PHYSICAL REVIEW A 75, 032712 ͑2007͒ Level shifts of rubidium Rydberg states due to binary interactions A. Reinhard, T. Cubel Liebisch, B. Knuffman, and G. Raithel FOCUS Center and Michigan Center for Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA ͑Received 16 August 2006; published 14 March 2007; corrected 19 March 2007͒ We use perturbation theory to directly calculate the van der Waals and dipole-dipole energy shifts of pairs of interacting Rb Rydberg atoms for different quantum numbers n, ᐉ, j, and mj, taking into account a large number of perturbing states. Our results can be used to identify good experimental parameters and illuminate important considerations for applications of the “Rydberg-excitation blockade.” We also use the results of the calculation to explain features of previous experimental data on the Rydberg-excitation blockade. To explore control methods for the blockade, we discuss energy shifts due to atom-atom interaction in an external electric field. DOI: 10.1103/PhysRevA.75.032712 PACS number͑s͒: 34.20.Cf, 31.15.Md, 34.60.ϩz, 03.67.Ϫa I. INTRODUCTION external electric field has also been reported ͓15͔. The Ryd- berg blockade has been modeled using many-body quantum Due to their large sizes and polarizabilities, Rydberg at- simulations ͓16,17͔, and the application of the blockade to oms excited from ensembles of laser-cooled ground state at- quantum phase gates ͓18,19͔ has been analyzed. Additional oms interact strongly via resonant dipole-dipole or off- work has been done to characterize Rydberg-atom interac- resonant van der Waals interactions. These interactions can tions, including spectroscopic studies of resonant dipolar in- lead to a number of interesting collisional effects including teractions ͓5,20,21͔, observation and characterization of mo- ͓ ͔ ͓ ͔ state mixing 1,2 , the formation of ultracold plasmas 3–5 , lecular resonances ͓22,23͔, and a study of the angular and the conversion of internal energy to center-of-mass en- ͓ ͔ ͓ ͔ dependence of dipolar Rydberg atom interactions 24 . ergy 6,7 . For relatively low densities and short interaction In Refs. ͓12–14,16,17͔, the Rydberg blockade is based times, however, a window of experimental parameters exists primarily on level shifts caused by binary interactions be- in which these incoherent processes are sufficiently sup- tween Rydberg excitations. While it is straightforward to es- pressed so that fragile, coherent dynamics is observable. A timate the order of magnitude of such interactions, knowl- coherent phenomenon that has recently attracted attention is the so-called “Rydberg-excitation blockade,” or “Rydberg edge of their detailed dependence on the atomic species, blockade.” If a small atomic sample is uniformly excited into quantum numbers, interatomic separations, etc., is important Rydberg levels, collective states are created in which Ryd- to understand the exact behavior of systems of interacting berg excitations are coherently shared among all atoms. For Rydberg atoms. In this paper, we present calculations of van example, the first excited state of an N-atom ensemble is der Waals and dipole-dipole energy level shifts, VInt, of pairs ͉ ͘ ͑ ͱ ͒⌺N ͉ ͘ of Rb Rydberg atoms for different quantum numbers n, ᐉ, j, N,1 = 1/ N i=1 g1 ,g2 ,...,ri ,...,gN , where the sub- scripts are the atom labels and ͉g͘ and ͉r͘ denote an atom in and mj, accounting for a large number of perturbing states. the ground state and the Rydberg state, respectively. The en- We identify parameter ranges suitable for applications of the ergy levels of the system deviate from an equally spaced Rydberg blockade and explain some features of recent ex- ͓ ͔ ladder of energies because of coherent interactions between perimental data 14 . We then discuss mechanisms for con- Rydberg atoms. This is schematically represented in Fig. 1. trol of the Rydberg blockade based on calculations of inter- The energy of the second excited state, ͉N,2͘, is shifted by action energies in external electric fields. These control schemes may become useful in the proposed applications the interaction energy, VInt, from its interaction-free value. If the laser is tuned to the lowest transition in the system discussed above. ͉N,0͘→͉N,1͘ and has a frequency bandwidth less than VInt/h, excitation of all quantum states higher in energy than V ͉N,1͘ is suppressed. This “blockade” of higher excitations Int can potentially be applied to quantum information processing 2W ͓8,9͔, quantum cryptography ͓10͔, atomic clocks ͓11͔, the Ryd N, k = 2 generation of mesoscopic entanglement ͓8͔, and the im- ͓ ͔ provement of spectroscopic resolution 8 . WRyd N, k = 1 Significant progress has been made towards demonstrat- Energy ing and understanding the blockade. Evidence of the block- ade effect has been observed in the suppression of bulk Ry- 0 N, k = 0 dberg excitation in large samples ͓12,13͔ and in the counting statistics of the number of Rydberg excitations in small FIG. 1. Lowest collective states of an ensemble of N atoms Rydberg-atom samples ͓14͔. Experimental evidence of a containing k Rydberg excitations. Binary Rydberg-Rydberg interac- ͉ ͘ blockade realized by the tuning of a Förster resonance in an tion causes an energy shift VInt of the state N,k=2 . 1050-2947/2007/75͑3͒/032712͑12͒ 032712-1 ©2007 The American Physical Society REINHARD et al. PHYSICAL REVIEW A 75, 032712 ͑2007͒ II. COMPUTATION OF BINARY-INTERACTION p2 LEVEL SHIFTS z, E We use perturbation theory to compute the second-order, binary interaction-induced energy level shifts due to cou- θ Atom 2 plings of the type ͗B͉ ͗C͉V ͉A͘ ͉A͘, where ͉A͘, ͉B͘, and n, , j, m Int j ͉C͘ are single-particle Rydberg states. Generally, the interac- p R tion operator, VInt, can be expanded in multipole terms that 1 scale with increasing negative powers of the interatomic separation, R ͓25–27͔. This paper primarily addresses situa- tions in which the typical Rydberg-atom separation, R,is about an order of magnitude larger than the atom size ͑Refs. Atom 1 n, , j, m ͓12–14,16,17͔͒. In first order, the electric-dipole interaction j ϳ 4 3 scales as VInt n /R . If there is no external electric field applied, the electric-dipole interaction produces level shifts FIG. 2. Two interacting Rydberg atoms separated by a vector R only in second order, because low-angular-momentum Ryd- which defines an angle ␪ with the quantization ͑zˆ͒ axis. In cases berg states of alkali atoms in zero electric field do not have where an electric field E is applied, the field is parallel to the quan- permanent dipole moments. For this case, the level shifts tization axis. scale as ϳn11/͑sR6͒, where the parameter s is a measure for ͑ 2 ␪͒ the magnitude of the energy detunings that occur in second- p1+p2− + p1−p2+ + p1zp2z 1−3cos ͓ ͔ V = order perturbation theory and is of order 0.1 or less 5 . For dd R3 P and D states, the quadrupole-quadrupole interaction occurs in first order and leads to shifts that scale as ϳn8 /R5. The 3 sin2 ␪͑p p + p p + p p + p p ͒ ratio of the quadrupole-quadrupole interaction to the second- 2 1+ 2+ 1+ 2− 1− 2+ 1− 2− 3 − order dipole-dipole interaction scales as sR/n . For the inter- R3 atomic separations of interest in this paper ͑Rϳ10n2͒, this ϳ 3 ratio is 1/n. We therefore restrict our calculations to the ␪ ␪͑ ͒ sin cos p1+p2z + p1−p2z + p1zp2+ + p1zp2− dipole-dipole interaction, which represents the dominant in- ͱ2 teraction at the atomic separations of interest. It is noted that − , R3 in more general cases quadrupolar and higher interactions will also become important ͓25–29͔. ͑2͒ In the case of zero external field, computation of the van der Waals level shifts due to the electric-dipole interaction where the first subscript of the p components identifies the 1 ͑ ͒ ͑ ͒ requires the evaluation of matrix elements of the form atom number, p± = ͱ2 x±iy , and pz =z in atomic units . The ͗ Љ ᐉЉ Љ Љ͉ ͗ Ј ᐉЈ Ј Ј͉ ͉ ᐉ ͘ ͉ ᐉ ͘ matrix elements of V are independent of the azimuthal n , , j ,mj n , , j ,mj Vdd n, , j,mj n, , j,mj , dd angle of R. From this expression, we see that calculating where Vdd is the potential energy operator for two interacting electric dipoles. The operator V is given, in atomic units, two-particle matrix elements of Vdd involves evaluating the dd ͗ Ј ᐉЈ Ј Ј͉ ͉ ᐉ ͘ by single-particle matrix elements n , , j ,mj p␣,␤ n, , j,mj where ␣ is the particle number and ␤෈͕+,−,z͖. In the following, we explain the essential details of the p · p −3͑n · p ͒͑n · p ͒ numerical method used to obtain the single-particle radial V = 1 2 1 2 , ͑1͒ dd R3 wave functions, ␺͑r͒, required to calculate the matrix ͗ Ј ᐉЈ Ј Ј͉ ͉ ᐉ ͘ elements n , , j ,mj p␣,␤ n, , j,mj . The radial wave functions are integrated using an algorithm that continuously where the operators p1 and p2 denote the individual electric- adjusts the step size such that it is smaller than, but dipole moments, the vector R is the separation between the close to, 2␲/˜v divided by a constant, f, where classical center-of-mass coordinates of the interacting atoms, ˜v=ͱ͉2E+2/r−l͑l+1͒/r2͉͑in atomic units; E is the single- and n is a unit vector pointing in the direction of R.