Eur. Phys. J. D 7, 279–284 (1999) T HE EUROPEAN P HYSICAL JOURNAL D c  EDP Sciences Societ` a Italiana di Fisica Springer-Verlag 1999 Exotic quantum dark states S. Kulin a , Y. Castin b , M. Ol’shanii c , E. Peik d , B. Saubam´ ea, M. Leduc, and C. Cohen-Tannoudji Coll` ege de France and Laboratoire Kastler Brossel, ´ Ecole Normale Sup´ erieure, 24 rue Lhomond, 75231 Paris Cedex 05, France Received 18 December 1998 and Received in final form 29 January 1999 Abstract. We extend studies of velocity selective coherent population trapping to atoms having a J = 1 → J = 0 transition. When placed in a two-dimensional laser field these atoms are optically pumped into different velocity selective nonabsorbing states. Each of these distinct energy eigenstates exhibits a unique entanglement between its internal and external degrees of freedom. We use a graphical method that makes easier the description of these states. We confirm our predictions experimentally. PACS. 32.80.Pj Optical cooling of atoms; trapping – 34.50.Rk Laser-modified scattering and reactions – 42.50.Vk Mechanical effects of light on atoms, molecules, electrons, and ions 1 Introduction Laser cooling and trapping techniques allow the creation of a sample of atoms or ions in well-specified states. The selection of atoms in the lowest vibrational state in a far detuned optical lattice [1] and the ion crystal in a ra- diofrequency trap are two examples [2,3]. Velocity selec- tive coherent population trapping (VSCPT) is a manner of preparing atoms in stationary states of very well-defined momentum in which the atoms do not absorb photons (“dark states”). To date VSCPT has been studied in de- tail on the J =1 → J = 1 transition in both helium and rubidium [4–10]. In this case there exists a dark state that is isomorphic to the laser field [5,6]. Exploiting the coherence properties of this state has led to a series of interesting experiments [11,12]. It is then natural to inquire about the existence of dark states on transitions other than J → J . It is well-known, that a J → J + 1 transition is a “cycling” transition and does not support nonabsorbing states. On the other hand, a J → J - 1 transition does allow non-coupled states [13,14]. The simplest J → J - 1 transition is the 1 → 0 transition, which is the one we study here. For instance, light polarized σ + pumps the atoms into both the |g 0 i and |g +1 i ground states, where they no longer absorb photons. But in a one-dimensional light field these non-coupled states are not velocity selective, and thus not of interest to a Present address: National Institute of Standards and Tech- nology, Gaithersburg, MD 20899-8424, USA. b e-mail: castin@physique.ens.fr c Present address: Lyman Laboratory, Physics Department Harvard University, Cambridge, MA 02138, USA. d Present address: Max Planck Institut f¨ ur Quantenoptik, Hans Kopfermann Straße 1, 85748 Garching, Germany. us. In this work, we discuss the existence and the nature of velocity selective dark states on the J =1 → J = 0 tran- sition in the presence of a two-dimensional light field. As opposed to the case of the J =1 → J = 1 transition, sev- eral dark states appear at different energies. These issues have been addressed theoretically in [5,13,15]. Here, we choose to use the cooling scheme of [5], the simplest one to implement experimentally. We use a graphical analysis inspired by that introduced in [5,13,16] that allows us to determine the minimal number of dark states that we can populate in each energy class, taking into account solely the symmetry of the laser field and the atomic transition. We then compare these predictions with the experimental results. 2 Characterization of dark states The idea of VSCPT is to optically pump the atoms into states |Ψ i where they can remain indefinitely without scat- tering any photons. We can thus characterize such dark states by two conditions: first, they have to be decoupled from the light, and second, they must be stationary. Mathematically, the condition for a non-coupled state is expressed by requiring that the transition amplitude A from a ground state |Ψ i to the excited state |Φi via the atom-laser interaction potential vanishes: A = hΦ|V AL |Ψ i =0, ∀|Φi . (1) Here, following the notation of [17], V AL = -d · E L (r), with d the raising dipole moment operator and E L (r) the positive frequency part of the electric field of the laser at the position r of the atom. The most general ground state |Ψ i is a superposition of all three Zeeman sublevels