BBrazilian Journal of Physics manuscript No. (will be inserted by the editor)
Cold atoms beyond atomic physics
Lucas Madeira · Vanderlei S. Bagnato , Received: date / Accepted: date
Abstract
In the last 25 years, much progress has beenmade producing and controlling Bose-Einstein conden-sates (BECs) and degenerate Fermi gases. The advancesin trapping, cooling and tuning the interparticle inter-actions in these cold atom systems lead to an unprece-dented amount of control that one can exert over them.This work aims to show that knowledge acquired study-ing cold atom systems can be applied to other fieldsthat share similarities and analogies with them, pro-vided that the differences are also known and takeninto account. We focus on two specific fields, nuclearphysics and statistical optics. The nuclear physics dis-cussion occurs with the BCS-BEC crossover in mind,in which we compare cold Fermi gases with nuclear andneutron matter and nuclei. We connect BECs and atomlasers through both systems’ matter-wave character forthe analogy with statistical optics. Finally, we presentsome challenges that, if solved, would increase our un-derstanding of cold atom systems and, thus, the relatedareas.
Keywords
Cold atoms · Atomic physics · Nuclearphysics
Much progress has been made producing and control-ling Bose-Einstein condensates (BECs) [1,2,3] and de-generate Fermi gases [4,5] of dilute atomic clouds in
Lucas MadeiraE-mail: [email protected] Instituto de F´ısica de S˜ao Carlos, Universidade de S˜aoPaulo, CP 369, S˜ao Carlos, S˜ao Paulo, 13560-970, Brazil. Hagler Fellow, Department of Biomedical Engineering,Texas A&M University, College Station, TX 77843, USA. the last 25 years [6]. The advances in trapping, cool-ing, and tuning the interparticle interactions in thesecold atom systems make them excellent candidates forstudying microscopic interactions due to the amountof control that one can exert over these systems. Thephysics learned in them could lead to progress in othersystems and fields, as long as we can understand thesimilarities and take into account the differences be-tween them. In this work, we considered two fields thatcan be related to cold atoms: nuclear physics and sta-tistical optics.First, we start with nuclear physics. The main rea-son nuclear and atomic physics share many conceptsand techniques is because of the short-range characterof the interactions. If we take a characteristic lengthunit (cid:96) (for example, the typical interparticle distance)and the mass m , we can construct an energy scale E = (cid:126) / ( m(cid:96) ). Casting all distances in (cid:96) units and all en-ergies in E units allows us to focus on the scale-independent differences [7,8,9]. An important quantityis the number density n multiplied by the scatteringlength a to the third power, na . A mean-field approachyields good results when na (cid:28)
1. However, strongcorrelations appear when na (cid:38) − for atoms with a/(cid:96) ∼
100 and nuclei with a/(cid:96) ∼
1, provided they areaway from resonance.The focus of the comparison is centered around fermi-onic cold gases since protons and neutrons are fermions.Moreover, we chose to discuss the similarities and differ-ences under the light of the smooth connection betweenBardeen-Cooper-Schrieffer (BCS) superfluidity [10] andBose-Einstein condensation [11], the BCS-BEC crossover.As we will see, this formulation is quite convenient tocompare both areas [7,12]. Special attention is devotedto neutron-neutron pairing since the parallel with two- a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Lucas Madeira, Vanderlei S. Bagnato component atomic Fermi gases can be readily estab-lished.One of the goals of this review was to connect thecold atoms and nuclear physics communities. Hence, wetried to keep the discussion at a level interesting to read-ers from both backgrounds, and we provided referencesto more in-depth discussions of specific topics.The second comparison is made with statistical op-tics. Reference [13] found that a ground-state BEC anda turbulent BEC [14,15] share an analogy with thepropagation of an optical Gaussian beam and ellipticalspeckle light map. This occurs because both are exam-ples of coherent matter-wave systems. In principle, thiscould be used to increase our understanding of quantumturbulence by looking at statistical atom optics.This work is structured as follows. Section 2 con-tains the comparison between cold atoms and nuclearphysics. First, we briefly discuss two-component atomicFermi gases, Sec. 2.1, since the concepts will be neededlater to present the BCS-BEC crossover and discussneutron matter. In Sec. 2.2, we introduce the BCS-BECcrossover and the unitary regime with cold atoms inmind. The BCS-BEC crossover from the nuclear physicsperspective is discussed in Sec. 2.3, including nuclearand neutron matter, alpha particle condensation, andnuclei. In Sec. 3, we show how BECs and statisticaloptics share some similarities. Finally, in Sec. 4, we listsome challenges that permeate both cold atoms and therelated areas. s -wave inter-actions between the species can be used to produce adegenerate gas [19]. The typical Fermi temperature isapproximately 1 µ K, while experiments reach hundredsof nK temperatures. The number density is close to 10 cm − , which means that the typical interparticle spac-ing is close to 500 nm, much larger than the interatomicpotential ranges (1-10 nm) and atom sizes (0.1 nm).The quantum numbers related to the atoms are ( J , I , F , m F ), J being the addition of the orbital angularmomentum L and spin S of the electrons, I the totalnuclear spin, and F the total spin (addition of J and I ) with projection m F . In the presence of a magnetic field,only m F is conserved, although good asymptotic quan-tum numbers exist in the limits of very low or high fieldstrengths [6]. The hyperfine energy splitting due to themagnetic field is typically of the order of 10 − eV, whichis much smaller than any possible electronic transition.Two-component Fermi gases correspond to populationsof the lowest two hyperfine states (two different valuesof m F ). It is also worth noting that three-componentfermionic gases have been produced [20,21,22], whichcorrespond to three m F values.The atomic clouds are trapped by external poten-tials, which adds an energy scale to the system, thespacing between the trap eigenstates. Let us consider,for simplicity, the case of an isotropic harmonic oscilla-tor. Typically, the energy (cid:126) ω is much smaller than thehyperfine splitting, sometimes by five orders of mag-nitude. Hence, only two different internal states (cor-responding to different m F values) can effectively de-scribe a two-component gas of fermions. After this dis-cussion, we hope that it is clear why the theoreticaloversimplification of treating a two-component Fermigas as a mixture of spin-up and spin-down componentsproduces the desired results.2.2 BCS-BEC crossover in cold atomsBesides the one-body external field discussed in the pre-vious section, two-body interactions between the atomsalso introduce an energy and length scale into thesesystems. Interparticle interactions in ultracold atomicgases can be tuned via Feshbach resonances, thus re-alizing the BCS-BEC crossover, a problem of signifi-cant interest [19]. The investigation of Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC)crossover arises in an attempt to better understandsuperfluidity and superconductivity beyond the stan-dard paradigms [23,24]. The crossover concepts can betraced back to considerations regarding strongly inter-acting superconductors by Eagles [25] and fermioniccold atoms by Leggett [26]. The key idea is that the in-terparticle (electrons or alkali atoms) two-body interac-tion can be varied so that increasing its strength allowsthe system to go from a BCS paired state to a molec-ular BEC, where the two-body bound state is the rele-vant characteristic, Fig. 1. The most exciting results arerelated to a very strongly interacting state of matter,the unitary Fermi gas, which is right at the crossover’sheart. Until recently, all superconductors and superflu-ids fell into one of two classes, bosonic and fermionic.This led to two different paradigms, BEC and BCS, forunderstanding the properties of quantum fluids. old atoms beyond atomic physics 3 Fig. 1
Phase diagram of the BCS-BEC crossover as a func-tion of temperature
T/E F and interaction strength (1 /k F a ).The schematic evolution from the BCS limit with largeCooper pairs to the BEC limit with tightly bound moleculesis shown at the bottom. The unitary point, 1 / ( k F a ) = 0, cor-responds to strongly interacting pairs with size comparableto k − F . Taken from Ref. [24]. In the BEC scheme, first developed for noninteract-ing bosons and later generalized to include repulsive in-teractions, it is possible to describe bosonic fluids, suchas He, and ultracold Bose gases, Rb for instance.The condensate is a macroscopic occupation of a singlequantum state that occurs below a transition tempera-ture T c , which is of the same order of magnitude as thequantum degeneracy temperature, at which the inter-particle spacing is of the order of the thermal de Brogliewavelength.The BCS paradigm, first conceived for metallic su-perconductors, describes a pairing instability arisingfrom a weak attractive interaction in a highly degen-erate system of fermions. Both the formation of pairsand their condensation occur at the same temperature T c , which is orders of magnitude smaller than E F /k B ,which sets the degeneracy temperature. The BCS the-ory is very successful in describing conventional super-conductors and it has been generalized to describe var-ious systems, such as superfluidity in He and pairingin nuclei.Early theoretical work on the crossover was concep-tually interesting, but real enthusiasm came from itsexperimental realization [27]. The most important dif-ference between ultracold Fermi gases and all previouslystudied superfluids is that the interaction between spin-up and spin-down can be tuned in the laboratory. Theaverage separation between atoms, k − F , is much largerthan the range of the interatomic potential r . For a di-lute gas with k F r (cid:28)
1, the interaction is described by a single parameter, the s-wave scattering length a . Allthermodynamic properties of the gas can be written ina universal scaling form. For example, the free-energy F takes the form F = N E F F ( k B T /E F , / ( k F a )) , (1)where F is a scaling function. This result is universal inthe sense that it is independent of microscopic details,as long as k F r → Li, of electronic spin S = 1 / I = 1. The electric spin is fully polarized, usually themagnetic field B (cid:62) G , and aligned in the samedirection for each of the three lowest hyperfine states.Hence, two colliding Li atoms are in a continuum spin-triplet state in the open channel. The closed channelhas a singlet-bound state that can resonantly mix withthe open channel due to the hyperfine interaction thatcouples the electron spin to the nuclear spin.The difference in the magnetic moments in the closedand open channels allows experimentalists to use an ex-ternal magnetic field B to tune a Feshbach resonance.The resulting interatomic interaction in the open chan-nel can be described by a B -dependent scattering lengthwhich, near the resonance, is [28] a ( B ) = a BG (cid:20) − ∆BB − B (cid:21) , (2)where a BG is the background value, in the absence ofthe coupling to the closed channel, B is the locationof the resonance, and | ∆B | is the width.Let us consider the problem of two fermions withspin-up and spin-down interacting with a two-body po-tential with range r . The low-energy properties as afunction of the momentum k , such that kr (cid:28)
1, aredescribed by the s-wave scattering amplitude f ( k ) = 1 k cot( δ ( k )) − ik ≈ − /a + ik . (3)The scattering length a completely determines δ ( k →
0) = − tan − ( ka ), the s-wave scattering phase shiftwith low-energy. The effective interaction is indepen-dent of the shape of the potential, thus we can choosethe simplest one: a square well of depth V and range r . If we start with a very shallow square-well, a < V , and diverges to −∞ at the for-mation of a bound state in vacuum ( V = (cid:126) π / ( mr ),where the reduced mass is m/ a > Lucas Madeira, Vanderlei S. Bagnato and decreases from + ∞ with increasing V . For positive a , the scattering length a is the bound state’s size, withenergy − (cid:126) / ( ma ).The threshold for bound-state formation, where | a | →∞ , is called unitary point. The phase shift is δ ( k =0) = π/
2, and the scattering amplitude takes the maxi-mum value f ≈ − / ( ik ). The unitary regime is the moststrongly interacting regime in the BCS-BEC crossover.The unitary point is of great interest since that | a | → ∞ means that the system loses a length scale, and the scat-tering length cannot be used as an expansion parame-ter, as noted by Bertsch [29]. For a fermionic system, itis expected that the energy is proportional to the onlyscale in the problem, the non-interacting Fermi gas en-ergy E F G , E = ξE F G = ξ (cid:126) k F m , (4)where the constant ξ is known as the Bertsch param-eter. In the limit ak F → −∞ , quantum Monte Carlo(QMC) results give the exact value of ξ = 0 . Li- K, between the twospecies [55]. The challenge proposed by Bertsch to the participantsof the Tenth International Conference on Recent Progress inMany-Body Theories can be stated as: what are the groundstate properties of the many-body system composed of spin-1/2 fermions interacting via a zero-range, infinite scatteringlength contact interaction? p ↑ , p ↓ , n ↑ , and n ↓ ). The interactions between two protons( pp ), two neutrons ( nn ), or a proton and a neutron( pn ) are nearly identical. A convenient formulation thattakes advantage of this charge independence consistsof adding a degree of freedom to each nucleon calledisospin. If we consider protons to be the analogous stateof a spin-up particle, and neutrons to be the counterpartof spin-down particles, we can benefit from the algebraalready developed for spin-1/2. Protons and neutronsare isospin-1/2 particles, and a common basis choiceis T z | p (cid:105) = +(1 / | p (cid:105) and T z | n (cid:105) = − (1 / | n (cid:105) . Our dis-cussion of the BCS-BEC crossover in nuclear physicsfollows Ref. [12] closely, where the reader can also findsome topics not covered here, such as finite temperatureresults.The only two-nucleon bound state is the deuteron,which corresponds to the pn channel. With Sec. 2.2in mind, we can imagine a crossover from a BEC ofdeuterons to a BCS state of pn Cooper pairs. The in-teraction between nucleons is fixed, contrary to coldatoms, where the scattering length can be tuned withgreat precision. However, the nucleon systems’ densitycan be varied, thus changing the value of k F a . For ex-ample, at low densities, the deuteron is bound. Increas-ing the density decreases the binding energy to a pointwhere the nucleus turns into a pn Cooper pair.In nuclei, which would be best described as “clus-ters” or “droplets” in the cold atoms context, nn and pp pairing are favored over pn simply because there areusually more neutrons than protons. Although there isno equivalent of the BEC regime, due to pp and nn be-ing unbound, the nn scattering length is much largerthan the interparticle spacing and effective range, thusleading to a scenario similar to the unitary limit.Another possibility for a BCS-BEC crossover in nu-clear physics involves the α particle, the bound stateof two protons and two neutrons. Its binding energy ismuch larger than the deuteron one, suggesting that aBEC of α particles would be energetically more favor-able than a deuteron BEC. However, the α particle ismuch more sensitive to finite density effects. Increasingthe density makes the α particle BEC turn into a BCSstate of pn , or pp and nn , Cooper pairs.In the following sections, we explore each of the pos-sibilities above in detail. old atoms beyond atomic physics 5 The only bound state of two nucleons is the deuteron.The pn that compose the deuteron are in a spin-tripletstate ( S = 1), isospin singlet ( T = 0), and total angularmomentum J = 1. It has angular momentum compo-nents L = 0 and 2, the latter being due to the non-central tensor force. It has only one bound-state, allof its excited states are scattering states. Another evi-dence that the deuteron is very weakly bound is that itsbinding energy is approximately 2.23 MeV. That is only0.1% of its rest mass, while heavier nuclei have bindingenergies of ≈ .
8% of their rest masses. Another char-acteristic of the deuteron is its anomalous size. The rmsradius, defined as r rms = (cid:104) r (cid:105) / , of the deuteron is 2.8fm. That is close to the value for Ne, which has twicethe number of protons and neutrons.Despite being very weakly bound, pn pairing canundergo the BCS-BEC crossover [56,57,58]. For exam-ple, this is relevant in heavy-ion collisions where low-density nuclear matter can approximate the state aftera collision [57]. Another example of pn pairing beingrelevant is in the case of not completely cooled neutronstars where the fraction of protons is not negligible.In Ref. [57], the BCS-BEC crossover in nuclear mat-ter was investigated within the BCS framework. TheBr¨uckner-Hartree-Fock approach was used to computesingle-particle energies ε k , which then can be used tocompute the the gap as a function of the wave-vector,the BCS occupation numbers n k , and the anomalousdensity (also known as pairing tensor in nuclear physics)in a self-consistent way. In the low-density limit ( ε k corresponds to free particles and n k vanishes) and zerotemperature, the gap equation is the Schr¨odinger equa-tion, and the pairing tensor reduces to the deuteronwave function, with the deuteron binding energy as aneigenvalue.These results can be used to determine the size ofthe deuteron in matter [59]. If we start at the low-density limit and increase the density, its size decreasesuntil a minimum at n ∼ .
036 fm − , after which itgrows. In Sec. 2.3.1, we were dealing with an equal numberof protons and neutrons. However, there are situationswhere the number of neutrons dramatically exceeds thenumber of protons, such as proto-neutron stars. Asym-metric nuclear matter refers to an isospin-imbalancedsystem, a situation analogous to pairing between a spin-up and spin-down in a polarized (spin-imbalanced) atomicgas. In Ref. [60] the pn pairing in bulk nuclear matterwas studied as a function of the density and isospinasymmetry within the BCS framework. The authorsfound that in the high-density (weak-coupling) regime,the pn paired state is strongly suppressed even by a mi-nor neutron excess. As density decreases, the BCS statewith large Cooper pairs evolves smoothly into a BEC ofdeuterons. In the resulting low-density state, a neutronexcess is not enough to quench the pair correlations be-cause of the large spatial separation of the deuteronsand neutrons. As a result, the deuteron BEC is weaklyaffected by an additional Fermi sea of the remainingneutrons, even at substantial asymmetries. In Secs. 2.3.1 and 2.3.2 we considered systems with twokinds of nucleons, protons and neutrons. When con-sidering only nn pairing, one is restricted to the BCSside of the crossover since there is no bound state oftwo neutrons to form a BEC. The comparison with theBCS side of the crossover of a two-component Fermigas, Sec. 2.1, is straightforward in this case.The nn s -wave scattering length, a nn = − . r enn =2 . r enn (cid:28) k − F (cid:28) | a nn | ,which is the equivalent of the unitary Fermi gas intrapped cold atoms. However, the finite effective rangeeffects play an important role as we move away fromthe low-density regime.The equation of state of a Fermi gas with resonantinteractions when the effective range is appreciable wasstudied using a t -matrix approach in Ref. [62], whichwas then applied to neutron matter.One of the essential applications of nn pairing isthe study of neutron stars, especially the crust. For areview of the physics of neutron star crusts, at the mi-croscopic and macroscopic levels, the reader is referredto Ref. [63].It is believed that in the inner crust of a neutronstar, there is a dilute gas of neutrons together with aCoulomb lattice of nuclear clusters. The lattice is com-posed of ionized atomic nuclei immersed in a nearlyconstant and uniform charge compensating backgroundof electrons [64]. As we move toward the center of thestar, the neutron gas density goes from 0 to 0.08 fm − .The spacing between the nuclear clusters can be con-siderable, ≈
100 fm, compared to other relevant lengthscales. Hence, the neutron gas can be approximated asuniform matter with corrections due to the Coulomblattice being of higher order. The microscopic structureof nn pairing impacts properties that can be observed,such as the cooling process of neutron stars [65]. Lucas Madeira, Vanderlei S. Bagnato
Much progress has been made in the low-densityregime due to the intrinsic interest and approximationsmaking it more tractable. The BCS gap equation hasbeen solved in Ref. [66], where the authors computedthe pairing tensor for several densities. The wave func-tion resembles a bound state in the low-density limit,but with only a resonance close to zero energy. The nn pair size was also investigated as a function of the den-sity, which diverges in the vanishing density limit sincethere is no nn bound state.Although we discussed many-body systems, up tothis point, we centered our discussion around proper-ties of two-body systems: the scattering length and ef-fective range. Besides the spin/isospin degrees of free-dom and the fact that nucleons are fermions, dealingwith the nuclear force is complicated because it con-tains an appreciable three-body component that is notwell-understood (actually, thinking about the nuclearforce only in terms of nucleon degrees of freedom is nothelpful since mesons play an essential role, the lightestof them being the pions [67]). In Ref. [68] the authorsemployed forces obtained from chiral perturbation the-ory, which include two-body terms and repulsive three-body forces. Although repulsion was introduced in thesystem, they did not find a very strong reduction ofthe gap in neutron matter. A later study using chiraleffective theory also observed the same behavior [69].Quantum Monte Carlo methods were also used tostudy neutron matter, not only restricted to the low-density regime. For a detailed review of QMC meth-ods applied to nuclear physics, the reader is referred toRefs. [35] and [36].One quantity that can be computed straightforwardlyin QMC simulations is the pairing gap. In Ref. [70] theauthors used Green’s function Monte Carlo (GFMC) tocompute the gap using two potential models: the AV4’,a simplified form of the AV18 potential [71], which in-cludes s - and p -wave contributions, and the s -wave com-ponent of AV18 (which makes the potential sphericallysymmetric and ignores spin-flip terms). They found thatin the low-density regime, the gap’s behavior is almostentirely determined by the s -wave interaction. The ra-tio of the gap to the Fermi energy, ∆/E F , was consis-tently lower than the BCS prediction. Its maximum, of ∆/E F ≈ − / ( k F a nn ) = 0 .
2, whichis very close to the unitary limit (1 / ( k F | a | ) = 0). Athigher densities, although the value k F a nn increases,the finite effective range of the interaction strongly sup-presses the gap. These results agree with the ones ob-tained with other methods and low-momentum poten-tials, for example, determinantal QMC simulations us-ing pionless EFT [72]. The suppression of the gap concerning the BCS pre-diction may be a consequence of particle-hole fluctu-ations. These screening effects are not present in theBCS formalism but are included in the QMC calcula-tions. The auxiliary-field diffusion Monte Carlo resultsof Ref. [73], which employed a more sophisticated po-tential (AV8’ plus a three-body force [35]), found almostno suppression of the gap with respect to the BCS re-sult. This is probably due to their choice of the trialwave function. Although we only discussed a few refer-ences, many works that compute neutron matter prop-erties using QMC methods are available [74,75,76,77,78].Accounting for screening in theoretical studies is adifficult task. While the nucleon systems are strongly-interacting, polarization is usually included in a pertur-bative manner. Despite these complications, progresshas been made [79,80,81,82].The possibility to connect the results of both two-component atomic Fermi gases and low-density neutronmatter comes from the low-energy behavior of the phaseshift δ ( k ), which can be related to a and the effectiverange r e [83], k cot δ ( k ) = − a + r e k O ( k ) . (5)This equation is often called shape independent approx-imation because different potentials that reproduce thesame scattering length and effective range yield thesame low-energy phase-shift behavior. In dilute coldgases, the effective range r e between atoms is muchsmaller than the interatomic spacing r , and can betaken to be zero. The diluteness can guarantee that thescattering length a is much larger than r . Compari-son with other systems is meaningful if they also obey | a | (cid:29) r (cid:29) r e . The scattering length of neutron matteris substantially larger than the interparticle distanceand the effective range, such that | r nne /a nn | ≈ . k F a .In Ref. [84], the authors used QMC methods to com-pute the equation of state of both cold atoms and low-density neutron matter, and they found both to be verysimilar, Fig. 2. They also computed the neutron matterpairing gap, which is significantly suppressed relativeto cold atoms. The difference was attributed to the fi-nite effective range in the neutron-neutron interaction.Reference [53], besides confirming these results, con-sidered vortex-line excitations to these fluids. The au-thors also found agreement in some vortex properties old atoms beyond atomic physics 7 between cold gases and neutron matter for very lowdensities. However, the density depletion at the vortexcore, which depends strongly on the short-ranged inter-action cold atomic gases, is approximately constant forneutron matter in the low-density regime. Fig. 2
Equation of state for cold atoms and neutron mat-ter at zero temperature computed using QMC methods. Thetriangle shows the cold atoms result at unitarity. The bottom x -axis corresponds to the interaction strength − k F a , whilethe top x -axis shows the value of k F if we substitute thevalue for the nn scattering length, a nn = − . In Sec. 2.3.1, we covered the BCS-BEC crossover in sys-tems with an equal number of protons and neutrons,where the Cooper pairs were pn and the BEC statecorresponded to the deuteron. This is the descriptionthat comes to mind when we draw the parallel withtwo-component Fermi gases. However, nucleons corre-spond to four states due to the spin/isospin degrees offreedom. Hence we can consider the possibility of quar-tet condensation, where the four-fermion object is the α particle (a bound state of two protons and two neu-trons, the He nucleus).This topic has not been so extensively covered as theconventional BCS-BEC crossover because of the tech-nical difficulties that arise, both in theoretical and nu-merical approaches. However, a quartet phase has beenpredicted in theoretical studies [85,86]. Also, QMC sim-ulations of four-component fermions with unitary inter-actions found a ground state of the eight-particle sys-tem whose energy is almost equal to that of two four-particle systems [87]. Also, the BCS-BEC crossover for quartets is notqualitatively the same as the usual one: for high densi-ties the quartets break up into two Cooper pairs. Thetheoretical description of quartet condensation in Fermisystems, with applications to nuclei and nuclear matter,has been discussed in Refs. [88,89,90,91,92].
Nuclei are self-bound, which means that the attrac-tive interactions inside a nucleus overcome the repulsiveones to form a cluster. In cold atoms, an external poten-tial can be applied to confine systems even with purelyrepulsive interactions [16,17]. Although in Secs. 2.3.1to 2.3.4 we talked about nuclei, they were immersedin infinite matter. We can also discuss the BCS-BECcrossover and alpha condensation in nuclei in vacuum.For heavy nuclei, where the number of neutrons exceedsthe number of protons, pn pairing is suppressed becauseof the same reasons we listed in Sec. 2.3.2. However,in nuclei where the number of neutrons and protons iscomparable, pn could be possible because this channel’sattraction exceeds the pp or nn ones. References [93]and [94] address this subject.We can also consider viewing some light nuclei as acluster plus a single Cooper pair to investigate pn pair-ing. A few words about the terminology are in order.It may seem strange to refer to a single Cooper pair.Still, we intend to discuss pairing between nucleons asa function of the interaction strength and draw a paral-lel with what we already saw in infinite nuclear matterand cold atoms. In this sense, the BEC state is simplythe deuteron. Going back to our discussion, Li, F,and Sc are examples of nuclei with a structure “clus-ter plus deuteron” (the clusters being α , O, Ca,respectively). Because of the relatively small number ofnucleons, results under this perspective are only avail-able for Li [95]. Much like the infinite matter case, thesize of the deuteron (as a function of the distance tothe α ) diminishes before it enters the α particle.Neutron-neutron pairing is also important to under-stand the properties of neutron-rich nuclei. In heavy nu-clei, close to the neutron drip line, a neutron-rich outerlayer called neutron skin is present. Interestingly, the nn Cooper pairs are localized on the surface of thesenuclei. Details can be found in Refs. [96,97,98,99].As for the BEC of α particles in nuclei, usually theground-state is too small to allow α condensation (anexception is Be, which shows a two α cluster structure[100]). However, there are long-lived excited states oflight nuclei that show clustering of α particles. One ofthese states is the famous Hoyle state [101], an excitedstate of C produced by a triple- α process, essential Lucas Madeira, Vanderlei S. Bagnato for the nucleosynthesis of carbon in stars. For a morein-depth explanation, and some other candidates for α condensation, the reader is referred to Refs. [102,103,104]. Statistical optics is a field of physics that has attractedmuch interest recently. Several forms of analysis re-garding photons and their distributions are part of thisarea’s interests, among these, the so-called speckle fields.Light waves belonging to a coherent beam, such as alaser beam, scattered over surfaces containing randomlydistributed scatters, generate interference patterns inthe light fields called speckles. Speckles constitute arandom collection of amplitudes and phases that reveala wide variety of light beams’ properties. They haveapplications in several fields of science and technology.Although the speckles’ field is largely focused onlight waves, we can equivalently represent them as matter-waves. This would undoubtedly represent the inclusionof matter-waves in an optical statistics framework. Infact, with the advent of atomic BECs, we can thinkabout these possibilities more easily than before. Whenwe produce a condensate and let it expand freely inspace, what we have is a coherent beam of matter, pre-cisely equivalent to a laser beam for propagating elec-tromagnetic waves. If we introduce a collection of ran-dom amplitudes and phases into this condensate, wewill have the equivalent of a field of speckle of matter-waves there.To establish this parallel, Hussein and collaborators[13] envisioned a turbulent condensate production andexpansion. An unperturbed condensate is a superfluid.With the introduction of excitations, which can gen-erate vortices and waves in this superfluid, quantumturbulence can be reached [14,15]. Once in this state,we have many amplitude and phase fluctuations, reach-ing a situation very similar to that expected in specklefields.The turbulent regime was achieved experimentallyand characterized. Once in this state, the system wasreleased from the trap, corresponding to a matter-wavepropagating in space. Still, instead of having the char-acteristics of a coherent and well-behaved wave, it wasa collection of coherent domains, much like in specklefields.The researchers analyzed mainly two characteris-tics of the spatial disorder of the systems. First, mea-surements of the aspect ratios of regular and turbulent BECs were performed. It is known that for ground-stateBECs there is an inversion of the aspect ratio of thecloud in time-of-flight (TOF) measurements, whereasin the turbulent case there is a self-similar expansion,without ever inverting its aspect ratio [105]. For a co-herent Gaussian beam, there is also an inversion of theaspect ratio of the waists, whereas it is preserved in thepropagation of the elliptical speckle light map.The analogy between the two systems is constructedthrough the state of disorder that characterizes the matter-waves. In Fig. 3, we show the behavior of the propaga-tion of an unperturbed condensate and its characteris-tics. When expanding, the behavior is like that foundin a beam of Gaussian laser light: as it propagates, thesmaller dimensions undergo a more significant diver-gence due to diffraction, resulting in an inversion ofthe so-called aspect-ratio as the propagation progresses.This is exactly what is found in the propagation of abeam of light containing a speckle field. Due to the do-mains, diffraction is no longer dominated by the wave-length to the beam ratio, but by its correlation length.The second property that was investigated was thecoherence in both systems. It was found that the corre-lations in regular BECs resemble the ones in the Gaus-sian beam, while the same is true for the turbulent BECand speckle beam pair.This was the first experimental evidence of a three-dimensional speckle matter-wave. The production ofthree-dimensional speckle fields is an exciting problemto be considered, and here, with the turbulent condi-tions and expansion, we naturally have this connection.Even more interestingly is the fact that the study ofspeckle fields as matter-waves and their statistical be-havior can now be considered. This certainly opens theopportunity of promoting the merging of quantum tur-bulence with statistical optics, now in the context ofmatter-waves.
In conclusion, we pointed out similarities and differ-ences between cold atom systems and nuclear physics,and also statistical optics. These topics where chosenbecause of the interesting comparisons that can be madebetween cold atoms and them. A full list of all the sys-tems that similarities could be investigated would in-clude superconductors, quantum chromodynamics (QCD),and many other topics beyond this review’s scope.Concerning nuclear physics, we chose to center thediscussion around the BCS-BEC crossover because webelieve it is a very clear and appropriate comparison.However, we did not cover all the related topics, forexample, finite temperature effects [12]. old atoms beyond atomic physics 9
Fig. 3
Comparison between BECs and optical beams. In the left panel, the BEC expansion during TOF is investigated. TheThomas-Fermi radii as a function of the TOF are plotted for standard (A) and turbulent (B) BECs. Then, three differentTOF snapshots showing the expansion of standard (C) and turbulent BECs (D) are shown. The panel on the right concernsthe optical beam propagation. The optical beam waists as a function of the axial distance are plotted for a coherent Gaussianbeam (A) and the speckle beam (B). A sequence of three different propagation distances shows the expansion of a Gaussianbeam (C) and a speckle beam (D). At first glance, the similarity between the plots is intriguing, but they can be understoodunder the light of both being coherent matter-wave systems. Taken from Ref. [13].
Besides the crossover, we could have centered thediscussion on other characteristics that connect coldatoms and nuclear physics, such as: the role of three-body forces [106] (specially under the perspective of ef-fective field theories [107,108,109,110]), the finite-sizecharacter [7], the Efimov effect [111,112], and halo nu-clei [113], to cite a few examples.Also, much of our discussion concerned two-componentFermi gases because of the clear comparison with neu-tron matter. Three-component fermionic gases have beenproduced [20,21,22], which increases our understand-ing of multi-component Fermi systems. However, webelieve a significant breakthrough will occur when four-component fermionic gases can be produced experimen-tally and properties calculated from theoretical models.Nuclear physics is a four-state fermionic problem (pro-tons and neutrons with two possibilities for the spin),and being able to produce analogous systems in the coldatoms scenario would be illuminating.The second parallel we drew was between BECs andstatistical optics, and it was used to study quantum tur-bulence. Turbulence is characterized by a large numberof degrees of freedom, distributed over several lengthscales, that result in a disordered state of a fluid. Thefield of quantum turbulence deals with the manifesta-tion of turbulence in quantum fluids, such as trappedcold gases. There are some inherent challenges in deter-mining and characterizing the turbulent state in thesesystems [14,15]. Hence, different approaches have beenused to clarify aspects of quantum turbulence [114,115,116]. What we showed in Sec. 3 can be used to look atthe phenomena from a different perspective, which is of paramount importance when dealing with quantumturbulence.It is not uncommon to find works where the connec-tion between cold atoms and another field is mentionedin the introduction or outlook. However, there are notmany that make a serious effort to compare and con-trast them. If that were to change, we believe that thiswould be beneficial to both areas.
Acknowledgements
At first glance, the parallel betweenBECs and statistical optics may seem slightly out of place.However, one of the authors of Ref. [13] is Mahir Saleh Hus-sein. Two of his greatest interests were nuclear physics andBose-Einstein condensation [117]. Since this is a special is-sue in his honor, we thought this two-fold tribute would beappropriate. This work was supported by the S˜ao Paulo Re-search Foundation (FAPESP) under the grants 2013/07276-1,2014/50857-8, and 2018/09191-7, and by the National Coun-cil for Scientific and Technological Development (CNPq) un-der the grant 465360/2014-9.
Conflict of interest
The authors declare that they have no conflict of inter-est.
References
1. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E.Wieman, E.A. Cornell, Science (5221), 198 (1995)2. C.C. Bradley, C.A. Sackett, J.J. Tollett, R.G. Hulet,Phys. Rev. Lett. (9), 1687 (1995)3. K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. VanDruten, D.S. Durfee, D.M. Kurn, W. Ketterle, Phys.Rev. Lett. (22), 3969 (1995)0 Lucas Madeira, Vanderlei S. Bagnato4. K.M. O’Hara, S.L. Hemmer, M.E. Gehm, S.R. Granade,J.E. Thomas, Science (5601), 2179 (2002)5. M.W. Zwierlein, J.R. Abo-Shaeer, A. Schirotzek, C.H.Schunck, W. Ketterle, Nature (7045), 1047 (2005)6. C.J. Pethick, H. Smith, Bose-Einstein condensation indilute gases , vol. 9780521846 (Cambridge UniversityPress, 2008)7. N.T. Zinner, A.S. Jensen, J. Phys. G Nucl. Part. Phys. (5), 053101 (2013)8. A.E. Amorim, T. Frederico, L. Tomio, Phys. Rev. C -Nucl. Phys. (5), R2378 (1997)9. K. Riisager, D.V. Fedorov, A.S. Jensen, Europhys. Lett. (5), 547 (2000)10. J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. (5), 1175 (1957)11. Bose, Zeitschrift f¨ur Phys. (1), 178 (1924)12. G.C. Strinati, P. Pieri, G. R¨opke, P. Schuck, M. Urban,Phys. Rep. , 1 (2018)13. P.E.S. Tavares, A.R. Fritsch, G.D. Telles, M.S. Hussein,F. Impens, R. Kaiser, V.S. Bagnato, Proc. Natl. Acad.Sci. (48), 12691 (2017)14. L. Madeira, M. Caracanhas, F. dos Santos, V. Bagnato,Annu. Rev. Condens. Matter Phys. (1), 37 (2020)15. L. Madeira, A. Cidrim, M. Hemmerling, M.A. Caracan-has, F.E.A. dos Santos, V.S. Bagnato, AVS QuantumSci. (3), 035901 (2020)16. I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. (3), 885 (2008)17. S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod.Phys. (4), 1215 (2008)18. R. Onofrio, Physics-Uspekhi (11), 1129 (2016)19. W. Ketterle, M.W. Zwierlein, Riv. del Nuovo Cim.(2008)20. T.B. Ottenstein, T. Lompe, M. Kohnen, A.N. Wenz,S. Jochim, Phys. Rev. Lett. (20), 203202 (2008)21. J.H. Huckans, J.R. Williams, E.L. Hazlett, R.W. Stites,K.M. O’Hara, Phys. Rev. Lett. (16), 165302 (2009)22. S. Nakajima, M. Horikoshi, T. Mukaiyama, P. Naidon,M. Ueda, Phys. Rev. Lett. (2), 023201 (2010)23. W. Zwerger, The BCS-BEC Crossover and the UnitaryFermi Gas , Lecture Notes in Physics , vol. 836 (SpringerBerlin Heidelberg, Berlin, Heidelberg, 2012)24. M. Randeria, E. Taylor, Annu. Rev. Condens. MatterPhys. (1), 209 (2014)25. D.M. Eagles, Phys. Rev. (2), 456 (1969)26. A.J. Leggett, in Mod. Trends Theory Condens. Matter (Springer Berlin Heidelberg, 2008), pp. 13–2727. C.A. Regal, M. Greiner, D.S. Jin, Phys. Rev. Lett. (4), 4 (2004)28. A.J. Moerdijk, B.J. Verhaar, A. Axelsson, Phys. Rev. A (6), 4852 (1995)29. G.A. Baker, Phys. Rev. C (5), 054311 (1999)30. J. Carlson, S. Gandolfi, K.E. Schmidt, S. Zhang, Phys.Rev. A - At. Mol. Opt. Phys. (6), 061602 (2011)31. M.J. Ku, A.T. Sommer, L.W. Cheuk, M.W. Zwierlein,Science (6068), 563 (2012)32. G. Z¨urn, T. Lompe, A.N. Wenz, S. Jochim, P.S. Juli-enne, J.M. Hutson, Phys. Rev. Lett. (13), 135301(2013)33. W.M. Foulkes, L. Mitas, R.J. Needs, G. Rajagopal, Rev.Mod. Phys. (1), 33 (2001)34. J. Carlson, S. Gandolfi, A. Gezerlis, Prog. Theor. Exp.Phys. (1) (2012)35. J. Carlson, S. Gandolfi, F. Pederiva, S.C. Pieper,R. Schiavilla, K.E. Schmidt, R.B. Wiringa, Rev. Mod.Phys. (3), 1067 (2015) 36. J. Lynn, I. Tews, S. Gandolfi, A. Lovato, Annu. Rev.Nucl. Part. Sci. (1), 279 (2019)37. J. Carlson, S.Y. Chang, V.R. Pandharipande, K.E.Schmidt, Phys. Rev. Lett. (5), 050401 (2003)38. G.E. Astrakharchik, J. Boronat, J. Casulleras,S. Giorgini, Phys. Rev. Lett. (20), 200404 (2004)39. S.Y. Chang, V.R. Pandharipande, J. Carlson, K.E.Schmidt. Quantum Monte Carlo studies of superfluidFermi gases (2004)40. S. Gandolfi, K.E. Schmidt, J. Carlson, Phys. Rev. A (4), 041601 (2011)41. A. Bulgac, M.M. Forbes, P. Magierski, in Lect. NotesPhys. , vol. 836 (Springer, Berlin, Heidelberg, 2012), pp.305–37342. J. Carlson, S. Gandolfi, A. Gezerlis, in
Fifty Years Nucl.BCS (World Scientific, 2013), pp. 348–35943. S. Gandolfi, J. Phys. Conf. Ser. , 012011 (2014)44. M.M. Forbes, S. Gandolfi, A. Gezerlis, Phys. Rev. Lett. (23), 235303 (2011)45. M.M. Forbes, S. Gandolfi, A. Gezerlis, Phys. Rev. A (5), 053603 (2012)46. R. Pessoa, S. Gandolfi, S.A. Vitiello, K.E. Schmidt,Phys. Rev. A (6), 063625 (2015)47. R. Pessoa, S.A. Vitiello, K.E. Schmidt, J. Low Temp.Phys. (1-2), 168 (2015)48. R. Pessoa, S.A. Vitiello, K.E. Schmidt, Phys. Rev. A (5), 053601 (2019)49. S. Hoinka, M. Lingham, K. Fenech, H. Hu, C.J. Vale,J.E. Drut, S. Gandolfi, Phys. Rev. Lett. (5), 055305(2013)50. A. Galea, H. Dawkins, S. Gandolfi, A. Gezerlis, Phys.Rev. A (2), 023602 (2016)51. A. Galea, T. Zielinski, S. Gandolfi, A. Gezerlis, J. LowTemp. Phys. (5-6), 451 (2017)52. L. Madeira, S.A. Vitiello, S. Gandolfi, K.E. Schmidt,Phys. Rev. A (4), 043604 (2016)53. L. Madeira, S. Gandolfi, K.E. Schmidt, V.S. Bagnato,Phys. Rev. C (1), 014001 (2019)54. L. Madeira, S. Gandolfi, K.E. Schmidt, Phys. Rev. A (5), 053603 (2017)55. A. Gezerlis, S. Gandolfi, K.E. Schmidt, J. Carlson, Phys.Rev. Lett. (6), 060403 (2009)56. T. Alm, B.L. Friman, G. R¨opke, H. Schulz, Nucl.Physics, Sect. A (1), 45 (1993)57. M. Baldo, U. Lombardo, P. Schuck, Phys. Rev. C (2),975 (1995)58. H. Stein, A. Schnell, T. Alm, G. R¨opk, Zeitschrift f¨urPhys. A Hadron. Nucl. (3), 295 (1995)59. F. Pistolesi, G.C. Strinati, Phys. Rev. B (9), 6356(1994)60. U. Lombardo, P. Nozi`eres, P. Schuck, H.J. Schulze,A. Sedrakian, Phys. Rev. C - Nucl. Phys. (6), 643141(2001)61. A. G˚ardestig, J. Phys. G Nucl. Part. Phys. (5),053001 (2009)62. A. Schwenk, C.J. Pethick, Phys. Rev. Lett. (16),160401 (2005)63. N. Chamel, P. Haensel, Living Rev. Relativ. (1), 10(2008)64. D.A. Baiko, J. Phys. Conf. Ser. (1), 012010 (2014)65. M. Fortin, F. Grill, J. Margueron, D. Page, N. Sand-ulescu, Phys. Rev. C - Nucl. Phys. (6), 065804 (2010)66. M. Matsuo, Phys. Rev. C - Nucl. Phys. (4), 044309(2006)67. L. Madeira, A. Lovato, F. Pederiva, K.E. Schmidt, Phys.Rev. C (3), 034005 (2018)old atoms beyond atomic physics 1168. S. Maurizio, J.W. Holt, P. Finelli, Phys. Rev. C - Nucl.Phys. (4), 044003 (2014)69. C. Drischler, T. Kr¨uger, K. Hebeler, A. Schwenk, Phys.Rev. C (2), 024302 (2017)70. A. Gezerlis, J. Carlson, Phys. Rev. C (2), 025803(2010)71. R.B. Wiringa, S.C. Pieper, Phys. Rev. Lett. (18),182501 (2002)72. T. Abe, R. Seki, Phys. Rev. C - Nucl. Phys. (5),054002 (2009)73. S. Gandolfi, A.Y. Illarionov, S. Fantoni, F. Pederiva,K.E. Schmidt, Phys. Rev. Lett. (13), 132501 (2008)74. A. Gezerlis, I. Tews, E. Epelbaum, S. Gandolfi,K. Hebeler, A. Nogga, A. Schwenk, Phys. Rev. Lett. (3), 032501 (2013)75. A. Roggero, A. Mukherjee, F. Pederiva, Phys. Rev. Lett. (22), 221103 (2014)76. G. Wlaz(cid:32)lowski, J.W. Holt, S. Moroz, A. Bulgac, K.J.Roche, Phys. Rev. Lett. (18), 182503 (2014)77. I. Tews, S. Gandolfi, A. Gezerlis, A. Schwenk, Phys.Rev. C (2), 024305 (2016)78. M. Piarulli, I. Bombaci, D. Logoteta, A. Lovato, R.B.Wiringa, Phys. Rev. C (4), 045801 (2020)79. A. Schwenk, B. Friman, G.E. Brown, Nucl. Phys. A (1-2), 191 (2003)80. A. Schwenk, B. Friman, Phys. Rev. Lett. (8), 082501(2004)81. L.G. Cao, U. Lombardo, P. Schuck, Phys. Rev. C - Nucl.Phys. (6), 064301 (2006)82. S.S. Zhang, L.G. Cao, U. Lombardo, P. Schuck, Phys.Rev. C (4), 044329 (2016)83. H.A. Bethe, Phys. Rev. (1), 38 (1949)84. A. Gezerlis, J. Carlson, Phys. Rev. C (3), 032801(2008)85. S. Capponi, G. Roux, P. Azaria, E. Boulat, P. Lechem-inant, Phys. Rev. B - Condens. Matter Mater. Phys. (10), 100503 (2007)86. S. Capponi, G. Roux, P. Lecheminant, P. Azaria,E. Boulat, S.R. White, Phys. Rev. A - At. Mol. Opt.Phys. (1), 013624 (2008)87. W.G. Dawkins, J. Carlson, U. Van Kolck, A. Gezerlis,Phys. Rev. Lett. (14), 143402 (2020)88. H. Kamei, K. Miyake, J. Phys. Soc. Japan (7), 1911(2005)89. T. Sogo, R. Lazauskas, G. R¨opke, P. Schuck, Phys. Rev.C - Nucl. Phys. (5), 051301 (2009)90. T. Sogo, G. R¨opke, P. Schuck, Phys. Rev. C - Nucl.Phys. (6), 064310 (2010)91. T. Sogo, G. R¨opke, P. Schuck, Phys. Rev. C - Nucl.Phys. (3), 034322 (2010)92. P. Schuck, Y. Funaki, H. Horiuchi, G. R¨opke,A. Tohsaki, T. Yamada, J. Phys. Conf. Ser. (1),012014 (2014)93. A. Gezerlis, G.F. Bertsch, Y.L. Luo, Phys. Rev. Lett. (25), 252502 (2011)94. B. Bulthuis, A. Gezerlis, Phys. Rev. C (1), 014312(2016)95. M. Kamimura, Nucl. Physics, Sect. A (3), 456(1981)96. M. Matsuo, K. Mizuyama, Y. Serizawa, Phys. Rev. C -Nucl. Phys. (6), 064326 (2005)97. N. Pillet, N. Sandulescu, P. Schuck, Phys. Rev. C - Nucl.Phys. (2), 024310 (2007)98. N. Pillet, N. Sandulescu, P. Schuck, J.F. Berger, Phys.Rev. C - Nucl. Phys. (3), 034307 (2010)99. K. Hagino, H. Sagawa, P. Schuck, J. Phys. G Nucl. Part.Phys. (6), 064040 (2010) 100. R.B. Wiringa, S.C. Pieper, J. Carlson, V.R. Pandhari-pande, Phys. Rev. C - Nucl. Phys. (1), 23 (2000)101. F. Hoyle, Astrophys. J. Suppl. Ser. , 121 (1954)102. T. Yamada, Y. Funaki, H. Horiuchi, G. R¨opke,P. Schuck, A. Tohsaki, Nuclear Alpha-Particle Conden-sates (Springer Berlin Heidelberg, Berlin, Heidelberg,2012), pp. 229–298103. P. Schuck, Y. Funaki, H. Horiuchi, G. R¨opke,A. Tohsaki, T. Yamada, Phys. Scr. (12), 123001(2016)104. A. Tohsaki, H. Horiuchi, P. Schuck, G. R¨opke, Rev.Mod. Phys. (1), 011002 (2017)105. M. Caracanhas, A.L. Fetter, G. Baym, S.R. Muniz, V.S.Bagnato, J. Low Temp. Phys. (3-4), 133 (2013)106. H.W. Hammer, A. Nogga, A. Schwenk, Rev. Mod. Phys. (1), 197 (2013)107. U. Van Kolck, Phys. Rev. C (6), 2932 (1994)108. U. Van Kolck, Nucl. Phys. A (2), 273 (1999)109. J.L. Friar, D. H¨uber, U. van Kolck, Phys. Rev. C - Nucl.Phys. (1), 53 (1999)110. P.F. Bedaque, U. van Kolck, Annu. Rev. Nucl. Part. Sci. (1), 339 (2002)111. E. Braaten, H.W. Hammer, Ann. Phys. (N. Y). (1),120 (2007)112. P. Naidon, S. Endo, Reports Prog. Phys. (5), 056001(2017)113. T. Frederico, A. Delfino, L. Tomio, M.T. Yamashita,Prog. Part. Nucl. Phys. (4), 939 (2012)114. A. Daniel Garc´ıa-Orozco, L. Madeira, L. Galantucci,C.F. Barenghi, V.S. Bagnato, EPL (Europhysics Lett. (4), 46001 (2020)115. ´A.V.M. Marino, L. Madeira, A. Cidrim, F.E.A. dos San-tos, V.S. Bagnato, arXiv Prepr. p. 2005.11286 (2020)116. L. Madeira, A.D. Garc´ıa-Orozco, F.E.A. dos Santos,V.S. Bagnato, Entropy (9), 956 (2020)117. B. Balantekin, C. Bertulani, V. Zelevinsky, Nucl. Phys.News29