# Quantum Integrability and Chaos in periodic Toda Lattice with Balanced Loss-
Gain

Supriyo Ghosh<EMAIL_ADDRESS>Pijush K. Ghosh
<EMAIL_ADDRESS>Department of Physics, Siksha-Bhavana,
Visva-Bharati, Santiniketan, PIN 731 235, India.

###### Abstract

We consider equal-mass quantum Toda lattice with balanced loss-gain for two
and three particles. The two-particle Toda lattice is integrable and two
integrals of motion which are in involution have been found. The bound-state
energy and the corresponding eigenfunctions have been obtained numerically for
a few low-lying states. The three-particle quantum Toda lattice with balanced
loss-gain and velocity mediated coupling admits mixed phases of integrability
and chaos depending on the value of the loss-gain parameter. We have obtained
analytic expressions for two integrals of motion which are in involution.
Although an analytic expression for the third integral has not been found, the
numerical investigation suggests integrability below a critical value of the
loss-gain strength and chaos above this critical value. The level spacing
distribution changes from the Wigner-Dyson to the Poisson distribution as the
loss-gain parameter passes through this critical value and approaches zero. An
identical behaviour is seen in terms of the gap-ratio distribution of the
energy levels. The existence of mixed phases of quantum integrability and
chaos in the specified ranges of the loss-gain parameter has also been
confirmed independently via the study of level repulsion and complexity in
higher order excited states.

###### Contents

  1. I Introduction
  2. II Two Particle Toda lattice
  3. III Periodic Toda lattice with three particles
    1. III.1 A Limiting Case
    2. III.2 General Hamiltonian
      1. III.2.1 Avoided level crossing and probability distribution
      2. III.2.2 Level statistics & Gap-ratio distribution
  4. IV Conclusions & Discussions
  5. V Acknowledgements

##  I Introduction

The Toda lattice, a system of particles interacting via exponential
potentials, has long been the subject of extensive researchtoda1 ; toda2 ;
toda3 . Its remarkable properties, such as solitonic solutionsdate1976  and
the integrability of its equations of motionflaschka ; henon ; agrotis , have
made it a pivotal topic in the study of integrable systems and mathematical
physics. Toda lattice is important in the context of mathematical modelling of
many physical phenomena like heat propagation in lattice systemtoda4 ; sataric
, dynamics of DNAmuto , peptide bonds in the $\alpha$-helixdovivio , laser
dynamicsoppo ; lien ; cialdi . Various generalizations and modification of
Toda lattice having realistic physical applications and mathematical
importance have been consideredhabib ; casati ; vergara ; ezawa ; proy .
Truncated Toda potential perturbed by weak friction and noise is important in
galactic dynamicshabib . Chaotic behaviour is seen for Toda lattice with
unequal massescasati .

The Balanced Loss-Gain(BLG) system is defined as the one in which the flow
preserves the volume in the position-velocity state space, although the
individual degrees of freedom may be subjected to gain or losspkg . The system
is non-dissipative and may admit a Hamiltonian. A novel feature of such
systems is the existence of (quasi-)periodic solutions within some regions of
the parameter-space, and have been studied extensively in the context of
$\mathcal{PT}$-symmetrybender ; bender2 ; cuevas ; barashenkov ; ds1 ; khare1
; ds2 ; ds3 ; ds4 ; proy1 ; proy2 ; proy ; sg1 ; sg2 ; pkg2 . The Hamiltonian
formulation of generic BLG systems has been discussed in Ref. ds2 ; ds3 ; ds4
; pkg1  and a few such examples include systems with nonlinear
interactionbarashenkov ; ds1 ; khare1 ; ds2 ; ds3 ; ds4 , many particle
systemsds2 ; ds3 ; ds4 , systems with space dependent loss-gain termds3 ,
systems with Lorentz interactionpkg1 , Hamiltonian chaosproy1 ; proy2 ; proy ,
oligomerspkg  etc. The formalism has been extended beyond the mechanical
systems —nonlinear Schrödinger equationpkg2 ; sg1 ; sg2  and nonlinear Dirac
equationpkg  with BLG have been studied from the viewpoint of exact
solvability and existence of solutions bounded in time.

A recent addition to the growing list of generalized Toda systems is a Toda
lattice with BLGproy . It has been shown that a two-particle Toda system with
BLG is integrable and analytic expressions for two integrals of motion which
are in involution have been found. Periodic solutions of the equations of
motion have been found numerically. A three particle Hamiltonian Toda system
with BLG and Velocity Mediated Coupling(VMC) has been shown to admit mixed
phases of integrability and chaos. Two of the integrals of motion which are in
involution have been found analytically. Although the third integral of motion
has not been found analytically, which is required for showing integrability,
the numerical studies reveal integrability below a critical value of the loss-
gain parameter and chaos above this critical value. The existence of mixed
phases of integrability and chaos in the system has been established through
the study of sensitivity to the initial conditions, Poincaré Sections,
Lyapunov exponent, power-spectra and auto-correlation functionsproy . A non-
Hamiltonian Toda system with BLG has also been shown to admit chaotic
behaviour and chaos in the system is solely induced due to BLG.

The purpose of the article is to study the Hamiltonian Toda system with BLG
and VMC of Ref. proy  in the quantum region. The two-particle quantum Toda
system is shown to be integrable via the construction of two integrals of
motion which are in involution. The translation invariance of the system
allows to separate the center of mass motion and the Schödinger equation in
the relative coordinate may be interpreted as that of a particle moving in an
inverted harmonic oscillator plus a cosine hyperbolic potential. The effective
potential is of a single-well or a symmetric/asymmetric double-well depending
on the parameters of the system. The quantum bound states have been obtained
numerically.

The three particle Hamiltonian Toda lattice is translation invariant. The
center of mass motion is separated out in the Jacobi coordinate and the
effective Hamiltonian may be interpreted as that of a particle moving in a two
dimensional potential, consisting of anisotropic harmonic oscillators and the
Toda Potential, and subjected to an external uniform magnetic field with its
magnitude being proportional to the loss-gain parameter. The system is exactly
solvable in the limit, in which the Toda potential reduces to that of coupled
oscillators, such that the starting Hamiltonian describes coupled oscillators
with BLG and VMC. The eigenvalues and eigenfunctions are obtained analytically
in this particular limit which are given by two decoupled anisotropic
oscillators.

In general, the quantum problem is not amenable to analytic solution. The
conserved quantity corresponding to the translation invariance may be
interpreted as a generalized momentum which commutes with the Hamiltonian,
thereby, two of the quantum integrals of motion are found analytically. The
third integral of motion that is required to establish integrability has not
been found analytically so far. However, the numerical investigations suggest
integrability below a critical value of the loss-gain parameter and chaos
above this critical value, as is the case for the corresponding classical
system. It may be recalled in this context that a link between the classical
and quantum chaos may be explored by studying statistical properties of energy
levels based on Random Matrix Theory(RMT)haake ; bgs1984 . In particular, for
Gaussian Orthogonal Ensemble(GOE), the level statistics of quantum Hamiltonian
with integrable classical counterpart follows Poisson distribution berry1977 ;
montambaux ; poilblanc ; gaspard ; georgeot ; kudo ; deguchi ; santos ; gubin
, while in the chaotic region it follows the Wigner distribution. The level
statistics is described in terms of the spacing of nearest-neighbour eigen-
energies, and unfolded data is used in the process instead of the raw data.
There is an alternative method to circumvent the problems associated with
unfolding procedure in some cases where the statistics of ratios of the
energy-gaps for consecutive levels is studied. The statistics of gap-ratio of
quantum Hamiltonian with integrable classical counterpart follows Poisson
distribution, while it follows Wigner distribution in the chaotic
regionoganesyan .

In this article, we establish the mixed phases of integrability and chaos by
studying the level spacing distribution as well as gap-ratio distribution. The
quantum transition from the chaotic to the integrable region is observed when
the loss-gain strength crosses a critical value and goes to zero —the level
spacing as well as the gap-ratio distributions smoothly changes from the
Wigner-Dyson distribution and tends to follow the Poisson distribution. We
also show the level repulsion phenomena in the energy spectra in both the two
and three particle Toda lattice. It is shown through the graphical
presentations that the degree of level repulsion is large in the case of the
three particle system. The quantum transition from chaotic to integrable
region is also confirmed independently by studying the complex behaviour of
higher order excited state wave functions.

The plan of this article is the following. The two-particle Toda system is
studied in the next section. The three particle Toda system is presented in
Sec. III and the exactly solvable limiting case is described in Sec. III.A.
The numerical results for the general quantum problem is described in Sec.
III.B. Finally, the results are summarized with discussions in Sec. IV.

##  II Two Particle Toda lattice

The Hamiltonian of the Periodic Toda lattice is given asproy

$\displaystyle H$ $\displaystyle=$ $\displaystyle
2\left(p_{1}+\frac{\gamma}{2}x_{2}\right)\left(p_{2}-\frac{\gamma}{2}x_{1}\right)$
(1) $\displaystyle+$
$\displaystyle\frac{a}{b}\left\\{e^{b(x_{1}-x_{2})}+e^{b(x_{2}-x_{1})}-2\right\\},$

where $\gamma$ is the strength of the BLG term and $a$,$b$ are the strengths
of nonlinear interaction. The canonical conjugate momenta to the coordinates
$(x_{1},x_{2})$ are defined as $(p_{1},p_{2})$. The Hamiltonian (1) is
$\mathcal{PT}$-symmetric where the parity transformation $\mathcal{P}:$
$x_{1}\leftrightarrow x_{2},p_{1}\leftrightarrow p_{2}$ and the time-reversal
operation $\mathcal{T}:(x_{1},x_{2})\rightarrow(x_{1},x_{2})$,
$(p_{1},p_{2})\rightarrow(-p_{1},-p_{2})$. In the limit of
$a\rightarrow\infty,b\rightarrow 0$ such that $ab\equiv\omega^{2}$, the above
Hamiltonian reduces to that of coupled oscillators with BLG which is a variant
of the model considered in Ref. bender . The generic system is translation
invariant, and in order to separate out the center of mass motion, we consider
the following coordinate transformation:

$\displaystyle x_{1}=\frac{x+y}{\sqrt{2}},\ x_{2}=\frac{x-y}{\sqrt{2}},\
p_{1}=\frac{p_{x}+p_{y}}{\sqrt{2}},\ p_{2}=\frac{p_{x}-p_{y}}{\sqrt{2}}.$ (2)

The Hamiltonian in terms of the new coordinates is expressed as,

$\displaystyle H$ $\displaystyle=$ $\displaystyle
p_{x}^{2}-p_{y}^{2}-\gamma(xp_{y}+yp_{x})-\frac{\gamma^{2}}{4}(x^{2}-y^{2})$
(3) $\displaystyle+$
$\displaystyle\frac{a}{b}\left\\{e^{b\sqrt{2}y}+e^{-b\sqrt{2}y}-2\right\\}.$

The $\mathcal{P}$ and $\mathcal{T}$ have the standard definition in this
coordinate, namely,
$\mathcal{P}:(x,y)\rightarrow(x,-y),(p_{x},p_{y})\rightarrow(p_{x},-p_{y})$
and
$\mathcal{T}:(x,y)\rightarrow(x,y),(p_{x},p_{y})\rightarrow(-p_{x},-p_{y})$.
The Hamiltonian is invariant under the operation of $\mathcal{PT}$. The
Hamiltonian is not positive-definite.

The eigen value equation corresponding to the Hamiltonian (3) is given as,

$\displaystyle\left[-\left(\partial_{x}^{2}-\partial_{y}^{2}\right)+i\gamma\left(x\partial_{y}+y\partial_{x}\right)-\frac{\gamma^{2}}{4}(x^{2}-y^{2})\right.$
$\displaystyle+\left.\frac{a}{b}\left\\{e^{b\sqrt{2}y}+e^{-b\sqrt{2}y}-2\right\\}\right]\psi=\tilde{E}\psi$
(4)

where we have used the standard coordinate representation of the Heisenberg
equation and denoted $p_{x}:=-i\partial_{x},\ p_{y}:=-i\partial_{y}$. The
system is translation invariant and the associated conserved operator,

$\displaystyle\widehat{\Pi}=\left(-2i\frac{\partial}{\partial x}+\gamma
y\right),$ (5)

commutes with the Hamiltonian (3). Thus, the quantum system is integrable as
is the case for the corresponding classical system.

The simultaneous wave function of $\widehat{\Pi}$ and $H$ is considered as,

$\displaystyle\psi(x,y)=\exp\left[\frac{i}{2}x(k-\gamma y)\right]\phi(y),$ (6)

which when substituted into Eq. (4) gives the following eigen value equation,

$\displaystyle\phi^{\prime\prime}(y)+\gamma^{2}y^{2}\phi(y)+\frac{k}{4}(k-2\gamma
y)\phi$
$\displaystyle+\frac{2a}{b}\left\\{\cosh(\sqrt{2}by)-1\right\\}\phi(y)=\tilde{E}\phi(y).$
(7)

We make a transformation $y\rightarrow y-\frac{k}{4\gamma}$ followed by a
scale transformation $y\rightarrow\sqrt{2}by$, and the resulting eigen value
equation takes the following form,

$\displaystyle-\phi^{\prime\prime}+\left[-\lambda
y^{2}-\beta\left\\{\cosh\left(y+y_{0}\right)-1\right\\}\right]\phi=E\phi$ (8)

where $\lambda=\frac{\gamma^{2}}{4b^{4}}$, $\beta=\frac{a}{b^{3}}$,
$E=-\frac{1}{2b^{2}}(\tilde{E}-\frac{3k^{2}}{16})$,
$y_{0}=\frac{bk}{2\sqrt{2}\gamma}$. It may be noted that $\lambda$ is always
positive and $\beta$ can be both positive as well as negative. The eigen value
equation in Eq. (8) may be interpreted as that of a particle moving in the one
dimensional potential,

$\displaystyle V(y)=\beta\left(1-\cosh(y+y_{0})\right)-\lambda y^{2}$ (9)

The potential $V(y)$ has two localized minima for $-2\lambda<\beta<0$
representing a symmetric double-well potential for $y_{0}=0$ and an asymmetric
one for $y_{0}\neq 0$. The potential has only one minima at $y=0$,
representing a single well, in the remaining region of the parameter-space.
The quantum bound states are allowed only for $\beta<0$. The potential and
wave functions for the ground and 1st excited states of the system for
$\lambda=1,k=0$ and different values of $\beta$ are given in the Fig. 1.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 1:  (Color online) Plot of wave functions and potential for different
values of $\beta$ and $\lambda=1$,. Green, Red and Orange solid lines
represent potential, ground state and 1st excited state wave functions,
respectively. Fig.(a), Fig.(c) $\beta=-0.5$; Fig.(b),Fig.(d) $\beta=-0.9$;
Fig.(e),Fig.(g) $\beta=-1.7$; Fig.(f),Fig.(h) $\beta=-2.1$

The barrier between the two wells increases with the decreasing value of
$|\beta|$, and the system decomposes into two independent components separated
from each other —the wave function is the set of two split up wave functions.
The tunnelling probability of the particle from a well to the other increases
with the decrease of the barrier height, and the wave-function of the left and
right regions tend to overlap. The standard behaviour of wave-function in a
double-well represented by a quatric potential is seen in the present context.
The eigen values corresponding to the quantum bound states of the system is
shown in the Fig. 2 as a function of $\beta$ for a few low lying states.

Figure 2: (Color online) Plot of the lowest six energy levels of 2D Toda
lattice. Parametric value : $\lambda=1$

We observe that the eigen values come close to each other as $\beta$
increases, although the levels do not cross each other. The degree of level
repulsion is low which is obvious for an integrable system. The ground state
energy for different values of $\beta$ and $y_{0}$ is shown in the Fig. 3.

Figure 3: (Color online) Plot of ground state energy of 2D Toda lattice for
different values of $\beta$ and $y_{0}$. Parametric value : $\lambda=1$

##  III Periodic Toda lattice with three particles

We consider the following Hamiltonian of a Toda lattice with BLG and VMC for
three particles,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\Pi_{1}\Pi_{2}+\Pi_{1}\Pi_{3}+\Pi_{2}\Pi_{3}+\frac{a}{b}\left[1-e^{b(q_{2}-q_{1})}\right.$
(10) $\displaystyle-$
$\displaystyle\left.e^{b(q_{3}-q_{2})}-e^{b(q_{1}-q_{3})}\right]$

The generalized momenta appearing in $H$ have the following expressions,

$\displaystyle\Pi_{1}=p_{1}+\frac{\gamma}{4}(q_{2}+q_{3}),\ \
\Pi_{2}=p_{2}-\frac{\gamma}{4}(q_{1}-q_{3}),$
$\displaystyle\Pi_{3}=p_{3}-\frac{\gamma}{4}(q_{1}+q_{2}).$ (11)

The $\gamma$ dependent terms in $\Pi_{i}$’s produce BLG and VMC in the
respective Hamilton’s equations of motion. In particular, the system is
governed by the equations of motion,

$\displaystyle\ddot{q}_{1}+\gamma\dot{q}_{1}+\frac{\gamma}{2}\left(\dot{q}_{2}-\dot{q}_{3}\right)-a\left[e^{b\left(q_{2}-q_{1}\right)}-e^{b\left(q_{1}-q_{3}\right)}\right]=0$
$\displaystyle\ddot{q}_{2}+\frac{\gamma}{2}\left(\dot{q}_{1}-\dot{q}_{3}\right)-a\left[e^{b\left(q_{3}-q_{2}\right)}-e^{b\left(q_{2}-q_{1}\right)}\right]=0$
$\displaystyle\ddot{q}_{3}-\gamma\dot{q}_{3}+\frac{\gamma}{2}\left(\dot{q}_{1}-\dot{q}_{2}\right)-a\left[e^{b\left(q_{1}-q_{3}\right)}-e^{b\left(q_{3}-q_{2}\right)}\right]=0,$

where particle-1 and particle-3 are subjected to BLG. Each particle interacts
with the remaining two particles via the Toda potential and the VMC. The above
classical system has been studied in detail in Ref. proy  and shown to admit
mixed phases of integrability and chaos —two of the integrals of motion have
been obtained analytically. In this article, we study the quantum Hamiltonian
$H$ from the view point of its integrability and chaos.

The system is translation invariant. We introduce the Jacobi coordinates
$Q_{1},Q_{2},Q_{3}$ in order to separate out the center of mass motion and to
work with only two coordinates,

$\displaystyle
Q_{j}=\frac{1}{\sqrt{j(j+1)}}\sum_{k=1}^{j}(q_{k}-jq_{j+1}),Q_{3}=\frac{1}{\sqrt{3}}\sum_{k=1}^{3}q_{k},$

where $j=1,2$. The momenta $P_{j}$ in the Jacobi coordinates are related to
$p_{j}$’s through the same relations —replace
$(Q_{k},q_{k})\rightarrow(P_{k},p_{k})$. The Hamiltonian can be rewritten in
terms of $(Q_{k},P_{k})$ as,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{{\tilde{\Pi}_{3}}^{2}}{4}-\frac{\tilde{\Pi}_{1}^{2}}{2}-\frac{\tilde{\Pi}_{2}^{2}}{2}-\frac{a}{b}\left[e^{-b\sqrt{2}Q_{1}}+e^{\frac{b}{\sqrt{2}}(Q_{1}+\sqrt{3}Q_{2})}\right.$
(12) $\displaystyle+$
$\displaystyle\left.e^{\frac{b}{\sqrt{2}}(Q_{1}-\sqrt{3}Q_{2})}-1\right]$

where $\tilde{\Pi}_{1}$,$\tilde{\Pi}_{2}$ and $\tilde{\Pi}_{3}$ are the
modified generalized momenta:

$\displaystyle\tilde{\Pi}_{1}$ $\displaystyle=$ $\displaystyle-
P_{1}-\frac{\gamma}{4\sqrt{3}}\left(Q_{2}+\sqrt{2}Q_{3}\right)$
$\displaystyle\tilde{\Pi}_{2}$ $\displaystyle=$ $\displaystyle-
P_{2}+\frac{\gamma}{4\sqrt{3}}\left(Q_{1}-\sqrt{6}Q_{3}\right)$
$\displaystyle\tilde{\Pi}_{3}$ $\displaystyle=$ $\displaystyle
2P_{3}-\frac{\gamma}{\sqrt{6}}\left(Q_{1}+\sqrt{3}Q_{2}\right)$ (13)

The Hamiltonian is not positive-definite even for $\frac{a}{b}<0$.

We quantize the Hamiltonian (12) by replacing $(Q_{j},P_{j})$ with the
corresponding operators $(Q_{j},-i\partial_{Q_{j}})$ satisfying the
commutation relations $[Q_{j},P_{l}]=i\delta_{jl}$, $[Q_{j},Q_{l}]=0$,
$[P_{j},P_{l}]=0$, and denote the resulting Hamiltonian as $\widehat{H}$. We
are working with $\hbar=1$. The translation invariance leads to a conserved
quantity and the corresponding operator

$\displaystyle\widehat{\Pi}=-2\sqrt{3}i\frac{\partial}{\partial
Q_{3}}+\frac{\gamma}{\sqrt{2}}(Q_{1}+\sqrt{3}Q_{2})$ (14)

commutes with the Hamiltonian $\widehat{H}$. The complete integrability
requires three integrals of motion. However, we have found analytic
expressions for two conserved quantities, namely, $\widehat{H}$ and
$\widehat{\Pi}$. It will be seen later that numerical investigations indicate
integrability of the system in some ranges of the parameter $\gamma$. The
commutation relation $[\widehat{H},\widehat{\Pi}]=0$ allows us to find
simultaneous wave function of these two operators,

$\displaystyle\psi(Q_{1},Q_{2},Q_{3})$ $\displaystyle=$
$\displaystyle\exp\left[\frac{i}{2\sqrt{3}}Q_{3}\left\\{k-\frac{\gamma}{\sqrt{2}}\left(Q_{1}\right.\right.\right.$
(15) $\displaystyle+$
$\displaystyle\left.\left.\left.\sqrt{3}Q_{2}\right)\right\\}\right]\phi(Q_{1},Q_{2}),$

where $k$ is the eigen value of the operator $\widehat{\Pi}$. Inserting the
expression of $\psi(Q_{1},Q_{2},Q_{3})$ into the time-independent schrödinger
equation $\widehat{H}\psi=\epsilon\psi$, we get an eigen value equation in
terms of an effective Hamiltonian $H_{\textrm{eff}}$ and energy E as
$H_{\textrm{eff}}\phi=E\phi$, where $E=-(\epsilon-\frac{a}{b})$. The
expression of effective Hamiltonian is as follows,

$\displaystyle H_{\textrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(P_{1}+\frac{\gamma}{4\sqrt{3}}Q_{2}\right)^{2}+\frac{1}{2}\left(P_{2}-\frac{\gamma}{4\sqrt{3}}Q_{1}\right)^{2}+V_{\textrm{eff}}$
$\displaystyle V_{\textrm{eff}}$ $\displaystyle=$
$\displaystyle-\frac{\gamma^{2}}{6}\left(Q_{1}+\sqrt{3}Q_{2}+\frac{k}{\sqrt{2}\gamma}\right)^{2}+\frac{a}{b}\left[e^{-b\sqrt{2}Q_{1}}\right.$
(16) $\displaystyle+$
$\displaystyle\left.e^{\frac{b}{\sqrt{2}}(Q_{1}+\sqrt{3}Q_{2})}+e^{\frac{b}{\sqrt{2}}(Q_{1}-\sqrt{3}Q_{2})}\right],$

where $P_{1},P_{2},Q_{1},Q_{2}$ are to be treated as quantum mechanical
operators as mentioned earlier. The effective Hamiltonian may be interpreted
as describing a particle moving in a two-dimensional potential
$V_{\textrm{eff}}$ and subjected to uniform “fictitious magnetic field”ds2 ;
ds3 ; pkg1  perpendicular to the “$Q_{1}-Q_{2}$”-plane with the absolute
magnitude $\frac{|\gamma|}{2\sqrt{3}}$. In the terminology of the quantization
of system with uniform magnetic field, the “fictitious gauge-potential” in
$H_{\textrm{eff}}$ is written in the symmetric gaugeds3 . One may choose the
Landau gauge and the corresponding Hamiltonian $H_{L}$ is obtained through a
gauge transformation as,

$\displaystyle H_{L}$ $\displaystyle=$ $\displaystyle
exp(-\frac{i\gamma}{4\sqrt{3}}Q_{1}Q_{2})\ H_{\textrm{eff}}\
exp(\frac{i\gamma}{4\sqrt{3}}Q_{1}Q_{2})$ (17) $\displaystyle=$
$\displaystyle\frac{1}{2}(P_{1}+\frac{\gamma}{2\sqrt{3}}Q_{2})^{2}+\frac{P_{2}^{2}}{2}+V_{\textrm{eff}}$

A quadratic potential arises in $V_{\textrm{eff}}$ as the effect of the BLG
and VMC, and the standard Toda potential gets modified. The potential is
bounded for $\frac{a}{b}>0$ and contour-plots of the same are shown in Fig.
(4) for a few choices of the parameters. We study bound states of the quantum
Hamiltonian $H_{\textrm{eff}}$. The analytical treatment of the general eigen
value problem appears to be non-trivial, and will be studied numerically.
However, exact solutions may be obtained in the limiting case in which the
Toda potential is approximated as that of coupled oscillators.

(a)

(b)

(c)

(d)

Figure 4: (Color online) Contour plot of $log(V_{\textrm{eff}})$ for $k=0$.
(a) $a=b=1$,$\gamma=0.2$; (b) $a=b=1$,$\gamma=1.6$; (c)
$a=0.8$,$b=1$,$\gamma=1.4$; (d) $a=1$,$b=0.8$,$\gamma=1.4$

###  III.1 A Limiting Case

The Toda potential reduces to that of coupled oscillators in the limit
$b\rightarrow 0,a\rightarrow\infty$ such that $ab\equiv\omega^{2}$. The
Hamiltonian $H_{\textrm{eff}}$ in this limit reduces to a two-dimensional
anisotropic oscillator in an external uniform magnetic field,

$\displaystyle\tilde{H}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(P_{1}+\frac{\gamma}{4\sqrt{3}}Q_{2}\right)^{2}+\frac{1}{2}\left(P_{2}-\frac{\gamma}{4\sqrt{3}}Q_{1}\right)^{2}$
(18) $\displaystyle+$
$\displaystyle\frac{9\omega^{2}-\gamma^{2}}{6}Q_{1}^{2}+\frac{3\omega^{2}-\gamma^{2}}{2}Q_{2}^{2}-\frac{\gamma^{2}}{\sqrt{3}}Q_{1}Q_{2},$

where we have taken $k=0$ and dropped the constant term $\frac{3a}{b}$. The
term $Q_{1}Q_{2}$ can be removed through a rotation by an angle thirty degree
in the anti-clockwise direction. In particular, the transformation

$\displaystyle Q_{1}=\frac{\sqrt{3}}{2}X+\frac{1}{2}Y,\
Q_{2}=-\frac{1}{2}X+\frac{\sqrt{3}}{2}Y$ (19)

transforms $\tilde{H}$ to the following form:

$\displaystyle\tilde{H}=\frac{1}{2}\left(\Pi_{X}^{2}+\Pi_{Y}^{2}\right)+\frac{1}{2}\omega_{1}^{2}X^{2}+\frac{1}{2}\omega_{2}^{2}Y^{2}$
$\displaystyle\Pi_{X}={P_{X}+\frac{\gamma}{4\sqrt{3}}Y,\
\Pi_{Y}=P_{Y}-\frac{\gamma}{4\sqrt{3}}X},$ (20)

where $P_{X}=-i\partial_{x},P_{Y}=-i\partial_{Y}$ and
$\omega_{1}^{2}=3\omega^{2},\omega_{2}^{2}=\omega_{1}^{2}-\frac{4\gamma^{2}}{3}$.
The Hamiltonian $\tilde{H}$ is $\mathcal{PT}$-symmetric where the parity
transformation
$\mathcal{P}:(X,Y)\rightarrow(X,-Y),(P_{X},P_{Y})\rightarrow(P_{X},-P_{Y})$
and time reversal operation
$\mathcal{T}:(X,Y)\rightarrow(X,Y),(P_{X},P_{Y})\rightarrow(-P_{X},-P_{Y})$.
Phase transition occurs in this systems at $9\omega^{2}=4\gamma^{2}$. The
eigen values of the systems are real for $\omega^{2}>\frac{4}{9}\gamma^{2}$
and the system is in unbroken $\mathcal{PT}$-symmetric phase. The
$\mathcal{PT}$-symmetry is broken for $\omega^{2}<\frac{4}{9}\gamma^{2}$ and
the eigen values become imaginary. The Hamiltonian $\tilde{H}$ describes an
anisotropic two dimensional harmonic oscillator in an external uniform
magnetic field which has been studied earliergui ; rebane . We follow the
method outlined in Ref. susy  to obtain the eigenspectra. Define a four
dimensional vector $U=\left(\omega_{1}X,\Pi_{X},\omega_{2}Y,\Pi_{Y}\right)$ in
the phase-space such that $\tilde{H}=\frac{1}{2}UU^{T}$, where $U^{T}$ denotes
the transpose of $U$. It may be noted that $\left[U_{i},U_{j}\right]=iM_{ij}$
where the $4\times 4$ matrix $M$ is given by,

$\displaystyle M=\begin{pmatrix}0&\omega_{1}&0&0\\\
-\omega_{1}&0&0&\frac{\gamma}{2\sqrt{3}}\\\ 0&0&0&\omega_{2}\\\
0&-\frac{\gamma}{2\sqrt{3}}&-\omega_{2}&0\end{pmatrix}.$ (21)

The eigenvalues of $iM$ are $(\Omega_{+},-\Omega_{+},\Omega_{-},-\Omega_{-})$
where

$\displaystyle\Omega_{\pm}=\sqrt{3\left[\omega^{2}\pm\frac{5\gamma}{24}\sqrt{\gamma^{2}+\frac{16}{25}\omega^{2}}-\frac{5}{24}\gamma^{2}\right]},$
(22)

and reality of $\Omega_{\pm}$ is ensured for
$\omega^{2}>\frac{4}{9}\gamma^{2}$. The determinant of the matrix $iM$ is zero
for $\omega^{2}=\frac{4}{9}\gamma^{2}$ and $\omega=\pm\frac{2}{3}\gamma$
characterizes a critical phase for which $\Omega_{-}=0$. There exists an
orthogonal transformation $V=O^{T}U$ such that the matrix $M$ can be block-
diagonalized asstopo ,

$\displaystyle M_{B}=O^{T}MO=\begin{pmatrix}0&\Omega_{+}&0&0\\\
-\Omega_{+}&0&0&0\\\ 0&0&0&\Omega_{-}\\\ 0&0&-\Omega_{-}&0\end{pmatrix}.$ (23)

where $O$ is an $O(4)$ rotation matrix and $O^{T}$ is the transpose of $O$.
The expression of $O$ is given as,

$\displaystyle
O=\begin{pmatrix}\frac{\sqrt{2}\omega_{1}a_{+}}{\Omega_{+}}({\omega_{2}}^{2}-{\Omega_{+}}^{2})&0&\frac{\sqrt{2}\omega_{1}a_{-}}{\Omega_{-}}({\omega_{2}}^{2}-{\Omega_{-}}^{2})&0\\\
0&\sqrt{2}a_{+}({\omega_{2}}^{2}-{\Omega_{+}}^{2})&0&\sqrt{2}a_{-}({\omega_{2}}^{2}-{\Omega_{-}}^{2})\\\
0&-\frac{\omega_{2}\gamma a_{+}}{\sqrt{6}}&0&-\frac{\omega_{2}\gamma
a_{-}}{\sqrt{6}}\\\ \frac{\gamma a_{+}\Omega_{+}}{\sqrt{6}}&0&\frac{\gamma
a_{-}\Omega_{-}}{\sqrt{6}}&0\end{pmatrix},$ (24)

where

$\displaystyle
a_{\pm}=\frac{\Omega_{\pm}}{\sqrt{(\omega_{2}^{2}-{\Omega_{\pm}}^{2})^{2}({\omega_{1}}^{2}+{\Omega_{\pm}}^{2})+\frac{\gamma^{2}{\Omega_{\pm}}^{2}}{12}({\omega_{2}}^{2}+{\Omega_{\pm}}^{2})}}.$

The matrix $O$ is unique modulo $O(4)$ rotations. We denote
$M_{D}=\textrm{diag}(\Omega_{+},-\Omega_{+},\Omega_{-},-\Omega_{-})$ and let
$S$ diagonalizes $iM$, i. e. $M_{D}=S^{\dagger}(iM)S$. The similarity
transformation can not change eigenvalues and let $T$ diagonalizes $iM_{B}$,
i.e. $M_{D}=T^{\dagger}(iM_{B})T$. The matrix $O$ is constructed as
$O=ST^{\dagger}$. The transformed variables in the phase space
$(x,p_{x},y,p_{y})\equiv(v_{1}/{\sqrt{\Omega_{+}}},v_{2}/{\sqrt{\Omega_{+}}},v_{3}/{\sqrt{\Omega_{-}}},v_{4}/{\sqrt{\Omega_{-}}})$
where $\Omega_{-}\neq 0$, satisfy the standard commutation relations,

$\displaystyle[x,p_{x}]=i,\ [y,p_{y}]=i,\ [x,y]=0=[p_{x},p_{y}].$ (25)

The Hamiltonian is expressed in terms of new variables as two decoupled
anisotropic harmonic oscillators,

$\displaystyle\tilde{H}=\frac{\Omega_{+}}{2}\left(p_{x}^{2}+x^{2}\right)+\frac{\Omega_{-}}{2}\left(p_{y}^{2}+y^{2}\right).$
(26)

The Hamiltonian can be expressed in terms of two sets of annihilation and
creation operators as,

$\displaystyle\tilde{H}=\Omega_{+}\left(a^{\dagger}a+\frac{1}{2}\right)+\Omega_{-}\left(b^{\dagger}b+\frac{1}{2}\right),$
$\displaystyle a=\frac{1}{\sqrt{2}}\left(p_{x}-ix\right),\ \
b=\frac{1}{\sqrt{2}}\left(p_{y}-iy\right)$ (27)

The ground state is determined from the conditions $a\Psi_{0}=0,b\Psi_{0}=0$,

$\displaystyle\Psi_{0}(x,y)=\frac{2}{\sqrt{\pi}}\
e^{-\frac{1}{2}\left(x^{2}+y^{2}\right)},\ \
E_{0,0}=\frac{\Omega_{+}+\Omega_{-}}{2}.$ (28)

The energy eigenvalues and the eigenfunctions are,

$\displaystyle
E_{n,m}=\left(n+\frac{1}{2}\right)\Omega_{+}+\left(m+\frac{1}{2}\right)\Omega_{-},\
n,m\in\mathbb{Z}^{\geq 0}$
$\displaystyle\Psi_{n,m}(x,y)=\frac{(a^{\dagger})^{n}}{\sqrt{n!}}\frac{(b^{\dagger})^{m}}{\sqrt{m!}}\Psi_{0}(x,y).$
(29)

The wave-function $\Psi_{n,m}(x,y)$ may be expressed in terms of the original
variables through a series of inverse transformations which we do not pursue
here. It may be noted that there is a reduction in the phase-space for
$\omega^{2}=\frac{4}{9}\gamma^{2}$ for which $\Omega_{-}=0$ and the
eigenvalues become infinitely degenerate. The energy eigen values are complex
for $\omega^{2}<\frac{4}{9}\gamma^{2}$ and entirely real for
$\omega^{2}>\frac{4}{9}\gamma^{2}$.

###  III.2 General Hamiltonian

The eigen value problem for the Hamiltonian $H_{eff}$ in its full generality
seems to be unamenable for an analytical treatment. We study the quantum bound
states of $H_{eff}$ numerically, and obtain the energy spectra and wave
functions by using finite difference method. The scattering states of
$H_{eff}$, if any, have been excluded from our purview of study. We have
considered $\gamma>0,k\geq 0$ and $a=b=1$ in the numerical calculation unless
otherwise stated explicitly. The eigen values and eigenfunctions thus obtained
are analyzed to investigate quantum integrability and chaos in the system. In
particular, we study avoid level crossing, probability distribution of wave
functions in the semi-classical region, level statistics and gap-ratio
distribution in order to study quantum integrability and chaos. It is worth
recalling that the system in Ref. proy  is integrable for $|\gamma|\lessapprox
1.5$ and chaotic above this critical value. We present numerical results in
the following for $\gamma$ varying from zero to values above this critical in
order to search for quantum integrable and chaotic region.

####  III.2.1 Avoided level crossing and probability distribution

The lowest ten energy eigen values of $H_{\textrm{eff}}$ for different values
of $\gamma$, and $k=0,a=b=1$ are shown in Fig. 5. The energies corresponding
to each state decrease with increasing $\gamma$. The energy levels are highly
correlated and repel each other. The avoided level crossing is a signature of
quantum chaoshaake , since level-crossing gives rise to the degeneracy in the
spectrum, implying symmetry and associated conserved quantities in the system.
The avoided level crossing is seen for larger values of $\gamma$ in Fig. 5.
The level repulsion is not obvious from the figure for smaller values of
$\gamma$, and is shown separately in the inset of Fig. 5 for $1\leq\gamma\leq
1.5$. The dependence of the ground state energy of
$H_{\textrm{eff}}+\frac{k^{2}}{12}$ on $\gamma$ and $k$ is shown in Fig. 6.
The ground state energy is shifted for $k\neq 0$ by an amount of the order of
$-\gamma^{2}k$ and the same behaviour is seen.

Figure 5: (Color online) Plot of lowest ten energy eigen value of 3-particle
system as a function of BLG parameter $\gamma$ for $k=0$ and $a=b=1$ Figure 6:
(Color online) Plot of ground state energy for different values of $\gamma$
and $k$. Here $a=b=1$

The Bohr’s correspondence principle allows to make a link between the
classical physics and the predictions of quantum mechanics for very high
excited states. Thus, a chaotic solution in classical physics should bear some
signature in the quantized version of the same system. One of the objects to
look for such signatures is the probability distribution of highly excited
states. The probability distribution of quantum states of a classically
integrable system is expected to be localized in space for low as well as very
high quantum numbers. On the other hand, the probability distribution of
quantum states corresponding to a classically chaotic system is expected to
spread out in space for large values quantum number. The spreading out of the
probability distribution for highly excited states can be thought of as the
quantum analogue of the chaotic path taken in classical region. We plot the
probability distribution of the ground state and a few excited states in Fig.
7 and Fig. 8 for $\gamma=.2$ and $\gamma=1.6$, respectively. It is seen that
the probability distribution for $\gamma=0.2$ is localized with uniform
distribution for low as well as highly excited states. On the other hand, the
complex behaviour arises in the highly excited wave-functions, and manifested
in the nonuniform probability distribution for $\gamma=1.6$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: (Color online) Plot of $|\Phi|^{2}$ for $k=0$,$a=b=1$ and
$\gamma=0.2$. (a) Ground state; (b) 1st excited state; (c) 2nd excited state;
(d) 200th excited state; (e) 300th excited state; (f) 400th excited state.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8: (Color online) Plot of $|\Phi|^{2}$ for $k=0$,$a=b=1$ and
$\gamma=1.6$. (a) Ground state; (b) 1st excited state; (c) 2nd excited state;
(d) 200th excited state; (e) 300th excited state; (f) 400th excited state.

####  III.2.2 Level statistics & Gap-ratio distribution

The chaotic dynamics in classical physics is determined by studying the phase
space trajectories, Lyapunov exponents, Poincaré sections etc. The Schrödinger
equation being a linear system, the notion of quantum chaos is not uniform,
and a quantification of the same is still an active area of research. One of
the standard approaches to explore signature of quantum chaos in a classically
chaotic system is to study the level statistics. According to the BGS
conjecture, the quantum Hamiltonian with chaotic classical dynamics must fall
into one of the three classical ensembles of RMTbgs1984  —GOE, Gaussian
unitary ensembles(GUE) and Gaussian symplectic ensembles(GSE). The level
statistics of quantum Hamiltonian for GOE with integrable classical
counterpart follow Poisson law $P_{P}(s)=exp(-s)$, while it follows Wigner
distribution $P_{W}(s)=\frac{\pi s}{2}exp\left(-\frac{\pi s^{2}}{4}\right)$ in
the classical chaotic region, where $s$ is the spacing of nearest-neighbour
eigen-energies berry1977 .

We study level statistics of the eigen spectrum of the Hamiltonian
$H_{\textrm{eff}}$. The level spacing distribution is shown by the probability
density function $\rho(s)$ of the nearest neighbour spacing of unfolded eigen
value. The procedure of unfolding the raw eigen value is a way of locally
rescaling the energy spectra such that the mean level density is one. The
cumulative spectral function or the staircase function is defined as the
number of levels with energy less than or equal to a certain value E. In
particular, staircase function is defined as, $N(E)=\sum_{n}\Theta(E-E_{n})$,
where $\Theta$ is the unit step function. The function $N(E)$ consists of
smooth part $N_{sm}(E)$ and a fluctuating part $N_{fl}(E)$, i. e.
$N(E)=N_{sm}+N_{fl}$. We obtain $N_{sm}$ numerically, by fitting the staircase
function with a polynomial of degree 15. The unfolded eigen values $x_{i}$ are
obtained from raw eigen values $E_{i}$ as $x_{i}=N_{sm}(E_{i})$. Finally The
level-spacing distributions are shown by the probability density function
$\rho(s)$, where $s=x_{i+1}-x_{i}$.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Figure 9:  (Color online) Plot of the level-spacing statistics of Hamiltonian
for different values of $\gamma$ and $k=0$, $a=b=1$. (a) $\gamma=0$; (b)
$\gamma=0.2$; (c) $\gamma=0.4$; (d) $\gamma=0.6$; (e) $\gamma=0.8$, (f)
$\gamma=1.0$, (g) $\gamma=1.2$, (h) $\gamma=1.4$, (i) $\gamma=1.6$, (j)
$\gamma=1.8$, (k) $\gamma=2.0$.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Figure 10: (Color online) Plot of the level-spacing statistics of Hamiltonian
for different values of $\gamma$ and $k=1$ and $a=b=1$. (a) $\gamma=0$; (b)
$\gamma=0.2$; (c) $\gamma=0.4$; (d) $\gamma=0.6$; (e) $\gamma=0.8$, (f)
$\gamma=1.0$, (g) $\gamma=1.2$, (h) $\gamma=1.4$, (i) $\gamma=1.6$, (j)
$\gamma=1.8$, (k) $\gamma=2.0$.

The level spacing distributions of the set of eigen values of the Hamiltonian
$H_{\textrm{eff}}$ for different values of $\gamma$ are shown in Fig. 9 and
Fig. 10. The blue and red solid lines represent the theoretical graphs for the
Poisson and Wigner distributions, respectively. It is seen by comparing Figs.
9 and 10 that the nature of the level spacing distributions are similar for
$k=0$ and $k=1$. Further, $\rho(s)$ changes smoothly from the Poisson to
Wigner distribution via some intermediate distributions as $\gamma$ is varied
from $\gamma=0$ to $\gamma=2$. We define a quantity $\eta$ as follows,

$\displaystyle\eta=\left|\frac{\int_{0}^{s_{0}}\left[P(s)-P_{W}(s)\right]ds}{\int_{0}^{s_{0}}\left[P_{P}(s)-P_{W}(s)\right]ds}\right|,$
(30)

where $s_{0}=0.4729......$ is the intersection point of $P_{P}(s)$ and
$P_{W}(s)$. It may be noted that $\eta=1$ for $P(s)=P_{P}(s)$, while $\eta=0$
for $P(s)=P_{W}(s)$. Thus, $\eta$ is an indicator of the nature of the
distribution, and the plot of the $\eta$ with respect to the parameter
$\gamma$ is shown in Fig. 11(a). This diagram in a sense acts as the
bifurcation diagram with respect to $\gamma$. The value of $\eta$ approaches
one near $\gamma=0$ and goes to zero as $\gamma$ passes through the value
$\gamma\approx 1.5$ —the system transits from the integrable to the chaotic
region.

(a)

(b)

Figure 11: (Color Online) (a) Plot of $\eta$ with respect to $\gamma$, for
$k=0,\ 1$; (b) Plot of $\langle\tilde{r}\rangle$ with respect to $\gamma$ for
$k=0$.

The unfolding procedure during the calculation of level statistics is a bit
cumbersome and sometime system dependent. The statistics of the ratio of two
consecutive energy-gapsoganesyan  has been proposed as an alternative measure
for the same purpose. The spacing between adjacent energy levels
$\delta_{n}=E_{n+1}-E_{n}$, and the ratio between adjacent gaps is defined as,

$\displaystyle
0\leq\tilde{r}_{n}=min\\{\delta_{n},\delta_{n-1}\\}/max\\{\delta_{n},\delta_{n-1}\\}\leq
1.$

The probability distribution of this ratio $\tilde{r}$ for integrable
Hamiltonian, $P_{P}(\tilde{r})=\frac{2}{(1+\tilde{r})^{2}}$ and the mean value
of this is $\langle\tilde{r}\rangle_{P}=2\ ln2-1\cong 0.386$. The theoretical
expression of probability distribution of $\tilde{r}$ in case of $GOE$
ensembles is,
$P_{GOE}(\tilde{r})=\frac{27}{4}\frac{\tilde{r}+\tilde{r}^{2}}{(1+\tilde{r}+\tilde{r}^{2})^{2}}$.
The mean value in this case, $\langle\tilde{r}\rangle_{GOE}=0.5295\pm 0.0006$.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Figure 12:  (Color online) Plot of the Gap Ration $\tilde{r}$ statistics of
Hamiltonian with $k=0$,$a=b=1$ for different values of $\gamma$. (a)
$\gamma=0$; (b) $\gamma=0.2$; (c) $\gamma=0.4$; (d) $\gamma=0.6$; (e)
$\gamma=0.8$, (f) $\gamma=1.0$, (g) $\gamma=1.2$, (h) $\gamma=1.4$, (i)
$\gamma=1.6$, (j) $\gamma=1.8$, (k) $\gamma=2.0$.

We study the gap ratio distribution for $k=0$ only because we have already
shown that the change in $k$ does not alter the nature of level spacing
distribution. The Fig. 12 shows that the probability density distribution of
gap ratio $\tilde{r}$ is close to the Poisson distribution up to
$\gamma\approx 1.0$. This signifies integrable system. When $\gamma\approx
1.4$ the distribution goes close to the theoretical value of the probability
density distribution of GOE ensembles. This signifies the non-integrable
region. The Fig. (11(b)) is the bifurcation diagram for this gap ratio
distribution. It is see from the bifurcation diagram that while
$\gamma\gtrapprox 1.4$ the average value of gap ratio is close to 0.52 i.e.
the system is chaotic. We show the quantum transition from integrable to
chaotic region by studying nearest neighbour level spacing and gap ratio
distribution.

##  IV Conclusions & Discussions

We have studied the periodic quantum Toda lattice with BLG for two and three
particles. The two-particle Toda lattice is integrable and we have constructed
two integrals of motion which are in involution. The translation invariance of
the system has been used to separate out the center of mass motion, and the
effective Hamiltonian in the relative coordinate is described by a particle
moving in a potential consisting of harmonic plus cosine hyperbolic. The
effective potential describes single or double wells depending on the strength
of the loss-gain and the exponential terms. The eigen value equation has been
solved numerically, and eigen energies as well as eigenfuntion corresponding
to quantum bound states for a few low lying states have been presented. The
qualitative behaviour is similar to that of single of double well arising out
of a Quartic potential.

The quantum Toda lattice with BLG and VMC for three particles is translation
invariant. The effective Hamiltonian after the separation of the center of
mass may be interpreted as a particle moving in a two dimensional potential,
consisting of anisotropic harmonic oscillators plus Toda potential, and
subjected to external uniform magnetic field with its strength being linearly
proportional to the strength of the BLG. The angular frequencies of the
harmonic oscillators depend quadratically on the strength of the BLG. It has
been shown that the effective Hamiltonian is exactly solvable in the limit in
which the Toda potential reduces to coupled oscillators. The eigenspectra and
eigenfunctions have been obtained analytically by mapping the effective
Hamiltonian to that of decoupled anisotropic oscillators in two dimensions via
a similarity transformation. There is a limit in which reduction of a degree
of freedom occur in phase space and spectra becomes infinitely degenerate.

We have obtained two integrals of motion which are in involution. Although we
have not found the third integral of motion which is required to show the
complete integrability, the numerical investigations indicates that the three-
particle quantum Toda system is integrable below a critical value of the BLG
strength and chaotic above this critical value. The quantum integrability and
chaos have been investigated via level statistics as well as gap-ratio
distribution of the level-spacing of nearest-neighbour eigen values. In
particular, we have observed the quantum transition from the chaotic to the
integrable region, when the loss-gain strength crosses a critical value and
goes to zero —the level spacing as well as the gap-ratio distributions
smoothly changes from the Wigner-Dyson distribution and tends to follow the
Poisson distribution. We have also studied the level repulsion phenomena in
the energy spectra in both the two and three particle Toda lattices. It has
been shown through the graphical presentations that the degree of level
repulsion is large in the case of the three particle system. The quantum
transition from chaotic to integrable region has also been independently
confirmed by studying the complexity in higher order excited state wave
functions.

There are some immediate questions which may be pursued for future
investigations. For example, the analytic expression for the third integral of
motion is yet unknown. The standard technique of LAX pair formalism or
Painelevé analysis may be employed to find an analytic expression for the
same. Further, the generic problem for $N>3$ particles is worth pursuing to
see whether or not the mixed phases of intergrability and chaos persist for
arbitrary number of particles. It is possible that the system may not be
integrable for any range of the strength of the BLG for large $N$. Finally, a
semi-classical approach to understand the integrability and chaos is
desirable. Some of these issues will be addressed in future publications.

##  V Acknowledgements

The work of SG is supported by Inspire fellowship(Inspire Code: IF190276) of
Govt. of India

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