Causality constraints and their symmetry
classification in passive devices
Henning Schomerus, Lancaster University & RIKENNH QM workshop, Tokyo University, 4 July 2022

NH effects
complex frequencies

exceptional points

nonorthogonal modes

NH skin effectcomplications
finite life times
enhanced sensitivity
instability
quantum noiseExample I: quantum noiseΓΓgainPTloss

exact PT symmetric phase:𝐼𝐼 𝜔𝜔 = tr (𝑆𝑆 † 𝑆𝑆 − 1)/2𝜋𝜋
remains finite for Γ → 0
HS, PRL 2010

EP to broken phase𝐼𝐼 𝜔𝜔 → (Lorentzian)2
Yoo, Sim & HS, PRA 2011

Petermann factor 𝐾𝐾𝑘𝑘 = 𝑈𝑈 † 𝑈𝑈 𝑘𝑘𝑘𝑘 𝑈𝑈 −1 𝑈𝑈 −† 𝑘𝑘𝑘𝑘 → ∞Example I: quantum noise

EP to broken phase𝐼𝐼 𝜔𝜔 → (Lorentzian)2Yoo, Sim & HS, PRA 2011

Petermann factor 𝐾𝐾𝑘𝑘 = 𝑈𝑈 † 𝑈𝑈 𝑘𝑘𝑘𝑘 𝑈𝑈 −1 𝑈𝑈 −† 𝑘𝑘𝑘𝑘 → ∞Example II: mode selection
topological zero mode

selected by gain and loss
HS 2013

midgapstate

microwaves

with Poli et al 2015

𝐻𝐻 = −𝑍𝑍𝐻𝐻 ∗ 𝑍𝑍, tr 𝑍𝑍 = 1 (C symmetry)lasersexcitons

with Zhao et al 2018with Whittaker et al 2018

Example II: mode selection
Nonlinear symmetry

Symmetry-protected Topological excitations
oscillations

Goldstone modes:
U(1) and time translation

Malzard, Cancellieri & HS 2018

microwaves

with Poli et al 2015

lasersexcitons

with Zhao et al 2018with Whittaker et al 2018

Example III: skin effectzero mode relocalises

Brandenbourger et al 2019transition to directed amplification

HS, PRR 2020

also in transport effects•

reflectionless transport depending on
topological phase•invisibility coinciding at
skin-effect phase transition•

CPA and one-sided transparency

Ghaemi-Dizicheh & HS, PRA 2021

Here: passive systems• physical limits from causality• symmetry constraints• visibility of effects:
EPs, skin effect, edge states

Causality
recall output intensity 𝐼𝐼 𝜔𝜔 = tr (𝑆𝑆 † 𝑆𝑆 − 1)/2𝜋𝜋passive system: 1 − 𝑆𝑆 † 𝑆𝑆 is positive definite
generic wide-band limit: 𝑆𝑆 𝜔𝜔 =1−𝑖𝑖Γ𝒢𝒢 𝜔𝜔1+𝑖𝑖Γ𝒢𝒢 𝜔𝜔with 𝒢𝒢 𝜔𝜔 =1
𝜔𝜔−𝐻𝐻⇒ 1 − 𝑆𝑆 † 𝑆𝑆 = 2 Γ Q with Q = 2𝒢𝒢 † (𝜔𝜔 + 𝑖𝑖Γ) (𝛾𝛾 − 𝐹𝐹)𝒢𝒢(𝜔𝜔 + 𝑖𝑖Γ)•••time-delay op., gives density of states 𝜌𝜌 𝜔𝜔 = tr 𝑄𝑄/2𝜋𝜋
𝐻𝐻 = 𝐻𝐻0 + 𝑖𝑖𝑖𝑖 − 𝑖𝑖𝑖𝑖 : nontrivial NH in 𝐹𝐹, background losses 𝛾𝛾
𝑄𝑄 positive definite: causality threshold 𝛾𝛾

causality > Lee-Wolfenstein (Wiersig 2019) > positive lifetimes

Symmetry classification
𝐻𝐻 = 𝐻𝐻 ∗(TRS)

𝐻𝐻 = 𝑃𝑃𝐻𝐻 ∗ 𝑃𝑃(PT)𝐻𝐻 = 𝐻𝐻 𝑇𝑇(reciprocal)

𝐻𝐻 = 𝑃𝑃𝐻𝐻 † 𝑃𝑃(PTT’)

𝐻𝐻 = −𝑃𝑃𝐻𝐻 ∗ 𝑃𝑃 (C)𝐻𝐻 = −𝑃𝑃𝐻𝐻 † 𝑃𝑃 (CT’)⇒

𝐹𝐹 = −𝐹𝐹 𝑇𝑇⇒

𝐹𝐹 = −𝑃𝑃𝐹𝐹 ∗ 𝑃𝑃 (charge conj, C)⇒⇒⇒⇒(Majorana basis)

𝐹𝐹 = 𝐹𝐹 ∗(TRS)𝐹𝐹 = −𝑃𝑃𝑃𝑃𝑃𝑃(chiral)

𝐹𝐹 = 𝑃𝑃𝑃𝑃𝑃𝑃(parity)

𝐹𝐹 = 𝑃𝑃𝐹𝐹 ∗ 𝑃𝑃(gen TRS)

• systematic pairing of NH and H symmetry classes• classes of 𝐻𝐻 and 𝑖𝑖𝑖𝑖 differ

Visibility of EPs
𝑎𝑎 − 𝑎𝑎∗
𝑐𝑐 − 𝑏𝑏 ∗𝑏𝑏 − 𝑐𝑐 ∗𝑎𝑎∗ − 𝑎𝑎𝑎𝑎 − 𝑖𝑖𝑖𝑖
EP normal form 𝐻𝐻 =𝑐𝑐𝑏𝑏−𝑎𝑎 − 𝑖𝑖𝑖𝑖

EP 𝑎𝑎2 + 𝑏𝑏𝑏𝑏 = 0: dos 𝜌𝜌𝐸𝐸𝐸𝐸 𝜔𝜔 =1𝑏𝑏 + 𝑐𝑐is a simple Lorentzian!
𝜋𝜋 𝜔𝜔2 + 𝑏𝑏 + 𝑐𝑐 2 /4

causality threshold 𝛾𝛾𝑐𝑐 =1⇒ 𝐹𝐹 =2𝑖𝑖(Im 𝑎𝑎)2 + 𝑏𝑏 − 𝑐𝑐 ∗ 2 /4

Visibility of skin effect and edge states
nonreciprocal dimer chain
𝑣𝑣 + 2𝑤𝑤1 cos 𝑘𝑘𝐻𝐻(𝑘𝑘) = 1 +𝑢𝑢1 + 𝑢𝑢2− 𝑒𝑒 𝑖𝑖𝑖𝑖

periodicopen⇒ 𝐹𝐹 𝑘𝑘 ⇒ 𝛾𝛾𝑐𝑐 =⇒ skin effect disguised, edge states visible

𝑢𝑢1− + 𝑢𝑢2+ 𝑒𝑒 −𝑖𝑖𝑖𝑖
𝑣𝑣2 + 2𝑤𝑤2 cos 𝑘𝑘

Δ𝑣𝑣 2 + Δu1 + Δu22

acknowledgements/references

Intro:
Quantum noise in PT exact phase:𝐼𝐼 𝜔𝜔 → (Lorentzian)2 at EP:
Topological mode selection theory:
Microwave demonstration:
Topological laser:
Exciton condensate:
Nonlinear extensions:
Directed amplification & sensing:Nonreciprocal transport signatures:

HS, PRL 104, 233601 (2010)
G Yoo, H-S Sim & HS, PRA 84, 063833 (2011)
HS, Opt Lett 38, 1912-1914 (2013)
C Poli et al, Nat Commun 6, 6710 (2015)
H Zhao et al, Nat Commun 9, 981 (2018)
C Whittaker et al, PRL 120, 097401 (2018)
S Malzard et al, Opt Exp 26, 22506 (2018)
S Malzard & HS, New J Phys 20, 063044 (2018)
HS, Phys Rev Research 2, 013058 (2020)
H Ghaemi-Dizicheh & HS, PRA 104, 023515 (2021)

Causality constraints:

HS, in preparation

Summary: NH topology from gain & loss
quantum noisemode selection

passive devices:causality constraints ⇒threshold lossesscattering theory ⇒EP’s & NHSE disguised,edge states visible

directed ampl.

⇒ lasers,
sensors …