We study the short-time dynamics of a degenerate Fermi gas positioned near a
Feshbach resonance following an abrupt jump in the atomic interaction resulting
from a change of external magnetic field. We investigate the dynamics of the
condensate order parameter and pair wavefunction for a range of field
strengths. When the abrupt jump is sufficient to span the BCS to BEC crossover,
we show that the rigidity of the momentum distribution precludes any
atom-molecule oscillations in the entrance channel dominated resonances
observed in the 40K and 6Li. Focusing on material parameters tailored to the
40K Feshbach resonance system at 202.1 gauss, we comment on the integrity of
the fast sweet projection technique as a vehicle to explore the condensed phase
in the crossover regionComment: 5 pages, 4 figure

Burnett, K.

Simons, B. D.

Szymanska, M. H.

English

arXiv

We study the short-time dynamics of a degenerate Fermi gas positioned near a Feshbach resonance following an abrupt jump in the atomic interaction resulting from a change of magnetic field. We investigate the dynamics of the condensate order parameter and pair wave function for a range of field strengths. When the jump is sufficient to span the BCS to Bose-Einstein condensation crossover, we show that the rigidity of the momentum distribution precludes any atom-molecule oscillations in the entrance channel dominated resonances observed in 40K and 6Li. Focusing on material parameters tailored to the 40K Feshbach resonance at 202.1 G, we comment on the integrity of the fast sweep projection technique as a vehicle to explore the condensed phase in the crossover region

Szymańska, MH

Simons, BD

Burnett, K

UCL Discovery

PRL 94, 170402 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending6 MAY 2005Dynamics of the BCS-BEC Crossover in a Degenerate Fermi GasM. H. Szyman´ska,1 B. D. Simons,1 and K. Burnett21Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 OHE, United Kingdom2Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, United Kingdom(Received 16 December 2004; published 3 May 2005)0031-9007=We study the short-time dynamics of a degenerate Fermi gas positioned near a Feshbach resonancefollowing an abrupt jump in the atomic interaction resulting from a change of magnetic field. Weinvestigate the dynamics of the condensate order parameter and pair wave function for a range of fieldstrengths. When the jump is sufficient to span the BCS to Bose-Einstein condensation crossover, we showthat the rigidity of the momentum distribution precludes any atom-molecule oscillations in the entrancechannel dominated resonances observed in 40K and 6Li. Focusing on material parameters tailored to the40K Feshbach resonance at 202.1 G, we comment on the integrity of the fast sweep projection technique asa vehicle to explore the condensed phase in the crossover region.DOI: 10.1103/PhysRevLett.94.170402 PACS numbers: 03.75.Kk, 03.75.Ss, 05.30.FkUltracold alkali atomic gases provide a valuable arena inwhich to explore molecular Bose-Einstein condensation(BEC) [1] and fermionic pair condensation [2,3]. Further,the facility to control interparticle interactions via a mag-netically tuned Feshbach resonance (FR) provides a uniqueopportunity to investigate the BCS-BEC crossover and thedynamics of condensate formation. As well as the adiabaticassociation of molecules [1], both fast sweep ‘‘projec-tions‘‘ of fermionic pair condensates onto the molecularBEC [2,3] and atom-molecule Ramsey fringes [4] havebeen reported in the recent literature. Lately, motivated byearlier work on the BCS system [5,6], it was shown [7,8]that the mean-field equations of motion of a Bose-Fermi(BF) model, commonly used to describe the FR system, arecharacterized by an integrable nonlinear dynamics. Fromthese works, three striking predictions emerged: First, afteran abrupt change in the strength of the pair interaction, thecondensate order parameter exhibits substantial oscilla-tions which range in magnitude between some initial statevalue, I, and that expected for the equilibrium final stateeq. Second, in the absence of energy relaxation processes,these oscillations remain undamped, suggesting the poten-tial to observe coherent atom-molecule oscillations in theFR system. Third, a spectral ‘‘hole-burning’’ phenomenonin the atomic momentum distribution, at one-half of themolecular binding energy, provides a signature of suchoscillations [8].In the following, we argue that this behavior rests on anauxiliary constraint that, in the present system, seems hardto justify. Drawing on the results of numerical analysis ofthe unconstrained dynamics, we show that, in the absenceof relaxational processes, the oscillations of the orderparameter are damped substantially, even at the level ofmean field. When the abrupt change of the interactionspans the BCS-BEC crossover, for both the single-channeland BF systems, the magnitude of oscillations are smalland die out, leaving  close to the initial, and much smallerthan the equilibrium value, eq. The distribution remainsessentially ‘‘frozen’’ and the hole-burning oscillations pre-05=94(17)=170402(4)$23.00 17040dicted in Ref. [8] do not occur. This rigidity of the initialstate prevents BCS-BEC atom-molecule oscillations butvalidates fast sweep techniques [2] as a probe of the cross-over region.A full microscopic theory of FR phenomena relies on allmatrix elements connecting different spin states. In prac-tice, an accurate description of the resonance can be ob-tained by using a magnetic field dependent effectiveinteraction between atoms in the entrance channel or,more generally, in the two most relevant channels. Forlow relative momenta, relevant to cold atom physics, thefull form of complex atomic potentials are not resolved.Indeed, separable potentials with parameters drawn fromexperiment and exact multichannel calculations can beused to recover all low-energy binary scattering observ-ables [9,10]. As often only one bound state of the closedchannel potential is relevant, it has been traditionally re-placed by a fictitious Bose particle and FR phenomenacaptured by a BF Hamiltonian [11].For the entrance channel dominated resonances ob-served in 40K and 6Li formal calculations [9,10] supporta picture in which the BCS-BEC crossover is mediated byonly a small admixture of closed channel states—in the40K FR at 202.1 G, the admixture of the closed channel isless than 8% of the total [10]. In fact, the weakly boundmolecular state appears at a detuning E0 which lies farfrom the value where the resonance state of the closedchannel crosses the dissociation threshold. Since the two-body observables drawn from the exact numerical solutionof the Shro¨dinger equation within finite-range single- andtwo-channel models do not differ over a wide range offields [10], FR phenomena in 40K can be equally welldescribed by a single-channel theory,H^ Xkskayksaks Xkk0qVkk0aykq"ayk#ak0#ak0q"; (1)involving Fermi operators ayks and aks. In the following, toaccount for the entire region of the crossover (and not only2-1  2005 The American Physical SocietyFIG. 1. Variation of the chemical potential  with field B atT  0 for a mixture of fermionic 40K atoms prepared in the f 9=2; mf  9=2	 and f  9=2; mf  7=2	 Zeeman states ata density of n  1:5 1013 cm3 (i.e., TF  0:35 K). The FRtakes place at B0  202:1 G while the BEC-BCS crossover(  0) occurs when B B0 ’ 0:3 G. Note that, since thedensity is nonzero,  reaches zero below the two-body FR. Theinset indicates the scattering length aB	  abg1 BBB0	 in theuniversal regime, where abg is the background potential scatter-ing length and B the width of the resonance. Note that, in thepresent theory, we use finite-range potentials with parameterswhich also describe the FR far beyond the universal region [10].PRL 94, 170402 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending6 MAY 2005the universal regime), we take as matrix elements Vkk0 V0B	kbg	k0 bg	 with kbg	  expkbg	2=2and the parameters V0B	 and bg chosen to recover thecorrect magnetic field dependence of the scattering lengthand the energy of the highest vibrational bound state [10].The Heisenberg equations of motion arei _k  2kk k2k  1	;i _k  k?k  ?kk;(2)where k  12Pshayksaksi, k  hak#ak;"i, and k V0B	kbg	Pk0k0 bg	k0 denotes the complex orderparameter. Similarly for a BF theory, defining gk g0k	 as the coupling of the entrance channel to thebosonic field bk associated with the Feshbach resonancelevel of the closed channel, the equations acquire thesame form as (2) with k  g0k	b0 Vbgkbg	Pk0k0 bg	k0 and b0  hbk0i obeying thesupplementary equation i _b0  E0B	b0  g0Pkk	k.As with the single-channel theory, the five parameterswhich characterize the resonance, the background poten-tial strength Vbg and range bg (which define the entrancechannel scattering length and its highest vibrational boundstate), the interchannel coupling g0 and its range , and thedetuning E0B	 (which specify the position and width ofthe resonance), are determined from experiment and exactmultichannel calculations [10].Before turning to numerics, it is instructive to contrastour approach to that adopted in [5,7,8]. Given an initialcondition, the Heisenberg equations (2) present a determi-nistic time evolution of the density distributions. Indeed,conservation of total density n, implicit in the dynamics(2), provides a check on the integrity of the numericalintegration. By contrast, the integrability of the equationsof motion as described by Refs. [5,7,8] relies on an addi-tional constraint involving density and an auxiliary pa-rameter playing the role of a ‘‘chemical potential.’’ Theconstraint is needed in this case to fix the value of amomentum-dependent sign in the solution which shouldproperly be determined by the initial conditions. A similarphenomenology describes the effect of a classical lasersource on a semiconductor electron-hole system wherethe laser frequency ‘‘imprints’’ a chemical potential ontothe system [12]: While equilibration processes are suffi-ciently small, the electrons and holes assume a nonequi-librium distribution around the externally imposedchemical potential resulting in a phenomenon of‘‘spectral-hole burning’’ in which the density distributionis depleted at the laser frequency. In the atomic gas, wherethe degrees of freedom remain internal, it is difficult to seehow such a choice is motivated or justified. Crucially, wewill see that the unconstrained dynamics (2) lead to be-havior very different from that obtained from the con-strained [5,7,8].With this background, let us now turn to the results of thenumerical investigation of the dynamics (2). Although we17040find that the qualitative behavior of BCS and BF dynamicsis generic, we focus specifically on potentials tailored tothe 40K resonance at B0  202:1 G with a density of n 1:5 1013 cm3 comparable to that used in experiment[2]. With these parameters, the equilibrium properties ofthe effective single-channel and BF models essentiallycoincide (for further details and values of parameters, werefer to Ref. [10]). Therefore, to keep our discussion con-cise, we focus on the single-channel theory, noting that theparallel application to the BF model with appropriatephysical parameters generates quantitatively similarresults.At a field of ca. 1.0 G above the FR, the condensate hasan essentially BCS-like character while the experimentusing the fast sweep technique observed the condensatestarting from 0.5 G above B0. To explore the entire regionof interest, we choose as initial conditions field values BIwhich span the entire crossover region (marked by stars inFig. 1). Starting from the ground state T  0 distribution,we follow the dynamics of the condensate after an abruptswitch to some different value of magnetic field BF.Figure 2 shows the time evolution of jk0j, normalizedby eq (the value that it would acquire were the system toreach the T  0 ground state at the final field BF). Here wehave chosen a field BF  B0  10 G deep within theBEC phase where the large binding energy of moleculesallows their momentum distribution to be inferred fromtime of flight measurements [2]. For comparison, insets of2-20.40.50.12 |κ k=0|PRL 94, 170402 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending6 MAY 2005Fig. 2 show BF  B0  1:0 G. In contrast to the predic-tions of the constrained dynamics [5,7,8], these resultsshow that (a) the coherent oscillations are substantiallydamped even at the mean-field level, (b) the amplitude ofthe oscillations is small, and (c) the order parameter jk0jasymptotes to a value much less than the expected finalstate equilibrium value eq. Referring to the insets ofFig. 2, one may note that, when the initial and final con-ditions are drawn closer, the period of the oscillationsbecomes longer and the effects of the damping morepronounced. Moreover, although the period of the oscilla-tions decreases monotonically with eq, the dependence isnonlinear and, referring to the bottom inset of Fig. 3, onemay note that eq does not provide a ceiling for themagnitude of the oscillation.To interpret the generic behavior, it is instructive toaccess the time dependence of the pair wave function k0 10 20 30 40 50Time [µs]00.10.20.30.40.50.6|∆ k=0|/∆eq0 200 4000.80.91BI = B0 - 0.5 GBF = B0 - 1.0 G0 200 400 600 8000.20.4BI = B0 + 0.5 G BF = B0 - 1.0 G BI = B0 - 0.5 GBI = B0BI = B0 + 0.5 GBI = B0 + 1.0 GBF = B0 - 10 GFIG. 2 (color online). Time dependence of the order parameterjk0j=eq following an abrupt switch from a field of BI B0	=G  0:5 (top), 0:0, 0:5, 1:0 (bottom) to BF  B0	=G 10, and from BI  B0	=G  0:5 (top inset) and BI B0	=G  0:5 (bottom inset) to BF  B0	=G  1:0. In additionto the constant phase velocity of k0 (found numerically to beset by the final state equilibrium ), there is an additional time-dependent phase modulation whose characteristics mirrorclosely that of the amplitude oscillations.17040and the distribution k. Figure 3 shows that, when theabrupt switch takes place from BI  B0  0:5 G to BF B0  10 G, although there is a slight tendency to shifttowards the final state equilibrium distribution, the pairwave function and the density distribution remain essen-tially frozen close to the initial BCS-like distribution,exhibiting only small oscillations in time. In the absenceof energy relaxational processes, the system is unableto significantly redistribute weight. By contrast, whenthe switch takes place from BI  B0  0:5 G to00.10.20.3|κ k|0 1 2 3Time [ms]0.80.911.1|∆ k=0|/∆eq0 0.5 1 1.5 2 2.5 3k/kF00.20.40.60.81Φk0 40 80Time [µs]0.10 1 2 3 k/kF00.20.4 |κ k|BI = B0 + 0.3 GBF = B0 + 0.2 GBI = B0 + 0.5 GBF = B0 - 10 G, -1 GFIG. 3 (color online). The pair wave function jkj (upperpanel) and the density distribution k (lower panel) shown asa function of k  jkj with BI  B0  0:5 G (dashed lines) andBF  B0  10 G after 50 s (dash-dotted line) and BF B0  1:0 G after 800 s (solid line) following the abruptswitch as in Fig. 2. The dotted lines signify the ground stateequilibrium distributions at BF  B0  1:0 G included forcomparison. The upper inset shows oscillations of jk0j forBF  B0  10 G. The lower two insets refer to a weak per-turbation on the BCS side from BI  B0  0:3 G (dashed line) toBF  B0  0:2 G. The dotted line shows the equilibrium distri-bution while the solid line provides a snapshot of the distributionat 3 ms after the switch. Note that the harmonic modulationsvisible in the distribution functions translate to a single energyscale of the same order of magnitude as the period of oscillationsseen in jk0j.2-30 10 20 30 40 50Time [µs]00.040.080.12(n mc(t)-nmc(0))/n mc(0) [%]FIG. 4 (color online). Time dependence of the condensedmolecule density nmct	  nmc0	=nmc0	. Here we have usedthe same field values as that used in Fig. 2 (main) with BI B0	=G  0:5 (top), 0:0, 0:5, and 1:0 (bottom) and BF B0	=G  10. When normalized to one-half of the total atomicdensity, nmc0	  0:3, 0:1, 0:012, and 0:0005, respectively.PRL 94, 170402 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending6 MAY 2005BF  B0  1:0 G, the proximity of the two phases al-lows the system to converge to a (nonstationary) modulateddistribution whose (stationary) envelope reflects moreclosely the final state equilibrium distribution. The lowerinsets in Fig. 3 show that a weakly perturbed condensate onthe BCS side approaches equilibrium with the pair distri-bution showing small modulations around the equilibriumone in accord with the linear stability analysis of theweakly perturbed BCS system (2) (Refs. [13,14]). In par-ticular, one may note that here (and, indeed, for other initialand final conditions) the hole-burning predicted by theconstrained dynamics [8] does not appear.To assess the potential of observing coherent atom-molecule oscillations, we now look at the time evolutionof the number of condensed molecules, nmct	 jR d3kkt	Bk; BF	j2. Here Bk; BF	 is the wave func-tion of the highest bound eigenstate of the two-body prob-lem [10]. Referring to Fig. 4, we see that the amplitude ofoscillations is negligible. Although one may adjust the finalfield BF to lie closer to the FR, the amplitude of theoscillations increases only slightly while the dampingrate is enhanced. Moreover, the inclusion of processesbeyond mean field would simply increase the dampingrate and not enhance the oscillations. We therefore con-clude that the observation of atom-molecule oscillations,after an abrupt change in the interaction strength, is infea-sible for the entrance channel dominated resonances cur-rently studied in 40K and 6Li.To conclude, we have presented a numerical analysis ofthe dynamical mean-field equations for the single-channeltheory of the FR following an abrupt field change. In therange of physical parameters appropriate to the 40K sys-tem, consideration of the two-channel BF theory does notchange the results. When applied to a theoretical regimewhere the population of the closed channel states below17040resonance is high, the numerical findings do not changequalitatively. Relying on the deterministic time evolutionof the initial state according to the Heisenberg equations,our results differ substantially from the findings of theconstrained dynamics [5,7,8]. Over a wide range of initialconditions, we observe substantially damped oscillationswith an amplitude strongly dependent on initial conditionsand a frequency set by the interaction strength after theswitch. To assess the capacity for BCS-BEC-like atom-molecule oscillations following an abrupt change in theinteraction, we have chosen initial and final conditions tospan the crossover from the BCS to the BEC limits. Wehave found that the amplitude of atom-molecule oscilla-tions is negligible and the distribution is essentially frozento the initial one, a behavior reminiscent of an orthogonal-ity catastrophe. We conclude that, in the entrance channeldominated resonances observed in 40K and 6Li, BCS-BECatom-molecule oscillations cannot be observed. The rigid-ity of the condensate wave function distribution at short-time scales, where the time evolution is mean field incharacter, supports the method of the fast sweep as areliable technique to probe the fermionic pair condensate.We are grateful to Krzysztof Go´ral and Thorsten Ko¨hlerfor numerous discussions concerning the FR, to PeterLittlewood and Emil Yuzbashyan for stimulating discus-sions, and to Sławomir Matyjas´kiewicz for advice concern-ing the numerical techniques. This research has beensupported by Gonville and Caius College (M. H. S.) andthe Wolfson Foundation (K. B.).2-4[1] M. Greiner et al., Nature (London) 426, 537 (2003); S.Jochim et al., Science 302, 2101 (2003); M. W. Zwierleinet al., Phys. Rev. Lett. 91, 250401 (2003).[2] C. A. Regal et al., Phys. Rev. Lett. 92, 040403 (2004).[3] M. W. Zwierlein et al., Phys. Rev. Lett. 92, 120403 (2004).[4] E. A. Donley et al., Nature (London) 417, 529 (2002).[5] R. A. Barankov et al., Phys. Rev. Lett. 93, 160401 (2004).[6] E. A. Yuzbashyan et al., cond-mat/0407501.[7] R. A. Barankov et al., Phys. Rev. Lett. 93, 130403 (2004).[8] A. V. Andreev et al., Phys. Rev. Lett. 93, 130402 (2004).[9] K. Go´ral et al., J. Phys. B 37, 3457 (2004).[10] M. H. Szyman´ska et al., cond-mat/0501728 [Phys. Rev. A(to be published)].[11] M. Holland et al., Phys. Rev. Lett. 87, 120406 (2001); E.Timmermans et al., Phys. Lett. A 285, 228 (2001); Y.Ohashi et al., Phys. Rev. Lett. 89, 130402 (2002).[12] S. Schmitt-Rink et al. Phys. Rev. B 37, 941 (1988).[13] A. F. Volkov and Sh. M. Kogan, Sov. Phys. JETP 38, 1018(1974).[14] In this context, it would be interesting to explore thepotential connection between the weak coherent oscilla-tions and equilibrium quantum fluctuations [15] foundabove and below Tc in the BF system.[15] Y. Ohashi and A. Griffin, Phys. Rev. A 67, 063612 (2003);T. Domanski and J. Ranninger, Phys. Rev. B 70, 184503(2004).

Dynamics of the BCS-BEC crossover in a degenerate Fermi gas

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