One-dimensional Rydberg Gas in a Magnetoelectric Trap
Michael Mayle, Bernd Hezel, Igor Lesanovsky, Peter Schmelcher
aa r X i v : . [ phy s i c s . a t o m - ph ] S e p One-dimensional Rydberg Gas in a Magnetoelectric Trap
Michael Mayle, Bernd Hezel, Igor Lesanovsky, and Peter Schmelcher
1, 2 Theoretische Chemie, Physikalisch–Chemisches Institut, Universit¨at Heidelberg,Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany Physikalisches Institut, Universit¨at Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany Institut f¨ur Theoretische Physik, Universit¨at Innsbruck, A-6020 Innsbruck, Austria (Dated: October 28, 2018)We study the quantum properties of Rydberg atoms in a magnetic Ioffe-Pritchard trap which issuperimposed by a homogeneous electric field. Trapped Rydberg atoms can be created in long-livedelectronic states exhibiting a permanent electric dipole moment of several hundred Debye. Theresulting dipole-dipole interaction in conjunction with the radial confinement is demonstrated togive rise to an effectively one-dimensional ultracold Rydberg gas with a macroscopic interparticledistance. We derive analytical expressions for the electric dipole moment and the required lineardensity of Rydberg atoms.
PACS numbers: 32.60.+i,32.10.Dk,34.20.Cf,32.80.Pj
Because of their widely tunable properties, ultracoldatomic gases provide the ideal playground to model andstudy complex many body systems. The interatomic in-teraction can be tailored using Feshbach-resonances andmagnetic, optical, and electric fields can be applied inorder to generate virtually any external potential. Fa-mous examples for the versatility are the demonstrationof the Mott-Insulator to superfluid phase-transition ofultracold atoms in an optical lattice [1], the BEC-BCScrossover in a gas of Li [2], or the Kosterlitz-Thoulessphase transition studied within a two-dimensional Bose-Einstein condensate [3]. Traditionally, there is a greatinterest in studying systems with reduced spatial dimen-sions, such as the latter example. One paradigm is con-stituted by the work of Lieb and Liniger who were thefirst to solve the system of pointlike interacting bosonsin one dimension using Bethe’s ansatz [4]. In the limitof an infinitely strong interparticle interaction strength,a so-called Tonks-Girardeau gas emerges [5].Besides gases of ground state atoms, particularly Ryd-berg gases represent excellent systems to study the influ-ence of a strong interparticle interaction on the dynamicsof many-particle systems. Due to the large displacementof the ionic core and the valence electron, Rydberg atomscan develop a large electric dipole moment leading to astrong and long-ranged dipole-dipole interaction amongthem [6]. However, unlike ground state atoms Rydbergatoms suffer from radiative decay and hence the mutualinteraction time is limited by the lifetime of the electron-ically excited state. But still, in so-called frozen Ryd-berg gases, where the timescale of the atomic motion ismuch longer than the radiative lifetime, exciting effectslike the dipole-blockade [7] or resonant population trans-fer [8] have been observed.Several works have focused on the important issue oftrapping Rydberg atoms based on electric [9], optical[10], or strong magnetic fields [11]. Due to the high leveldensity and the strong spectral fluctuations with spa- tially varying fields, trapping or manipulation in generalis a delicate task. This holds particularly for the casewhere both the center of mass and internal motion areof quantum nature. Moreover, the inhomogeneous exter-nal fields lead to an inherent coupling of these motions.Addressing this regime, it has been recently theoreti-cally shown that Rydberg atoms can be tightly confinedand prepared in long-lived electronic states in a magneticIoffe-Pritchard (IP) trap [12] which can be miniaturizedusing so-called atomchips [13]. Here we use the IP con-figuration as a key ingredient in order to ’prepare’ andstudy a one-dimensional (1D) Rydberg gas. Specifically,we propose a modified IP trap, a magneto-electric trap,which offers confining potential energy surfaces for theatomic center of mass (c.m.) motion in which the atomspossess an oriented permanent electric dipole moment.We derive analytical expressions for the dipole-dipole in-teraction among trapped Rydberg atoms and estimatebelow which Rydberg atom density a 1D Rydberg gas isexpected to form. Moreover, we estimate the lifetime ofsuch a gas.Proceeding along the lines of Refs. [12, 14], we em-ploy a two-body approach in order to model an alkalimetal atom in a Rydberg state. We assume the singlevalence electron and the ionic core (mass M c ) to inter-act via a pure Coulomb potential. While the inclusionof the fine-structure and quantum defects can be read-ily done, it turns out not to be necessary for high an-gular momentum electronic states in the regime we arefocussing on [12]. The IP field configuration is given by B ( x ) = B e + B lin ( x ) with B lin ( x ) = G [ x e − x e ]and the vector potential reads A ( x ) = A c ( x ) + A lin ( x )with A c ( x ) = B [ x e − x e ] and A lin ( x ) = Gx x e ,where B and G are the Ioffe field strength and the gra-dient, respectively. In addition, we apply a homogeneouselectric field pointing in the x -direction of the labora-tory frame F = F e . After introducing relative andc.m. coordinates ( r and R ) and employing the unitarytransformation U = exp (cid:2) i B e × r · R (cid:3) , the Hamiltoniandescribing the Rydberg atom becomes (atomic units areused unless stated otherwise) H IPE = H A + A c ( r ) · p + P M c − µ N · B ( R ) + F · r − µ e · B ( R + r ) + A lin ( R + r ) · p . (1)Here, H A = p − r is the Hamiltonian of a hydrogen atompossessing the energies E n = − n − . The second termdenotes the energy of the electron in the homogeneousIoffe field due to its orbital motion. The following twoterms of H IPE describe the motion of a point-like parti-cle possessing the magnetic moment µ N in the presenceof the field B . The magnetic moments are connectedto the electronic spin S and the nuclear spin Σ accord-ing to µ e = − S and µ N = − g N M c Σ , with g N being thenuclear g -factor. We neglect the term involving µ N inthe following due to the large nuclear mass. The electricfield interaction, which in case of a neutral two-body sys-tem couples only to the relative coordinates, gives rise tothe fifth term. The last two terms of H IPE are spin-fieldand motionally induced terms coupling the electronic andc.m. dynamics. We focus on a parameter regime whichallows us to neglect the diamagnetic interactions [12].In order to find the stationary states of the Hamilto-nian (1), we assume that neither the magnetic nor theelectric field causes couplings between electronic stateswith different principal quantum number n . In this casewe can consider each n -manifold separately and may rep-resent the Hamiltonian (1) in the space of the 2 n stateswhich span the n -manifold under investigation. The pa-rameter range in which this approximation is valid hasbeen thoroughly discussed in Refs. [12, 15]. Because ofthe translational symmetry of the IP and the electricfield, the axial c.m. motion along Z can be separated fromthe transversal motion in the X - Y plane. If we omit theenergy offset E n and introduce scaled c.m. coordinates( R → γ − R with γ = GM c ) while scaling the energyunit with ǫ scale = γ /M c , we arrive at the Hamiltonian H = P + P µ · G + γ − M c [ H m + H e ] . (2)This Hamiltonian governs the transversal c.m. as wellas the electronic dynamics and involves the effectivemagnetic field G = X e − Y e + γ − M c B e . Thesymbols µ , H m , and H e are the 2 n -dimensional ma-trix representations of the operators [ L + 2 S ], H m = A lin ( r ) · p + B lin ( r ) · S , and the electric field interaction H e = F x , respectively (we introduced L = r × p ). Athorough interpretation of this Hamiltonian in the ab-sence of H e is provided in Ref. [12]. In order to solvethe corresponding Schr¨odinger equation we employ anadiabatic approach. To this end an unitary transforma-tion U ( X, Y ) which diagonalizes the last two (matrix)terms of the Hamiltonian is applied, U † ( X, Y )( µ · G + FIG. 1: (Color online) Potential energy surfaces of the c.m.motion of a Rb atom ( n = 30) in an IP trap with B =10 G, G = 10 Tm − . Dashed lines: F = 0, solid lines: F =5 .
14 Vm − . An overview of the seven energetically highestpotential curves is shown in panel (a). Magnified views of theuppermost (b,c) and next lower ones (d,e) are also provided.The range of the X -coordinate, corresponding to 2 . µ m, is thesame for each subfigure (a)-(e). The total field configurationis sketched in panel (f) where the circles depict the locations ofthe minima of the uppermost (big circle) and the two adjacentlower-lying (small circles) c.m. surfaces. The magnetic fieldlines are indicated in gray while the electric field is sketchedby black arrows. γ − M c [ H m + H e ]) U ( X, Y ) = E α ( X, Y ). Since U ( X, Y )depends on the c.m. coordinates, the transformed ki-netic energy term involves non-adiabatic (off-diagonal)coupling terms which can be neglected in our parameterregime [12]. We are thereby led to a set of 2 n decoupleddifferential equations governing the adiabatic c.m. mo-tion within the individual two-dimensional energy sur-faces E α ( X, Y ), i.e., the surfaces E α ( X, Y ) serve as po-tentials for the c.m. motion of the atom. In Fig. 1 wepresent intersections along the X -direction of such poten-tial surfaces for B = 10 G, G = 10 Tm − and n = 30 inthe case of Rb. For zero electric field strength (dashedlines), the potential curves are organized in groups whichare energetically well-separated by a gap of γ − M c B =89 . ω = G p n/ BM c = 13 . . µ K. The two adjacent lower surfaces aredegenerate and also approximately harmonic. As soonas an electric field is applied, all surfaces are shiftedconsiderably in energy. This is visible from the solidcurves in Fig. 1 for which an electric field of strength F = 5 .
14 Vm − is applied. The shapes of the potentialsare barely affected by the electric field such that Rydbergstates which were trapped in a pure IP configuration re-main confined also in the magneto-electric trap. More-over, adding the electric field leads to non-trivial effects:The second and third surface, which were almost degen-erate in the absence of the electric field, are now shifted inopposite ways along the x -direction. All surfaces shownprovide a harmonic confinement with a trap frequency ω also in the x -direction. We remark that the chosenparameter set does not generate an extreme constella-tion hence an even stronger confinement can be achievedwithout invalidating the applied approximations [12].Let us now investigate the electronic properties of aRydberg atom being trapped in the uppermost potentialsurface. For F = 0 and sufficiently large values of B , thissurface is formed almost exclusively by the highest pos-sible electronic angular momentum state, i.e., l = n − F is increased, electronic states with smaller l willbe inevitably admixed to the electronic state belongingto this energy surface. An interesting property to in-vestigate is hence the electric dipole moment of trappedRydberg atoms: While for F = 0 the electronic statesare almost pure parity eigenstates and therefore exhibitalmost no electric dipole, the admixture of lower l statesto the uppermost surface in the presence of the field isexpected to give rise to a non-vanishing expectation valueof the dipole operator. Indeed, this becomes evident inFig. 2 where the uppermost potential surface and thethree components of the expectation value of the elec-tric dipole operator D ( R ) = h r i ( R ) are shown (sameparameters as in Fig. 1). It can be clearly seen that apermanent dipole moment is established whose dominantcontribution points along the electric field vector.In order to study the dependence of D ( R ) on the fieldstrengths F and B as well as on the degree of elec-tronic excitation, we use perturbation theory. In thelimit of a large Ioffe field strength B the unitary trans-formation which diagonalizes the Hamiltonian (2) can bewritten explicitly as U r = e − iα ( L x + S x ) e − iβ ( L y + S y ) withsin α = − Y | G | − , cos α = p | G | − Y | G | − , sin β = X ( | G | − Y ) − and cos β = γ − M c B ( | G | − Y ) − .This transformation rotates the z -axis into the local mag-netic field direction where α and β denote the rotationangles. Using this result we find up to first order in F/B the electric dipole moment (in atomic units) D ( R ) = 92 FB n ( n − cos β sin β sin α sin β cos α . (3)We note that D ( R ) scales proportional to the thirdpower of the principal quantum number and can there-fore gain a significant magnitude even if the ratio F/B is small. Good agreement of Eq. (3) with the calculateddata presented in Fig. 2 is found; e.g., in the vicinity ofthe minimum of the potential surface ( X = Y = 0) wefind an exact value of D x = 270 whereas the expression FIG. 2: (Color online) (a) Uppermost electronic potential sur-face for the c.m. motion of Rb in the n = 30 multipletand the parameters used in Fig. 1. (b-d) Components of theelectronic dipole moment D ( R ) in atomic units. One recog-nizes the clear alignment of the electric dipole moment alongthe electric field vector. The numerically calculated values of D ( R ) are to good accuracy reproduced by Eq. (3). (3) yields 276. For the remaining components Eq. (3)yields zero at the origin. For smaller ratios of F/B , evenbetter agreement can be achieved.Because of the dependence on the angles α and β , thedipole moment depends weakly on the quantum state ofthe c.m. motion. However, since the field configurationis translationally symmetric the electric dipole momentis independent of the Z -position of the Rydberg atoms inthe trap. If we now consider two transversally confinedatoms in the same trap at the longitudinal positions Z A and Z B , we can write for their dipole-dipole interaction V D ( R A , R B ) = 1 | R A − R B | (cid:16) D ( R A ) · D ( R B ) − D ( R A ) · e ] [ D ( R B ) · e ] (cid:17) ≈ D ( R A ) · D ( R B ) | Z A − Z B | (4)where e denotes the interparticle unit vector. The ap-proximation in Eq. (4) holds due to the orientation ofthe dipoles and the assumption that | Z A − Z B | is largecompared to the transversal oscillator length of the trap.These conditions moreover ensure a minimal coupling ofthe transversal and longitudinal motion. Using this ap-proximation one can estimate the interaction energy ofone atom being part of an infinite atomic chain with aninterparticle spacing a . One finds E int = 2 D (0) a ∞ X k =1 k − = 812 a F B n ( n − ζ (3) (5)with the Riemann zeta function ζ ( x ). Here we have ap-proximated D ( R ) ≈ D (0) since the dipole momentbarely varies in the vicinity of X = Y = 0. If the in-teraction energy E int is smaller than the transversal trapfrequency ω , we can assume that the interacting atomsremain in the transversal ground state: This is consid-ered the 1D regime. The linear density below which a 1DRydberg gas is expected to form is then given by N = √ B " r M c F G ζ (3) n / ( n − − . (6)Above this density, excited transversal c.m. states mightbe populated resulting in a quasi 1D Rydberg gas whichis certainly of interest on its own. For our parameter set,we obtain a minimal interparticle spacing a = 43 µ m;hence a chain of 1 mm in length contains 23 particles.This density can be further increased by either increasingthe magnetic field gradient and/or decreasing the electricfield strength: At B = 10 G, G = 100 Tm − , and F =0 .
514 Vm − a chain of the same length would contain 230Rydberg atoms.Finally, the issue of radiative decay has to be ad-dressed. Since the electric field admixes merely a few l = m states ( l < n −
1) to the electronic wave func-tion of the uppermost surface, its circular character re-mains dominant resulting in one prevalent decay chan-nel. For an atom being confined to the energy surfacewhich is shown in Fig. 2, we have calculated a lifetimeof τ ≈ . τ ( n, n − ≈ c (cid:0) nα (cid:1) [16].Corrections to this bare decay rate are found to be of theorder of ( F/B ) n . Due to the scaling proportional to n , the lifetime can be significantly enhanced by excitingto a higher principal quantum number n . In addition,it can be further prolonged by establishing an adaptedexperimental setup which inhibits the electromagneticfield mode at the dominant transition frequency [17].At the same time, a cryogenic environment will dimin-ish the undesirable effect of stimulated (de-)excitationby blackbody radiation. The timescale of the dynam-ics of the Rydberg chain on the other hand depends onthe field strengths via the dipole moment and the in-terparticle spacing: A harmonic approximation of thedipole-dipole interaction yields the one-particle oscilla-tor frequency ω dd = p D (0) /M c a . As an example,the field configuration B = 10 G, G = 100 Tm − , and F = 0 .
514 Vm − yields a timescale of less than 1 ms.Let us now briefly comment on the realization of such aRydberg gas which is certainly a challenging experimen-tal task. One could start from an extremely dilute ultra-cold atomic gas prepared in an elongated IP trap. Fortransferring ground state atoms to high angular momen-tum Rydberg states, techniques such as crossed electricand magnetic fields or rotating microwave fields can beemployed, see Ref. [18] and references therein. For lowangular momentum states, trapping and the formationof a permanent dipole represent still an open questionsince quantum defects, spin-orbit coupling and reducedradiative lifetimes have to be taken into account. Duringthe preparation, the excitation lasers have to be focussedsuch that Rydberg atoms emerge only at positions sepa-rated by the interparticle spacing a which is required tomeet the criterion (6). Since a is in the order of several µ m, which can be resolved optically, this should be feasi-ble. The large value of a moreover ensures that the mu-tual ionization due to the overlap of the electronic cloudsof two atoms does not occur. For our circular states with n = 30, the atomic extension can be estimated by h r i ≈ n = 48 nm and is thus orders of magnitude smallerthan the corresponding value of a for our field configu-ration. In order to probe the dynamics of the resultantRydberg chain, one can field-ionize the atoms: From thespatially resolved electron signal a direct mapping to thepositions of the Rydberg atoms should be possible.In conclusion, we demonstrated that in a magneto-electric trap Rydberg states can be confined in electronicstates exhibiting a permanent electric dipole moment ofhundreds of Debyes. Analytical expressions for the den-sity which is required to enter the 1D regime were calcu-lated. Moreover, we pointed out that the lifetime of theRydberg states is sufficiently long to probe the dynamicsof the interacting gas. This regime is complementary tothe well-studied frozen Rydberg gases where mechanicalatom-atom interaction effects can hardly be probed. Thepotential of the proposed magneto-electric trap is by farnot entirely exhausted; e.g., one could think of using thedouble-well structure which is visible in Fig. 1(d) in orderto realize two coupled dipolar Rydberg chains.This work was supported by the German ResearchFoundation (DFG) under the contract SCHM 885/10-2and within the framework of the Excellence Initiativethrough the Heidelberg Graduate School of FundamentalPhysics (GSC 129/1). M.M. acknowledges support fromthe Landesgraduiertenf¨orderung Baden-W¨urttemberg. [1] M. Greiner et al. , Nature , 39 (2002)[2] C. Chin et al. , Science , 1128 (2004)[3] Z. Hadzibabic et al. , Nature , 1118 (2006)[4] E. H. Lieb and W. Liniger, Phys. Rev. , 1605 (1963);E. H. Lieb, ibid. , 1616 (1963)[5] M. Girardeau, J. Math. Phys. , 516 (1960); B. Paredes et al. , Nature , 277 (2004)[6] T.F. Gallagher, Rydberg Atoms , Cambridge UniversityPress 1994[7] M. D. Lukin et al. , Phys. Rev. Lett. , 037901 (2001);D. Tong et al. , ibid. , 063001 (2004)[8] W. R. Anderson, J. R. Veale, and T. F. Gallagher Phys.Rev. Lett. , 249 (1998); I. Mourachko et al. , ibid. ,253 (1998)[9] P. Hyafil et al. , Phys. Rev. Lett. , 103001 (2004)[10] S.K. Dutta et al. , Phys. Rev. Lett. , 5551 (2000)[11] J.-H. Choi et al. , Phys. Rev. Lett. , 243001 (2005)[12] B. Hezel, I. Lesanovsky, and P. Schmelcher, Phys. Rev.Lett. , 223001 (2006); arXiv:0705.1299v2[13] J. Fortagh and C. Zimmermann, Rev. Mod. Phys. ,235 (2007); R. Folman et al. , Adv. At. Mol. Opt. Phys. , 263 (2002)[14] I. Lesanovsky and P. Schmelcher, Phys. Rev. Lett. ,053001 (2005)[15] U. Schmidt, I. Lesanovsky, and P. Schmelcher, J. Phys.B , 1003 (2007)[16] U. D. Jentschura et al. , J. Phys. B , S97 (2005)[17] R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev.Lett. , 2137 (1985)[18] R. Lutwak et al. , Phys. Rev. A56