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Controlling the Ratchet Effect for Cold Atoms Anatole Kenfack,1 Jiangbin Gong,2 and Arjendu K. Pattanayak3 1MaxPlanckInstitut fu¨r Physik Komplexer Systeme, No¨thnitzer Strasse 38, D01187 Dresden, Germany 2Department of Physics and Center for Computational Science and Engineering, National University of Singapore, 117542, Republic of Singapore 3Department of Physics and Astronomy, Carleton College, Northfield, Minnesota 55057, USA (Received 5 September 2007; published 31 January 2008) Loworder quantum resonances manifested by directed currents have been realized with cold atoms. Here we show that by increasing the strength of an experimentally achievable deltakicking ratchet potential, quantum resonances of a very high order may naturally emerge and can induce larger ratchet currents than loworder resonances, with the underlying classical limit being fully chaotic. The results offer a means of controlling quantum transport of cold atoms. DOI: 10.1103/PhysRevLett.100.044104 PACS numbers: 05.45. a, 05.60.Gg, 37.10.Jk The ratchet effect, i.e., the possibility to derive directed transport without bias in periodic systems with broken symmetries, was originally proposed by Feynman. This effect, prohibited in systems at equilibrium by the second law of thermodynamics [1], has recently gained renewed interest [2,3] as a model for the physics of molecular motors [4]. Directed transport is possible when particles are driven out of equilibrium and relevant spatiotemporal symmetries are broken [5]. This has motivated the construction of nanoscale devices in which artificial ratchets may serve as new electrons pumps, molecular switches, and particle selectors, among other applications [4,6,7]. Other studies have shown that when the noise is absent, its role can be replaced by deterministic chaos induced by the inertial term [8]. In such inertial ratchets, the issue of current reversal was intuitively addressed [9] and later carefully reformulated [10]. Purely Hamiltonian ratchets, where noise and friction are eliminated, have received notable attention as well [5,11]. Besides these classical ratchets, quantum Hamiltonian ratchet effects arising from purely unitary evolution is also possible. These are very important, for example, for the design of coherent nanoscale devices [12]. Exploring quantum coherence phenomena in chaotic Hamiltonian ratchets hence becomes necessary. The quantum deltakicked rotor (QKR), a paradigm of quantum chaos [13], is a convenient and experimentally realizable model for such explorations, possessing dynamical localization [14], quantum accelerator modes [15], tunneling [12,16], as well as quantum resonances [17–23]. The ‘‘quantum ratchet accelerator,’’ where the coherent ratchet current accelerates linearly, was first studied with a modified QKR and later in the kicked Harper model [24]. Since the pioneering experiment of coldatom ratchets [25], new designs looking for ratchet effects in nonlinear Hamiltonian systems [26] have emerged. Motivated by the first experimental realizations of sawtoothlike asymmetric potentials [27,28] as well as quantum resonance ratchets [29], in this Letter we revisit the quantum flashing ratchet model in Ref. [30], with the perspective of detecting and controlling quantum resonance dynamics of very high orders. The ultimate goal is to help design powerful means for the coherent control of the dynamics of cold atoms with driven but dissipationless optical lattices. For other coldatom control scenarios using also Hamiltonian ratchet effects, see Refs. [31,32]. A quantum resonance occurs when the flashing period is commensurate with the recoil frequency and is related to the arithmetic nature of the effective Planck constant ~@ of kicked systems, occurring specifically if ~@ 4 r=s; (1) with r, s being mutually prime integers. Cases with small s and large s values can be called loworder quantum resonance (LOQR) and highorder quantum resonance (HOQR), respectively. We show below that HOQRs can manifest themselves strongly in the ratchet current behavior, with their corresponding classical phase space being fully chaotic. This further enhances the view that quantum control techniques can be applied to classically chaotic systems [33]. The system we consider is described, in dimensionless units, by the following Schro¨dinger equation [30]: i~@ @ @t ~@ 2 2 @2 @x2 v x X 1 l 0 t l ; (2) where x is the position, and the potential v x K sin x sin 2x is assumed to be periodically flashed off and on with delta kicks. Here t is the time variable and l an integer that counts the number of kicks. By superimposing a conventional standing wave potential of =2 spatial periodicity with a fourthorder lattice potential of =4 periodicity, such a dissipationless ratchet potential v x has been successfully engineered [27,28]. The scheme, satisfying the RamanNath transition processes [27,28,34], uses three level atoms with two stable ground states and one electronically excited state (for more ex PRL 100, 044104 (2008) PHYSICAL REVIEW LETTERS week ending 1 FEBRUARY 2008 00319007=08=100(4)=044104(4) 0441041 © 2008 The American Physical Society
Object Description
Collection Title  Scholarly Publications by Carleton Faculty and Staff 
Journal Title  Physical Review Letters 
Article Title  Controlling the ratchet effect for cold atoms 
Article Author 
Pattanayak, Arjendu Kenfack, Anatole Gong, Jiangbin 
Carleton Author 
Pattanayak, Arjendu 
Department  Physics 
Field  Science and Mathematics 
Year  2008 
Volume  100 
Publisher  American Physical Society 
File Name  037_PattanayakArjendu_ControllingTheRatchetEffectForColdAtoms.pdf; 037_PattanayakArjendu_ControllingTheRatchetEffectForColdAtoms.pdf 
Rights Management  This document is authorized for selfarchiving and distribution online by the author(s) and is free for use by researchers. 
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Publisher PDF Archiving  Yes 
Paid OA Option  Yes 
Contributing Organization  Carleton College 
Type  Text 
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Language  English 
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Article Title  Page 1 
FullText  Controlling the Ratchet Effect for Cold Atoms Anatole Kenfack,1 Jiangbin Gong,2 and Arjendu K. Pattanayak3 1MaxPlanckInstitut fu¨r Physik Komplexer Systeme, No¨thnitzer Strasse 38, D01187 Dresden, Germany 2Department of Physics and Center for Computational Science and Engineering, National University of Singapore, 117542, Republic of Singapore 3Department of Physics and Astronomy, Carleton College, Northfield, Minnesota 55057, USA (Received 5 September 2007; published 31 January 2008) Loworder quantum resonances manifested by directed currents have been realized with cold atoms. Here we show that by increasing the strength of an experimentally achievable deltakicking ratchet potential, quantum resonances of a very high order may naturally emerge and can induce larger ratchet currents than loworder resonances, with the underlying classical limit being fully chaotic. The results offer a means of controlling quantum transport of cold atoms. DOI: 10.1103/PhysRevLett.100.044104 PACS numbers: 05.45. a, 05.60.Gg, 37.10.Jk The ratchet effect, i.e., the possibility to derive directed transport without bias in periodic systems with broken symmetries, was originally proposed by Feynman. This effect, prohibited in systems at equilibrium by the second law of thermodynamics [1], has recently gained renewed interest [2,3] as a model for the physics of molecular motors [4]. Directed transport is possible when particles are driven out of equilibrium and relevant spatiotemporal symmetries are broken [5]. This has motivated the construction of nanoscale devices in which artificial ratchets may serve as new electrons pumps, molecular switches, and particle selectors, among other applications [4,6,7]. Other studies have shown that when the noise is absent, its role can be replaced by deterministic chaos induced by the inertial term [8]. In such inertial ratchets, the issue of current reversal was intuitively addressed [9] and later carefully reformulated [10]. Purely Hamiltonian ratchets, where noise and friction are eliminated, have received notable attention as well [5,11]. Besides these classical ratchets, quantum Hamiltonian ratchet effects arising from purely unitary evolution is also possible. These are very important, for example, for the design of coherent nanoscale devices [12]. Exploring quantum coherence phenomena in chaotic Hamiltonian ratchets hence becomes necessary. The quantum deltakicked rotor (QKR), a paradigm of quantum chaos [13], is a convenient and experimentally realizable model for such explorations, possessing dynamical localization [14], quantum accelerator modes [15], tunneling [12,16], as well as quantum resonances [17–23]. The ‘‘quantum ratchet accelerator,’’ where the coherent ratchet current accelerates linearly, was first studied with a modified QKR and later in the kicked Harper model [24]. Since the pioneering experiment of coldatom ratchets [25], new designs looking for ratchet effects in nonlinear Hamiltonian systems [26] have emerged. Motivated by the first experimental realizations of sawtoothlike asymmetric potentials [27,28] as well as quantum resonance ratchets [29], in this Letter we revisit the quantum flashing ratchet model in Ref. [30], with the perspective of detecting and controlling quantum resonance dynamics of very high orders. The ultimate goal is to help design powerful means for the coherent control of the dynamics of cold atoms with driven but dissipationless optical lattices. For other coldatom control scenarios using also Hamiltonian ratchet effects, see Refs. [31,32]. A quantum resonance occurs when the flashing period is commensurate with the recoil frequency and is related to the arithmetic nature of the effective Planck constant ~@ of kicked systems, occurring specifically if ~@ 4 r=s; (1) with r, s being mutually prime integers. Cases with small s and large s values can be called loworder quantum resonance (LOQR) and highorder quantum resonance (HOQR), respectively. We show below that HOQRs can manifest themselves strongly in the ratchet current behavior, with their corresponding classical phase space being fully chaotic. This further enhances the view that quantum control techniques can be applied to classically chaotic systems [33]. The system we consider is described, in dimensionless units, by the following Schro¨dinger equation [30]: i~@ @ @t ~@ 2 2 @2 @x2 v x X 1 l 0 t l ; (2) where x is the position, and the potential v x K sin x sin 2x is assumed to be periodically flashed off and on with delta kicks. Here t is the time variable and l an integer that counts the number of kicks. By superimposing a conventional standing wave potential of =2 spatial periodicity with a fourthorder lattice potential of =4 periodicity, such a dissipationless ratchet potential v x has been successfully engineered [27,28]. The scheme, satisfying the RamanNath transition processes [27,28,34], uses three level atoms with two stable ground states and one electronically excited state (for more ex PRL 100, 044104 (2008) PHYSICAL REVIEW LETTERS week ending 1 FEBRUARY 2008 00319007=08=100(4)=044104(4) 0441041 © 2008 The American Physical Society 