Proposal for interferometric detection of topological defects in modulated superfluids
PProposal for interferometric detection of topological defects in modulated superfluids
Mason Swanson, Yen Lee Loh, and Nandini Trivedi
Department of Physics, The Ohio State University, 191 W Woodruff Avenue, Columbus, OH 43210 (Dated: June 20, 2011)
Attractive interactions between fermions canproduce a superfluid ground state, in which pairsof up and down spins swirl together in a coordi-nated, coherent dance. How is this dance affectedby an imbalance in the population of up and downfermions? Do the extra fermions stand on thesides, or do they disrupt the dance? The mostintriguing possibility is the formation of a modu-lated superfluid state, known as an LO phase, inwhich the excess fermions self-organize into do-main walls where the pairing amplitude changessign. Despite fifty years of theoretical and exper-imental work, there has so far been no direct ob-servation of an LO phase. Here we propose an ex-periment in which two fermion clouds, preparedwith unequal population imbalances, are allowedto expand and interfere. A zipper pattern in theinterference fringes is unequivocal evidence of LOphysics. Furthermore, because the experimentis resolved in time and in two spatial directions,we expect an observable signature even at finitetemperatures (when thermal fluctuations destroylong-range LO order averaged over time).
The study of modulated superfluidity has its roots inthe context of a superconductor in a parallel magneticfield. The inter-electron attraction favors a superfluid(SF) state consisting of pairs of up- and down-spin elec-trons, whereas the field favors a polarized Fermi liq-uid (FL) state with a lower Zeeman energy. The sim-plest theories predict a first-order SF-FL transition .However, the competition between pairing and polariza-tion can produce far more subtle physics – an interme-diate Larkin-Ovchinnikov (LO) state , as depicted inFig. 1. As the field is increased beyond h c it forces ex-cess fermions into the superfluid in the form of domainwalls – topological defects – at which the order param-eter changes sign between positive and negative values.(This is analogous to how a perpendicular field forcesvortices into a superconducting film.) The wavelength ofthese modulations decreases with increasing field and ul-timately the system gives way to a polarized Fermi liquid,shown in Fig. 1.As a fascinating example of self-organized quantummatter, LO states have long been sought after in su-perconductors, in ultracold atomic gases , and even inneutron stars . In the context of cold atoms, there havebeen proposals to detect LO states through (a) modula-tions of the polarization in real space; (b) peaks in thepair momentum distribution at the modulation wavevec-tor; (c) shadow features in the single-particle momentumdistribution; and (d) Andreev bound states in the density of states. A recent experiment on an array of tubes founddensity profiles in agreement with Bethe ansatz calcula-tions, which predict power-law LO correlations at zerotemperature. However, direct evidence of the modula-tions of the order parameter – the defining feature of anLO state – is still lacking. Pairing Δ Polarization m h c1 h c2 h = ( µ ↑ − µ ↓ ) / Paired SF Strong LO Weak LO Polarized FL x x x x Δ m FIG. 1: (Top) Depictions of a fully paired superfluid, an LOstate with excess fermions in domain walls, and a polarizedFermi liquid. (Center) Pairing amplitude ∆ and magnetiza-tion m as a function of the Zeeman field h , which is the differ-ence between the chemical potentials of up and down spins.The LO phase exists in a field range h c < h < h c . (Bottom)Real space behavior of ∆( x ) and m ( x ) in each phase. Experimental Proposal:
In this Letter, we propose aninterference experiment in which two isolated superflu-ids expand into each other, as illustrated in Fig. 2. Theideal situation is shown in the lower panel of this fig-ure, where one layer is a uniform fully-paired superfluid,which serves as a reference phase, and the other layer isa modulated (LO) superfluid. The resulting interferencepattern is directly sensitive to real-space modulations ofthe order parameter, and should provide an unequivocalsignature of the elusive LO phase.LO states have been predicted to exist in varioussituations . The most likely of these to be realizedin the near future is an array of tubes with a small inter-tube coupling.
This quasi-one-dimensional geometryprovides good Fermi surface nesting at the LO wavevec- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J un x y z LO / SF Δ FIG. 2: Proposed experiment and interferometric LO signa-ture. An optical lattice is used to prepare two independentlayers. These clouds are subsequently allowed to expand andinterfere and are then imaged along one of the in-plane direc-tions (here y ). If both clouds are in the fully paired SF phase,the interference pattern is similar to the familiar double-slitexperiment (top). However, if one of the layers contains anLO state (the change in sign of ∆( x ) is indicated by color),the interference pattern contains sharp dislocations of the in-terference fringes (bottom). tor q LO = k F ↑ − k F ↓ , together with Josephson couplingbetween the order parameter in adjacent tubes which isnecessary to stabilize true long-range order.Thus, we propose the following experiment. Twospecies of fermions (referred to as ↑ and ↓ ) are loaded intoan optical trap. The cloud is separated into two indepen-dent quasi-2D “pancake” layers using an optical latticewith a wide spacing in the vertical direction z (createdwith two laser beams intersecting at a shallow angle). The two layers are caused to have different populationimbalances, such that n top ↑ (cid:54) = n top ↓ but n bottom ↑ = n bottom ↓ .This may happen by chance due to natural number fluc-tuations; alternatively, one can induce hyperfine tran-sitions at the beat frequency between two laser beams,using a shallow-angle interference technique with appro-priate phase shifts to address each layer separately. Anadditional optical lattice is further used to create a 2Darray of weakly-coupled tubes, conducive to the forma-tion of an LO state. The resulting geometry is shown inFig. 3. x y z Imaging beam d a
FIG. 3: The fermion gas is confined in a harmonic trap. Anoptical lattice with a large spacing in the z -direction is usedto separate the gas into two independent quasi-2D layers. Asecond optical lattice in the y -direction cuts each layer into aseries of weakly coupled tubes – the optimal geometry for LOphysics. The trap and lattices are turned off abruptly, allow-ing the two layers to expand and interfere with one another.Finally, the shadow of a probe laser in the y -direction givesthe interference pattern projected onto the x - z plane. Once the gases are allowed to equilibrate for sufficienttime, the interactions in the system are quickly rampedto the BEC side of the Feshbach resonance to “freeze”or “project” the pair wavefunction into a boson wave-function, so that from this point onwards the pairs moveas independent bosons (instead of disintegrating intofermions). Then, the confining potentials are abruptlyturned off. As the clouds expand into one another, theyinterfere and form a 3D matter wave interference pattern.The projection of this interference pattern onto the x - z plane can be measured by absorption imaging along the y direction. Any LO phase modulation features will becaptured in these projected interference patterns. Interference between coupled tubes:
We now discuss an-alytically the salient features of the interference patternsof a 2D array of coupled tubes. We begin by consider-ing two layers, each containing N coupled tubes, withseparation d in the z -direction. After ramping up the in-teraction to produce a molecular BEC (of fermion pairs),the wavefunction is∆( x, y, z ) = e − ( z − d/ / σ z N (cid:88) n =1 ∆ Tn ( x ) e − ( y − an ) / σ y (cid:124) (cid:123)(cid:122) (cid:125) top layer+ e − ( z + d/ / σ z N (cid:88) n =1 ∆ Bn ( x ) e − ( y − an ) / σ y (cid:124) (cid:123)(cid:122) (cid:125) bottom layer (1)where a is the separation between in-plane tubes and σ y and σ z are the Gaussian confinements in the respec-tive directions. The wavefunction in the n th tube is de-noted by ∆ Tn ( x ) in the top layer and by ∆ Bn ( x ) in thebottom layer. When the trap and lattices are switched x z x z x z Fully paired SF Phase locked LO Phase unlocked LO Top layer pairing amplitude t = 0 Integrated interference pattern t = t tof x y x y x y -1 ! FIG. 4: Interference patterns for three different configura-tions: fully paired SF state (top), locked LO state (middle),and unlocked LO state (bottom). In each case we considera 2 × x ) in each of the five top layer tubes isshown to the left of the interference patternt. The locked LOstates were taken to be ∆( x ) = e − x / σ sn( x/L | k ), for k (cid:46) L , where sn( x | k ) is the Jacobi sine function. Inthe unlocked case we added a random displacement of the do-main walls ∆( x ) = e − x / σ sn(( x + δ ) /L | k ) where δ ∈ [ − L, L ].This interference pattern still contains signatures of the LOphase even when the domain wall locations fluctuate betweentubes. off, the clouds expand predominantly in the tightly con-fined directions ( y and z ). After a suitable time of flight t , the wavefunction is effectively Fourier-transformed inthe y and z directions. That is, the final wavefunction∆( x, y, z, t ) is approximately proportional to the initialmomentum distribution in the y and z directions, i.e.,∆( x, y, z, t ) ≈ ∆( x, k y , k z , t = 0) where y = tk y /m and z = tk z /m :∆( x, k y , k z ) = σ y σ z e − z σ z / e − y σ y / (2) × (cid:34) e ik z d/ N (cid:88) n =1 ∆ Tn ( x ) e ik y an + e − ik z d/ N (cid:88) n =1 ∆ Bn ( x ) e ik y an (cid:35) . The 3D density of the cloud, after expansion, is givenby I ( x, k y , k z ) ∼ | ∆( x, k y , k z ) | . The imaging processmeasures the integrated density along the y direction, I ( x, k z ) ∼ (cid:82) dk y | ∆( x, k y , k z ) | : I ( x, k z ) ∼ e − σ z z N (cid:88) n =1 N (cid:88) m =1 e − ( n − m ) a / σ y (3) × (cid:2) ∆ Tn ( x )∆ Tm ( x ) + ∆ Bn ( x )∆ Bm ( x )+ 2∆ Tn ( x )∆ Bm ( x ) cos k z d (cid:3) . We can consider the behavior of the above interfer-ence formula in its two limits: widely separated tubes( a/σ y → ∞ ) and overlapping tubes ( a/σ y → / (cid:112) N layers .Figure 4 illustrates the projected interference pattern,described by Eq. 4, for three configurations of the up-per layer: a uniform SF; an LO phase with domain wallslocked between tubes; and an LO phase with domainwalls fluctuating between tubes. In each case we assumethat the lower layer has been prepared in a uniform SFstate. We represent the LO pairing amplitude modula-tions using Jacobi sine functions sn( x | k ) multiplied byGaussian envelopes in the x , y , and z directions. The“zipper” pattern in the lower two panels is a clear sig-nature of oscillations of the relative phase between theSF and LO layers, in contrast to the straight interferencefringes in the top panel.We have also obtained similar results from Bogoliubov-de Gennes (BdG) simulations. The BdG simulations con-verge to “locked LO” states even when the intertube cou-pling is small. Thermal phase fluctuations beyond BdGmay be expected to produce domain wall fluctuations be-tween tubes, as in the lowest panel of Fig. 4, but even FIG. 5: Illustration of imaginary time worldlines of LO do-main walls and corresponding magnetizations with (bottom)and without (top) quantum fluctuations. Pluses and minusescorrespond to the sign of the pairing amplitude ∆( x ) in eachregion. In the bottom panel, the quantum fluctuations en-force a spatial profile for the magnetization that is less sharp,but which still clearly maintains the features of domain walls. then a zipper-like pattern is still visible provided that thefluctuations are not too severe. Discussion:
In an experiment previously proposed bythe cold atom group at Rice, an LO state is prepared ina 2D array of tubes (with a spacing of about 532nm),the interaction is ramped to the BEC side, the trap isswitched off, and the final density profile is a measureof the initial longitudinal pair momentum distribution P ( q x ) (along the tube axis). The function P ( q x ) is ex-pected to have cusps at ± q LO , but these features tend toget washed out by temperature and spatial inhomogene-ity, and their interpretation requires detailed quantitativeanalysis. The key advantage of our proposal is that theclouds expand in the transverse ( z ) direction, giving an interference pattern resolved along the x -axis, which pro-vides a much more direct way to probe the LO pairingamplitude modulations.The penetration of domain walls in an LO state re-sembles the penetration of vortices into a rotating BEC,and indeed, the zipper interference pattern we predict issimilar to patterns that have been seen in experimentson vortices in 2D Bose systems . Those experimentsinvolved two pancake BECs separated by 3 µ m.A remarkable feature of cold atom experiments is thatcontrol parameters (interaction, lattice depth, and trapdepth) can be turned off very quickly, much faster thanthe typical timescale of domain wall movement. Thisallows us to take a snapshot of the wavefunction (resolvedin real time), in a way not possible in condensed matterexperiments. Thus, even above the critical temperaturewhere thermal fluctuations destroy long-range order, itmay still be possible to detect LO physics in the form of“temporary” domain walls!Quantum fluctuations of an LO state are more subtle.Their effect can be thought of as diffusion of the domainwalls in imaginary time τ ; measurements are necessarilyaveraged over τ . For isolated tubes, quantum fluctua-tions of the domain walls prevent long-range LO order,and there is only quasi-long-range order at zero tempera-ture; hence, the interference pattern will be washed out.This is why we recommend using sufficient coupling be-tween the tubes to stabilize long-range order, so that thepairing amplitude modulations remain even after averag-ing over quantum fluctuations (see Fig. 5).In a larger perspective, interferometric techniques haveproven to be powerful methods to detect the relativephase of the pair wavefunction in cuprates, ruthenates,pnictides, and other unconventional superconductors. The interferometric signature we propose is different inthat it involves a phase change in the center-of-mass mod-ulations of the order parameter that occur across the do-main walls and not in the relative phase of the pair.We gratefully acknowledge support from DARPA grantno. W911NF-08-1-0338 (YLL), ARO W911NF-08-1-0338(NT), and the DARPA OLE Program (NT). MS acknowl-edges support from the NSF Graduate Research Fellow-ship Program. We are grateful to Randy Hulet for usefuldiscussions. B. S. Chandrasekhar,
Appl. Phys. Lett. , , 7 (1962). A. M. Clogston,
Phys. Rev. Lett. , , 266 (1962). P. Fulde and R. A. Ferrell,
Phys. Rev. , , A550 (1964). A. I. Larkin and Y. N. Ovchinnikov,
Zh. Eksp. Teor. Fiz. , , 1136 (1964), also Sov. Phys. JETP 20, 762 (1965). K. Machida and H. Nakanishi,
Phys. Rev. B , , 122(1984). H. Burkhardt and D. Rainer,
Ann. Physik , , 181 (1994). N. Yoshida and S.-K. Yip,
Phys. Rev. A , , 063601 (2007). Y. L. Loh and N. Trivedi,
Phys. Rev. Lett. , , 165302(2010). Y. A. Liao, et al. , Nature , , 567 (2010). M. Alford, J. A. Bowers, and K. Rajagopal,
Phys. Rev. D , , 074016 (2001). A. Bulgac and M. M. Forbes,
Phys. Rev. Lett. , , 215301(2008). Z. Cai, Y. Wang, and C. Wu,
Phys. Rev. A , , 063621(2011). M. M. Parish, S. K. Baur, E. J. Mueller, and D. A. Huse,
Phys. Rev. Lett. , , 250403 (2007). E. Zhao and W. V. Liu,
Phys. Rev. A , , 063605 (2008). M. Rizzi, et al. , Phys. Rev. B , , 245105 (2008). Z. Hadzibabic, et al. , Nature , , 1118 (2006). C. A. Regal, M. Greiner, and D. S. Jin,
Phys. Rev. Lett. , , 040403 (2004). R. M. Lutchyn, M. Dzero, and V. M. Yakovenko,
ArXive-prints (2010). D. A. Wollman, et al. , Phys. Rev. Lett. , , 2134 (1993). J. D. Strand, D. J. Van Harlingen, J. B. Kycia, and W. P.Halperin,
Phys. Rev. Lett. ,103