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Atoms in Optical Lattices


Atoms in Optical Lattices

Tech Area / Field

  • PHY-ANU/Atomic and Nuclear Physics/Physics
  • PHY-OPL/Optics and Lasers/Physics

3 Approved without Funding

Registration date

Leading Institute
Engineering Center, Armenia, Ashtarak-2


  • CNRS / Laboratoire Aime Cotton, France, Orsay\nUniversitat Kaiserslautern / Fachberaich Physik, Germany, Kaiserlautern\nUniversité Paul Verlain-Metz / Laboratoire de Physique Moleculaire et des Collisions, France, Metz\nHelsinki Institute of Physics, Finland, Helsinki

Project summary

Project's purpose and the state of the art in the field. Quantum information is currently one of the main fields in Atomic, Molecular and Optical physics [1,2]. There are good technology reasons. For instance, secure transfer of information, say, credit card numbers, over the internet is based on the difficulty of factoring large integers. A quantum computer could in principle make factoring a tractable problem, with devastating implications on the society. Quantum cryptography, in its turn, could restore the security that quantum computers take away. Aside from applications, how to make technology out of quantum mechanics is also a study of the nature of quantum mechanics. Quantum mechanics, while the foundation of our understanding of the Universe, nevertheless remains something of a mystery.

Much of the novelty of quantum information derives from entanglement [3], the property that the state vector of a joint system cannot be written as a tensor product of the state vectors of its constituents [for simplicity we only speak of pure states characterized by a state vector]. This observation has spawned many studies into characterization and properties of entanglement. The general observation about entanglement is that it is fragile, and can be utilized only in exceptionally well-controlled quantum systems. One candidate system is a Bose-Einstein condensate (BEC) in an optical lattice [4, 5], i.e., in a standing wave of light that is capable of trapping the atoms. The general theme of our work is the motion of atoms in an optical lattice on a level of inpidual atoms and quanta, under conditions when one could conceivably use atoms on an optical lattice as a basis for a scalable quantum computer.

The main topic of the project is adiabaticity of the Mott metal-insulator transition in a gas of cold atoms that takes place upon varying the depth of a confining optical lattice. We study it by computer simulations, and, if possible, analytically. This phase transition could be used to prepare an optical lattice with exactly one atom per site for use in quantum information processing.

Adiabaticity in an optical lattice containing a string of Bose-Einstein condensates is turning into a prominent issue both experimentally [4, 5] and theoretically [6-10]. Nonetheless, surprisingly, the problem we have just formulated, probably the simplest and most fundamental case one can envisage, has not been addressed before. Our analysis of the model arises from the realization that this is an issue of adiabaticity in linear quantum mechanics in a Hilbert space spanned by three many-body states: both atoms left, or right, or one atom in each trap. We study adiabaticity for various temporal profiles of the tunneling matrix element both numerically and analytically.

From the two-trap system we proceed to multi-well systems. In the limit of a large number of traps, a Mott metal-insulator phase transition has been predicted that could put exactly one atom to each lattice site [11]. However, a prediction of a phase transition a priori says nothing about how long it will take. We are back to the question of adiabaticity. For the most part, the problem has been already solved in the case when there are many atoms in each well [8,9]. As appropriate to the question of loading an optical lattice with inpidual atoms, here the focus is on the limit with about one atom per lattice site [11]. We will do direct numerical simulations first in order to try and guess the right scaling with the number of the wells. Unfortunately, the state space grows with the number of wells roughly like, so the simulations will run into trouble already for a few wells. Our hope is to eventually find an analytical approximation. We will try to go at least to high enough atom numbers to infer the adiabaticity condition for an n-well system on the basis of the numerical data and a detailed understanding of the two-well problem. The known fact that in the case of a large average occupation number the result for the multi-well problem is an obvious extension of the two-well result lends credence to our plan.

Overall, our first assignment is to find out how one can distribute atoms as evenly as possible in the optical wells for use as qubits in a quantum computer. An obvious variation of the question is what fraction of the atoms will end up not in the ground states of the potential wells of the optical lattice, but in excited states. An excited atom is spread out over a large wave function, which smears out the interaction between the atoms used to operate the quantum computer. Contamination by excited states limits coherence and entanglement, and merits an investigation. Finally, as is usual in theoretical basic research, we will probably be led to questions that are more interesting and important than the problems we have initially envisaged.

Scope of activities and expected results.

With recent developments in atom optics and Bose-Einstein condensation, it has become increasingly important to understand the various mechanisms that one can use to modify coherently the center-of-mass motion of atoms. We will undertake a theoretical study of nonlinear atom-field interactions that are particularly designed to achieve targeted states of atomic motion. This topic is among those at the forefront of current research in atomic, molecular and optical physics.

We hope to establish analytical and efficient numerical techniques that have a wide range of applicability in problems related to the motion of cold atoms in optical lattices. All the problems discussed above have this interaction as a common theme. There is no doubt that advances in our understanding of the manipulation of cold atoms by optical fields will serve as the basis for important applications in the future. In view of present developments in atom optics and quantum information technology the possible technical developments from this work may be quite substantial.

Among the topics we plan to investigate are:

- Potential shape and phase modulation effects including several linear level crossing and avoided crossing as well as non-linear or multiple or periodic level crossing models
- Nonlinear pulse shape and phase modulation effects including several level crossing models
- Cold atom few-mode photoassociation theory on the basis of above models
- Effects of initial state preparation at quantum motion of atoms in standing wave fields
- Temporal dynamics of the two-trap system loading
- Loading a condensate to a multi-well or periodic potential.

The proposed project consists of 3 blocks of tasks including 6 particular issues on the above-mentioned problems that include the research work and the dissemination of the results. All the tasks are closely related and are to be performed both numerically and analytically.

Since the work is original then all the work will be presented at International conferences and written up and published in the open scientific literature. With the present paper production rate of the groups it is likely that the final outcome of the project will be at least 12 research publications in the leading refereed journals (Science, Nature, Phys. Rev. Lett., Phys. Rev. A, J. Phys. A, etc.).

Meeting ISTC objectives. In accordance with the purposes of ISTC, the proposed Project will assist to form a new research team from scientists and engineers, working in the Engineering Center, who were previously engaged in different defense programs of USSR. This is another dimension of the proposed collaboration, i.e., aside to the pure scientific factors it is to assist the Armenian group in making transition from weapons-related technology research to non-weapon based research. Due to the strife that followed the break-up of the Soviet Union, Armenia was for quite a while also virtually cut off from mainstream physics. The influx of new thinking that comes with collaborations will undoubtedly steer the Armenian side in the allocation of resources, and guide inpidual scientists so that they can again focus their effort on topics that will make a maximal impact. This is further endorsed by appreciating the ages of the junior members involved in Armenian group. The project will promote for constant increasing of the professional level and for the intimacy of the scientists, collaboration in their scientific activity efforts. It will promote complete integration of the members of the group into the international scientific community.

Competence of the project team. Our group has a strong background in the field of theoretical atomic and quantum optics. The members of our group are the authors of two important articles on the problem of the coherent multiphoton scattering of neutral atoms by a standing wave optical field. Dr. A. Ishkhanyan is a theorist who proposed a model explanation, based on the quantum interference, of the experimentally observed anomalies in the pattern of coherent diffraction of atoms by short counterpropagating pulses of laser radiation. Thus, he has much experience in studies of the interactions of an atom with a standing-wave light field [12], the prototypical optical lattice and, most notably, in the elaborate mathematics that goes into the analytical studies of adiabaticity in quantum systems [13]. As an item that is closely related in methodology, Dr. A. Ishkhanyan and his collaborators have recently introduced and solved both analytically and numerically a nonlinear generalization of the Landau-Zener model that governs adiabaticity in photoassociation of an atomic Bose-Einstein condensate to a molecular condensate [14].

Role of Foreign Collaborators. The proposed collaboration between the western groups and the Atom Optics group of the Engineering Center of the Armenian National Academy of Sciences is anticipated to be beneficial for all participating partners since it combines high level of expertise for the problems at hand.

Prof. Peter Zoller from the Institute for Theoretical Physics, Leopold Franzens University of Innsbruck, Austria leads the Center for Quantum Optics and Quantum Information. His world leading team has been involved in theoretical research related to the interaction of radiation with matter for many years. Some specific directions of their current research on condensates include: quantum kinetics (the theory of how condensates grow as trapped gases are cooled); condensate state engineering (how to control condensates by shining lasers on them); ultra-cold atoms in optical lattices (grid-like force fields made with lasers); vortices, monopoles, and solitons in condensates [11], [15].

Professor Pierre Pillet is the director of the Laboratoire Aime Cotton in Orsay, France, that is an internationally recognized leader in the field of cold molecules both experimentally and theoretically. The first realization, via photoassociation, and identification of ultracold molecules made of Cs2 dimers has been obtained for the first time in 1997 by his group [16]. Apart from the perspective for very good interactions with several theoretical researchers in the Laboratoire Aime Cotton, the collaboration with Professor Pillet gives possibility to compare the results of the developed models to experimental data. The latter adds a new, very important, dimension to the present proposal.

Professor K.A. Suominen from the Helsinki Institute of Physics, University of Helsinki, leads a well-known research group actively working on several topics of contemporary atomic, molecular and optical physics: interactions between laser-cooled and trapped atoms, Bose-Einstein condensation, creation and time evolution of molecular wave packets, time-dependent quantum systems, quantum information, etc. [17].

Professor J. Hanssen heads the Laboratory of Molecular Physics and Collisions of the University of Metz, France. His group for a long time works on matters concerning the basic concepts of radiation-matter interaction in the microscopic level, collision mechanisms involving clusters, molecules and atoms [18]. These topics are highly complementary to the problems to be considered by the present project.

Professor M. Fleischhauer from the University of Kaiserslautern, Germany, leads an actively working group which main subjects of research coincide with the central topics of current proposal: cold atoms in periodic potentials and quantum information [19]. The group is currently working on the numerical stochastic methods to model the equilibrium quantum properties of the Mott-insulator-superfluid transition of cold bosonic atoms in periodic optical lattices.

Evidently, the contemplated project represents an ideal mix of the modelling and analytic expertise of the collaborators and the mathematical qualifications of the Armenian team. Undoubtedly, the Armenian group will benefit from contacts with the collaborating western groups. We have worked with these groups before and have several joint publications and thus we have confidence in our ability to be able to cooperate successfully on this project.

The collaborators to this project will participate in the following ways:

a) curry out joint research on all the topics of the project,
b) conduct regular reviews of progress of work throughout the project effort, review project publications,
c) provide technical assistance, conduct joint seminars and workshops.
d) help host project personnel visits to the partner countries.

Since the current research interests of the involved groups substantially overlap, it is our clear intention to coordinate closely the work of the teams by means of intensive information exchange. Communication and intensive collaboration between the groups has been established already and can be continued via electronic means. We foresee a promising long-term collaboration in cutting edge science which, we hope, will find deepening continuation in future under internationally funded projects.

Technical approach and methodology. All the tasks of the project are very complicated and will require a combination of numerical and analytical methods for their solution. The purposes of the project will be arrived with the help of known and well-developed methods of mathematical physics, both analytical and numerical. (In the meantime, the theoretical and mathematical methods for the solution of some of the proposed tasks are essentially based on the new approaches developed by us for the previous investigations of these problems.) However, a special attention will be focused on the analytical methods, though the results will be illustrated also by various graphs and tables obtained by computer simulation. Such approach is stimulated by desire to make clear, in first turn, the qualitative aspects of the occurring physical processes. The members of the group have sufficiently high qualification and skills in working with both analytical and numerical methods.


[1] APS and AIP have even compiled an online virtual journal on quantum information, see

[2] M.A. Nielsen and I.L. Chuang, Quantum computation and quantum information (Cambridge University Press, New York, 2000).

[3] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935).

]4] B. P. Anderson and M. A. Kasevich, “Macroscopic quantum interference from atomic tunnel arrays”, Science 282, 1686 (1998).

]5] C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, and M. A. Kasevich, “Squeezed states in a Bose-Einstein condensate”, Science 291, 2301 (2001).

[6] A. J. Leggett and F. Sols, “Comment on ’Phase and Phase Diffusion of a Split Bose-Einstein Condensate’”, Phys. Rev. Lett. 81,1344 (1998).

[7] J. Javanainen and M. Wilkens, “Javanainen and Wilkens Reply”, Phys. Rev. Lett. 81, 1345 (1998).

[8] J. Javanainen and M. Yu. Ivanov, “Splitting a trap containing a Bose-Einstein condensate: Atom number fluctuations”, Phys. Rev. A 60, 2351 (1999).

[9] J. Javanainen, “Phonon approach to an array of traps containing Bose-Einstein condensate”, Phys. Rev. A 60, 4902 (1999); J. Javanainen, “Co-operation includes all atoms”, Nature 412, 689 (2001); M. Mackie and J. Javanainen, “Role of Bose enhancement in photoassociation”, J. Mod. Opt. 47, 2645 (2000); J. Javanainen, Phys. Rev. Lett. 57, 3164 (1986); J. Javanainen and S.M. Yoo, Phys. Rev. Lett. 76, 161 (1996); M.Mackie, E. Timmermans, R. Cote, and J. Javanainen, “Driving super.uidity with photoassociation”, Optics Express 8, 118 (2001); J. Javanainen and M. Mackie, “Rate limit for photoassociation of a Bose-Einstein condensate”, Phys. Rev. Lett. 88, 090403 (2002).

[10] Y. B. Band and M. Trippenbach, unpublished.

[11] D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, and P. Zoller, “Cold bosonic atoms in optical lattices”, Phys. Rev. Lett. 81, 3108 (1998).

[12] A.M. Ishkhanyan, "Narrowing of interference fringes in diffraction of prepared atoms by standing waves", Phys. Rev. A 61, 063609 (2000); A.M. Ishkhanyan, "Diffraction of atoms by a standing wave at Gaussian initial momentum distribution of amplitudes", Phys. Rev. A 61, 063611 (2000); A. M. Ishkhanyan, "Anomalous diffraction of atoms in the field of a standing wave", Laser Physics 7, 1225 (1997); A.M. Ishkhanyan, J. Cont. Phys. 32, 1 (1997).

[13] A.M. Ishkhanyan and K.-A. Suominen, "Three-level system driven by delayed pulses of finite duration", Phys. Rev. A 65, 051403(R) (2002); A.M. Ishkhanyan, "New analytically integrable models of the two-state problem", Opt. Commun. 176, 155 (2000); A.M. Ishkhanyan, "New classes of analytic solutions of the two-level problem", J. Phys. A: Math. Gen. 33, 5539 (2000); A.M. Ishkhanyan, "New classes of analytic solutions of the three-state problem", J. Phys. A: Math. Gen. 33, 5041 (2000); A.M. Ishkhanyan and K.-A. Suominen, "Solutions of the two-level problem in terms of biconfluent Heun functions", J. Phys. A: Math. Gen. 34, 6301 (2001); A.M. Ishkhanyan and K.-A. Suominen, "Analytic treatment of the polariton problem for a smooth interface", J. Phys. A: Math. Gen. 34, L591 (2001); A.M. Ishkhanyan and G.P. Chernikov, "Polariton spectra for two specific symmetric structures", Reports (Armenian Nat’l Ac. Sci.), 102(2), 127 (2002); A.M. Ishkhanyan and A.M. Manukyan, “Exact solutions of the three-level problem in terms of Goursat functions”, J. Contemp. Physics (Armenian Nat’l Ac. Sci.), 37, no.4, 1 (2002); A.M. Ishkhanyan, "Solutions of the three-level problem in terms of generalized hypergeometric functions 3F2", Reports (Armenian Nat’l Ac. Sci.), 102(4), 320 (2002); A.M. Ishkhanyan and K.-A. Suominen, "New solutions of Heun's general equation", J. Phys. A: Math. Gen. 36, L81 (2003).

[14] A. Ishkhanyan, M. Mackie, A. Carmichael, Ph. Gould, and J. Javanainen, “Landau-Zener Problem in Nonlinear Quantum Mechanics”, LANL e-print, physics/0205018 (2002).

[15] P. Zoller, “Making it with molecules”, Nature 417, 493 (2002); L. Duan, M. Lukin, J. I. Cirac, P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics”, Nature 414, 413 (2001); A. Sшrensen, Lu-Ming Duan, J. I. Cirac, P. Zoller, “Many-particle entanglement with Bose-Einstein condensates”, Nature 409, 63 (2001); D. Jaksch, J.I. Cirac, P. Zoller, S.L. Rolston, R. Cote, M.D. Lukin, “Fast quantum gates for neutral atoms”, Phys. Rev. Lett. 85, 2208 (2000); J. I. Cirac, and P. Zoller, “A scalable quantum computer with ions in an array of microtraps”, Nature 404, 579 (April 06. 2000); S.J. van Enk, J.I. Cirac, and P. Zoller, “Photonic Channels for Quantum Communication”, Science 279, 205 (1998); R. Dum, J.I. Cirac, M. Lewenstein and P. Zoller, “Creation of Dark Solitons and Vortices in Bose-Einstein Condensates”, Phys. Rev. Lett. 80, 2972 (1998).

[16] A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws et P. Pillet, “Formation of cold Cs2 molecules through photoassociation “, Phys. Rev. Lett. 80, 4402-4405 (1998); D. Comparat, C. Drag, A. Fioretti, O. Dulieu and P. Pillet, “Photoassociative spectroscopy and formation of cold molecules in cold cesium vapor: trap-loss spectrum versus ion spectrum”, Journal of Molec. Spectrosc. 195, 229 (1999); P. Pillet, A. Crubellier, A. Bleton, O. Dulieu, P. Nosbaum, I. Mourachko et F. Masnou-Seeuws, “Photoassociation in a gas of cold alkali atoms: I - Perturbative quantum approach”, J. Phys.B 30, 2801-2820 (1997); M. Vatasescu, O. Dulieu, R. Kosloff and F. Masnou-Seeuws, “Optimal control of photoassociation of cold atoms and photodissociation of long-range molecules: characteristic times for wave-packet propagation”, Phys. Rev. A 63, 033407 (2001).

[17] J. Piilo, K.-A. Suominen, and K. Berg-Sшrensen,”Cold collisions between atoms in optical lattices”, J. Phys. B: At. Mol. Opt. Phys. 34, L231 (2001); J.-P. Martikainen, K.-A. Suominen, L. Santos, T. Schulte, and A. Sanpera, “Generation and evolution of vortex-antivortex pairs in Bose-Einstein condensates”, Phys. Rev. A 64, 063602 (2001); J. Calsamiglia, M. Mackie, and K.-A. Suominen, “Superposition of macroscopic numbers of atoms and molecules”, Phys. Rev. Lett. 87, 160403 (2001).

[18] P. Weck, O.A. Fojon, J Hanssen, B. Joulakian and R.D. Rivarola, ''A two-effective center approximation for the single ionisation of molecular hydrogen by fast electron impact''; Phys. Rev. A 63, 042709 (2001); V.V. Serov, V.L. Derbov, B. Joulakian and S.I. Vinitsky ''Wave packet evolution approach to ionization of hydrogen molecular ion by fast electrons, Phys. Rev. A 63, 062711 (2001); P. Weck, B. Joulakian, J Hanssen, O.A. Fojon and R.D. Rivarola, ''Multiple differential cross sections for single ionisation of H2, D2 and T2 molecules by fast electron impact. Influence of vibrational states'', Phys. Rev. A 62, 014701 (2000).

[19] M. Fleischhauer and M.D. Lukin, “Dark-State Polaritons in Electromagnetically Induced Transparency”, Phys. Rev. Lett. 84, 5094 (2000); M.D. Lukin, M. Fleischhauer, R. Cote, L. Duang, D. Jaksch, I. Cirac and P. Zoller, “Dipole Blockade and Quantum Information Processing in Mesoscopic Atomic Ensembles”, Phys. Rev. Lett. 87, 037901 (2001); L.I. Plimak, M.K. Olsen, M. Fleischhauer, and M.J. Collett, “Beyond the Fokker-Planck equation: Stochastic Simulation of Complete Wigner representation for the Optical Parametric Oscillator”, Europhys. Lett. 56, 372 (2001).


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