Journal of Modern Physics, 2010, 1, 303-311
doi:10.4236/jmp.2010.15043 Published Online November 2010 (
Copyright © 2010 SciRes. JMP
Dynamical Phase Transitions in Quantum Systems
Ingrid Rotter
Max Planck Institute for the Physics of Complex Systems, D-01187Dresden, Germany
Received September 16, 2010; revised October 5, 2010; accepted October 20, 2010
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
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
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
Copyright © 2010 SciRes. JMP
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
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
the exceptional points. An example is the operator
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()= .
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 22Hamiltonian describing a two-level
system with the unperturbed energies 1
and 2
the interaction
between the two levels,
()= .H
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
1,21 2
 
The two eigenvalue trajectories cross when =0Z, i.e.
At these crossing points, the two eigenvalues coalesce,
= .
The crossing points may be called therefore exceptional
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 ()
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
()( )
. 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)
|=| ; ||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
 
  (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
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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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
 
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,
iffj iBj
hr ij
eH H
iffj ij
Here, the
are eigenfunctions of
HE , and the scattering wave functions
follow from () =0
. Further, P denotes
the principal value integral. The ,
VV stand for the
interaction between the two subspaces B and C and
ˆ=2 ||
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
but also Re
H, and that by including the
principal value integral and without any statistical
assumptions. The Schrödinger equation reads
ffk k
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
in general, and the eigenfunctions k
are complex and
ij ij
 (23)
compare (7). The coupling matrix elements between the
and the
=2 ||
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
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from the different positions of the energetically lowest
decay channel (being a neutron (proton) decay channel)
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-
) and non-Hermitian (with the Hamiltonian
) quantum physics. While the phases are rigid
) in the first case, they may vary according to (26)
in the second case. It is possible therefore that some
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
 
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
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-
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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
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
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
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.
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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
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
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
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
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
Copyright © 2010 SciRes. JMP
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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