Ultracold rare-earth magnetic atoms with an electric dipole moment Maxence Lepers1,2, Hui Li1, Jean-Fran¸cois Wyart1,3, Goulven Qu´em´ener1 and Olivier Dulieu1 1Laboratoire Aim´eCotton, CNRS, Universit´eParis-Sud, ENS Paris-Saclay, Universit´eParis-Saclay, 91405 Orsay, France 2Laboratoire Interdisciplinaire Carnot de Bourgogne, CNRS, Universit´ede Bourgogne Franche-Comt´e, 21078 Dijon, France∗ and 3LERMA, Observatoire de Paris-Meudon, PSL Research University, Sorbonne Universit´es, UPMC Univ. Paris 6, CNRS UMR8112, 92195 Meudon, France (Dated: March 13, 2018) We propose a new method to produce an electric and magnetic dipolar gas of ultracold dysprosium − atoms. The pair of nearly degenerate energy levels of opposite parity, at 17513.33 cm 1 with − electronic angular momentum J = 10, and at 17514.50 cm 1 with J = 9, can be mixed with an external electric field, thus inducing an electric dipole moment in the laboratory frame. For field amplitudes relevant to current-day experiments, we predict a magnetic dipole moment up to 13 Bohr magnetons, and an electric dipole moment up to 0.22 Debye, which is similar to the values obtained for alkali-metal diatomics. When a magnetic field is present, we show that the electric dipole moment is strongly dependent on the angle between the fields. The lifetime of the field- mixed levels is found in the millisecond range, thus allowing for suitable experimental detection and manipulation. Introduction. In a classical neutral charge distribu- with an electric and a magnetic dipole moment, which up tion, a dipole moment appears with a separation be- to now consist of paramagnetic polar diatomics [23–32]. tween the barycenter of positive and negative charges In this Letter, we propose a new method to produce [1]. An obvious example is provided by an heteronuclear an electric and magnetic dipolar gas of ultracold dys- diatomic molecule, which possesses a permanent dipole prosium atoms. Our method is based on the electric- moment along its interatomic axis. It will manifest in field mixing of quasi-degenerate opposite-parity energy the laboratory frame when such a molecule is placed in levels, which appear accidentally in the rich spectra of an external electric field, acquiring a preferred orienta- lanthanides. Historically, the pair of levels at 19797.96 tion along the direction of the field. Moreover, a neutral cm−1 with electronic angular momenta J = 10 has been atom placed in an external electric field acquires a small employed for fundamental measurements [33–35]. How- dipole moment, as the spherical symmetry of space is ever, their reduced transition dipole moment, equal to broken. This effect is spectacularly maximized in Ry- 0.015 atomic units (a.u.) [36], is not sufficient to ob- dberg atoms, where the induced dipole moment scales serve dipolar effects. On the contrary, the odd-parity 2 −1 as n , where n is the principal quantum number of the level |ai at Ea = 17513.33 cm with Ja = 10 and the −1 considered Rydberg state [2]. even-parity level |bi at Eb = 17514.50 cm with Jb = 9, At the single-particle scale, the external electric field which present a reduced transition dipole moment of 3.21 mixes even and odd-parity levels of the energy spectrum: a.u., are very promising for dipolar gases [37]. rotational levels for a diatomic molecule (see e.g. [3]), or We calculate the energies, electric (EDMs) and mag- levels with different orbital angular momenta for Rydberg netic dipole moments (MDMs) of a dysprosium atom in atoms (see e.g. [4]). In both cases, this leads to a pro- a superposition of levels |ai and |bi, and submitted to an nounced linear Stark shift on the energy levels, revealing electric and a magnetic field with an arbitrary respective the existence of a permanent dipole moment in the lab- orientation. For field amplitudes relevant to current-day oratory frame. More surprisingly, it has been observed experiments, we predict a MDM of µmax = 13 Bohr mag- that a homonuclear diatomic molecule can exhibit a per- netons, to our knowledge the largest value observed in arXiv:1803.04159v1 [physics.atom-ph] 12 Mar 2018 manent dipole moment in the laboratory frame, when it ultracold experiments, and an EDM of dmax = 0.22 De- combines a ground state atom bound inside the spatial bye, which is similar to the values of diatomic molecules extension of a Rydberg atom [5, 6]. [38]. We also demonstrate a strong control of the electric The search for such dipolar systems, involving espe- dipole moment, which ranges from 0 to dmax as a func- cially lanthanide atoms, is currently very active in the tion of the angle between the fields. Because |ai and |bi context of ultracold dilute gases [7–17]. Indeed, the par- are excited levels, we also calculate the atomic radiative ticles of the gas interact through a highly anisotropic lifetime as functions of the fields parameters, and obtain long-range potential energy varying as the inverse cu- a few millisecond for the level characterized by µmax and bic power of their spatial separation [18, 19]. Prospects dmax. Finally, we show that our method is applicable for related to many-body physics, quantum simulation and all bosonic and fermionic isotopes. ultracold chemistry are nowadays within reach experi- Model. We consider an atom lying in two energy lev- mentally [20–22]. A particular attention is paid on gases els |ai and |bi, of energies Ei and total angular momen- 2 tum Ji (i = a, b). Firstly, we consider bosonic isotopes 17518 ε θ which have no nuclear spin, I = 0. In absence of field, = = 0 (a) each level |ii is (2Ji +1)-time degenerate, and the corre- ) 17516 -1 |b〉 sponding Zeeman subslevels are labeled with their mag- 17514 netic quantum number Mi. The atom is submitted both B e e to a magnetic field = B z, with z the unit vector 〉 Energy (cm 17512 |a in the z direction, taken as quantization axis, and to E u u electric field = E , with a unit vector in the di- 17510 rection given by the polar angles θ and φ = 0. In the 0 1000 2000 3000 4000 5000 basis {|Ma = −Jai, ..., | + Jai, |Mb = −Jbi, ..., | + Jbi} Magnetic field (G) spanned by the Zeeman sublevels of |ai and |bi, the ) -0.5 -1 o Hamiltonian can be written B = 100 G ; θ = 0 (b) Ji -0.55 Hˆ = E |M ihM | + Wˆ + Wˆ . (1) i i i Z S |a〉 i=a,b Mi=−Ji –17513.92 (cm X X n -0.6 E The Zeeman Hamiltonian Wˆ Z only contains diagonal terms equal to MigiµBB, with gi the Land´eg-factor of -0.65 Energy level |ii. The last term of Eq. (1) is the Stark Hamilto- 0 2 4 6 8 10 nian, which couples sublevels |Mai with sublevels |Mbi Electric field (kV/cm) as ) -0.5 -1 B = 100 G ; ε = 5 kV/cm (c) ˆ 4π dˆ hMa|WS|Mbi = − hak kbiE -0.55 s3(2Ja + 1) ∗ JaMa |a〉 × Y1 − (θ, 0) C − , (2) ,Ma Mb JbMb,1,Ma Mb –17513.92 (cm n -0.6 E where hakdˆkbi is the reduced transition dipole moment, cγ -0.65 Ykq (θ, φ) a spherical harmonics and Caαbβ a Clebsch- Energy Gordan coefficient [39]. For given values of E, B and 0 30 60 90 120 150 180 Angle (degrees) θ, we calculate the eigenvalues En and eigenvectors Ji FIG. 1. Eigenvalues of the atom-field Hamiltonian (1) as func- |Ψni = cn,Mi |Mii (3) tions of: (a) the magnetic field B for vanishing electric field i=a,b Mi=−Ji and angle E = θ = 0; (b) the electric field E for B = 100 X X ◦ G and θ = 0 ; (c) the angle θ for B = 100 G and E = 5 of the Hamiltonian in Eq. (1). kV/cm. In panels (b) and (c), the origin of energies is taken at −1 The energy levels that we consider here are Ea = (Ea + Eb)/2 = 17513.92 cm . The blue curve with crosses −1 −1 17513.33 cm , Ja = 10 and Eb = 17514.50 cm , corresponds to the eigenstate converging towards |Ma = −10i Jb = 9. Their Land´eg-factors ga = 1.30 and gb = 1.32 when θ → 0. are experimental values taken from Ref. [40]. The re- duced transition dipole moment hakdˆkbi is calculated us- ing the method developed in our previous works [41–44]. Energies in electric and magnetic fields. Figure 1(a) Firstly, odd-level energies are taken from Ref. [43] which shows the eigenvalues of the Hamiltonian (1) as func- include the electronic configurations [Xe]4f 106s6p and tions of the magnetic field for E = θ = 0. The field splits [Xe]4f 95d6s2, [Xe] being the xenon core. Even-level en- levels |ai and |bi into 21 and 19 sublevels respectively, 10 2 ergies are calculated with the configurations [Xe]4f 6s , each one associated with a given Ma or Mb. On fig. 1(a), 10 9 2 [Xe]4f 5d6s and [Xe]4f 6s 6p [45]. Secondly, following we emphasize the lowest sublevel |Ma = −10i, in which Ref. [43], we adjust the mono-electronic transition dipole ultracold atoms are usually prepared. Due to the close moments by multiplying their ab initio values by appro- Land´eg-factors, the two Zeeman manifolds look very sim- priate scaling factors [42], equal to 0.794 for h6s|rˆ|6pi, ilar, i.e. the branches characterized by the same values 0.97 for h4f|rˆ|5di and 0.80 for h5d|rˆ|6pi. From the result- Ma = Mb are almost parallel. For B ≥ 1000 Gauss, the ing Einstein coefficients, we can extract hakdˆkbi = 3.21 two Zeeman manifolds overlap; but because the magnetic 4 −1 a.u., as well as the natural linewidth γb =2.98 × 10 s .