# Effects of quantum fluctuations on macroscopic quantum tunneling and self-
trapping of BEC in a double well trap

Fatkhulla Kh. Abdullaev1, Ravil M. Galimzyanov1 and Akbar M. Shermakhmatov1,2
1Physical-Technical Institute of the Uzbekistan Academy of Sciences, Ch.
Aytmatov str., 2-B, 100084, Tashkent-84, Uzbekistan.
2”TIIAME” National Research University, Kari Niyazov 39, 100000, Tashkent,
Uzbekistan

###### Abstract

We study the influence of quantum fluctuations on the macroscopic quantum
tunneling and self-trapping of a two-component Bose-Einstein condensate in a
double-well trap. Quantum fluctuations are described by the Lee-Huang-Yang
term in the modified Gross-Pitaevskii equation. Employing the modified Gross-
Pitaevskii equation in scalar approximation, we derive the dimer model using a
two-mode approximation. The frequencies of Josephson oscillations and self-
trapping conditions under quantum fluctuations are found analytically and
proven by numerical simulations of the modified Gross-Pitaevskii equation. The
tunneling and localization phenomena are investigated also for the case of the
Lee-Huang-Yang fluid loaded in the double-well potential.

###### pacs:

67.85.Hj, 03.75.Kk, 03.75.Lm, 03.75.Nt

##  1 Introduction

Recently much attention has been paid to the study of the beyond mean-field
effects. The contribution to the energy of a BEC due to quantum fluctuations
around the Bogolyubov ground state has been found by Lee, Huang and Yang (LHY)
[1]. While these corrections are small, in many cases they lead to nontrivial
effects. One of important is their role in the collapsing condensate. It has
been shown, that in the case of two-component condensate, quantum
fluctuations(QF) can stabilize collapsing regime and lead to the appearance of
quantum droplets(QD) [2, 3]. Generation of the quantum droplets in the Bose
mixture is possible due to the joint action of the attractive residual mean-
field interaction and the effective repulsion introduced by QF. Another system
is a dipolar BEC, where quantum droplets are generated by the balance between
the small residual mean-field attraction of a combination of local two-body
interactions, repulsive dipolar interactions and repulsive LHY term [5]. The
existence of QD in other systems as the Bose-Fermi mixtures and Rabi-coupled
BEC is discussed in works [6, 7].

The influence of the beyond mean-field (BMF) contribution has been studied for
the collective excitations spectrum in Ref. [9],while modulational instability
of matter waves has been studied in Ref. [10].

In this regard, it is interesting to study the influence of BMF contribution
to another phenomena in the BEC. An important case is the processes of
macroscopic quantum tunneling (MQT) and self-trapping(ST) in BEC loaded in a
double-well trap potential. In this system, the nonlinearity due to the
interaction between atoms plays an important role. In particular, it induces
the transition from the Josephson oscillations(JO) of atomic imbalance to the
nonlinear self-trapping (ST) regime [12]. The quantum fluctuations induce an
additional nonlinear interaction in the form of effective repulsion. Thus in
some region of parameters, we can expect the processes of tunneling in the
double-well potential to be very sensitive to the action of quantum
fluctuations. In this problem, we have a few small parameters: the tunneling
parameter(describing the amplitude of tunneling between condensates), the
strength of QF given by the LHY correction, and the residual mean field
nonlinearity between components of BEC. When they are comparable, we can
expect new regimes in the switching of the matter waves in the double-well
potential. The important limiting case of the vanishing of residual mean-field
interactions is a LHY fluid [11] loaded in the double-well potential. Note
that recently the LHY fluid has been observed in Ref. [13]

In this work, we study the MQT and ST processes in a two-component quasi-one-
dimensional BEC loaded into a double-well trap, when quantum fluctuations are
taken into account.

The structure of the paper is as follows.

In Section 2, we describe the modified quasi-one-dimensional GP equation,
obtained for two-component BEC in a scalar approximation and including the LHY
term. In Section 3, two-mode(dimer) model for BEC loaded into the double-well
trap is derived. The Hamiltonian, fixed points, the Josephson oscillations are
analyzed in Section 4. The nonlinear self-trapped regime is discussed in
Section 4.2. In Section 5, results of full numerical simulations of the
modified Gross-Pitaevskii(GP) equation with the double-well potential are
presented. The conditions for experimental observation of the investigated
effects are discussed.

##  2 The beyond mean field model for a BEC in a double-well potential

We consider the dynamics of a two-component BEC loaded in a cigar-type trap,
when the beyond mean-field effects, describing quantum fluctuations, are taken
into account. Assuming that the wave functions are related by:

$\Psi_{1}=\sqrt{\frac{g_{11}}{g_{22}}}\Psi_{2}=\Psi,$

where $g_{ij}=4\pi\hbar^{2}a_{ij}/m$, $a_{ij}$ are inter- and intra-species
atomic scattering lengths, and supposing $g_{11}=g_{22}=g$ and $g_{12}=\delta
g+g$, we reduce the coupled GP equations to the scalar GP equation for $\Psi$.
Then the governing equation, describing the wave function of a condensate with
the Lee-Huang-Yang term, is [2]:

$i\hbar\Psi_{t}(r,t)=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi(r,t)+V(r)\Psi(r,t)-\delta
g|\Psi|^{2}\Psi+\gamma_{LHY}|\Psi|^{3}\Psi,$ (1)

where

$\gamma_{LHY}=\frac{128\sqrt{\pi}\hbar^{2}}{3m}|a|^{5/2}.$

We study below a quasi-one-dimensional geometry given by a cigar-type trap,
when the transversal ($\omega_{\perp}$) and longitudinal ($\omega_{x}$) trap
frequencies are satisfied to the condition $\omega_{\perp}\gg\omega_{x}$. Here
we use the factorization [14]:

$\Psi=\Phi(x,t)R(\rho),\ R(\rho)=Ae^{-\rho^{2}/2l_{\perp}^{2}},\
l_{\perp}=\sqrt{\hbar/(m\omega_{\perp})}.\ $ (2)

After integration over the transverse distribution, we obtain the quasi-one-
dimensional MGP equation:

$i\hbar\Psi_{t}(x,t)=-\frac{\hbar^{2}}{2m}\Psi_{xx}(x,t)+V(x)_{ext}\Psi(x,t)-\frac{\delta
g}{2\pi
l_{\perp}^{2}}|\Psi|^{2}\Psi+\frac{2\gamma_{LHY}}{5\pi^{3/2}l_{\perp}^{3}}|\Psi|^{3}\Psi,$
(3)

where $V_{ext}(x)$ is a double well potential chosen as

$V_{ext}(x)=\frac{mw_{x}^{2}}{2}x^{2}+U_{0}\exp(-(x/l_{0})^{2}).$ (4)

It is useful to introduce dimensionless variables according to:

$t=\omega_{x}t,x=\frac{x}{l_{x}},\
l_{x}=\sqrt{\frac{\hbar}{m\omega_{x}}},\psi=\sqrt{\frac{l_{x}}{N}}\psi.$

Then we can rewrite the governing equation as follows:

$i\psi_{t}=-\frac{1}{2}\psi_{xx}+V_{ext}(x)\psi+\delta
g|\psi|^{2}\psi+\gamma|\psi|^{3}\psi,$ (5)

where

$\displaystyle\delta g=\frac{4\delta
a}{l_{x}}\frac{\omega_{\perp}}{\omega_{x}}N,\gamma=\frac{128}{15\pi}(\frac{a}{l_{x}})^{5/2}(\frac{\omega_{\perp}}{\omega_{x}})^{3/2}N^{3/2},$
$\displaystyle
V_{ext}=\frac{1}{2}x^{2}+V_{0}e^{-(x/l)^{2}},l=\frac{l_{0}}{l_{x}},V_{0}=\frac{U_{0}}{\hbar\omega_{x}}.$

This normalization of the wave function $\psi$ corresponds to:

$\int_{-\infty}^{\infty}|\psi|^{2}dx=1.$

Let us estimate the parameters for the experiments, as performed in Ref.[13].
The atomic scattering length is about $100a_{0}$, where $a_{0}$ is the Bohr
radius, $l_{x}\approx 300\mu m$, $N=10^{2}\div 10^{3}$. The values of
parameters $g=0\div 10,\ \gamma=0.01\div 1$.

##  3 Two-mode model

To describe the dynamics of the BEC we will use the two-mode model [12]. Let
us consider the solution in the form:

$\psi=u(t)\Phi_{1}(x)+v(t)\Phi_{2}(x),$ (6)

where $\Phi_{1,2}(x)$ are determined by the solutions of the stationary GP
equation. Firstly we obtain two solutions of the stationary equation, (ground
state with the norm $N_{gr}=1$ and an excited state with the same norm
$N_{exc}=1$)

$\displaystyle[-\frac{1}{2}\frac{d^{2}}{dx^{2}}+V_{ext}(x)+\delta
g|\Phi_{gr/exc}|^{2}+$
$\displaystyle\gamma|\Phi_{gr/exc}|^{3}-\mu]\Phi_{gr/exc}=0.$ (7)

It should be noted that a ground state function and an excited one are chosen
to be symmetric and antisymmetric, respectively, $\Phi_{gr}(-x)=\Phi_{gr}(x)$
and $\Phi_{exc}(-x)=-\Phi_{exc}(x)$. Then, functions $\Phi_{1,2}(x)$,
localized in corresponding wells (1,2) of the double-well potential are
expressed as

$\displaystyle\Phi_{1}(x)=(\Phi_{gr}(x)+\Phi_{exc}(x))/\sqrt{2},$ (8)
$\displaystyle\Phi_{2}(x)=(\Phi_{gr}(x)-\Phi_{exc}(x))/\sqrt{2},$ (9)
$\displaystyle I=\int_{-\infty}^{\infty}\Phi_{i}(x)\Phi_{j}(x)dx,\ i,j=1,2.$
(10)

where $I$ is the overlap integral.

In Fig. 1 the double-well potential with wave functions $\Phi_{1,2}(x)$ are
depicted. Note that the two-mode approximation works well for the small mean
field nonlinearity and overlaps [15, 16]

Figure 1: The double-well potential with parameters $V_{0}=5,\ l=1.5$. The
solid line is for the double-well potential, - dash-dot-dot and dot- lines are
for $\Phi_{2}(x)$ and $\Phi_{1}(x)$ wave functions, respectively.

Let us substitute the solution 6 into Eq. (3), and multiply by $\Phi_{1}$, and
integrate over $x$ in the interval $(-\infty,+\infty)$. Supposing a weak
overlap criterion and ignoring nonlinear overlap integrals, we obtain the
following equation:

$iu_{t}=E_{1}^{0}u-Kv-\alpha|u|^{2}u+\beta|u|^{3}u.$ (11)

In a similar way, the equation for $v(t)$ can be obtained

$iv_{t}=E_{2}^{0}v-Ku-\alpha|v|^{2}v+\beta|v|^{3}v.$ (12)

Here $E_{1,2}^{0}$ are the zero-point energies in each well, for the symmetric
trap $E_{1}^{0}=E_{2}^{0}$. K is the amplitude of the tunneling, $\alpha$ and
$\beta$ are other nonlinearity parameters

$E_{1}^{0}=\frac{1}{2}\int_{-\infty}^{\infty}|\Phi_{1,x}|^{2}dx+\int_{-\infty}^{\infty}\Phi_{1}V(x)\Phi_{1}dx,$
(13)
$K=-\left(\frac{1}{2}\int_{-\infty}^{\infty}\Phi_{1,x}\Phi_{2,x}dx+\int_{-\infty}^{\infty}\Phi_{1}V(x)\Phi_{2}dx\right),$
(14) $\alpha=\delta g\int|\Phi_{1}|^{4}dx,\ \beta=\gamma\int\Phi_{1}^{5}dx.$
(15)

The condition of the applicability of the two-mode model is $\mu<V_{0}$, where
$\mu$ is the chemical potential. For more details see the work [15], where the
different extensions of the two-mode model are considered.

##  4 Dimer model with LHY terms

We consider below the symmetric dimer case, when $\Delta
E=E_{1}^{0}-E_{2}^{0}=0$. Then eqs.(11),(12) are written as:

$\displaystyle iu_{t}+Kv+\alpha|u|^{2}u-\beta|u|^{3}u-E_{1}^{0}u=0,$ (16)
$\displaystyle iv_{t}+Ku+\alpha|v|^{2}v-\beta|v|^{3}v-E_{1}^{0}v=0.$ (17)

Introducing variables:

$u=\sqrt{N_{1}}e^{i\phi_{1}},\ v=\sqrt{N_{2}}e^{i\phi_{2}},$

we obtain coupled equations for the relative imbalance $Z=(N_{1}-N_{2})/N$ and
the relative phase $\psi=\phi_{2}-\phi_{1}$:

$\displaystyle Z_{t}=-2K\sqrt{1-Z^{2}}\sin{\psi},$ (18)
$\displaystyle\psi_{t}=2K\frac{Z}{\sqrt{1-Z^{2}}}\cos{\psi}+\alpha
NZ-\frac{\beta N^{3/2}}{2^{3/2}}[(1+Z)^{3/2}-(1-Z)^{3/2}].$ (19)

Introducing variables:

$\Lambda=\frac{\alpha N}{2K},\ \epsilon=\frac{\beta N^{3/2}}{2K},\ t=2Kt,$

we get:

$\displaystyle Z_{t}=-\sqrt{1-Z^{2}}\sin{\psi},$ (20)
$\displaystyle\psi_{t}=\frac{Z}{\sqrt{1-Z^{2}}}\cos{\psi}+\Lambda
Z-\frac{\epsilon}{2^{3/2}}[(1-Z)^{3/2}-(1+Z)^{3/2}].$ (21)

This system has the Hamiltonian form:

$Z_{t}=-\frac{\partial H}{\partial\phi},\ \psi_{t}=\frac{\partial H}{\partial
Z},$ (22)

with the Hamiltonian

$H=-\sqrt{1-Z^{2}}\cos{\psi}+\frac{\Lambda}{2}Z^{2}+\frac{\epsilon}{5\sqrt{2}}a,$
(23)

where

$a=[(1+Z)^{5/2}+(1-Z)^{5/2}].$

The case, when the residual mean field interaction is equal to
zero,$\Lambda=0$, corresponds to the Lee-Huang-Yang fluid, loaded into a
double-well potential [11].

###  4.1 Josephson oscillations

To analyze different regimes in the dynamics of the atomic population
imbalance and the relative phase, let us find the stationary states of the
system. Equating the derivatives on time to zero, we have:

$\psi_{c}=2\pi n,\ z_{c}=0,n=0,1,2,...\ \mbox{with}\
E_{+}=-1+\frac{\sqrt{2}}{5}\epsilon,$

and

$\psi_{c}=(2n+1)\pi,\ z_{c}=0,n=0,1,2,...\ \mbox{with}\
E_{-}=1+\frac{\sqrt{2}}{5}\epsilon,$

where $E_{+/-}$ are the values of Hamiltonian (23) at the critical points
$Z_{c},\ \psi_{c}$. From Eq. (22), we can find the frequencies of Josephson
oscillations $\omega_{J}$ of the atomic population imbalance $Z$. Two cases of
the relative phase difference can be studied.

i).The case of a zero-phase mode: $\psi(0)=0,<Z>=0$. Then we obtain for the
frequency of small oscillations Z, i.e. the JO frequency $\omega_{J}$:

$\omega_{J}=\sqrt{1+\Lambda+\frac{3\sqrt{2}\epsilon}{4}}.$ (24)

ii).The case of a $\pi$-phase mode: $\psi(0)=\pi,\ <z>=0.$ The frequency of
the JO is:

$\omega_{J}=\sqrt{1-\Lambda-\frac{3\sqrt{2}\epsilon}{4}}.$ (25)

If the strength of the residual mean field nonlinearity and the LHY
corrections are satisfied to the relation:

$\Lambda=-\frac{3\sqrt{2}\epsilon}{4},\ \alpha=\frac{3\beta
N^{1/2}}{2^{3/2}},$ (26)

then the frequency of Josephson oscillations is close to the frequency of
oscillations of the unperturbed system (the Rabi oscillations case). It
corresponds to the quasi-linear regime. Thus it is open the possibility to
measure the strength of quantum fluctuations strength by properly detuning the
residual mean-field scattering length $\delta a=-|a_{12}|+a.$

If we linearize Eq. (22) on $Z$ only, the next equation for the phase is
obtained:

$\psi_{tt}+(\Lambda+\frac{3\epsilon}{2^{3/2}})\sin(\psi)+\frac{1}{2}\sin(2\psi)=0.$
(27)

This equation has the form of an equation of motion of the unit mass particle
under the action of the periodic potential. When $\epsilon=0$, the result
obtained in Ref. [12] is reproduced. At $\psi=\pi$($\pi$-phase mode), in the
potential a valley exists, where the effective particle oscillates. By
properly detuning the parameter $\Lambda\rightarrow 1-3\epsilon/2^{3/2}$, we
can diminish this valley to zero. The nonlinearity breaks the $z$-symmetry and
leads to the appearance of states with $\langle Z\rangle\neq 0$.

###  4.2 Self-trapped regime

The self-trapped regime corresponds to the localization of atoms in one of the
wells, i.e. to $\langle Z\rangle\neq 0$. If the population imbalance is fixed,
then we can obtain the condition on the nonlinearity parameter. It can be find
from the condition:

$H>1+\frac{\sqrt{2}\epsilon}{5}.$ (28)

Using Eq. (23), we obtain the condition for self-trapping:

$\Lambda>\Lambda_{c}=\frac{2}{Z(0)^{2}}[1+\frac{\epsilon}{5\sqrt{2}}(2-a)+\sqrt{1-Z(0)^{2}}\cos(\psi(0))].$
(29)

For the LHY fluid case, the localization criterion is:

$\epsilon_{c}=\frac{5\sqrt{2}(1+\sqrt{1-Z(0)^{2}}\cos(\psi(0)))}{a-2}.$ (30)

Figure 2:  a) The dynamics of the relative imbalance $Z(t)$ for the LHY fluid
($\delta g=0$). Dot-line is for $\epsilon=1$ and the solid line is for
$\epsilon=0.96$. Critical value $\epsilon_{c}=0.9872$. For both cases,
oscillations are in the $\pi$-phase regime and the initial imbalance
$Z(0)=0.4$. b) Phase portrait for the LHY fluid, where $\epsilon=1.96$.

In Fig. 2a the dynamics of the relative imbalance $Z$ for the LHY fluid
($\Lambda=0$)in $\pi$-phase mode is shown. From Eq.(30) we obtain
$\epsilon_{c}=0.9872$. Here the dotted line is for the case $\epsilon=1$ and
the solid line for $\epsilon=0.96$. One can see that can see that at
$\epsilon>\epsilon_{c}$, the self-trapping regime appears.

In Fig. 2b the phase-portrait in ($Z,\psi$) plane for the LHY fluid case is
presented. Is clearly seen the self-trapping of condensate in one of the wells
under the action of quantum fluctuations. Note, that at these values of
parameters, the self-trapping exists in the $\pi$-phase mode.

##  5 Numerical simulations

We also perform full numerical simulations of the modified GP equation with
the double-well potential to the study the macroscopic quantum tunneling and
self-trapping phenomena.

We start from the calculation of the ground state and exited state solutions
with $N_{gr}=N_{ex}=1$ for given $\gamma$ and $\delta g$. Then using Eq. (8)
we construct initial wave function for governing equation Eq. (5):

$\Psi(x,t=0)=u(0)\Phi_{1}(x)+v(0)\Phi_{2}(x),$ (31)

where $u(0)^{2}-v(0)^{2}=Z(t=0)$, By computing the full equation Eq. (5) we
get the dependencies $Z(t),\phi(t)$.

For obtaining the relative imbalance $Z(t)$ and relative phase $\psi(t)$ one
needs to calculate $N_{1,2}$ and $\phi_{1,2}$ as follows:

$\displaystyle N_{1}(t)=\int_{-\infty}^{0}|\Psi(x,t)|^{2}dx,$ (32)
$\displaystyle N_{2}(t)=\int_{0}^{+\infty}|\Psi(x,t)|^{2}dx,$ (33)
$\displaystyle\phi_{1}(t)=\arctan{\frac{\int_{-\infty}^{0}\Im(\Psi(x,t))|\Psi(x,t)|^{2}dx}{\int_{-\infty}^{0}\Re(\Psi(x,t))|\Psi(x,t)|^{2}dx}},$
(34)
$\displaystyle\phi_{2}(t)=\arctan{\frac{\int_{0}^{+\infty}\Im(\Psi(x,t))|\Psi(x,t)|^{2}dx}{\int_{0}^{+\infty}\Re(\Psi(x,t))|\Psi(x,t)|^{2}dx}}$
(35)

where $\Im$ and $\Re$ are the imaginary and real parts. For the following
calculations the double-well potential parameters are $V_{0}=4,6,7$ with
$l=1.5$.

In Fig. 3 the dependencies of the JO frequency $\omega_{J}$ : a) on $\gamma$
(the case of LHY-dimer at $\delta g=0$) and b) on $\delta g$ (the case when
the LHY-term is neglected and $\delta g\neq 0$) are presented. There are two
curves in this figure. The upper curve is for the frequency dependence of
$Z(t)$ oscillations in the zero-phase regime, when $\psi=0$, and the lower one
is for the case of $\pi$-phase regime $\psi=\pi$. One can see that at the very
beginning of the two curves, they are described by equations Eq. (24) and Eq.
(25) correspondingly. For both cases $Z(t=0)=0.02$.

As the LHY-term strength $\gamma$ increases, a localization phenomenon with
$\langle Z\rangle\neq 0$ for $\gamma\approx 0.03$ occurs breaking the smooth
behavior of the lower $\pi$-phase mode curve.

Figure 3: The frequency dependencies of the Josephson oscillations for a) the
LHY-dimer, $\delta g=0$ (left graph) and b) the case when LHY-term is
neglected. In both graphs upper solid lines are for zero-phase cases and lower
dotted lines are for $\pi$-phase cases. In all cases the initial imbalance of
Josephson oscillations $Z(t=0)=0.02$ for $V_{0}=7,\ l=1.5$.

The frequency dependencies for the same cases, but for larger intervals of
$\gamma$ and $\delta g$ are presented in Fig. 4.

It should be noted that as seen in the left panel (the LHY fluid case) for
$0<\gamma<1.5$ the mean value of the imbalance $\langle Z\rangle=0$, and
$\langle Z\rangle\neq 0$ for $\gamma>1.6$. In the right panel (when the LHY
term is neglected) for $0<\delta g<1.5$, the mean value of the imbalance
$\langle Z\rangle=0$, and $\langle Z\rangle\neq 0$ for $\delta g>1.5$.

Figure 4: The frequency dependencies of the Josephson oscillations in the
zero-phase mode: a) for different $LHY$-term $\gamma$ when $\delta g=0$; b)
for different nonlinear term $\delta g$ when the LHY-term is neglected.
Parameter $Z(t=0)=0.4$. Double-well potential parameters $V_{0}=6,\ l=1.5$ in
both cases.

Figure 5: The time dependencies of $Z(t)$ for the LHY-fluid (with $\delta
g=0$): a) $\gamma=0.02\ $, b) $\gamma=0.025\ $, c) $\gamma=0.03$. Solid lines
are for the zero-phase and dot-lines are for the $\pi$-phase modes. The
parameters of the trap potential are $V_{0}=7,\ l=1.5$. Everywhere initial
imbalance $Z(t=0)=0.02$.

It is interesting to examine the evolution of the time dependencies of the
population imbalance $Z(t)$ of the LHY-fluid near the symmetry breaking point
that one can see at the beginning of the lower curve depicted in Fig. 3. In
Fig. 5 the time dependencies of $Z(t)$ for three values of $\gamma$ are
presented. One can see a transition of the $\pi$-phase mode into near the
bifurcation point located between $\gamma=0.2$ and $\gamma=0.25$.

We consider Josephson oscillations of the atomic population imbalance $Z(t)$
($\langle Z\rangle=0$) in the quasi-linear and self-localization regimes
($<Z>\neq 0$).

In Fig. 6, Josephson oscillations are presented based on the results obtained
from the two-modes model and the numerical simulations. In all graphs, one can
see phase shifts that increase with time.

In Fig.6a, the largest phase shift is observed when the nonlinear parameter
$\delta g=1$ is large enough (at $\gamma=0$). For the LHY-fluid case, in Figs.
6b and 6c with the nonlinear parameters $\delta g=0$ and $\gamma=0.01$ (small
enough), the phase shifts decrease. So, the accuracy of the two-modes method
decreases at large times and large values of nonlinear parameters.

Figure 6: The time dependencies of the imbalance $Z(t)$ obtained from the two-
mode model (solid lines) and the full numerical solution of Eq (5) (dot lines)
for: a) $\delta g=1$, $\gamma=0$, (zero-phase); b) $\delta g=0$,
$\gamma=0.01$, (zero-phase); c) $\delta g=0$, $\gamma=0.01$ ($\pi$-phase).
Everywhere initial imbalance $Z(0)=0.4$. The parameters of the trap potential
are $V_{0}=4,\ l=1.5$.

From Eq. (5) we see that if two nonlinear parameters are taken in opposite
signs, they can suppress each other and the quasilinear (the Rabi) regime can
be obtained. To create this regime we should calculate necessary values of the
nonlinear parameters using Eq. (26). Thus we need also to calculate parameters
$\alpha=\int_{-\infty}^{\infty}|\Phi_{1}(x)|^{4}dx$ and
$\beta=\int_{-\infty}^{\infty}|\Phi_{1}(x)|^{5}dx$ that depend on values of
the nonlinear parameters $\delta g$ and $\gamma$. In Figs. 7a and 7b, in order
to observe effect of suppression, we choose the nonlinear parameters with
opposite signs, $\delta g=-0.1,\ \gamma=0.1$ at the zero-phase and the
$\pi$-phase modes. In Fig. 7c we choose corrected value for $\delta g$:
$\delta g=-0.076,\ \gamma=0.1$ at the zero-phase mode. calculated from
Eq.(26). One can see from Figs. 7a and 7b that there remain the phase shift
between the solutions calculated based on the two-mode model and numerical
simulations of the GP equation. Fig. 7c shows that the phase shifts of
solutions obtained on the basis of the two-mode method and the numerical
method have almost disappeared which indicates that the nonlinear effects have
decreased and the mode of quasi-linear oscillations has been reached.

Figure 7: The time dependencies of the imbalance $Z(t)$ obtained from the two-
modes model (solid lines) and the full numerical solution of Eq (5) (dot
lines) for: a) $\delta g=-0.1$, $\gamma=0.1$ (zero-phase); b) $\delta g=-0.1$,
$\gamma=0.1$ ($\pi$-phase), c) $\delta\ g=-0.076$, $\gamma=0.1$ (zero-phase)
with initial imbalance $Z(0)=0.4$. The parameters of the trap potential are
$V_{0}=4,\ l=1.5$.

To study the localization phenomenon, we consider the case of the LHY-fluid,
when $\gamma\neq 0$ at $\delta g=0$. The results of calculation are given in
Fig. 8 for three values of $\gamma=1.5;\ 1.6;\ 1.8$. As seen, the self-
localization phenomenon is observed at $\gamma\simeq 1.6$. Two-mode model
predict the existence of the ST transition, but the agreement with full
numerical simulations of the MGP equation is only qualitative. This
discrepancy is connected with neglect of nonlinear overlap integrals and thus
to the error in the calculation of the tunneling amplitude $K$ [15].

Figure 8: The time dependencies of $Z(t)$ for the LHY-fluid case (with $\delta
g=0$) when the LHY-term is a) $\gamma=1.5$; b) $\gamma=1.6$; c) $\gamma=1.8$.
The parameters of the trap potential $V_{0}=6,\ l=1.5$.

In Figs. 9a, 9b and 9c the time dependencies of the imbalance $Z(t)$ at zero-
phase and $\pi$-phase modes for $\gamma=0.5$, $\delta g=-0.1$ are presented
for different values of the initial imbalance. As seen, an increase in the
initial imbalance $Z(t=0)$ causes a change in the behavior of the dependencies
$Z(t)$ and the emergence of the localization in both phase modes.

Figure 9: Time dependencies of $Z(t)$ for different initial imbalances: a)
$Z(0)=0.4$; b) $Z(0)=0.6$; c) $Z(0)=0.6$. All graphs are presented for $\delta
g=-0.1$, $\gamma=0.5$. Solid lines are for the $\pi$-phase $\psi(0)=\pi$ and
dot lines are for the zero-phase $\psi(0)=0$ modes. Parameters of the trap
potential $V_{0}=7,\ l_{0}=1.5$.

Let us estimate the parameters for the experiment with 26Ka BEC, as performed
in Ref. [13]. Transverse and longitudinal frequencies of the trap can be taken
as, $\omega_{\perp}=2\pi\cdot 200$Hz, $\omega_{x}=2\pi\cdot 40$Hz and the
number of atoms is $N=10^{3}\div 10^{4}$. Double-well potential can be
generated by the superposition of the harmonic trap and periodic
potential[17]. The barrier values for this double-well potential are
$2\pi\cdot(160-240)$Hz when $V_{0}=4-6$. The atomic scattering length is about
$a=50a_{0}$ and the scattering length between two components is about $\delta
a=a_{0}$, where $a_{0}$ is the Bohr radius. The values of the scattering
length can be varied using the Feshbach resonance technics. If we take
$a=57.5a_{0}$, $\delta a=0$ and $N=10^{4}$, we get the values $\gamma=1.5$ and
$\delta g=0$ and JO frequencies is $\omega_{J}=2\pi\cdot 2.8$Hz (Josephson
oscillation around $\langle Z\rangle=0$). If we take $a=61.9a_{0}$, $\delta
a=0$ and $N=10^{4}$, we get the values $\gamma=1.8$ and $\delta g=0$ and
frequencies of atomic population imbalance $Z(t)$ is $\omega_{J}=2\pi\cdot
6$Hz(the frequency of self-localization mode) (see Fig.8). For the case
$a=148a_{0}$, $\delta a=-0.25a_{0}$ and $N=10^{3}$, we get the values
$\gamma=0.5$ and $\delta g=-0.1$ and thus frequencies of atomic population
imbalance $Z(t)$ is $\omega_{J}=2\pi\cdot 1.35$Hz for zero-phase mode
(Josephson oscillation around $\langle Z\rangle=0$) and $\omega_{J}=2\pi\cdot
3.6$Hz for $\pi$-phase mode(self-localization mode) (see Fig.9a).

##  6 Conclusion.

In conclusion, we have investigated the influence of quantum fluctuations on
the dynamics of a two-component BEC in the double-well potential. In the
scalar approximation and in a quasi-one-dimensional geometry, the problem is
described by the modified GP equation with the additional LHY terms,
corresponding to the beyond mean-field effects of quantum fluctuations.

We have derived a two-mode (dimer) model to describe the macroscopic quantum
tunneling and self-trapping phenomena, when the quantum fluctuations are taken
into account. The frequencies of the Josephson oscillations for zero- and
$\pi$\- phase modes are calculated. It is shown that the quasi-linear (i.e.
the Rabi oscillations) regime of oscillations can be observed by a proper
choice of the residual mean-field interactions between components of the BEC.
It opens the possibility to measure the strength of quantum fluctuations.

The criterion for switching from the MQT regime to the ST one is obtained and
applications for the observation of the LHY fluid have been discussed. Direct
simulations of the modified GP equation with the Lee-Huang-Yang term, confirm
predictions of the two-mode model for the JO frequencies. For the self-
trapping regime, the agreement is only qualitative, due to limits of the two-
mode model. Also, full simulations predict the self-trapping regime of the LHY
fluid in the double-well potential.

In future will be interested to investigate the MQT and MST phenomena in the
crossover regime from 3D and 2D geometry to one-dimensional geometry.

##  7 Acknowledgments

We thank B.Baizakov and E. N. Tsoy for fruitful discussions. The work was
supported by the grant FA-F2-004 of Ministry of Innovative Development of the
Republic of Uzbekistan.

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