚(𝜋r 𝑇 q − Δ , 𝐵l )N-bandFundamentalgroup of𝑇C − Δ

N-strandbraid group

C. Wojcik, K. Wang, A. Dutt, J. Zhong and S. Fan, Physical Review B 106,
L161401 (2022).

2D example

𝐻𝑜𝑚 𝜋r 𝑇 s − Δ , 𝐵l = 𝑎, 𝑏, 𝑒r, 𝑒s, … , 𝑒t ∈ 𝐵l : 𝑎, 𝑏 = 𝑒r𝑒s ⋅⋅⋅ 𝑒t

C. Wojcik, K. Wang, A. Dutt, J. Zhong and S. Fan, Physical Review B 106,
L161401 (2022).

3D examples

C. Wojcik, K. Wang, A. Dutt, J. Zhong and S. Fan, Physical Review B 106,
L161401 (2022).

Summary
Background

Hermitian Topology:
experiments

L0

Non-Hermitian Topology:
experiments

Non-Hermitian Topology:
theory