Wave packet dynamics in a non-Hermitian exciton-polariton system Y.-M. Robin Hu1Elena A. Ostrovskaya1and Eliezer Estrecho1 1ARC Centre of Excellence in Future Low-Energy Electronics Technologies and

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Wave packet dynamics in a non-Hermitian exciton-polariton system

Y.-M. Robin Hu,1Elena A. Ostrovskaya,1and Eliezer Estrecho1

1ARC Centre of Excellence in Future Low-Energy Electronics Technologies and

Department of Quantum Science and Technology, Research School of Physics,

The Australian National University, Canberra, ACT 2601 Australia

(Dated: October 13, 2022)

We theoretically investigate the dynamics of wave packets in a generic, non-Hermitian, optically

anisotropic exciton-polariton system that exhibits degeneracies of its complex-valued eigenenergies

in the form of pairs of exceptional points in momentum space. We observe the self-acceleration and

reshaping of the wave packets governed by their eigenenergies. We further ﬁnd that the exciton-

polariton wave packets tend to self-organize into the eigenstate with the smaller decay rate, then

propagate towards the minima of the decay rates in momentum space, resulting in directional

transport in real space. We also describe the formation of pseudospin topological defects on the

imaginary Fermi arc, where the decay rates of the two eigenstate coincide in momentum space. These

eﬀects of non-Hermiticity on the dynamics of exciton polaritons can be observed experimentally in

a microcavity with optically anisotropic cavity spacer or exciton-hosting materials.

I. INTRODUCTION

Open dissipative systems described by non-Hermitian

Hamiltonian operators exhibit a special type of spectral

degeneracies called exceptional points [1–6], leading to

the emergence of novel topological invariants [1,2,6], new

topological states [1–3,7–9], nontrivial lasing [4,10,11],

non-reciprocal transmission [4,10] and unidirectional

transport [12,13]. Microcavity exciton polaritons, cre-

ated when the electron-hole pairs (excitons) are strongly

coupled to photons in an optical microcavity [6,14–16],

represent an accessible solid-state platform for studies

of non-Hermitian physics due to their inherent open-

dissipative character. Non-Hermitian spectral degenera-

cies, both in parameter and momentum space, as well as

the associated topological invariant, have been observed

in this system [5,6,17].

In a Hermitian system, the motion of a wave packet

is described by a pair of semi-classical equations of mo-

tion, and its centre-of-mass motion in momentum space

is governed by external forces [18–20]. Recently, it was

discovered that a wave packet in a system described by

a non-Hermitian Dirac model can move in momentum

space without the presence of an external force as a re-

sult of the growths and decay of its components corre-

sponding to diﬀerent eigenenergy branches [21]. The tra-

jectories of the wave packets under this self-acceleration

are polarization-dependent and the centre-of-mass mo-

menta for certain initial conditions follow the gradient

of the imaginary part of the eigenenergy. Similar eﬀects

were also described in the context of an one-dimensional

non-Hermitian lattice [22]. In the absence of an out-

of-plane magnetic ﬁeld, the model of exciton polaritons

in a planar microcavity has a band structure similar to

the non-Hermitian Dirac model investigated in Ref. [21],

which is characterized by pairs of exceptional points con-

nected by the so-called Fermi arcs [6]. Therefore, we

can expect similar wave packet dynamics to emerge in

a non-Hermitian optically anisotropic exciton-polariton

system.

In this work, we theoretically investigate the non-

Hermitian wave packet dynamics in a microcavity

exciton-polariton system. Apart from the motion in the

absence of an external force, we also ﬁnd that for some

initial conditions, the wave packets tend to split into mul-

tiple components that propagate towards diﬀerent direc-

tions. Moreover, these wave packets tend to self-organize

into diﬀerent eigenstates as they evolve and propagate

towards the maxima of the imaginary part of the corre-

sponding eigenenergy. Finally, we examine the exciton-

polariton pseudospin textures resulting from the wave

packet evolution and describe the emergent pseudospin

anti-merons [23–28] on the imaginary Fermi arc in mo-

mentum space. The anti-merons are non-singular topo-

logical point defects that are characterized by half-integer

topological invariant. They have been studied both in

real space [23,28–30] and in momentum space [25,31] in a

variety of physical systems. Their detection on the imag-

inary Fermi arc in an exciton-polariton system would sig-

nify a clear signature of the non-Hermitian wave packet

dynamics.

This work is organized as follows. In Section II, we

present the non-Hermitian exciton-polariton model con-

sidered in this work. In Section III A, we discuss the

wave-packet self-acceleration and splitting in momentum

space. In Section III B, we describe the asymptotic be-

haviour of the exciton-polariton wave packets in momen-

tum space and the unidirectional propagation in real

space. Finally, in Section III C, we present our inves-

tigation on the dynamics of the exciton-polariton pseu-

dospins, including the formation of the pseudospin de-

fects on the imaginary Fermi arc without (Section III C 1)

and with (Section III C 2) the presence of an out-of-plane

ﬁeld, and the defects formation in real space (Section

III C 3). In Appendix F, we compare the results for

the exciton-polariton model presented here to those of

the non-Hermitian Dirac model and the non-Hermitian

Chern insulator.

arXiv:2210.05860v1 [cond-mat.mes-hall] 12 Oct 2022

II. THEORETICAL MODEL

In this work, we focus on a non-Hermitian exciton-

polariton model exhibiting an eﬀective non-Abelian

gauge ﬁeld. The general form of this 2×2, two-band

Hamiltonian in momentum space is H(k) = H0(k)I+

−→

G(k)·−→

σ , where −→

G(k) = [Gx(k), Gy(k), Gz(k)] is the

gauge ﬁeld, −→

σis a vector of Pauli matrices, Iis the 2 ×2

identity matrix, and k= (kx, ky). The Hamiltonian is

written in the basis of circular polarization of the cav-

ity photon or the spin projection of the excitons on the

z-direction normal to the plane of the exciton-hosting

material embedded in the microcavity. This basis forms

the psuedospin of the exciton polaritons. Dropping the

kdependence for brevity, the complex energy spectrum

can be written simply as E±=H0±G.

We model the exciton polaritons in an optical micro-

cavity using the non-Hermitian Hamiltonian presented in

Ref. [6], which has the components:

H0(k) = E0−iγ0+~2k2

2m−iχk4

−→

G(k) = [α−ia + (β−ib)(k2

x−k2

y),2kxky(β−ib),∆].

(1)

The expression for the complex energy can be written as:

E±=E0−iγ0+~2k2

2m−iχk4±G

G=α−ia + (β−ib)(kx+iky)2

×α−ia + (β−ib)(kx−iky)2+ ∆21/2

(2)

where Gdenotes the mean-subtracted eigenenergies.

This model is a non-Hermitian generalization of the one

describing the microcavity exciton polaritons in Ref. [32].

Here, αdenotes the microcavity anisotropy which leads

to energy splitting between the linearly-polarized modes,

βdescribes the photonic spin-orbit coupling which splits

the TE (transverse electric) and TM (transverse mag-

netic) modes [6,32]. The parameters a,bdescribe the

photonic losses that are polarization-dependent due to

microcavity anisotropy and the TE-TM splitting [6]. The

presence of these parameters results in the non-Hermitian

nature of the Hamiltonian. The term ∆σzin the Hamil-

tonian describes the eﬀects of Zeeman splitting induced

by an out-of-plane magnetic ﬁeld which can be intro-

duced into the experimental setup [6,33]. The term

~2k2

2mapproximates the kinetic energies of the exciton po-

laritons, where mis the eﬀective polariton mass. The

two terms −iγ0−iχk4ensure that the imaginary part of

the eigenenergies Im E±are always negative. This is re-

quired since this model describes the eﬀects of both gain

(e.g., an optical pump) and loss (e.g., radiative decay) by

an eﬀective loss term which determines the linewidth of

the exciton-polariton spectrum. Positive values of Im E±

(c)

(b)

(a)

(d)

(e)

FIG. 1. The (a) real part and (b) imaginary part of the

eigenenergies of exciton-polariton model with cuts (denoted

by the orange and blue lines) to show the degeneracies and

the branch cuts. The green dots mark the exceptional points,

the pink lines are the bulk Fermi arc where the two Re E

surfaces touch and the purple lines are the imaginary Fermi

arcs where the two Im Esurfaces cross. (c): The simpliﬁed

structure of the bulk Fermi arcs, the imaginary Fermi arcs

and the exceptional points in momentum space. (d,e): The

imaginary parts of the eigenenergies at ky= 0 and kx= 0,

respectively. The black dots highlight the maxima of Im E±.

The values of the parameters used here and the rest of the

work are described in Appendix Aunless speciﬁed otherwise.

would suggest a net gain, which is not physical for this

system.

The eigenenergies E±are plotted in Fig. 1(a,b). The

two exciton-polariton energy bands exhibit four excep-

tional points, each pair connected by the bulk Fermi arc

where the real energy surfaces cross, and by the imag-

inary Fermi arc where the linewidth surfaces cross [see

Fig. 1(a-c)] [6,34]. A non-zero Zeeman splitting ∆ would

shrink the bulk Fermi arc, moving the exceptional points

in a pair towards each other. A suﬃciently strong Zee-

man splitting, |∆|>|(aβ −bα)/β|, will destroy the ex-

ceptional points and open a gap. This shows that the

exceptional points are more stable than the Dirac points

(Hermitian spectral degeneracies) which annihilate for

any non-zero value of ∆.

The imaginary part of each eigenenergy branch has

two local maxima. As shown in Fig. 1(b) and high-

lighted in Fig. 1(d,e), Im E+(represented by the orange

surfaces and the orange lines) has two maxima lying on

the ky-axis, while Im E−(represented by the blue sur-

faces and the blue lines) has two maxima at the kx-axis.

These points play an important role in the dynamics of

the exciton-polariton wave packets as discussed in the

next Section.

III. RESULTS AND DISCUSSION

Inspired by previous works [21,22], we investigate the

dynamics of Gaussian wave packets in a non-Hermitian

exciton-polariton systems in the absence of an exter-

nal potential. Since the Hamiltonian (1) contains k-

components only, i.e., no external potential or spatial

components, the time evolution of wave packets in real

and momentum space and the resulting pseudospin tex-

tures can be calculated exactly (see Appendix Bfor de-

tails).

The initial exciton-polariton wave packet is Gaussian

in momentum space and can be represented as a su-

perposition of multiple k-components of the pseudospin

eigenstates. The time evolution of the contributions from

these components can be described as:

|ψR

±(k, t)i=e−iRe E±(k)teIm E±(k)t|ψR

±(k)i,(3)

where the superscript Rdenotes the right-eigenstates (see

Appendix Cfor details). The real part of energies Re E±

describes the motion of the wave packet centre of mass

in real space while the imaginary part Im E±governs

the growth and decay of the corresponding eigenstate.

If Im E±>0, Im E±corresponds to the growth rate of

the corresponding eigenstate. Similarly, if Im E±<0,

−Im E±corresponds to the decay rate. The variation

in the imaginary part, as shown below, is responsible for

the peculiar eﬀects described in this work.

A. Self-Acceleration and Splitting of

Exciton-Polariton Wave Packets

Similar to the dynamics of the wave packets in a non-

Hermitian Dirac model [21], we observe the acceleration

of the exciton-polariton wave packets in the absence of

an external potential. Furthermore, the self-accelerating

wave packet trajectories are sensitive to the initial po-

larization as shown by the centre-of-mass motion in Fig.

2(a-c). Note, however, that regardless of the initial po-

larization, self-accelerating wave packets tend to move

towards the same point in momentum space or towards

the same direction in real space. This acceleration arises

from the gradient of the imaginary part of eigenenergy

∇Im E±[see Fig. 1(b,d,e)]. Namely, some k-components

are decaying faster than others, resulting in an eﬀective

displacement of the wave packet center of mass towards

(b)

(a)

(g)

(h)

t=5 ps

(c)

FIG. 2. The polarization-dependent trajectories of the

exciton-polariton wave-packet centre-of-mass in (a,b) the mo-

mentum space and (b) the real space, where (b) is the zoom-in

of (a). Note that the trajectories of the horizontally-polarized

and the vertically-polarized modes overlap. The white dots

denote the initial wave packet centre of mass in both momen-

tum and real space and the black dots denote the maxima of

Im E±in momentum space.

k-components with larger imaginary part (or lesser decay

rates).

Surprisingly, when the exciton-polariton wave packet is

initialized on the imaginary Fermi arc, it splits into two

components that propagate away from each other as seen

in Fig. 3(a-c). Furthermore, when the wave packet is

initialized at the origin k= 0, while overlapping with the

two imaginary Fermi arcs, it splits into four components.

The four components accelerate away from each other

along the ±kxand ±kydirections [see Fig. 3(d-f)].

The splitting of the wave packets is due to the diﬀer-

ent imaginary parts (or decay rates) of the two pseu-

dospin components. Since Im E±corresponds to the

growth/decay rates of the corresponding eigenstate, as

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Wavepacketdynamicsinanon-Hermitianexciton-polaritonsystemY.-M.RobinHu,1ElenaA.Ostrovskaya,1andEliezerEstrecho11ARCCentreofExcellenceinFutureLow-EnergyElectronicsTechnologiesandDepartmentofQuantumScienceandTechnology,ResearchSchoolofPhysics,TheAustralianNationalUniversity,Canberra,ACT2601Australia(Da...

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Wave packet dynamics in a non-Hermitian exciton-polariton system Y.-M. Robin Hu1Elena A. Ostrovskaya1and Eliezer Estrecho1 1ARC Centre of Excellence in Future Low-Energy Electronics Technologies and

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