Abstract

We review here the studies performed about interactions in an assembly of cold Rydberg atoms. We focus especially on the review of the dipole–dipole interactions and on the effect of the dipole blockade in the laser Rydberg excitation, which offers attractive possibilities for quantum engineering. We present first the various interactions between Rydberg atoms. The laser Rydberg excitation of such an assembly is then described with the introduction of the dipole blockade phenomenon. We report recent experiments performed in this subject by starting with the case of a pair of atoms allowing the entanglement of the wave-functions of the atoms and opening a fascinating way for the realization of quantum bits and quantum gates. We consider then several works on the blockade effect in a large assembly of atoms for three different configurations: blockade through electric-field induced dipole, through Förster resonance, and in van der Waals interaction. The properties of coherence and cooperativity are analyzed. Finally, we treat the role of dipole–dipole interactions between Rydberg atoms responsible for Penning ionization. The perturbation of the dipole blockade by ions and the evolution of the Rydberg towards an ultracold plasma are discussed.

© 2010 Optical Society of America

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2010 (7)

L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-NOT quantum gate,” Phys. Rev. Lett. 104, 010503 (2010).
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T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, “Entanglement of two individual neutral atoms using Rydberg blockade,” Phys. Rev. Lett. 104, 010502 (2010).
[CrossRef] [PubMed]

I. I. Ryabtsev, D. B. Tretyakov, I. I. Beterov, and V. M. Entin, “Observation of the Stark-tuned Förster resonance between two Rydberg atoms,” Phys. Rev. Lett. 104, 073003 (2010).
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T. Amthor, C. Giese, C. S. Hofmann, and M. Weidemüller, “Evidence of antiblockade in an ultracold Rydberg gas,” Phys. Rev. Lett. 104, 013001 (2010).
[CrossRef] [PubMed]

S. Wüster, J. Stanojevic, C. Ates, T. Pohl, P. Deuar, J. F. Corney, and J. M. Rost, “Correlations of Rydberg excitations in an ultracold gas after an echo sequence,” Phys. Rev. A 81, 023406 (2010).
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N. Saquet, A. Cournol, J. Beugnon, J. Robert, P. Pillet, and N. Vanhaecke, “Landau-Zener transitions in frozen pairs of Rydberg atoms,” Phys. Rev. Lett. 104, 133003 (2010).
[CrossRef] [PubMed]

H. Weimer, M. Müller, I. Lesanovsky, P. Zoller, and H. P. Büchler, “A Rydberg quantum simulator,” Nat. Phys. 6, 382–388 (2010).
[CrossRef]

2009 (27)

M. Müller, I. Lesanovsky, H. Weimer, H. P. Büchler, and P. Zoller, “Mesoscopic Rydberg gate based on electromagnetically induced transparency,” Phys. Rev. Lett. 102, 170502 (2009).
[CrossRef] [PubMed]

I. C. H. Liu, J. Stanojevic, and J. M. Rost, “Ultra-long-range Rydberg trimers with a repulsive two-body interaction,” Phys. Rev. Lett. 102, 173001 (2009).
[CrossRef] [PubMed]

K. R. Overstreet, A. Schwettmann, J. Tallant, D. Booth, and J. P. Shaffer, “Observation of electric-field-induced cs Rydberg atom macrodimers,” Nat. Phys. 5, 581–585 (2009).
[CrossRef]

B. Olmos, R. González-Férez, and I. Lesanovsky, “Collective Rydberg excitations of an atomic gas confined in a ring lattice,” Phys. Rev. A 79, 043419 (2009).
[CrossRef]

B. Olmos, R. González-Férez, and I. Lesanovsky, “Fermionic collective excitations in a lattice gas of Rydberg atoms,” Phys. Rev. Lett. 103, 185302 (2009).
[CrossRef] [PubMed]

J. Qian, Y. Qian, M. Ke, X.-L. Feng, C. H. Oh, and Y. Wang, “Breakdown of the dipole blockade with a zero-area phase-jump pulse,” Phys. Rev. A 80, 053413 (2009).
[CrossRef]

C. S. E. van Ditzhuijzen, A. Tauschinsky, and H. B. van Linden van den Heuvell, “Observation of Stückelberg oscillations in dipole–dipole interactions,” Phys. Rev. A 80, 063407 (2009).
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T. J. Carroll, C. Daniel, L. Hoover, T. Sidie, and M. W. Noel, “Simulations of the dipole–dipole interaction between two spatially separated groups of Rydberg atoms,” Phys. Rev. A 80, 052712 (2009).
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U. Raitzsch, R. Heidemann, H. Weimer, B. Butscher, P. Kollmann, R. Löw, H. P. Büchler, and T. Pfau, “Investigation of dephasing rates in an interacting Rydberg gas,” New J. Phys. 11, 055014 (2009).
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K. C. Younge and G. Raithel, “Rotary echo tests of coherence in Rydberg-atom excitation,” New J. Phys. 11, 043006 (2009).
[CrossRef]

M. T. Bell and T. P. Softley, “Ultracold molecules and ultracold chemistry,” Mol. Phys. 107, 99 (2009).
[CrossRef]

L. D. Carr, D. DeMille, R. V. Krems, and J. Ye, “Cold and ultracold molecules: science, technology and applications,” New J. Phys. 11, 055049 (2009).
[CrossRef]

O. Dulieu and C. Gabbanini, “The formation and interactions of cold and ultracold molecules: new challenges for interdisciplinary physics,” Rep. Prog. Phys. 72, 086401 (2009).
[CrossRef]

T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, “The physics of dipolar bosonic quantum gases,” Rep. Prog. Phys. 72, 126401 (2009).
[CrossRef]

K. C. Younge, A. Reinhard, T. Pohl, P. R. Berman, and G. Raithel, “Mesoscopic Rydberg ensembles: beyond the pairwise-interaction approximation,” Phys. Rev. A 79, 043420 (2009).
[CrossRef]

T. Amthor, J. Denskat, C. Giese, N. N. Bezuglov, A. Ekers, L. S. Cederbaum, and M. Weidemüller, “Autoionization of an ultracold Rydberg gas through resonant dipole coupling,” Eur. Phys. J. D 53, 329–335 (2009).
[CrossRef]

V. A. Nascimento, L. L. Caliri, A. Schwettmann, J. P. Shaffer, and L. G. Marcassa, “Electric field effects in the excitation of cold Rydberg-atom pairs,” Phys. Rev. Lett. 102, 213201 (2009).
[CrossRef] [PubMed]

C. Vadla, V. Horvatic, and K. Niemax, “Line-broadening studies of excited diatomic homoatomic and heteroatomic Rydberg molecules formed by potassium, rubidium, and cesium atoms,” Phys. Rev. A 80, 052506 (2009).
[CrossRef]

V. Bendkowsky, B. Butscher, J. Nipper, J. P. Shaffer, R. Löw, and T. Pfau, “Observation of ultralong-range Rydberg molecules,” Nature 458, 1005–1008 (2009).
[CrossRef] [PubMed]

D. B. Tretyakov, I. I. Beterov, V. M. Entin, I. I. Ryabtsev, and P. L. Chapovsky, “Investigation of cold rubidium Rydberg atoms in a magneto-optical trap,” J. Exp. Theor. Phys. 108, 374–383 (2009).
[CrossRef]

E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Observation of Rydberg blockade between two atoms,” Nat. Phys. 5, 110–114 (2009).
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J. Stanojevic and R. Côté, “Many-body Rabi oscillations of Rydberg excitation in small mesoscopic samples,” Phys. Rev. A 80, 033418 (2009).
[CrossRef]

R. Löw, H. Weimer, U. Krohn, R. Heidemann, V. Bendkowsky, B. Butscher, H. P. Büchler, and T. Pfau, “Universal scaling in a strongly interacting Rydberg gas,” Phys. Rev. A 80, 033422 (2009).
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T. Pohl and P. R. Berman, “Breaking the dipole blockade: nearly resonant dipole interactions in few-atom systems,” Phys. Rev. Lett. 102, 013004 (2009).
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I. Buluta and F. Nori, “Quantum simulators,” Science 326, 108–111 (2009).
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I. I. Beterov, I. I. Ryabtsev, D. B. Tretyakov, and V. M. Entin, “Quasiclassical calculations of blackbody-radiation-induced depopulation rates and effective lifetimes of Rydberg ns, np, and nd alkali-metal atoms with n<80,” Phys. Rev. A 79, 052504 (2009).
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A. Gaëtan, Y. Miroshnychenko, T. Wilk, A. Chotia, M. Viteau, D. Comparat, P. Pillet, A. Browaeys, and P. Grangier, “Observation of collective excitation of two individual atoms in the Rydberg blockade regime,” Nat. Phys. 5, 115–118 (2009).
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2008 (29)

P. J. Tanner, J. Han, E. S. Shuman, and T. F. Gallagher, “Many-body ionization in a frozen Rydberg gas,” Phys. Rev. Lett. 100, 043002 (2008).
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A. Reinhard, T. Cubel Liebisch, K. C. Younge, P. R. Berman, and G. Raithel, “Rydberg–Rydberg collisions: resonant enhancement of state mixing and penning ionization,” Phys. Rev. Lett. 100, 123007 (2008).
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S. D. Hogan and F. Merkt, “Demonstration of three-dimensional electrostatic trapping of state-selected Rydberg atoms,” Phys. Rev. Lett. 100, 043001 (2008).
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Th. F. Gallagher and P. Pillet, “Dipole–dipole interactions of Rydberg atoms,” Adv. At., Mol., Opt. Phys. 56, 161 (2008).

T. G. Walker and M. Saffman, “Consequences of zeeman degeneracy for the van der Waals blockade between Rydberg atoms,” Phys. Rev. A 77, 032723 (2008).
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P. G. Brooke, K.-P. Marzlin, J. D. Cresser, and B. C. Sanders, “Super- and subradiant emission of two-level systems in the near-Dicke limit,” Phys. Rev. A 77, 033844 (2008).
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M. Reetz-Lamour, J. Deiglmayr, T. Amthor, and M. Weidemüller, “Rabi oscillations between ground and Rydberg states and van der Waals blockade in a mesoscopic frozen Rydberg gas,” New J. Phys. 10, 045026 (2008).
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T. A. Johnson, E. Urban, T. Henage, L. Isenhower, D. D. Yavuz, T. G. Walker, and M. Saffman, “Rabi oscillations between ground and Rydberg states with dipole–dipole atomic interactions,” Phys. Rev. Lett. 100, 113003 (2008).
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A. Chotia, M. Viteau, T. Vogt, D. Comparat, and P. Pillet, “Kinetic Monte Carlo modeling of dipole blockade in Rydberg excitation experiment,” New J. Phys. 10, 045031 (2008).
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J. V. Hernández and F. Robicheaux, “Simulation of a strong van der Waals blockade in a dense ultracold gas,” J. Phys. B 41, 5301 (2008).
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H. Weimer, R. Löw, T. Pfau, and H. P. Büchler, “Quantum critical behavior in strongly interacting Rydberg gases,” Phys. Rev. Lett. 101, 250601 (2008).
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A. Reinhard, K. C. Younge, T. C. Liebisch, B. Knuffman, P. R. Berman, and G. Raithel, “Double-resonance spectroscopy of interacting Rydberg-atom systems,” Phys. Rev. Lett. 100, 233201 (2008).
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C. Ates, A. Eisfeld, and J. M. Rost, “Motion of Rydberg atoms induced by resonant dipole dipole interactions,” New J. Phys. 10, 045030 (2008).
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T. Koch, T. Lahaye, J. Metz, B. Fröhlich, A. Griesmaier, and T. Pfau, “Stabilization of a purely dipolar quantum gas against collapse,” Nat. Phys. 4, 218–222 (2008).
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2007 (22)

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A. K. Mohapatra, T. R. Jackson, and C. S. Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98, 113003 (2007).
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T. Wang, S. F. Yelin, R. Côté, E. E. Eyler, S. M. Farooqi, P. L. Gould, M. Koštrun, D. Tong, and D. Vrinceanu, “Superradiance in ultracold Rydberg gases,” Phys. Rev. A 75, 033802 (2007).
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A. Reinhard, T. C. Liebisch, B. Knuffman, and G. Raithel, “Publisher’s note: Level shifts of rubidium Rydberg states due to binary interactions,” Phys. Rev. A 75, 039902 (2007).
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2006 (14)

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V. Vuletic, “Quantum networks: when superatoms talk photons,” Nat. Phys. 2, 801–802 (2006).
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T. Vogt, M. Viteau, J. Zhao, A. Chotia, D. Comparat, and P. Pillet, “Dipole blockade at Förster resonances in high resolution laser excitation of Rydberg states of cesium atoms,” Phys. Rev. Lett. 97, 083003 (2006).
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B. Knuffman and G. Raithel, “Emission of fast atoms from a cold Rydberg gas,” Phys. Rev. A 73, 020704 (2006).
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W. Li, P. J. Tanner, Y. Jamil, and T. F. Gallagher, “Ionization and plasma formation in high n cold Rydberg samples,” Eur. Phys. J. D 40, 27–35 (2006).
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2005 (14)

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M. Mudrich, N. Zahzam, T. Vogt, D. Comparat, and P. Pillet, “Back and forth transfer and coherent coupling in a cold Rydberg dipole gas,” Phys. Rev. Lett. 95, 233002 (2005).
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T. C. Liebisch, A. Reinhard, P. R. Berman, and G. Raithel, “Atom counting statistics in ensembles of interacting Rydberg atoms,” Phys. Rev. Lett. 95, 253002 (2005).
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2004 (10)

T. J. Carroll, K. Claringbould, A. Goodsell, M. J. Lim, and M. W. Noel, “Angular dependence of the dipole–dipole interaction in a nearly one-dimensional sample of Rydberg atoms,” Phys. Rev. Lett. 93, 153001 (2004).
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I. Mourachko, W. Li, and T. F. Gallagher, “Controlled many-body interactions in a frozen Rydberg gas,” Phys. Rev. A 70, 031401 (2004).
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D. Tong, S. M. Farooqi, J. Stanojevic, S. Krishnan, Y. P. Zhang, R. Côté, E. E. Eyler, and P. L. Gould, “Local blockade of Rydberg excitation in an ultracold gas,” Phys. Rev. Lett. 93, 063001 (2004).
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K. Singer, M. Reetz-Lamour, T. Amthor, L. G. Marcassa, and M. Weidemüller, “Suppression of excitation and spectral broadening induced by interactions in a cold gas of Rydberg atoms,” Phys. Rev. Lett. 93, 163001 (2004).
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K. Afrousheh, P. Bohlouli-Zanjani, D. Vagale, A. Mugford, M. Fedorov, and J. D. Martin, “Spectroscopic observation of resonant electric dipole–dipole interactions between cold Rydberg atoms,” Phys. Rev. Lett. 93, 233001 (2004).
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W. Li, M. P. Robinson, M. W. Noel, P. J. Tanner, T. F. Gallagher, D. Comparat, B. Laburthe-Tolra, N. Vanhaecke, T. Vogt, N. Zahzam, P. Pillet, and D. A. Tate, “Evolution dynamics of a dense, frozen Rydberg gas to plasma,” Phys. Rev. A 70, 042713 (2004).
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2003 (4)

T. C. Killian, V. S. Ashoka, P. Gupta, S. Laha, S. B. Nagel, C. E. Simien, S. Kulin, S. L. Rolston, and S. D. Bergeson, “Ultracold neutral plasmas: recent experiments and new prospect,” J. Phys. A: Math. Theor. 36, 6077 (2003).
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T. F. Gallagher, P. Pillet, M. P. Robinson, B. Laburthe-Tolra, and M. W. Noel, “Back and forth between Rydberg atoms and ultracold plasmas,” J. Opt. Soc. Am. B 20, 1091–1097 (2003).
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A. L. de Oliveira, M. W. Mancini, V. S. Bagnato, and L. G. Marcassa, “Rydberg cold collisions dominated by ultralong range potential,” Phys. Rev. Lett. 90, 143002 (2003).
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2002 (6)

C. Boisseau, I. Simbotin, and R. Côté, “Macrodimers: Ultralong range Rydberg molecules,” Phys. Rev. Lett. 88, 133004 (2002).
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A. A. Khuskivadze, M. I. Chibisov, and I. I. Fabrikant, “Adiabatic energy levels and electric dipole moments of Rydberg states of rb2 and cs2 dimers,” Phys. Rev. A 66, 042709 (2002).
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B. E. Granger, E. L. Hamilton, and C. H. Greene, “Quantum and semiclassical analysis of long-range Rydberg molecules,” Phys. Rev. A 64, 042508 (2001).
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2000 (6)

M. P. Robinson, B. L. Tolra, M. W. Noel, T. F. Gallagher, and P. Pillet, “Spontaneous evolution of Rydberg atoms into an ultracold plasma,” Phys. Rev. Lett. 85, 4466 (2000).
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Other (15)

This ought not be confused with the reduced matrix element ⟨nl‖r‖n′l±1⟩=2l′+1Cl+10,10l0⟨nl|r|n′l±1⟩, Cj1m1,j2m2jm being the standard Clebch–Gordan coefficient. For instance, with l′=l±1, ⟨nlm|qez|n′l′m⟩=(l+l′+1−2m)(l+l′+1+2m)∕(4(2l+1)(2l′+1))Rnln′l′, where z is the coordinate along the quantization axis.

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The probability to find a kth nearest neighbor at a distance R is given by the Erlang distribution (4πR2)*(3∕(4πk!))*((R3)(k−1)∕(R03)k)*e{−(R∕R0)3} and R0=(4πnRyd∕3)−1∕3.

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S. T. Rittenhouse and H. R. Sadeghpour, “Ultracold giant polyatomic Rydberg molecules: coherent control of molecular orientation,” ArXiv e-prints, March 2010.

V. Bendkowsky, B. Butscher, J. Nipper, J. Balewski, J. P. Shaffer, R. Löw, T. Pfau, W. Li, J. Stanojevic, T. Pohl, and J. M. Rost, “Rydberg trimers and excited dimers bound by internal quantum reflection,” ArXiv e-prints, December 2009.

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Figures (23)

Fig. 1
Fig. 1

Energy levels of the cesium Rydberg state n l j m j with | m j | = 1 2 versus the electric field. The fine structure of the 23 p level is visible. The Förster resonance 23 p + 23 p 23 s + 24 s is resonant for a field of 80 V cm .

Fig. 2
Fig. 2

Potential energies for the interaction channel 43 d 5 2 + 43 d 5 2 45 p 3 2 + 41 f in rubidium Rydberg atoms. The cut-off radius R c = R v d W represents the distance scale for the transition from resonant dipole–dipole to van der Waals behavior. From [3].

Fig. 3
Fig. 3

Application of the dipole blockade between two atoms. Left: the shift created by the dipole–dipole interaction allow to control the excitation of the second atom B by the first one A by preventing its excitation when the first atom A has been excited. Right: when generalized to a larger atomic sample, the shift created by the dipole–dipole interaction allows a single excitation creating a Fock state. The short-lived Rydberg Fock state can then be transfered by laser to a more stable one.

Fig. 4
Fig. 4

Population of the Rydberg level for the rubidium two-step excitation. Solid line is the result of the rate equation and dashed line of the full optical Bloch equation. Each column shows results with constant Rabi frequency, but the second column describes results when a higher laser intensity is used for the first excitation step, as schematically indicated in the inset of (a) and (b). (a) and (b) display the Rydberg excitation for a single atom (for different pulse length of 0.3, 1.0, and 2.0 μ s . (c)–(f) results for two interacting atoms case (scaled to the case of two n = 48 (s) atoms separated by 5 μ m ). ρ e e is the population of the Rydberg level, f e is the fraction of excited atoms, ρ e e , e e is the probability that both atoms are in the Rydberg state. The blockade effect is visible for n > 60 in (e), and the anti-blockade is visible for n 65 in (f). Adapted from [52].

Fig. 5
Fig. 5

Calculations of the conditional excitation versus the laser-excitation, δ, and the Förster resonances detuning Δ. The calculation is performed at small time Ω t = 1 and in a strong coupling ( V = 3 Ω ) regime. At Förster resonance, Δ = 0 , we observe a blockade effect of the excitation at resonance δ = 0 .

Fig. 6
Fig. 6

Conditional excitation of a second atom into a Rydberg state | r 2 when a neighboring atom is already in Rydberg state. Same notations as in Fig. 5. Excitation probability in function of the strength of the dipole–dipole interaction V, of the interaction time t (in Rabi Ω-frequency units). Rabi oscillation are visible in the time evolution. The blockade of the excitation is obtained as soon as V Ω .

Fig. 7
Fig. 7

Single atom Rabi oscillation and evidence for blockade of the Rydberg excitation when a second atom is present. Experimental data for Rydberg excitation of the target atom with and without a second (control) atom present. From [55].

Fig. 8
Fig. 8

Principle and experimental realization of the Rydberg blockade. (a) Principle of the blockade between two atoms, in the regime of conditional excitation. When both atoms are simultaneously excited in the blockade regime, the symmetrical state | Ψ + , described in the text, is only coupled to the ground state | g , g with a strength 2 Ω while the state | Ψ is not coupled by the laser to the states | g , g and | r , r . (c) Collective excitation of the two atoms separated by 3.6 μ m . The circles represent the probability to excite one atom when the second atom is absent. The squares represent the probability to excite only one atom when the two atoms are trapped and are exposed to the same excitation pulse. From [20].

Fig. 9
Fig. 9

Collective excitation of two atoms, i.e., creation of a single but delocalized Rydberg excitation. The laser of single atom Rabi frequency Ω is tuned at the resonance δ = 0 , and the Rydberg state is energy shifted by a resonant Förster dipole–dipole interaction of strength V = F | H 12 | r , r equals to 0, 1, and 3 times Ω . We present the probability that one atom (number 1 or 2) is excited in the Rydberg state | r , that two atoms are excited but in the Förster state | F , and that the two atoms are excited in any Rydberg states | r , r or | F . The blockade occurs when the Rydberg state is shifted more than the Rabi frequency, i.e., for V Ω . Rabi oscillation are visible and, in the blockade regime, with a 2 speed enhancement for the collective excitation (single delocalized Rydberg excitation) compared to the single atom one.

Fig. 10
Fig. 10

Van der Waals blockade of the excitation. Dependence of the Rydberg excitation fraction in function of the laser power exciting Rydberg state for several n principle quantum numbers. For large n the saturation of the Rydberg excitation clearly indicates a blockade effect. From [58].

Fig. 11
Fig. 11

Scaling of the Rydberg number N R versus the initial atom number N (or density n g ). Left theory (adapted from [65]), the three sets are for three different value of the (single atom) Rabi frequency Ω, the line are the simple law N R ( ( 4 π 3 ) * n g C 6 Ω ) 4 5 α 0.4 . Right (adapted from [67]). Experimental points (black) and theoretical points (green) with fit N R N α 0.45 for the experiment and N R N α 0.4 for the theory.

Fig. 12
Fig. 12

Small atom number Rydberg excitation. (a) Experimental spectra of the Förster reaction in rubidium 37 p 3 2 + 37 p 3 2 37 s 1 2 + 38 s 1 2 for atoms randomly positioned in a ( 18 μ m ) 3 cubic volume. (b) Theoretical corresponding probability distributions for the number of actually interacting Rydberg atoms. (c,d) Monte Carlo Theory and experiment for the resonance amplitude and resonance width. From [72].

Fig. 13
Fig. 13

Experimental study of energy levels band spectrum in the van der Waals and in a pure dipolar interaction case. The (left) low panel shows a shadow image of atoms in an optical dipole trap. The right panel shows the blockade radius R b as well as an excitation domain with zero, one, or two interacting Rydberg excitations. Upper figures: spectra for zero applied field (a) and for an applied field to reach the Förster resonance (b). S 1 ( ν 1 ) (black squares; left axis) scan corresponds to the transition | 0 r | 1 r . In order to probe the transition | 1 r | 2 r , ν 1 is kept at resonance ν res while a second pulse ν 2 is scaned and extra signal is recorded in the S 1 + 2 ( ν 2 ) scan (red circles; right axis). Adapted from [75].

Fig. 14
Fig. 14

Experimental investigation of the broadening of the 45 d 5 2 Rydberg energy levels when dipole–dipole interaction (i.e., when 46 p 3 2 atoms) is added to the system. The probing of the 45 d 5 2 Rydberg energy levels is performed using a two-photon 45 d 5 2 46 d 5 2 microwave transition. Adapted from [73].

Fig. 15
Fig. 15

Observation of the antiblockade effect. Comparison between calculated (a) and measured (b) 62 d Rydberg excitation spectra (upper graphs, black) and Penning ionization spectra (lower graphs, red) taken at different time delay. Model and experiments have an Autler–Townes splitting of 100 MHz and a density of the trapped ground-state atoms of 7 × 10 9 cm 3 . From [77].

Fig. 16
Fig. 16

Evidence for the dipole blockade induced by electric field. (a) Excitation of the spectral lines of the 75 p 3 2 level for different electric fields. The shift of the lines is due to the Stark effect and the reduced signal is mainly due to the dipole blockade effect. (b) Comparison of spectral lines of the 70 p 3 2 for two different electric fields. (c) Kinetic Monte Carlo simulation. Adapted from [79].

Fig. 17
Fig. 17

Spatial and time resolution of dipole–dipole interaction between Rydberg atoms. Upper part, schematic of the Förster ( 41 d 3 2 + 49 s 1 2 42 p 1 2 + 49 p 3 2 ) experiment. Lower part: measured production of the 49 p state as a function of the interaction time for several separation of the cylinders: from d = 0 μ m (red) to 20, 30, 40, 50 μ m and (purple). A full many-body quantum calculation, performed by Robicheaux, (solid line) reproduces the main features of the experiments, mainly the fact that the transfer rate scaled as d 5 2 with the effective separation distance d. Adapted from [83].

Fig. 18
Fig. 18

Dipole blockade at Förster resonance. The Cs resonance 38 p 3 2 + 38 p 3 2 38 s 1 2 + 39 s 1 2 is studies versus the applied electric field. Upper curve: total number of Rydberg atoms (a, black) and number of 38 s atoms (b, green). Lower curve: number of formed ions (c, blue). Adapted from [78].

Fig. 19
Fig. 19

Ground state estimates of many-Rydberg-atoms system. In a possible experimental setup [scheme in (a)]: Rydberg-dressed atoms are confined to 2D by a strong confining laser beam, with dipoles polarized perpendicular to the plane. In-plane harmonic confinement is provided, e.g., by the beam waist. (b)–(f) Monte Carlo snapshots of the density of particles for N = 13 dipoles, for several strength τ of the dipole–dipole interactions. (b) superfluid; (c) supersolid; (d)–(e) ring-like crystals; (f) classical crystal. From [116] see also [119].

Fig. 20
Fig. 20

Evidence of many-body effects and their dependence with dimensionality. (a) results of the simulations of the 31 d + 31 d 33 p + n = 29 transfer. The interaction strength (fraction of atoms that interacts) behavior reproduces qualitatively the experimental features of the data shown in part (b) taken at two different excited volumes. Even though the density is higher in the smaller, one-dimensional volume, the interaction is stronger in the sparser but more three-dimensional volumes. Adapted from [6].

Fig. 21
Fig. 21

Penning ionization due to attractive potential curves. (a) Energy levels for the Rb pair states 39 s 39 s and 39 s 39 p as a function of internuclear separation (b) Plasma electron signal observed as a function of microwave frequency driving the 39 s 39 s 39 s 39 p transition; only transitions to the attractive curve of (a) are observed. From [137].

Fig. 22
Fig. 22

Observation of dynamical Stark effect during the laser excitation. Quadratic (solid line) dependence of n p 3 2 Rydberg signal after excitation of cold rubidium n s Rydberg atoms for several n values. Because the interaction between n s atoms is repulsive, the presence of n p 3 2 atoms is not coming from binary collision but comes from direct laser excitation facilitated by the ac Stark shift created during the laser excitation and by the dipole–dipole mixing between the n s + n s and the n p + n p levels. This hypothesis is attested by the inset showing the expected (for the theory see [143]) K ( n ) n 8.5 dependence of the rate N n p = K ( n ) N n s 2 . Adapted from [143].

Fig. 23
Fig. 23

Penning ionization on attractive or repulsive potential curves. (a) Schematic potential curves of a pair of cesium atoms representing, as a function of internuclear distance, the energy difference (at zero electric field) between a pair of atoms in state n p , n p ( p p ) and n s , ( n + 1 ) s ( s s ) . (b) Rydberg and Ion signal with two different laser intensities ( 42 mW and 278 mW ) for (upper part) 40 p state (repulsive) and (lower part) 43 p (attractive). Data are taken after 10 μ s of free evolution. (c) Ion density as function of initial Rydberg density for different n states and two delay time (upper part) 0.45 μ s and (lower part) 10 μ s the “40p BB” is taken with no 6 p atoms present. Adapted from [103].

Equations (25)

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R n l n l ± 1 a 0 s n c 5 ( n * n * ) 3 2 [ J s ( e s ) ± e 2 1 ( J s ( e s ) sin ( π s ) π s ) + ( 1 e ) sin ( π s ) π ] ,
( E r μ r r F μ r r F E r ) ,
H 12 = e 2 R 3 ( r A 1 . r B 2 3 ( r A 1 . n ) ( r B 2 . n ) ) = μ 1 . μ 2 3 ( μ 1 . n ) ( μ 2 . n ) 4 π ϵ 0 R 3 .
V 12 = μ 1 μ 2 4 π ϵ 0 R 3 ( 1 3 cos 2 θ ) .
E ( R ) = exp ( i k R ) 4 π ϵ 0 { [ k 2 ( n × μ ) × n ] 1 R + [ 3 ( n . μ ) n μ ] ( 1 R 3 i k R 2 ) } ,
H 12 = Re [ E 1 ( R ) . μ 2 ] = 1 4 π ϵ 0 { [ μ 1 . μ 2 3 ( n . μ 1 ) ( n . μ 2 ) ] ( cos ( k R ) R 3 + k sin ( k R ) R 2 ) + [ μ 1 . μ 2 ( n . μ 1 ) ( n . μ 2 ) ] k 2 cos ( k R ) R } ,
H 1 + 2 = ω r 1 | r 1 r 1 | + ω r 1 | r 1 r 1 | + ω r 2 | r 2 r 2 | + ω r 2 | r 2 r 2 | + H 12 r 1 r 1 ; r 2 r 2 + H 12 r 1 r 1 ; r 2 r 2 ,
[ | n p , n p ± ( | n s , ( n + 1 ) s + | ( n + 1 ) s , n s ) 2 ] 2 .
ψ ( t ) = a g ( t ) | r 1 , g 2 + e i k . R 2 e i ω L t [ a r ( t ) | r 1 , r 2 + a F ( t ) | r 1 r 2 ] .
i d a g d t = δ a g + Ω 2 a r ,
i d a r d t = Ω * 2 a g + ( V * ) a F ,
i d a F d t = ( V ) a r + Δ a F ,
| g , g
| Ψ + = 1 2 ( | g , r e i k R 2 + | r , g e i k R 1 ) ,
| Ψ = 1 2 ( | g , r e i k R 2 | r , g e i k R 1 ) ,
| r , r
ψ ( t ) = b g ( t ) | g , g + b + ( t ) e i ω L t | Ψ + + e 2 i ω L t e i k R 1 e i k R 2 [ b r ( t ) | r , r + b F ( t ) | F ] .
i d b g d t = 2 δ b g + Ω 2 2 b + ,
i d b + d t = Ω * 2 2 b g + δ b + + Ω 2 2 b r ,
i d b r d t = Ω * 2 2 b + + ( V * ) b F ,
i d b F d t = ( V ) b r + Δ b F .
| Ψ + = 1 N b i | 1 ( g ) , , ( i 1 ) ( g ) , i ( r ) , ( i + 1 ) ( g ) , , N b ( g ) e i k . R i ,
H 12 = μ 1 μ 2 4 π ϵ 0 { k 2 cos ( k R ) sin 2 θ R + ( 1 3 cos 2 θ ) [ cos ( k R ) R 3 + k sin ( k R ) R 2 ] } .
H = i n ω r n | r i ( n ) r i ( n ) | + i , j n , n ; n , n H 12 r i ( n ) r i ( n ) ; r i ( n ) r i ( n ) .
j i Tr j [ H i j , ρ i , j ] = ( j i V i j ρ j r r ) ( ρ i g r | g i i r | ρ i e g | r i i r | ) ,

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