## Abstract

We describe the time-dependent formation of a molecular Bose–Einstein condensate from a BCS state of fermionic atoms as a result of slow sweeping through a Feshbach resonance. We apply a path integral approach for the molecules, and use two-body adiabatic approximations to solve the atomic evolution in the presence of the classical molecular fields, obtaining an effective action for the molecules. In the narrow resonance limit, the problem becomes semiclassical, and we discuss the growth of the molecular condensate in the saddle point approximation. Considering this time-dependent process as an analogue of the cosmological Zurek scenario, we compare the way condensate growth is driven in this rigorous theory with its phenomenological description via time-dependent Ginzburg–Landau theory.

## 1. Introduction

The ground state of a gas of weakly attractively interacting fermions is a Bardeen–Cooper–Schrieffer (BCS) state, in which most of the fermions behave ideally, but some of those near the Fermi surface form something like a Bose condensate of Cooper pairs. As the extreme example of a gas of protons and electrons shows, however, stronger attractions instead produce a true condensate of tightly bound composite bosons. Current laboratory techniques exploit the atomic physics of collisional Feshbach resonances to produce real ultracold gases in which attractions can be varied dramatically, admitting experimental study of the entire BCS–BEC crossover (Chin *et al*. 2004; Regal *et al*. 2004; Zwierlein *et al*. 2004; Köhler *et al*. 2006). Several laboratories can now produce molecular condensates from weakly interacting degenerate Fermi gases, by adiabatically changing a control parameter (typically a magnetic field). Prospects for more precise and detailed measurements of this process and its products are good.

Theoretical studies of this problem have so far focused mainly on equilibrium properties of systems with various fixed values of the interaction parameter. Along with wide tunability of equilibrium parameters, however, controllable and observable non-equilibrium dynamics are a major advantage of cold quantum gases as experimental systems. Essentially, non-equilibrium phenomena in quantum gases, such as the originally cosmological Zurek scenario (Kibble 1976; Zurek 1985, 1996), can therefore serve as ‘colliders for many-body physics’. Like the quanta scattered from a high-energy particle collision, excitations generated by a brief non-equilibrium epoch can provide rich data about the non-trivial dynamics that spawned them.

The dynamics of cold, dilute quantum gases are, of course, comparatively simple. On the one hand, this dynamical simplicity may ultimately allow mesoscopic quantum gas systems to shed light on problems, such as the emergences of irreversibility and classicality, which are even more fundamental than the issue of which effective Hamiltonian is correct. On the other hand, even for cold dilute gases, quantum many-body theory can be difficult enough that understanding will require comparisons among experiments, phenomenological models and first-principle calculations for simple cases.

In the present work, we therefore develop a rigorous theory for adiabatic evolution of a dilute quantum gas from the BCS to the BEC regime, via a time-dependent Feshbach resonance, in the tractable limit of a narrow resonance. We show that for a finite resonance width, adiabaticity can in principle be maintained, as far as two-body dynamics are concerned, through the entire evolution. Long-wavelength collective excitations will still presumably be generated, so that many-body adiabaticity is not expected to be possible in an infinite system. Unlike the case in the phenomenological time-dependent Ginzburg–Landau model, however, the emergence of long-wavelength excitations will not be governed by the same time scales relevant to the growth of the mean order parameter itself. On the other hand, we will show that proximity to the second order quantum phase transition that occurs for zero resonance width means that extreme slowness is required even for two-body adiabaticity. Experimental accessibility of Kibble–Zurek-like phenomena (Kibble 1976; Zurek 1985, 1996) in this system, related to critical slowing down of the order parameter dynamics, can therefore not yet be ruled out.

## 2. The time-dependent BCS–BEC crossover

We use here a Hamiltonian similar to those described in Timmermans *et al*. (1999), Holland *et al*. (2001, 2004), Milstein *et al*. (2002), Javanainen *et al*. (2004, 2005), Romans & Stoof (2006) and Lee *et al*. (2007), which is based on a delta-like pseudopotential for a collisional interaction in which two fermionic atoms unite into a bosonic molecule (or conversely, in which a molecule splits into two atoms). We will consider our evolution to start from the ground state at a point close enough to the resonance that the resonant interaction already predominates over background scattering in other channels. So our many-body Hamiltonian appears in second-quantized notation as(2.1)where *M* is the fermion mass; *Δ*(*t*) is the external control parameter that is ramped up linearly in time ; and *γ* is the interaction strength, determined by atomic collision physics and indicating the width of the Feshbach resonance. The annihilation and creation fields and are bosonic operators, whereas and are the field operators of the fermions, with two spin states denoted by ±. The key feature of this model is that for large |*Δ*| either the fermions or the bosons may be adiabatically eliminated from each other's effective dynamics, providing weak interactions between either type of particle, mediated by virtual production and destruction of the other type. For large negative *Δ*, the ground state consists mainly of fermions, interacting weakly with an effective scattering length *∝γ*^{2}/*Δ*. For large positive *Δ*, bosons dominate low-energy states. For small *Δ*, however, the mediated interactions between either species alone formally diverge in strength. What this means is that the dynamics involves both species non-trivially; if *γ* is small enough, no truly strong interactions are necessarily involved. We will therefore consider this small-*γ* limit, which is not typical for experiments so far conducted, but is also attainable with current techniques (many different atomic species are trappable today, and each typically has several collisional Feshbach resonances, some of which are very narrow).

### (a) Path integral for molecules with effective action from atoms

We will formulate the time evolution of the molecules in terms of a coherent state path integral, which we will only begin to evaluate after solving the atomic problem with canonical operator methods. In this framework the dynamics for the fermions is given by a Hamiltonian that contains the molecules as *c*-number driving fields *α*(** x**,

*t*), :(2.2)Because this Hamiltonian is quadratic in the fermionic operators, it may in general be diagonalized into pairs of modes. As a simplification that should become realistic in the limit of a very slow evolution, we examine the case where

*α*and are constant in space, so that the decoupling fermion mode pairs are simply opposite momentum and spin modes. With the substitutions

*α*(

**,**

*x**t*)→

*V*

^{−1/2}

*α*

_{0}(

*t*) and , we switch to Fourier notation, and the Hamiltonian can be written as a sum where(2.3)with . Because only the states and are coupled by , the fermionic dynamics effectively factorizes into that of independent two-state systems, with time-dependent driving. The sum of all the quantum phases of these time-dependent systems, which is in general a functional of

*α*

_{0}(

*t*) and , provides the effective action for the bosonic path integral.

### (b) Adiabatic time evolution and geometric phase

This fermionic evolution may be solved explicitly as long as *Δ*(*t*) is slow in comparison with the avoided crossing width *γ*|*α*(*t*)| for *t*>*t*_{i}, and as long as the bosonic fields *α*_{0}(*t*), consist of one component that is similarly slow, plus faster fluctuations that remain small at all times. For sufficiently slow *Δ*(*t*), however, one expects to be able to show retrospectively that the bosonic path integral is indeed dominated by such paths. We can therefore evolve under adiabatically, and then add perturbative corrections from the ‘small, fast’ components of arbitrarily fluctuating bosonic fields.

In fact we can handle something more general than a strictly slow *α*_{0}(*t*), and as it turns out, we must do so. We need to consider *α*_{0}(*t*) to have slowly varying modulus and frequency; but the rate of change of its phase need not be small. That is, we make the ansatz(2.4)(2.5)where *N* is the conserved eigenvalue equal to the total number of bosons, plus one-half of the total number of fermions. *θ*(*t*), *ω*(*t*) and *r*_{0}(*t*) are assumed to be slowly time dependent, but none is required to be small. For any particular given path *α*_{0}(*t*), slow time dependence of *θ*(*t*) is identical to a small perturbation on *ω*(*t*), so *for any single path* we could take *θ* to be constant without loss of generality. It is important to realize, however, that *r*_{0}(*t*), *θ*(*t*) are the integration variables in the path integral; to include *ω*(*t*) as well, as a dynamical variable for the bosons, would be overcounting of degrees of freedom. So *ω*(*t*) is a fixed function, not a fluctuating path integration variable. We can freely choose *any* slow *ω*(*t*), and we will need to choose a particular one, in order to ensure that *r*_{0} and *θ* are self-consistently slow, as we assume, for all paths that dominate the path integral. However *ω*(*t*) cannot be chosen independently for different *r*_{0}(*t*), *θ*(*t*) trajectories. So it is important to retain *θ*(*t*) as an arbitrary slow function of *t*.

If the state of each opposite momenta mode pair isthen our many independent two-level problems take the form(2.6)Here we have also made a final change of variables to make all quantities dimensionless, by expressing them in the natural Fermi units (Fermi energy, *k*_{F}, and so on). In particular, we have defined .

As long as all the explicitly time-dependent parameters in these equations vary slowly in comparison with *γ*_{F}*r*_{0}(*t*_{i}), the two-state evolution will be adiabatic. This limit is attainable in current experiments, in which practically all the initial fermions are adiabatically converted into bosons. Note that this two-body adiabaticity does *not* imply many-body adiabaticity, because there is always a gap ≥*γ*_{F}*r*_{0}(*t*_{i}) between the two-state fermionic sector energies, but the full many-body spectrum is gapless (owing to the presence of bosonic collective modes that we represent through our path integral). This means that even when the adiabatic conversion of fermions into bosons is total, the correlation length of the final bosonic state will approach infinity only in the limit of infinite slowness.

The result of solving all these two-state problems in the adiabatic approximation is the bosonic path integral's effective action(2.7)The *k*-dependent lower adiabatic energy eigenvalue of each two level system is(2.8)The main result we report in this paper is the unusual extra contribution to the kinetic term in the effective Lagrangian,(2.9)This extra term arises as a geometric phase in the adiabatic fermion evolution under driving by the slow bosonic fields. It is thus trivial in equilibrium calculations but very important in the time-dependent problem. It is in fact the mechanism by which the real-time bosonic path integral expresses the creation of bosons as the already solved fermions disappear.

### (c) Classical equations of motion

In principle the action for small fast fluctuations around the slow *α*_{0}(*t*) assumed above still remains to be computed perturbatively, and generalization to *α*(** x**,

*t*) that vary (slowly) in space remains as well. It has recently been shown, however, that for small

*γ*

_{F}the BCS–BEC crossover can be treated semi-classically (see Diehl & Wetterich 2006; Diehl

*et al*. 2007), and the path integral can be evaluated in saddle point approximation. We will therefore restrict our attention to the saddle point of our action, which for small enough sweep rate

*ν*will indeed have

*α*(

**,**

*x**t*) varying very slowly in both space and time. We therefore simply extremize our action with respect to variations in

*r*

_{0}(

*t*) and

*θ*(

*t*). Tuning

*ω*(

*t*) to enforce that

*θ*(

*t*) is indeed slow fixes it uniquely. We report the nonetheless cumbersome details elsewhere, and here simply quote the results for small

*γ*

_{F}. A full numerical solution for

*γ*

_{F}=0.1 is shown in figure 1. The point on the horizontal axis is the moment when the Feshbach resonance meets the Fermi surface and begins sweeping inwards, adiabatically pairing all fermions into bosons by the time it reaches the centre at

*t*=0.

## 3. Discussion

As we would expect, at early times we recover BCS physics. The asymptotic solution of our equations of motion at early times becomes(3.1)which remains non-perturbatively small for small *γ*_{F} for and coincides with the solution to the BCS gap equation for the fermion–fermion interaction mediated by the off-resonance bosons. At late times, in a similar way, we recover Gross–Pitaevskii behaviour. In between, it is the growth in the bosonic population prescribed by the geometric phase contribution in our action's kinetic term which is mainly of interest. In the interval −2<*ν*^{2}*t*<0, grows from very close to 0 to very close to 1; it then approaches 1 ever more closely for all *t*>0. The transitions between these three time regimes for *r*_{0} are gradual on the time scale .

What is important to note is that for *γ*_{F}→0, it is not just that the growth in has a simple solution, but that once the fixed function *ω*(*t*) has been self-consistently determined, the crucial effective action term that fixes the growth of *r*_{0}(*t*) becomes trivial:(3.2)Variation of the action with respect to the boson phase *θ* produces the constraint , so that the time dependence of *f* drives boson production; but this driving term in the action does not even depend on *r*_{0} (to leading order in ). For the small *γ*_{F} adiabatic experiment we model, condensate growth is prescribed by a constraint, rather than dynamically.

This is no unreasonable feature in an adiabatic process, but it provides a significant caveat for attempts to estimate fluctuations in condensate number or phase by referring to time scales for mean condensate growth. What we have is a counterexample, showing explicitly that in a qualitatively Kibble–Zurek-like process (Kibble 1976; Zurek 1985, 1996), the equilibration or growth of the order parameter's phase fluctuations may be quite unrelated to the growth of the order parameter's modulus. For as we have mentioned, the gapless many-body spectrum implies that phase fluctuations on some sufficiently long wavelengths will indeed be excited by any finite speed sweep through a Feshbach resonance. Yet the rate and temporal profile of mean condensate growth is fixed entirely by the adiabatic two-fermion physics, and therefore appears in the bosonic path integral as a geometric constraint. The conclusion is that the fluctuations and mean field cannot in general be expected to evolve on the same time scales, as they are assumed to do in simple theories like the so-called time-dependent Ginzburg–Landau theory. A more rigorous theory of fluctuations will in general be required. The semiclassical many-body theory with the path integral we have described will provide such a theory for the extremely slow time-dependent BCS–BEC crossover problem at small *γ*_{F} and zero temperature.

The fully adiabatic generation of molecules is possible because for finite *γ*_{F} there is no phase transition involved in the BCS–BEC crossover. For *γ*_{F}=0, however, there is a quite trivial second-order quantum phase transition: the free particle ground state energy for fixed *N* has a discontinuity in at *Δ*=−2 (the Fermi surface). The fact that the adiabaticity threshold for *ν* is proportional to *γ*_{F} even for fixed initial gas density and atom–atom effective scattering length may be considered related to critical slowing down at the (*γ*_{F}, *Δ*)=(0, −2) critical point. If this extra-slow slowness is not attained, the adiabatic effective action we have found becomes self-inconsistent, and in fact adiabaticity will only be restored after a sufficient molecular population has been formed non-adiabatically. This regime may well prove to resemble more closely the original Ginzburg–Landau-based conception of Zurek.

Obtaining rigorous results for this non-adiabatic regime will be less trivial, for finite temperatures it will be more difficult, and for larger *γ*_{F} will be much more difficult still. Indeed, experiments to look for Zurek-scenario vortex generation via Feshbach sweeps with large *γ*_{F} could offer an important new type of data for motivating and checking theoretical ideas about a difficult many-body regime of strongly interacting fermions. An idea that was originally intended to use known terrestrial physics for explaining cosmological structures may yet turn out to be an important contribution from cosmology to deeper understanding of many-body physics here on Earth.

## Acknowledgments

Support from the Deutsche Forschungsgemeinschaft via the Graduiertenkolleg ‘Nichtlineare Optik und Ultrakurzzeitphysik’ is gratefully acknowledged.

## Footnotes

One contribution of 16 to a Discussion Meeting Issue ‘Cosmology meets condensed matter’.

- © 2008 The Royal Society