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de Haas van Alphen oscillations for neutral atoms in electric fields B. Farias∗ Centro de Ciências e Tecnologia Agroalimentar, Universidade Federal de Campina Grande, 58840-000, Pombal, PB, Brazil. C. Furtado† Departamento de F´ısica, Universidade Federal da Para´ıba, Caixa Postal 5008, 58051-970, Joaõ Pessoa, PB, Brazil. ThedeHaasvanAlphen(dHvA)effectiswellknownasanoscillatoryvariationofthemagnetiza- tionofconductorsasafunctionoftheinversemagneticfieldandthefrequencyisproportionaltothe area of the Fermi surface. Here, we show that an analog effect can occur for neutral atoms with a nonvanishingmagneticmomentinteractingwithanelectricfield. Underanappropriatefield-dipole configuration,theneutralatomssubjecttoasyntheticmagneticfieldarrangethemselvesinLandau levels. Using the Landau-Aharonov-Casher (LAC) theory, we obtain the energy eigenfunctions and 6 1 eigenvalues as well as the degeneracy of the system. In a strong effective magnetic field regime we 0 present the quantum oscillations in the energy and effective magnetization of a two-dimensional 2 (2D) atomic gas. From the dHvA period we determine the area of the Fermi circle of the atomic cloud. l u J PACSnumbers: 67.85.-d0,03.75.-b,03.65.G Keywords: deHaasvanAlpheneffect,Landau-Aharonov-Cashertheory,neutralatoms 9 1 INTRODUCTION eitherbyputtingthegasinrotationorbyusingartificial ] h gauge fields. In another recent work we propose an ex- p perimental scheme for the realization of the dHvA effect - Inthelastfewdecades,thestudyoftheartificialmag- in a 2D ultracold atomic cloud which uses the coupling t n netism for neutral atomic systems have grown [1–9]. In between the internal states of tripod-type atoms and an a these systems, neutral particles interacting with a suit- appropriatespatiallyvaryinglaserfieldarrangement[19]. u able configuration either of electric [10], magnetic [11] or q In this work, we show that the dHvA effect can be laser [12] fields behave themselves as charged particles [ induced in a neutral atomic system by the interaction in presence of a synthetic magnetic field. The quest for between atoms with magnetic dipole moment and an 2 artificial magnetism is to realize situations where a neu- v electric field (AC interaction). From the Ericsson and tralparticleacquiresageometricalphasewhenitfollows 0 Sjöqvist theory we describe how a symmetric gauge and a closed contour. In a seminal work [1], Aharonov and 7 consequentlyauniformmagneticfieldcanberealizedina 6 Casher showed that a particle with a magnetic moment 2Datomicgasusingasuitablefield-dipoleconfiguration. 1 moving in an electric field accumulates a quantum phase This leads to the LAC quantization. We show that the 0 whichisrelatedwithavectorpotential. Thisinteraction confinement of the neutral particles in an atomic cloud . 1 atom-electric field through a nonvanishing magnetic mo- restricts the magnetic field strength. In view of the fact 0 ment (known as Aharonov-Casher (AC) effect) coincides that Rydberg atoms are very sensitive to electric fields, 6 formally in the nonrelativistic limit with that of mini- 1 we consider an atomic gas composed by 87Rb ultracold mal coupling, where AC vector potential is determined : RydbergatomsandwecalculatetheLACdegeneracyfor v by the electric field and the direction of the magnetic thissystem. Inaregimeofstrongmagneticfieldandzero i X dipole. BasedontheACeffectEricssonandSjöqvist[10] temperature we display the quantum oscillations in the have demonstrated the existence of a certain field-dipole r energyandeffectivemagnetizationofthegas. Finally,as a configuration in which an atomic analog of the standard a result of the dHvA oscillations, we determine the area Landau effect [13] occurs. This result has paved the way of the Fermi circle of the atomic cloud. fortheatomicrealizationofthequantumHalleffectand Shubnikov-de Haas effect as well as de Haas-van Alphen effect using electric fields. LANDAU-AHARONOV-CASHER QUANTIZATION Inthesolid-statecontext,dHvAoscillationshavebeen usedtostudytheshapeoftheFermisurfaceoftheclean- According to the AC theory the Hamiltonian operator enoughmaterials[14–16]. Suchuseisalsopossibleinthe that describes the interaction between a neutral particle context of atomic gases as an alternative technique to with nonvanishing magnetic moment µ and an electric the adiabatic band mapping [17]. However, the atomic field E, in the non-relativistic limit, is given by [10] dHvA effect is still poorly studied. In [18], Grenier et al. have explored the possibility of observing like-dHvA 1 (cid:16) µ (cid:17)2 µ(cid:126) oscillations for a non-interacting gas of fermionic atoms, H = p− n×E + ∇·E, (1) 2M c2 2Mc2 2 where n is the direction of the magnetic dipole moment momentum Lˆ = i(cid:126) ∂ is a quantum integral of motion z ∂φ µ, M is the mass of the particle and c is the speed of and so the wavefunction Ψ can be factorized to separate light. The Hamiltonian (1) presents an analogy to the the variables minimal coupling, where p− µn×E is the homologous ofthekinematicmomentumfoc2rachargedparticleinthe Ψ=eimφR(r). (9) presence of the magnetic field. In this context, the AC Here eimφ is the eigenfunction of the operator Lˆ , with vector potential is defined as z the eigenvalue (cid:126)m where m is an integer. 1 Substituting the solution (9) into Eq.(7) the A = (n×E), (2) AC c2 Schrödinger equation assumes the form and the associated magnetic field as (cid:126)2 (cid:20) d2 1 d m2(cid:21) (cid:20) σ(cid:126)|ω | + − R+ E− AC (m+1) 1 2M dr2 rdr r2 2 BAC = c2∇×(n×E). (3) Mω2 (cid:21) − ACr2 R=0.(10) 8 In order to obtain the LAC quantization we assume the magnetic dipole moment µ aligned parallel with the Introducing the dimensionless variable ξ = M|ωAC|r2 direction z, i.e., n=zˆ, and we adopt the following cylin- 2(cid:126) we rewrite Eq. (10) in a dimensionless form drical electric field configuration (cid:20) d2 d (cid:21) (cid:20) m2 ξ(cid:21) E= ρ0 rrˆ, (4) ξdξ2 + dξ R+ −4ξ +β− 4 R=0, (11) 2(cid:15) 0 where (cid:15)0 is the electric vacuum permittivity and ρ0 is a where β = (cid:126)|ωEAC| − σ2 (m+1). uniform volume charge density. The asymptotic analysis of Eq. (11) prompts us to Then the corresponding AC vector potential becomes write a solution for R(ξ) as ρ A = 0 rφˆ, (5) R(ξ)=e−ξ/2ξ|m|/2W(ξ). (12) AC 2(cid:15) 0 It is instructive to note that in case of the standard and the AC magnetic field takes the form of the uniform Landauproblemforfreeelectronsinanuniformmagnetic magnetic field field the solution (12) is obtained taking r → ∞ (which ρ B = 0zˆ. (6) is to say ξ → ∞). On the other hand, in our system AC (cid:15)0 we consider the particles confined in a 2D atomic cloud which implies that the value of r is limited. In this case, Note that the field-dipole configuration presented above wecanobtaintheanalyticalsolution(12), inthelimitof obeys the Ericsson and Sjöqvist conditions for the emer- [12, 19–21] gence of an AC analog of the Landau effect, namely: (i) condition for vanishing torque on the dipole, n×((cid:104)p− M|ω | µn×E(cid:105)×E)=0,where(cid:104)·(cid:105)denotesexpectationvalue; AC (cid:29)1 m−2, (13) c2 2(cid:126) (ii)conditionsforelectrostatics; ∂ E=0and∇×E=0; t (iii) B uniform. which leads to the following restriction on the effective AC Under the field-dipole configuration presented above magnetic field strength we can write the Schrödinger equation for the system, in 2(cid:126)c2 cylindrical coordinates, as B (cid:29) . (14) AC |µ| (cid:126)2 (cid:20)1 ∂ (cid:18) ∂ (cid:19) 1 ∂2 (cid:21) (cid:20) i(cid:126)ω ∂ − r + Ψ+ − AC 2M r∂r ∂r r2∂φ2 2 ∂φ Bysubstitutingsolution(12)intoEq. (11),onearrives Mω2 (cid:126)ω (cid:21) in the confluent hypergeometric equation + ACr2+ AC Ψ=EΨ,(7) 8 2 d2 (cid:20) (cid:21) d ξ W(ξ)+ |m|+1−ξ W(ξ) with the cyclotron frequency dξ2 dξ (cid:20) (cid:21) |m|+1 µρ σ|µρ | + β− W(ξ)=0, (15) ω = 0 = 0 =σ|ω |, (8) 2 AC Mc2(cid:15) Mc2(cid:15) AC 0 0 which is satisfied by the confluent hypergeometric func- forwhichthesignσ =±1describestherevolutiondirec- tion tion of the corresponding classical motion. (cid:18) (cid:19) Once the coefficients in the differential Eq. (7) are |m|+1 W(ξ)=F −β+ ,|m|+1,ξ , (16) independent of the azimuth coordinate φ, the angular 2 3 with the energy eigenvalues in the form of |5S ,F = 2,m = 2(cid:105) ground state [24]. This would 1/2 F lead to a stronger AC interaction for the Rydberg atom (cid:18) (cid:19) |m| σm σ 1 E(σ) =(cid:126)|ω | n + + + + . (17) inanelectricfield,ascomparedtoanatomintheground nξ,m AC ξ 2 2 2 2 state. Furthermore,theregimeofultracoldtemperatures allows reaching a significant LAC quantization without Here n is a nonzero negative integer. These levels are ξ the requirement of too extreme electric fields. equivalent to the Landau levels of the charged system. In what follows we consider a 2D ultracold atomic The radial eigenstates for these LAC states are given by cloudwithanareaofA∼150 µm2andcontainsN ∼104 (cid:115) 87Rb atoms in the n = 51 excited state [25–27]. Under Rnξ,m(r)= a|m1|+1 2||mm|n|+ξ!|nmξ|!!2e−4aρ2A2Cρ|m| these conditions, the expression (14) becomes AC (cid:18) ρ2 (cid:19) BAC (cid:29)40.93 Teff, (23) ×F −n ,|m|+1, , (18) ξ 2a2 AC where T = N is an effective unit. (cid:113) eff C·m where a = (cid:126) . In addition, the degeneracy is written as AC MωAC Finally, introducing a new quantum number D =1.17×10−15B . (24) AC AC |m|+σm n=nξ+ 2 , (19) As a result, the lowest landau level regime of the system, i.e., D = 104, is achieved when B = 8.55× AC AC then the energy spectrum acquires the standard form of 1018 T . eff the LAC spectrum (cid:18) (cid:19) 1 E(σ) =(cid:126)|ω | n+ (1+σ) , (20) QUANTUM OSCILLATIONS FOR NEUTRAL n AC 2 PARTICLES SUBJECT TO AN ELECTRIC FIELD where n=0,1,2,.... Fromnowon,weinvestigatethephenomenonofdHvA The quantities that characterize this LAC system can oscillations for the neutral atomic system presented in be obtained by using the AC duality [1, 10] the previous section. We consider the system at zero- µλ temperature limit and containing a fixed number of N qΦ↔ , (21) atoms. We do not take into account the temperature c2(cid:15) 0 smearing of the quantum oscillations. We assume that where Φ is a magnetic flux and λ is an uniform linear the lowest p LAC levels (where p is a positive integer) charge density in the direction of the magnetic dipole. are completely filled with pD atoms each one and the In addition, we have that λ = ρ0A, with A being the highest (p+1)th level is partly occupied with N −pD area of the atomic cloud. atoms. In this case, the Fermi level lies in the (p+1)th Inthisway,thespacingbetweentheenergylevelsfora level. fixedσ is∆E =(cid:126) |µρ0| , theeffectivemagneticlength From Eq. (22) we can see that by decreasing the is l=(cid:113)(cid:126)c2(cid:15)0AaCnd theMdc2e(cid:15)g0eneracy is magnetic field BAC, it decreases the degeneracy D of µ(cid:15)0 the system. As a consequence, fewer atoms can be ac- commodated on each LAC level and the atomic popu- D =ρB (22) AC AC lation of the highest (p + 1)th energy level will range where ρ = µA and B = ρ0. Note that the degener- from completely full to entirely empty. Fig.(1a) illus- c2h AC (cid:15)0 trates the transfer of atoms to the highest partly occu- acy linearly depends on the magnetic field B and it is AC pied LAC level of a 2D ultracold cloud with N = 104 limited by the fact that we have a finite atomic trap. 87Rb atoms when the magnetic field is slept in the range Fromanexperimentalpointofview,ultracoldRydberg 8.55×1018 T ≤ B ≤ 8.55×1017 T or equiva- atomic clouds are good candidates for the realization of eff AC eff lently 1.17×10−19 T−1 ≤ 1 ≤1.17×10−18 T−1. At theLACquantization. Thisisbecausetheextremeprop- eff BAC eff erties of the highly excited atoms compared to atoms in 1 = 1.17×10−19 T−1 only the lowest level p = 0 is BAC eff ground state, such as very high dipole polarizabilities, populatedwith104 atomsandtheupperlevel(p+1)th= magneticmomentsandatom-atomstrengthsbecomethe 1 is empty. As the reciprocal magnetic field is increased Rydberg atoms very sensitive to electric fields [22–24]. the level p = 1 starts to accommodate atoms until it is Forinstance,ifan87Rbatomisexcitedtoastaten=51 fully occupied with 5×103 atoms. Then a new upper (n is the principal quantum number), the atom can have level (p+1)th = 2 becomes populated and so on. Note a magnetic moment of µ = 50 µ , which is a factor of thatsingularitiesappearatstrengthofthemagneticfield B 50 higher than the magnetic moment of an atom in a where a new LAC level becomes occupied. The distance 4 between such singularities are regularly spaced and de- is expressed as finesthedHvAperiod∆( 1 )= ρ =1.17×10−19 T−1. BAC N eff (cid:20) N(cid:21)(cid:20) N(cid:21) Fig.(1b)displayshowthepopulationofthelowestpcom- ε(cid:48) =−µeffρ pB − (p+1)B − . (27) pletely filled levels varies with 1 . It is this jump of B AC ρ AC ρ BAC atoms to a higher energy level that causes the oscilla- As Fig. (2) shows, the energy ε(cid:48) oscillates with a tions in the energy and in the effective magnetization of quadratic dependence on each dHvA period. Note that the atomic gas as a function of the inverse AC magnetic if we set either 1 = pρ or 1 = (p+1)ρ there is no field. BApC N BAp+C1 N LAClevelpartlypopulatedandtheEq. (27)iszero. On N(cid:45)pD the other hand, at 1 = ρ p(p+1) the energy ε(cid:48) has a 5000 BAC N p+12 local maximum. The amplitude of the oscillations falls because a new higher level passes to be occupied each time with fewer particles as 1 varies. 3000 BAC Energy 10 1000 8 1(cid:144)B 2 4 6 8 10 (a) 6 pD 4 10000 2 8000 6000 1(cid:144)B 2 4 6 8 10 4000 Figure2. Variationoftheenergy(inunitsof10−27 J)ofthe 2000 partiallyoccupiedLAClevelwithrespecttotheinverseeffec- tivemagneticfield(inunitsof10−19 T−1). Forcalculations, 1(cid:144)B we takeM =1.443×10−25 kg andµe=ff 4.64×10−22 J/T. 2 4 6 8 10 Rb (b) The effective magnetization M is obtained taking − ∂ε(cid:48) , then we have Figure 1. (a) (Color online) Number of N −pD atoms in ∂BAC the LAC level partially filled as a function of the reciprocal (cid:20) N (cid:21) magnetic field (in units of 10−19 T−eff1). (b) Number of sD M=µeBffρ 2BACp(p+1)− ρ (2p+1) , (28) atoms in energy levels which are completely occupied as a function of the inverse magnetic field. where pD < N ≤ (p+1)D. In Fig. (3) we display the dHvA oscillations of the AC magnetization as a function Summing the energy of the atoms in fully occupied of the inverse magnetic field calculated from Eq. (3). levels with the energy of the atoms in the partly filled As expected, at T = 0 these oscillations have a saw- levelwehavethetotalenergyofthesystemthatisgiven tooth shape with a constant amplitude. M is linear in by 1 except in points for which a new LAC level starts BAC to be filled. In these points the Fermi level makes an (cid:88)p−1 1 1 abrupt jump between two LAC levels and consequently ε= (n+2)(cid:126)|ωAC|D+(N−pD)(cid:126)|ωAC|(p+2) (25) Mexperiencesjumpsof2NµeBff ≈3.76×10−44 JT·s atthe n=0 end of each dHvA period. The effective magnetization which can be rewritten in a most appropriate form as is −NµeBff (Landau diamagnetism) when just the first p LAC levels are occupied, and it jumps to +Nµeff as the (cid:18) (cid:19) B N (p+1)thlevelstartstobepopulated,returningsmoothly ε=−µeffρ B2 p(p+1)− B (2p+1) , (26) B AC ρ AC to −Nµeff again when this new level is completely pop- B ulated. for ρs < 1 < ρ(s+1). Here µeff = (cid:126)µ is an effective It is important to observe that since the AC magnetic N BAC N B 2Mc2 Bohr magneton. fieldinEq. (6)isartificial,theeffectivemagnetizationM However, for simplicity, we consider just the total en- is not directly observable. However, as discussed in [18] ergyoftheatomsinthepartlyoccupiedLAClevelwhich there are indications that the dHvA oscillations should 5 beobservedintheangularmomentum(cid:104)L (cid:105)oftheatomic ACKNOWLEDGMENTS z gas, which is directly analogous to the magnetization in the solid-state context. WewouldliketothankCNPqandCAPESforfinancial Magnetization support. 1.5 1 0.5 1(cid:144)B ∗ [email protected] 2 4 6 8 10 † furtado@fisica.ufpb.br (cid:45)0.5 [1] Y. Aharonov, and A. Casher, Phys. Rev. Lett. 53, 319 (cid:45)1 (1984). (cid:45)1.5 [2] J. Anandan, Phys. Rev. Lett. 48, 1660 1982. [3] X.-G. He, B.H.J. McKellar, Phys. Rev. A 47, 3424 (1993). Figure3. EffectiveMagnetization(inunitsof10−44 J·s)asa [4] M. Wilkens, Phys. Rev. Lett. 72, 5 (1994). T functionoftheinversemagneticfield(inunitsof10−19 T−1). [5] H.Wei,R.Han,X.Wei,Phys.Rev.Lett.75(1995)2071. [6] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [7] A. L. Fetter, Rev. Mod. Phys. 81, 647 (2009). [8] J. Ruseckas, G. Juzeliu¯nas, P. O¨hberg, and M. Fleis- As a natural consequence of the dHvA oscillations, we chhauer, Phys. Rev. Lett. 95, 010404 (2005). can determinate the area S of the extremal cross-section [9] Y. J. Lin et al., Nature (London) 462, 628 (2009). of the Fermi surface of the atomic cloud through the [10] M. Ericsson, E. Sjo¨qvistt, Phys. Rev. A 65, 013607 Onsager-like relation (2001). [11] L.R. Ribeiro, C. Furtado, and J.R. Nascimento, Phys. Lett. A 348, 135 (2006). 2πN S = [12] B.Farias,J.LemosdeMelo,andC.Furtado,Eur.Phys. (cid:126)ρ J. D 68, 77 (2014). [13] L.D. Landau, Z. Phys. 64, 629 (1930). [14] W.deHaas,andP.vanAlphen,Comm.Phys.Lab.Lei- Usingthephysicalparametersofthesystemweestimate den 212a (1930). the area of the Fermi circle in S ≈5×1053 m−2. [15] A. Abrikosov, Introduction to the theory of normal metals, Solid State Physics: Supplement (Academic Press,1972). [16] N. Ashcroft, and N. Mermin, Solid State Physics (Saun- ders College, Philadelphia, 1976). CONCLUSION [17] M. Ko¨hl, H. Moritz, T. Sto¨ferle, K. Go¨nter, and T. Esslinger, Phys. Rev. Lett. 94, 080403 (2005). [18] Ch. Grenier, C. Kollath, and A. Georges, Phys. Rev. A An atomic analog of the de Haas van Alphen effect 87, 033603 (2013). based on the Aharonov-Casher interaction has been pro- [19] B. Farias, and C. Furtado, Physica B 481, 19 (2016). posed. In this context, neutral atoms may interact with [20] P. O¨hberg, G. Juzeliu¯nas, J. Ruseckas, and M. Fleis- an electric field via a nonvanishing magnetic moment. chhauer, Phys. Rev. A 72, 053632 (2005). Using the Ericsson and Sjöqvist theory we have shown [21] G. Juzeliu¯nas, and P. O¨hberg, Phys. Rev. Lett. 93, how a uniform magnetic field can be induced by an elec- 033602 (2004). tric field in a 2D ultracold atomic gas. We have shown [22] JongseokLim,Han-gyeolLee,andJaewookAhn,Journal that the magnetic field strength is limited by the fact of the Korean Physical Society, 63, No. 4, 867 (2013). [23] T. Pohl, H. R. Sadeghpour, and P. Schmelcher, Physics that the gas is finite. Applying the Schrödinger equa- Reports 484 181 (2009). tion approach the LAC energy levels and eigenfunctions [24] R. R. Mhaskar. Toward an Atom Laser: Cold Atoms in are obtained. Due to the high sensitivity of the Rydberg aLong,HighgradientMagneticGuide.PhDthesis,Uni- atomstotheelectricfieldswehaveconsideredanatomic versity of Michigan, (2008). cloud composed by 87Rb ultracold Rydberg atoms and [25] Another important point to be considered in an exper- we have calculated the LAC degeneracy for this system. imental arrangement is that the electric field, given by In the limit of high magnetic field and zero temperature Eq. (4), requires a uniform charge density inside of the atomiccloud.Althoughexperimentallychallenging,such we have presented the oscillatory variation of the energy anarrangementcouldbecomposedofa2Dtoroidaltrap andeffectivemagnetizationofthegasasafunctionofthe and a perpendicular line of charge passing through the magnetic field strength. As a consequence of the dHvA centre of the toroid [26]. oscillations,wehaveestimatedtheareaoftheFermisur- [26] A. A. Wood, B.H.J. McKellar, and A. M. Martin, face of the atomic gas. arXiv:1604.04996v1. 6 [27] O. Morizot, Y. Colombe, V. Lorent, and H. Perrin and B. M. Garraway, Phys. Rev. A 74, 023617 (2006).