View source for BEC-BCS Crossover

== BCS superfluidity ==
Superfluidity of boson was first discovered in <math>^4 He</math> system at a critical temperature of <math>T_C \sim 2.2K</math>. This was connected to the formation of <math>^4 He</math> condensates. Superfluidity of fermions, the electrons, was first discovered in Mecury at a transition temperature <math>T_C \sim 4.2K</math>, which is known as the `superconductivity' of metals. \\
In the early age, there are two major confusions about the fermionic superfluidity

* what is the mechanism for superfluidity of fermions (electrons)?
**It is intuitive to suggest that two electrons could form tightly bounded pairs (Schafroth pairs) and then form condensates. However, there was no known interaction which is strong enough to overcome the ''Coulomb'' repulsion.
* why does it happen at such low temperature compared with <math>T_F </math> (typically <math>\sim 10^4 K</math> in metal)?
**For bosonic case in <math>^4 He</math>, we can estimate the transition temperature <math>T_{BEC} </math> (assuming phase space density 1 and typical Helium density) to be <math>T_{BEC} \sim 3K</math> which is consistent with the experimental findings. However, the fermi temperature in a fermionic system in Mercury is much higher (10^4) than the observed superfluidity transition temperature. 

The two puzzles remain unresolved until 1956 when Bardeen, Cooper and Schrieffer proposed the BCS theory. In short:
* It is correct to think of fermion (electron) pairs. However, instead of the tightly bound pairs, the pair here is the loosely bound BCS pair of electrons formed due to the effective attractive interaction mediated by the hosting lattice.
* The temperature scale is the ''Debye temperature'' because of the involvement of the hosting lattice in the pairing mechanism. This temperature is further modified by the pairing energy and the density of states on the Fermi sea.

== Cooper instability in a Fermi gas with attractive interactions ==
In contrast to bosons, the non-interacting Fermi gas does not show any phase transition down to zero temperature. One might assume that this qualitative fact should not change as interactions are introduced, at least as long as they are weak. This is essentially true in the case of repulsive interactions. For attractive interactions, the situation is, however, dramatically different. Even for very weak attraction, the fermions form pairs and become superfluid, due to a generalized form of pair condensation.

The idea of pairing might be natural, as tightly bound pairs of fermions can be regarded as point-like bosons, which should form a Bose-Einstein condensate. However, for weak attractive interaction -- as is the case for the residual, phonon-induced electron-electron interaction in metals -- it is not evident that a paired state exists. Indeed, we will see in the following that in three dimensions there is no bound state for two isolated particles and arbitrarily weak interaction. However, by discussing exact solutions in 1D and 2D, where bound states exist for weak interactions, we gain insight into how a modified density of states will lead to bound states even in 3D --  this is the famous Cooper instability.

=== Two-body bound states in 1D, 2D and 3D ===
Localizing a quantum-mechanical particle of mass <math>\mu = m/2</math> to a certain range <math>R</math> leads to an increased momentum uncertainty of <math>p \sim \hbar/R</math> at a kinetic energy cost of about <math>E_{R} = p^2/m = \hbar^2 / mR^2</math>. Clearly, a shallow potential well of size <math>R</math> and depth <math>V</math> with <math>V/E_R \equiv \epsilon \ll 1</math> cannot confine the particle within its borders. But we can search for a bound state at energy <math>|E_B| \ll E_R</math> of much larger size <math>r_B = 1/\kappa \equiv\sqrt{\hbar^2/m |E_B|}  \gg R</math>.
* In 1D and 2D, there is always a bound state for an arbitrarily weak attractive interaction.
* In 3D: For a spherically symmetric well, the Schr\"odinger equation for the wave function transforms into an effective one-dimensional problem for the wave function <math>u = r \psi</math> c). We might now be tempted to think that there must always be a bound state in 3D, asthe  case in 1D. However, the boundary condition on <math>u(r)</math> is now to vanish linearly at <math>r=0</math>, in order for <math>\psi(0)</math> to be finite. Outside of the potential well, we still have <math>u \propto e^{-\kappa r}</math> for a bound state. Inside the well the wave function must fall off to zero at $r=0$ and necessarily has to change its slope from <math>-\kappa</math> outside to <math>\sim 1/R</math> inside the well over a distance <math>R</math>. This costs the large kinetic energy <math>\sim\hbar^2 u''/2m u \approx \hbar^2 /m R^2 = E_R</math>. If the well depth is smaller than a critical depth <math>V_c</math> on the order of <math>E_R</math>, the particle cannot be bound. 

The results hold for quite general shapes <math>V(r)</math> of the (purely attractive) potential well.

Applying these results to the equivalent problem of two interacting particles colliding in their center-of-mass frame, we see that in 1D and 2D, two isolated particles can bind for an arbitrarily weak purely attractive interaction. Hence in 1D and 2D, pairing of fermions can be understood already at the two-particle level. Indeed, one can show that the existence of a two-body bound state for isolated particles in 2D is a necessary and sufficient condition for the instability of the many-body Fermi sea (Cooper instability, see below)~\cite{rand89bound}. In 3D, however, there is a threshold interaction below which two isolated particles are unbound. We conclude that if pairing and condensation occur for arbitrarily weak interactions in 3D, then this must entirely be due to many-body effects.

=== Density of states ===
We show that the "density of states" in the different dimensions as the decisive factor for the existence of bound states.

Searching for a shallow bound state of energy <math>E = -\frac{\hbar^2\kappa^2}{m}</math> (<math>m/2</math> is the reduced mass), we start by writing the Schrodinger equation for the relative wave function momentum space (<math>n</math>-dimensional):
:<math>
    \psi_\kappa(\vec{q}) = - \frac{m}{\hbar^2}\frac{1}{q^2 + \kappa^2} \int \frac{d^n q'}{(2\pi)^n} V(\vec{q}-\vec{q}') \psi_\kappa(\vec{q}')
\,.
</math>
For a short-range potential, <math>V(\vec{q})</math> is practically constant <math>V_0</math> for all relevant <math>q</math>, and falls off to zero on a large <math>q</math>-scale of </amth>\approx 1/R</math>.  Thus
:<math>
    \psi_\kappa(\vec{q}) \approx  - \frac{mV_0}{\hbar^2}\frac{1}{q^2 + \kappa^2} \int_{q'\lesssim\frac{1}{R}} \frac{d^n q'}{(2\pi)^n} \psi_\kappa(\vec{q}')
</math>
We integrate once more over $\vect{q}$, applying the same cut-off $1/R$, and then divide by the common factor $\int_{q\lesssim\frac{1}{R}}\frac{d^n q}{(2\pi)^n} \psi_\kappa(\vect{q})$. We obtain the following equation for the bound state energy $E$:
\begin{equation}
   - \frac{1}{V_0} \;=\; \frac{m}{\hbar^2}\int_{q\lesssim\frac{1}{R}} \frac{d^n q}{(2\pi)^n} \frac{1}{q^2 + \kappa^2}\; =\; \frac{1}{\Omega}\int_{\epsilon<E_R} d\epsilon \frac{\rho_n(\epsilon)}{2\epsilon+\left|E\right|}
\end{equation}
with the density of states in <math>\rho_n(\epsilon)</math>,the energy cut-off <math>E_R = \hbar^2/m R^2</math> and the volume <math>\Omega</math> of the system. The question on the existence of bound states for arbitrarily weak interaction has now been reformulated: As <math>|V_0| \rightarrow 0</math>, the left hand side diverges. This equation has a solution for small <math>|V_0|</math> only if the right hand side also diverges for vanishing bound state energy <math>|E| \rightarrow 0</math>, and this involves an integral over the density of states.  In 1D and 2D, there is always bound states. However, in 3D the integral is finite for vanishing <math>|E|</math>, and there is a threshold for the interaction potential to bind the two particles.

These results give us an idea why there might be a paired state for two fermions immersed in a 3D medium, even for arbitrarily weak interactions: It could be that the density of available states to the two fermions is altered due to the presence of the other atoms. 

=== Pairing of fermions -- The Cooper problem ===
Consider now two weakly interacting spin 1/2 fermions not in vacuum, but on top of a (non-interacting) filled Fermi sea, the Cooper problem.  For weak
interactions, the particles' momenta are essentially confined to a narrow shell above the Fermi surface. The density of states at the Fermi surface is <math>\rho_{\rm 3D}(E_F)</math>, which is a constant just like in two dimensions. We should thus find a "bound state" for the two-particle system for
"arbitrarily weak attractive interaction".

In principle, the two fermions could form a pair at any finite momentum. However, considering the discussion in the previous section, the largest binding energy can be expected for the pairs with the largest density of scattering states. For zero-momentum pairs, the entire Fermi surface is available for
scattering. If the pairs have finite center-of-mass momentum <math>\vec{q}</math>, the number of contributing states is strongly reduced, as they are confined to a circle. Consequently, pairs at rest experience the strongest binding. In the following, we will calculate this energy.

We can write the Schr\"odinger equation for the two interacting particles as before, but now we need to search for a small binding
energy <math>E_B = E-2E_F<0</math> on top of the large Fermi energy of the two particles. The equation for <math>E_B</math> is
:<math>
    -\frac{1}{V_0} = \frac{1}{\Omega}\int_{E_F<\epsilon<E_F+E_R} d\epsilon \frac{\rho_{\rm 3D}(\epsilon)}{2(\epsilon-E_F)+\left|E_B\right|}
</math>
The effect of Pauli blocking of momentum states below the Fermi surface is explicitly included by only integrating over energies
<math>\epsilon > E_F</math>.

In conventional superconductors, the natural cut-off energy <math>E_R</math> is given by the Debye frequency <math>E_R = \hbar \omega_D</math>, corresponding to the highest frequency at which ions in the crystal lattice can respond to a bypassing electron. Since we have
<math>\hbar \omega_D \ll E_F</math>, we can approximate <math>\rho_{\rm 3D}(\epsilon) \approx \rho_{\rm 3D}(E_F)</math> and find:
:<math>
    E_B = - 2 \hbar \omega_D e^{-2 \Omega/\rho_{\rm 3D}(E_F) \left|V_0\right|}\\
</math>

In the case of an atomic Fermi gas, we should replace $1/V_0$ by the physically relevant scattering length $a < 0$. The equation for the
bound state becomes
:<math>
    -\frac{m}{4\pi\hbar^2 a} = \frac{1}{\Omega}\int_{E_F}^{E_F+E_R} d\epsilon \frac{\rho_{\rm 3D}(\epsilon)}{2(\epsilon-E_F)+\left|E_B\right|}
- \frac{1}{\Omega}\int_0^{E_F+E_R} d\epsilon \frac{\rho_{\rm
3D}(\epsilon)}{2\epsilon}
</math>
The right hand expression is now finite as we let the cut-off <math>E_R \rightarrow \infty</math>, the result being (one assumes <math>\left|E_B\right|\ll E_F</math>)
:<math>
    -\frac{m}{4\pi\hbar^2 a} = \frac{\rho_{\rm 3D}(E_F)}{2\Omega}
\left(-\log\left(\frac{\left|E_B\right|}{8E_F}\right) - 2\right)
</math>
Inserting <math>\rho_{\rm 3D}(E_F) = \frac{\Omega m k_F}{2\pi^2\hbar^2}</math> with the Fermi wave vector <math>k_F = \sqrt{2mE_F/\hbar^2}</math>, one arrives at
:<math>
    E_B = - \frac{8}{e^2} E_F\, e^{-\pi/k_F \left|a\right|}
\,.
</math>
The binding energies can be compared with the result for the bound state of two particles in 2D. The role of the constant density of states <math>\rho_{\rm 2D}</math> is here played by the 3D density of states at the Fermi surface, <math>\rho_{\rm 3D}(E_F)</math>.

We therefore see that two weakly interacting fermions on top of a Fermi sea form a bound state due to Pauli blocking. However, in this artificial problem we neglected the interactions between particles "inside" the Fermi sea.  As we ``switch on'' the interactions for all particles from top to the bottom of the Fermi sea, the preceding discussion indicates that the gas will reorder itself into a completely new, paired state. The Fermi sea is thus unstable towards pairing (Cooper instability). The full many-body description of such a paired state, including the necessary anti-symmetrization of the
full wave function, was achieved by Bardeen, Cooper and Schrieffer (BCS) in 1957. As we will see in the next section, the self-consistent inclusion of all fermion pairs leads to more available momentum space for pairing. The effective density of states is then twice as large, giving a superfluid gap <math>\Delta</math> that differs from above result by a factor of 2 in the exponent.

It should be noted that the crucial difference to the situation of two particles in vacuum in 3D is the "constant density" of states at the Fermi energy (and not the 2D character of the Fermi surface). Therefore, if we were to consider the Cooper problem in higher dimensions and have two
fermions scatter on the <math>(n-1)</math> dimensional Fermi surface, the result would be similar to the 2D case (due to the constant density of states), and not to the case of <math>(n-1)</math> dimensions.

The conclusion of this section is that Cooper pairing is a many-body phenomenon, but the binding of two fermions can still be understood by two-body quantum mechanics, as it is similar to two isolated particles in two dimensions.  To first order, the many-body physics is not the modification of interactions, but rather the modification of the density of states due to Pauli blocking.

=== Crossover wave function ===
We know that in 3D, two fermions in isolation can form a molecule for strong enough attractive interaction. The ground state of the system should be a
Bose-Einstein condensate of these tightly bound pairs.  However, if we increase the density of particles in the system, we will ultimately reach the point where the Pauli pressure of the fermionic constituents becomes important and modifies the properties of the system.  When the Fermi energy of the
constituents exceeds the binding energy of the molecules, we
expect that the equation of state will be fermionic, i.e. the
chemical potential will be proportional to the density to the
power of 2/3. Only when the size of the molecules is much smaller than the interparticle spacing, i.e. when the binding energy largely exceeds the Fermi energy, is the fermionic nature of the
constituents irrelevant -- tightly bound fermions are spread out widely in momentum space and do not run into the Pauli limitation of unity occupation per momentum state.

For too weak an attraction there is no bound state for two
isolated fermions, but Cooper pairs can form in the medium as discussed
above. The ground state of the system turns out to be a
condensate of Cooper pairs as described by BCS theory. In contrast to the physics of molecular condensates, however,
the binding energy of these pairs is much less than the Fermi energy
and therefore Pauli pressure plays a major role.

The crossover from the BCS- to the BEC-regime is smooth. This is somewhat surprising since the two-body physics shows a threshold behavior at a critical interaction strength, below which there is no bound state for two particles. In the presence of the Fermi sea, however, we simply cross over from a regime of tightly bound molecules to a regime where the pairs are of much larger size than the interparticle spacing. Closely following Leggett's work~\cite{legg80}, and its extension to finite temperatures by Nozi\`eres and Schmitt-Rink~\cite{nozi85}, we will describe the BEC-BCS
crossover in a simple ``one-channel'' model of a potential well.
Rather than the interaction strength $V_0$ as in section~\ref{s:boundstates}, we will take the scattering length $a$ as the parameter that ``tunes'' the interaction. The relation between $V_0$ and $a$ is given by Eq.~\ref{e:renormalize} and its explicit form Eq.~\ref{e:acutoff}.
For positive $a>0$, there is a two-body bound state available at $E_B
= -\hbar^2/m a^2$ (see table~\ref{t:momentumboundstates}), while
small and negative $a<0$ corresponds to weak attraction where Cooper pairs can form in the medium. In either case, for $s$-wave interactions, the orbital part of the pair wave function
$\varphi(\vect{r}_1,\vect{r}_2)$ will be symmetric under exchange
of the paired particles' coordinates and, in a uniform system, will only depend on their
distance $\left|\vect{r}_1-\vect{r}_2\right|$. We will explore the many-body wave function
\begin{equation}
    \Psi\left(\vect{r}_1,\dots,\vect{r}_N\right) = \mathcal{A}\left\{\varphi(\left|\vect{r}_1-\vect{r}_2\right|)\chi_{12}\dots \varphi(\left|\vect{r}_{N-1}-\vect{r}_N\right|)\chi_{N-1,N}\right\}
\label{e:fermicondensatepsi}
\end{equation}
that describes a condensate of such fermion pairs, with the
operator $\mathcal{A}$ denoting the correct antisymmetrization of
all fermion coordinates, and the spin singlet $\chi_{ij} =
\frac{1}{\sqrt{2}}(\left|\uparrow\right>_i \left|\downarrow\right>_j - \left|\downarrow\right>_i \left|\uparrow\right>_j)$. In the experiment, ``spin up'' and ``spin down'' will correspond to two atomic hyperfine states.

In second quantization notation we write
\begin{equation}
    \left|\Psi\right>_N = \int \prod_i d^3 r_i \,\varphi(\vect{r}_1 - \vect{r}_2) \Psi_\uparrow^\dagger(\vect{r}_1) \Psi_\downarrow^\dagger(\vect{r}_2) \dots \varphi(\vect{r}_{N-1} - \vect{r}_N) \Psi_\uparrow^\dagger(\vect{r}_{N-1}) \Psi_\downarrow^\dagger(\vect{r}_N) \left|0\right>
\end{equation}
where the fields $\Psi_\sigma^\dagger(\vect{r}) = \sum_k
c_{k\sigma}^\dagger \frac{e^{-i \vect{k} \cdot \vect{r}}}{\sqrt{\Omega}}$. With the
Fourier transform $\varphi(\vect{r}_1-\vect{r}_2) = \sum_k
\varphi_k \frac{e^{i \vect{k} \cdot (\vect{r}_1- \vect{r}_2)}}{\sqrt{\Omega}}$ we can introduce the pair creation operator
\begin{equation}
    b^\dagger = \sum_k \varphi_k c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger
\end{equation} and find
\begin{equation}
    \left|\Psi\right>_N = {b^\dagger}^{N/2} \left|0\right>
\end{equation}
This expression for  $\left|\Psi\right>_N$ is formally identical to
the Gross-Pitaevskii ground state of a condensate of bosonic particles. However, the
operators $b^\dagger$  obey bosonic commutation relations only in
the limit of tightly bound pairs.  For the commutators, we obtain
\begin{eqnarray}
\label{e:commutators}
    \left[b^\dagger,b^\dagger\right]_- &= \sum_{k k'} \varphi_k \varphi_{k'} \left[ c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger,c_{k'\uparrow}^\dagger c_{-k'\downarrow}^\dagger \right]_- &= 0  \\
    \left[b,b\right]_- &= \sum_{k k'} \varphi^*_k \varphi^*_{k'} \left[c_{-k\downarrow} c_{k\uparrow},c_{-k'\downarrow} c_{k'\uparrow}\right]_- &= 0 \nonumber\\
\left[b,b^\dagger\right]_- &= \sum_{k k'} \varphi^*_k \varphi_{k'}
\left[c_{-k\downarrow} c_{k\uparrow},c_{k'\uparrow}^\dagger
c_{-k'\downarrow}^\dagger \right]_- &= \sum_k |\varphi_k|^2 (1 -
n_{k\uparrow} - n_{k\downarrow}) \nonumber
\end{eqnarray}

The third commutator is equal to one only in the limit where the
pairs are tightly bound and occupy a wide region in momentum
space.  In this case, the occupation numbers $n_k$ of any momentum
state $k$ are very small (see section~\ref{s:evolution} below),
and $\left[b,b^\dagger\right]_- \approx \sum_k |\varphi_k|^2 = \int {\rm d}^3 r_1\int {\rm d}^3 r_2 \,|\varphi(\vect{r}_1,\vect{r}_2)|^2 = 1$.

Working with the $N$-particle state $\left|\Psi\right>_N$ is
inconvenient, as one would face a complicated combinatoric problem
in manipulating the sum over all the $c_k^\dagger$'s (as one
chooses a certain $k$ for the first fermion, the choices for the
second depend on this $k$, etc.). It is preferable to use the
grand canonical formalism, not fixing the number of atoms but the
chemical potential $\mu$. A separate, crucial step is to define a many-body state which is
a superposition of states with different atom numbers.  In the BEC
limit, this is analogous to the use of coherent states (vs. Fock
states) in quantum optics. Let $N_p = N/2$ be the number of pairs. Then,
\begin{eqnarray}
    \label{e:coherentstate}
    \mathcal{N}\left|\Psi\right> &= \sum_{J_{\rm even}} \frac{N_p^{J/4}}{(J/2)!} \left|\Psi\right>_J &= \sum_M \frac{1}{M!} {N_p^{M/2}\; b^\dagger}^M \left|0\right> = e^{\sqrt{N_p} \;b^\dagger} \left|0\right>\nonumber \\
&= \prod_k e^{\sqrt{N_p}\;  \varphi_k c_{k\uparrow}^\dagger
c_{-k\downarrow}^\dagger} \left|0\right> &= \prod_k (1 + \sqrt{N_p}\;
\varphi_k\, c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger)
\left|0\right>
\end{eqnarray}
The second to last equation follows since the operators $c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger$ commute for different $k$, and the last equation follows from $c_k^{\dagger 2} = 0$. If we choose the constant $\mathcal{N}
= \prod_k \frac{1}{u_k} = \prod_k \sqrt{1 + N_p |\varphi_k|^2}$,
then $\left|\Psi\right>$ becomes a properly normalized state
\begin{equation}
    \left|\Psi_{\rm BCS}\right> = \prod_k (u_k + v_k c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger) \left|0\right>
    \label{e:BCSstate}
\end{equation}
with $v_k = \sqrt{N_p}\,\varphi_k u_k$ and $|u_k|^2 + |v_k|^2 = 1$. This is
the famous BCS wave function, first introduced as a variational Ansatz, later shown to be the exact solution of the simplified Hamiltonian Eq.~\ref{e:Hsimplified} (below). It is a product of wave functions referring to the occupation of pairs of single-particle momentum states, $(\vect{k},\uparrow,-\vect{k},\downarrow)$.
As a special case, it describes a non-interacting Fermi sea, with all momentum pairs occupied up to the Fermi momentum ($u_k=0,
v_k=1$ for $k<k_F$ and $u_k=1, v_k=0$ for $k>k_F$). In general, for
a suitable choice of the $v_k$'s and $u_k$'s, it describes a ``molten'' Fermi sea, modified
by the coherent scattering of pairs with zero total momentum. Pairs of momentum states are seen to be in a superposition of being fully empty and fully occupied.
The above derivation makes it clear that this wave function
encompasses the entire regime of pairing, from point bosons
(small molecules) to weakly and non-interacting fermions.

=== Gap and number equation ===

The variational parameters $u_k$ and $v_k$ are derived in the
standard way by minimizing the free energy $E - \mu N =
\left<\hat{H} - \mu \hat{N}\right>$. The many-body Hamiltonian for
the system is
\begin{equation}
    \hat{H} = \sum_{k,\sigma} \epsilon_k c_{k\sigma}^\dagger c_{k\sigma} + \frac{V_0}{\Omega}\sum_{k,k',q} c_{k+\frac{q}{2} \uparrow}^\dagger c_{-k+\frac{q}{2}\downarrow}^\dagger c_{k'+\frac{q}{2}\downarrow} c_{-k'+\frac{q}{2}\uparrow}
\label{e:Hamiltonian}
\end{equation}

The dominant role in superfluidity is played by fermion pairs with
zero total momentum.   Indeed, as we have seen in
section~\ref{s:cooperproblem},  Cooper pairs with zero momentum
have the largest binding energy. Therefore, we simplify the
mathematical description by neglecting interactions between pairs
at finite momentum, i.e. we only keep the terms for $\vect{q} =
0$.
This is a very drastic simplification, as hereby density fluctuations are eliminated.  It is less critical for charged superfluids, where density fluctuations are suppressed by Coulomb interactions.  However, for neutral superfluids, sound waves  (the Bogoliubov-Anderson mode, see section~\ref{s:collexcitations}) are eliminated by this approximation.
The approximate Hamiltonian (``BCS Hamiltonian'') reads
\begin{equation}
    \hat{H} = \sum_{k,\sigma} \epsilon_k c_{k\sigma}^\dagger c_{k\sigma} + \frac{V_0}{\Omega}\sum_{k,k'} c_{k \uparrow}^\dagger c_{-k\downarrow}^\dagger c_{k'\downarrow} c_{-k'\uparrow}
\label{e:Hsimplified}
\end{equation}
The free energy becomes
\begin{eqnarray}
    \label{e:free-energy}
   \mathcal{F} = \left<\hat{H} - \mu \hat{N}\right> &=& \sum_k 2 \xi_k v_k^2 + \frac{V_0}{\Omega}\sum_{k,k'} u_k v_k u_{k'} v_{k'}\\
   \mbox{with }\;\xi_k &=& \epsilon_k - \mu \nonumber
\end{eqnarray}
Minimizing $E-\mu N$ leads to
\begin{eqnarray}
    v_k^2 &=& \frac{1}{2}\left(1 - \frac{\xi_k}{E_k}\right) \nonumber \\
    u_k^2 &=& \frac{1}{2}\left(1 + \frac{\xi_k}{E_k}\right) \nonumber \\
    \mbox{with }\;E_k &=& \sqrt{\xi_k^2 + \Delta^2}
    \label{e:ukvk}
\end{eqnarray}
where $\Delta$ is given by the {\it gap equation} $\Delta \equiv
\frac{V_0}{\Omega} \sum_k \left<c_{k\uparrow}
c_{-k\downarrow}\right> = - \frac{V_0}{\Omega} \sum_k u_k v_k = -
\frac{V_0}{\Omega} \sum_k \frac{\Delta}{2 E_k}$ or
\begin{equation}
    -\frac{1}{V_0} = \int \frac{d^3 k}{\left(2\pi\right)^3} \;\frac{1}{2E_{k}}
\end{equation}
Note the similarity to the bound state equation in free space,
Eq.~\ref{e:densityboundstates}, and in the simplified Cooper
problem, Eq.~\ref{e:Cooper}. An additional constraint is given by the {\it number equation} for the
total particle density $n = N / \Omega$
\begin{equation}
n = 2 \int \frac{d^3 k}{\left(2\pi\right)^3} \; v_k^2
\end{equation}
Gap and number equations have to be solved simultaneously to yield the two unknowns
$\mu$ and $\Delta$. We will once more replace $V_0$ by the
scattering length $a$ using prescription Eq.~\ref{e:renormalize},
so that the gap equation becomes (compare
Eq.~\ref{e:cooperrenorm})
\begin{equation}
-\frac{m}{4\pi\hbar^2 a} = \int \frac{d^3 k}{\left(2\pi\right)^3}
\left(\frac{1}{2 E_k} - \frac{1}{2 \epsilon_k}\right)
\end{equation}
where the integral is now well-defined. The equations can be
rewritten in dimensionless form with the Fermi energy $E_F =
\hbar^2 k_F^2 / 2m$ and wave vector $k_F = (3 \pi^2 n)^{1/3}$~\cite{orti05bcs}
\begin{eqnarray}
-\frac{1}{k_F a} &= &\frac{2}{\pi} \sqrt{\frac{\Delta}{E_F}}\; I_1\left(\frac{\mu}{\Delta}\right) \\
1 &=& \frac{3}{2}\left(\frac{\Delta}{E_F}\right)^{3/2} I_2\left(\frac{\mu}{\Delta}\right)\\
\mbox{with }\; I_1(z) &=& \int_0^\infty dx \;x^2 \left(\frac{1}{\sqrt{\left(x^2 - z\right)^2 + 1}} - \frac{1}{x^2}\right)\\
\mbox{and }\;  I_2(z) &=& \int_0^\infty dx \;x^2 \left(1 -
\frac{x^2 - z}{\sqrt{\left(x^2-z\right)^2 + 1}}\right)
\end{eqnarray}
This gives
\begin{eqnarray}
    -\frac{1}{k_F a} &=& \frac{2}{\pi} \left(\frac{2}{3 I_2\left(\frac{\mu}{\Delta}\right)}\right)^{1/3} I_1\left(\frac{\mu}{\Delta}\right)\\
    \frac{\Delta}{E_F} &=& \left(\frac{2}{3 I_2\left(\frac{\mu}{\Delta}\right)}\right)^{2/3}
\end{eqnarray}
The first equation can be inverted to obtain $\mu / \Delta$ as a
function of the {\it interaction parameter} $1/k_F a$, which can
then be inserted into the second equation to yield the gap $\Delta$.
The result for $\mu$ and $\Delta$ as a function of $1/k_F a$ is
shown in Fig.~\ref{f:Deltamu}. It is possible to obtain analytic
expressions for the solutions in terms of complete elliptic
integrals~\cite{mari98becbcs}.

\begin{figure}
\centering
  \includegraphics[width=3in]{figs_BECBCSCrossover/deltamu.eps}\\
  \caption[Chemical potential and gap in the BEC-BCS crossover]{Chemical potential (dotted line) and gap (straight line, red) in the BEC-BCS crossover as a function of the interaction parameter $1/k_Fa$. The BCS-limit of negative $1/k_F a$ is to the right on the graph. The resonance where $1/k_F a = 0$ is indicated by the dashed line.}\label{f:Deltamu}
\end{figure}

In this derivation, we have combined the simplified Hamiltonian,
Eq.~\ref{e:Hsimplified} with the BCS variational Ansatz.  Alternatively one can
apply a decoupling (mean field) approximation to the Hamiltonian~\cite{peth02bec}. Expecting that there will be
some form of pair condensate, we assume that the pair creation and
annihilation operator only weakly fluctuates around its non-zero expectation value
\begin{eqnarray}
    C_k=  \left<c_{k\uparrow} c_{-k\downarrow}\right>= -\left< c_{k\uparrow}^\dagger
    c_{-k\downarrow}^\dagger\right>
\end{eqnarray}
chosen to be real (since the relative phase of states which differ
in particle number by two can be arbitrarily chosen).
That is, we write
\begin{equation}
c_{k\uparrow} c_{-k\downarrow} = C_k + (c_{k\uparrow} c_{-k\downarrow} - C_k)
\end{equation}
with the operator in parentheses giving rise to fluctuations that are small on the scale of $C_k$.
The gap parameter $\Delta$ is now defined as
\begin{equation}
\Delta=\frac{V_0}{\Omega} \sum_{k} C_k
\end{equation}
We only include terms in the interaction part of the Hamiltonian which
involve the $C_k$'s at least once. That is, we neglect the
correlation of fluctuations of the pair creation and annihilation
operators. One obtains
\begin{equation}
    \hat{H} = \sum_{k} \epsilon_k (c_{k\uparrow}^\dagger c_{k\uparrow}+ c_{k\downarrow}^\dagger c_{k\downarrow}) -\Delta \sum_k\left( c_{k \uparrow}^\dagger c_{-k\downarrow}^\dagger + c_{k \downarrow}
    c_{-k\uparrow} + \sum_{k'} C_{k'}\right)
\end{equation}

This Hamiltonian  is bilinear in the creation and annihilation
operators and can easily be solved by a Bogoliubov transformation~\cite{bogo58,vala58,peth02bec} from the
particle operators $c_{k\downarrow}$ and $c_{k\uparrow}$ to new
quasi-particle operators $\gamma_{k\uparrow}$ and
$\gamma_{k\downarrow}$:
\begin{eqnarray}
    \gamma_{k\uparrow} &=& u_k c_{k\uparrow} - v_k c_{-k\downarrow}^\dagger \\
    \gamma_{-k\downarrow}^\dagger &=& u_k c_{-k\downarrow}^\dagger + v_k c_{k\uparrow} \nonumber
    \label{e:quasiparticles}
\end{eqnarray}
The $u_k$ and $v_k$ are determined from the requirements that the
new operators fulfill fermionic commutation relations and that the
transformed Hamiltonian is diagonal with respect to the
quasiparticle operators.  This solution is identical to the one
obtained before for the $u_k$ and $v_k$, and the transformed
Hamiltonian becomes
\begin{equation}
\hat{H} - \mu \hat{N} = - \frac{\Delta^2}{V_0/\Omega} + \sum_k (\xi_k - E_k) + \sum_k E_k
(\gamma_{k\uparrow}^\dagger \gamma_{k\uparrow} +
\gamma_{k\downarrow}^\dagger \gamma_{k\downarrow})
\label{e:BogoliubovH}
\end{equation}
The first two terms give the free
energy $E - \mu N$ of the pair condensate, identical to Eq.~\ref{e:free-energy} when the correct $u_k$ and $v_k$ are inserted. The third term represents the
energy of excited quasi-particles, and we identify $E_k$ as
excitation energy of a quasi-particle.  The superfluid ground
state is the quasi-particle vacuum: $\gamma_{k\uparrow}
\left|\Psi\right> = 0 = \gamma_{k\downarrow} \left|\Psi\right>$.

This approach via the pairing field is analogous to the Bogoliubov
treatment of an interacting Bose-Einstein condensate: There, the
creation and annihilation operators for atoms with zero momentum
are replaced by $\sqrt{N_0}$, the square root of the number of
condensed atoms (i.e.~a coherent field). In the interaction term
of the Hamiltonian all terms are dropped that contain less than
two factors of $\sqrt{N_0}$.  In other words, the Hamiltonian
(Eq.~\ref{e:Hsimplified}) is solved by keeping only certain pair
interactions, either by using a variational pairing wave function,
or by introducing a mean pairing field. It should be noted that
these approximations are not even necessary, as the BCS
wave function can be shown to be the {\it exact} solution to the
reduced Hamiltonian $Eq.~\ref{e:Hsimplified}$~\cite{duke04review}.

== Discussion of the three regimes -- BCS, BEC and crossover ==
=== BCS limit ===
\label{s:BCSlimit}
In the BCS-limit of weak attractive interaction, $k_F a
\rightarrow 0_-$, we have\footnote{This follows by substituting
$\xi = x^2 - z$ in the integrals and taking the limit
$z\rightarrow \infty$. One has $I_1(z) \approx
\sqrt{z}\left(\log(8z) - 2\right)$ and $I_2(z) = \frac{2}{3}
z^{3/2}$.}
\begin{eqnarray}
    \mu &\approx& E_F \\
    \Delta &\approx& \frac{8}{e^2} e^{-\pi/2k_F\left|a\right|}
    \label{e:BCSLimit}
\end{eqnarray}

The first equation tells us that adding a spin up and spin down
particle to the system costs a Fermi energy per particle (with the
implicit assumption that both a spin up and a spin down particle
are added, raising the total energy by $2 \mu$).  In the weakly
interacting BCS limit Pauli blocking still dominates over
interactions, and hence the particles can only be added at the
Fermi surface. The second equation is the classic result of BCS
theory for the superfluid gap\footnote{The present mean field
treatment does not include density fluctuations, which modify the
prefactor in the expression for the gap $\Delta$ ~\cite{gork61,peth02bec}.}.
Compared to the bound state energy for a single Cooper pair on top
of a non-interacting Fermi sea, Eq.~\ref{e:cooperproblem}, the gap
is larger (the negative exponent is smaller by a factor of two),
as the entire collection of particles now takes part in the
pairing\footnote{In the self-consistent BCS solution, not only the
momentum states above the Fermi surface contribute to pairing, but
also those {\it below} it, in a symmetric shell around the Fermi
momentum. In the Cooper problem the states below the Fermi surface
were excluded, reducing the effective density of states by a
factor of two.}. However, the gap is still exponentially small
compared to the Fermi energy: Cooper pairing is fragile.

The ground state energy of the BCS state can be calculated from Eq.~\ref{e:free-energy} and is
\begin{equation}
    E_{\rm G,\, BCS} = \frac{3}{5} N E_F - \frac{1}{2}\,\rho(E_F)\, \Delta^2
\end{equation}

The first term is the energy of the non-interacting normal state,
where $\frac{3}{5}E_F$ is the average kinetic energy per fermion
in the Fermi sea. The second term is the energy due to
condensation, negative as it should be, indicating that the BCS
state is energetically favorable compared to the normal state.

Although the total kinetic energy of the Fermi gas has been
increased (by populating momentum states above $E_F$), the total
energy is lower due to the gain in potential energy.  This is
valid for any kind of pairing (i.e. proton and electron forming a
hydrogen atom), since the localization of the pair wave function
costs kinetic energy.

The energy of the BCS state, $- \frac{1}{2}\,\rho(E_F)\, \Delta^2$
can be interpreted in two ways. One way refers to the wave function Eq.~\ref{e:fermicondensatepsi}, which consists of $N/2$ identical fermion pairs.  The energy per
pair is then $- \frac{3}{4} \Delta^2/E_F$.
The other interpretation refers to the BCS wave function Eq.~\ref{e:BCSstate}. It is essentially a product of a ``frozen'' Fermi sea (as $v_k \approx 1$, $u_k \approx 0$ for low values of $k$)
with a paired component consisting of $\sim \rho(E_F) \, \Delta
\sim N \Delta/E_F$  pairs, located in an energy shell of width
$\Delta$ around the Fermi energy.  They each contribute a pairing
energy on the order of $\Delta$.  The second interpretation
justifies the picture of a Cooper pair condensate. In the solution
of the Cooper problem (section~\ref{s:cooperproblem}), the pair wave function
has a peak occupation per momentum state of $\sim 1/ \rho(E_F)
\Delta$. Therefore, one can stack up  $\sim  \rho(E_F) \Delta$
pairs with zero total momentum without getting into serious
trouble with the Pauli exclusion principle and construct a
Bose-Einstein condensate consisting of $\sim \rho(E_F) \Delta$
Cooper pairs~\footnote{Similarly to the fermion pairs described by the operator $b^\dagger$, the Cooper pairs from section~\ref{s:cooperproblem} are not bosons, as shown by the equivalent of Eq.~\ref{e:commutators}. However, if there were only a few Cooper pairs, much less than $\rho(E_F) \Delta$, the occupation of momentum states $n_k$ would still be very small compared to 1 and these pairs would be to a good approximation bosons.}.

It depends on the experiment whether it reveals a pairing energy
of $\frac{1}{2} \Delta^2/E_F$ or of $\Delta$.  In RF
spectroscopy, all momentum states can be excited (see section~\ref{e:chap2RFspectroscopy}),
and the spectrum shows a gap of $\frac{1}{2} \Delta^2/E_F$ (see section~\ref{s:RFspectrum}).
Tunnelling experiments in superconductors probe the region
close to the Fermi surface, and show a pairing gap of $\Delta$.

The two interpretations for the BCS energy carry along two possible choices of the pairing wave function (see section~\ref{s:evolution}). The first one is $\varphi_k = u_k/v_k\sqrt{N_p}$, which can be shown to extend throughout the whole Fermi sea from zero to slightly above $k_F$, whereas the second one, $\psi(k) = u_k v_k$, is concentrated around the Fermi surface (see Fig.~\ref{f:excitation}).

To give a sense of scale, Fermi energies in dilute atomic gases
are on the order of a $\mu\rm K$, corresponding to $1/k_F \sim
4\,000\, a_0$. In the absence of scattering resonances, a typical
scattering length would be about $50-100\; a_0$ (on
the order of the van der Waals-range). Even if $a < 0$, this would result
in a  vanishingly small gap $\Delta/k_B \approx 10^{-30}\dots
10^{-60}\, \rm K$. Therefore, the realization of superfluidity in
Fermi gases requires scattering or Feshbach resonances to increase
the scattering length, bringing the gas into the strongly
interacting regime where $k_F \left|a\right| > 1$ (see
chapter~\ref{c:feshbach}). In this case, the above mean field
theory predicts $\Delta > 0.22 \;E_F$ or $\Delta/k_B > 200\, \rm
nK$ for $k_F |a| > 1$, and this is the regime where current
experiments are operating.

=== BEC limit ===
\label{s:BEClimit}

In the BEC limit of tightly bound pairs, for $k_F a \rightarrow
0_+$, one finds\footnote{This result follows from the expansion of
the integrals for $z<0$ and $|z|\rightarrow \infty$. One finds
$I_1(z) = -\frac{\pi}{2}\sqrt{|z|}
-\frac{\pi}{32}\frac{1}{|z|^{3/2}}$ and $I_2(z) =
\frac{\pi}{8}\frac{1}{\sqrt{|z|}}$.}
\begin{eqnarray}
    \label{e:BEClimit}
    \mu = -\frac{\hbar^2}{2 m a^2} + \frac{\pi \hbar^2 a n}{m}\\
    \Delta \approx \sqrt{\frac{16}{3\pi}} \frac{E_F}{\sqrt{k_F a}}
\label{e:BEClimitgap}
\end{eqnarray}

The first term in the expression for the chemical potential is the
binding energy per fermion in a tightly bound molecule (see
table~\ref{t:momentumboundstates}). This reflects again the
implicit assumption (made by using the wave function in
Eq.~\ref{e:fermicondensatepsi}) that we always add {\it two}
fermions of opposite spin at the same time to the system.

The second term is a mean field contribution describing the
repulsive interaction between molecules in the gas. Indeed, a
condensate of molecules of mass $m_M = 2m$, density $n_M = n/2$
and a molecule-molecule scattering length $a_M$ will have a
chemical potential $\mu_M = \frac{4\pi \hbar^2 a_M n_M}{m_M}$.
Since $\mu_M$ is twice the chemical potential for each fermion, we
obtain from the above expression the molecule-molecule scattering
length $a_M = 2 a$. However, this result is not exact. Petrov,
Shlyapnikov and Salomon~\cite{petr04dimers} have performed an
exact calculation for the interaction between four fermions and
shown that $a_M = 0.6\, a$. The present mean field approach neglects
correlations between different pairs, or between one fermion and a
pair. If those are included, the correct few-body physics is
recovered~\cite{pier00becbcs,holl04bosefermi,hu06becbcs}.

The expression for the quantity $\Delta$ signifies neither the binding energy of molecules nor does it correspond to a gap in the excitation spectrum. Indeed, in the BEC-regime, as soon as $\mu <0$, there is no longer a gap at non-zero $k$ in the single-fermion excitation spectrum (see Fig.~\ref{f:pairwavefunction} below). Instead, we have for the quasi-particle energies $E_k = \sqrt{(\epsilon_k - \mu)^2 + \Delta^2} \approx |\mu| + \epsilon_k + \frac{\Delta^2}{2|\mu|}$. So $\Delta$ itself does not play a role in the BEC-regime, but only the combination $\Delta^2/|\mu|$ is important. As we see from Eq.~\ref{e:BEClimit},
\begin{equation}
    \frac{\Delta^2}{2|\mu|} = \frac{8}{3\pi} \frac{E_F^2}{k_F a} \frac{2 m a^2}{\hbar^2} = \frac{4}{3\pi}\frac{\hbar^2}{m} k_F^3 a = \frac{4\pi\hbar^2}{m} n\; a
\label{e:Deltasquared}
\end{equation}
which is two times the molecular mean field. In fact, we will show in section~\ref{s:excitations} that it can be interpreted here as the mean field energy experienced by a single fermion in a gas of molecules.

It might surprise that the simplified Hamiltonian Eq.~\ref{e:Hsimplified} contains interactions between two molecules or between a molecule and a single fermion at all. In fact, a crucial part of the simplification has been to explicitly {\it neglect} such three- and four-body interactions. The solution to this puzzle lies in the Pauli principle, which acts as an effective repulsive interaction: In a molecule, each constituent fermion is confined to a region of size $\sim a$ around the molecule's center of mass (see next section). The probability to find another like fermion in that region is strongly reduced due to Pauli blocking. Thus, effectively, the motion of molecules is constrained to a reduced volume $\Omega' = \Omega - c N_M a_M^3$, with the number of molecules $N_M$ and $c$ on the order of 1. This is the same effect one has for a gas of hard-sphere bosons of size $a_M$, and generally for a Bose gas with scattering length $a_M$. An analogous argument leads to the effective interaction between a single fermion and a molecule.
We see that the only way interactions between pairs, or between a pair and a single fermion, enter in the simplified description of the BEC-BCS crossover is via the anti-symmetry of the many-body wave function.

=== Evolution from BCS to BEC ===
\label{s:evolution}
\begin{figure}
\centering
  \includegraphics[width=3in]{figs_BECBCSCrossover/nk.eps}\\
  \caption[Occupation $n_k$ of momentum states $k$ in the BEC-BCS crossover]{Occupation $n_k$ of momentum states $k$ in the BEC-BCS crossover. The numbers give the interaction parameter $1/k_F a$. After~\cite{nozi85}.}\label{f:momoccupation}
\end{figure}

Our variational approach smoothly interpolates between the two
known regimes of a BCS-type superfluid and a BEC of molecules. It is a crossover, which occurs approximately between $1/k_F a = -1$ and +1 and is fully continuous. The occupation of momentum states
$n_k = v_k^2$ evolves smoothly from the step-function $\Theta(k_F
- k)$ of a degenerate Fermi gas, broadened over a width $\Delta
\ll E_F$ due to pairing, to that of $N_p$ molecules, namely the number of molecules $N_p$ times the probability $|\varphi_k|^2$ to find a molecule with momentum $k$ (we have $\varphi_k = \frac{(2 \pi a)^{3/2}}{\sqrt{\Omega}}\frac{1}{\pi}\frac{1}{1+k^2 a^2}$) (see
Fig.~\ref{f:momoccupation}). It is also interesting to follow the
evolution of the ``Cooper pair'' wave function\footnote{Note that
this definition is not equal to the Fourier transform of the pair
wave function $\varphi(\vect{r})$ introduced in
Eq.~\ref{e:fermicondensatepsi}, which would be $v_k/u_k\sqrt{N_p}$. The
definition given here is the two-point correlation function. Both
definitions for the Cooper pair wave function show a sharp feature,
either a peak or an edge at the Fermi surface, of width $\sim
\delta k$, thus giving similar behavior for the real space
wave function.} both in $k$-space, where it is given by
$\left<\Psi_{BCS}\right|c^\dagger_{k\uparrow}
c^\dagger_{-k\downarrow}\left|\Psi_{BCS}\right> = u_k v_k$, and in
real space, where it is
\begin{eqnarray}
\psi(\vect{r}_1,\vect{r}_2) &=&
\left<\Psi_{BCS}\right|\Psi_{\uparrow}^\dagger(\vect{r}_1)\Psi_{\downarrow}^\dagger(\vect{r}_2)\left|\Psi_{BCS}\right>
= \frac{1}{\Omega}\sum_k u_k v_k e^{-i \vect{k} \cdot (\vect{r}_1-\vect{r}_2)} \nonumber \\
&=&\frac{1}{\Omega}\sum_k \frac{\Delta}{2E_k}\, e^{-i \vect{k} \cdot
(\vect{r}_1-\vect{r}_2)} \label{e:cooperpairwavefunction}
\end{eqnarray}


\begin{figure}
\centering
 \includegraphics[width=5.5in]{figs_BECBCSCrossover/pairwavefunction.eps}\\
  \caption[Evolution of the spatial pair wave function $\psi(r)$ in the BEC-BCS crossover]{Evolution of the spatial pair wave function $\psi(r)$ in the BEC-BCS crossover. The inset shows the Fourier transform $\psi(k)$, showing clearly that in the BCS-limit, momentum states around the Fermi surface make the dominant contribution to the wave function. In the crossover, the entire Fermi sphere takes part in the pairing. In the BEC-limit, $\psi(k)$ broadens as the pairs become more and more tightly bound. $\psi(r)$ was obtained via numerical integration of $\int_{-\mu}^\infty d\xi \frac{\sin(r \sqrt{\xi + \mu})}{\sqrt{\xi^2 + \Delta^2}}$ (here, $\hbar = 1 = m$), an expression that follows from Eq.~\ref{e:cooperpairwavefunction}.}\label{f:pairwavefunction}
\end{figure}

\begin{figure}
\centering
 \includegraphics[width=3in]{figs_BECBCSCrossover/pairsize.eps}\\
  \caption[From tightly bound molecules to long-range Cooper pairs]{From tightly bound molecules to long-range Cooper pairs. Evolution of the pair size $\xi_0  =\sqrt{\frac{\left<\psi(\vect{r})\right|r^2\left|\psi(\vect{r})\right>}{\left<\psi(\vect{r})|\psi(\vect{r})\right>}}$ as a function of the interaction parameter $1/k_F a$. On resonance (dashed line), the pair size is on the order of the inverse wave vector, $\xi_0(0) \sim \frac{1}{k_F}$, about a third of the interparticle spacing.}\label{f:pairsize}
\end{figure}

In the BCS limit, the pairing occurs near the Fermi surface $k=
k_F$, in a region of width $\delta k \sim \frac{\partial
k}{\partial \epsilon} \delta \epsilon \approx \frac{\Delta}{\hbar
v_F}$, where $v_F$ is the velocity of fermions at the Fermi
surface. Therefore, the spatial wave function of Cooper pairs has
a strong modulation at the inverse wave vector $1/k_F$, and an
overall extent of the inverse width of the pairing region, $\sim
1/\delta k \sim \frac{\hbar v_F}{\Delta} \gg 1/k_F$. More
quantitatively, Eq.~\ref{e:cooperpairwavefunction} gives (setting
$r = \left|\vect{r}_1-\vect{r}_2\right|$)~\cite{bard57}
\begin{equation}
    \psi(r) = \frac{k_F}{\pi^2 r}\frac{\Delta}{\hbar v_F} \sin(k_F r)\, K_0\left(\frac{r}{\pi \xi_{BCS}}\right)\; \stackrel{r\rightarrow \infty}{\sim}\;\sin\left(k_F r\right)\, e^{-r/(\pi \xi_{BCS})}
\end{equation}
where $K_0(k r)$ is the modified Bessel function that falls off as
$e^{-k r}$ at infinity. We have encountered a similar exponential
envelope function for a two-body bound state (see
table~\ref{t:boundstate}). The characteristic size of the Cooper
pair, or the {\it two-particle correlation length} $\xi_0$, can be
defined as $\xi_0^2
=\frac{\left<\psi(\vect{r})\right|r^2\left|\psi(\vect{r})\right>}{\left<\psi(\vect{r})|\psi(\vect{r})\right>}$,
and this gives indeed $\xi_0 \sim 1/\delta k$,
\begin{equation}
    \xi_0 \approx \xi_{BCS} \equiv \frac{\hbar v_F}{\pi \Delta} \gg 1/k_F \quad \mbox{in the BCS-limit}
\end{equation}

In the BEC limit, $u_k v_k \propto \frac{1}{1+(k a)^2}$, and
so
\begin{equation}
    \psi(\vect{r}_1,\vect{r}_2) \sim \frac{e^{- \left|\vect{r}_1 - \vect{r}_2\right|/a}}{\left|\vect{r}_1 - \vect{r}_2\right|}
\end{equation}
which is simply the wave function of a molecule of size $\sim a$
(see table~\ref{t:boundstate}). The two-particle correlation
length\footnote{This length scale should be distinguished from the
{\it coherence length} $\xi_{phase}$ that is associated with
spatial fluctuations of the order parameter. The two length scales
coincide in the BCS-limit, but differ in the BEC-limit, where
$\xi_{phase}$ is given by the healing length $\propto
\frac{1}{\sqrt{n a}}$. See~\cite{pist94xi} for a detailed
discussion.} is thus $\xi_0 \sim a$.
Figs.~\ref{f:pairwavefunction} and~\ref{f:pairsize} summarize the
evolution of the pair wave function and pair size throughout the
crossover.
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