Dynamical Phase Transitions in Quantum Systems

doi:10.4236/jmp.2010.15043

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Journal of Modern Physics, 2010, 1, 303-311

doi:10.4236/jmp.2010.15043 Published Online November 2010 (http://www.SciRP.org/journal/jmp)

Dynamical Phase Transitions in Quantum Systems

Ingrid Rotter

Max Planck Institute for the Physics of Complex Systems, D-01187Dresden, Germany

E-mail: rotter@mpipks-dresden.mpg.de

Received September 16, 2010; revised October 5, 2010; accepted October 20, 2010

Abstract

Many years ago Bohr characterized the fundamental differences between the two extreme cases of quantum

mechanical many-body problems known at that time: between the compound states in nuclei at extremely

high level density and the shell-model states in atoms at low level density. It is shown in the present paper

that the compound nucleus states at high level density are the result of a dynamical phase transition due to

which they have lost any spectroscopic relation to the individual states of the nucleus. The last ones are

shell-model states which are of the same type as the shell-model states in atoms. Mathematically, dynamical

phase transitions are caused by singular (exceptional) points at which the trajectories of the eigenvalues of

the non-Hermitian Hamilton operator cross. In the neighborhood of these singular points, the phases of the

eigenfunctions are not rigid. It is possible therefore that some eigenfunctions of the system align to the scat-

tering wavefunctions of the environment by decoupling (trapping) the remaining ones from the environment.

In the Schrödinger equation, nonlinear terms appear in the neighborhood of the singular points.

Keywords: Non-Hermiti an Quantum Physics, Dynamical Phase Transitions, Phase Rigidity of Eigenfunctions,

Nonlinear Schrodinger Equation, Exceptional Points

1. Introduction

In 1936, Niels Bohr wrote in the address delivered on

January 27 befor e th e Copenhag en Acad emy [1,2 ]: In the

atom and in the nucleus we have indeed to do with two

extreme cases of mechanical many-body problems for

which a procedure of approximation resting on a

combination of one-body problems, so effective in the

former case, loses any validity in the latter where we,

from the very beginning, have to do with essential

collective aspects of the in terplay between the constituent

particles. About 20 years later, it could be shown that the

spectra of nuclei at low excitation energy are described

well on the basis of the shell model [3], i.e. on the basis

of an approximation resting on a combination of one-

body problems. The shell closures in nuclei differ from

those in atoms since the symmetries of nuclear forces

differ fro m those of the forces in atoms. In the fo llowing

years, the collective aspects of the interplay between the

constituent particles were considered in nuclear physics

studies only partly, i.e. they were reduced, above all, to

the two-body residual interaction in the framework of the

shell model. These studies prov ided very good results fo r

light and medium-mass nuclei as well as for the lo w-lyi ng

states of heavy nuclei. In the case of the narrow com-

pound-nucleus resonances, the contradictions to some

basic assumptions of the statistical description were

ignored. Here, most studies are performed by using the

Gaussian orthogonal ensemble although the differences

of this ensemble to a two-body (random) ensemble are

not discussed thoroughly. Also the influence of particle

decay thresholds is almost not considered in these

papers.

As a result of nuclear physics studies during many

years, we have to accept today that the resonance states

at low and high level density differ fundamentally from

one another. In light nuclei, most resonance states are at

low excitation energy of the nucleus, where the level

density is small. The lifetimes of the resonance states are

often near to the limit for single-particle (or alpha) decay.

All resonance states show individual spectroscopic

features.

The situation in heavy nuclei is completely different.

The first (elastic) threshold for particle decay is at about

8 MeV excitation energy of the nucleus where the level

density is extremely high. In a small energy region above

this threshold, the so-called neutron (compound nucleus)

resonances are identified. They are extremely long-lived

I. ROTTER

304

corresponding to decay widths of the order of eV. The

central part of the spectrum is described well by stat isti cal

methods, e.g. by the Gaussian orthogonal ensemble. The

single resonance states decay according to a power law

[4,5] and show chaotic features [6]. Much less discussed

in literature are the so-called single-particle resonances in

heavy nuclei the widths of which are of the order of

magnitude of MeV. Their energy is mostly just below the

elastic decay threshold, and their width at energies above

the threshold (see [7] for the energy dependence of the

widths) is much larger than the widths of the long-lived

states. In the cross section, they appear as a smooth

background for the very narrow neutron resonances. The

time scales of these two different types of resonance

states are well separated from one another: up to6

10 neut ron

resonances are overlapped by one single-particle resonance.

The Feshbach unified theory of nuclear reactions gives a

good description of this situation [8,9].

In medium-mass nuclei , t he firs t (el asti c ) dec ay threshold

is at a comparably low excitation energy of the nucleus

where the level density is still relatively low. These

nuclei are characterized by overlapping resonance states

with different lifetimes. An example are the isobaric

analogue resonances which are described well by the

doorway picture. According to this picture, the doorway

states coexist with long-lived co mpound nucleus r e so n a n c e

states (see also [10]). The doorway states being comparably

short-lived, are coupled directly to the decay channels

and to the narrow compound nucleus resonance states.

The narrow resonance states, however, are assumed to be

coupled to the continuum only via the doorway states.

This model characterizes the transition from the regime

at low level density to that at high level density. Un-

fortunately, this transition can not be controlled by a

parameter since the nuclear forces are too strong. Ac-

cording to [7], it may be considered to be a dynamical

phase transition.

This interpretation is supported by experimental results

obtained some years ago for the mean compound-

nucleus lifetime in proton induced reactions on Ni

isotopes by using the crystal blocking technique [11].

The mean lifetime is determined at the energies =E

5.65 and 6.50 MeV. It is significantly longer at the

higher bombarding energy than at the lower one,

contrary to expectations of a pu rely statistical th eo ry, and

is much smaller than expected. Furthermore, the data

suggest an interpretation in terms of some form of

intermediate structure resulting from the local spreading

of a comparatively simple configuration [11]. This picture

obtained experimentally, fits in with the spectroscopic

redistribution processes appearing (according to the theory

of open quantum systems) in the regime of overlapping

resonances [7,12] .

In the following, dynamical phase transitions in quan-

tum systems will be discussed. They are directly related

to the existence of exceptional points the mathematical

properties of which are known for more than 40 years

[13]. Their role in physical systems is discussed only

recently. They are identified also in theoretical studies

for nuclei under realistic conditions [14]. It will be

shown in the present paper that the above cited statement

by Niels Bohr is true in spite of its seeming contradiction

to the shell structure of nuclei. The narrow compound

nucleus resonances in heavy nuclei (well known at that

time) are the result of a dynamical phase transitio n. They

are characterized by essential collective aspects of the

interplay between the constituent particles and not by a

combination of one-body problems. Exceptional points

play an important role in this transition.

In Section 2, the mathematical properties of ex cep tiona l

points are sketched. They are singular points appearing at

the crossing points of eigenvalue trajectories. The scattering

on many-level systems is considered in Section 3. The

exceptional points influence not only resonance states

but also discrete states the energy of which is beyond the

window coupled directly to the continuum of scattering

wavefunctions. Resonance trapping and dynamical phase

transitions are discussed in Section 4. Here, also the

experimental verification of the resonance trapping phe-

nomenon is mentioned. In contrast to nuclei, governed by

the strong nuclear forces, the appearance of a dynamical

phase transition can be controlled by means of a para-

meter in many other quantum systems. In Section 5,

three examples will be considered, in which a dynamical

phase transition is experimentally shown to exist. Some

outlook is given in the last section.

2. Definition and Mathematical Properties of

Exceptional Points

Many years ago, Kato [13] introduced the notation ex-

ceptional points for singularities appearing in the per-

turbation theory for linear operators. Consider a family

of operators of the form

()= (0)TT T





 (1)

where



is a scalar parameter, (0)T is the unperturbed

operator and T





is the perturbation. Then the number

of eigenvalues of ()T



is independent of



with the

exception of some special values of



where (at least)

two eigenvalues coalesce. These special values of



are

the exceptional points. An example is the operator

I. ROTTER

305

()= .













(2)

In this case, the two values =



 i give the same

eigenvalue 0.

Operators of the type (2) appear in the description of

physical systems, for example in the theory of open

quantum systems [7]. In this case, they represent a

symmetric 22Hamiltonian describing a two-level

system with the unperturbed energies 1



and 2



and

the interaction



between the two levels,

()= .H













(3)

In an open quantum system, two states can interact

directly (corresponding to a first-order term) as well as

via an environment (second-order term) [7]. In the

present paper, we consider the case that the direct in-

teraction is contained in th e energies (=1,2)



. Then



contains exclusively the coupling of the states via the

environment which, in the case of an open quantum

system, consists of the continuum of scattering wave-

functions into which the system is embedded. This allows

us to study environmentally induced effects in open quan-

tum systems in a very clear manner.

The eigenvalues of the operator ()H



are

1,21 2

=;=()4.





 



(4)

The two eigenvalue trajectories cross when =0Z, i.e.

when





 (5)

At these crossing points, the two eigenvalues coalesce,

120

= .







(6)

The crossing points may be called therefore exceptional

points.

However, there are some essential differences between

the exceptional points considered in the mathematical

literature and the crossing points which appear in phy-

sical systems. The differences arise from the fact that the

crossing points are points in the continuum of scattering

wavefunctions (which represents the environment). They

are therefore of measure 0 and can not be observed

directly. However, they influence the behavior of the

eigenvalue trajectories ()





(where



is a certain

parameter) in their neighborhood in a non-negligible

manner. Thus, the most interesting features of the ex-

ceptional (crossing) points in physical systems are not

the properties at the crossing points themselves. Much

more interesting are their effects onto the eigenvalue

trajectories ()





in a finite parameter range around

the critical value c



(at which two trajectories cross)

and, above all, the behavior of the eigenvalue trajectories

in approaching the crossing point,c

()( )

kkr





. The

phenomenon of avoided level crossing is known in

physical systems since many years [15,16]. It occurs not

only for discrete states but also for narrow resonance

states [17]. In the scattering theory, the crossing points

appear as double poles of theSmatrix.

The Hamilton operator

describing an open qu antum

system is non-Hermitian. The eigenfunctionsk



of such

an operator are biorthogonal,

|= .

kl kl

 

 (7)

From these equations follows [7]

kk k





 (8)

and

|=| ; ||0.

klklkkk k

 





 (9)

At the crossing point

() ()

crl cr

AB  (10)

and the relation between the eigenfunctions 1



and 2



of the operator (3) is

cccc

1221

rrrr



 

  (11)

According to the last relation (11), the two eigen-

functions are linearly dependent of one another at the

crossing point such that the number of eigenfunctions of

is redu ced at this point. This result shows on ce more

that the crossing point is an exceptional point in the

sense defined by Kato [1 3 ] .

Let us now consider the consequences of the biortho-

gonality relations (7) and (8) for the two borderlin e cases

characteristic of neighboring resonance states.

1) The two levels are distant from one another. Then

the eigenfunctions are (almost) orthogonal

*||=1.

kkkk k

 



 (12)

2) The two levels cross. Then the two eigenfunctions

are linearly dependent according to (11) and

|= .

kk k





 (13)

according to (10).

The two relations (12) and (13) show that the phases

of the two eigenfunctions relative to one another change

when the crossing point is approached. This can be ex-

pressed quantitatively by defining the phase rigidity k

of the eigenfunctions k



|=.







 (14)

According to (12) and (13) hold s

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306

1 0.

r (15)

The non-rigidityk

rof the phases of the eigen-functions of

follows also from the fact that





is a complex number (in difference to the norm



which is a real number) such that the nor-

malization condition (7) can be fulfilled only by the

additional postulation Im *|=0





(what corres-

ponds to a rotation).

It should be mentioned finally, that the Schrödinger

equation describing an open quantum system, becomes

nonlinear near an exceptional point [7,17]. The most

important part of the nonlinear contributions is contained

() ||| || |=0;

kkkkk k

 













(16)

with 0

HW in (3). This equation is a nonlinear

Schrödinger equation. The degree of nonlinearity is

determined by the value 2



, Equation (8), i.e. by

the biorthogonality of the eigenfunctions of the non-

Hermitian Hamilton operator e

. It is the larger the

closer the crossing (exceptional) point is approached.

3. Scattering on Many-Level Systems

In an open many-level quantum system, the states of the

system Bcan interact via the common environment C.

Hence, the Hamilton operator consists of a first-order

and a second-order interaction term,

fBBC CB

HV V



 (17)

with



R||=||

ˆˆ

BBBB

iffj iBj

Ecc

hr ij

Ehr

eH H











 (18)



e1ˆˆ

I||=.

Bcc

iffj ij





 (19)

Here, the

are eigenfunctions of

()=0

Bkk

HE , and the scattering wave functions



follow from () =0



. Further, P denotes

the principal value integral. The ,

CCB

VV stand for the

interaction between the two subspaces B and C and

ˆ=2 ||

ccB

kEk



 (20)

is the coupling matrix element between the wavefunction

 of the discrete state k of the system and the

scattering wavefunction c



of the environment. The

direct (first-order) interaction V between two states is

included in

and its eigenfunctions

.

According to [7 ], we are lookin g for the exact solution

of the problem. The Ham i ltonian e

is non-Hermitian,

generally. We calculate not only Im



iffj

H,

but also Re





iffj

H, and that by including the

principal value integral and without any statistical

assumptions. The Schrödinger equation reads

()=0

ffk k



 (21)

with the eigenvalues k

z and eigenfunctions k



of the

Hamilton operator e

. In detail:

1) The states inside the energy window are coupled

directly to the environment such that the effective

Hamilton operatore

is non-Hermitian, i.e. the prin-

cipal value integral in (18) as well as the residuum (19)

have to be calculated. The eigenvalues are complex,

kk k

zE

(22)

in general, and the eigenfunctions k

 are complex and

biorthogonal,

|=,

ij ij



 (23)

compare (7). The coupling matrix elements between the



and the



are

=2 ||

kck



 (24)

in analogy to (20).

2) The states outside the energy window are not

coupled directly to the environment such that the

effective Hamiltonian e

is Hermitian at the energy

of the states, i.e. only the principal value integral in (18)

has to be calculated. At the energy of the states, the

eigenvalues =

zE are real, i.e. =0

, and the k



are orthogonal in the standard manner,

|=.

ij ij



 (25)

The coupling matrix elements between the k

 and the



vanish at the energy of the state. They are, however,

different from zero at energies inside the window

coupled directly to the environment and contribute to the

principle value integral.

Thus, the non-Hermitian Hamilton operator e

the open system provides, according to the boundary

conditions, resonance or discrete states. It is interesting

to remark that the spectroscopic properties of mirror

nuclei differ from one another. An example are the

nuclei 17 Oand 17

with 8 (9) protons and 9 (8)

neutrons in the first (second) case. The differences arise

I. ROTTER

307

from the different positions of the energetically lowest

decay channel (being a neutron (proton) decay channel)

[12].

The individual states k

 of the many-level system

have different spectroscopic properties and hence depend

on parameters in a different manner. They may therefore

cross or avoid crossing. The most interesting effects

appear in the very neighborhood of the crossing points

where the contributions of all the other states to the

crossing phenomenon need not to be considered. Hence,

the exceptional points defined in (2) and (3) play an

important role also in the many-level system. The phases

of the eigenfunctionsk

of e

are not rigid in ap-

proaching a crossing point, and the phase rigidity



the eigenfunctions k

 may be defined. From (15)

follows 10,



 (26)

for details see [7].

The value



of the phase rigidity is surely the most

interesting difference between Hermitian (with the Hami-

ltonian

) and non-Hermitian (with the Hamiltonian

) quantum physics. While the phases are rigid

(=1



) in the first case, they may vary according to (26)

in the second case. It is possible therefore that some

wavefunctionsk

of the system align with the scattering

wavefunctions of the environment while the other states

decouple (more or less) from the environment. This

phenomenon, called resonance trapping, is nothing but

width bifurcation caused by exceptional points (see

Section 2). In this manner, the non-Hermitian quantum

physics is able to describe environmentally induced effects.

Also the nonlinearities in the neighborhood of ex-

ceptional points can be seen when the scattering problem

on a many-level system is considered [7]. For example,

the S matrix at a double pole (corresponding to an

exceptional point) in the one-continuum case reads

00 00

=1 2()

Si ii

EE EE





 

(27)

where the notation (6) is used and 00 0



. At the

exceptional point, the cross section vanishes due to

interferences. The minimum is however washed out in

the neighborhood of the double pole. In any case, the

resonance is broader than a Breit-Wigner resonance

according to (27).

4. Resonance Trapping and Dynamical

Phase Transitions

Some years ago, the question has been studied [18]

whether or not the resonance trapping phenomenon is

related to some type of phase transition. The study is

performed by using the toy model

HiVV





 (28)

in the one-channel case and with the assumption that

(almost) all crossing (exceptional) points accumulate in

one point [19]. The control parameter



is a real

number. It has been found that resonance trapping may

be understood, in this case, as a second-order phase

transition. The calculations are performed for a linear

chain consisting of a finite number N of states. The

state in the center of the spectrum traps the other ones

and becomes a collective state in a global sense: it

contains components of almost all basic states of the

system, also of those which are not overlapped by it. The

normalized width 0/N



of this state can be considered

as the order parameter: it increases linearly as a function



, and the first derivative of 0/N jumps at the

critical value c



. The two phases of the system

differ by the number of localized states. In the case

considered, this number is N at c



, and 1N



at c



Much more interesting is the realistic case with the

Hamiltonian (17). In this case, trapping of resonance

states occurs in the regime of overlapping resonances

hierarchically, i.e. one by one [7]. The crossing points do

not accumulate in one point, but are distributed over a

certain range of the parameter: a dynamical phase

transition takes place in a finite parameter range inside

the regime of overlapping resonances [7]. Also in this

case, almost all resonance states are involved in the

phase transition of the system and, furthermore, the

number N of localized states is reduced. That means,

the subspace of localized states splits into two parts

under the influence of the environment. One part

contains the few short-lived states which are (more or

less) aligned to the scattering states of the environment,

while the other part contains the trapped, long-lived and

well localized states. Both time scales are well separated

from one another. A theoretical example are the short-

lived whispering gallery modes in a small microwave

cavity with convex boundary which coexist with many

long-lived states, for details see [20-22]. An experimental

example are the isobaric analogue resonances in medium

-mass nuclei.

The dynamical phase transitions are surely the most

interesting feature of non-Hermitian quantum physics.

Physically, they are environmentally induced, as can be

seen from the Hamiltonian (3) or (17), see also [7,23].

Mathematically, this phenomenon is directly related to

the existence of exceptional (crossing) points. In detail:

 The phases of the eigenfunctions of the non-

I. ROTTER

308

Hermitian Hamilton operator are not rigid in ap-

proaching the exceptional (crossing) point: <1

in the regime of overlappin g re sonances.

 Due to <1

r, some resonance states may align

with the scattering states of the environment while

other ones decouple from the environment (width

bifurcation).

 The short-lived aligned resonance states lose, to a

great deal, their localization and make the system

(almost) transparent.

 The long-lived trapped resonance states are well

localized and show chaotic features.

 The spectroscopic relation between the localized

states at low level density (without resonance over-

lapping) and those at high level density (with over-

lapping short-lived and long-lived resonances) is

lost.

The two phases of the system below and beyond the

dynamical phase transition are characterized by the

following properties. In one of the phases, the states have

individual spectroscopic features. Here, the real parts

(energies) of the eigenvalue trajectories avoid crossing as

function of a certain control parameter, while the ima-

ginary ones (widths) can cross. In the other phase, the

narrow resonance states are superimposed with a smooth

background and the individual spectroscopic features of

the states are lost. The narrow resonance states and,

respectively, the corresponding discrete states show

chaotic features. They do not cross in energy, but show

level repulsion. The real parts (energies) of the eigen-

value trajectories of narrow resonance states can cross

with those of the broad states since the narrow and broad

states exist at well separated time scales. In the transition

region, the different time scales corresponding to the

short-lived and long-lived resonance states are formed.

In this regime, the cross section is enhanced due to the

(at least partial) alignment of some states with the

scattering states of the environment [7]. An example is

the enhanced transmission through quantum dots in the

regime of ove r l apping resonances [2 4, 2 5].

About 10 years ago, the counterintuitive resonance

trapping phenomenon is tested experimentally [26]. The

experiment is based on the equivalence of the elec-

tromagnetic spectrum for flat cavities to the quantum

mechanical spectrum of the corresponding system. This

equivalence holds also when the system is opened by

coupling the discrete states of the cavity to an attached

waveguide. In the experiment [26], a microwave Sinai

cavity with an attached waveguide with variable slit

width was used.

The result of this experimental study agrees with th eo r y:

the widths of all resonance states first increase with in-

creasing coupling strength to the channels (continuum of

scattering wavefunctions) but finally decrease again for

most of the states. Thus, the dynamical phase transition

has been directly traced in this experiment.

The appearance of dynamical phase transitions can

explain some puzzles that are observed experimentally

and cannot be explained theoretically in the framework

of conventional Hermitian quantum theory. Some ex-

amples will be sketched in Section 5. The dynamical

phase transitions are responsible also for the observation

of the two extreme cases of mechanical many-body

problems mentioned by Niels Bohr [1,2]. Shell model

states with individual spectroscopic features may appear

only at low level density in nuclei as well as in atoms. At

high level density, however, the long-lived resonance

states cannot be described by a combination of one-body

problems. They are the result of resonance trapping

(characteristic of the dynamical phase transition) and

have almost nothing in common with shell model states.

They rather are states of an ensemble of long-lived

resonance states that is overlapped by a short-lived

resonance state.

5. Dynamical Phase Transitions in

Experimental Results

5.1. Phase Lapses

In experiments [27-29] on Aharonov-Bohm rings con-

taining a quantum dot in one arm, both the phase and the

magnitude of the transmission amplitude =|| i

TTe



the dot can be extracted. The obtained results caused

much discussion since they do not fit into the standard

understanding of the transmission process. As a function

of the plunger gate voltage

V, a series of well-

separated transmission peaks of rather similar width and

height has been observed in many-electron dots and,

according to expectations, the transmission phases

()



increase continuously by



across every resonance.

In contrast to expectations, however,



always jumps

sharply downwards by



in each valley between any

two successive peaks. These jumps called phase lapses,

were observed in a large succession of valleys for every

many-electron dot studied. Only in few-electron dots, the

expected so-called mesoscopic behavior is observed, i.e.

the phases are sensitive to details of the dot configuration.

The problem is considered theoretically, in the framework

of conventional Hermitian quantum physics, in many

papers over many years, however without solving it.

In [30], the phase lapses observed experimentally at

high level density are related to the trapped resonance

states resulting from the dynamical phase transition. In

accordance to this picture, only the resonance states at

low level density show individual spectroscopic features.

I. ROTTER

309

At high level density, the observed resonances arise from

trapped states. They show level repulsion, have vanishing

spectroscopic relation to the open decay channels (i.e.

small decay widths), and phase lapses appear. It follows

further, that any theoretical study on the basis of

conventional Hermitian quantum physics is unable to

explain the experimental results convincingly. In other

words: the experimentally observed phase lapses can be

con sidere d to b e a pr oof for the dyn amical ph as e transitions

occurring in mesoscopic systems.

5.2. Quantum Dynamical Phase Transition

in the Spin Swapping Operation

A swapping gate in a two-spin system exchanges the

degenerate states |, and |,. Experimentally,

this is achieved by turning on and off the spin-spin

interaction b that splits the energy levels and induces

an oscillation with a natural frequency



. An interaction

/SE



 with an environment of neighboring spins

degrades this oscillation within a decoherence time scale





. The experimental frequency



is expected to be

roughly proportional to /b and the decoherence time





proportional to SE



. In [31], experimental data are

presented that show drastic deviations in both



and





from this expectation. Beyond a critical interaction

with the environment, the swapping freezes and the

decoherence rate drops as 2

1/( / )SE





. That

means, the relaxation decreases when the coupling to the

environment increases. The transition between these two

quantum dynamical phases occurs when

(/) (/)



becomes imaginary (where k

depends only on the anisotropy of the system-environ-

ment interaction, 01k). The experimental results

are interpreted by the authors as an environmentally

induced quantum dynamical phase transition occurring in

the spin swapping operation [31-34].

Further theoretical studies within the Keldysh forma-

lism showed that





is a non-trivial function of the

system-environment interaction rate SE



, indeed: it is

1/1/SE





 at low SE



(according to the Fermi

golden rule) but 1/ SE





 at large SE



. This

theoretical result is in (qualitative) agreement with the

experimental results. In [35], the dynamical phase

transition in the spin swapping operation is related to the

existence of an exceptional point.

The dynamical phase transition observed experimen-

tally in the spin swapping operation and described

theoretically within the Keldysh formalism shows quali-

tatively the same features as the dynamical phase tran-

sitions discussed in the present paper on the basis of the

resonance trapping phe n omenon (width bifurcation).

5.3. Loss Induced Optical Transparency in

Complex Optical Potentials

Recently, the prospect of realizing complex PT sym-

metric potentials within the framework of optics has

been suggested [36-38]. It is based on the fact that the

optical wave equation is formally equivalent to the quan-

tum mechanical Schrödinger equation. One expects there-

fore that PT symmetric optical lattices show a behavior

which is qualitatively similar to that discussed for open

quantum systems in the present paper.

Experimental studies showed, indeed, a phase transition

that leads to a loss induced optical transparency in

specially designed non-Hermitian guiding potentials [39

-41]: the output transmission first decreases, attains a

minimum and then increases with increasing loss. The

phase transition is related, in these papers, toPT sym-

metry breaking. I n a following theoretical paper [42], the

Floquet-Bloch modes are investigated inPTsymmetric

complex periodic potentials. As a result, the modes are

skewed (nonorthogonal) and nonreciprocal. That means,

they show the same features as modes of an open quan-

tum system under the influence of exceptional points. A

detailed discussion of this analogy is given in [23]. The

optical realization of relativistic non-Hermitian quantum

mechanics is considered in [43]. Here, thePT symmetry

breaking of the Dirac Hamiltonian is shown to be related

to resonance narrowing what is nothing but resonance

trapping.

The title of one of the papers published in Nature

Physics [41] to this topic reads: Broken symmetry makes

light work. It is exactly this property which characterizes

the phase transition in complex optical potentials. How-

ever, the situation in open quantum systems is quali-

tatively the same: in the dynamical phase transition, the

spectroscopic relation to the individual resonance states

(including all symmetries) is broken and the system

becomes transparent, see e.g. Section 4.

6. Summary

Dynamical phase transitions are a phenomenon cha-

racteristic of quantum systems at high level density.

Mathematically, this phenomenon can be described in

the non-Hermitian quantum physics since the phases of

the eigenfunctions of a non-Hermitian operator are, in

general, not rigid. The non-rigidity is large in the neigh-

borhood of exceptional points (crossing points of the

eigenvalue trajectories). Here, the Schrödinger equation is

I. ROTTER

310

nonlinear. Physically, the dynamical phase transitions are

environmentally induced.

It is interesting to see that quantum systems behave

according to expectations only at low level density. Here,

the states show individual spectroscopic features. After

passing the transition region with overlapping resonances

by further variation of the control parameter, the behavior

of the system becomes counterintuitive: the narrow reso-

nance states decouple more or less from the continuum

of scattering states, and the number of localized states

decreases. The decoupling (resonance trapping) occurs

due to the alignment of a few states of the system to the

scattering states of the environment. This is an effect to

which all states of the system contribute, see Section 4

and [18].

In his address [1,2], Niels Bohr compared the trapped

resonance states in nuclei (compound nucleus resonances)

at high level density with the low-lying resonance states

of single-particle (shell model) nature in atoms. These

states differ from one another exactly in the manner

described by him. In the first case, the states are beyond

the dynamical phase transition of the system while they

are below the transition in the second case. The re-

sonance states at low level density (below the dyna-

mical phase transition) in nuclei as well as in atoms have

individual spectroscopic properties described well by the

shell model. The narrow states in nuclei at high level

density, however, are described with adequate accuracy

by a statistical ensemble containing the interaction be-

tween all particles, e.g. by the Gaussian orthogonal

ensemble. There is no need to consider, in the center of

the spectrum, the relation to a two-body random ensemble.

The short-lived and long-lived resonance states are f ormed

under the influence of the environment in the transition

region with many overlapping resonances.

According to the results represented in the present

paper, dynamical phase transitions in quantum systems

occur due to the existence of ex ceptional (crossing) p oi nt s.

They are therefore a generic property emerging in the

regime of overlapping resonances where the resonance

states lose any spectroscopic relation to the individual

resonance states of the system. Correspondingly, dyna-

mical phase transitions are found experimentally in

different systems.

Knowing the mathematical properties of the excep-

tional points it is possible, on the one hand, to explain (at

least qualitatively) some experimental results which

could not be understood in the framework of the con-

ventional Hermitian quantum physics in spite of much

effort. On the other hand, quantum systems can be ma-

nipulated system atically for applications. This includes also

the interesting topic of non-Hermitian quantum physics

which results from the formal equivalence of the optical

wave equation in PT symmetric optical lattices to the

quantum mechanical Schrödinger equation. This equi-

valence allows to receive much new information on

quantum systems. In any case, further theoretical and

experimental studies in the field of non-Hermitian quan-

tum physics will broaden our understanding of quantum

mechanics. Moreover, the results are expected to be of

great value for applications.

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