Dynamics of quantum quenching for BCS-BEC systems in the shallow BEC regime
The problem of coupled Fermi-Bose mixtures of an ultracold gas near a narrow Feshbach resonance is approached through the time-dependent and complex Ginzburg-Landau (TDGL) theory. The dynamical system is constructed using Ginzburg-Landau-Abrikosov-Gor’kov (GLAG) path integral methods with the single mode approximation for the composite Bosons. The equilibrium states are obtained in the BEC regime for adiabatic variations of the Feshbach detuning along the stationary solutions of the dynamical system. Investigations into the rich superfluid dynamics of this system in the shallow BEC regime yield the onset of multiple interference patterns in the dynamics as the system is quenched from the deep-BEC regime. This results in a partial collapse and revival of the coherent matter wave field of the BEC, whose temporal profile is reported.
Over the course of the last two decades, there has been a groundswell of theoretical and experimental interest in ultracold gases of alkali metals confined in optical and/or magnetic traps. These systems of ultracold gases have proved to be extremely robust and tunable systems for studying condensed matter physics in regimes that are inaccessible in solid state systems . During this time, physicists also became specifically interested in condensates of Fermionic alkali atoms (such as ), and obtaining a BCS superfluid of Cooper pairs similar to those seen in solid state superconductors (BCS theory). If the effective attraction between Cooper pairs can be rendered sufficiently strong, Cooper pairs of Fermions are no longer merely correlated and far apart as in traditional BCS systems, but have much smaller correlation lengths approaching the interparticle spacing. Thus, they can be treated as composite Bosons, causing the system to undergo a BCS-BEC crossover to a BEC superfluid.
Theoretical work on the single channel model of the BCS-BEC crossover over the course of the late s and s [2, 3, 4, 5, 6] motivated numerous experiments with laser cooling and trapping of Fermions, culminating in the observation of this crossover in the early s [7, 8, 9, 10]. In the late s, the more informative dual-channel model of resonant superfluidity was proposed, by Holland et al , and by Timmermans et al . This model, built from the Timmermans’ Hamiltonian, is a generalization of the Dicke (Tavis - Cummings) model in quantum optics . The attractive interaction between the Fermions can be controlled by tuning a homogeneous magnetic field to their Feshbach resonances , which are caused in a two-particle system by coupling bound states in a close channel with states in the scattering continuum. The effective scattering length can be tuned simply by varying the external magnetic field that controls the net magnetic moments of the different channels. In this dual channel model, sufficiently strong resonances cause bound states to form in the closed channel. Thus, Cooper pairs of Fermions can physically combine into Bosonic molecules , entities referred to variously as composite Bosons, diatomic molecules, or quasimolecular states in the literature. These composite Bosons have extremely long lifetimes near a Feshbach resonance [15, 16], and repel each other , facilitating their condensation to a BEC superfluid [18, 19]. This area of research has gained enormous interest over the last two decades. An overview of the phase portrait of a population balanced Fermi gas can be found in , and reviews of the current status of research can be found in [21, 22].
The possibility of observing the nonequilibrium dynamics of coherent quantum states via quenching (diabatic variations of the system parameters) is one that is unique to systems of ultracold gases, and unavailable in other similar condensed matter systems. Post-quenched dynamics in cold atom systems have been studied and reported. In particular, the collapse and revival of a post-quenched coherent BEC state in an optical lattice  generated considerable interest. Quenched dynamics have also been investigated for Fermi-Bose mixtures, experimentally , as well as theoretically [26, 27, 28, 29, 13, 30]. The possibility of observing a collapse and revival phenomenon as the system is quenched past the BCS-BEC crossover has been raised . More recently, nonequilibrium dynamics in condensed matter physics have also been described by the complex Ginzburg-Landau equation . The nonlinear damping in the dynamics produces very rich and diverse behavior in Fermi-Bose mixtures . The mean field dynamics of BCS superconductivity have been obtained from microscopic models via Ginzburg-Landau-Abrikosov-Gor’kov (GLAG) theory. A similar approach has proven successful for BCS-BEC systems as well, thus leading to their description by the TDGL equation for the single channel case . More recently, the applicability of TDGL dynamics have been demonstrated in the dual channel case . This motivates the use of this treatment to study the dynamics of coherent matter waves in BCS-BEC systems in this report.
This paper focuses on the dynamics of the TDGL equation in coupled BCS-BEC systems, and the dynamics of quantum quenching therein. The most general state of this system across the phase diagram is that of a Fermi-Bose mixture, where the composite Bosons coexist with correlated Fermions, and their dynamics are linked in the mean field by the two-channel scattering process through coupled Ginzburg-Landau and Gross-Pitaevski equations. The mixture is characterized by two distinct phases, the Fermi (BCS) superfluid phase given by the order parameter , and the Bose (BEC) superfluid phase given by the order parameter . The Fermi superfluid consists of distinct Cooper pairs, and the Bose superfluid consists of quasimolecular Bosons. This 2-channel model is more faithful to the microscopic nature of the system than the single-channel model, especially when Feshbach resonances are involved. The dynamics of the coupled BCS-BEC phases is richer than that described by a single-phase model in the single channel case, since the Ginzburg-Landau dynamics of the latter is only the damped one except in the BEC regime. This is especially true if the Feshbach resonance is narrow, which is the case being discussed in this work. Though many current experiments correspond to broad resonances, the consideration of a narrow Feshbach resonance here is not unrealistic at all [35, 36]. The paper reconstructs the time dependent Ginzburg Landau equations for the dynamics of a Fermi-Bose mixture in the two channel case starting from the many-body functional field integral for the Timmermans’ Hamiltonian. This Hamiltonian describes the Fermions by the BCS Hamiltonian and the composite Bosons by the Bose-Hubbard Hamiltonian , and includes a resonant coupling between the two species. The path integral is written as a functional of the superfluid gap parameter and the boson coherent state (all of them are taken to lie in the zero momentum state) , taken to be a c-number. The mean-field dynamics is subsequently derived, and the equilibrium densities evaluated at the stationary solutions. The paper then details investigations of the dynamics of quantum quenching, and reports the possibility of collapse and revival in the full matter wave of the BEC at large times. Section 2 begins by outlying the formalism that obtains the TDGL dynamics from the Timmermans’ Hamiltonian. Section 3 reports the study of the stationary solutions of the TDGL dynamical system in 3 dimensions. The chemical potential and condensate fractions are evaluated as the Feshbach detuning is varied adiabatically. Section 4 looks at the dynamical evolution of the BEC (which can be seen directly in the lab via time-of-flight absorption) as the Feshbach detuning is varied diabatically (ie ’quenched’) to the shallow BEC regime. Concluding remarks are made in Section 5.
2 Dynamical Equations of Motion
The treatment that obtains the coupled TDGL dynamics of this system closely follows that which obtains the conventional TDGL dynamics of the single channel model, as applied by Huang, Yu and Yin . The formalism is applied to the dual channel case in a manner similar to the treatments by Machida and Koyama , except for the nature of the Bosonic states, which is approximated by a single mode. It is assumed that, at , the time-scales of the dynamics are sufficiently weak so as to not induce Boson excitations above the ground state, an assumption justified in the context of dynamics in greater detail in the literature . This approximation greatly simplifies the dynamics by removing any spatial information in the composite Boson amplitudes from the beginning. This approximation also allows for rapid transitions near unitarity.
The dimensional Fermion and zero momentum composite Boson fields in this system are represented by the operators and respectively, with the index representing the Fermion pseudospin. The dual-channel Timmermans’ Hamiltonian  for a Fermi-Bose mixture at for a unit volume is
Here, . Equation 2 describes a system of ultracold electrically neutral two-component Fermions interacting attractively. The first line in equation 2 represents the Fermi-BCS part of the Hamiltonian. Here, is the single particle Hamiltonian, and is the Fermion mass (the mass of the composite Bosons is thus ). The second line represents the Hamiltonian of the composite Bosons [12, 32]. Here, the Feshbach threshold energy (also called the Feshbach ’detuning’ from the molecular channel to the continuum ) is represented by , and represents the amplitude of the repulsion between the composite Bosons. The final line describes the Feshbach resonance that leads to the Fermions binding to (or dissociating from) the composite Bosons, with representing the atom-molecule coupling. Note that the chemical potentials satisfy . The path integral grand partition function is defined by
where and are the path integral measures of the Fermion and Boson fields respectively, and is the imaginary time. The action is given by
This integral can be evaluated by introducing the macroscopic gap parameter into equation 3 via the Gaussian identity
where the Fermion field (characterized by the gap parameter) is spatially homogeneous due to it’s coupling to the spatially homogeneous Bose field by momentum conservation. Performing the Hubbard-Stratonovich transformation, and , cancels out the four-Fermion term from in equation 4. Now, integrating out the Fermion fields remaining in equation 3 using formal Grassman calculus yields
The action is split into a field independent part , where
and a field dependent part which vanishes when and do. Expanding to the fourth order in  and performing gradient expansion results in
the coefficient is obtained from ,
Here, , where is the measure of integration over all spatial degrees of freedom, and
with , and the Gor’kov Green’s function, , which is obtained from the Green’s function of a noninteracting Fermi gas i.e.
Note that the product in the expression for above is over the spatial dimensions only, whereas the measure is over the space-time dimensions . The mean field equations of motion of the order parameters can be obtained by equating the functional derivatives and to after analytically continuing the time to the real axis by substituting . This yields the final dynamical equations for this system
where is the total (Fermion) particle number. The constants expressed as Greek letters in equation 2 are given by
Finally, note that time has been rendered dimensionless via the transformation , and the chemical potential is presumed to be negative. Thus, the dynamics of the Fermi-Bose mixture is that of a system where the Fermi and Bose fields evolve according to coupled Ginzburg-Landau Gross-Pitaevski-Bogoliubov dynamics. The coupling is caused in the Ginzburg-Landau case by the order parameter getting nonlinearly dressed by the Bose field , and in the Gross-Pitaevski case by a harmonic coupling to .
It is also noted that the dynamical system bears a resemblance to that of a molecular BEC of atomic Bosons in a resonance effective field theory as proposed by Kokkelmans and Holland in 2002 . The role of the pairing field of noncondensed atoms (represented by the ’anomalous density’ of noncondensed pairs) in that system is assumed by the Fermion gap parameter (related to the anomalous Cooper pair density by ) in this one. This strengthens the analogy between Cooper pairs in Fermi-Bose systems and noncondensed atoms in Bose-Bose systems.
The dynamics is investigated for a system confined in a 3 dimensional box where the confinement is much larger than the inter-particle spacing, effectively treating the trap as homogeneous. The constants and can be evaluated from equations 2 and 11 by using the Green’s function for a free particle. This yields [34, 37]
Here, is the Fermion energy and is the renormalization cutoff in momentum, placed to counter ultraviolet divergences. An important caveat here is that the integrals in equation 17 contain singularities if . Thus, this formalism breaks down in that regime, which corresponds to regions where the BCS state dominates over the BEC state, well beyond the current region of interest at a negative . Going to the continuum limit by substituting for the formal sum in equations 17 and performing the integrals results in
where the dimensionless constants in Greek letters are now functions of the chemical potential
has been used to obtain the expression for . Here, , is the bare interaction , and is the s-wave scattering length controlled by Feshbach resonance. This relation comes about as a consequence of renormalizing the BCS gap equation so as to counter ultraviolet divergences in the single channel model [3, 5, 37]. Finally, note that the inability to renormalize this theory analytically disallows taking the theoretical limit of for some of the constants above. This problem will be discussed further in the subsequent section.
3 Stationary solutions and Chemical Potential
For a complete phenomenological description of the dynamics, number conservation has to be satisfied and used as a constraint at to obtain the chemical potential where . In order to do this, the action from equation 9 is investigated at small temperature . Since we only consider regimes with negative chemical potential, all the Fermions in the gas are correlated and no ’free Fermions’ remain
where the quartic contribution has been neglected. The temperature dependence of all constants can also be neglected and the expression above simplified to get
The Helmholtz free energy at is calculated from
Simplifying equation 23 by taking the mean field (ignoring fluctuations) yields
Thus, the Helmholtz free energy at is given by
Imposing number conservation by using ,
where , and the dependence of on is obtained from equations 17. Writing the equation above in terms of dimensionless variables,
where are the stationary solutions of the dynamical system. The trivial stationary solutions of the dynamics in the phase space, , are realized at temperatures above the critical temperature . The nontrivial ones are the locus of points satisfying the equations above plus the chemical potential equation 27. Thus, the 3 unknowns are solved from the 3 simultaneous equations
As explained in section 2, the BCS-dominant regime is not entirely accessible in this formalism. However, the predominantly BEC gapless regime can be obtained from equations 3. In that regime, ie . In the case of noninteracting Bosons, ie (), the second equation of 3 necessitates that ie , in agreement with mean field results . This is also consistent with the physics of the system, since the chemical potential is the energy required to remove one Fermion from the system. In the gapless and noninteracting BEC dominant case, the majority of the Fermions are dimerised with binding energy , and a dimerised Fermion requires energy to dissociate from the dimer and free itself. For the more general case,
Clearly, and so has to be negative for this to be true ( is always positive). Also
Since , so is . Therefore, for the equation above to hold, and therefore must be large and negative for a gapless BEC dominant state to exist. This criterion is in agreement with the physics of the system. For large negative , the binding energy and therefore the molecular affinity of the Boson dimers will be large and negative, facilitating the dimerisation of the majority of the Fermions . In order to calculate the chemical potential in this regime, the relation is substituted in equation 32. Then, equations 2 are applied, yielding
Solving this quadratic equation in and rejecting the unphysical root,
In the case of noninteracting Bosons, equation 34 gives as above. Furthermore, in this regime (no BCS state, all Fermions are dimerised), or . This simplifies equation 27 to or . Figure 1 shows plots of as functions of (where ) for , and several large values of . Note that and therefore indeed vanishes in the limit , which is where the BEC regime is expected.
Continuing with the case of noninteracting Bosons ( ie ), equations 3 reduce to
with . Solving the first two equations of 3 yields
where . The problem of renormalizability of this dual-channel theory mentioned in the previous section manifests here. The inability to renormalize happens because of the quantity in the equations above. In the single channel case, when and , this expression converges as , as can be seen using equations 2, yielding
However, finite nonzero leads to vanishing as , yielding which diverges for arbitrarily large (see equations 2)
The difficulty described above can be worked around by noting that the allowed momenta need to have a finite cutoff , which is governed by the physics of the system. For instance, in the case of solid state BCS systems, the cutoff is given by the Debye frequency on account of the physical origins of the electron-electron attraction in the slowly relaxing lattice vibrations. Ordinarily, the choice of cutoff is decided by the range of the interatomic potential, with if the bare repulsion between the Fermions is neglected [12, 38]. However, for small confinements and small detuning, the maximum momentum allowed in these cold atom systems cannot be greater than , where is the size of the atoms. Experimentally, these atoms are very cold, and never move so fast as to cause inelastic scattering beyond the Feshbach resonance, since such excitations will dynamically alter the internal degrees of freedom of the atoms themselves. Thus, this choice of keeps a reasonable upper bound in all of the integrations. For Fermionic atoms like , is , which needs to be rescaled with respect to the unit length viz. the confinement dimensions. The confinement, however, is highly tunable in cold atom systems, with a fairly wide range of permitted values. Thus, using a wide variety of ’s to get quantitatively correct results should be permissible, once the trap sizes are adjusted accordingly. In this report, are chosen to be in units of inverse trap size. These should be attainable in small volume systems of atoms (obtained experimentally via laser culling by Chuu et al ) confined to length scales of (obtained experimentally by confining ultracold gases using atom chips ).
Equations 3 are nothing more than the gap equation for the BCS-BEC system in position space. In all these equations, are functions of . Note from the above that if ie , then , and , which will give a pure BCS until goes to a regime where can no longer be evaluated without running into singularities (as per section 2), causing this formalism to break down. Plugging the values of above to the final equation of 3 yields
Equation 38 is a transcendental equation and needs to be solved numerically for (). Figure 2 contains the numerical results of evaluating the condensate fractions and chemical potentials. These have been obtained by setting to and using representative values of the interaction parameters. The chemical potential was obtained by numerically solving equation 38 using Newton-Raphson methods. The default working precision was kept to the eighth place of decimal. Figures 2 (a), (b) and (c) contain plots for , and figures 2 (d), (e) and (f) contain plots for . Figures 2(a) and (d) are plots for as a function of . Note from these figures that, for sufficiently large , in accordance with the analytical results in the BEC regime. Figures 2(b) and (e) are plots for the condensate fractions (Fermions in BCS) and (molecular composite Bosons in BEC). The fractions are computed after solving equation 38, and substituting the corresponding values of (obtained from equations 3) into the relations (see equation 27)
The rise in the BCS superfluid density can be clearly seen in figure 2(b) around , and in figure 2(e) of . These results agree qualitatively with more sophisticated theories of BCS-BEC systems , considering that numbers are highly sensitive to the choice of the renormalization cutoff . In the case of interacting bosons, equations 3 are simplified by operating in a regime where the chemical potential is weak enough so that can be ignored. This yields
Solving the first two equations of 3 yields the gap equations in this regime. The solutions are
where a necessary condition is that be negative. Substituting these results into the final equation of 3 yields the transcendental equation for in this case,
Equation 42 is solved numerically using the same algorithms and tolerances as equation 38 for representative values of the parameters. The results are shown in figures 3. Figure 3 (a) contains plots of the condensate fractions (red) and (blue) for , , , and . Here, is chosen to be . Note the shallow BEC point at in Figure 3 (a). In this case, the fermion population appears to increase much faster than in the case of noninteracting Bosons as is varied adiabatically past this regime. The system should populate to a full BCS state shortly after unitarity.
4 Quenched Time Evolution
Equations 2 provide a complete description of mean field dynamics of the Timmermans’ Hamiltonian in equation 2. The s will not, in general, represent densities when the system departs from equilibrium. The definition of in equation 2 indicates that while will be proportional to the Boson density, will no longer represent the Fermion density out of equilibrium (which will only depend on ), since equation 26 breaks down away from equilibrium. The general quasiharmonic solutions of have been discussed by Machida and Koyama , and the global existence of weak solutions have been established recently by Chen et al . This paper focuses on the dynamics in the shallow BEC regime close to the BCS-BEC crossover as the system is quenched from the deep-BEC regime.
The continued use of the single mode approximation in this case can be justified by evaluating the maximum characteristic time scale for the quench below which sizable excitations of the closed channel Bosons and free Fermions will occur. In order to do so, the Timmermans’ Hamiltonian in equation 2 is written in momentum space with the closed channel excitations included, yielding
Here, the first line represents the Fermions in momentum space, with the creation (annihilation) operators () obtained by Fourier transforms of () from equation 2, and the free Fermion energies are represented by . The second line represents the Hamiltonian for the closed channel quasimolecular bosons () with energies . Finally, the last line represents the atom-molecule coupling discussed in section 2. The Fermi and Bose condensates exchange momentum through this coupling. The Hamiltonian above can be written in terms of the generators of the Lie algebra . Assuming that most of the Bosons are in the ground state given by , treating as a c-number, retaining only terms up to second order in () and neglecting fluctuations about the mean field for all generators, the Hamiltonian can be diagonalized in a manner similar to that done for a pure Boson system in Bogoliubov theory , with the eigenstates given by the generalized coherent state. If the system is in a pure quantum state, the energy eigenvalues are given by 
Here, are the energies of the Bogoliubov quasiparticles from conventional BCS theory, are the energies of the quasiparticle excitations of the pure Boson system from Bogoliubov theory , and / are indices that correspond to the ladder operators of the / Lie algebras respectively, with and . If the energies are referenced from the ground state , then the excitation energies are given by