#  Phase diffusion and fluctuations in a dissipative Bose-Josephson junction

Abhik Kumar Saha  School of Physical Sciences, Indian Association for the
Cultivation of Science, Jadavpur, Kolkata 700032, India.  Deb Shankar Ray
<EMAIL_ADDRESS>School of Chemical Sciences, Indian Association for the
Cultivation of Science, Jadavpur, Kolkata 700032, India.  Bimalendu Deb
Corresponding author<EMAIL_ADDRESS>School of Physical Sciences, Indian
Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India.

###### Abstract

We analyze the phase diffusion, quantum fluctuations and their spectral
features of an one-dimensional Bose-Josephson junction (BJJ) coupled to a
bosonic heat bath. We show the dependence of the phase diffusion coefficient
on the on-site interaction parameter $U$ and the temperature in zero-phase and
$\pi$-phase modes. We find that in the $\pi$-phase mode, the phase diffusion
co-efficient as a function of $U$ decreases so long as $U$ is below a critical
value while it increases above the critical value. This criticality of on-site
interaction reflects a transition between Josephson oscillation and
macroscopic quantum self-trapping (MQST) regime. Based on the thermal
canonical Wigner distribution, we calculate the coherence factor to understand
its dependence on temperature and on-site interaction energy in Josephson
oscillation and MQST regime. Furthermore, we discuss coherent and incoherent
spectral properties in connection with the fluctuations of the relative phase
and the population imbalance in both zero and $\pi$-phase modes from weak to
strong dissipation regime.

##  I Introduction

A dynamical system in contact with a reservoir has been a subject of wide
attention in dissipative dynamics [1, 2]. Over the years, the dissipative
quantum systems under the effect of random noise have been extensively
investigated both theoretically and experimentally [3, 4, 5, 6, 7] in widely
different areas such as condensed matter physics, quantum optics, magnetic
resonance spectroscopy etc. In recent years, the study of quantum dissipation
in ultracold atomic gases has attracted much attention due to the presence of
various loss processes affecting the coherence of the atomic matter waves [8,
9, 10]. Also, an ultracold atomic system has become a test bed to study
dissipation in an out-of-equilibrium quantum many-body system [11, 12, 13, 14,
15, 16]. Apart from dephasing and relaxation dynamics, the combined effect of
interaction and dissipation in open quantum systems can give rise to non-
equilibrium steady states and transitions between them [17, 18, 19, 20]. The
BJJ is an ideal system where both the effects of interaction and dissipation
can be explored to understand coherent and incoherent quantum dynamics of
matter waves [21, 22].

There are various ways in which the dissipation can be introduced in the
system. The presence of intrinsic coupling between the Josephson mode and the
quasi-particle mode often termed as sound mode leads to damping of oscillation
of population imbalance in a one-dimensional (1D) BJJ [23, 24]. Moreover,
impurity models can be realized in an ultracold atomic system by coupling the
impurity atom with the phonon bath of a condensate [25, 26, 27, 28], resulting
in dissipation in the system. Dissipation also originates from the finite
temperature effects [29, 30, 31], and the coupling of the system with the
external environment [32]. In this context, phase diffusion plays a
fundamental role in the dynamical behaviour of [33, 34, 35, 36] cold atoms in
optical lattices and Bose-Einstein condensates (BECs) [37, 38, 39, 40]. There
are many theoretical and experimental works suggesting that the interaction
between the particles can lead to phase diffusion [41, 42, 43, 44, 45, 46, 47,
33, 34, 35, 36]. Phase diffusion and heating effect in a BJJ due to thermal
fluctuations have already been observed in experiment [48]. Motivated by these
recent advances, we examine the phase diffusion, the coherent and incoherent
phenomena in a 1D dissipative BJJ. The question we address here is how the
phase diffusion coefficient depends on the interaction parameter and
temperature in both zero and $\pi$-phase modes of 1D dissipative BJJ.

In this paper, we consider a two-mode Hamiltonian of BJJ coupled to bosonic
heat baths. The nonlinear interaction between the system and an external
bosonic bath induces the dissipation and the phase diffusion. This is distinct
from the treatment of the system having intrinsic coupling between Josephson
mode and the quasi-particle mode. The aim of the present work is to derive
analytically the dissipative BJJ equations in presence of quantum noise to
probe the Langevin dynamics of relative phase. We derive the analytical
formula for the phase diffusion coefficient for zero and $\pi$-phase modes and
observe that the phase diffusion coefficient depends on the on-site
interaction and temperature of the system. In the zero-phase mode, it becomes
comparatively large for small dissipation in the high temperature regime. On
the other hand, in the $\pi$-phase mode it exhibits an interesting phase
transition-like behaviour between the Josephson oscillation and the MQST
regime. We observe that in the $\pi$-phase mode, the phase diffusion co-
efficient decreases so long as the on-site interaction is below a critical
value while it increases above the critical value. Making use of the thermal
canonical Wigner distribution [49], we calculate the coherence factor [50] to
capture its dependence on interaction energy and temperature. Our analysis of
coherence factor reveals that the 1D dissipative BJJ has a higher degree of
coherence which is in good agreement with the experimental observation [48].
The underlying quantum fluctuations in population imbalance and relative phase
have been analyzed in terms of fluctuation spectra to demonstrate the coherent
and incoherent behaviour of the system from weak to strong dissipation limit
in both zero and $\pi$-phase modes.

The paper is organized in the following way. In section II, we derive the
dissipative BJJ equations by coupling the two-mode Hamiltonian to the bosonic
baths in presence of noise to study the quantum and thermal properties of
noise. In section III, we analyze theoretically phase diffusion coefficient in
1D dissipative BJJ. In section IV, we derive the analytical formula for the
fluctuation spectrum of population imbalance and phase difference. In section
V, we present and discuss our results on numerical simulations of coherence
factor, phase diffusion coefficient and the spectra of the fluctuation to
corroborate the theoretical scheme. The paper is concluded in section VI.

##  II Dissipative Bose-Josephson junction: A phase diffusion model

The Gross-Pitaevskii Hamiltonian for a system of $N$ bosons at zero
temperature is given by

$\displaystyle\hat{H}_{\rm GP}=\hat{H}_{0}+\hat{H}_{\rm int}$ (1)

where

$\displaystyle\hat{H}_{0}=\int d{\bf
r}\left[-\frac{\hbar^{2}}{2m}\hat{\psi}^{\dagger}\nabla^{2}\hat{\psi}+\hat{\psi}^{\dagger}V_{\rm
ext}\hat{\psi}\right]$ (2) $\displaystyle\hat{H}_{\rm
int}=\frac{2\pi\hbar^{2}a_{s}}{m}\int d{\bf
r}\hat{\psi}^{\dagger}\hat{\psi}^{\dagger}\hat{\psi}\hat{\psi}$ (3)

$\hat{\psi}$ and $\hat{\psi}^{\dagger}$ represent bosonic fields and $V_{\rm
ext}$ is the external trap potential [51, 52] of the form $V_{\rm
ext}=V(\rho)+V_{\rm dw}(x)$, where
$V(\rho)=\frac{1}{2}m\omega_{\rho}^{2}\rho^{2}$ which permits harmonic
oscillations with frequency $\omega_{\rho}$ along radial directions i.e, $y$\-
and z\- axes and a symmetric double-well (DW) potential $V_{\rm dw}(x)$ along
$x$\- axis. Here $\rho^{2}=y^{2}+\textrm{z}^{2}$, $a_{s}$ denotes the $s$-wave
scattering length and $m$ is the atomic mass. In the strong radial confinement
regime ($\omega_{\rho}\gg\omega_{x}$) where $\omega_{x}$ be the axial
frequency, we assume that all the atoms occupy the ground state of the radial
harmonic potential. To proceed further, we integrate over the radial harmonic
oscillator states and obtain an effective 1D Hamiltonian for the system. The
lowest two energy eigen-functions are quasi-degenerate. For symmetric DW, the
lowest eigenstate $\phi_{\rm g}$ is space-symmetric $(\phi_{\rm
g}(x)=\phi_{\rm g}(-x))$ and the other quasi-degenerate state $\phi_{\rm
e}(x)$ is anti-symmetric $(\phi_{\rm e}(x)=-\phi_{\rm e}(-x))$.

The wave function $\hat{\psi}$ can be written as

$\displaystyle\hat{\psi}=\hat{a}_{g}\phi_{g}+\hat{a}_{e}\phi_{e}$ (4)

with $\hat{a}_{g}$ and $\hat{a}_{e}$ ($\hat{a}^{\dagger}_{g}$ and
$\hat{a}^{\dagger}_{e}$ ) being the annihilation (creation) operators for a
particle in the ground and first excited states, respectively. The operators
obey the standard bosonic commutation relation
$[\hat{a}_{i},\hat{a}^{\dagger}_{j}]=\delta_{ij}$. By defining further two
operators $\hat{a}_{L}=\frac{1}{\sqrt{2}}(\hat{a}_{g}+\hat{a}_{e})$ and
$\hat{a}_{R}=\frac{1}{\sqrt{2}}(\hat{a}_{g}-\hat{a}_{e})$ and their hermitian
counterparts, the effective 1D wave-function becomes

$\displaystyle\hat{\psi}=\hat{a}_{L}\phi_{+}+\hat{a}_{R}\phi_{-}$ (5)

where, $\phi_{+}=\frac{1}{\sqrt{2}}(\phi_{g}+\phi_{e})$ and
$\phi_{-}=\frac{1}{\sqrt{2}}(\phi_{g}-\phi_{e})$. Making use of this wave
function in the Gross-Pitaevskii Hamiltonian of Eq. (1), we obtain a two-mode
Hamiltonian of the form

$\displaystyle\hat{H}_{\rm TM}$ $\displaystyle=$
$\displaystyle\hat{a}^{\dagger}_{L}\hat{a}_{L}E_{1}+\hat{a}^{\dagger}_{R}\hat{a}_{R}E_{2}-(\hat{a}^{\dagger}_{L}\hat{a}_{R}+\hat{a}^{\dagger}_{R}\hat{a}_{L})K$
(6) $\displaystyle+$
$\displaystyle\frac{U_{+}}{2}\hat{a}^{\dagger}_{L}\hat{a}^{\dagger}_{L}\hat{a}_{L}\hat{a}_{L}+\frac{U_{-}}{2}\hat{a}^{\dagger}_{R}\hat{a}^{\dagger}_{R}\hat{a}_{R}\hat{a}_{R}$

where

$\displaystyle
K=-\int\left[\frac{\hbar^{2}}{2m}(\nabla_{x}\phi_{+}\nabla_{x}\phi_{-})+\phi_{+}V_{\rm
dw}(x)\phi_{-}\right]dx$ $\displaystyle
E_{1(2)}=\int\left[\frac{\hbar^{2}}{2m}|\nabla_{x}\phi_{+(-)}|^{2}+|\phi_{+(-)}|^{2}V_{dw}(x)\right]dx$

and

$\displaystyle U_{+(-)}=\frac{4\pi\hbar^{2}a_{s}}{m}\int|\phi_{+(-)}|^{4}dx$

where, $\nabla_{x}\equiv\frac{\partial}{\partial x}$, $K$ is the tunneling
amplitude between two sites of the DW and $U_{+(-)}$ is the on-site
interaction strength for left(right) side of the DW arising out of
nonlinearity. For a symmetric DW potential, we write $E_{1}=E_{2}=E$.

###  II.1 Two-mode model coupled to bosonic heat baths

Usually, Josephson oscillations in a DW potential is nondissipative meaning
that the dynamics of the atom number imbalance and relative phase remains
undamped over time [53, 54, 55]. However, in recent times several studies
reported a dissipative BJJ which is analogous to a pendulum with friction [56,
57, 58, 64]. In order to study the effects of dissipation in BJJ, we consider
a model of a BJJ coupled to two bosonic baths described by the total
Hamiltonian

$\displaystyle\hat{H}_{\rm T}$ $\displaystyle=$ $\displaystyle\hat{H}_{\rm
TM}+\hbar\sum_{k,m=L,R}\omega_{k}(\hat{b}^{\dagger}_{\rm mk}\hat{b}_{\rm mk})$
(7) $\displaystyle+$
$\displaystyle\hbar\sum_{k,m=L,R}g_{k}\hat{a}^{\dagger}_{m}\hat{a}_{m}(\hat{b}^{\dagger}_{\rm
mk}+\hat{b}_{\rm mk})$

where $\hat{b}_{mk}$ and $\hat{b}^{\dagger}_{mk}$ are the bosonic annihilation
and creation operators respectively, corresponding to $k$-th bath mode and
$m$-th well. $\omega_{k}$ represents the frequency of the $k$-th bath mode.
$g_{k}$ represents the coupling between the $k$-th bath mode and the on-site
boson number. The bath-BJJ coupling constants for the $k$-th bath mode for
both the wells are assumed to be same. It is important to emphasize that the
coupling between the system and the bath modes are nonlinear [32]. Also the
excitation in the bath modes is not accompanied by energetic deexcitation of
the system as in the usual linear system-bath coupling [52]. The influence of
bath and the non-linear coupling of the form (7) have been used earlier [32]
and this gives rise to dissipation and the phase diffusion of the system. The
Heisenberg equations of motion for the system and the bath operators are given
by the following equations

$\displaystyle\dot{\hat{a}}_{L}$ $\displaystyle=$
$\displaystyle-\frac{iE}{\hbar}\hat{a}_{L}+\frac{iK}{\hbar}\hat{a}_{R}-\frac{iU_{+}}{\hbar}\hat{a}^{\dagger}_{L}\hat{a}_{L}\hat{a}_{L}-i\sum_{k}g_{k}\hat{b}^{\dagger}_{Lk}(t)a_{L}$
(8) $\displaystyle-$ $\displaystyle i\sum_{k}g_{k}\hat{b}_{Lk}(t)a_{L}$
$\displaystyle\dot{\hat{a}}_{R}$ $\displaystyle=$
$\displaystyle-\frac{iE}{\hbar}\hat{a}_{R}+\frac{iK}{\hbar}\hat{a}_{L}-\frac{iU_{-}}{\hbar}\hat{a}^{\dagger}_{R}\hat{a}_{R}\hat{a}_{R}-i\sum_{k}g_{k}\hat{b}^{\dagger}_{Rk}(t)a_{R}$
(9) $\displaystyle-$ $\displaystyle i\sum_{k}g_{k}\hat{b}_{Rk}(t)a_{R}$
$\displaystyle\dot{\hat{b}}_{Lk}=-i\omega_{k}\hat{b}_{Lk}-ig_{k}\hat{a}^{\dagger}_{L}\hat{a}_{L}(t)$
(10)
$\displaystyle\dot{\hat{b}}_{Rk}=-i\omega_{k}\hat{b}_{Rk}-ig_{k}\hat{a}^{\dagger}_{R}\hat{a}_{R}(t)$
(11)

Formal integration of Eqs. (10) and (11) yields

$\displaystyle\hat{b}_{Lk}(t)$ $\displaystyle=$
$\displaystyle\hat{b}_{Lk}(t_{0})e^{-i\omega_{k}(t-t_{0})}$ (12)
$\displaystyle-$ $\displaystyle
ig_{k}\int_{t_{0}}^{t}\hat{a}^{\dagger}_{L}(t^{\prime})\hat{a}_{L}(t^{\prime})e^{-i\omega_{k}(t-t^{\prime})}dt^{\prime}$
$\displaystyle\hat{b}_{Rk}(t)$ $\displaystyle=$
$\displaystyle\hat{b}_{Rk}(t_{0})e^{-i\omega_{k}(t-t_{0})}$ (13)
$\displaystyle-$ $\displaystyle
ig_{k}\int_{t_{0}}^{t}\hat{a}^{\dagger}_{R}(t^{\prime})\hat{a}_{R}(t^{\prime})e^{-i\omega_{k}(t-t^{\prime})}dt^{\prime}$

where the first term is the free evolution of bath operators whereas the
second term arises due to the interaction with the system. Inserting Eqs. (12)
and (13) in Eqs. (8) and (9) we get

$\displaystyle\dot{\hat{a}}_{L}$ $\displaystyle=$
$\displaystyle-\frac{iE}{\hbar}\hat{a}_{L}+\frac{iK}{\hbar}\hat{a}_{R}-\frac{iU_{+}}{\hbar}\hat{a}^{\dagger}_{L}\hat{a}_{L}\hat{a}_{L}$
(14) $\displaystyle-$ $\displaystyle
i\sum_{k}g_{k}\hat{b}_{Lk}(t_{0})e^{-i\omega_{k}(t-t_{0})}\hat{a}_{L}$
$\displaystyle-$
$\displaystyle\sum_{k}g^{2}_{k}\int_{t_{0}}^{t}\hat{a}^{\dagger}_{L}(t^{\prime})\hat{a}_{L}(t^{\prime})\hat{a}_{L}(t)e^{-i\omega_{k}(t-t^{\prime})}dt^{\prime}$
$\displaystyle\dot{\hat{a}}_{R}$ $\displaystyle=$
$\displaystyle-\frac{iE}{\hbar}\hat{a}_{R}+\frac{iK}{\hbar}\hat{a}_{L}-\frac{iU_{-}}{\hbar}\hat{a}^{\dagger}_{R}\hat{a}_{R}\hat{a}_{R}$
(15) $\displaystyle-$ $\displaystyle
i\sum_{k}g_{k}\hat{b}_{Rk}(t_{0})e^{-i\omega_{k}(t-t_{0})}\hat{a}_{R}$
$\displaystyle-$
$\displaystyle\sum_{k}g^{2}_{k}\int_{t_{0}}^{t}\hat{a}^{\dagger}_{R}(t^{\prime})\hat{a}_{R}(t^{\prime})\hat{a}_{R}(t)e^{-i\omega_{k}(t-t^{\prime})}dt^{\prime}$

Changing the integration variable from $t^{\prime}$ to $\tau=t-t^{\prime}$ in
Eq. (14), we may write the last term in Eq. (14) as
$\sum_{k}g^{2}_{k}\int_{0}^{t-t_{0}}\hat{a}^{\dagger}_{L}(t-\tau)\hat{a}_{L}(t-\tau)\hat{a}_{L}(t-\tau)e^{-i\omega_{k}\tau}d\tau$,
where we have approximated $a_{L}(t)=a_{L}(t-\tau)$ for the last annihilation
operator, since the interference time $\tau_{c}$ of
$\sum_{k}g^{2}_{k}e^{-i\omega_{k}\tau}$ is much smaller than the time over
which the amplitude and the phase modulation of $a_{L}(t)$ take place. Thus
for times $t-t_{0}>\tau_{c}$, $\tau_{c}\rightarrow 0$, the summation acts as a
delta function so that we may write the integral approximately as
$\hat{a}^{\dagger}_{L}\hat{a}_{L}\hat{a}_{L}\int_{0}^{\infty}d\tau\sum_{k}g^{2}_{k}e^{-i(E/\hbar-\omega_{k})\tau}$.
Assuming that the bath modes are closely spaced in frequency we replace the
summation over $k$ by an integral over $\omega$, i.e $\sum_{k}\rightarrow\int
d\omega\Theta(\omega)$, where $\Theta(\omega)$ is the density of states. This
density and $g(\omega)$ are proportional to the powers of $\omega$ and vary
very little in the frequency interval $\tau^{-1}$ over $\omega$. This leads us
to following two equations for the reduced dynamics (same procedure is
followed for Eq. (15))

$\displaystyle\dot{\hat{a}}_{L}$ $\displaystyle=$
$\displaystyle-\frac{iE}{\hbar}\hat{a}_{L}+\frac{iK}{\hbar}\hat{a}_{R}-\frac{iU_{+}}{\hbar}\hat{a}^{\dagger}_{L}\hat{a}_{L}\hat{a}_{L}-\frac{\gamma}{2}\hat{a}^{\dagger}_{L}\hat{a}_{L}\hat{a}_{L}$
(16) $\displaystyle+$ $\displaystyle\hat{f}_{L}(t)\hat{a}_{L}$
$\displaystyle\dot{\hat{a}}_{R}$ $\displaystyle=$
$\displaystyle-\frac{iE}{\hbar}\hat{a}_{R}+\frac{iK}{\hbar}\hat{a}_{L}-\frac{iU_{-}}{\hbar}\hat{a}^{\dagger}_{R}\hat{a}_{R}\hat{a}_{R}-\frac{\gamma}{2}\hat{a}^{\dagger}_{R}\hat{a}_{R}\hat{a}_{R}$
(17) $\displaystyle+$ $\displaystyle\hat{f}_{R}(t)\hat{a}_{R}$

where, $\gamma=2\pi g^{2}(\Omega)\Theta(\Omega)$ represents the dissipation of
the modes and $\Omega=E/\hbar$. $\Theta(\Omega)$ is the density of the bath
modes. The terms
$\hat{f}_{L}(t)=-i\sum_{k}g_{k}(t_{0})\hat{b}_{Lk}(t_{0})e^{-i\omega_{k}(t-t_{0})}$
and
$\hat{f}_{R}(t)=-i\sum_{k}g_{k}(t_{0})\hat{b}_{Rk}(t_{0})e^{-i\omega_{k}(t-t_{0})}$
refer to quantum noise due to the heat baths for the $L$ and $R$ modes. Now,
eliminating the high frequency oscillation term using the transformation
$\hat{A}_{L,R}=\hat{a}_{L,R}e^{i\Omega(t-t_{0})}$, the above equations becomes

$\displaystyle\dot{\hat{A}}_{L}$ $\displaystyle=$
$\displaystyle\frac{iK}{\hbar}\hat{A}_{R}-\frac{iU_{+}}{\hbar}\hat{A}^{\dagger}_{L}\hat{A}_{L}\hat{A}_{L}-\frac{\gamma}{2}\hat{A}^{\dagger}_{L}\hat{A}_{L}\hat{A}_{L}$
(18) $\displaystyle+$ $\displaystyle\hat{F}_{L}(t)\hat{a}_{L}$
$\displaystyle\dot{\hat{A}}_{R}$ $\displaystyle=$
$\displaystyle\frac{iK}{\hbar}\hat{A}_{L}-\frac{iU_{-}}{\hbar}\hat{A}^{\dagger}_{R}\hat{A}_{R}\hat{A}_{R}-\frac{\gamma}{2}\hat{A}^{\dagger}_{R}\hat{A}_{R}\hat{A}_{R}$
(19) $\displaystyle+$ $\displaystyle\hat{F}_{R}(t)\hat{a}_{R}$

where ,

$\displaystyle\hat{F}_{L}(t)=-i\sum_{k}g_{k}(t_{0})\hat{b}_{Lk}(t_{0})e^{-i(\omega_{k}-\Omega)(t-t_{0})}$
(20)
$\displaystyle\hat{F}_{R}(t)=-i\sum_{k}g_{k}(t_{0})\hat{b}_{Rk}(t_{0})e^{-i(\omega_{k}-\Omega)(t-t_{0})}$
(21)

refer to the quantum noise due to the heat baths modulated by oscillation of
the system for the $L$ and $R$ modes. Eqs. (18) and (19) and the noise
operators in Eqs. (20) and (21) appear as a natural consequence of system-
reservoir coupling, Born-Markov and secular approximation [59, 60, 61].

To construct quantum Langevin equation with $c$-number noise [62, 63], we
return to Eq. (18) and (19) and carry out quantum mechanical average over the
initial product separable quantum states of the system oscillator and the bath
oscillator at $t_{0}=0$
$|\alpha\rangle|\mu_{1}\rangle|\mu_{2}\rangle....|\mu_{k}\rangle...|\mu_{N}\rangle$.
Here $|\alpha\rangle$ refers to the initial coherent state of the system and
$\\{|\mu_{k}\rangle\\}$ corresponds to the initial coherent states of the bath
operators. We denote the quantum mechanical averages for the system and the
bath operators as $\langle\hat{A}_{L,R}\rangle=\alpha_{L,R}$,
$\langle\hat{A}^{\dagger}_{L,R}\rangle=\alpha^{*}_{L,R}$, and
$\langle\hat{F}_{L,R}\rangle=\xi_{L,R}$. Here
$\xi_{L,R}=-i\sum_{k}g_{k}(t_{0})\mu_{Lk,Rk}e^{-i(\omega_{k}-\Omega)(t-t_{0})}$.
The $c$-number amplitudes may now be written as
$\alpha_{L}=\sqrt{N_{L}}e^{i\theta_{L}}$ and
$\alpha_{R}=\sqrt{N_{R}}e^{i\theta_{R}}$. $N_{L(R)}$ is the number of atoms in
the left (right) well. $\theta_{L(R)}$ is the phase of the atoms in the left
(right) well. The complex amplitude equations for two wells are given by

$\displaystyle\dot{\alpha}_{L}=\frac{iK}{\hbar}\alpha_{R}-\frac{iU_{+}}{\hbar}N_{L}\alpha_{L}-\frac{\gamma}{2}N_{L}\alpha_{L}+\xi_{L}(t)\alpha_{L}$
(22)
$\displaystyle\dot{\alpha}_{R}=\frac{iK}{\hbar}\alpha_{L}-\frac{iU_{-}}{\hbar}N_{R}\alpha_{R}-\frac{\gamma}{2}N_{R}\alpha_{R}+\xi_{R}(t)\alpha_{R}$
(23)

Separating the real and imaginary parts of the above equations we obtain after
some algebra the following equations in terms of the normalized atom number
imbalance $z(t)$ and phase difference $\theta(t)$ as follows:

$\displaystyle\dot{z}(t)=-\frac{2K}{\hbar}{\sqrt{1-z^{2}(t)}}\sin\theta(t)-\gamma
Nz(t)+\xi_{z}$ (24) $\displaystyle\dot{\theta}(t)$ $\displaystyle=$
$\displaystyle\frac{2K}{\hbar}\left[\frac{z(t)}{\sqrt{1-z^{2}(t)}}\cos\theta(t)+\frac{U_{+}-U_{-}}{4K}N+\frac{UN}{2K}z(t)\right]+\xi_{\theta}$
(25)

where, $\xi_{z}=2\xi_{L}(t)N_{L}/N-2\xi_{R}(t)N_{R}/N$,
$\xi_{\theta}=\xi_{R}-\xi_{L}$ and

$\displaystyle
z=\frac{|\alpha_{L}|^{2}-|\alpha_{R}|^{2}}{|\alpha_{L}|^{2}+|\alpha_{R}|^{2}}=\frac{N_{L}-N_{R}}{N_{L}+N_{R}}$
(26)

The conjugate variable is relative phase defined by

$\displaystyle\theta=\theta_{R}-\theta_{L}$ (27)

where, $U=\frac{U_{+}+U_{-}}{2}$. Now for symmetric DW, we have
$U_{+}=U_{-}=U$ and the phase equation reduces to

$\displaystyle\dot{\theta}(t)=\frac{2K}{\hbar}\left[\frac{z(t)}{\sqrt{1-z^{2}(t)}}\cos\theta(t)+\Lambda_{0}z(t)\right]+\xi_{\theta}$
(28)

where, $\Lambda_{0}=\frac{NU}{2K}$ characterizes the many-body interaction
parameter with $U$ being the on-site mean two-body interaction energy.
Equations (24) and (28) represent the dissipative BJJ equation with noise. In
the absence of noise ($\xi_{L,R}(t)=0$) the equations reduce to the
dissipative BJJ equations in which the dissipative coefficient term can be
modified by the linear contribution from $\dot{\theta}(t)$ which is usually
studied in the standard dissipative BJJ [52, 56, 57, 64]. However, the damping
coefficient of the form $\dot{z}\sim\gamma Nz$ can be useful for the study of
two-state system, such as, two condensates in different hyperfine levels in a
single harmonic trap, connected by the tunneling transition [64]. In absence
of dissipation ($\gamma=0$) the equations (24) and (28) reduce to the standard
BJJ equations.

###  II.2 Quantum and thermal properties of noise

The noise properties of the operator $\hat{F}_{L,R}$ and
$\hat{F}^{\dagger}_{L,R}$ in equations (20) and (21) can be derived using a
suitable canonical thermal distribution of the bath operators at $t_{0}=0$. To
this end we define the quantum statistical average of any reservoir operator
$\hat{O}$

$\displaystyle\langle\hat{O}\rangle_{\rm qs}=\frac{{\rm
Tr}[\hat{O}\exp{(-\hat{H}_{R}/k_{B}T)}]}{{\rm
Tr}[\exp{(-\hat{H}_{R}/k_{B}T)}]}$ (29)

where $\hat{H}_{R}=\sum_{j}\hbar\omega_{j}\hat{n}_{j}$ at $t_{0}=0$ and
$\hat{n}_{j}$ denotes the number operator in $j$-th bath mode. Based on the
above considerations the noise properties of the operator may be calculated
using the canonical distribution of Eqs. (20) and (21). This immediately gives
$\langle\hat{F}_{L,R}(t)\rangle_{\rm qs}=0$ and
$\langle\hat{F}^{\dagger}_{L,R}(t)\rangle_{\rm qs}=0$ and
$\langle\hat{F}^{\dagger}_{L,R}(t)\hat{F}_{L,R}(t^{\prime})\rangle_{\rm
qs}=\gamma\bar{n}_{L,R}(\Omega)\delta(t-t^{\prime})$ and
$\langle\hat{F}_{L,R}(t)\hat{F}^{\dagger}_{L,R}(t^{\prime})\rangle_{\rm
qs}=\gamma[\bar{n}_{L,R}(\Omega)+1]\delta(t-t^{\prime})$. The fluctuation-
dissipation relation gives

$\displaystyle\langle\hat{F}^{\dagger}_{L,R}(t)\hat{F}_{L,R}(t^{\prime})+\hat{F}_{L,R}(t)\hat{F}^{\dagger}_{L,R}(t^{\prime})\rangle_{\rm
qs}$ $\displaystyle=$
$\displaystyle\gamma[2\bar{n}_{L,R}(\Omega)+1]\delta(t-t^{\prime})$
$\displaystyle=$
$\displaystyle\gamma\coth(\hbar\Omega/2k_{B}T)\delta(t-t^{\prime})$

where the contangent hyperbolic factor in Eq. (II.2) can be identified with
Bose-Einstein distribution

$\displaystyle\bar{n}_{L,R}(\Omega)=\frac{1}{e^{\hbar\Omega/k_{B}T}-1}$ (31)

using the following relation
$2\bar{n}_{L,R}(\Omega)+1=\coth(\hbar\Omega/2k_{B}T)$ and the plus one factor
is responsible for the vacuum fluctuation which is always present on quantum
scale even at absolute zero temperature.

Now to realize $\xi_{L,R}(t)$ as an effective $c$-number noise, we now
introduce the ansatz that $\mu_{Lk,Rk}(0)$ and $\mu^{*}_{Lk,Rk}(0)$ are
distributed according to Wigner thermal canonical distribution of Gaussian
form [49] as follows:

$\displaystyle
W_{Lk,Rk}[\mu_{Lk,Rk}(0),\mu^{*}_{Lk,Rk}(0)]=N_{BL,BR}\exp\left[-\frac{|\mu_{Lk,Rk}(0)|^{2}}{2\coth(\frac{\hbar\Omega}{2k_{B}T})}\right]$
(32)

Here $N_{BL,BR}$ is the normalization constant for $L$ and $R$ wells.
$\coth(\frac{\hbar\Omega}{2k_{B}T})$ is the width of the distribution. For any
arbitrary quantum mechanical mean value of the bath operator
$\langle\hat{B}_{Lk,Rk}\rangle$ which is a function of
$\mu_{Lk,Rk}(0),\mu^{*}_{Lk,Rk}(0)$, its statistical average can then be
calculated as

$\displaystyle\langle\langle\hat{B}_{Lk,Rk}\rangle\rangle_{s}=\int\langle\hat{B}_{Lk,Rk}\rangle
W_{Lk,Rk}[\mu_{Lk,Rk}(0),\mu^{*}_{Lk,Rk}(0)]d\mu_{Lk,Rk}(0)d\mu^{*}_{Lk,Rk}(0)$
(33)

Using the ansatz (32) and the definition of statistical average of Eq. (33),
one can show that $c$-number noise satisfy the following relations

$\displaystyle\langle\xi_{L,R}(t)\rangle_{s}=0$ (34)
$\displaystyle\langle\xi^{*}_{L,R}(t)\rangle_{s}=0$

and

$\displaystyle\langle\xi^{*}_{L,R}(t)\xi_{L,R}(t^{\prime})\rangle_{s}=\gamma\coth(\hbar\Omega/2k_{B}T)\delta(t-t^{\prime})$
(35)
$\displaystyle\langle\xi_{L,R}(t)\hat{\xi}^{*}_{L,R}(t^{\prime})\rangle_{s}=\gamma\coth(\hbar\Omega/2k_{B}T)\delta(t-t^{\prime})$

Eqs. (34) and (35) implies that the $c$-number noise $\xi_{L,R}(t)$ is
characterized by zero mean and follows the fluctuation-dissipation relation.
By using the $c$-number formalism, we may thus bypass the operator ordering
prescription for the derivation of noise properties. The $c$-number noise
$\xi_{L,R}(t)$ as characterized by Eqs. (34) and (35) is classical looking in
form but essentially quantum mechanical in nature [65].

##  III Phase Diffusion

The Eqs. (24) and (28) describe the dissipative BJJ equations with quantum
noise whose properties are governed by Eqs. (34) and (35). These are nonlinear
Langevin equations which can not be solved by any direct analytical method.
The traditional way to circumvent this difficulty is to take resort to the
weak noise limit. To this end we consider first the steady state of the system
in absence of noise and linearize the dynamics around it. We are then led a
multivariate Ornstein–Uhlenbeck (OU) process as considered below.

To proceed further, we begin with the steady state of the dynamical system
($z_{s},\theta_{s}$). Now linearizing the system around it with
$z=z_{s}+\delta z$ and $\theta=\theta_{s}+\delta\theta$, where $\delta z$ and
$\delta\theta$ are small perturbations, we obtain the linearized Langevin
equations in $c$-numbers

$\delta\dot{z}=-\frac{2K}{\hbar}{\sqrt{1-z_{s}^{2}}}\cos\theta_{s}\delta\theta+\frac{2K}{\hbar}\frac{z_{s}\delta
z}{\sqrt{1-z_{s}^{2}}}\sin\theta_{s}-\gamma N\delta z+\xi_{z}(t)$ (36)
$\delta\dot{\theta}=\frac{2K}{\hbar}\left[\Lambda_{0}\delta
z-\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}\delta\theta+\frac{\cos\theta_{s}}{(1-z_{s}^{2})^{\frac{3}{2}}}\delta
z\right]+\xi_{\theta}(t)$ (37)

where $\xi_{z}(t)=\xi_{L}(t)-\xi_{R}(t)+z_{s}[\xi_{L}(t)+\xi_{R}(t)]$ and
$\xi_{\theta}(t)=\xi_{R}(t)-\xi_{L}(t)$. Here, we have used the relations
$N_{Ls,Rs}=\frac{N}{2}(1\pm z_{s})$. $N_{Ls,Rs}$ are the steady state values
of the number of atoms in left and right wells. For $\gamma>0$, $\delta z$
gets equilibrated at a fast rate. Therefore adiabatic elimination of the fast
variable $z$ results in :

$\displaystyle\delta
z=\frac{\frac{2K}{\hbar}\sqrt{1-z_{s}^{2}}\cos\theta_{s}\delta\theta-\xi_{z}}{\frac{2K}{\hbar}\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}-\gamma
N}$ (38)

Now inserting $\delta z$ in Eq. (37), we finally obtain after some algebra an
equation in the form of linear Langevin dynamics for phase $\delta\theta(t)$

$\displaystyle\delta\dot{\theta}=-\eta\delta\theta+C\xi_{z}(t)+\xi_{\theta}(t)$
(39)

where $\eta$ is the linear phase drift. The general expressions for $\eta$ and
$C$ are

$\displaystyle\eta=\frac{2K}{\hbar}\left[\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}-\frac{\cos\theta_{s}}{(1-z_{s}^{2})^{\frac{3}{2}}}\left\\{\frac{\sqrt{1-z_{s}^{2}}\cos\theta_{s}}{\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}-\frac{\gamma
N\hbar}{2K}}\right\\}-\Lambda_{0}\left\\{\frac{\sqrt{1-z_{s}^{2}}\cos\theta_{s}}{\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}-\frac{\gamma
N\hbar}{2K}}\right\\}\right]$ (40)

and

$\displaystyle
C=-\frac{\cos\theta_{s}}{(1-z_{s}^{2})^{\frac{3}{2}}}\left\\{\frac{1}{\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}-\frac{\gamma
N\hbar}{2K}}\right\\}-\Lambda_{0}\left\\{\frac{1}{\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}-\frac{\gamma
N\hbar}{2K}}\right\\}$ (41)

,respectively. In the subsequent sections, we will use the expressions of
$\eta$ and $C$ to calculate the phase diffusion coefficient for zero-phase
mode ($z_{s}=0,\theta_{s}=0$) and $\pi$-phase modes ($z_{s}=0,\theta_{s}=\pi$;
$z_{s}=\pm\sqrt{1-\frac{1}{\Lambda_{0}^{2}}},\theta_{s}=\pi$). Making use of
Eqs. (34) and (35) one can show

$\displaystyle\langle\xi(t)\xi(t^{\prime})\rangle=C^{2}(1+z_{s}^{2})2\gamma\coth(\hbar\Omega/2k_{B}T)\delta(t-t^{\prime})$
(42)

where $\xi(t)=C\xi_{z}(t)+\xi_{\theta}(t)$. The Fokker-Planck equation
corresponding to the linear Langevin dynamics (Eq. (39)) is given by [60]

$\displaystyle\frac{\partial P(\Psi,t)}{\partial
t}=-\frac{\partial}{\partial\Psi}(\eta\Psi)P(\Psi,t)+\mathcal{D}\frac{\partial^{2}P(\Psi,t)}{\partial\Psi^{2}}$
(43)

Here $P(\Psi,t)$ is the probability of finding $\Psi$ at time $t$; we have put
$\delta\theta\equiv\Psi$. $\mathcal{D}$ is the phase diffusion coefficient as
given by

$\displaystyle\mathcal{D}=C^{2}\gamma(1+z_{s}^{2})\coth(\hbar\Omega/2k_{B}T)$
(44)

A clear separation of the statistical part $\coth(\hbar\Omega/2k_{B}T)$ from
the dynamical pre-factor is quite apparent in the above expression. The phase
diffusion coefficient of Eq. (44) is one of the main results of this section.
Eq. (43) shows that phase perturbations can be described as Brownian motion of
a particle that are characterized by phase drift and diffusion. At finite
temperature, the macroscopic quantum tunneling across 1D dissipative BJJ is
thus significantly affected by number- and phase- fluctuations.

Before closing this section, we digress a little bit about the consistency
check of the calculation. Putting the Fokker-Planck Eq. (43) in the form of a
continuity equation $\frac{\partial P}{\partial t}+\frac{\partial
F}{\partial\Psi}=0$, we identify the flux [60]

$\displaystyle F=-\left[\eta\Psi P+\mathcal{D}\frac{\partial
P}{\partial\Psi}\right]$ (45)

At equilibrium $F=0$, we obtain from Eq. (45) the equilibrium distribution
function in the zero-phase mode, $\pi$-phase mode ($z_{s}=0,\theta_{s}=\pi$)
and $\pi$-phase self-trapping mode
($z_{s}=\pm\sqrt{1-\frac{1}{\Lambda_{0}^{2}}},\theta_{s}=\pi$)

$\displaystyle
P_{0}(\Psi)=A\exp\left[-\left\\{\frac{N(1+\Lambda_{0})^{-1}}{2\coth(\frac{\hbar\Omega}{2k_{B}T})}\right\\}\Psi^{2}\right]$
(46)

$\displaystyle
P_{\pi}(\Psi)=A\exp\left[-\left\\{\frac{N(1-\Lambda_{0})^{-1}}{2\coth(\frac{\hbar\Omega}{2k_{B}T})}\right\\}\Psi^{2}\right]$
(47)

$\displaystyle P_{\rm
ST}(\Psi)=A\exp\left[-\left\\{\frac{N[\Lambda_{0}^{2}(\Lambda_{0}^{2}-1)]^{-1}}{2\coth(\frac{\hbar\Omega}{2k_{B}T})}\right\\}\Psi^{2}\right]$
(48)

at a finite temperature $T$ where $A$ is the normalization constant,
respectively. The distribution does not depend on $\gamma$ as it should be for
the attainment of equilibrium. Second, the width is governed by
$\coth$-function which carries the signature of Wigner canonical thermal
distribution (32) employed in course of construction of ensemble for
$c$-number noise in the present treatment. Third, the occurrence of the
interaction parameter $\Lambda_{0}$ in the $\Psi^{2}$ term of the parenthesis
of the distributions of Eqs. (46), (47) and (48) for both zero and $\pi$-phase
modes is reminiscent of the potential energy term of an equilibrium
distribution.

##  IV Spectrum of fluctuations

We now go beyond the adiabatic elimination of fast variable to calculate the
spectra of fluctuation of number imbalance and phase associated with the
dissipative BJJ dynamics. To proceed we recast Eqs. (36) and (37) in the
matrix form as follows

$\displaystyle\dot{\beta}(t)=-B\beta(t)+M(t)$ (49)

where

$\beta=\begin{pmatrix}\delta z\\\ \delta\theta\end{pmatrix},\\\
B=\begin{pmatrix}\gamma
N-\frac{2K}{\hbar}\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}&\frac{2K}{\hbar}{\sqrt{1-z_{s}^{2}}}\cos\theta_{s}\\\
-\frac{2K}{\hbar}\left\\{\Lambda_{0}+\frac{\cos\theta_{s}}{(1-z_{s}^{2})^{\frac{3}{2}}}\right\\}&\frac{2K}{\hbar}\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}\end{pmatrix},\\\
M=\begin{pmatrix}\xi_{z}\\\ \xi_{\theta}\end{pmatrix}$ (50)

such that $\beta\beta^{T}\equiv\beta\times\beta$ is a direct product of the
matrix

$\beta\times\beta=\begin{pmatrix}\delta z(t)\delta z(t)&\delta
z(t)\delta\theta(t)\\\ \delta\theta(t)\delta
z(t)&\delta\theta(t)\delta\theta(t)\end{pmatrix}$ (51)

Taking average on the both sides of the Eq. (49) we obtain

$\langle\dot{\beta}(t)\rangle=-B\langle\beta(t)\rangle+\langle M(t)\rangle$
(52)

By virtue of Eq. (34), we have $\langle M(t)\rangle=0$. Direct integration
yields

$\displaystyle\langle\beta(t)\rangle=e^{-Bt}\langle\beta(0)\rangle$ (53)

where $\langle\beta(0)\rangle$ gives the average of the initial value
$\langle\beta(t)\rangle$. Now, according to the regression theorem the
correlation function decays in the same way as the average decay which
suggests that

$\displaystyle\langle\beta(t)\beta(0)\rangle=e^{-Bt}\langle\beta(0)\beta(0)\rangle$
(54)
$\displaystyle\langle\beta(0)\beta(t)\rangle=\langle\beta(0)\beta(0)\rangle
e^{-B^{T}t}$

To obtain the low frequency spectrum of various modes of correlation, we
calculate the Fourier transform of $\langle\beta(t)\beta(0)\rangle$ and
define,

$\displaystyle\mathcal{S}(\Delta)=\int_{-\infty}^{+\infty}e^{-i\Delta
It}\langle\beta(t)\beta(0)\rangle dt$ (55)

where, $\Delta$ refers to the detuning around $\Omega$, $I$ is the identity
matrix and $t=0$ refers to the stationary state, i.e, we calculate the
correlation of fluctuation around the stationary state. On further
manipulation of Eqs. (55) and (54), we obtain

$\displaystyle\mathcal{S}(\Delta)=(B+i\Delta
I)^{-1}\langle\beta(0)\beta(0)\rangle+\langle\beta(0)\beta(0)\rangle(B^{T}-i\Delta
I)^{-1}$ (56)

The stationary state contribution in the above equation can be expressed in
terms of the diffusion matrix of the form

$\displaystyle\mathcal{D}_{z,\theta}=\begin{pmatrix}\gamma(1+z_{s}^{2})\coth(\hbar\Omega/2k_{B}T)&0\\\
0&\gamma\coth(\hbar\Omega/2k_{B}T)\end{pmatrix}$ (57)

so that $2\times 2$ fluctuation spectrum matrix becomes

$\displaystyle\mathcal{S}(\Delta)=(B+i\Delta
I)^{-1}2\mathcal{D}_{z,\theta}(B^{T}-i\Delta I)^{-1}$ (58)

Explicit evaluation of the matrix elements results in the fluctuations of
number imbalance as $\mathcal{S}_{11}$ element, while $\mathcal{S}_{22}$
element represents the contribution due to phase fluctuation [59].

The number fluctuation spectrum ($\mathcal{S}_{11}$ element) can be written as

$\displaystyle\mathcal{S}_{z}(\Delta)=\frac{2\mathcal{D}_{11}(B_{22}^{2}+\Delta^{2})+2\mathcal{D}_{22}B_{12}^{2}}{(B_{11}B_{22}-B_{12}B_{21}-\Delta^{2})^{2}+\Delta^{2}(B_{11}+B_{22})^{2}}$
(59)

Similarly, the phase fluctuation term ($\mathcal{S}_{22}$ element) can be
written as

$\displaystyle\mathcal{S}_{\theta}(\Delta)=\frac{2\mathcal{D}_{11}B_{21}^{2}+2\mathcal{D}_{22}(B_{11}^{2}+\Delta^{2})}{(B_{11}B_{22}-B_{12}B_{21}-\Delta^{2})^{2}+\Delta^{2}(B_{11}+B_{22})^{2}}$
(60)

where, $B_{11}=\gamma
N-\frac{2K}{\hbar}\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}$,
$B_{12}=\frac{2K}{\hbar}{\sqrt{1-z_{s}^{2}}}\cos\theta_{s}$,
$B_{21}=-\frac{2K}{\hbar}\left\\{\Lambda_{0}+\frac{\cos\theta_{s}}{(1-z_{s}^{2})^{\frac{3}{2}}}\right\\}$,
and $B_{22}=\frac{2K}{\hbar}\frac{z_{s}\sin\theta_{s}}{\sqrt{1-z_{s}^{2}}}$.
The diffusion matrix elements are defined as
$\mathcal{D}_{11}=\gamma(1+z_{s}^{2})\coth(\hbar\Omega/2k_{B}T)$ and
$\mathcal{D}_{22}=\gamma\coth(\hbar\Omega/2k_{B}T)$.

##  V Results and discussions

###  V.1 Coherence factor

In order to study the coherent and incoherent regimes of the dissipative BJJ,
we now define $\langle\cos\Psi\rangle$ as a coherence factor [50] as it
provides the degree of coherence of the system. If the value of the linearized
phase is localized around zero, the value of the coherence factor is close to
unity. If instead the phase is fully delocalized and all its values are
equally probable, then the value of the coherence factor is close to zero
implying that the system is in the incoherent state. Since at equilibrium, the
linearized phase follow the Wigner thermal canonical distribution [49] as
described by Eqs. (46), (47) and (48), one may define explicitly the coherence
factor as follows:

$\displaystyle\langle\cos\Psi\rangle=\frac{\int_{-\pi}^{\pi}d\Psi\cos\Psi\exp\left[{\frac{\cos\Psi}{\kappa\coth(\frac{\hbar\Omega}{2k_{B}T})}}\right]}{\int_{-\pi}^{\pi}d\Psi\exp\left[{\frac{\cos\Psi}{\kappa\coth(\frac{\hbar\Omega}{2k_{B}T})}}\right]}$
(61)

where $\kappa$ is defined as $\kappa=\frac{1+\Lambda_{0}}{N}$ for zero-phase
mode. The appearance of the factor $\kappa$ in the canonical distribution
makes the coherence factor dependent on the on-site interaction energy $U$ and
the tunneling energy $K$. In what follows we examine the coherence factor in
the light of these parameters. For our numerical calculation, we choose the
value of the total number of atoms $N=1000$.

In Fig. 1(a), we show that the coherence factor $\langle\cos\Psi\rangle$ as a
function of the ratio $U/K$ for the low temperature limit
$k_{B}T/\hbar\Omega=1/10$ in zero-phase mode. It is apparent that in the limit
of strong tunneling $U/K\ll 1$, the value of the coherence factor becomes
close to unity because under this condition the system undergoes small
oscillation around the equilibrium zero-phase value. In this limit the
fluctuations of the phase is also small. In the opposite limit, when $U/K\gg
1$, the amplitude of the oscillation around the equilibrium increases as a
result of delocalization of the linearized phase due to the large on-site
interaction. The phase fluctuations are not small and the coherence factor
gradually decreases. This is similar to the prediction of the coherence factor
calculated using Josephson Hamiltonian [50].

---
Figure 1: (Color online) (a) Variation of coherence factor
$\langle\cos\Psi\rangle$ as a function of $U/K$ at low temperature
$k_{B}T/\hbar\Omega=1/10$ in zero-phase mode. (b) Variation of coherence
factor $\langle\cos\Psi\rangle$ as a function of $2k_{B}T/\hbar\Omega$ for
different energies $U/K=1$ (blue solid line), $U/K=2$ (red solid line),
$U/K=3$ (black solid line). Note the logarithmic scale on the horizontal axis.

In Fig. 1(b), we plot the temperature dependence of the coherence factor for
fixed interaction energies in zero-phase mode. It is clear that for a
particular value of the interaction energy, the coherence factor for low
temperature is almost constant and then with increase of the temperature the
coherence factor decreases. At high temperature ($2k_{B}T/\hbar\Omega\approx
100$), the curves coincide for different interaction energies and becomes
close to zero which implies that at large temperature the system becomes
incoherent. It is also noted that the slope of the coherence factor decreases
with increase of the on-site interaction energy which implies that in the
limit of negligible tunneling $U/K\gg 1$ for a particular temperature the
coherence factor decreases which is consistent with Fig. 1(a). This general
behaviour of the coherence factor is observed over a three orders of magnitude
of $2k_{B}T/\hbar\Omega$, and is in good agreement with the experimental
observation [48].

---
Figure 2: (Color online) (a) Variation of coherence factor
$\langle\cos\Psi\rangle$ as a function of $U/K$ at low temperature
$k_{B}T/\hbar\Omega=1/10$ in the $\pi$-phase self-trapping regime. The
vertical dashed-dotted magenta line corresponds to $U/K=0.004$. See text for
details. (b) Variation of coherence factor $\langle\cos\Psi\rangle$ as a
function of $2k_{B}T/\hbar\Omega$ for different energies $U/K=0.006$ (blue
solid line), $U/K=0.01$ (black solid line). Note the logarithmic scale on the
horizontal axis.

In the $\pi$-phase mode there are two possible regimes where one can study the
effect of coherence with the help of the many-body interaction parameter
$\Lambda_{0}$. The regime $\Lambda_{0}<1$ corresponds to the $\pi$-phase
Josephson oscillation regime and $\Lambda_{0}>1$ refers to the $\pi$-phase
self-trapping regime. The regime $1<\Lambda_{0}<2$ specifies the $\pi$-phase
self-trapping mode and $\Lambda_{0}>2$ refers to the running phase self-
trapping regime which is similar to the zero-phase mode self-trapping regime.
We have checked that in the limit of $\pi$-phase Josephson oscillation regime
the value of $\kappa$ is defined as $\kappa=\frac{1-\Lambda_{0}}{N}$ for
$\Lambda_{0}<1$ which implies that $U/K\ll 1$ and as a result the value of the
coherence factor is positive and lies in between zero and unity. However, the
nature of the coherence factor as a function of $U/K$ and temperature remains
same as in zero-phase mode.

In Fig. 2(a), we plot the coherence factor $\langle\cos\Psi\rangle$ as a
function of the ratio $U/K$ for the low temperature limit
$k_{B}T/\hbar\Omega=1/10$ in $\pi$-phase self-trapping regime. In this regime
$\kappa$ is defined as $\kappa=\frac{\Lambda_{0}^{2}[\Lambda_{0}^{2}-1]}{N}$
with $\Lambda_{0}>1$. We see that the coherence factor quickly falls from
unity to zero with small change in the $U/K$ ratio. We also see that when
$U/K<0.004$, the value of the coherence factor is close to unity. This regime
signifies the $\pi$-phase self-trapping regime where the system oscillates
around a non zero-value of the population imbalance and phase difference. Also
in this regime the system undergoes oscillation around the equilibrium
$\pi$-phase value and the fluctuation of the phase remains low. We also note
that when $U/K>0.004$ the coherence factor suddenly falls from unity to zero.
In Fig. 2(a), the vertical dashed-dotted magenta line corresponds to
$U/K=0.004$ which basically separates the $\pi$-phase self-trapping regime to
running $\pi$-phase self-trapping regime. For $U/K>0.004$, in the running
$\pi$-phase self-trapping regime where the population imbalance oscillates
around a non-zero mean value but the phase difference between the two BECs in
the left and right well evolves unbound as a result of which the relative
phase increases monotonically. So, the phase is delocalized and the phase
fluctuations are not small. The system becomes incoherent. We also see that
when $U/K>0.025$, the value of the coherence factor is almost zero which
signifies that the system is in the incoherent regime.

In Fig. 2(b), we show that the temperature dependence of coherence factor for
fixed interaction energies in $\pi$-phase self-trapping regime. To study this
aspect we choose the value of $U/K$ in the nearly coherent regime where the
system is in the running phase self-trapping mode. We observe similar
behaviour when compared to the case of zero phase mode where in the low
temperature limit the coherence factor is almost constant. Then with increase
of the temperature the coherence factor decreases and also at high temperature
($2k_{B}T/\hbar\Omega\approx 1000$), the curves coincide for the different
interaction energies and become close to zero which implies that the system is
in the incoherent regime. However the degree of coherence increases by one
order of magnitude compared to the zero-phase mode. This study of coherence
reveals that the 1D dissipative BJJ has a higher degree of coherence.

###  V.2 Phase diffusion coefficient; phase-transition-like behaviour

Here we present our results for the phase diffusion coefficient in zero-phase
mode and $\pi$-phase modes of 1D dissipative BJJ. We first analyze how phase
diffusion coefficient $\mathcal{D}$ depends on the system interaction
parameter and temperature. For this we choose the zero-phase mode, i.e
$z_{s}=0,\theta_{s}=0$ which is well studied in BJJ. From equation (44), in
zero-phase mode, the phase diffusion coefficient is given by

$\displaystyle\mathcal{D}=C^{2}\gamma\coth(\hbar\Omega/2k_{B}T)$ (62)

where $C$ in the zero-phase mode is given by
$C=\frac{2K}{\hbar\tilde{\gamma}}\left[1+\Lambda_{0}\right]$ where
$\tilde{\gamma}=\gamma N$. A closer look at the analytical expression makes it
clear that $\mathcal{D}\propto 1/\gamma$ (as $C^{2}\propto
1/\tilde{\gamma}^{2}$). To investigate how the phase diffusion coefficient
$\mathcal{D}$ changes with temperature for weak to strong dissipation limit,
we plot $\mathcal{D}$ as a function of $2k_{B}T/\hbar\Omega$ for several
dissipation coefficient $\tilde{\gamma}$ in Fig. 3. For numerical calculation,
we choose the total number of atoms $N=1000$ and $U/K=0.001$ such that the
many body interaction parameter becomes $\Lambda_{0}=0.5$, i.e, weak
interaction limit or the strong tunneling regime. The reason behind this
choice of parameter is that we want to investigate the nature of the phase
diffusion coefficient in the coherent regime ($U/K\ll 1$) by introducing
dissipation in the system. Fig. 3 reveals that the $\mathcal{D}$ remains
constant upto $2k_{B}T/\hbar\Omega\approx 0.4$ which is determined by the
system parameters $\Lambda_{0}$ and $\tilde{\gamma}$. However, if we further
increase the temperature i.e ($2k_{B}T/\hbar\Omega>0.4$) $\mathcal{D}$
increases but for lower value of $\tilde{\gamma}$ as shown by the blue dashed-
dotted line in Fig. 3. If $\tilde{\gamma}$ is comparatively large the change
of $\mathcal{D}$ with temperature occurs slowly which are shown by the magenta
and red dashed-dotted lines in Fig. 3. Since $\mathcal{D}$ is proportional to
$C^{2}$ which is determined by the many body interaction parameter
$\Lambda_{0}$, the increase in $\Lambda_{0}$ does not change effectively the
nature of the phase diffusion coefficient.

---
Figure 3: (Color online) Variation of the phase diffusion co-efficient
$\mathcal{D}$ as a function of temperature $2k_{B}T/\hbar\Omega$ for different
dissipation co-efficient $\tilde{\gamma}$ with small many body interaction
parameter $\Lambda_{0}=0.5$ in zero-phase mode.

In the $\pi$-phase mode, there are three steady state solutions (a) $z_{s}=0$,
$\theta_{s}=\pi$ (b) $z_{s}=\sqrt{1-\frac{1}{\Lambda_{0}^{2}}}$,
$\theta_{s}=\pi$ and (c) $z_{s}=-\sqrt{1-\frac{1}{\Lambda_{0}^{2}}}$,
$\theta_{s}=\pi$. The first solution arises for $\Lambda_{0}<1$ and the second
and third solutions arise when $\Lambda_{0}>1$ and at $\Lambda_{0}=1$ a
bifurcation of population imbalance occurs which is plotted in the inset of
Fig. 4. This bifurcation in the $\pi$-phase mode has been observed
experimentally in internal BJJ [66]. To analyze the steady states in terms of
the phase diffusion co-efficient, we plot the phase diffusion coefficient as a
function of the interaction parameter $\Lambda_{0}$ in Fig. 4. We see that for
$\Lambda_{0}<1$, i.e, when the steady state of the population imbalance is
expressed as $z_{s}=0$, $\theta_{s}=\pi$, the phase diffusion coefficient
$\mathcal{D}\propto C^{2}\gamma$ where
$C=\frac{2K}{\hbar\tilde{\gamma}}\left[\Lambda_{0}-1\right]$. The phase
diffusion coefficient behaves similarly with dissipation ($\mathcal{D}\propto
1/\gamma$) as in zero-phase mode. But there is a critical value of
$\Lambda_{0}$ for which $C$ and $\mathcal{D}$ become zero. This phase-
transition like behaviour originates due to the symmetry breaking of BJJ in
the $\pi$-phase mode. So with increase of $\Lambda_{0}$, the value of $C$
decreases and as a result the phase diffusion coefficient $\mathcal{D}$
decreases as represented by the solid magenta line in Fig. 4. At
$\Lambda_{0}=1$, the value of $C$ and in turn the phase diffusion coefficient
becomes zero. But for $\Lambda_{0}>1$, there are two possible solutions of
$z_{s}$ and for both cases $C$ is given by
$C=\frac{2K}{\hbar\tilde{\gamma}}\left[\Lambda_{0}-\frac{1}{\Lambda_{0}^{3}}\right]$.
For this, we see that $\mathcal{D}$ increases sharply with increase of
$\Lambda_{0}$ represented by the blue solid line and red dashed line in Fig.
4. However, the bifurcation shown for the steady state population imbalance is
missing for the phase diffusion coefficient in the regime $\Lambda_{0}>1$.
This can be understood from the phase diffusion equation (44) as $\mathcal{D}$
varies as $(1+z_{s}^{2})$. As we know from BJJ analysis that, when
$\Lambda_{0}<1$ the dynamics of the population imbalance shows Josephson
oscillation. However when the critical value of $\Lambda_{0}$ is crossed,
there are two types of MQST as observed depending on the time average of
population imbalance such as $\langle z\rangle<|z_{s}|\neq 0$ and $\langle
z\rangle>|z_{s}|\neq 0$. These two types of MQST are known as running phase
and $\pi$-phase modes of MQST. We also know that the range, $1<\Lambda_{0}<2$
specifies the regime of $\pi$-phase mode MQST and $\Lambda_{0}>2$ refers to
the running phase mode MQST which is similar to the zero-phase mode MQST.

---
Figure 4: (Color online) Variation of the phase diffusion coefficient
$\mathcal{D}$ as a function of the many body interaction parameter
$\Lambda_{0}$ at low temperature $k_{B}T/\hbar\Omega=1/10$ with
$\tilde{\gamma}=1$ in the $\pi$-phase mode. The phase diffusion coefficient
$\mathcal{D}$ for $z_{s}=0,\theta_{s}=\pi$ represented by magenta solid line
and for $z_{s}=\pm\sqrt{1-\frac{1}{\Lambda_{0}^{2}}},\theta_{s}=\pi$,
$\mathcal{D}$ as a function of $\Lambda_{0}$ represented by solid blue and red
dashed lines. The vertical black dashed line represent the critical
$\Lambda_{0}$ for the transition between Josephson oscillation to MQST in
$\pi$-phase mode. Inset: Variation of the steady state values of the
population imbalance as a function of the many body interaction parameter
$\Lambda_{0}$. The three different colors signify three different steady
states of the population imbalance. The vertical black dashed line represents
$\Lambda_{0}$ where the bifurcation of $z_{s}$ occurs.

For small dissipation coefficient ($\tilde{\gamma}=1$) at low temperature
($k_{B}T/\hbar\Omega=1/5$), one may assume that the dynamics of the
dissipative BJJ follows the standard results of the BJJ and the Josephson
Hamiltonian [53] gives negative energy for the zero-phase mode and positive
energy for the $\pi$-phase mode [53]. Now, in the $\pi$-phase mode, the system
initially has large energy; so increasing $\Lambda_{0}$ does not change the
flow of the relative phase between two wells in the DW potential as a result
of which $\mathcal{D}$ decreases. But at the critical $\Lambda_{0}$, the
interaction energy becomes equal to the steady state Josephson Hamiltonian
energy which implies that at the critical point there is no flow of the
relative phase between the wells. Further increase of $\Lambda_{0}$, implies
that many body interaction energy becomes large compared to the Josephson
Hamiltonian energy so that the flow is reversed and as a result $\mathcal{D}$
increases with increase of $\Lambda_{0}$. This is also similar to the case of
zero-phase mode because in zero-phase mode initial Josephson Hamiltonian
energy is negative and with increase of positive $\Lambda_{0}$, $\mathcal{D}$
increases. Although the behaviour of $\mathcal{D}$ as a function of
$\Lambda_{0}$ in zero-phase mode is not shown in the text (since
$\mathcal{D}\propto C^{2}\gamma\propto(1+\Lambda_{0})^{2}$), it is apparent
that $\mathcal{D}$ increases with increase of $\Lambda_{0}$.

###  V.3 Quantum fluctuation spectra

For the study of the quantum fluctuation of number and phase we first choose
the region from Fig. 1(a) where $U/K\ll 1$ to explore the effect of the
dissipation on fluctuation spectra in the coherent and near coherent regime.
Our primary focus lies on the interplay of interaction and dissipation to
study in these regimes of 1D dissipative BJJ.

From Eqs (59) and (60), in the zero-phase mode, the analytical expression for
the spectrum of number fluctuation becomes

$\displaystyle
S_{z}(\Delta)=\frac{\frac{2\tilde{\gamma}}{N}\coth({\frac{\hbar\Omega}{2k_{B}T})[4+\Delta^{2}]}}{\left[\Delta^{2}-\omega_{J}^{2}+\frac{\tilde{\gamma}^{2}}{2}\right]^{2}+\left[\omega_{J}^{2}\tilde{\gamma}^{2}-\frac{\tilde{\gamma}^{4}}{4}\right]}$
(63)

and the spectrum of phase fluctuation can be written as

$\displaystyle
S_{\theta}(\Delta)=\frac{\frac{2\tilde{\gamma}}{N}\coth({\frac{\hbar\Omega}{2k_{B}T})[\omega_{J}^{2}(1+\Lambda_{0})+\tilde{\gamma}^{2}+\Delta^{2}]}}{\left[\Delta^{2}-\omega_{J}^{2}+\frac{\tilde{\gamma}^{2}}{2}\right]^{2}+\left[\omega_{J}^{2}\tilde{\gamma}^{2}-\frac{\tilde{\gamma}^{4}}{4}\right]}$
(64)

where $\tilde{\gamma}=\gamma N$, and Josephson frequency
$\omega_{J}=\frac{2K}{\hbar}\sqrt{1+\Lambda_{0}}$. In the $\pi$-phase mode
Josephson oscillation and the $\pi$-phase mode self-trapping regime the
analytical expression for the spectrum of number fluctuation becomes

$\displaystyle
S_{z}(\Delta)=\frac{\frac{2\tilde{\gamma}}{N}\coth({\frac{\hbar\Omega}{2k_{B}T})[4+\Delta^{2}]}}{\left[\Delta^{2}-\omega_{\pi}^{2}+\frac{\tilde{\gamma}^{2}}{2}\right]^{2}+\left[\omega_{\pi}^{2}\tilde{\gamma}^{2}-\frac{\tilde{\gamma}^{4}}{4}\right]}$
(65)

$\displaystyle
S_{z}(\Delta)=\frac{\frac{2\tilde{\gamma}}{N}\coth({\frac{\hbar\Omega}{2k_{B}T})[\frac{4}{\Lambda_{0}^{2}}+2\Delta^{2}-\frac{\Delta^{2}}{\Lambda_{0}^{2}}]}}{\left[\Delta^{2}-\omega_{\rm
ST}^{2}+\frac{\tilde{\gamma}^{2}}{2}\right]^{2}+\left[\omega_{\rm
ST}^{2}\tilde{\gamma}^{2}-\frac{\tilde{\gamma}^{4}}{4}\right]}$ (66)

respectively and the corresponding spectrum of phase fluctuation are given by

$\displaystyle
S_{\theta}(\Delta)=\frac{\frac{2\tilde{\gamma}}{N}\coth({\frac{\hbar\Omega}{2k_{B}T})[\omega_{\pi}^{2}(1-\Lambda_{0})+\tilde{\gamma}^{2}+\Delta^{2}]}}{\left[\Delta^{2}-\omega_{\pi}^{2}+\frac{\tilde{\gamma}^{2}}{2}\right]^{2}+\left[\omega_{\pi}^{2}\tilde{\gamma}^{2}-\frac{\tilde{\gamma}^{4}}{4}\right]}$
(67)

$\displaystyle
S_{\theta}(\Delta)=\frac{\frac{2\tilde{\gamma}}{N}\coth({\frac{\hbar\Omega}{2k_{B}T})[\omega_{\rm
ST}^{2}(\Lambda_{0}^{2}-1)(2\Lambda_{0}^{2}-1)+\tilde{\gamma}^{2}+\Delta^{2}]}}{\left[\Delta^{2}-\omega_{\rm
ST}^{2}+\frac{\tilde{\gamma}^{2}}{2}\right]^{2}+\left[\omega_{\rm
ST}^{2}\tilde{\gamma}^{2}-\frac{\tilde{\gamma}^{4}}{4}\right]}$

where $\omega_{\pi}=\frac{2K}{\hbar}\sqrt{1-\Lambda_{0}}$ and $\omega_{\rm
ST}=\frac{2K}{\hbar}\sqrt{\Lambda_{0}^{2}-1}$ signifies the Josephson
frequency and self-trapping frequency in $\pi$-phase mode of standard BJJ.

From the above expressions for the number and phase fluctuation spectra, we
observe the following

1\. When the detuning $\Delta=0$, both number and phase fluctuations persist
$(S_{z},S_{\theta}\neq 0)$ in zero and $\pi$-phase modes of 1D dissipative
BJJ. Quantum fluctuations remain even at absolute zero temperature.

2\. It is also clear that the term defining the peak position depends on the
Josephson frequency $\omega_{J}$ and $\omega_{\pi}$ for zero and $\pi$-phase
modes and $\omega_{\rm ST}$ for $\pi$-phase mode self-trapping regime with
dissipation parameter $\tilde{\gamma}$. The contribution of $\tilde{\gamma}$
arises due to the induced bath effect on the system.

For our numerical calculation, we consider first the low temperature limit
$k_{B}T/\hbar\Omega=1/10$ and the weak interaction limit $U/K=0.001$. It is
interesting to examine how the system behaves in the weak interaction regime
in presence of weak dissipation. For that we choose $\tilde{\gamma}=1$ with
total number of atoms $N=1000$. In Fig. 5(a) and 5(b), we plot the number and
phase fluctuation spectra as a function of detuning for weak interaction
$(U/K=0.001)$ and weak dissipation limit with $\tilde{\gamma}=1$ in the low
temperature regime for zero phase mode denoted by the solid blue line and the
$\pi$-phase Josephson oscillation regime indicated by the solid black line.
For each mode, we observe that two peaks appear where the dashed-dotted lines
indicate the Josephson frequency in the respective mode. It is clear that
exactly at the Josephson frequency the amplitude of the number and phase
fluctuation spectra is maximum and at $\Delta=0$ both number fluctuation and
phase fluctuation persist in the system for both zero and $\pi$-phase mode.
However, we observe that in the $\pi$-phase mode the amplitude of the number
fluctuation spectrum increases compared to the amplitude of the number
fluctuation spectrum in the zero phase mode and the opposite behaviour is
observed for the phase fluctuation spectra. We also observe that the depth
which implies that the difference between the maximum and minimum of the
amplitude of the spectra at $\Delta=0$ increases for the $\pi$-phase number
fluctuation spectra compared to that for the zero phase. The reason behind
this behaviour is that for a fixed many body interaction parameter the
Josephson frequency for the zero phase mode is always greater than the
Josephson frequency in $\pi$-phase mode. As a result when the Josephson
frequency decreases the amplitude increases as well as the depth increases.
However, we also note similar but opposite behaviour for the phase fluctuation
spectra. It is also apparent that due to the presence of $\tilde{\gamma}$ in
the analytical as well as numerical calculation of spectra the peak values of
the number and phase fluctuation spectra are not exactly at the Josephson
frequency in the $\pi$-phase mode of the standard BJJ. This also signifies
that the peak frequency is dressed by $\tilde{\gamma}$. For both cases we
observe that the number and phase fluctuation becomes close to zero for large
detuning. This picture qualitatively describes the coherent behaviour of the
1D dissipative BJJ in the small interaction and the small dissipation limit
where both the number and the phase of the atoms in the well oscillate with
the Josephson frequency.

Figure 5: (Color online) a) Variation of number fluctuation spectrum
$\mathcal{S}_{z}(\Delta)$ as a function of detuning $\Delta$ for zero phase
mode (solid blue line) and $\pi$-phase mode (solid black line). b) Variation
of the phase fluctuation spectrum $\mathcal{S}_{\theta}(\Delta)$ as a function
of the detuning $\Delta$ for zero phase mode (solid blue line) and $\pi$-phase
mode (solid black line). Both the curves are plotted for small interaction
limit $U/K=0.001$, small dissipation $\tilde{\gamma}=1$ limit with $N=1000$
and low temperature regime $k_{B}T/\hbar\Omega=1/10$. The blue and black
dashed-dotted lines indicate the Josephson frequency for zero and $\pi$-phase
mode.

We now examine how this phase and number fluctuations change on increasing the
dissipation in the system. In Fig. 6(a) and 6(b), we plot the number and phase
fluctuation spectra as a function of detuning for weak interaction
$(U/K=0.001)$ in the weak dissipation $(\tilde{\gamma}=2)$ to strong
dissipation limit $(\tilde{\gamma}=5)$ in the low temperature regime for zero-
phase mode. Here we also choose the total number of atoms to be $N=1000$. For
$\tilde{\gamma}=2$ (solid blue line), we observe that the number and phase
fluctuation spectra exhibit coherent behaviour. If we further increase
$\tilde{\gamma}(=3$, solid red line) the amplitude of the number fluctuation
spectrum is reduced and also the depth decreases. We also note that the peak
frequency is shifted away from the Josephson frequency which implies that the
peak frequency is dressed by the $\omega_{J}$ and $\tilde{\gamma}$. Similar
behaviour is observed in the phase fluctuation spectrum with a difference that
the amplitude of the phase fluctuation spectrum slightly increases and also
the peak frequency gets shifted slightly from the Josephson frequency. With
further increase of $\tilde{\gamma}(=4$, solid magenta line), the amplitude of
the number fluctuation spectrum is reduced and the depth at $\Delta=0$ is also
reduced. However the two peaks originate in the number fluctuation spectrum
which implies that the system still oscillates with a dressed Josephson
frequency whereas the phase fluctuation spectrum shows a single peak
originated at $\Delta=0$ and describes the phase fluctuation is in incoherent
regime because the phase oscillates in each well between the ground and first
excited state (of a single-particle state). In this case the tunneling of the
particle is prohibited. For further increase of $\tilde{\gamma}(=5$, solid
black line), both number and phase fluctuation spectra show the growth of a
single peak at $\Delta=0$ which implies both number and phase of the atoms in
the well oscillate with a frequency which is equal to the energy gap of the
first excited state and the ground state (of a single particle state). So, by
changing the dissipation coefficient in the low temperature regime with weak
interaction one can realize a transition from coherent to incoherent regime.
The coherent regime implies the standard BJJ picture where the system
oscillates between each well with a frequency close to the Josephson frequency
whereas for the incoherent regime the tunneling of the particles are
prohibited.

Figure 6: (Color online) a) Variation of number fluctuation spectrum
$\mathcal{S}_{z}(\Delta)$ as a function of detuning $\Delta$ with different
dissipation coefficient $\tilde{\gamma}$ for zero phase mode. b) Variation of
the phase fluctuation spectrum $\mathcal{S}_{\theta}(\Delta)$ as a function of
the detuning $\Delta$ with different dissipation coefficient $\tilde{\gamma}$
for zero phase mode. Both the curves are plotted for small interaction limit
$U/K=0.001$, with $N=1000$ and low temperature regime
$k_{B}T/\hbar\Omega=1/10$. The green dashed-dotted line indicates the
Josephson frequency $\omega_{J}$ in zero phase mode in absence of dissipation.

We now examine the effect of dissipation in number and phase fluctuation
spectra in the $\pi$-phase mode Josephson oscillation regime. In Fig. 7(a) and
7(b), we plot the number and phase fluctuation spectra as a function of
detuning for weak interaction $(U/K=0.001)$ in the low temperature regime for
several values of dissipation strength. We also choose that the total number
of atom $N=1000$. For $\tilde{\gamma}=2$ (solid blue line), we see that the
number and phase fluctuation spectra exhibit two peaks with small amplitude
which are distinct from the $\pi$-phase mode Josephson frequency as
represented by the green dashed-dotted line in Fig. 7. With further increase
of dissipation parameter $\tilde{\gamma}=3$, the number and the phase
fluctuation spectra show a single peak at $\Delta=0$ which implies that the
system is in the incoherent regime. With further increase of
$\tilde{\gamma}=4$ and $\tilde{\gamma}=5$, the number and phase fluctuation
spectra remain in the incoherent regime but the amplitude of the fluctuation
spectra increases with increasing dissipation strength. Here almost for
$\tilde{\gamma}=2$, the system makes a transition to the incoherent state
whereas in the zero phase mode with $\tilde{\gamma}=4$ the system becomes
incoherent in phase. We also observe that for large detuning both number and
phase fluctuation become close to zero. In the $\pi$-phase mode with weak
dissipation one can always realize the incoherent regime whereas in the zero
phase mode one may observe a transition from coherent to incoherent regime for
comparatively large dissipation.

Figure 7: (Color online) a) Variation of number fluctuation spectrum
$\mathcal{S}_{z}(\Delta)$ as a function of detuning $\Delta$ with different
dissipation coefficient $\tilde{\gamma}$ for $\pi$-phase Josephson oscillation
regime. b) Variation of the phase fluctuation spectrum
$\mathcal{S}_{\theta}(\Delta)$ as a function of the detuning $\Delta$ with
different dissipation coefficient $\tilde{\gamma}$ for $\pi$-phase Josephson
oscillation regime. Both the curves are plotted for small interaction limit
$U/K=0.001$, with $N=1000$ and low temperature regime
$k_{B}T/\hbar\Omega=1/10$. The green dashed-dotted line indicates the
Josephson frequency $\omega_{\pi}$ in $\pi$-phase mode Josephson oscillation
regime in absence of damping. Figure 8: (Color online) a) Variation of number
fluctuation spectrum $\mathcal{S}_{z}(\Delta)$ as a function of detuning
$\Delta$ for weak dissipation coefficient $\tilde{\gamma}=1$ (solid blue
color) and strong dissipation coefficient $\tilde{\gamma}=8$ for $\pi$-phase
mode self-trapping regime. b) Variation of the phase fluctuation spectrum
$\mathcal{S}_{\theta}(\Delta)$ as a function of the detuning $\Delta$ for weak
dissipation coefficient $\tilde{\gamma}=1$ (solid blue color) and strong
dissipation coefficient $\tilde{\gamma}=8$ for $\pi$-phase mode self-trapping
regime. Both the curves are plotted for interaction value $U/K=0.003$, with
$N=1000$ and low temperature regime $k_{B}T/\hbar\Omega=1/10$. The blue
dashed-dotted line indicates the self-trapping frequency $\omega_{\rm ST}$ in
$\pi$-phase mode.

In Fig. 8(a) and 8(b), we plot the number and phase fluctuation spectra as a
function of detuning for interaction $U/K=0.003$ in the low temperature regime
for weak dissipation $(\tilde{\gamma}=1)$ and strong dissipation limit
$(\tilde{\gamma}=8)$ in $\pi$-phase self-trapping regime. For our numerical
calculation, we choose total number of atoms $N=1000$. In the $\pi$-phase
self-trapping regime, it is necessary to choose the value of interaction
energy $U/K>0.002$ such that the many-body interaction parameter $\Lambda_{0}$
becomes greater than unity. We observe that for weak dissipation
$(\tilde{\gamma}=1$ the solid blue line) the number and phase fluctuation
spectra show the two peaks implying the coherent behaviour where the peak
amplitude of the number and phase fluctuation spectra is at $\pi$-phase mode
self-trapping frequency denoted by the dashed-dotted blue line in Fig. 8. This
picture qualitatively describes the coherent behaviour of the 1D dissipative
BJJ in the small dissipation limit where both number and phase of the atoms in
the wells oscillate with non-zero average value and the frequency of the
oscillation is determined by the $\pi$-phase mode self-trapping frequency. Now
for strong dissipation $(\tilde{\gamma}=8)$, we observe that a single peak
appear at $\Delta=0$ both in number and phase fluctuation spectra. However the
amplitude of the phase fluctuation spectra is always greater than that for the
number fluctuation spectra. This picture qualitatively describes the effect of
dissipation and the transition from coherent to incoherent regime in the
$\pi$-phase mode self-trapping regime of 1D dissipative BJJ.

##  VI Conclusion

In this paper, we have considered a nonlinear dissipative BJJ in presence of
quantum noise. We have shown that the dissipative BJJ equations in presence of
weak noise follow linear Langevin dynamics of relative phase. From this
Langevin description of the relative phase, we derive the analytical formula
for the phase diffusion coefficient for zero and $\pi$-phase modes of BJJ. We
further derive the analytical formula of the number fluctuation spectra and
the phase fluctuation spectra to analyze its dependence on interaction energy
and dissipation. The main conclusions of this study can be summarized as
follows:

  * •

We have formulated the dissipative BJJ equations (24) and (28) within the
framework of a $c$-number description of noise within Born-Markov
approximation, where the equilibrium distribution follows Wigner thermal
canonical distribution. The equations are classical looking in form but
quantum mechanical in content.

  * •

From the dissipative BJJ dynamics around the steady states, the phase
diffusion coefficient and the spectra of number and phase fluctuation are
derived.

  * •

Our numerical simulations in zero-phase mode suggest that $\mathcal{D}$
depends on the system interaction parameter $\Lambda_{0}$, dissipation
$\gamma$ and temperature. We show that $\mathcal{D}$ increases with increase
in temperature for small dissipation. However, if dissipation is comparatively
large the variation of $\mathcal{D}$ with temperature slows down.

  * •

Our numerical simulations in $\pi$-phase modes for small dissipation and low
temperature regime suggest that the phase diffusion coefficient $\mathcal{D}$
exhibits an interesting phase transition like behaviour between the Josephson
oscillation and the MQST regime. In the Josephson oscillation regime
$\mathcal{D}$ decreases with increase of $\Lambda_{0}$ and in the MQST regime
above the critical $\Lambda_{0}$, $\mathcal{D}$ increases with increase of
$\Lambda_{0}$. $\mathcal{D}$ does not show bifurcation behaviour although the
steady state population imbalance shows the bifurcation behaviour.

  * •

We have also analyzed the coherence factor of the dissipative BJJ as defined
in equation (61) where the averaging is done using Wigner thermal canonical
distribution. We analyze numerically the coherence factor as a function of
$U/K$ and show that in the strong tunneling regime $U/K\ll 1$, the system
becomes coherent and for the opposite case $U/K\gg 1$, the system goes into
the incoherent state. We also analyze the coherent and incoherent behaviour of
the system in terms of temperature for the fixed $U/K$ ratios where for low
temperature the system remains in the coherent state and for high temperature
the system goes into a incoherent state. In the $\pi$-phase mode self-trapping
regime, the coherence factor suddenly falls from unity to zero with small
change in $U/K$ ratio but it exhibits similar temperature dependence as in
zero-phase mode. Our analysis of coherence factor reveals that the BJJ has a
higher degree of coherence which is in good agreement with the experimental
observation [48].

  * •

We have also numerically analyzed the coherent and incoherent behaviour of the
system in terms of quantum fluctuation spectra of population imbalance and
phase difference in presence of dissipation. We observe that in the weak
dissipation limit the fluctuation spectra show two peak at the Josephson
frequency whereas for the large dissipation limit the fluctuation spectra show
a single peak which signifies the incoherent state of the system. This is
reminiscent of Autler-Townes effect in quantum optics.

Although the calculations in this work are specific to a scalar BECs, but the
concepts and ideas presented in this work are also applicable to the two-
component BEC where the dynamics of the phase diffusion coefficient can be
studied in terms of internal BJJ in presence of noise and dissipation. It
would be interesting to investigate how phase diffusion coefficient behave
with change of the population transfer between the modes of the two-component
BEC.

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