VOLUME 70 21 JUNE 1993 Rydberg Atoms in Curved Space-Time NUMBER 25 Fabrizio Pinto Department of Physics, Boise State University, Boise, Idah, o 88785 (Received 17 September 1992) The possible use of loosely bound Rydberg atoms for remote gravimetric measurements is explored. The first-order corrections to the nonrelativistic nS and nP states for n & 2 are obtained for the first time. A procedure to evaluate corrections of any order is outlined and applied to the IS state in a spherical symmetry. It is shovrn that observations of the eKects described in this Letter near objects of neutron-star-like densities are possible in principle only in the absence of significant magnetic fields. PACS numbers: 95.30. Sf, 32.80.Rm, 95.30. Dr, 97. 60.Jd It has been known for quite some time that the en- ergy eigenstates of an atom are afFected by the local space-time curvature [1]. These energy shifts would be detected by a locally inertial, co-moving observer close to the atom and are exclusive of further gravitational, Doppler, and cosmological shifts possibly apparent to far- away observers [2]. The typical treatment of the prob- lem found in the literature [1] consists of using Fermi normal coordinates [3] to write the Dirac equation for a one-electron atom in curved space-time and of deriving corrective terms to the Bat-space Hamiltonian. In what follows we shall be mainly concerned with the nonrela- tivistic limit of this approach (that is, v/c = — P &( 1, not the limit of vanishing curvature). This yields a time- independent Schrodinger equation containing an effective classical geodesic deviation potential describing the tidal interaction of the atom with the gravitational field. The nonrelativistic Hamiltonian HNR, to first order in the Riemann tensor, reads 1 g Zc 1 m ~NR = P + P+OlOm& & ) 2p where p, is the reduced mass, Ao~o~ are the components of the Riemann tensor, and 2. " is the position of the electron in the nucleus-centered reference frame. It should be no- ticed that nuclear e8ects, radiative corrections, and the electron-nucleus gravitational interaction are neglected, and that cgs units are used. In the case of a hydrogenlike atom of principal quan- turn number n in free fall at the surface of a spheri- cal object of mass M and radius R, the nonvanishing components of the Riemann tensor are GM/Rs and 2;" ~ r aon, where ao is the Bohr radius. Thus the order of magnitude of the energy shift is, to first order, 4E (GMti/B )aon = (4vr/3)Gppaon It was fi. rst shown by Parker that, in order for the energy shift of a hydrogen atom in a tightly bound state at the hori- zon of a black hole to be of the same order as the I amb shift (4. 4 x 10 eV), the Schwarzschild radius should be 10 cm, typical of unobserved cosmological black holes [1]. The situation changes if, instead of concentrating on the first few tightly bound states of hydrogenlike atoms, one considers highly excited (n )) 1), loosely bound ones, commonly referred to as Rydberg states [4]. Since the av- erage distance of the electron from the nucleus is macro- scopic in these cases (r )) ac), the electron is much more sensitive to small external perturbations. Since several radio and optical lines from Rydberg atoms with princi- pal quantum number up to n 350 have been detected in various astrophysical environments (see, for instance, [5)), it is interesting to see how the above conclusions are modified in this case and if any observations of this pro- cess are possible. By using the above expression it can be seen that, for this gravitational tidal shift to be even just a small fraction of 1 peV, it is necessary that n be of the order of or larger than 103 — 104. It is straightforward to verify, again from Eq. (1), 0031-9007/93/70(25)/3839(5) $06.00 1993 The American Physical Society 3839