Berry phase caused by nondissipative drag of superflow in a Bose qubit
A microscopic theory of superfluid drag in a two-component Bose gas is developed. The drag factor is shown to be proportional to the square root of the gas parameter. Basing on the similarity between the "drag force"\ and vector potential of magnetic field we propose how the superfluid drag can be used to detect a Berry phase in a Bose counterpart of the Josephson charge qubit. We consider a system in which the drag component, situated inside the drive component, is confined in two half-ring traps separated by two Josephson barriers. Under cyclic adiabatic change of Josephson coupling parameters and an asymmetry of two traps a Berry phase is generated. This phase can be observed though measurements of relative number of atoms in two traps. The Berry phase depends on the drag factor and its detection can be used for determining the value of the drag.
fil-drag.tex
\documentclass{iopart}
\usepackage{epsf}
%\usepackage{iopams}
\usepackage{bm}

\begin{document}
\title
{Berry phase caused by nondissipative drag of superflow in a Bose
qubit}

\author{D V Fil\dag \footnote[3]{fil@isc.kharkov.com}
 and S I Shevchenko\ddag}



\address{\dag Institute for Single Crystals National Academy of
Sciences of Ukraine, Lenin av. 60, Kharkov 61001, Ukraine}
\address{\ddag Verkin Institute for Low Temperature Physics and
Engineering National Academy of Sciences of Ukraine, Lenin av. 47,
Kharkov 61103, Ukraine}



\begin{abstract}A microscopic theory of superfluid drag in a
two-component Bose gas is developed. The drag factor is shown to
be proportional to the square root of the gas parameter. Basing on
the  similarity between the "drag force"\ and  vector potential of
magnetic field we propose how the superfluid drag  can be used to
detect a Berry phase in a Bose counterpart of the Josephson charge
qubit. We consider a system in which the drag component, situated
inside the drive component, is confined in two half-ring traps
separated by two Josephson barriers. Under cyclic adiabatic change
of Josephson coupling parameters and an asymmetry of two traps a
Berry phase is generated. This phase can be observed though
measurements of relative number of atoms in two traps. The Berry
phase depends on the drag factor and its detection can be used for
determining the value of the drag. \end{abstract}

\pacs{03.75.Kk, 03.75.Lm, 03.67.Lx}
% \submitto{\jpb}
\maketitle
%
\section{Introduction}
\label{intro} The interest in studying of macroscopic quantum
phenomena in Josephson superconductive devices has renewed
recently in connection with the possibility of using such devices
for performing quantum logic operations \cite{1}. Related problem
- Josephson-type effects in Bose systems was received considerable
attention \cite{2} after experimental realization of Bose-Einstein
condensation in ultracold atomic gases. It was shown in \cite{3}
that Josephson Bose system can also be used for quantum computing.
In \cite{3} a simple model of a Bose counterpart of the Josephson
charge qubit has been formulated.


The supercurrent in superconductors is coupled to the vector
potential of electromagnetic fields that allows to perform quantum
logic operations  by external magnetic fields. External magnetic
fields couple with Bose atoms only through Zeeman and hyperfine
interactions. Therefore, in difference with the case of
superconductors, magnetic field cannot be used to control
supercurrents in uncharged Bose systems.

In this paper we consider the effect that can be used for quantum
state engineering in Bose Josephson systems in the same manner as
magnetic field in superconductive Josephson qubits. This effect is
a nondissipative drag of supercurrent in a two-component
superfluid system.

Nondissipative drag is caused by interaction between the atoms and
consists in  a redistribution of a supercurrent between two
superfluid components. For the first time the nondissipative drag
was considered by Andreev and Bashkin \cite{4}. In  \cite{4} a
three-velocity hydrodynamic model of a $^3$He-$^4$He mixture was
developed. It was shown that superfluid behavior of such a system
can be described under accounting the "drag"\ term in the free
energy. This term is proportional to the scalar product of the
superfluid velocities of two components times the difference
between the effective and the bare masses of $^3$He atoms. A
microscopic theory of the nondissipative drag for a special case
of a bilayer system of charged bosons was developed in Refs.
\cite{5,6}.

Nondissitive drag is a fundamental property of  systems with a
macroscopic quantum coherence and takes place  also in
superconductors \cite{7}, quantum Hall systems \cite{8} and
quantum wires \cite{9}.  But for the system of uncharged bosons
this effect is especially important since the "drag force" plays
the role similar to the role of the vector potential of a magnetic
field in superconductors \cite{4}.

An important problem of quantum computation is to minimize the
influence of random noise coming from the environment. One of the
ways of solutions of this problem consists in using of geometrical
phases to implement quantum gates \cite{10}. As was shown in
\cite{11} a geometrical Berry phase can be generated in a SQUID
loop with two Josephson junctions in an external magnetic field.
In this paper we propose a model of Bose qubit that may
demonstrate analogous effect.

In Sec. \ref{sec2} the theory of nondissipative drag in
two-component Bose gases is developed. In Sec. \ref{sec3} the
model of Bose qubit is formulated. The scheme of generating of the
Berry phase is proposed in Sec. \ref{sec4}. Conclusions are given
in Sec. \ref{sec5}.




\section{Nondissipative drag of superflow in a two-component Bose system}
\label{sec2}

In this section we develop a microscopic theory  of the
nondissipative drag in a two-component atomic Bose system with
point two-particle interaction.

Let  us consider an infinite uniform two-component atomic Bose gas
in a Bose-Einstein condensed state. In general case the densities
of atoms in each component are different from one another ($n_1\ne
n_2$), the atoms of each components  have different masses
($m_1\ne m_2$) and the interaction between atoms is described by
three different scattering lengths ($a_{11}\ne a_{22}\ne a_{12}$).
The Hamiltonian of the system can be presented in the form
\begin{equation}\label{1}
  H=\sum_{i=1,2} (E_i - \mu_i N_i)+\frac{1}{2}\sum_{i,i'=1,2}
  E_{ii'}^{int},
\end{equation}
where
\begin{equation}\label{2}
  E_i=\int d^3 r \frac{\hbar^2}{2 m_i}
  [\nabla\hat{\Psi}^+_i({\bf r})]\nabla\hat{\Psi}_i({\bf r})
 \end{equation}
is the kinetic energy,
\begin{equation}\label{2a}
  E_{i i'}^{int}=\int d^3  r
 \hat{\Psi}^+_i({\bf r})\hat{\Psi}^+_{i'}({\bf r})\gamma_{i i'}
 \hat{\Psi}_{i'}({\bf r})
 \hat{\Psi}_{i}({\bf r})
 \end{equation}
is the energy of interaction,   $\gamma_{ii}=4\pi\hbar^2
a_{ii}/m_i$ and $\gamma_{12}=2\pi\hbar^2(m_1+m_2)a_{12}/(m_1m_2)$
are the interaction parameters.

For further analysis it is convenient to use the density and phase
operator approach (see, for instance, \cite{12,13}). The approach
is based on the following representation for the Bose field
operators
\begin{equation}\label{3}
  \hat{\Psi}_i({\bf r})=\exp\left[i\varphi_{i}({\bf r})+i\hat{\varphi}_i({\bf r})\right]
  \sqrt{n_i+\hat{n}_i({\bf r})},
\end{equation}
\begin{equation}\label{4}
  \hat{\Psi}_i^+({\bf r})=\sqrt{n_i+\hat{n}_i({\bf r})}
  \exp\left[-i\varphi_{i}({\bf r})-i\hat{\varphi}_i({\bf
  r})\right],
\end{equation}
where $\hat{n}_i$ and $\hat{\varphi}_i$ are the density and phase
fluctuation operators, $\varphi_{i}({\bf r})$ is the $c$-number
term of the phase operator, which are connected with the
superfluid velocity  by common relation ${\bf
v}_i=\hbar\nabla\varphi_i/m_i$. In what follows we specify the
case of superfluid velocities independent of ${\bf r}$.

Substituting Eqs.  (\ref{3}), (\ref{4}) into Eq. (\ref{1}) and
expanding it in series in powers of $\hat{n}_i$ and
$\nabla\hat{\varphi}_i$ we present the Hamiltonian of the system
in the following form
\begin{equation}\label{5}
  H=H_{0}+H_{2}+\ldots
\end{equation}
In (\ref{5}) the term
\begin{equation}\label{5a}
  H_0=V\Bigg(\sum_{i=1,2} \left[\frac{1}{2} m_i n_i
  {\bf v}_i^2
  +\frac{\gamma_{ii}}{2}n_{i}^2-\mu_in_i\right] +\gamma_{12}n_1
  n_2\Bigg)
\end{equation}
does not contain the operator part ($V$ is the volume of the
system). The chemical potentials are determined from  the
minimization condition of the Hamiltonian $H_0$:
\begin{equation}\label{5d}
\frac{1}{2}m _i {\bf v}_i^2+\gamma_{ii} n_i+\gamma_{12} n_{3-i}-
  \mu_i=0.
\end{equation}
Fulfillment of Eq. (\ref{5d}) results in  vanishing of linear in
the density fluctuation operators terms  in the Hamiltonian. The
terms linear in the phase fluctuation operators are absent in the
Hamiltonian due to continuity conditions.

The quadratic in $\nabla\hat{\varphi}_i$ and $\hat{n}_i$
 operator term in $H$ reads as
\begin{eqnarray}\label{5c}
   H_2=\int d {\bf r} \Bigg(\sum_i\Bigg\{\frac{\hbar^2}{2
  m_i}\Bigg[\frac{\Big(\nabla\hat{n}_i({\bf r})\Big)^2}{4 n_i}
  +n_i\Big(\nabla\hat{\varphi}_i({\bf r})\Big)^2\Bigg] \cr  +
  \frac{\hbar {\bf v}_i}{2}\Big(\hat{n}_i({\bf r})\nabla\hat{\varphi}_i({\bf r})
  +[\nabla\hat{\varphi}_i({\bf r})]\hat{n}_i({\bf r})\Big)\cr \fl
  +
  \frac{i\hbar^2}{2m_i}\Big([\nabla\hat{n}_i({\bf r})]\nabla\hat{\varphi}_i({\bf r})-
  [\nabla\hat{\varphi}_i({\bf r})]\nabla\hat{n}_i({\bf r})\Big)
  +\frac{\gamma_{ii}}{2}\Big(\hat{n}_i({\bf r})\Big)^2\Bigg\}
   + \gamma_{12}\hat{n}_1({\bf r})\hat{n}_2({\bf r})
  \Bigg).
\end{eqnarray}

The quadratic part of the Hamiltonian determines the spectrum of
the elementary excitations. Hereafter we will neglect the higher
order terms in the Hamiltonian (\ref{5}). These terms describe the
scattering of the quasiparticles and they can be omitted if the
temperature is much smaller than the temperature of Bose-Einstein
condensation.

Let us rewrite the quadratic part of the Hamiltonian in terms of
the operators of creation and annihilation of the elementary
excitations. As the first step we use the substitution
\begin{equation}\label{6}
  \hat{n}_i({\bf r})=\sqrt{\frac{n_i}{V}}\sum_{\bf k}
  e^{i{\bf k r}}\sqrt{\frac{\epsilon_{i k}}{E_{i k}}}
  \left[ b_{i}({\bf k})+b^+_{i}(-{\bf
  k})\right],
\end{equation}
\begin{equation}\label{7}
  \hat{\varphi}_i({\bf r})=\frac{1}{2i}\sqrt{\frac{1}{n_i V}}\sum_{\bf k}
  e^{i{\bf k r}}\sqrt{\frac{E_{i k}}{\epsilon_{i k}}}
  \left[ b_{i}({\bf k})-b^+_i(-{\bf
  k})\right],
\end{equation}
where operators $b_i^+$, $b_i$ satisfy the Bose commutative
relations. Here $\epsilon_{i k}=\hbar^2 k^2/2 m_i$ is the spectrum
of free atoms, and $E_{lk}=\sqrt{\epsilon_{i k}(\epsilon_{i
k}+2\gamma_{ii} n_i)}$ is the spectrum of the elementary
excitations at $\gamma_{12}=0$ and ${\bf v}_i=0$. The substitution
(\ref{6}), (\ref{7}) reduces the Hamiltonian (\ref{5c}) to the
form, quadratic in operators $b_i^+$, $b_i$:
\begin{eqnarray}\label{8}
  H_2=\sum_{i{\bf k}} \left[{\cal E}_{i}({\bf k}) \left(
b^+_{i}({\bf k})  b_{i}({\bf
  k})+\frac{1}{2}\right)-\frac{1}{2}\epsilon_{i k}\right]
 \cr +\sum_{\bf k} g_k\left[b^+_{1}({\bf k})  b_{2}({\bf k})+b_{1}({\bf k})
   b_{2}({\bf
  -k})+h.c.\right].
\end{eqnarray}
Here
\begin{equation}\label{401}
  {\cal E}_{i}({\bf k})=E_{i k}+\hbar  {\bf
k v}_i
\end{equation}
and
\begin{equation}\label{402}
  g_k=\gamma_{12}  \sqrt{\frac{\epsilon_{1 k}\epsilon_{2 k}n_1 n_2}{E_{1 k}E_{2
k}}}.
\end{equation}

The Hamiltonian (\ref{8}) contains non-diagonal in Bose creation
and annihilation operator terms and  it can be diagonalized using
the standard procedure of u-v transformation \cite{14}. The result
is
\begin{equation}\label{14}
 H_2= \sum_{\bf k}\Bigg[\sum_{\lambda=\alpha,\beta} {\cal E}_{\lambda}({\bf k})\Bigg(
   \beta_\lambda^+({\bf k})  \beta_\lambda({\bf
  k})+\frac{1}{2}\Bigg)-\frac{1}{2}\sum_{i=1,2}\epsilon_{i
  k}\Bigg],
\end{equation}
where $\beta_{\lambda}(k)^+$  and $\beta_{\lambda}(k)$ are the
operators of creation and annihilation of elementary excitations.

The energies ${\cal E}_{\lambda}({\bf k})$ are found as solutions
of the equation
\begin{equation}\label{n1}
  \det\left(\matrix{{\bf A}-{\cal E}{\bf I}&{\bf B}\cr{\bf B}&{\bf A}+{\cal
  E}{\bf I}}\right)=0,
\end{equation}
where
\begin{equation}\label{n2}
  {\bf A}=\left(\matrix{{\cal E}_1({\bf k})&0&g_k&0\cr
  0&{\cal E}_1(-{\bf k})&0&g_k\cr
  g_k&0&{\cal E}_2({\bf k})&0\cr
  0&g_k&0&{\cal E}_2(-{\bf k})}\right)
\end{equation}
\begin{equation}\label{n2a}
  {\bf B}=\left(\matrix{0&0&0&g_k\cr
  0&0&g_k&0\cr
  0&g_k&0&0\cr
  g_k&0&0&0}\right)
\end{equation}
and ${\bf I}$ is the identity matrix. Having the spectra of
elementary  excitations  one can obtain the relation between the
supercurrents in each component are their superfluid velocities.

We will find the current densities  using the relation
\begin{equation}\label{101n}
  {\bf j}_i=\frac{1}{V}\frac{\partial  F}{\partial {\bf
  v}_i},
\end{equation}
where $F$ is the free energy of the system. Here the quantity
${\bf j}_i$ is defined as density of the mass current.

The free energy of the system, described by the Hamiltonian
(\ref{5}), is given by the formula
\begin{equation}\label{201n}
\fl  F=H_0 +
 \frac{1}{2}\sum_{{\bf k}}\left[\sum_{\lambda=\alpha,\beta}
  {\cal E}_{\lambda}( {\bf k})
   - \sum_{i=1,2}\epsilon_{i k}\right]
   + T \sum_{{\bf k}}\sum_{\lambda=\alpha,\beta}
   \ln\left[1-\exp\left(-\frac{{\cal
  E}_{\lambda}( {\bf k})}{T}\right) \right].
 \end{equation}
 The second term in (\ref{201n}) is
the energy of the zero-point fluctuations and the third term is
the standard temperature dependent part of the free energy for the
gas of noninteracting  elementary excitations.

To proceed further we consider the case of small superfluid
velocities, much smaller than the velocities of elementary
excitations, and seek for the free energy as series in ${\bf
v}_i$.

At  ${\bf v}_1={\bf v}_2=0$ the equation (\ref{n1}) is easily
solved and the spectra are found to be
\begin{equation}\label{n3}
  E_{\alpha(\beta) k}=
  \Bigg(\frac{E_{1k}^2+E_{2k}^2}{2}\pm\sqrt{\frac{(E_{1k}^2-E_{2k}^2)^2}{4}+
  4 \gamma_{12}^2 n_1 n_2 \epsilon_{1 k} \epsilon_{2
  k}}\Bigg)^{1/2}.
\end{equation}
As required in the procedure \cite{14} we take positive valued
solutions of Eq. (\ref{n1}). We should emphasize that  the
energies (\ref{n3}) should be real valued. This requirement yields
the common condition of stability of a two-component system:
$\gamma_{12}^2\leq \gamma_{11}\gamma_{22}$. If this condition were
not fulfilled spatial separation of two components (at positive
$\gamma_{12})$ or a collapse (at negative $\gamma_{12}$) would
take place.


At nonzero superfluid velocities we present the solutions  of Eq.
(\ref{n1}) as series in ${\bf v}_i$, neglecting the terms higher
than quadratic:
\begin{eqnarray}\label{n4}
\fl {\cal E}_{\alpha}({\bf k})=E_{\alpha k}+\frac{1}{2}\hbar {\bf
k v}_1 \left(1+\frac{E_{1k}^2-E_{2k}^2}{E_{\alpha k} ^2-E_{\beta
k}^2}\right) + \frac{1}{2}\hbar {\bf k v}_2\left(1-\frac{E_{1
k}^2-E_{2 k} ^2}{E_{\alpha k} ^2-E_{\beta k}^2}\right)\cr \fl +
\frac{2 \gamma_{12}^2 n_1 n_2 \epsilon_{1 k} \epsilon_{2
k}\left(3E_{\alpha k}^2+E_{\beta k}^2\right)}{E_{\alpha
k}\left(E_{\alpha k}^2-E_{\beta k}^2\right)^3}\hbar^2\left({\bf k
v}_1-{\bf k v}_2\right)^2,
\end{eqnarray}

\begin{eqnarray}\label{n5}
\fl {\cal E}_{\beta}({\bf k})=E_{\beta k}+\frac{1}{2}\hbar {\bf k
v}_1\left(1-\frac{E_{1k}^2-E_{2k}^2}{E_{\alpha k} ^2-E_{\beta
k}^2}\right)+ \frac{1}{2}\hbar {\bf k v}_2\left(1+\frac{E_{1
k}^2-E_{2 k} ^2}{E_{\alpha k} ^2-E_{\beta k}^2}\right)\cr \fl -
\frac{2 \gamma_{12}^2 n_1 n_2\epsilon_{1 k} \epsilon_{2
k}\left(E_{\alpha k}^2+3E_{\beta k}^2\right)}{E_{\alpha
k}\left(E_{\alpha k}^2-E_{\beta k}^2\right)^3}\hbar^2\left({\bf k
v}_1-{\bf k v}_2\right)^2.
\end{eqnarray}
 Note that at ${\bf v}_1={\bf v}_2={\bf v}$ the spectra (\ref{n4}), (\ref{n5}) are
 reduced to common expressions for the energies of quasiparticles in a moving
 condensate:
 ${\cal E}_{\alpha(\beta)}({\bf k})=E_{\alpha(\beta) k}+\hbar {\bf k
v}$.

Using Eqs. (\ref{201n}),(\ref{n4}) and (\ref{n5}) we obtain the
following expression for the free energy
\begin{equation}\label{28a}
  F= F_0+  \frac{V}{2}\Bigg[(\rho_{1}-\rho_{n1}){\bf v}_1^2
  + (\rho_{2}-\rho_{n2}){\bf v}_2^2 -\rho_{\rm dr}
  ({\bf v}_1-{\bf v}_2)^2\Bigg],
\end{equation}
where $F_0$ does not depend on ${\bf v}_i$. In (\ref{28a})
$\rho_i=m_i n_i$ are the mass densities, the quantities
\begin{equation}\label{n6}
\fl  \rho_{n1}=-\frac{2 m_1}{3 V }\sum_{\bf
k}\epsilon_{1k}\Bigg[\frac{d N_{\alpha
  k}}{d E_{\alpha k}}+\frac{d N_{\beta
  k}}{d E_{\beta k}}+ \frac{E_{1 k}^2-E_{2 k} ^2}{E_{\alpha k}
^2-E_{\beta k}^2} \left(\frac{d N_{\alpha
  k}}{d E_{\alpha k}}-\frac{d N_{\beta
  k}}{d E_{\beta k}}\right)\Bigg],
\end{equation}
\begin{equation}\label{n6a}
\fl  \rho_{n2}=-\frac{2 m_2}{3 V}\sum_{\bf
k}\epsilon_{2k}\Bigg[\frac{d N_{\alpha
  k}}{d E_{\alpha k}}+\frac{d N_{\beta
  k}}{d E_{\beta k}}- \frac{E_{1 k}^2-E_{2 k} ^2}{E_{\alpha k}
^2-E_{\beta k}^2} \left(\frac{d N_{\alpha
  k}}{d E_{\alpha k}}-\frac{d N_{\beta
  k}}{d E_{\beta k}}\right)\Bigg]
\end{equation}
describe the thermal reduction of the superfluid densities, and
the quantity
\begin{eqnarray}\label{n7}
 \fl \rho_{dr}=\frac{4}{3 V }\sqrt{m_1 m_2} \sum_{\bf k}\frac{\gamma_{12}^2 n_1 n_2
  \left(\epsilon_{1 k}\epsilon_{2 k}\right)^{3/2}}
  {E_{\alpha k} E_{\beta k}} \Bigg[\frac{1+N_{\alpha k} +N_{\beta
  k}}{(E_{\alpha k}+E_{\beta k})^3}- \frac{N_{\alpha k} -N_{\beta
  k}}{(E_{\alpha k}-E_{\beta k})^3}\cr+\frac{2 E_{\alpha k} E_{\beta
  k}}{\left(E_{\alpha k}^2-E_{\beta k}^2\right)^2}\left(\frac{d N_{\alpha
  k}}{d E_{\alpha k}}+\frac{d N_{\beta
  k}}{d E_{\beta k}}\right)\Bigg],
\end{eqnarray}
which we call the "drag density," yields the value of
redistribution of the superfluid densities between components. In
Eqs. (\ref{n6})-(\ref{n7}) $N_{\alpha(\beta) k} =[\exp(
E_{\alpha(\beta) k}/T)-1]^{-1}$ is the Bose distribution function.

Using Eqs. (\ref{101n}), (\ref{28a}) we arrive to the following
expressions for the supercurrents
\begin{equation}\label{22}
  {\bf j}_1=(\rho_{1}-\rho_{n1}-\rho_{dr}){\bf v}_1
  +\rho_{{\rm dr}}{\bf v}_2,
\end{equation}
\begin{equation}\label{23}
  {\bf j}_2=(\rho_{2}-\rho_{n2}-\rho_{dr}){\bf v}_2
  +\rho_{{\rm dr}}{\bf v}_1.
\end{equation}
One can see that  at nonzero $\rho_{dr}$ the current of one
component contains the term proportional to the superfluid
velocity of the other component. It means that there is a transfer
of motion between the components, or the drag effect. Since
$\rho_{dr}\propto \gamma_{12}^2$ the effect occurs only at nonzero
interaction between the components and it does not depend on the
sign of this interaction.

Eq. (\ref{n7}) is the main result of this section.  This equation
yields the value of the drag for a general case of  two-component
Bose system with components having different densities, different
masses of atoms, different interaction parameters, and for zero as
well as for nonzero temperatures.

To estimate the absolute value of the drag we, for simplicity,
specify the case of $m_1=m_2$, that is realized when two
components are two hyperfine states of the same atoms.

At  $m_1=m_2=m$ and $T=0$ Eq. (\ref{n7}) is reduced to the form
\begin{equation}\label{n11}
  \rho_{dr}=\frac{4 m}{3}  \int_0^\infty d \epsilon
\frac{\gamma_{12}^2 n_1  n_2 \nu(\epsilon)
\epsilon^{1/2}}{\sqrt{(\epsilon+w_1)
  (\epsilon+w_2)}\left(\sqrt{\epsilon+w_1}+\sqrt{\epsilon+w_2}\right)^3},
\end{equation}
where
\begin{equation}\label{b1}
w_{1(2)}=\gamma_{11}n_1+\gamma_{22}n_2\pm\sqrt{(\gamma_{11}
n_1-\gamma_{22} n_2)^2+ 4\gamma_{12}^2 n_1 n_2 }
\end{equation}
and
 $$\nu(\epsilon)= \frac{m^{3/2}}{\sqrt{2}\pi^2\hbar^3}
  \sqrt{\epsilon}$$ is the density of states of elementary
 excitations.

 The integral in (\ref{n11}) is
evaluated analytically. To present the answer in a compact form it
is convenient to introduce dimensionless parameters $$
\eta=\frac{\gamma_{12}^2}{\gamma_{11} \gamma_{22}}\quad \textrm{
and}\quad \kappa=\sqrt{\frac{n_1 \gamma_{11}}{n_2 \gamma_{22}}}.$$
Using these notations we have
\begin{equation}\label{n11b}
\rho_{dr}= \sqrt{\rho_1 \rho_2}\sqrt[4]{ n_1 a_{11}^3 n_2
a_{22}^3}\eta^2 F(\kappa,\eta),
\end{equation}
where
\begin{eqnarray}\label{n11c}
 \fl F(\kappa,\eta)=\frac{256}{45\sqrt{2\pi}}
  \Big(\kappa+\kappa^{-1}+3\sqrt{1-\eta^2}\Big)\cr \fl \times
  \Bigg(\sqrt{\kappa+\kappa^{-1}+\sqrt{(\kappa-\kappa^{-1})^2+4
  \eta^2}}+\sqrt{\kappa+\kappa^{-1}-\sqrt{(\kappa-\kappa^{-1})^2+4
  \eta^2}}\Bigg)^{-3}.
\end{eqnarray}


At $a_{11}n_1=a_{22}n_2$ we obtain from (\ref{n11b}) the following
approximate relation
\begin{equation}\label{1001}
  \frac{\rho_{dr}}{\rho_1}\approx
  \frac{1}{2}\frac{a_{12}^2}{a_{11}a_{22}}\sqrt{n_1 a_{11}^3}.
\end{equation}
For $n_2\gg n_1$ and $a_{11}\sim a_{22}$ we find from (\ref{n11b})
that
\begin{equation}\label{1002}
\frac{\rho_{dr}}{\rho_1}\approx 0.8
\frac{a_{12}^2}{a_{22}^2}\sqrt{n_2 a_{22}^3}.
\end{equation}
 Since the scattering lengths
in two-component systems are of the same order and the gas
parameters $n_i a_{ii}^3$ can be as much as $10^{-4}$ the quantity
$\rho_{dr}/\rho_1$ can be of order of $10^{-2}$.


Eq. (\ref{n7}) shows that the "drag density" decreases under
increasing the temperature. Evaluating the temperature dependence
of $\rho_{dr}$ numerically we conclude that at the temperatures of
order of the interaction energy this decreasing is not crucial. As
an example, the dependence of $\rho_{dr}$ on the temperature at
$n_1=n_2=n$ and $\gamma_{11}=\gamma_{22}=\gamma$ is shown in Fig.
\ref{fig1}. Results, presented in Fig. \ref{fig1}, correspond to
$\eta=0.5$, but, actually, temperature dependence of $\rho_{dr}$
is not very sensitive to this parameter.

In conclusion, let us discuss  to which extend the results,
obtained in this section, can be applied to Bose gases confined in
a trap. In nonuniform systems the spectrum of elementary
excitations is modified \cite{13,15,16,17},  but for healing
lengths much smaller than the linear size of the Bose clouds the
spectrum of excitations  at wave vectors of order or higher than
inverse healing length is well described by the quasi uniform
approximation. The main contribution to the sum in Eq. (\ref{n7})
comes from the excitations  with such wave vectors. Therefore, the
drag effect in nonuniform system can be described by the same
equations, as in the uniform case with the only modification that
the quantities $n_1$ and $n_2$, and, correspondingly, $\rho_i$,
$\rho_{ni}$, $\rho_{dr}$ and $j_i$ in Eqs. (\ref{n6})-(\ref{23})
are understood as functions of coordinates.



\begin{figure}
\begin{center}
\epsfbox{fig1.ps}
\end{center}
\caption{\label{fig1}Dependence of the "drag density" on the
temperature.}
\end{figure}




\section{Model of BEC qubit with external drag force}
\label{sec3}

Let our two-component system is confined in a toroidal trap and
the Bose cloud of component 1 (drag component) is situated inside
and overlaps with the Bose clouds of component 2 (drive
component). Such a situation can be realized if
$|\gamma_{12}|<\min(\gamma_{11},\gamma_{22})$. Then the confining
potential is deformed to cut the drag component in two clouds of a
half-torus shape (separated by two Josephson links), but the
toroidal shape of the Bose cloud of the drive component is
survived  under such a deformation (see Fig. \ref{fig2}). In what
follows we imply that circumference $L=2\pi R_t$ of the toroidal
clouds is much larger than radiuses of their cross-sections
$r_{t1}$ and $r_{t2}$.

\begin{figure}
\begin{center}
\epsfbox{fig2.eps}
\end{center}
\caption{\label{fig2}Schematic shapes of Bose clouds for the drag
(top figure) and drive (bottom figure) components. In  system
considered the drag component overlaps with the drive component.}
\end{figure}




Rotating this trap one can excite a superflow in the drive
component. After the rotation be switched off there will be a
circulating superflow in the drive component and no superflow in
the drag component (at negligible small Josephson coupling).
 The superfluid velocity of the drive component satisfies the
Feynman-Onsager quantization conditions and its value is
proportional to the vorticity parameter $N_v$ for the drive
component:
\begin{equation}\label{vor}
  v_2=\frac{N_v \hbar} {m_2 R_t}
\end{equation}
($N_v$ is integer). In (\ref{vor}) we, for simplicity, neglect the
effect caused by variation of $r_{t2}$.

Since $j_1=0$, the phase gradient $\nabla\varphi_1$ should be
nonzero to compensate the drag effect. In polar coordinates the
$\phi$ component of the phase gradient is given by the relation
\begin{equation}\label{1004}
  (\nabla \varphi_1)_\phi=-\frac{N_v}{ R_t} f_{dr},
 \end{equation}
where
\begin{equation}\label{fdr}
  f_{dr}=\frac{m_1}{m_2}\frac{\rho_{dr}}{\rho_1-\rho_{n1}-\rho_{dr}}
\end{equation}
is the drag factor. We imply that $r_{t1}$ and $r_{t2}$ are much
higher than healing lengths that allows to describe the drag
effect in quasi-uniform approximation. For definiteness, we
specify the case of $\rho_{1}\ll \rho_2$ and $\rho_2\approx const$
in the overlapping region. In such a case one can neglect space
dependence the drag factor (see Eq. (\ref{1002})).

At nonzero Josephson coupling the current $j_1$ can be nonzero,
but it cannot exceed maximum Josephson current $J_m$. Relation
(\ref{1004}) remains approximately correct at nonzero Josephson
coupling, if a strong inequality $J_m\ll \hbar\rho_1/(m R_t)$ is
satisfied. Here we specify just such a case.

The drag force  can be considered as external "magnetic field"
applied to the drag component. The latter component in the
geometry considered forms a superfluid circuit with two  Josephson
links. The situation described is analogous to the one that takes
place in superconductive Josephson qubits \cite{1,11} and our
system can be treated as a Bose qubit controlled by the drag
force.

We will formulate the model of "dragged" Bose qubit using an
approach similar to one of Ref. \cite{3}.

Let us present the Bose field operators for the component 1 in the
two mode approximation:
\begin{equation}\label{1006}
 \eqalign{ \hat{\Psi}_1({\bf r},t)=\sum_{l={L},{R}} a_l(t) \Psi_{l}({\bf r}-{\bf
  r}_l),\\
 \hat{\Psi}_1^+({\bf r},t)=\sum_{l=L,R} a_l^+(t) \Psi_{l}^*({\bf r}-{\bf
  r}_l),}
\end{equation}
where $a_{{L}({R})}^+$  and $a_{{L}({R})}$ are the operators of
creation and annihilation of bosons in the condensate confined in
the left(right) half-torus, and $\Psi_L$, $\Psi_R$ are two almost
orthogonal local mode functions $$ \int d^3 r \Psi_{l}^*({\bf r})
\Psi_{l'}({\bf r})\approx\delta_{ll'}, \quad l,l'=L,R$$ that
describe the condensate in the left and right traps \cite{rag}.


Substituting (\ref{1006}) into Hamiltonian (\ref{1}), we obtain
the following expression for the parts of the Hamiltonian that
depends on the operators $a^+_l$, $a_l$:
\begin{equation}\label{3f}
  H_a=\sum_{l={L},{R}} \left(K_l a_l^+ a_l + \lambda_l a_l^+a_l^+ a_l a_l\right)+
  (J a_{ L}^+ a_{R} + J^* a_{R}^+ a_{L}).
\end{equation}
Here
\begin{equation}\label{4f}
  K_l=\int d^3 r  \Psi^*_{l}\left[-\frac{\hbar^2}{2 m} \nabla^2
  +V_{tr}+\gamma_{12} \Psi^*_2\Psi_2\right]\Psi_{l},
\end{equation}
\begin{equation}\label{5f}
  \lambda_l =\frac{\gamma_{11}}{2} \int d^3 r |\Psi_{l}|^4,
\end{equation}
\begin{equation}\label{6f}
  J=\int d^3 r \left[\frac{\hbar^2}{2 m}
  \nabla\Psi_{{L}}^*\nabla\Psi_{{R}}+V_{tr} \Psi_{{L}}^*
  \Psi_{{R}}\right].
\end{equation}


Functions $\Psi_L$ and $\Psi_R$ contain the phase factors
$e^{i\varphi_L(\bf r)}$ and $e^{i\varphi_R(\bf r)}$  with the
phases satisfying Eq. (\ref{1004}). Taking these factors into
account, we choose the following basis for the one mode wave
functions
\begin{equation}\label{1007}
\Psi_{L(R)}({\bf r})=|\Psi_{L(R)}({\bf r})|
\exp\left[i\phi_{L(R)}({\bf r})\Phi_{dr}\right],
\end{equation}
where
\begin{equation}\label{1009}
\Phi_{dr}=-N_v f_{dr}
\end{equation}
and $\phi_{L}$,$\phi_{R}$ are the polar angles counted from the
centers of $L$ and $R$ Bose cloud, correspondingly (see Fig.
\ref{fig3}). The angles $\phi_{L(R)}(\bf r)$, defined as shown in
Fig. \ref{fig3}, satisfy the conditions
\begin{equation}\label{fff}
  \phi_{R}({\bf r}_A)-\phi_{L}({\bf r}_A)=\phi_{L}({\bf
r}_B)-\phi_{R}({\bf r}_B)=\pi,
\end{equation}
where ${\bf r}_A$ and ${\bf r}_B$ are the radius-vectors of
Josephson links.

\begin{figure}
\begin{center}
\epsfbox{fig3.ps}
\end{center}
\caption{\label{fig3} Left (L) and right(R) half-torus of the drag
component, separated by Josephson links A and B.}
\end{figure}






Substituting (\ref{1007}) into Eq. (\ref{6f}), using Eq.
(\ref{fff}) and taking into account, that the functions $\Psi_L$
and $\Psi_R$ overlap in a small vicinity of A and B links, we
obtain the following expression for the Josephson coupling
parameter:
\begin{equation}\label{1009a}
  J=(J_A+J_B)\cos(\pi
\Phi_{dr})+i(J_A-J_B)\sin(\pi \Phi_{dr}),
\end{equation}
where
\begin{equation}\label{1010}
  J_{A(B)}\approx \int_{V_{A(B)}} d^3 r \left[\frac{\hbar^2}{2 m}
  \nabla|\Psi_ L|\nabla|\Psi_R|+V_{tr} |\Psi_L|
  |\Psi_R|\right].
\end{equation}
Here $V_{A}$ and $V_{B}$ are the areas of overlapping of two one
mode functions at links $A$ and $B$, correspondingly.



Considering the Hilbert space in which the total number operator
\begin{equation}\label{1012}
  \hat{N}=a_{\cal L}^+a_{\cal L}+a_{\cal R}^+a_{\cal R}
\end{equation}
is a conservative quantity ($\hat{N}=N$) we present the
Hamiltonian (\ref{3f}) in the following form
\begin{equation}\label{1013}
  H_a=E_c (\hat{n}_{RL}+n_g)^2+(Ja_{L}^+ a_{R} +
  h.c.) + const ,
\end{equation}
where
\begin{equation}\label{1011}
  \hat{n}_{RL}=\frac{a_{R}^+a_{R}-a_{L}^+a_{L}}{2}
\end{equation}
is the number difference operator,
\begin{equation}\label{1014}
  E_c=\lambda_{R}+\lambda_{L}
\end{equation}
is the interaction energy, and the quantity
\begin{equation}\label{1015}
  n_g=\frac{1}{2 E_c}\left[K_{R}-K_{L}+(N-1)(\lambda_{R}-
  \lambda_{L})\right]
\end{equation}
describes an asymmetry of L and R half-torus.

In what follows we imply that the system is in the Fock regime
($|J|N\ll E_c$) and use the number representation
 $$ |n_{RL}\rangle\equiv|n_
R,n_L\rangle \equiv |\frac{N}{2}+n_{RL},\frac{N}{2}-n_{RL}\rangle.
$$ In this basis the first term in (\ref{1013}) is diagonal. The
second term in (\ref{1013}) can be considered as small nondiagonal
correction. But if  $n_g$ is biased  near  one of degeneracy
points
\begin{equation}\label{1016}
  n_{deg}=\cases{M+\frac{1}{2}& for even $N$\cr M& for odd $N$}
\end{equation}
(where $M$ is integer and $|M|<N/2$) the second term in
(\ref{1013}) results in a strong mixing of two lowest states
($|\uparrow\rangle=|n_{deg}+ 1/2\rangle$ and
$|\downarrow\rangle=|n_{deg}-
  1/2\rangle$)
and the low energy dynamics of the system can be described by a
pseudospin Hamiltonian
\begin{equation}\label{1019}
  H_{eff}=-
  \frac{\Omega_x}{2}\hat{\sigma}_x-\frac{\Omega_y}{2}\hat{\sigma}_y-
  \frac{\Omega_z}{2}\hat{\sigma}_z,
\end{equation}
where
\begin{equation}\label{1020}
\eqalign{
\hat{\sigma}_x=|\uparrow\rangle\langle\downarrow|+|\uparrow\rangle\langle\downarrow|\\
\hat{\sigma}_y=-i(|\uparrow\rangle\langle\downarrow|-|\uparrow\rangle\langle\downarrow|)\\
\hat{\sigma}_z=|\uparrow\rangle\langle\uparrow|-|\downarrow\rangle\langle\downarrow|}
\end{equation}
are the Pauli operators, and
\begin{equation}\label{1020a}
\eqalign{
\Omega_x=-(J_A+J_B)\sqrt{(N+1)^2-n_{deg}^2}\cos(\pi\Phi_{dr}),\\
\Omega_y=-(J_A-J_B)\sqrt{(N+1)^2-n_{deg}^2}\sin(\pi\Phi_{dr}),\\
\Omega_z=2E_c(n_g-n_{deg})}
\end{equation}
are the components of the pseudomagnetic field.

If the quantities $n_g$, $J_A$ and $J_B$  depend on time and the
conditions $N(J_A+J_B)\ll E_c$, $|n_g-n_{deg}|\ll 1$ are
satisfied, our system can be used as qubit. In difference with the
situation, considered in \cite{3}, our effective qubit Hamiltonian
(\ref{1019}) contains the $\hat{\sigma}_y$ term. The appearance of
this term in the Hamiltonian is caused by the drag effect. If the
ratio $J_A/J_B$ depends on time and the value of $2\Phi_{dr}$ is
not integer this term cannot be excluded from the Hamiltonian by
gauge transformation.

\section{Detection of Berry phase in the Bose qubit}
\label{sec4}

In experiments one can control the parameters $n_g$, $J_A$ and
$J_B$ independently.  It allows to realize an adiabatic cyclic
evolution of the parameters of the Hamiltonian (\ref{1019}), for
which  vector ${\bm \Omega}$ subtends nonzero solid angle at the
origin. Such an evolution results in appearance of a geometrical
phase of the wave function known as Berry phase \cite{ber}.  For
the system (\ref{1019}) the value of this phase is  the half of
the solid angle subtending by the contour of evolution of ${\bm
\Omega}$.

In this section we propose a scheme for generating Berry phase in
the Bose qubit and show that in our scheme  this phase is
proportional to the value of $\Phi_{dr}$, and, consequently, to
the drag factor.

 Let initially $J_A=J_B=0$ and $n_{g}>n_{deg}$. The
systems relaxes to the ground state $|\uparrow\rangle$. After that
one switches on the equal Josephson coupling at links A and B
($J_{A}=J_{B}=J_{in}$) and sets the asymmetry parameter $n_g$
equals to $n_{deg}$. The initial state of the qubit is the
superposition of two instantaneous eigenstates
$|e_a\rangle=(|\uparrow\rangle+|\downarrow\rangle)/\sqrt{2}$ and
$|e_b\rangle=(|\uparrow\rangle -|\downarrow\rangle)/\sqrt{2}$:
\begin{equation}\label{301}
|\psi_{in}\rangle=\frac{1}{\sqrt{2}}(|e_a\rangle+|e_b\rangle).
\end{equation}
The expectation value of the relative number operator in the
initial state
$\langle\psi_{in}|\hat{n}_{RL}|\psi_{in}\rangle=1/2$.

At  the starting point  the pseudomagnetic field is directed along
the (-1,0,0) axes. Adiabatic evolution includes 4 stages.

At the 1-st stage   the Josephson coupling $J_B$ is switched off.
After this stage  pseudomagnetic field becomes aligned along the
$(-\cos\pi\Phi_{dr}, -\sin\pi\Phi_{dr},0)$ direction.

At the 2-nd stage the coupling $J_A$ is switched off and
simultaneously the value of $n_g$ is deviated from $n_{deg}$.
After this stage ${\bm \Omega}$ is directed along the (0,0,1)
axis.

At the 3-d stage the value of $n_g$ is returned to the same
degeneracy point and the coupling $J_B$ is switched on. ${\bm
\Omega}$ becomes directed in $(-\cos\pi\Phi_{dr},
\sin\pi\Phi_{dr},0)$ direction.

At the 4-th stage the coupling $J_A$ is switched on and vector
${\bm \Omega}$ returns to  its starting value.

After such an evolution the system arrives at the state
\begin{equation}\label{302}
  |\psi_{m}\rangle=\frac{1}{\sqrt{2}}\left(e^{i\delta_a+i\gamma}|e_a\rangle+
e^{i\delta_b-i\gamma}|e_b\rangle\right),
\end{equation}
where $\gamma=\pi\Phi_{dr}$ is the Berry phase equals to half of
the solid angle subtending by ${\bm \Omega}$ and $\delta_a$,
$\delta_b$ are the dynamical phases.

Dynamical phases depends on time of evolution, while the Berry
phase does not. Elimination of the dynamical phases is performed
by swapping the eigenstates ($\pi$-transformation) and repeating
the same cycle of evolution in a reverse direction \cite{10}.

 The
$\pi$-transformation can be done if one switches of the Josephson
coupling and deviates the asymmetry parameter $n_g$ from $n_{deg}$
value during the time interval $t_\pi=2\pi E_c
(n_{deg}-n_g)/\hbar$. After the  $\pi$-transformation the state of
the qubit becomes
\begin{equation}\label{303}
  |\psi_{m\pi}\rangle=-\frac{i}{\sqrt{2}}(e^{i\delta_a+i\gamma}|e_b\rangle+
e^{i\delta_b-i\gamma}|e_a\rangle).
\end{equation}
After the cyclic evolution in the reverse direction the state
becomes
\begin{equation}\label{304}
  |\psi_{f}\rangle=-\frac{i}{\sqrt{2}}e^{i(\delta_a+\delta_b)}(e^{2i\gamma}|e_b\rangle+
e^{-2i\gamma}|e_a\rangle).
\end{equation}

The expectation value of the difference number operator in the
final state (\ref{304}) differs from this value in the initial
state (\ref{301}) to the quantity that depends on the Berry phase
and, consequently, on $\Phi_{dr}$:
\begin{equation}\label{305}
\langle\psi_{f}|\hat{n}_{RL}|\psi_{f}\rangle=\frac{1}{2}-\sin^2(2\gamma)=\frac{1}{2}-
\sin^2(2\pi\Phi_{dr}).
\end{equation}
Thus, the value of the drag  can be determined from measurements
of relative number of atoms in left and right condensates.


\section{Conclusions}
\label{sec5}

We derived  a microscopic model of nondissipative drag in a
two-component superfluid Bose gas and proposed possible
implementation of a Bose qubit in which the drag effect is used to
control quantum states. We have described how the Berry phase can
be detected in such a qubit and found that this phase is
proportional to the drag factor.

Main idea of our proposal is based on a correspondence between the
drag force and a vector potential. One can found from Eq.
(\ref{28a}) that the effective vector potential, applied to the
drag component, is equal to ${\bf A}_{eff}=\hbar f_{dr}
\nabla\varphi_{drv}$ (in units of $e=c=1$), where $\varphi_{drv}$
is the phase of the drive component, and $f_{dr}$ is the drag
factor, which is of order of the square root of the gas parameter.
In a trap of a ring geometry the effective vector potential can be
associated with an effective flux of external field
$\Phi_{eff}=2\pi \hbar f_{dr} N_v$, where $N_v$ is the vorticity
of the drive component. It allows to realize the Bose charge
qubit, analogous to superconductive qubit, considered in
\cite{11}. Note that while $\Phi_{eff}$ is quantized, for
$f_{dr}\ll 1$ it can be treated as a continuous one.

In conclusion we would mention another possibility that opens the
drag effect.  Tuning the effective flux $\Phi_{eff}$ to the
$\Phi_{eff}^\pi=\pi\hbar$ value and increasing the Josephson
coupling one can achieved a regime in which the states with
different vorticities of the drag component are mixed. Such a
system can be considered as a Bose counterpart of superconductive
Josephson flux qubit.



\section*{Acknowledgements}
This work is supported by the INTAS grant No 01-2344.

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\end{thebibliography}


\end{document}

cond-mat0404567.extracted_bib
\begin{thebibliography}{99}
\bibitem{1}
Makhin Y, Schon G and Shnirman A 2001 {\it Rev. Mod. Phys.} {\bf
73} 357.
\bibitem{2}
Leggett A J 2001 {\it Rev. Mod. Phys.} {\bf 73} 307.
\bibitem{3}
Chen Z B and Zhang  Y D 2002 {\it Phys. Rev.} A {\bf 65} 022318.
\bibitem{4}Andreev A F and  Bashkin E P 1975 {\it Zh. Exp. Teor.
Fiz.} {\bf 69} 319 [Sov. Phys. JETP, {\bf 42} 164].

\bibitem{5}Tanatar B and Das A K 1996 {\it Phys. Rev.} B {\bf 54}
13827.

\bibitem{6}Terentjev S V and  Shevchenko S I 1999
{\it Fiz. Nizk. Temp.} {\bf 25} 664 [{\it Low Temp. Phys.} {\bf
25} 493].

\bibitem{7}Duan J M and Yip S 1993 {\it Phys. Rev. Lett.} {\bf 70}
3647.

\bibitem{8}Duan J M 1995  {\it Europhysics Lett.} {\bf 29} 489.

\bibitem{9}Rojo A G and Mahan G D 1992 {\it Phys. Rev. Lett.} {\bf 68}
2074.
\bibitem{10}Ekert A, Ericsson M, Hayden P, Inamori H,
Jones J A, Oi D K L and Vedral V 2000 {\it J. Mod. Opt.} {\bf 47}
2501.
\bibitem{11} Falci G, Fazio R, Palma G M,  Siewert J and
Vedral V 2000  {\it Nature}(London) {\bf 407} 355.

\bibitem{12}
Shevchenko S I 1992  {\it Fiz. Nizk. Temp.} {\bf 18} 328 [{\it
Sov. J. Low Temp. Phys.} {\bf 18} 223].

\bibitem{13}Fil D V and Shevchenko S I 2001 {\it Phys. Rev.} A {\bf 64}
013607.

\bibitem{14}Bogolyubov N N and  Bogolyubov N N (Jr) 1982 {\it Introduction
to quantum statistical mechanics} (Singapore: World Scientific).

\bibitem{15}Stringari S 1996 {\it Phys. Rev. Lett.} {\bf 77} 2360.

\bibitem{16}\"Ohberg P, Surkov E L, Tittonen I, Stenholm S,
Wilkens M and Shlyapnikov G V 1997  {\it Phys. Rev.} A {\bf 56}
R3346.

\bibitem{17}Khawaja U Al, Pethick C J and  Smith H 1999 {\it Phys.
Rev.} A {\bf 60} 1507.

\bibitem{rag} Raghavan S,  Smerzi A, Fantoni S and Shenoy S R 1999
{\it Phys. Rev.} A {\bf 59} 620.

\bibitem{ber}Berry M V 1984 {\it Proc. Roy. Soc.} A {\bf 392} 47.

\end{thebibliography}