Dynamically crossing diabolic points while encircling exceptional curves A programmable symmetric-asymmetric multimode switch Ievgen I. Arkhipov1Adam Miranowicz2 3Fabrizio Minganti4 5Şahin K. Özdemir6and Franco Nori7 8 9y

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Dynamically crossing diabolic points while encircling exceptional curves:

A programmable symmetric-asymmetric multimode switch

Ievgen I. Arkhipov,1, ∗Adam Miranowicz,2, 3 Fabrizio Minganti,4, 5 Şahin K. Özdemir,6and Franco Nori7, 8, 9, †

1Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS,

Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

2Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan

3Institute of Spintronics and Quantum Information, Faculty of Physics,

Adam Mickiewicz University, 61-614 Poznań, Poland

4Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

5Center for Quantum Science and Engineering, Ecole Polytechnique

Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland

6Department of Engineering Science and Mechanics, and Materials Research Institute (MRI),

The Pennsylvania State University, University Park, Pennsylvania 16802, USA

7Theoretical Quantum Physics Laboratory, Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan

8Quantum Information Physics Theory Research Team,

Quantum Computing Center, RIKEN, Wakoshi, Saitama 351-0198, Japan

9Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

(Dated: April 18, 2023)

Nontrivial spectral properties of non-Hermitian systems can lead to intriguing eﬀects with no

counterparts in Hermitian systems. For instance, in a two-mode photonic system, by dynamically

winding around an exceptional point (EP) a controlled asymmetric-symmetric mode switching can

be realized. That is, the system can either end up in one of its eigenstates, regardless of the initial

eigenmode, or it can switch between the two states on demand, by simply controlling the winding

direction. However, for multimode systems with higher-order EPs or multiple low-order EPs, the

situation can be more involved, and the ability to control asymmetric-symmetric mode switching

can be impeded, due to the breakdown of adiabaticity. Here we demonstrate that this diﬃculty

can be overcome by winding around exceptional curves by additionally crossing diabolic points.

We consider a four-mode PT -symmetric bosonic system as a platform for experimental realization

of such a multimode switch. Our work provides alternative routes for light manipulations in non-

Hermitian photonic setups.

INTRODUCTION

Physical systems that are described by non-Hermitian

Hamiltonians (NHHs) have attracted much research in-

terest during the last two decades thanks to their peculiar

spectral properties. Namely, such systems can possess ex-

otic spectral singularities referred to as exceptional points

(EPs). While in classical and semiclassical systems EPs

are associated with the coalesce of both the eigenvalues

and the corresponding eigenmodes of an NHH (thus, re-

ferred to as Hamiltonian EPs) [1,2], in quantum systems

they are associated with eigenvalue degeneracies and the

coalescence of the corresponding eigenmatrices of a Li-

ouvillian superoperator (hence, Liouvillian EPs) [3]. The

latter takes into account the eﬀects of decoherence, quan-

tum jumps, and associated quantum noise.

In addition to EPs, physical systems can also exhibit

diabolic point (DP) spectral degeneracies where eigen-

values coalesce but the corresponding eigenstates remain

orthogonal. Although they are often referred to as Her-

mitian spectral degeneracies and studied in Hermitian

systems, it is well-known that DPs can emerge in non-

Hermitian systems, too.

The term DP was coined in Ref. [4] referring to the de-

generacies of energy levels of two-parameter real Hamil-

tonians. Graphically, such a DP corresponds to a double-

cone connection between energy-level surfaces resembling

a diabolo toy, which justiﬁes the DP notion.

Analogously to EPs, this original deﬁnition of DPs was

later generalized to the eigenvalue degeneracies of non-

Hermitian Hamiltonians (see, e.g., [5]) as DPs of classi-

cal or semiclassical systems and DPs of Liouvillians [3] in

case of quantum systems. Note that quantum jumps are

responsible for a fundamental diﬀerence between semi-

classical and quantum EPs/DPs, and the eﬀect of quan-

tum jumps can be experimentally controlled by postse-

lection [6].

EPs have been predicted and observed in diﬀerent ex-

perimental platforms [1,6–22]. It seems that DPs in non-

Hermitian systems have been attracting relatively less in-

terest than EPs in recent years (see, e.g., [1,18,23,24]).

The reported demonstrations of a Berry phase (with a

controlled phase shift), acquired by encircling a DP [25–

27], can lead to applications in topological photonics [28],

quantum metrology [29], and geometric quantum compu-

tation in the spirit of Refs. [30–33]. Note that the Berry

curvature (i.e., the “curvature” of a certain subspace) can

be nonzero for non-Hermitian systems and, thus, can be

used for simulating eﬀects of general relativity [34–36].

The emergence of geometric Berry phases is quite com-

arXiv:2210.14840v2 [physics.optics] 16 Apr 2023

mon in non-Hermitian systems, but the acquired phases

can be largely enhanced by encircling DPs or EPs [37–39].

Moreover, DPs and EPs are useful in testing and classify-

ing phases and phase transitions [40,41]. For example, a

Liouvillian spectral collapse in the standard Scully-Lamb

laser model occurs at a quantum DP [42,43].

Recent studies on EPs have also shown that by ex-

ploiting a nontrivial topology in the vicinity of EPs in

the energy spectrum can lead to a swap-state eﬀect,

where the initial state does not come back to itself af-

ter a round trip around an EP. Such phenomenon has

been predicted theoretically [44,45] and observed exper-

imentally in [21,37,46–48], while performing ‘static’,

i.e., independent, measurements at various locations in

the system parameter space. However, when encircling

an EP dynamically, another intriguing eﬀect can be in-

voked; namely, a chiral mode behavior, such that a start-

ing state, after a full winding period, can eventually re-

turn to itself [49–52]. The latter eﬀect stems from the

breakdown of the adiabatic theorem in non-Hermitian

systems [49,53]. This asymmetric mode switching phe-

nomenon has also been experimentally conﬁrmed in var-

ious platforms [38,54–58]. A number of studies have

demonstrated the practical feasibility to observe the chi-

ral light behavior on a pure quantum level [59] and even

in a so-called hybrid mode [60], where by exploiting var-

ious measurement protocols, one can switch between the

system dynamics described by a quantum Liouvillian and

the corresponding classical-like eﬀective NHH.

Other works, both theoretical [61] and experimen-

tal [62], have pointed that a crucial ingredient in detect-

ing a dynamical ﬂip-state asymmetry is the very curved

topology near EPs. In other words, it is not necessary

to wind around EPs in order to observe such phenom-

ena. However, the dynamical contours must be in a close

proximity to EPs [61].

More recently, much eﬀort is put on studying the be-

havior of modes while encircling high-order or multiple

EPs in a parameter space of multimode systems. In-

deed, the presence of high-order or multiple low-order

EPs in a system spectrum, along with the non-Hermitian

breakdown of adiabaticity, can impose a substantial diﬃ-

culty to manipulate the mode-switching behavior on de-

mand [52,63,64]. That is, a system may end up only

in a few states out of many regardless of the encircling

direction and winding number.

In this work we demonstrate that dynamically wind-

ing around exceptional curves (ECs), whose trajecto-

ries can additionally cross diabolic curves (DCs), pro-

vides a feasible route to realize a programmable multi-

mode switch with controlled mode chirality. We use a

four-mode parity-time (PT )-symmetric bosonic system,

which is governed by an eﬀective NHH, as an exemplary

platform to demonstrate this programmable switch. At

the crossing of EC and DC a new type of a spectral sin-

gularity is formed, referred to as diabolically degener-

ate exceptional points (DDEPs) [65]. By exploiting the

presence of DDEPs in dynamical loops of the system pa-

rameter space, one can restore the swap-state symmetry,

which breaks down in two-mode non-Hermitian systems.

This implies that the initial state can eventually return to

itself after a state ﬂip in a double cycle. In other words,

the interplay between the topologies of EPs and DPs en-

ables one to restore (impose) mode symmetry (asymme-

try) on demand. These results are valid also for purely

dissipative systems (i.e., loss only systems without gain)

and can be extended to arbitrary multimode systems.

RESULTS

Theory

We start from the construction of a four-mode NHH,

possessing both exceptional and diabolic degeneracies.

For this, we follow the procedure described in [65], where

one can construct a matrix, whose spectrum is a combi-

nation of the spectra of two other matrices by exploiting

Kronecker sum properties. Namely, by taking two PT -

symmetric matrices

M1=i∆k

k−i∆, M2=0g

g0,(1)

one can form a PT -symmetric 4×4non-Hermitian matrix

H=M1⊗I+I⊗M2,(2)

where Iis the 2×2identity matrix. Explicitly, the matrix

Hreads

H=





i∆g k 0

g i∆ 0 k

k0−i∆g

0k g −i∆







.(3)

The symbols in Eq. (3) can have various physical mean-

ings, but in our context they may denote, e.g., cou-

pling (g, k) and dissipation (∆) strengths in a pho-

tonic system (see the text below). The PT -symmetry

operator is expressed via the parity operator P=

antidiag[1,1,1,1] and the time-reversal operator T, thus,

implying PT HPT −1=H. The matrix Hcan be related

to a linear four-mode NHH operator ˆ

H, written in the

mode representation, i.e.,

H=Xˆa†

jHˆak,

where ˆai(ˆa†

i) are the annihilation (creation) operators

of bosonic modes i= 1,...,4. Such an NHH can be

associated, e.g., with a system of four coupled cavities or

waveguides (see Fig. 1a). A similar scheme, based on two

lossy and two ampliﬁed subsystems, has been proposed

FIG. 1. Scheme and encircling trajectory space for a four-

mode system. aSchematic representation of a four-mode PT -

symmetric non-Hermitian Hamiltonian ˆ

H, given in Eq. (3).

The red (blue) balls represent cavities with gain (loss) rate

i∆(−i∆). Various mode couplings are depicted by double

arrows. bThe encircling trajectory is described by a loop in

the 3D parameter space deﬁned by the dissipation strength ∆,

perturbation δ, and coupling g. The winding direction can be

either counterclockwise or clockwise, as shown by arrows. The

encircling starts at t0at a point in the exact PT -phase (the

orange ball). The loop winds around an exceptional curve,

EC (red vertical line), determined by the condition ∆ = 1

and δ= 0. The trajectory may cross a diabolic curve, DC

(green horizontal line), at some point when g= 0, i.e., a dia-

bolic point, DP. Moreover, at g= 0, a diabolically degenerate

exceptional point, DDEP, is formed, at the intersection of EC

and DC. Note that in this 3D parameter space, the DC is

presented as a line.

in Ref. [66] to generate high-order EPs but with diﬀerent

coupling conﬁguration and spectrum with no DPs.

The peculiarity of such a non-Hermitian Hamiltonian

His that its eigenvalues are just sums of the eigenvalues

of M1(±√k2−∆2) and M2(±g) [65,67]. Namely,

E1,2,3,4=∓pk2−∆2∓g. (4)

In what follows, we always list eigenvalues in ascending

order, i.e.,

Re(E1)≤Re(E2)≤Re(E3)≤Re(E4).

The corresponding eigenvectors of Hare simply formed

by the tensor products of eigenvectors of ψM1

jand ψM1

(j, k = 1,2) of the two matrices M1and M2, respectively,

ψM1

1,2=±exp (±iφ)

1, ψM2

1,2=±1

1,(5)

where φ= arctan ∆/√k2−∆2. Namely, the eigen-

vector ψH

jk =ψM1

j⊗ψM2

kcorresponds to the eigenvalue

jk =EM1

j+EM2

kof the matrix H[65]. The spectrum

of this PT -symmetric ˆ

Hhas two types of degeneracies:

•a pair of second-order ECs at k= ∆, determined

by ±g(g6= 0),

•and a pair of DCs at g= 0, deﬁned by ±√k2−∆2.

FIG. 2. Spectrum of a four-mode PT -symmetric system.

Real aand imaginary bparts of the spectrum of the non-

Hermitian Hamiltonian, NHH, H(δ)in Eq. (11). For real-

valued energies, the spectrum of the NHH is formed by two

pairs of Riemann surfaces, whereas for the imaginary-valued

spectrum, those two pairs coincide. Each pair of Riemann

sheets, for a given value of g, has a branch cut at an excep-

tional point determined by the conditions ∆=1and δ= 0.

The system parameters are: k= 1 and g= 2.

System dynamics in modulated parameter space

In order to implement the dynamical winding around

ECs, one may apply a perturbation δ(t)to the NHH ˆ

in the following form:

H(δ) = 





i∆(t) + δ(t)g(t) 1 0

g(t)i∆(t) 0 1

1 0 −i∆(t)g(t)

0 1 g(t)−i∆(t)−δ(t)







(6)

where we set k= 1, i.e., the coupling kdetermines a unit

of the system energy. The time-dependent parameters

are:

∆(t) = 1 + cos(ωt +φ0),

g(t) = g0sin2(ωt/2 + φ0/2),

δ(t) = sin(ωt +φ0).(7)

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Dynamicallycrossingdiabolicpointswhileencirclingexceptionalcurves:Aprogrammablesymmetric-asymmetricmultimodeswitchIevgenI.Arkhipov,1,AdamMiranowicz,2,3FabrizioMinganti,4,5ahinK.Özdemir,6andFrancoNori7,8,9,y1JointLaboratoryofOpticsofPalackýUniversityandInstituteofPhysicsofCAS,FacultyofScience,Palac...

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Dynamically crossing diabolic points while encircling exceptional curves A programmable symmetric-asymmetric multimode switch Ievgen I. Arkhipov1Adam Miranowicz2 3Fabrizio Minganti4 5Şahin K. Özdemir6and Franco Nori7 8 9y

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