Dynamical projection of atoms to Feshbach molecules at strong coupling
R.\
A. Barankov and L.\
S. Levitov
The dynamical atom/molecule projection, %% at fermionic Feshbach resonance recently used to probe fermion pairing, is fast compared to collective fermion times, but slow on the Feshbach resonance width scale. Theory of detuning-induced dynamics of molecules coupled to resonantly associating atom pairs, employing a time-dependent many-body Green's function approach, is presented. %% While the conventional Landau-Zener level anticrossing approach %% is inadequate due to atom pairs forming a continuum, a %% time-dependent many-body Green's function provides an An exact solution is found, predicting a $1/3$ power law for molecule production efficiency at fast sweep. %% , and describing in a unified way the fast and slow sweep regimes. The results for $s$- and $p$-wave resonances are obtained and compared. %% The particle number dependence on the sweep rate, The predicted production efficiency agrees with experimental observations for both condensed and incoherent molecules away from saturation. %% , is found to be in agreement with observations.
Gtmolecule.tex
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\begin{document}
\title{Dynamical projection of atoms to Feshbach molecules at strong coupling}
\author{R.\,A. Barankov and L.\,S. Levitov}
\affiliation{Department of Physics, Massachusetts Institute of Technology,
77 Massachusetts Ave, Cambridge, MA 02139}
%%\date{\today}

\begin{abstract}
The dynamical atom/molecule projection,
%% at fermionic Feshbach resonance 
recently used to probe fermion pairing,
is fast compared to collective fermion times, but slow
on the Feshbach resonance width scale.
Theory of detuning-induced dynamics
of molecules coupled to
resonantly associating
atom pairs, employing a time-dependent many-body Green's function
approach, is presented. 
%% While the conventional Landau-Zener level anticrossing approach 
%% is inadequate due to atom pairs forming a continuum, a
%% time-dependent many-body Green's function provides an 
An exact solution is found, predicting a $1/3$ power law
for molecule production efficiency at fast sweep.
%% , and describing in a unified way the fast and slow sweep regimes.
The results for $s$- and $p$-wave resonances are obtained and compared.
%% The particle number dependence on the sweep rate, 
The predicted
production efficiency agrees with experimental observations
for both condensed
and incoherent molecules away from saturation.
%% , is found to be in agreement with observations.
\end{abstract} \pacs{} \keywords{}

\maketitle
\vspace{-10mm}
%\begin{multicols}{2}

%\narrowtext
Cold atomic Fermi gases, magnetically tuned to a Feshbach resonance
region \cite{fermion_expts}, host an intriguing strongly interacting many-body
system \cite{Timmermans01,Holland01,Timmermans99,Griffin02}. Recently, pairing
of fermions near the resonance was probed with the help of dynamical projection
of atomic state on the molecular state \cite{Regal04,Zwierlein04}, achieved by a
sweep
through the resonance, followed by the detection of molecular Bose-Einstein
condensate. The sweep could be made very fast compared to typical fermion time
scales, such as the collision frequency or inverse Fermi bandwidth and pairing
energy gap, making the process a ``snapshot probe'' with regard to the
collective fermion processes.

On a single particle level, however, broad Feshbach resonances studied in
Refs.\,\cite{Regal04,Zwierlein04}, exhibit strong atom-molecule
coupling in a relatively wide detuning range.
In this sense,
the sweep speed \cite{Regal04,Zwierlein04} corresponds to
essentially adiabatic atom/molecule conversion, slow on the scale of the
resonance width. For example, in the JILA experiment \cite{Regal04},
the Feshbach resonance width $\Delta B\simeq 10{\rm\,G}$ translates into
$\Delta\nu\simeq 180{\rm\,MHz}$ in detuning frequency, while
the characteristic time of
the magnetic field sweep, $\tau_*=(dt/d\nu)^{1/2}\simeq
1{\rm\,\mu s}$, is about
$10^2$ times longer than $\Delta\nu^{-1}$.
A similar estimate applies to the MIT fast projection
experiment \cite{Zwierlein04}.
%% The situation is somewhat paradoxical, as 
Somewhat paradoxically, the fermions involved in this
%% effectively adiabatic 
``slow'' molecule formation are the same whose
%% very same fermions whose
many-body state is being analyzed by the dynamical ``snapshot'' projection.
Thus a correct physical picture of the molecular state swept through the
resonance must combine the adiabatic single-particle and the ``snapshot''
many-body aspects in a seamless way.

Continuing efforts to use atom/molecule projection as investigative tool
call for better understanding of the driven molecular state.
%% The adiabatic aspects of the problem
%% have been described as generalized Landau-Zener process 
The Landau-Zener model \cite{Mies00,Goral04}, 
%% which seems 
which fits the data well near saturation \cite{Hodby04}, focuses
on the adiabatic aspects, ignoring molecule dissociation into
%% on the fermionic side of the resonance  
continuous spectrum of atom pairs.
%% describe with some support from observations \cite{Hodby04},
%% as well as using 
The dynamical mean field
approach \cite{Kokkelmans02,Kohler03,Yurovsky04,Duine04,Haque04,Javanainen04,Dicke_FR,Goral05}, which can be justified for bosons in the atomic BEC
regime, lacks firm foundation in the fermion case. 
%% The snapshot aspects have been analyzed using 
Recently, the many-body state
overlap models \cite{Diener04,Avdeenkov04,Perali05,Altman05}
were put forward. While providing some
guidance, these approaches do not account for the experimentally relevant
situation of broad resonance \cite{Regal04,Zwierlein04} when the ``snapshot''
many-body projection is slow on the scale of individual molecule formation.

%% {\Large Added text! LL: Why Here? Move?}
%% %[Correlations of $\vec p$ and $-\vec p$ in the BCS state]
%% The observed condensate fraction of molecules in the final state is associated
%% with the existence of the pairing correlations in the initial state of the
%% Fermi gas.
%% %An important aspect of
%% %the projection experiments is associated with the observed condensate fraction
%% %of molecules in the final state.
%% These correlations correspond to a finite probability for a pair of fermions
%% with opposite spins to have opposite momenta, and thus there is a macroscopic
%% number of fermion pairs that can create molecules with zero momentum. It is
%% this peculiarity of the initial superfluid state that makes the molecular
%% condensate so special in the process of molecular creation. Fast magnetic field
%% sweep through the Feshbach resonance provides, therefore, a method to probe the
%% pairing correlations in the initial state looking at the final momentum
%% distribution of molecules.

%% The goal of this paper
Our objective is to describe molecules at a sweep
fast compared
to the elastic collisions,
when only the quantum-mechanical
%% atom-molecule
processes are relevant.
We develop a theoretical framework which
accounts for resonance
dissociation/association
%% strong atom-molecule mixing near the resonance 
in the
presence of time-dependent detuning as well as for
fermion pairing correlations.
%% as well as the
%% strong atom-molecule mixing near the resonance in the
%% presence of time-dependent detuning.
%% The fundamental difference
%% %% of the atom/molecule transformation
%% from the Landau-Zener process of discrete levels
%% pair anticrossing arises due
%% to free relative motion of the atoms associating to form molecules. 
%% making it impossible
%% to describe the atom pairs by a single discrete level in the
%% Landau-Zener fashion.
%% Instead, we 
We describe the molecules swept through the resonance using a
time-dependent Green's function which fully accounts 
free relative motion of the atoms associating to form molecules.
%% for resonance dissociation/association processes. 
While our method is quite general and
applicable to Feshbach resonances with any angular momentum, here we focus, for
the sake of concreteness, on the $s$-wave case.
%% We are mainly interested in the
We consider the evolution from equilibrium
at $\nu=\nu_0$, followed by an abrupt linear sweep:
%% {\bf at the initial detuning $\nu_0$}:
%%%% after a long period of time-independent initial detuning:
%
\be\label{eq:omega_0(t)}
\nu(t)=\cases{\nu_0, & $t<0$\cr \nu_0-\alpha t, & $t>0$}
\ee
%
with $\alpha$ the sweep rate.
The generalization
%% of the results
to the $p$-wave
resonances \cite{Ticknor04,Zhang04,Schunck05,Gurarie04,Chevy04}
will be straightforward (see below).
%% , and will be discussed at the end.

Finding the time-dependent Green's function is a nontrivial mathematical
problem, here solved exactly using an idea similar to that of the Wiener-Hopf
method. The important time scale, characterizing the adiabaticity of the
sweep
%% time interval during which
%% the sweep through the resonance is nonadiabatic
(see Fig.\,\ref{fig1} inset), is found to be
%
\be\label{eq:time_scales} \tau_0=\lp \hbar\lambda^2/\alpha^2\rp^{1/3} ,\quad
\lambda=g^2m^{3/2}/4\pi\hbar^3,
%% \,,\quad
%% t_0=\nu_0/\alpha
\ee
%
with $g$ the atom-molecule coupling (see Eq.(\ref{H_feshbach})),
and $m$ the atom mass.
The time scale
$\tau_0$ can also be inferred, as noted by Altman and
Vishwanath \cite{Altman05}, from the adiabaticity
condition $\dot\omega\lesssim\omega^2$ for the time-dependent molecule energy
$\hbar\omega$.
Different regimes arise depending on the relation between
$\tau_0$ and $\nu_0/\alpha$, the time it takes the sweep to reach the resonance
(Fig.\,\ref{fig1}). The atom-to-molecule transformation
takes place at times less than $\tau_0$ after crossing the resonance, where
the evolution is nonadiabatic.
At later times, the molecules, dressed by atom pairs,
evolve adiabatically.
For a fast sweep, $\alpha\tau_0\gg\nu_0$, the number of produced molecules
scales with the sweep rate as $\alpha^{-1/3}$, while for slower sweep,
$\alpha\tau_0\ll\nu_0$, the number of molecules scales as $\alpha^{-1}$.

These results agree with the 
molecular number and condensate production efficiency
reported by JILA group \cite{Regal04}.
%% should be noted.
The sweep speeds $|dt/dB|\approx 10-80{\rm\,\mu s/G}$ \cite{Regal04} correspond
to $\nu_0/\alpha\approx 1-100{\rm\,\mu s}$ with $\nu_0=0.1-1\,{\rm G}$ in the
magnetic field units. The characteristic atom-molecule coupling
$\lambda^2\approx1{\rm\,GHz}$
%7{\rm\, m K}
gives the adiabaticity time $\tau_0\approx 10-20{\rm\,\mu s}$ depending on the
sweep speed. Thus with $0.2<\alpha\tau_0/\nu_0<10$ both the fast and the slow
regimes are realized. Indeed, the molecule number obtained for
different sweep speeds below saturation
(see Fig.\,5 in Ref.\,\cite{Regal04} displaying the data
for $\nu_0=0.12\,{\rm G}$) can be fitted quite accurately with the $1/3$ power law
dependence, $N_m\propto |dt/dB|^{1/3}$, in agreement with our results.
%% for fast sweep. 
%% The theory also predicts correctly, 
Also reasonable, by the order of magnitude, is the predicted
total number of produced molecules. 
Our conclusions regarding the incoherent molecule
production channel are consistent with the observed independence
%% The observed independence
of the condensate fraction \cite{Regal04} of the sweep speed.
%% also agrees with our theory which predicts 
We obtain the same production efficiency
for 
condensed and incoherent molecules (Eq.(\ref{eq:condensate_fraction})), 
except near saturation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\centerline{
\begin{minipage}[t]{3.5in}
\vspace{0pt}
\centering
\includegraphics[width=3.5in]{Gtmolecule_fig1.eps}
\end{minipage}
\hspace{-1.75in}
\begin{minipage}[t]{1.6in}
\vspace{1.4in}
\centering
\includegraphics[width=1.5in]{Gtmolecule_inset.eps}
\end{minipage}
} \vspace{-0.25cm} \caption[]{Molecule number $N_m$ \emph{vs.} the sweep rate
$\lp\lambda^4/\alpha\rp^{1/3}$ at different initial detuning $\nu_0$. The
asymptotic regimes, $N_m\propto \alpha^{-1},\,\alpha^{-1/3}$, correspond to
slow and fast sweep. Inset: Molecular energy
time dependence (\ref{eq:omega_0(t)})
with the time interval corresponding to nonadiabatic evolution marked.
%Notice that $t_0/\tau_0=\nu_0/\lambda^2\lp\lambda^4/\alpha\rp^{1/3}$.
}
%% step mixing parameter values $x=0.05, 0.1,...,0.9, 0.95$.}
\label{fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \vspace{-5mm}


%% The uncertainty of the Feshbach resonance position in the experiment is by two
%% orders of magnitude larger than the Fermi energy. This allows to neglect the
%% degeneracy effects associated with the Pauli principle in the dynamics of
%% molecular formation during magnetic field sweep across the resonance.

Let us recall the form of the two-channel Hamiltonian
%% atom-molecule Hamiltonian
\cite{Timmermans99}, describing pairs of fermions binding to form molecules at
the resonance:
%
\be\label{H_feshbach} \HH=\HH_a+\HH_m+ \sum_{\vec p,\vec p'}\lp g b^+_{\vec
p+\vec p'}a_{\vec p\uparrow}a_{\vec p'\downarrow} + {\rm h.c.}\rp
%% \int \lp g b^+(x)a_\uparrow(x)a_\downarrow(x) + {\rm h.c.}\rp d^3x
\ee
%
with $\HH_a= \sum_{\vec p\sigma} \frac{p^2}{2m} a^+_{\vec p\sigma}a_{\vec
p\sigma}$, $\HH_m= \sum_\vec k (\nu+\frac{k^2}{4m}) b_\vec k^+b_\vec k$,
%% E_kb_\vec k^+b_\vec k$,
%% +g\lp b_k^+c_k + {\rm h.c.}\rp
%
and $a_{\vec p\sigma}$, $b_\vec k$ the atom and molecule operators, $\sigma$
the spin ($\hbar=1$).
%%    atom pair creation and annihilation operators
%%    $c_k=\frac1{\sqrt{2}}\lp a_{-p_-\downarrow}a_{p_+\uparrow}
%%    +a_{p_+\downarrow}a_{-p_-\uparrow}\rp$,
%%    $c_k^+=\frac1{\sqrt{2}}\lp a_{p_+\uparrow}^+a_{-p_-\downarrow}^+
%%    +a_{-p_-\uparrow}^+a_{p_+\downarrow}^+\rp$, ($p_\pm=p\pm k/2$),
%%    describe
%%    pairs of fermions in a spin singlet state
%%    that associate to form a molecule at the resonance.
%% The molecule `bare' spectrum is described by $E_k = \nu+\frac{k^2}{4m}$, with the 
The detuning $\nu$ is
%% from the Feshbach resonance $B_0$ is
determined by
the molecule and two-atom Zeeman energy difference, $\nu=\Delta\mu\lp
B-B_0\rp$.

The single molecule Green's function, obtained from Dyson equation
\cite{Bruun04}, has the form
%
\be\label{eq:mol_green}
G(\omega,k)=\frac{1}{\tilde\omega-\nu-\Sigma(\tilde\omega)}
%% \lambda(-\tilde\omega)^{1/2}}
\,,\quad
\tilde\omega=\omega-\frac{k^2}{4m}+i0
\,,
\ee
%
where $\Sigma(\omega)=\lambda(-\omega)^{1/2}$ is the self-energy describing
molecule dissociation ($s$-wave),
%% . (Such $\Sigma(\tilde\omega)$
%% $\lambda=g^2m^{3/2}/4\pi\hbar^3$
which arises after integrating over the 3d density of atom pair states
$N(\epsilon)\propto \epsilon^{1/2}$ along with a suitable ultraviolet
regularization \cite{Griffin02}.


%%     The pole of Eq.(\ref{eq:mol_green}) defines
%%     molecular energy, corrected by the effects of resonant mixing
%%     with fermions, $\omega-\Sigma(\omega)=\nu_0$.
%%     %% is given by the poles of Eq.(\ref{eq:mol_green}).
%%     Near the resonance, at
%%     %
%%     \be
%%     0<-\nu_0\ll \Delta E_\ast=\lambda^2,
%%     \ee
%%     %
%%     since $|\omega|\ll |\Sigma(\omega)|$, the molecular energy is given by
%%     %
%%     \be\label{eq:Mdispersion}
%%     \omega(k)=-(\nu_0/\lambda)^2+k^2/4m
%%     \ee
%%     %
%%     This dispersion relation describes molecules dressed by fermion pairs
%%     at negative detuning from the resonance, $\nu_0<0$.
%%     At positive detuning $\nu_0>0$, the molecules are unstable,
%%     the dispersion equation $G^{-1}(\omega,k)=0$ solution
%%     is also given by (\ref{eq:Mdispersion}). The meaning of
%%     the \emph{negative} energy (\ref{eq:Mdispersion})
%%     at $\nu_0>0$, is that of a virtual level
%%     describing resonant two-fermion scattering.

For time-independent $\nu$, the molecular state dressed
%% by the effects of resonant mixing with fermions,
by atom pairs,
is described by the Green's function pole:
%%  of the Green's function (\ref{eq:mol_green}),
%%     $\tilde\omega-\Sigma(\tilde\omega)=\nu$,
%
\be\label{G,Z}
G_0(\omega)=\frac{Z(\omega)}{\omega-\omega(k)+i0}
%%    \,,\quad
%%    Z(\omega)=\lp \frac{dG^{-1}(\omega)}{d\omega}\rp^{-1}
%%    =\frac{(-\tilde\omega)^{1/2}}{(-\tilde\omega)^{1/2}+\lambda/2}
%% 1+\frac{\lambda}2(-\tilde\omega-i0)^{-1/2}
\ee
%
with $\omega(k)$ given by
$\tilde\omega-\Sigma(\tilde\omega)=\nu$.
%% (\ref{eq:Mdispersion}).
Near the resonance, at
%
$
|\nu|\ll \Delta E_\ast=\lambda^2,
$
%
neglecting $\omega$ compared to $\Sigma(\omega)$, one obtains molecular energy
quadratic in detuning:
%
\be\label{eq:Mdispersion}
\omega(k)=-(\nu/\lambda)^2+k^2/4m
.
\ee
%
At $\nu<0$, Eq.(\ref{eq:Mdispersion})
%% this relation describes
gives the energy of molecules, while at $\nu>0$
%% dressed by fermion pairs, while at $\nu>0$,
%% where the molecules are unstable,
%% the dispersion equation $G^{-1}(\omega,k)=0$ solution
%% is also given by (\ref{eq:Mdispersion}). The
it describes a resonance in the two-fermion scattering mediated by 
%% weakly unstable
virtual molecules \cite{virtual_resonance}.
%% found at \emph{negative} energy \cite{virtual_resonance}.
%% The Green's function
The residue $Z$ defines the bare molecule weight in the physical
molecule state,
$Z^{-1}(\omega)=dG^{-1}/d\omega=1+\frac{\lambda}{2}(-\tilde\omega)^{-1/2}$,
%
%%%%%%%%%%% save this eqn for long article %
%% \be\label{Z}
%% Z(\omega)=\lp\frac{dG^{-1}(\omega)}{d\omega}\rp^{-1}
%% =\frac{(-\tilde\omega)^{1/2}}{(-\tilde\omega)^{1/2}+\lambda/2}
%% \ee
%% %
%% The weight $Z(\omega)$
which varies from zero to one across the resonance, at $-\Delta E_\ast\lesssim
\nu<0$. At relatively small detuning, $|\nu/\Delta E_\ast|\ll 1$, $Z$
increases linearly: $Z(\omega)\approx 2|\nu|/\lambda^2$.


To investigate molecule formation at the resonance, we consider the Green's
function for the problem with time-dependent detuning $\nu(t)$. In this case,
due to nonlocal character of $\Sigma$ in the time domain, the molecule
evolution is described by an integral-differential equation~\cite{Duine04,Haque04}
%
\be\label{eq:int-diff}
\lp i\p_t-\nu(t)-{\textstyle \frac{k^2}{4m}}\rp b_k(t)
-\!\int\!\Sigma_k(t,t')b_k(t')dt'=\eta_k(t)
\ee
%
with $\eta_k(t)=g \int\! e^{-ikx}\psi_\uparrow(x,t)\psi_\downarrow(x,t)
d^3x$ the pairing
field,
%% depending only on time due to translational invariance, where
and $\psi_\sigma(x,t)=\sum_p a_{p,\sigma} e^{ip x-i\epsilon_p t}$. Here the self-energy is
%% operator
%
\be\label{eq:Sigma(t,t')}
\Sigma_k(t>t')=\int\! \Sigma_k(\tilde\omega)e^{-i\omega(t-t')}
\frac{d\omega}{2\pi}
=\frac{a e^{-i\frac{k^2}{4m}(t-t')}}{(t-t')^{3/2}}
%% =\frac{\lambda e^{-i\frac{k^2}{4m}\tau}}{2(i\pi)^{1/2} \tau^{3/2}}
,
%% (t-t')^{-3/2}\lambda(8\pi)^{-1/2}
\ee
%
%% ($\tau=t-t'$),
$a=\frac{\lambda}{2\sqrt{i\pi}}$, and $\Sigma(t<t')$ vanishes due to the causality.


The pairing field $\eta$, which acts as a source in
%% our problem,
Eq.(\ref{eq:int-diff}), should be taken as a c-number
for the condensed molecules
(with $k=0$), and as an operator for
the incoherent molecules. Generally,
its correlation function
%% , most generally,
includes
both the coherent and incoherent parts:
%
\be\label{eq:eta_correlator}
\la \bar\eta_{k,\omega}\eta_{k,\omega}\ra
=(2\pi)^4|\eta_0|^2\delta(\omega-\mu)\delta(k)
+K(\omega,k)
,
\ee
%
where $\eta_0$ describes a finite amplitude for two fermions
%% with opposite spins
to have opposite momenta in the paired state, with
$\mu\lesssim 2E_F$ the chemical potential of a pair,
and $K(\omega,k)=\la\!\la\bar\eta_{k,\omega}\eta_{k,\omega}\ra\!\ra$
the dynamical pair correlator which is nonzero even
for ideal Fermi gas.
We first consider the coherent molecule
production, treating both $\eta=\eta_0 e^{-i\mu t}$ and $b(t)$ as c-numbers.
%% The modifications appropriate for
The incoherent pair source
$K(\omega,k)$  will be discussed subsequently
below.

The evolution problem (\ref{eq:int-diff}) is non-elementary
due to nonlocality of $\Sigma(t,t')$.
%% in the time domain.
Our approach
%% will rely on the
employs an idea similar to that used in
the Wiener-Hopf method. We first handle
%% shall start with
an auxiliary problem in which the linear
sweep $\nu(t)=\nu_0-\alpha t$
%% takes place
extends from $-\infty$
to $\infty$, and then modify the solution to describe
the situation of interest (\ref{eq:omega_0(t)}).

The auxiliary problem in question is to find $b(t)$, $-\infty<t<\infty$, which
obeys a linear integral-differential equation
%
\be\label{eq:auxiliary}
\lp \hat\omega-\nu_0-\Sigma(\hat\omega)
%% \lambda(-\hat\omega)^{1/2}
+\alpha t\rp b(t)
=\eta(t)
\,,\quad
\hat\omega=i\p_t
\ee
%
with a source term $\eta(t)$ of a generic form. It is convenient to go to
Fourier representation, in which $t=-i\p_\omega$ and the problem is reduced to
an ordinary differential equation $\lp \omega-\Sigma(\omega)-\nu_0
-i\alpha\p_\omega\rp b(\omega)=\eta(\omega)$ for $b(\omega)=\int e^{i\omega
t}b(t)dt$. Solution of this equation, first order in $\p_\omega$, is found
using the gauge transformation
%% $ D_0(\omega)+i\alpha\p_\omega=i\alpha e^{i\phi(\omega)}\p_\omega e^{-i\phi(\omega)}$
$b(\omega)\to e^{i\phi(\omega)}b(\omega)$
with the phase $\phi$ satisfying
%
\be\label{eq:phi'=D0}
-\alpha\phi'(\omega)= D_0(\omega)\equiv \omega-\Sigma(\omega)-\nu_0
.
\ee
%
This problem is solved by the function
%
\be\label{eq:b_linear_answer}
b(\omega)=-i\alpha^{-1}e^{i\phi(\omega)}\int_{\omega}^{+\infty}e^{-i\phi(\omega')}\eta(\omega')d\omega'
. \ee
%
To verify (\ref{eq:b_linear_answer}), one can compare
its behavior to that of $b(\omega)=Ce^{i\phi(\omega)}$,
the solution to the homogeneous problem
(\ref{eq:auxiliary}).
%% $b(\omega)=Ce^{i\phi(\omega)}$.
For $\omega$ large and positive, since
$(-\omega-i0)^{1/2}=-i\sqrt{\omega}$, we obtain
the asymptotic behavior
%% of $D_0$, $\phi$ and $b$ will be
$D_0\approx i\lambda\omega^{1/2}$,
$\phi\approx -i(2\lambda/3\alpha)\omega^{3/2}$,
$e^{\pm i\phi}\propto e^{\pm a\omega^{3/2}}$ ($a=2\lambda/3\alpha$).
Thus $C=0$ is required to eliminate exponential growth.
Indeed, the asymptotic behavior
of the integral in (\ref{eq:b_linear_answer})
at large positive $\omega$ is non-exponential:
$\int_{\omega}^{+\infty}e^{-i\phi(\omega')}\eta(\omega')d\omega'
\approx e^{-i\phi(\omega)}\eta(\omega)/\phi'(\omega)$.
(For any physical source, $\eta(\omega\to\infty)$ is algebraic.)
At the same time, the behavior at large negative $\omega$
%% at $\omega\to-\infty$
does not require special attention:
%%% As for the large negative $\omega$, since
$\phi$ is real at $\omega<0$, and so the exponentials $e^{\pm i\phi}$
oscillate as  $e^{\pm ia(-\omega)^{3/2}}$
%% at large negative $\omega$,
without giving rise
to ``dangerous'' asymptotic behavior.

%%%%%%%%%%%%% save this paragraph for long article %%%%%%%%%%
%% This leads to a general solution of the form
%% %
%% \be\label{eq:b_linear_general}
%% b(\omega)=-i\alpha^{-1}e^{i\phi(\omega)}\lb C+\int_{-\infty}^{\omega}e^{-i\phi(\omega')}\eta(\omega')\frac{d\omega'}{2\pi}\rb
%% \ee
%% %
%% where $C$ is a constant to be found using initial conditions.
%% To obtain $C$, let us consider the behavior of $D_0$, $\phi$ and $b$
%% %% $b(\omega)$
%% at large $\omega$. For $\omega$ large and positive, since
%% $(-\omega-i0)^{1/2}=-i\sqrt{\omega}$, the asymptotic behavior will be
%% $D_0\approx i\lambda\omega^{1/2}$,
%% $\phi\approx -i(2\lambda/3\alpha)\omega^{3/2}$,
%% $e^{\pm i\phi}\propto e^{\pm a\omega^{3/2}}$ ($a=2\lambda/3\alpha$).
%% The function $b(\omega)$ would grow exponentially
%% at large positive $\omega$,
%% unless $C$ is chosen so that the expression in the brackets,
%% Eq.(\ref{eq:b_linear_general}),
%% vanishes at $\omega\to+\infty$.
%% Thus we must take $C$ equal $-\int_{-\infty}^{+\infty}e^{-i\phi(\omega')}d\omega'$,
%% in which case the solution (\ref{eq:b_linear_general}) takes the form
%% %
%% \be\label{eq:b_linear_answer}
%% b(\omega)=-\frac{\alpha^{-1}}{2\pi i}e^{i\phi(\omega)}\int_{\omega}^{+\infty}e^{-i\phi(\omega')}\eta(\omega')d\omega'
%% \ee
%% %
%% %% Taking into account
%% Given the asymptotic behavior of $e^{\pm i\phi}$ mentioned above,
%% it is easy to obtain asymptotic behavior
%% of the integral in (\ref{eq:b_linear_answer})
%% at large positive $\omega$:
%% $\int_{\omega}^{+\infty}e^{-i\phi(\omega')}\eta(\omega')d\omega'
%% \approx e^{-i\phi(\omega)}\eta(\omega)/\phi'(\omega)$.
%% Thus we
%% see that the function (\ref{eq:b_linear_answer})
%% behaves as $\eta(\omega)/D_0(\omega)$ at large positive $\omega$.
%% For any physical source, the behavior of $\eta(\omega\to\infty)$ is algebraic,
%% i.e. the described choice of $C$ indeed eliminates the nonphysical exponential
%% part.
%% (We note that the asymptotic behavior
%% at $\omega\to-\infty$ does not require special attention:
%% %%% As for the large negative $\omega$, since
%% $\phi$ is real at $\omega<0$, and thus the exponentials $e^{\pm i\phi}$
%% oscillate as  $e^{\pm ia(-\omega)^{3/2}}$
%% at large negative $\omega$, i.e. do not give rise
%% to ``dangerous'' asymptotic behavior.)

Now, having found the solution for the sweep spanning the entire range
$-\infty<t<+\infty$, let us consider the sweep trajectory
(\ref{eq:omega_0(t)}). In this case, it is convenient to represent
the function $b(t)$ as a sum $b_<(t)+b_>(t)$, with $b_{>,<}(t)$ nonzero
only at $t\ge 0$ ($t\le 0$), respectively,
obtained by restricting $b(t)$ on the half-line $t\ge 0$ ($t\le 0$).
Then the evolution equation, in operator form written as
$\lp \hat\omega-\nu(t)-\Sigma(\hat\omega)\rp b(t)=\eta(t)$,
can be represented as
%
\be
\hat D_0 b_<+ (\hat D_0+\alpha \hat t) b_> =
\eta_<+\eta_>
\ee
%
with $\eta_{>,<}=\theta(\pm t)\eta(t)$ having the same meaning as $b_{>,<}(t)$,
and $\hat D_0\equiv D_0(\hat \omega)$ defined in Eq.(\ref{eq:phi'=D0}).

Let us project the terms on the left hand side on the regions
$t\ge 0$, $t\le 0$, taking into account the constraints due to causality.
The integral operator $\Sigma$ acts only forward, not backward in time,
the property explicit in Eq.(\ref{eq:Sigma(t,t')}).
%% the kernel
%% $K(t-t')=\frac1{2\pi}\int e^{-i\omega(t-t')}(-\omega-i0)^{1/2}d\omega$
%% which vanishes at $t<t'$ due to analyticity of $(-\omega-i0)^{1/2}$
%% in the upper half-plane of complex $\omega$.
Thus the function $\lp D_0(\hat\omega)+\alpha \hat t\rp b_>$
is nonzero only at $t>0$, while the function
$D_0(\hat\omega) b_<$ has both the $t>0$ and the $t<0$ parts.
This observation allows to write the problem as two separate
problems for $b_{>,<}(t)$ as follows:
%
\be
\lb \hat D_0 b_<\rb_<=\eta_<
\,,\quad
( \hat D_0+\alpha \hat t) b_> +\lb \hat D_0 b_<\rb_>=\eta_>
,
\ee
%
where $[...]_{<,>}$ denotes the part of the function
at $t>0$ ($t<0$), with zero value on the opposite half-line.
Now, we can solve the first equation for $b_<$ and substitute the result in the
second equation, which (after some algebra) can be brought to the form
%
\be
( \hat D_0+\alpha \hat t) b_> = \hat D_0 \lb \hat D_0^{-1}\eta \rb_>
.
\ee
%
We note that $b_>$ and the function on the right-hand side
are both nonzero only at $t>0$.
This allows to treat this equation as Eq.(\ref{eq:auxiliary}),
formally extending the
linear time dependence $\alpha t$ to negative $t$. Using the above result,
we obtain the answer in Fourier representation
of the form (\ref{eq:b_linear_answer})
with $\eta$ replaced by
%
\[
\tilde\eta(\omega)=\hat D_0\lb \hat D_0^{-1}\eta \rb_>
=D_0(\omega)\int \frac{D_0^{-1}(\omega')\eta(\omega')}{\delta-i(\omega-\omega')}\frac{d\omega'}{2\pi}
.
\]
%
Now, let us consider the source $\eta(t)=\eta_0 e^{-i\mu t}$,
describing coherent fermion pairs with the
chemical potential $\mu/2$ per particle.
In this case, $\eta(\omega)=2\pi\eta_0 \delta(\omega-\mu)$ and
$\tilde\eta(\omega)=D_0^{-1}(\mu)\eta_0 D_0(\omega)/(\delta-i(\omega-\mu))$.
Inserting $\tilde\eta$ in Eq.(\ref{eq:b_linear_answer}),
and using the identity
$(\delta-i(\omega-\mu))^{-1}=\int_0^\infty e^{i(\omega-\mu)\tau}d\tau$,
we find a closed form representation
%
\bea\label{eq:b0+Delta b}
&& b(\omega)=\frac{A\eta_0 }{\delta-i(\omega-\mu)}+\Delta b(\omega),
%% i A\eta_0  e^{i\phi(\omega)}
\\\nonumber
%% && -iA e^{i\phi(\omega)}
&& %% \times
\Delta b(\omega)=i A\eta_0  e^{i\phi(\omega)}\int_0^\infty\!\! e^{-i\mu\tau}\tau
%%\lb
\int_{\omega}^{+\infty}\!\!\!\! e^{i\omega'\tau-i\phi(\omega')}d\omega'
%%\rb
d\tau
\eea
%
with $A=D_0^{-1}(\mu)$. (To obtain (\ref{eq:b0+Delta b}), we transformed
the integral over $\omega'$
by writing $D_0(\omega')
%% =-\alpha\phi'(\omega')
=\alpha d(\omega'\tau-\phi(\omega'))/d\omega'-\alpha\tau$
and integrating by parts.)
%
%% %
%% \be\label{eq:b_answer}
%% b(\omega)=
%% %% -\frac{\alpha^{-1}}{2\pi}
%% i\alpha^{-1}A e^{i\phi(\omega)}\int_0^\infty e^{-i\mu\tau}
%% \lp
%% \int_{\omega}^{+\infty}e^{i\omega'\tau-i\phi(\omega')}D_0(\omega')d\omega'
%% \rp
%% d\tau
%% \ee
%% %
%% with $A=D_0^{-1}(\mu)/2\pi$.
%%
%% A more convenient representation can be obtained by transforming the integral
%% over $\omega'$ via writing $D_0(\omega')=-\alpha\phi'(\omega')
%% =\alpha d(\omega'\tau-\phi(\omega'))/d\omega'-\alpha\tau$
%% and integrating by parts. The resulting expression is
%% %
%% \breakon
%% %
%% \be\label{eq:b0+Delta b}
%% b(\omega)=\frac{A}{\delta-i(\omega-\mu)}
%% -iA e^{i\phi(\omega)}\int_0^\infty e^{-i\mu\tau}\tau
%% \lp \int_{\omega}^{+\infty}e^{i\omega'\tau-i\phi(\omega')}d\omega'
%% \rp
%% d\tau
%% \ee
%% %
%% \breakoff
%% %
Since the first term of (\ref{eq:b0+Delta b}) gives the would be
$b(\omega)$ in the absence of the sweep,
%% at $\alpha=0$, the second term
$\Delta b(\omega)$
describes the effect of the sweep.

%% Up to here all transformations have been exact.
%% Now, let us recall that we are interested in
Now, let us analyze the asymptotic behavior of
$b(t)=\int e^{-i\omega t}b(\omega)d\omega/2\pi$ at large positive
$t\gg\tau_0,\nu_0/\alpha$. In this
case, the integral over $\omega$ is controlled by large negative $\omega$,
which can be seen
%% by evaluating it
with the help of the stationary phase
approximation.
%%%%%%%% $\int e^{if(x)}dx=\sqrt{2\pi i/f''}e^{if(x_\ast)}$.
Indeed, the saddle point $\omega_\ast$ of $-\omega t+\phi(\omega)$, obtained
from $\phi'=t$, at $t\to+\infty$ implies
%% large negative
$\omega\to-\infty$. With
that in mind, we obtain the asymptotic for $b(t)$ by setting the lower
integration limit
%% of integral over $\omega'$
in Eq.(\ref{eq:b0+Delta b}) at
$\omega=-\infty$, leading to the central result of this work:
%% {\bf check}
%
\bea\label{eq:Delta_b}
&&
\Delta b(t)=-\frac{A\eta_0}{2\pi i} F^\ast(t)\int_0^\infty e^{-i\mu\tau}\tau F(\tau)d\tau
,
%% \ee
%% with
%% \,,\quad
\\\label{eq:F(t)}
&&
F(t)=\int_{-\infty}^{+\infty}e^{i\omega't-i\phi(\omega')}d\omega'
.
%% =\int_{-\infty}^{+\infty}
%% \exp\lp i\omega'(t-t_0)+ia(-\omega'-i0)^{3/2}\rp d\omega'
\eea
%
%%%%%%% with $\phi(\omega)$ obtained from Eq.(\ref{eq:phi'=D0}).
%% The analytic answer
%% for molecule evolution, given in Eqs.(\ref{eq:Delta_b}),(\ref{eq:F(t)}),
%% represents the central result
%% of this work.
%
%% The phase $\omega't-\phi(\omega')$ is $\omega'(t-t_0)+a(-\omega'-i0)^{3/2}$
%% with $a=2\lambda/3\alpha$ and $t_0=\nu_0/\alpha$, the time
%% of sweep from $\nu_0$ to resonance.
%% It is convenient to nondimensionalize variables, using $w=\tau_0\omega$,
%% $x=(t-t_0)/\tau_0$, where the time $\tau_0=(\lambda/\alpha)^{2/3}$
%% describes the duration of
%% atom-molecule transformation at the transition through resonance. Then,
%% %
%% \be
%% F(t)=\tau_0^{-1}f(x)
%% \,,\quad
%% f(x)=\int_{-\infty}^{+\infty}e^{ ixw+2i(-w-i0)^{3/2}/3} dw
%% \ee
%% %
%% The function $f(x)$ can be evaluated numerically (...)
%
The qualitative behavior of $F(t)$ can be analyzed using
the stationary phase approximation.
%% seen with the help of the stationary phase approximation
%% in the integral over $\omega'$.
We obtain
$F(t)=\sqrt{-2\pi i/\phi''(\omega_\ast)}e^{i\omega_\ast\tau-i\phi(\omega_\ast)}$,
%% %
%% \be
%% F(t)=
%% \cases{
%% \sqrt{-2\pi i/\phi''(\omega_\ast)}e^{i\omega_\ast\tau-i\phi(\omega_\ast)}
%% , & $\omega_\ast<0$\cr 0, & else}
%% \ee
%% %
where the stationary phase equation for $\omega_\ast$, given by
$D_0(\omega_\ast)+\alpha t=0$, has a real solution
$\omega_\ast=-\alpha^2(t-t_0)^2/\lambda^2$
only
for $t>t_0=\nu_0/\alpha$.
%% We use the relation of
Relating the curvature $\phi''$ and the Green's function residue,
%% $Z=(D_0')^{-1}$, given by
$-\alpha\phi''=D_0'=Z^{-1}$, yields the asymptotic form
%% which allows to write the above asymptotic form as
%
\be\label{eq:Fstationary_phase}
F(t>t_0) = \lp 2\pi i\alpha Z(\omega_\ast)\rp^{1/2}
e^{ -i\frac{\alpha^2}{3\lambda^2}(t-t_0)^3}
%% e^{i\omega_\ast\tau-i\phi(\omega_\ast)}
\ee
%
with $Z(\omega_\ast)=2\alpha(t-t_0)/\lambda^2$.
(The self-energy-dominated $D_0(\omega)=-\nu_0-\Sigma(\omega)$,
appropriate for broad $s$-wave
Feshbach resonance,
was used in the
above estimates.)
%% %% \lambda(-\omega)^{1/2}$,
%% we obtain
%% %
%% \be
%% \omega_\ast=-\frac{\alpha^2(t-t_0)^2}{\lambda^2}
%% \,,\quad
%% t>t_0=\nu_0/\alpha
%% \ee
%% %
%% (assuming $\nu_0>0$). From that we find
%% %
%% \be
%% \omega_\ast t-\phi(\omega_\ast)=\frac13 \omega_\ast(t-t_0)=
%% -\frac{\alpha^2}{3\lambda^2}(t-t_0)^3
%% \,,\quad
%% \phi''(\omega_\ast)=- \frac{\lambda}{2\alpha(-\omega)^{1/2}}
%% = - \frac{\lambda^2}{2\alpha^2(t-t_0)}
%% \ee
%% %
Thus $F(t)$ grows as $(t-t_0)^{1/2}$
and oscillates at $(t-t_0)/\tau_0 \gg 1$,
decreasing exponentially at $t-t_0<0$.

To apply Eq.(\ref{eq:Delta_b}) to the experimental situation
%%  \cite{Regal04},
%% we note
we take into account that
$\mu\ll\hbar/\tau_0,\,\hbar/t_0$.
(Indeed,
$\mu\lesssim 2E_F$, with
$E_F=0.35\,{\rm \mu K}=50\,{\rm KHz}$ in Ref.\,\cite{Regal04}.)
We evaluate the integral in (\ref{eq:Delta_b}) using the
stationary phase form (\ref{eq:Fstationary_phase}):
%% for $F(t)$:
%% {\bf check}
%
\be
\Delta b(t)=
%% \frac{i\sqrt{2}}{3}\sqrt{Z(t)}\,A\eta_0\alpha^{1/2}
(2\alpha Z(t))^{1/2}A\eta_0
e^{ i\frac{\alpha^2}{3\lambda^2}(t-t_0)^3}\!\!
\lp  C_1 t_0+C_2\tau_0\rp
%% \int_0^\infty \tau F(\tau)d\tau
%% =
%% A\frac{(\alpha\tau_0)^{3/2}}{3\lambda}
%% \frac23(i\pi)^{1/2}
%% \lp  C_1 t_0+C_2\tau_0\rp
,
\ee
%
$C_1=(i/3)^{1/2}\Gamma(1/2)$, $C_2=(i/3)^{1/6}\Gamma(5/6)$.
The asymptotic number of molecules at time $t$
%% , shown in Fig.\ref{fig1},
is evaluated as
$N_m^{(0)}=|\Delta b(t)|^2$ (see Fig.\ref{fig1}).
%% , where $t$ is the measurement time.
%% (After division by $Z(t)$ this quantity is $t$-independent.)
The fast and slow sweep regimes can be identified:
at $t_0\ll\tau_0$ we obtain $N_m\propto\alpha^{-1/3}$,
while at $t_0\gg\tau_0$ we have $N_m\propto\alpha^{-1}$.

The incoherent molecule production can be studied in a similar manner. Using
%% an analogous manner, using
the operator $\eta_k$ as a source
in (\ref{eq:Sigma(t,t')}), and averaging over its dynamical
correlations (\ref{eq:eta_correlator})
we obtain the molecule momentum distribution
%% of noncondensed molecules
%
\be
N_m(k)=\sum_\omega \Big|\frac{A}{2\pi}F^\ast(t)\int\limits_0^\infty\!\! e^{-i\omega\tau}\tau F(\tau)d\tau\Big|^2 K(\omega,k)
.
\ee
%
with $A=D_0^{-1}(\omega)$.
As a function of frequency, $K$ is nonzero at $\omega\lesssim 2E_F$.
At $E_F\ll\hbar/\tau_0,\,\hbar/t_0$, the expression $|...|^2$ is $\omega$-independent,
as above.
Factoring it out, we conclude that the condensed and incoherent
molecule
production efficiencies are identical.
%% for the condensed and noncondensed
%% coherent and incoherent
%% molecules.
The molecule condensate fraction is then expressed
through the fermion pair fraction:
%
\be\label{eq:condensate_fraction}
\frac{N_m^{(0)}}{N_m^{(0)}+\sum_k N_m(k)}=
\frac{|\la\psi_\uparrow(x)\psi_\downarrow(x)\ra|^2}{\la\hat n_\uparrow(x)\hat n_\downarrow(x)\ra}
%% =
%% \frac{|\la\psi_\uparrow\psi_\downarrow\ra|^2}{
%% |\la\psi_\uparrow\psi_\downarrow\ra|^2+
%% (n/2)^2+\la\!\la\hat n_\uparrow\hat n_\downarrow\ra\!\ra}
\ee
%
$\hat n_\sigma(x)=\bar \psi_\sigma(x)\psi_\sigma(x)$.
%% It is interesting
We note that the incoherent contribution
exists even in
the absence of pairing. For ideal fermions at density $n$, we have $\sum_\omega
K(\omega,k)=\frac12 g^2n(1-u)^2(1+u/2) \theta(1-u)$, $u=k/2p_F$, which
%% directly translates into
corresponds to a broad molecule momentum distribution
with $k\le 2p_F$.

The approach presented above yields accurate results for the atom/molecule
projection in a wide range of sweep rates, fast and slow,
as long as the times $\tau_0$, $t_0$ are short on the scale of $E_F$.
The only limitation
stems from the assumption of a steady source,
%% . We use coherent $\eta_0 e^{-i\mu t}$, and incoherent
%% $K(t-t')=\int e^{-i\omega (t-t')}K(\omega)\frac{d\omega}{2\pi}$,
%% with the atom depletion effects ignored,
which describes the situation when
%% . In other words, we suppose that
the fraction of atoms converted into molecules is small.
The depletion effects, which are different for the condensed and
incoherent molecules, can be incorporated in the framework
of a quantum kinetic equation.

The above method is applicable to the $p$-wave Feshbach resonance case,
with the essential
modification in the self-energy form
$\Sigma(\omega)\propto (-\omega)^{3/2}$ \cite{Gurarie04,Chevy04}.
This self-energy is an irrelevant perturbation
near the resonance, so we have the atom/molecule conversion time
$\tau_0=\alpha^{-1/2}$ and
$Z=1$, as for weak coupling.
Thus one can use
$D_0(\omega)=\omega-\nu_0$ in (\ref{eq:phi'=D0}),
yielding the result identical to
(\ref{eq:Delta_b}) with
%
\[F(t)=\int e^{i\omega(t-t_0)+i\omega^2/2\alpha}d\omega=(2\pi i\alpha)^{1/2}
e^{-i\alpha(t-t_0)^2/2}.
\]
%
%% Unlike the $s$-channel case, t
The number of produced molecules, both condensed
and incoherent, scales as
inverse sweep rate
$\alpha^{-1}$. The production is less efficient than in the
$s$-channel case due to weaker coupling at resonance.
%% number scales as inverse sweep rate
%% $\alpha^{-1}$ for both the condensed
%% and incoherent molecules.
%% for slow sweep, and is $\alpha$-independent for fast sweep.
%% The conclusion about identical sweep dependence of the condensed
%% and noncondensed molecule numbers at fast sweep remains unchanged.

In summary, molecule production at Feshbach resonance
is considered as a many-body problem for which the exact Green's function
is obtained using Wiener-Hopf method.
The theory is applied to the $s$-wave and $p$-wave resonances.
The slow and fast sweep regimes are identified in the $s$-wave case,
controled by the adiabaticity time scale
(\ref{eq:time_scales}).
The predicted power law $1/3$ for the
molecule production, as well as the total molecule number, 
are found to be in agreement
%% excellent agreement
with observations away from saturation \cite{Regal04}.
The independence of the produced condensate fraction on the sweep rate
observed at fast sweep \cite{Regal04}
is also explained by this theory.



% %%%%%%%%%%%% save this for long article
% Turning to the integral over $\tau$ in Eq.(\ref{eq:Delta_b}), we note that
% from the structure of the phase $\omega_\ast t-\phi(\omega_\ast)$
% it is clear that the integral of $F(\tau)$ is determined by the interval
% of length $\tau_0$ centered at $\tau=t_0$.
% %% The time $\tau_0$ defines the duration of
% %% atom-molecule transformation at the transition through resonance.
% The above expression (\ref{eq:Delta_b}) holds asymptotically,
% at $t\gg\tau_0$.

% A simple estimate can be obtained for  slow sweep,
% such that $\tau_0\ll t_0$. In this case, the integral
% $\int_0^\infty...d\tau$ can be replaced by $\int_{-\infty}^\infty...d\tau$,
% which can be evaluated exactly as follows:
% %
% \bea
% &&
% \int_{-\infty}^\infty e^{-i\mu\tau}\tau F(\tau)d\tau
% =i\p_\mu\int_{-\infty}^\infty  e^{-i\mu\tau}F(\tau)d\tau
% \\
% &&
% =
% 2\pi i\p_\mu e^{-i\phi(\mu)}
% =-\frac{2\pi}{\alpha}D_0(\mu) e^{-i\phi(\mu)}
% =-(\alpha A)^{-1} e^{-i\phi(\mu)}
% \eea
% %
% Note that, since $\phi(\mu>0)$ is complex, the ``phase factor''
% $e^{-i\phi(\mu)}$ has exponentially decreasing magnitude
% at $\mu>0$.
% Combined with the stationary phase form of $F^\ast(t\gg\tau_0)$
% derived above, this gives
% %
% \be
% \Delta b(t)=
% \lp \frac{2\pi i Z(\omega_\ast)}{\alpha}\rp^{1/2}
% e^{-i\omega_\ast t+i\phi(\omega_\ast)} e^{-i\phi(\mu)}
% \ee
% %
% i.e. molecular amplitude
% growing at slow sweep as $\Delta b(t)\propto\alpha^{-1/2}$.
% The number of molecules formed by this (coherent) process
% is
% %
% \bea
% &&
% N_c=\frac{2\pi Z(\omega_\ast)}{\alpha}\exp\lp 2{\rm Im}\,\phi(\mu)\rp
% |\eta|^2
% \\
% &&
% =
% \frac{2\pi Z(\omega_\ast)}{\alpha}\exp\lp -(4\lambda/3\alpha)\mu^{3/2}\rp
% |\eta|^2
% \eea
% %
% %% \underline{$N_c=(2\pi Z(\omega_\ast)/\alpha)|\eta|^2$},
% where
% $\eta$ is the coherent pair amplitude at detuning $\nu_0$.

% (Accuracy?)


We are grateful to Dmitry Petrov for useful comments.
%% We benefited from useful discussion with Dmitry Petrov.


\vspace{-5mm}

% BEC
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%Fermi cooling
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%zero crossing of the scattering length in a Feshbach resonance

%% Experiments on creation of molecules at a Feshbach resonance
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%%Creation of ultracold molecules from a Fermi gas of atoms
%
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%%Emergence of a molecular Bose�Einstein condensate from a Fermi gas
%
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%
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%%Conversion of an Atomic Fermi Gas to a Long-Lived Molecular Bose Gas


% \bibitem{Bartenstein05} M. Bartenstein {\it et al.}, Phys. Rev. Lett. {\bf 94}, 103201 (2005)
%Precise Determination of 6Li Cold Collision Parameters by Radio-Frequency Spectroscopy
%on Weakly Bound Molecules

%%%%%%Theory, Feshbach resonance and pairing, two channel Hamiltonian

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% Fast projection experiments

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%Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance



%%%%% Adiabatic conversion of atoms into molecules

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%time-dependent magnetic fields

\bibitem{Goral04} K. G\'{o}ral, T. K\"{o}hler, S. A. Gardiner, E. Tiesinga,
and P. S. Julienne, J. Phys. B {\bf 37}, 3457 (2004).

%Atom/molecular oscillations on the molecular side of the FR
%Experiment
% \bibitem{Donley02} E. A. Donley, N. R. Claussen, S.T. Thompson, and
% C. E. Wieman, Nature (London) {\bf 417}, 529 (2002)

\bibitem{Hodby04} E. Hodby, \emph{et al.},
% S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S. Jin,
% E. A. Cornell, and C. E. Wieman,
%cond-mat/0411487;
Phys. Rev. Lett. {\bf 94}, 120402 (2005).
%Production Efficiency of Ultracold Feshbach Molecules in Bosonic and Fermionic Systems


%Theory
\bibitem{Kokkelmans02} S. J. J. M. F. Kokkelmans and M. J. Holland,
Phys. Rev. Lett. {\bf 89}, 180401 (2002).
%Ramsey Fringes in a Bose-Einstein Condensate between Atoms and Molecules

\bibitem{Kohler03} T. K\"{o}hler, T. Gasenzer, and K. Burnett,
Phys. Rev. A {\bf 67}, 013601 (2003).
%Microscopic theory of atom-molecule oscillations in a Bose-Einstein condensate

\bibitem{Yurovsky04}V. A. Yurovsky and A. Ben-Reuven, Phys. Rev. A
{\bf 70}, 013613 (2004).
%Formation of molecules in an expanding Bose-Einstein condensate

\bibitem{Javanainen04} J. Javanainen, \emph{et al.},
% M. Kostrun, Yi Zheng, A. Carmichael,
% U. Shrestha, P. J. Meinel, M. Mackie, O. Dannenberg, and
% K.-A. Suominen,
Phys. Rev. Lett. {\bf 92}, 200402 (2004).
%Collective Molecule Formation in a Degenerate Fermi Gas via a Feshbach Resonance

\bibitem{Dicke_FR} R. A. Barankov and L. S. Levitov,
Phys. Rev. Lett. {\bf 93}, 130403 (2004).
%Atom-molecule coexistence and collective dynamics near a FR

\bibitem{Goral05} K. G\'{o}ral, T. K\"{o}hler, and K. Burnett,
Phys. Rev. A {\bf 71}, 023603 (2005).
%Ramsey interferometry with atoms and molecules: Two-body versus
%many-body phenomena

\bibitem{Haque04}M. Haque and H.T.C. Stoof, cond-mat/0410648
%Sweeping a molecular Bose-Einstein condensate across a Feshbach resonance

\bibitem{Duine04}R. A. Duine and H. T. C. Stoof, Phys. Rep. {\bf 396}, 115 (2004).


%\bibitem{Holland01_bec}M. Holland, J. Park, and R. Walser,
%Phys. Rev. Lett. {\bf 86}, 1915 (2001)
%%Formation of Pairing Fields in Resonantly Coupled Atomic and
%%Molecular Bose-Einstein Condensates


%Experiment supporting Landau-Zener approach
%% \bibitem{Hodby04}E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson,
%% D. S. Jin, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. {\bf 94}, 120402 (2005)
%Production Efficiency of Ultracold Feshbach Molecules in
%Bosonic and Fermionic Systems


%Theory of projection experiments

\bibitem{Diener04} R. B. Diener and Tin-Lun Ho, cond-mat/0404517
%Projecting Fermion Pair Condensates into Molecular Condensates

\bibitem{Avdeenkov04} A. V. Avdeenkov and J. L. Bohn, cond-mat/0404653
%Pair Wave Functions in Atomic Fermi Condensates

\bibitem{Perali05} A. Perali, P. Pieri, and G.C. Strinati, cond-mat/0501631
%Extracting the condensate density from projection experiments with Fermi gases

\bibitem{Altman05} E. Altman and A. Vishwanath, cond-mat/0501683
%Dynamic projection on Feshbach molecules: a probe of pairing and phase fluctuations

\bibitem{Ticknor04} C. Ticknor, \emph{et al.},
% C. A. Regal, D. S. Jin, and J. L. Bohn,
Phys. Rev. A {\bf 69}, 042712 (2004).
%Multiplet structure of Feshbach resonances in nonzero partial waves

\bibitem{Zhang04} J. Zhang, \emph{et al.},
% E. G. M. van Kempen, T. Bourdel, L.
% Khaykovich, J. Cubizolles, F. Chevy,
% M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon,
Phys. Rev A {\bf 70}, 030702(R) (2004).
%P-wave Feshbach resonances of ultracold 6Li

\bibitem{Schunck05} C. H. Schunck, \emph{et al.},
% M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, and W. Ketterle,
Phys. Rev. A {\bf 71}, 045601 (2005).
%Feshbach resonances in fermionic 6Li

%END Experiment on p-wave Feshbach resonance

%Theory on p-wave Feshbach resonance

\bibitem{Gurarie04} V. Gurarie, L. Radzihovsky, and A. V. Andreev,
cond-mat/0410620
%Quantum phase transitions across a p-wave Feshbach resonance

\bibitem{Chevy04} F.\,Chevy, \emph{et al.},
% E.\,G.\,M. van Kempen, T. Bourdel, J. Zhang,
% L. Khaykovich,
% M. Teichmann, L. Tarruell, S.\,J.\,J.\,M.\,F. Kokkelmans, and C. Salomon,
cond-mat/0412393
%Resonant scattering properties close to a p-wave Feshbach resonance
%END Theory on p-wave Feshbach resonance

% Molecular Green's function at a FR
\bibitem{Bruun04}
G. M. Bruun and C. J. Pethick, Phys. Rev. Lett. {\bf 92}, 140404 (2004).
%Effective Theory of Feshbach Resonances and Many-Body Properties of Fermi Gases
%Experiment on p-wave Feshbach resonance



\bibitem{virtual_resonance}
L.\,D. Landau and E.\,M. Lifshitz,
%% virtual resonance
Quantum Mechanics, \S 133 (Pergamon, New York, 1987).

%\bibitem{Szymanska04}M. H. Szyma\'nska, B. D. Simons, and K. Burnett, arXiv:cond-mat/0412454
%%Dynamics of the BCS-BEC crossover in a degenerate Fermi gas


%\bibitem{Gorkov61}L.P. Gor'kov and T.K. Melik-Barkhudarov,
%Zh. Eksp. Teor. Fiz. {\bf 40}, 1452, (1961) [Sov. Phys. JETP, {\bf 13},  1018
%(1961)]


\end{thebibliography}

%\end{multicols}

\end{document}



\vspace{-5mm}

% BEC
\begin{thebibliography}{10}
%Fermi cooling
\bibitem{fermion_expts}
%% \bibitem{DeMarco991}
B. DeMarco, {\it et al.},
%% B. DeMarco, J. L. Bohn, J. P. Burke, Jr., M. Holland, and D. S. Jin,
Phys. Rev. Lett. {\bf 82}, 4208 (1999); %scattering length
%% \bibitem{DeMarco992}
%% B. DeMarco and D. S. Jin,
Science {\bf 285}, 1703 (1999); %onset of degeneracy
%% \bibitem{DeMarco01}
%% B. DeMarco, S. B. Papp, and D. S. Jin,
Phys. Rev. Lett. {\bf 86}, 5409 (2001);
%Pauli blocking in deg. Fermi gas
%% \bibitem{Gensemer01}
%% S. D. Gensemer and D. S. Jin,  Phys. Rev. Lett. {\bf 87}, 173201 (2001)
%switch from colissionless to hydrodyn. regime
%% \bibitem{DeMarco02}
%% B. DeMarco and D. S. Jin,
Phys. Rev. Lett. {\bf 88}, 040405 (2002);
%spin excitations in Fermi gas
%% \bibitem{Truscott01}
A. G. Truscott {\it et al.}, Science {\bf 291}, 2570 (2001);
%Fermi pressure, degeneracy
%% \bibitem{Loftus02}
T. Loftus {\it et al.}, Phys. Rev. Lett. {\bf 88}, 173201 (2002);
%Feshbach resonance and tuning of the scattering length
%% \bibitem{Ohara02}
K. M. O'Hara, {\it et al.},
%% S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas,
Science {\bf 298}, 2179 (2002);
%degenerate strongly inter. Fermi gas
%% \bibitem{Ohara022}
%% K. M. O�Hara, S. L. Hemmer, S. R. Granade, M. E. Gehm, and J. E. Thomas,
Phys. Rev. A {\bf 66}, 041401 (2002)
%zero crossing of the scattering length in a Feshbach resonance

%% Experiments on creation of molecules at a Feshbach resonance
%\bibitem{Regal03}C. A. Regal, C. Ticknor, J. L. Bohn and D. S. Jin, Nature, {\bf 424}, 47, (2003)
%%Creation of ultracold molecules from a Fermi gas of atoms
%
%\bibitem{Greiner03}M. Greiner, C. A. Regal and D. Jin, Nature {\bf 426}, 537 (2003)
%%Emergence of a molecular Bose�Einstein condensate from a Fermi gas
%
%\bibitem{Zwierlein03}M.W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle,
%Phys. Rev. Lett. {\bf 91}, 250401 (2003)
%%Observation of Bose-Einstein Condensation of Molecules
%
%\bibitem{Strecker03}K. E. Strecker, G. B. Partridge, and R. G. Hulet, Phys. Rev. Lett. {\bf 91}, 080406(2003)
%%Conversion of an Atomic Fermi Gas to a Long-Lived Molecular Bose Gas

\bibitem{Hodby04} E. Hodby, \emph{et al.},
% S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S. Jin,
% E. A. Cornell, and C. E. Wieman, cond-mat/0411487;
Phys. Rev. Lett. {\bf 94}, 120402 (2005)
%Production Efficiency of Ultracold Feshbach Molecules in Bosonic and Fermionic Systems

% \bibitem{Bartenstein05} M. Bartenstein {\it et al.}, Phys. Rev. Lett. {\bf 94}, 103201 (2005)
%Precise Determination of 6Li Cold Collision Parameters by Radio-Frequency Spectroscopy
%on Weakly Bound Molecules

%%%%%%Theory, Feshbach resonance and pairing, two channel Hamiltonian

\bibitem{Timmermans01} E. Timmermans, K. Furuya, P. W. Milonni, and A. K.
Kerman, Phys. Lett. A {\bf 285}, 228 (2001)
%Prospect of creating a composite Fermi-Bose superfluid

\bibitem{Holland01}M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser, Phys.
Rev. Lett. {\bf 87}, 120406 (2001)
%Resonance Superfluidity in a Quantum Degenerate Fermi Gas

% Theory of Feshbach resonance, a nice review
\bibitem{Timmermans99} E. Timmermans, P. Tommasini, M. Hussein,
and A. Kerman, Physics Reports {\bf 315}, 199 (1999)
%Feshbach resonances in atomic Bose-Einstein condensates

\bibitem{Griffin02} Y. Ohashi and A. Griffin, Phys. Rev. Lett. {\bf 89}, 130402 (2002)
%BCS-BEC crossover in a gas of Fermi atoms with a Feshbach resonance

%%%%% Adiabatic conversion of atoms into molecules

%Theory, Landau-Zener approach
\bibitem{Mies00} F.H. Mies, E. Tiesinga, and P. S. Julienne,
Phys. Rev. A {\bf 61}, 022721 (2000)
%Manipulation of Feshbach resonances in ultracold atomic collisions using
%time-dependent magnetic fields

\bibitem{Goral04} K. G\'oral, T. K\"ohler, S. A. Gardiner, E. Tiesinga
and P. S. Julienne, J. Phys. B {\bf 37}, 3457 (2004).

%Atom/molecular oscillations on the molecular side of the FR
%Experiment
% \bibitem{Donley02} E. A. Donley, N. R. Claussen, S.T. Thompson, and
% C. E. Wieman, Nature (London) {\bf 417}, 529 (2002)

%Theory
\bibitem{Kokkelmans02} S. J. J. M. F. Kokkelmans and M. J. Holland,
Phys. Rev. Lett. {\bf 89}, 180401 (2002)
%Ramsey Fringes in a Bose-Einstein Condensate between Atoms and Molecules

\bibitem{Kohler03} T. K\"{o}hler, T. Gasenzer, and K. Burnett,
Phys. Rev. A {\bf 67}, 013601 (2003)
%Microscopic theory of atom-molecule oscillations in a Bose-Einstein condensate

\bibitem{Javanainen04} J. Javanainen, \emph{et al.},
% M. Kostrun, Yi Zheng, A. Carmichael,
% U. Shrestha, P. J. Meinel, M. Mackie, O. Dannenberg, and
% K.-A. Suominen,
Phys. Rev. Lett. {\bf 92}, 200402 (2004)
%Collective Molecule Formation in a Degenerate Fermi Gas via a Feshbach Resonance

\bibitem{Dicke_FR} R. A. Barankov and L. S. Levitov,
Phys. Rev. Lett. {\bf 93}, 130403 (2004)
%Atom-molecule coexistence and collective dynamics near a FR

\bibitem{Goral05} K. G\'{o}ral, T. K\"{o}hler, and K. Burnett,
Phys. Rev. A {\bf 71}, 023603 (2005)
%Ramsey interferometry with atoms and molecules: Two-body versus
%many-body phenomena

\bibitem{Haque04}M. Haque and H.T.C. Stoof, cond-mat/0410648
%Sweeping a molecular Bose-Einstein condensate across a Feshbach resonance

\bibitem{Duine04}R. A. Duine and H. T. C. Stoof, Phys. Rep. 396, 115 (2004)


%\bibitem{Holland01_bec}M. Holland, J. Park, and R. Walser,
%Phys. Rev. Lett. {\bf 86}, 1915 (2001)
%%Formation of Pairing Fields in Resonantly Coupled Atomic and
%%Molecular Bose-Einstein Condensates


%Experiment supporting Landau-Zener approach
%% \bibitem{Hodby04}E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson,
%% D. S. Jin, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. {\bf 94}, 120402 (2005)
%Production Efficiency of Ultracold Feshbach Molecules in
%Bosonic and Fermionic Systems

% Fast projection experiments

\bibitem{Regal04}C. A. Regal, M. Greiner, and D. S. Jin,
Phys. Rev. Lett. {\bf 92}, 040403 (2004)
%Condensation of Pairs of Fermionic atoms near a Feshbach resonance

\bibitem{Zwierlein04}M. W. Zwierlein, \emph{et al.},
% C. A. Stan, C. H. Schunck,
% S. M. F. Raupach, A. J. Kerman, and W. Ketterle,
Phys. Rev. Lett. {\bf 92}, 120403 (2004)
%Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance

%Theory of projection experiments

\bibitem{Diener04} R. B. Diener and Tin-Lun Ho, cond-mat/0404517
%Projecting Fermion Pair Condensates into Molecular Condensates

\bibitem{Avdeenkov04} A. V. Avdeenkov, J. L. Bohn, cond-mat/0404653
%Pair Wave Functions in Atomic Fermi Condensates

\bibitem{Perali05} A. Perali, P. Pieri, and G.C. Strinati, cond-mat/0501631
%Extracting the condensate density from projection experiments with Fermi gases

\bibitem{Altman05} E. Altman and A. Vishwanath, cond-mat/0501683
%Dynamic projection on Feshbach molecules: a probe of pairing and phase fluctuations


% Molecular Green's function at a FR
\bibitem{Bruun04}
G. M. Bruun and C. J. Pethick, Phys. Rev. Lett. {\bf 92}, 140404 (2004)
%Effective Theory of Feshbach Resonances and Many-Body Properties of Fermi Gases
%Experiment on p-wave Feshbach resonance

\bibitem{Ticknor04} C. Ticknor, \emph{et al.},
% C. A. Regal, D. S. Jin, and J. L. Bohn,
Phys. Rev. A {\bf 69}, 042712 (2004)
%Multiplet structure of Feshbach resonances in nonzero partial waves

\bibitem{Zhang04} J. Zhang, \emph{et al.},
% E. G. M. van Kempen, T. Bourdel, L.
% Khaykovich, J. Cubizolles, F. Chevy,
% M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon,
Phys. Rev A {\bf 70}, 030702(R) (2004)
%P-wave Feshbach resonances of ultracold 6Li

\bibitem{Schunck05} C. H. Schunck, \emph{et al.},
% M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, and W. Ketterle,
Phys. Rev. A {\bf 71}, 045601 (2005)
%Feshbach resonances in fermionic 6Li

%END Experiment on p-wave Feshbach resonance

%Theory on p-wave Feshbach resonance

\bibitem{Gurarie04} V. Gurarie, L. Radzihovsky, and A. V. Andreev,
cond-mat/0410620
%Quantum phase transitions across a p-wave Feshbach resonance

\bibitem{Chevy04} F.\,Chevy, \emph{et al.},
% E.\,G.\,M. van Kempen, T. Bourdel, J. Zhang,
% L. Khaykovich,
% M. Teichmann, L. Tarruell, S.\,J.\,J.\,M.\,F. Kokkelmans, and C. Salomon,
cond-mat/0412393
%Resonant scattering properties close to a p-wave Feshbach resonance


%END Theory on p-wave Feshbach resonance

\bibitem{virtual_resonance}
L.\,D. Landau and E.\,M. Lifshitz,
%% virtual resonance
Quantum Mechanics, \S 133 (Pergamon, New York, 1987)

%\bibitem{Szymanska04}M. H. Szyma\'nska, B. D. Simons, and K. Burnett, arXiv:cond-mat/0412454
%%Dynamics of the BCS-BEC crossover in a degenerate Fermi gas


%\bibitem{Gorkov61}L.P. Gor'kov and T.K. Melik-Barkhudarov,
%Zh. Eksp. Teor. Fiz. {\bf 40}, 1452, (1961) [Sov. Phys. JETP, {\bf 13},  1018
%(1961)]


\end{thebibliography}

%\end{multicols}

\end{document}

cond-mat0506323.extracted_bib
\begin{thebibliography}{10}
%Fermi cooling
\bibitem{fermion_expts}
%% \bibitem{DeMarco991}
B. DeMarco, {\it et al.},
%% B. DeMarco, J. L. Bohn, J. P. Burke, Jr., M. Holland, and D. S. Jin,
Phys. Rev. Lett. {\bf 82}, 4208 (1999); %scattering length
%% \bibitem{DeMarco992}
%% B. DeMarco and D. S. Jin,
Science {\bf 285}, 1703 (1999); %onset of degeneracy
%% \bibitem{DeMarco01}
%% B. DeMarco, S. B. Papp, and D. S. Jin,
Phys. Rev. Lett. {\bf 86}, 5409 (2001);
%Pauli blocking in deg. Fermi gas
%% \bibitem{Gensemer01}
%% S. D. Gensemer and D. S. Jin,  Phys. Rev. Lett. {\bf 87}, 173201 (2001)
%switch from colissionless to hydrodyn. regime
%% \bibitem{DeMarco02}
%% B. DeMarco and D. S. Jin,
Phys. Rev. Lett. {\bf 88}, 040405 (2002);
%spin excitations in Fermi gas
%% \bibitem{Truscott01}
A. G. Truscott {\it et al.}, Science {\bf 291}, 2570 (2001);
%Fermi pressure, degeneracy
%% \bibitem{Loftus02}
T. Loftus {\it et al.}, Phys. Rev. Lett. {\bf 88}, 173201 (2002);
%Feshbach resonance and tuning of the scattering length
%% \bibitem{Ohara02}
K. M. O'Hara, {\it et al.},
%% S. L. Hemmer, M. E. Gehm, S. R. Granade, and J. E. Thomas,
Science {\bf 298}, 2179 (2002);
%degenerate strongly inter. Fermi gas
%% \bibitem{Ohara022}
%% K. M. O�Hara, S. L. Hemmer, S. R. Granade, M. E. Gehm, and J. E. Thomas,
Phys. Rev. A {\bf 66}, 041401 (2002).
%zero crossing of the scattering length in a Feshbach resonance

%% Experiments on creation of molecules at a Feshbach resonance
%\bibitem{Regal03}C. A. Regal, C. Ticknor, J. L. Bohn and D. S. Jin, Nature, {\bf 424}, 47, (2003)
%%Creation of ultracold molecules from a Fermi gas of atoms
%
%\bibitem{Greiner03}M. Greiner, C. A. Regal and D. Jin, Nature {\bf 426}, 537 (2003)
%%Emergence of a molecular Bose�Einstein condensate from a Fermi gas
%
%\bibitem{Zwierlein03}M.W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle,
%Phys. Rev. Lett. {\bf 91}, 250401 (2003)
%%Observation of Bose-Einstein Condensation of Molecules
%
%\bibitem{Strecker03}K. E. Strecker, G. B. Partridge, and R. G. Hulet, Phys. Rev. Lett. {\bf 91}, 080406(2003)
%%Conversion of an Atomic Fermi Gas to a Long-Lived Molecular Bose Gas


% \bibitem{Bartenstein05} M. Bartenstein {\it et al.}, Phys. Rev. Lett. {\bf 94}, 103201 (2005)
%Precise Determination of 6Li Cold Collision Parameters by Radio-Frequency Spectroscopy
%on Weakly Bound Molecules

%%%%%%Theory, Feshbach resonance and pairing, two channel Hamiltonian

\bibitem{Timmermans01} E. Timmermans, K. Furuya, P. W. Milonni, and A. K.
Kerman, Phys. Lett. A {\bf 285}, 228 (2001).
%Prospect of creating a composite Fermi-Bose superfluid

\bibitem{Holland01}M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser, Phys.
Rev. Lett. {\bf 87}, 120406 (2001).
%Resonance Superfluidity in a Quantum Degenerate Fermi Gas

% Theory of Feshbach resonance, a nice review
\bibitem{Timmermans99} E. Timmermans, P. Tommasini, M. Hussein,
and A. Kerman, Physics Reports {\bf 315}, 199 (1999).
%Feshbach resonances in atomic Bose-Einstein condensates

\bibitem{Griffin02} Y. Ohashi and A. Griffin, Phys. Rev. Lett. {\bf 89}, 130402 (2002).
%BCS-BEC crossover in a gas of Fermi atoms with a Feshbach resonance

% Fast projection experiments

\bibitem{Regal04}C. A. Regal, M. Greiner, and D. S. Jin,
Phys. Rev. Lett. {\bf 92}, 040403 (2004).
%Condensation of Pairs of Fermionic atoms near a Feshbach resonance

\bibitem{Zwierlein04}M. W. Zwierlein, \emph{et al.},
% C. A. Stan, C. H. Schunck,
% S. M. F. Raupach, A. J. Kerman, and W. Ketterle,
Phys. Rev. Lett. {\bf 92}, 120403 (2004).
%Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance



%%%%% Adiabatic conversion of atoms into molecules

%Theory, Landau-Zener approach
\bibitem{Mies00} F.H. Mies, E. Tiesinga, and P. S. Julienne,
Phys. Rev. A {\bf 61}, 022721 (2000).
%Manipulation of Feshbach resonances in ultracold atomic collisions using
%time-dependent magnetic fields

\bibitem{Goral04} K. G\'{o}ral, T. K\"{o}hler, S. A. Gardiner, E. Tiesinga,
and P. S. Julienne, J. Phys. B {\bf 37}, 3457 (2004).

%Atom/molecular oscillations on the molecular side of the FR
%Experiment
% \bibitem{Donley02} E. A. Donley, N. R. Claussen, S.T. Thompson, and
% C. E. Wieman, Nature (London) {\bf 417}, 529 (2002)

\bibitem{Hodby04} E. Hodby, \emph{et al.},
% S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S. Jin,
% E. A. Cornell, and C. E. Wieman,
%cond-mat/0411487;
Phys. Rev. Lett. {\bf 94}, 120402 (2005).
%Production Efficiency of Ultracold Feshbach Molecules in Bosonic and Fermionic Systems


%Theory
\bibitem{Kokkelmans02} S. J. J. M. F. Kokkelmans and M. J. Holland,
Phys. Rev. Lett. {\bf 89}, 180401 (2002).
%Ramsey Fringes in a Bose-Einstein Condensate between Atoms and Molecules

\bibitem{Kohler03} T. K\"{o}hler, T. Gasenzer, and K. Burnett,
Phys. Rev. A {\bf 67}, 013601 (2003).
%Microscopic theory of atom-molecule oscillations in a Bose-Einstein condensate

\bibitem{Yurovsky04}V. A. Yurovsky and A. Ben-Reuven, Phys. Rev. A
{\bf 70}, 013613 (2004).
%Formation of molecules in an expanding Bose-Einstein condensate

\bibitem{Javanainen04} J. Javanainen, \emph{et al.},
% M. Kostrun, Yi Zheng, A. Carmichael,
% U. Shrestha, P. J. Meinel, M. Mackie, O. Dannenberg, and
% K.-A. Suominen,
Phys. Rev. Lett. {\bf 92}, 200402 (2004).
%Collective Molecule Formation in a Degenerate Fermi Gas via a Feshbach Resonance

\bibitem{Dicke_FR} R. A. Barankov and L. S. Levitov,
Phys. Rev. Lett. {\bf 93}, 130403 (2004).
%Atom-molecule coexistence and collective dynamics near a FR

\bibitem{Goral05} K. G\'{o}ral, T. K\"{o}hler, and K. Burnett,
Phys. Rev. A {\bf 71}, 023603 (2005).
%Ramsey interferometry with atoms and molecules: Two-body versus
%many-body phenomena

\bibitem{Haque04}M. Haque and H.T.C. Stoof, cond-mat/0410648
%Sweeping a molecular Bose-Einstein condensate across a Feshbach resonance

\bibitem{Duine04}R. A. Duine and H. T. C. Stoof, Phys. Rep. {\bf 396}, 115 (2004).


%\bibitem{Holland01_bec}M. Holland, J. Park, and R. Walser,
%Phys. Rev. Lett. {\bf 86}, 1915 (2001)
%%Formation of Pairing Fields in Resonantly Coupled Atomic and
%%Molecular Bose-Einstein Condensates


%Experiment supporting Landau-Zener approach
%% \bibitem{Hodby04}E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson,
%% D. S. Jin, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. {\bf 94}, 120402 (2005)
%Production Efficiency of Ultracold Feshbach Molecules in
%Bosonic and Fermionic Systems


%Theory of projection experiments

\bibitem{Diener04} R. B. Diener and Tin-Lun Ho, cond-mat/0404517
%Projecting Fermion Pair Condensates into Molecular Condensates

\bibitem{Avdeenkov04} A. V. Avdeenkov and J. L. Bohn, cond-mat/0404653
%Pair Wave Functions in Atomic Fermi Condensates

\bibitem{Perali05} A. Perali, P. Pieri, and G.C. Strinati, cond-mat/0501631
%Extracting the condensate density from projection experiments with Fermi gases

\bibitem{Altman05} E. Altman and A. Vishwanath, cond-mat/0501683
%Dynamic projection on Feshbach molecules: a probe of pairing and phase fluctuations

\bibitem{Ticknor04} C. Ticknor, \emph{et al.},
% C. A. Regal, D. S. Jin, and J. L. Bohn,
Phys. Rev. A {\bf 69}, 042712 (2004).
%Multiplet structure of Feshbach resonances in nonzero partial waves

\bibitem{Zhang04} J. Zhang, \emph{et al.},
% E. G. M. van Kempen, T. Bourdel, L.
% Khaykovich, J. Cubizolles, F. Chevy,
% M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon,
Phys. Rev A {\bf 70}, 030702(R) (2004).
%P-wave Feshbach resonances of ultracold 6Li

\bibitem{Schunck05} C. H. Schunck, \emph{et al.},
% M. W. Zwierlein, C. A. Stan, S. M. F. Raupach, and W. Ketterle,
Phys. Rev. A {\bf 71}, 045601 (2005).
%Feshbach resonances in fermionic 6Li

%END Experiment on p-wave Feshbach resonance

%Theory on p-wave Feshbach resonance

\bibitem{Gurarie04} V. Gurarie, L. Radzihovsky, and A. V. Andreev,
cond-mat/0410620
%Quantum phase transitions across a p-wave Feshbach resonance

\bibitem{Chevy04} F.\,Chevy, \emph{et al.},
% E.\,G.\,M. van Kempen, T. Bourdel, J. Zhang,
% L. Khaykovich,
% M. Teichmann, L. Tarruell, S.\,J.\,J.\,M.\,F. Kokkelmans, and C. Salomon,
cond-mat/0412393
%Resonant scattering properties close to a p-wave Feshbach resonance
%END Theory on p-wave Feshbach resonance

% Molecular Green's function at a FR
\bibitem{Bruun04}
G. M. Bruun and C. J. Pethick, Phys. Rev. Lett. {\bf 92}, 140404 (2004).
%Effective Theory of Feshbach Resonances and Many-Body Properties of Fermi Gases
%Experiment on p-wave Feshbach resonance



\bibitem{virtual_resonance}
L.\,D. Landau and E.\,M. Lifshitz,
%% virtual resonance
Quantum Mechanics, \S 133 (Pergamon, New York, 1987).

%\bibitem{Szymanska04}M. H. Szyma\'nska, B. D. Simons, and K. Burnett, arXiv:cond-mat/0412454
%%Dynamics of the BCS-BEC crossover in a degenerate Fermi gas


%\bibitem{Gorkov61}L.P. Gor'kov and T.K. Melik-Barkhudarov,
%Zh. Eksp. Teor. Fiz. {\bf 40}, 1452, (1961) [Sov. Phys. JETP, {\bf 13},  1018
%(1961)]


\end{thebibliography}