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Yudkin Y, Elbaz R, D'Incao JP, Julienne PS, Khaykovich L. Reshaped three-body interactions and the observation of an Efimov state in the continuum. Nat Commun 2024; 15:2127. [PMID: 38459026 PMCID: PMC10923905 DOI: 10.1038/s41467-024-46353-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/23/2023] [Accepted: 02/23/2024] [Indexed: 03/10/2024] Open
Abstract
Efimov trimers are exotic three-body quantum states that emerge from the different types of three-body continua in the vicinity of two-atom Feshbach resonances. In particular, as the strength of the interaction is decreased to a critical point, an Efimov state merges into the atom-dimer threshold and eventually dissociates into an unbound atom-dimer pair. Here we explore the Efimov state in the vicinity of this critical point using coherent few-body spectroscopy in 7Li atoms using a narrow two-body Feshbach resonance. Contrary to the expectation, we find that the 7Li Efimov trimer does not immediately dissociate when passing the threshold, and survives as a metastable state embedded in the atom-dimer continuum. We identify this behavior with a universal phenomenon related to the emergence of a repulsive interaction in the atom-dimer channel which reshapes the three-body interactions in any system characterized by a narrow Feshbach resonance. Specifically, our results shed light on the nature of 7Li Efimov states and provide a path to understand various puzzling phenomena associated with them.
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Affiliation(s)
- Yaakov Yudkin
- Department of Physics, QUEST Center and Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 5290002, Israel.
| | - Roy Elbaz
- Department of Physics, QUEST Center and Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 5290002, Israel
| | - José P D'Incao
- JILA, University of Colorado and NIST, Boulder, CO, 80309-0440, USA.
- Department of Physics, University of Colorado, Boulder, CO, 80309-0440, USA.
| | - Paul S Julienne
- Joint Quantum Institute (JQI), University of Maryland and NIST, College Park, MD, 20742, USA
| | - Lev Khaykovich
- Department of Physics, QUEST Center and Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan, 5290002, Israel.
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2
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Liang A, Xie Y, Huang M, Wang B, Zhou S, Liu L. Compensating three-dimensional field inhomogeneity in cold atom Efimov-state search by a time-averaged optical potential. THE REVIEW OF SCIENTIFIC INSTRUMENTS 2023; 94:073203. [PMID: 37462458 DOI: 10.1063/5.0156075] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2023] [Accepted: 07/04/2023] [Indexed: 07/21/2023]
Abstract
The Efimov effect and its universal property are of paramount importance in quantum few-body physics. Despite this, the predicted ground state Efimov resonance has not yet been observed in 39,40,41K-87Rb mixtures within the currently available observation window. Cooling atoms in the microgravity environment of outer space might overcome this limitation, whereas the residual curvature of the strong magnetic fields may result in significant atom leakage. In this work, we propose an optical method based on far-detuned time-averaged dipole potential to counteract the three-dimensional inhomogeneous field. The target intensity distribution can be conveniently obtained by modulating the central position of the quasi-1D print beam using acoustic optical modulators. Within a volume of 300 × 300 × 400 µm3, the residual potential fluctuations can be reduced by two orders of magnitude to less than 100 pK. The proposed approach offers a realistic prospect of investigating the Efimov-type resonance in the 40K-87Rb Bose-Fermi mixture.
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Affiliation(s)
- Angang Liang
- Space Laser Engineering Center, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
- Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China
- Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
| | - Yu Xie
- Space Laser Engineering Center, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
- Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China
- Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
| | - Mingshan Huang
- Space Laser Engineering Center, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
- Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China
- Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
| | - Bin Wang
- Space Laser Engineering Center, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
| | - Shuyu Zhou
- Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
| | - Liang Liu
- Space Laser Engineering Center, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
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3
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Zhu Z, Yao H, Sanchez-Palencia L. Thermodynamic Phase Diagram of Two-Dimensional Bosons in a Quasicrystal Potential. PHYSICAL REVIEW LETTERS 2023; 130:220402. [PMID: 37327407 DOI: 10.1103/physrevlett.130.220402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2022] [Accepted: 05/10/2023] [Indexed: 06/18/2023]
Abstract
Quantum simulation of quasicrystals in synthetic bosonic matter now paves the way for the exploration of these intriguing systems in wide parameter ranges. Yet thermal fluctuations in such systems compete with quantum coherence and significantly affect the zero-temperature quantum phases. Here we determine the thermodynamic phase diagram of interacting bosons in a two-dimensional, homogeneous quasicrystal potential. We find our results using quantum Monte Carlo simulations. Finite-size effects are carefully taken into account and the quantum phases are systematically distinguished from thermal phases. In particular, we demonstrate stabilization of a genuine Bose glass phase against the normal fluid in sizable parameter ranges. We interpret our results for strong interactions using a fermionization picture and discuss experimental relevance.
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Affiliation(s)
- Zhaoxuan Zhu
- CPHT, CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France
| | - Hepeng Yao
- Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, CH-1211 Geneva, Switzerland
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4
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Bera J, Batin AQ, Ghosh S, Malomed B, Roy U. Generation of higher harmonics in dipolar Bose-Einstein condensates trapped in periodically modulated potentials. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220075. [PMID: 36842989 DOI: 10.1098/rsta.2022.0075] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2022] [Accepted: 09/26/2022] [Indexed: 06/18/2023]
Abstract
We consider a quasi-one-dimensional Bose-Einstein condensate with contact and long-range dipolar interactions, under the action of the time-periodic modulation applied to the harmonic-oscillator and optical-lattice trapping potentials. The modulation results in generation of a variety of harmonics in oscillations of the condensate's width and centre-of-mass coordinate. These include multiple and combinational harmonics, represented by sharp peaks in the system's spectra. Approximate analytical results are produced by the variational method, which are verified by systematic simulations of the underlying Gross-Pitaevskii equation. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.
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Affiliation(s)
- Jayanta Bera
- Department of Physics, C. V. Raman Global University, Bhubaneswar 752054, Odisha, India
| | - Abdul Q Batin
- Department of Physics, Indian Institute of Technology Patna,Patna 801106, Bihar, India
| | - Suranjana Ghosh
- Department of Physics, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India
| | - Boris Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Utpal Roy
- Department of Physics, Indian Institute of Technology Patna,Patna 801106, Bihar, India
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5
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Ding S, Drewsen M, Arlt JJ, Bruun GM. Mediated Interaction between Ions in Quantum Degenerate Gases. PHYSICAL REVIEW LETTERS 2022; 129:153401. [PMID: 36269954 DOI: 10.1103/physrevlett.129.153401] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2022] [Revised: 08/04/2022] [Accepted: 09/14/2022] [Indexed: 06/16/2023]
Abstract
We explore the interaction between two trapped ions mediated by a surrounding quantum degenerate Bose or Fermi gas. Using perturbation theory valid for weak atom-ion interaction, we show analytically that the interaction mediated by a Bose gas has a power-law behavior for large distances whereas it has a Yukawa form for intermediate distances. For a Fermi gas, the mediated interaction is given by a power law for large density and by a Ruderman-Kittel-Kasuya-Yosida form for low density. For strong atom-ion interaction, we use a diagrammatic theory to demonstrate that the mediated interaction can be a significant addition to the bare Coulomb interaction between the ions, when an atom-ion bound state is close to threshold. Finally, we show that the induced interaction leads to substantial and observable shifts in the ion phonon frequencies.
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Affiliation(s)
- Shanshan Ding
- Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark
| | - Michael Drewsen
- Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark
| | - Jan J Arlt
- Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark
| | - G M Bruun
- Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark
- Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
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6
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Kwon K, Mukherjee K, Huh SJ, Kim K, Mistakidis SI, Maity DK, Kevrekidis PG, Majumder S, Schmelcher P, Choi JY. Spontaneous Formation of Star-Shaped Surface Patterns in a Driven Bose-Einstein Condensate. PHYSICAL REVIEW LETTERS 2021; 127:113001. [PMID: 34558915 DOI: 10.1103/physrevlett.127.113001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2021] [Accepted: 08/03/2021] [Indexed: 06/13/2023]
Abstract
We observe experimentally the spontaneous formation of star-shaped surface patterns in driven Bose-Einstein condensates. Two-dimensional star-shaped patterns with l-fold symmetry, ranging from quadrupole (l=2) to heptagon modes (l=7), are parametrically excited by modulating the scattering length near the Feshbach resonance. An effective Mathieu equation and Floquet analysis are utilized, relating the instability conditions to the dispersion of the surface modes in a trapped superfluid. Identifying the resonant frequencies of the patterns, we precisely measure the dispersion relation of the collective excitations. The oscillation amplitude of the surface excitations increases exponentially during the modulation. We find that only the l=6 mode is unstable due to its emergent coupling with the dipole motion of the cloud. Our experimental results are in excellent agreement with the mean-field framework. Our work opens a new pathway for generating higher-lying collective excitations with applications, such as the probing of exotic properties of quantum fluids and providing a generation mechanism of quantum turbulence.
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Affiliation(s)
- K Kwon
- Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
| | - K Mukherjee
- Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
| | - S J Huh
- Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
| | - K Kim
- Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
| | - S I Mistakidis
- Center for Optical Quantum Technologies, Department of Physics,University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - D K Maity
- Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - S Majumder
- Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
| | - P Schmelcher
- Center for Optical Quantum Technologies, Department of Physics,University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
- The Hamburg Centre for Ultrafast Imaging, University of Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - J-Y Choi
- Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea
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7
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Chen CA, Hung CL. Observation of Scale Invariance in Two-Dimensional Matter-Wave Townes Solitons. PHYSICAL REVIEW LETTERS 2021; 127:023604. [PMID: 34296901 DOI: 10.1103/physrevlett.127.023604] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2021] [Accepted: 06/21/2021] [Indexed: 06/13/2023]
Abstract
We report near-deterministic generation of two-dimensional (2D) matter-wave Townes solitons and a precision test on scale invariance in attractive 2D Bose gases. We induce a shape-controlled modulational instability in an elongated 2D matter wave to create an array of isolated solitary waves of various sizes and peak densities. We confirm scale invariance by observing the collapse of solitary-wave density profiles onto a single curve in a dimensionless coordinate rescaled according to their peak densities and observe that the scale-invariant profiles measured at different coupling constants g can further collapse onto the universal profile of Townes solitons. The reported scaling behavior is tested with a nearly 60-fold difference in soliton interaction energies and allows us to discuss the impact of a non-negligible magnetic dipole-dipole interaction (MDDI) on 2D scale invariance. We confirm that the effect of MDDI in our alkali cesium quasi-2D samples effectively conforms to the same scaling law governed by a contact interaction to well within our experiment uncertainty.
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Affiliation(s)
- Cheng-An Chen
- Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
| | - Chen-Lung Hung
- Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA
- Purdue Quantum Science and Engineering Institute, Purdue University, West Lafayette, Indiana 47907, USA
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8
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Gautier R, Yao H, Sanchez-Palencia L. Strongly Interacting Bosons in a Two-Dimensional Quasicrystal Lattice. PHYSICAL REVIEW LETTERS 2021; 126:110401. [PMID: 33798372 DOI: 10.1103/physrevlett.126.110401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2020] [Accepted: 02/16/2021] [Indexed: 06/12/2023]
Abstract
Quasicrystals exhibit exotic properties inherited from the self-similarity of their long-range ordered, yet aperiodic, structure. The recent realization of optical quasicrystal lattices paves the way to the study of correlated Bose fluids in such structures, but the regime of strong interactions remains largely unexplored, both theoretically and experimentally. Here, we determine the quantum phase diagram of two-dimensional correlated bosons in an eightfold quasicrystal potential. Using large-scale quantum Monte Carlo calculations, we demonstrate a superfluid-to-Bose glass transition and determine the critical line. Moreover, we show that strong interactions stabilize Mott insulator phases, some of which have spontaneously broken eightfold symmetry. Our results are directly relevant to current generation experiments and, in particular, drive prospects to the observation of the still elusive Bose glass phase in two dimensions and exotic Mott phases.
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Affiliation(s)
- Ronan Gautier
- CPHT, CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France
| | - Hepeng Yao
- CPHT, CNRS, Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France
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9
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Zou YQ, Bakkali-Hassani B, Maury C, Le Cerf É, Nascimbene S, Dalibard J, Beugnon J. Magnetic Dipolar Interaction between Hyperfine Clock States in a Planar Alkali Bose Gas. PHYSICAL REVIEW LETTERS 2020; 125:233604. [PMID: 33337228 DOI: 10.1103/physrevlett.125.233604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2020] [Accepted: 11/04/2020] [Indexed: 06/12/2023]
Abstract
In atomic systems, clock states feature a zero projection of the total angular momentum and thus a low sensitivity to magnetic fields. This makes them widely used for metrological applications like atomic fountains or gravimeters. Here, we show that a mixture of two such nonmagnetic states still displays magnetic dipole-dipole interactions comparable to the one expected for the other Zeeman states of the same atomic species. Using high-resolution spectroscopy of a planar gas of ^{87}Rb atoms with a controlled in plane shape, we explore the effective isotropic and extensive character of these interactions and demonstrate their tunability. Our measurements set strong constraints on the relative values of the s-wave scattering lengths a_{ij} involving the two clock states.
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Affiliation(s)
- Y-Q Zou
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - B Bakkali-Hassani
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - C Maury
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - É Le Cerf
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - S Nascimbene
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - J Dalibard
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
| | - J Beugnon
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin Berthelot, 75005 Paris, France
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10
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Luo D, Jin Y, Nguyen JHV, Malomed BA, Marchukov OV, Yurovsky VA, Dunjko V, Olshanii M, Hulet RG. Creation and Characterization of Matter-Wave Breathers. PHYSICAL REVIEW LETTERS 2020; 125:183902. [PMID: 33196245 DOI: 10.1103/physrevlett.125.183902] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2020] [Revised: 09/03/2020] [Accepted: 09/28/2020] [Indexed: 06/11/2023]
Abstract
We report the creation of quasi-1D excited matter-wave solitons, "breathers," by quenching the strength of the interactions in a Bose-Einstein condensate with attractive interactions. We characterize the resulting breathing dynamics and quantify the effects of the aspect ratio of the confining potential, the strength of the quench, and the proximity of the 1D-3D crossover for the two-soliton breather. Furthermore, we demonstrate the complex dynamics of a three-soliton breather created by a stronger interaction quench. Our experimental results, which compare well with numerical simulations, provide a pathway for utilizing matter-wave breathers to explore quantum effects in large many-body systems.
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Affiliation(s)
- D Luo
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| | - Y Jin
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| | - J H V Nguyen
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| | - B A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, 6997801 Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - O V Marchukov
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, 6997801 Tel Aviv, Israel
- Institute for Applied Physics, Technical University of Darmstadt, 64289 Darmstadt, Germany
| | - V A Yurovsky
- School of Chemistry, Tel Aviv University, 6997801 Tel Aviv, Israel
| | - V Dunjko
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - M Olshanii
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - R G Hulet
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
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11
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Hulet RG, Nguyen JHV, Senaratne R. Methods for preparing quantum gases of lithium. THE REVIEW OF SCIENTIFIC INSTRUMENTS 2020; 91:011101. [PMID: 32012609 DOI: 10.1063/1.5131023] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2019] [Accepted: 12/24/2019] [Indexed: 06/10/2023]
Abstract
Lithium is an important element in atomic quantum gas experiments because its interactions are highly tunable due to broad Feshbach resonances and zero-crossings and because it has two stable isotopes: 6Li, a fermion, and 7Li, a boson. Although lithium has special value for these reasons, it also presents experimental challenges. In this article, we review some of the methods that have been developed or adapted to confront these challenges, including beam and vapor sources, Zeeman slowers, sub-Doppler laser cooling, laser sources at 671 nm, and all-optical methods for trapping and cooling. Additionally, we provide spectral diagrams of both 6Li and 7Li and present plots of Feshbach resonances for both isotopes.
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Affiliation(s)
- Randall G Hulet
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| | - Jason H V Nguyen
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| | - Ruwan Senaratne
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
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12
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Fujiwara CJ, Singh K, Geiger ZA, Senaratne R, Rajagopal SV, Lipatov M, Weld DM. Transport in Floquet-Bloch Bands. PHYSICAL REVIEW LETTERS 2019; 122:010402. [PMID: 31012654 DOI: 10.1103/physrevlett.122.010402] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/15/2018] [Indexed: 06/09/2023]
Abstract
We report Floquet band engineering of long-range transport and direct imaging of Floquet-Bloch bands in an amplitude-modulated optical lattice. In one variety of Floquet-Bloch bands we observe tunable rapid long-range high-fidelity transport of a Bose condensate across thousands of lattice sites. Quenching into an opposite-parity Floquet-hybridized band allows Wannier-Stark localization to be controllably turned on and off using modulation. A central result of this work is the use of transport dynamics to demonstrate direct imaging of a Floquet-Bloch band structure. These results demonstrate that transport in dynamical Floquet-Bloch bands can be mapped to transport in quasistatic effective bands, opening a path to cold atom quantum emulation of ultrafast multiband electronic dynamics.
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Affiliation(s)
- C J Fujiwara
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Kevin Singh
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Zachary A Geiger
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Ruwan Senaratne
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Shankari V Rajagopal
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Mikhail Lipatov
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - David M Weld
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
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13
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Fu H, Feng L, Anderson BM, Clark LW, Hu J, Andrade JW, Chin C, Levin K. Density Waves and Jet Emission Asymmetry in Bose Fireworks. PHYSICAL REVIEW LETTERS 2018; 121:243001. [PMID: 30608768 DOI: 10.1103/physrevlett.121.243001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2018] [Indexed: 06/09/2023]
Abstract
A Bose condensate, subject to periodic modulation of the two-body interactions, was recently observed to emit matter-wave jets resembling fireworks [Nature (London) 551, 356 (2017)NATUAS0028-083610.1038/nature24272]. In this Letter, combining experiment with numerical simulation, we demonstrate that these "Bose fireworks" represent a late stage in a complex time evolution of the driven condensate. We identify a "density wave" stage which precedes jet emission and results from the interference of matter waves. The density waves self-organize and self-amplify without breaking long range translational symmetry. This density wave structure deterministically establishes the template for the subsequent patterns of the emitted jets. Moreover, our simulations, in good agreement with experiment, address an apparent asymmetry in the jet pattern, and show that it is fully consistent with momentum conservation.
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Affiliation(s)
- Han Fu
- James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
| | - Lei Feng
- James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
- Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
| | - Brandon M Anderson
- Department of Computer Science, University of Chicago, Chicago, Illinois 60637, USA
| | - Logan W Clark
- James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
- Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
| | - Jiazhong Hu
- James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
- Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
| | - Jeffery W Andrade
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Cheng Chin
- James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
- Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, Illinois 60637, USA
| | - K Levin
- James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
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14
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Geiger ZA, Fujiwara KM, Singh K, Senaratne R, Rajagopal SV, Lipatov M, Shimasaki T, Driben R, Konotop VV, Meier T, Weld DM. Observation and Uses of Position-Space Bloch Oscillations in an Ultracold Gas. PHYSICAL REVIEW LETTERS 2018; 120:213201. [PMID: 29883162 DOI: 10.1103/physrevlett.120.213201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2018] [Indexed: 05/22/2023]
Abstract
We report the observation and characterization of position-space Bloch oscillations using cold atoms in a tilted optical lattice. While momentum-space Bloch oscillations are a common feature of optical lattice experiments, the real-space center-of-mass dynamics are typically unresolvable. In a regime of rapid tunneling and low force, we observe real-space Bloch oscillation amplitudes of hundreds of lattice sites, in both ground and excited bands. We demonstrate two unique capabilities enabled by tracking of Bloch dynamics in position space: measurement of the full position-momentum phase-space evolution during a Bloch cycle, and direct imaging of the lattice band structure. These techniques, along with the ability to exert long-distance coherent control of quantum gases without modulation, may open up new possibilities for quantum control and metrology.
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Affiliation(s)
- Zachary A Geiger
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Kurt M Fujiwara
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Kevin Singh
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Ruwan Senaratne
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Shankari V Rajagopal
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Mikhail Lipatov
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Toshihiko Shimasaki
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
| | - Rodislav Driben
- Department of Physics and CeOPP, University of Paderborn, D-33098 Paderborn, Germany
| | - Vladimir V Konotop
- Centro de Física Teórica e Computacional and Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Ed. C8, Lisboa 1749-016, Portugal
| | - Torsten Meier
- Department of Physics and CeOPP, University of Paderborn, D-33098 Paderborn, Germany
| | - David M Weld
- University of California and California Institute for Quantum Emulation, Santa Barbara, California 93105, USA
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15
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Nguyen JHV, Luo D, Hulet RG. Formation of matter-wave soliton trains by modulational instability. Science 2017; 356:422-426. [DOI: 10.1126/science.aal3220] [Citation(s) in RCA: 122] [Impact Index Per Article: 17.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2016] [Accepted: 03/23/2017] [Indexed: 11/02/2022]
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16
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Ma X, Driben R, Malomed BA, Meier T, Schumacher S. Two-dimensional symbiotic solitons and vortices in binary condensates with attractive cross-species interaction. Sci Rep 2016; 6:34847. [PMID: 27703235 PMCID: PMC5050495 DOI: 10.1038/srep34847] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2016] [Accepted: 09/12/2016] [Indexed: 11/09/2022] Open
Abstract
We consider a two-dimensional (2D) two-component spinor system with cubic attraction between the components and intra-species self-repulsion, which may be realized in atomic Bose-Einstein condensates, as well as in a quasi-equilibrium condensate of microcavity polaritons. Including a 2D spatially periodic potential, which is necessary for the stabilization of the system against the critical collapse, we use detailed numerical calculations and an analytical variational approximation (VA) to predict the existence and stability of several types of 2D symbiotic solitons in the spinor system. Stability ranges are found for symmetric and asymmetric symbiotic fundamental solitons and vortices, including hidden-vorticity (HV) modes, with opposite vorticities in the two components. The VA produces exceptionally accurate predictions for the fundamental solitons and vortices. The fundamental solitons, both symmetric and asymmetric ones, are completely stable, in either case when they exist as gap solitons or regular ones. The symmetric and asymmetric vortices are stable if the inter-component attraction is stronger than the intra-species repulsion, while the HV modes have their stability region in the opposite case.
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Affiliation(s)
- Xuekai Ma
- Department of Physics and Center for Optoelectronics and Photonics Paderborn (CeOPP), Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
| | - Rodislav Driben
- Department of Physics and Center for Optoelectronics and Photonics Paderborn (CeOPP), Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
| | - Torsten Meier
- Department of Physics and Center for Optoelectronics and Photonics Paderborn (CeOPP), Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
| | - Stefan Schumacher
- Department of Physics and Center for Optoelectronics and Photonics Paderborn (CeOPP), Universität Paderborn, Warburger Strasse 100, 33098 Paderborn, Germany
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17
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Sakaguchi H, Sherman EY, Malomed BA. Vortex solitons in two-dimensional spin-orbit coupled Bose-Einstein condensates: Effects of the Rashba-Dresselhaus coupling and Zeeman splitting. Phys Rev E 2016; 94:032202. [PMID: 27739749 DOI: 10.1103/physreve.94.032202] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/18/2016] [Indexed: 06/06/2023]
Abstract
We present an analysis of two-dimensional (2D) matter-wave solitons, governed by the pseudospinor system of Gross-Pitaevskii equations with self- and cross attraction, which includes the spin-orbit coupling (SOC) in the general Rashba-Dresselhaus form, and, separately, the Rashba coupling and the Zeeman splitting. Families of semivortex (SV) and mixed-mode (MM) solitons are constructed, which exist and are stable in free space, as the SOC terms prevent the onset of the critical collapse and create the otherwise missing ground states in the form of the solitons. The Dresselhaus SOC produces a destructive effect on the vortex solitons, while the Zeeman term tends to convert the MM states into the SV ones, which eventually suffer delocalization. Existence domains and stability boundaries are identified for the soliton families. For physically relevant parameters of the SOC system, the number of atoms in the 2D solitons is limited by ∼1.5×10^{4}. The results are obtained by means of combined analytical and numerical methods.
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Affiliation(s)
- Hidetsugu Sakaguchi
- Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
| | - E Ya Sherman
- Department of Physical Chemistry, University of the Basque Country UPV-EHU, 48940 Bilbao, Spain and IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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18
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Finite-temperature fluid-insulator transition of strongly interacting 1D disordered bosons. Proc Natl Acad Sci U S A 2016; 113:E4455-9. [PMID: 27436894 DOI: 10.1073/pnas.1606908113] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
We consider the many-body localization-delocalization transition for strongly interacting one-dimensional disordered bosons and construct the full picture of finite temperature behavior of this system. This picture shows two insulator-fluid transitions at any finite temperature when varying the interaction strength. At weak interactions, an increase in the interaction strength leads to insulator [Formula: see text] fluid transition, and, for large interactions, there is a reentrance to the insulator regime. It is feasible to experimentally verify these predictions by tuning the interaction strength with the use of Feshbach or confinement-induced resonances, for example, in (7)Li or (39)K.
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19
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Lebedev ME, Alfimov GL, Malomed BA. Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity. CHAOS (WOODBURY, N.Y.) 2016; 26:073110. [PMID: 27475070 DOI: 10.1063/1.4958710] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, being essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate that one branch of the DS family (namely, which obeys the Vakhitov-Kolokolov criterion) is stable, while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too.
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Affiliation(s)
- M E Lebedev
- National Research University of Electronic Technology MIET, Zelenograd, Moscow 124498, Russia
| | - G L Alfimov
- National Research University of Electronic Technology MIET, Zelenograd, Moscow 124498, Russia
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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20
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Zhang XF, Wen L, Dai CQ, Dong RF, Jiang HF, Chang H, Zhang SG. Exotic vortex lattices in a rotating binary dipolar Bose-Einstein condensate. Sci Rep 2016; 6:19380. [PMID: 26778736 PMCID: PMC4726089 DOI: 10.1038/srep19380] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2015] [Accepted: 12/08/2015] [Indexed: 11/29/2022] Open
Abstract
In the last decade, considerable advances have been made in the investigation of dipolar quantum gases. Previous theoretical investigations of a rotating binary dipolar Bose-Einstein condensate, where only one component possesses dipole moment, were mainly focused on two special orientations of the dipoles: perpendicular or parallel to the plane of motion. Here we study the ground-state and rotational properties of such a system for an arbitrary orientation of the dipoles. We demonstrate the ground-state vortex structures depend strongly on the relative strength between dipolar and contact interactions and the rotation frequency, as well as on the orientation of the dipoles. In the absence of rotation, the tunable dipolar interaction can be used to induce the squeezing or expansion of the cloud, and to derive the phase transition between phase coexistence and separation. Under finite rotation, the system is found to exhibit exotic ground-state vortex configurations, such as kernel-shell, vortex necklace, and compensating stripe vortex structures. We also check the validity of the Feynman relation, and find no significant deviations from it. The obtained results open up alternate ways for the quantum control of dipolar quantum gases.
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Affiliation(s)
- Xiao-Fei Zhang
- Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, People’s Republic of China
| | - Lin Wen
- College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 400047, People’s Republic of China
| | - Cai-Qing Dai
- School of Science, Zhejiang Agriculture and Forestry University, Lin’an, Zhejiang 311300, People’s Republic of China
| | - Rui-Fang Dong
- Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, People’s Republic of China
| | - Hai-Feng Jiang
- Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, People’s Republic of China
| | - Hong Chang
- Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, People’s Republic of China
| | - Shou-Gang Zhang
- Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, People’s Republic of China
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21
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Petrov DS. Quantum Mechanical Stabilization of a Collapsing Bose-Bose Mixture. PHYSICAL REVIEW LETTERS 2015; 115:155302. [PMID: 26550732 DOI: 10.1103/physrevlett.115.155302] [Citation(s) in RCA: 102] [Impact Index Per Article: 11.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2015] [Indexed: 06/05/2023]
Abstract
According to the mean-field theory a condensed Bose-Bose mixture collapses when the interspecies attraction becomes stronger than the geometrical average of the intraspecies repulsions, g_{12}^{2}>g_{11}g_{22}. We show that instead of collapsing such a mixture gets into a dilute liquidlike droplet state stabilized by quantum fluctuations thus providing a direct manifestation of beyond mean-field effects. We study various properties of the droplet and find, in particular, that in a wide range of parameters its excitation spectrum lies entirely above the particle emission threshold. The droplet thus automatically evaporates itself to zero temperature, the property potentially interesting by itself and from the viewpoint of sympathetic cooling of other systems.
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Affiliation(s)
- D S Petrov
- Université Paris-Sud, CNRS, LPTMS, UMR8626, Orsay, F-91405, France
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22
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Maier RAW, Eisele M, Tiemann E, Zimmermann C. Efimov Resonance and Three-Body Parameter in a Lithium-Rubidium Mixture. PHYSICAL REVIEW LETTERS 2015; 115:043201. [PMID: 26252682 DOI: 10.1103/physrevlett.115.043201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2014] [Indexed: 06/04/2023]
Abstract
We study collisional heating in a cold ^{7}Li-^{87}Rb mixture near a broad Feshbach resonance at 661 G. At the high field slope of the resonance, we find an enhanced three-body recombination rate that we interpret as a heteronuclear Efimov resonance. With improved Feshbach spectroscopy of two further resonances, a model for the molecular potentials has been developed that now consistently explains all known Feshbach resonances of the various Li-Rb isotope mixtures. The model is used to determine the scattering length of the observed Efimov state. Its value of -1870a_{0} Bohr radii supports the currently discussed assumption of universality of the three-body parameter also in heteronuclear mixtures.
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Affiliation(s)
- R A W Maier
- Physikalisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany
| | - M Eisele
- Physikalisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany
| | - E Tiemann
- Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
| | - C Zimmermann
- Physikalisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany
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23
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Creation of vortices by torque in multidimensional media with inhomogeneous defocusing nonlinearity. Sci Rep 2015; 5:9420. [PMID: 25800140 PMCID: PMC4371187 DOI: 10.1038/srep09420] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/02/2014] [Accepted: 03/02/2015] [Indexed: 11/08/2022] Open
Abstract
Recently, a new class of nonlinear systems was introduced, in which the self-trapping of fundamental and vortical localized modes in space of dimension D is supported by cubic self-repulsion with a strength growing as a function of the distance from the center, r, at any rate faster that rD. These systems support robust 2D and 3D modes which either do not exist or are unstable in other nonlinear systems. Here we demonstrate a possibility to create solitary vortices in this setting by applying a phase-imprinting torque to the ground state. Initially, a strong torque completely destroys the ground state. However, contrary to usual systems, where the destruction is irreversible, the present ones demonstrate a rapid restabilization and the creation of one or several shifted vortices orbiting the center. For the sake of comparison, we show analytically that, in the linear system with a 3D trapping potential, the action of a torque on the ground state is inefficient and creates only even-vorticity states with a small probability.
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24
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Kartashov YV, Malomed BA, Shnir Y, Torner L. Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity. PHYSICAL REVIEW LETTERS 2014; 113:264101. [PMID: 25615341 DOI: 10.1103/physrevlett.113.264101] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2014] [Indexed: 06/04/2023]
Abstract
Toroidal modes in the form of so-called Hopfions, with two independent winding numbers, a hidden one (twist s), which characterizes a circular vortex thread embedded into a three-dimensional soliton, and the vorticity around the vertical axis (m), appear in many fields, including field theory, ferromagnetics, and semi- and superconductors. Such topological states are normally generated in multicomponent systems, or as trapped quasilinear modes in toroidal potentials. We uncover that stable solitons with this structure can be created, without any linear potential, in the single-component setting with the strength of repulsive nonlinearity growing fast enough from the center to the periphery, for both steep and smooth modulation profiles. Toroidal modes with s=1 and vorticity m=0, 1, 2 are produced. They are stable for m≤1, and do not exist for s>1. An approximate analytical solution is obtained for the twisted ring with s=1, m=0. Under the application of an external torque, it rotates like a solid ring. The setting can be implemented in a Bose-Einstein condensate (BEC) by means of the Feshbach resonance controlled by inhomogeneous magnetic fields.
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Affiliation(s)
- Yaroslav V Kartashov
- ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya, Mediterranean Technology Park, E-08860 Castelldefels (Barcelona), Spain and Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, 142190 Moscow, Russia
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, 69978 Tel Aviv, Israel
| | - Yasha Shnir
- Bogoliubov Laboratory for Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Moscow Region, Russia and Institute of Physics, Carl von Ossietzky University Oldenburg, Oldenburg D-26111, Germany
| | - Lluis Torner
- ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya, Mediterranean Technology Park, E-08860 Castelldefels (Barcelona), Spain
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25
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Xie Q, Wang L, Wang Y, Shen Z, Fu J. Bright solitons in nonlinear media with a self-defocusing double-well nonlinearity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:063204. [PMID: 25615215 DOI: 10.1103/physreve.90.063204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/18/2014] [Indexed: 06/04/2023]
Abstract
We show that stable bright solitons can appear in a medium with spatially inhomogeneous self-defocusing (SDF) nonlinearity of a double-well structure. For a specific choice of the nonlinearity parameters, we obtain exact analytical solutions for the fundamental bright solitons. By making use of the linear stability analysis, the stability region in the parameter space for the exact fundamental bright soliton is obtained numerically. We also show the bifurcation from an antisymmetric to an asymmetric bright soliton for the SDF double-well nonlinearity.
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Affiliation(s)
- Qiongtao Xie
- College of Physics and Electronic Engineering, Hainan Normal University, Haikou 571158, China and Department of Physics, Hunan Normal University, Changsha 410081, China
| | - Linmao Wang
- College of Physics and Electronic Engineering, Hainan Normal University, Haikou 571158, China
| | - Yizhen Wang
- College of Physics and Electronic Engineering, Hainan Normal University, Haikou 571158, China
| | - Zhenjiang Shen
- College of Physics and Electronic Engineering, Hainan Normal University, Haikou 571158, China
| | - Jun Fu
- College of Physics and Electronic Engineering, Hainan Normal University, Haikou 571158, China
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26
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Mareeswaran RB, Charalampidis EG, Kanna T, Kevrekidis PG, Frantzeskakis DJ. Vector rogue waves and dark-bright boomeronic solitons in autonomous and nonautonomous settings. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042912. [PMID: 25375572 DOI: 10.1103/physreve.90.042912] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2014] [Indexed: 06/04/2023]
Abstract
In this work we consider the dynamics of vector rogue waves and dark-bright solitons in two-component nonlinear Schrödinger equations with various physically motivated time-dependent nonlinearity coefficients, as well as spatiotemporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark-bright boomeronlike soliton solutions of the latter are converted back into ones of the original nonautonomous model. Using direct numerical simulations we find that, in most cases, the rogue wave formation is rapidly followed by a modulational instability that leads to the emergence of an expanding soliton train. Scenarios different than this generic phenomenology are also reported.
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Affiliation(s)
- R Babu Mareeswaran
- Post Graduate and Research Department of Physics, Bishop Heber College, Tiruchirapalli-620 017, Tamil Nadu, India
| | - E G Charalampidis
- School of Civil Engineering, Faculty of Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece and Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - T Kanna
- Post Graduate and Research Department of Physics, Bishop Heber College, Tiruchirapalli-620 017, Tamil Nadu, India
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - D J Frantzeskakis
- Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
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27
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Hofmann J, Natu SS, Das Sarma S. Coarsening dynamics of binary Bose condensates. PHYSICAL REVIEW LETTERS 2014; 113:095702. [PMID: 25215993 DOI: 10.1103/physrevlett.113.095702] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/03/2014] [Indexed: 06/03/2023]
Abstract
We study the dynamics of domain formation and coarsening in a binary Bose-Einstein condensate that is quenched across a miscible-immiscible phase transition. The late-time evolution of the system is universal and governed by scaling laws for the correlation functions. We numerically determine the scaling forms and extract the critical exponents that describe the growth rate of domain size and autocorrelations. Our data are consistent with inviscid hydrodynamic domain growth, which is governed by a universal dynamical critical exponent of 1/z=0.68(2). In addition, we analyze the effect of domain wall configurations which introduce a nonanalytic term in the short-distance structure of the pair correlation function, leading to a high-momentum "Porod" tail in the static structure factor, which can be measured experimentally.
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Affiliation(s)
- Johannes Hofmann
- Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
| | - Stefan S Natu
- Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
| | - S Das Sarma
- Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA
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28
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Driben R, Kartashov YV, Malomed BA, Meier T, Torner L. Soliton gyroscopes in media with spatially growing repulsive nonlinearity. PHYSICAL REVIEW LETTERS 2014; 112:020404. [PMID: 24483996 DOI: 10.1103/physrevlett.112.020404] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/21/2013] [Indexed: 06/03/2023]
Abstract
We find that the recently introduced model of self-trapping supported by a spatially growing strength of a repulsive nonlinearity gives rise to robust vortex-soliton tori, i.e., three-dimensional vortex solitons, with topological charges S≥1. The family with S=1 is completely stable, while the one with S=2 has alternating regions of stability and instability. The families are nearly exactly reproduced in an analytical form by the Thomas-Fermi approximation. Unstable states with S=2 and 3 split into persistently rotating pairs or triangles of unitary vortices. Application of a moderate torque to the vortex torus initiates a persistent precession mode, with the torus' axle moving along a conical surface. A strong torque heavily deforms the vortex solitons, but, nonetheless, they restore themselves with the axle oriented according to the vectorial addition of angular momenta.
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Affiliation(s)
- Rodislav Driben
- Department of Physics & CeOPP, University of Paderborn, Warburger Straße 100, Paderborn D-33098, Germany and Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
| | - Yaroslav V Kartashov
- ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya, Mediterranean Technology Park, E-08860 Castelldefels (Barcelona), Spain and Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow 142190, Russia
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
| | - Torsten Meier
- Department of Physics & CeOPP, University of Paderborn, Warburger Straße 100, Paderborn D-33098, Germany
| | - Lluis Torner
- ICFO-Institut de Ciencies Fotoniques, and Universitat Politecnica de Catalunya, Mediterranean Technology Park, E-08860 Castelldefels (Barcelona), Spain
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29
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Armaitis J, Duine RA, Stoof HTC. Quantum rotor model for a Bose-Einstein condensate of dipolar molecules. PHYSICAL REVIEW LETTERS 2013; 111:215301. [PMID: 24313496 DOI: 10.1103/physrevlett.111.215301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/2013] [Revised: 09/02/2013] [Indexed: 06/02/2023]
Abstract
We show that a Bose-Einstein condensate of heteronuclear molecules in the regime of small and static electric fields is described by a quantum rotor model for the macroscopic electric dipole moment of the molecular gas cloud. We solve this model exactly and find the symmetric, i.e., rotationally invariant, and dipolar phases expected from the single-molecule problem, but also an axial and planar nematic phase due to many-body effects. Investigation of the wave function of the macroscopic dipole moment also reveals squeezing of the probability distribution for the angular momentum of the molecules.
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Affiliation(s)
- J Armaitis
- Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands
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30
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Abstract
We demonstrate that in mesoscopic type II superconductors with the lateral size commensurate with London penetration depth, the ground state of vortices pinned by homogeneously distributed columnar defects can form a hierarchical nested domain structure. Each domain is characterized by an average number of vortices trapped at a single pinning site within a given domain. Our study marks a radical departure from the current understanding of the ground state in disordered macroscopic systems and provides an insight into the interplay between disorder, vortex-vortex interaction, and confinement within finite system size. The observed vortex phase segregation implies the existence of the soliton solution for the vortex density in the finite superconductors and establishes a new class of nonlinear systems that exhibit the soliton phenomenon.
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31
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Rem BS, Grier AT, Ferrier-Barbut I, Eismann U, Langen T, Navon N, Khaykovich L, Werner F, Petrov DS, Chevy F, Salomon C. Lifetime of the Bose gas with resonant interactions. PHYSICAL REVIEW LETTERS 2013; 110:163202. [PMID: 23679599 DOI: 10.1103/physrevlett.110.163202] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/20/2012] [Indexed: 06/02/2023]
Abstract
We study the lifetime of a Bose gas at and around unitarity using a Feshbach resonance in lithium 7. At unitarity, we measure the temperature dependence of the three-body decay coefficient L(3). Our data follow a L(3)=λ(3)/T(2) law with λ(3)=2.5(3)(stat)(6)(syst)×10(-20) (μK)(2) cm(6) s(-1) and are in good agreement with our analytical result based on zero-range theory. Varying the scattering length a at fixed temperature, we investigate the crossover between the finite-temperature unitary region and the previously studied regime where |a| is smaller than the thermal wavelength. We find that L(3) is continuous across the resonance, and over the whole a<0 range our data quantitatively agree with our calculation.
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Affiliation(s)
- B S Rem
- Laboratoire Kastler-Brossel, École Normale Supérieure, CNRS and UPMC, 24 rue Lhomond, 75005 Paris, France
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Mashayekhi MS, Bernier JS, Borzov D, Song JL, Zhou F. Two-dimensional Bose gases near resonance: competing two-body and three-body effects. PHYSICAL REVIEW LETTERS 2013; 110:145301. [PMID: 25167002 DOI: 10.1103/physrevlett.110.145301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2012] [Revised: 01/23/2013] [Indexed: 06/03/2023]
Abstract
We report in this Letter the results of our investigation of 2D Bose gases beyond the dilute limit emphasizing the role played by three-body scattering events. We demonstrate that a competition between three-body attractive interactions and two-body repulsive forces results in the chemical potential of 2D Bose gases to exhibit a maximum at a critical scattering length beyond which these quantum gases possess a negative compressibility. For larger scattering lengths, the increasingly prominent role played by three-body attractive interactions leads to an onset instability at a second critical value. The three-body effects studied here are universal, fully characterized by the effective 2D scattering length a(2D) (or the size of the 2D bound states) and are, in comparison to the 3D case, independent of three-body ultraviolet physics. We find, within our approach, the ratios of the contribution to the chemical potential due to three-body interactions to the one due to two-body to be 0.27 near the maximum of the chemical potential and 0.73 in the vicinity of the onset instability.
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Affiliation(s)
- Mohammad S Mashayekhi
- Department of Physics and Astronomy, University of British Columbia, Vancouver V6T 1Z1, Canada
| | - Jean-Sébastien Bernier
- Department of Physics and Astronomy, University of British Columbia, Vancouver V6T 1Z1, Canada
| | - Dmitry Borzov
- Department of Physics and Astronomy, University of British Columbia, Vancouver V6T 1Z1, Canada
| | - Jun-Liang Song
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, A-6020 Innsbruck, Austria
| | - Fei Zhou
- Department of Physics and Astronomy, University of British Columbia, Vancouver V6T 1Z1, Canada
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Affiliation(s)
- Lincoln D. Carr
- Department of Physics, Colorado School of Mines, Boulder, CO 80401, USA
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Abstract
We study the properties of strongly interacting Bose gases at the density and temperature regime when the three-body recombination rate is substantially reduced. In this regime, one can have a Bose gas with all particles in scattering states (i.e., the "upper branch") with little loss even at unitarity over the duration of the experiment. We show that because of bosonic enhancement, pair formation is shifted to the atomic side of the original resonance (where scattering length a(s)<0), opposite to the fermionic case. In a trap, a repulsive Bose gas remains mechanically stable when brought across resonance to the atomic side until it reaches a critical scattering length a(s)*<0. For a(s)<a(s)*, the density consists of a core of upper branch bosons surrounded by an outer layer of equilibrium branch. The conditions of low three-body recombination require that the particle number N<1.024(T/ω)(5/2) in a harmonic trap with frequency ω.
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Affiliation(s)
- Weiran Li
- Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
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Fialko O, Hallwood DW. Isolated quantum heat engine. PHYSICAL REVIEW LETTERS 2012; 108:085303. [PMID: 22463540 DOI: 10.1103/physrevlett.108.085303] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2011] [Indexed: 05/31/2023]
Abstract
We present a theoretical and numerical analysis of a quantum system that is capable of functioning as a heat engine. This system could be realized experimentally using cold bosonic atoms confined to a double well potential that is created by splitting a harmonic trap with a focused laser. The system shows thermalization, and can model a reversible heat engine cycle. This is the first demonstration of the operation of a heat engine with a finite quantum heat bath.
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Affiliation(s)
- O Fialko
- Centre for Theoretical Chemistry and Physics and NZIAS, Massey University, North Shore, Auckland 0745, New Zealand
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36
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Malomed BA, Kaup DJ, Van Gorder RA. Unstaggered-staggered solitons in two-component discrete nonlinear Schrödinger lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:026604. [PMID: 22463346 DOI: 10.1103/physreve.85.026604] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/18/2011] [Indexed: 05/31/2023]
Abstract
We present stable bright solitons built of coupled unstaggered and staggered components in a symmetric system of two discrete nonlinear Schrödinger equations with the attractive self-phase-modulation nonlinearity, coupled by the repulsive cross-phase-modulation interaction. These mixed modes are of a "symbiotic" type, as each component in isolation may only carry ordinary unstaggered solitons. The results are obtained in an analytical form, using the variational and Thomas-Fermi approximations (VA and TFA), and the generalized Vakhitov-Kolokolov (VK) criterion for the evaluation of the stability. The analytical predictions are verified against numerical results. Almost all the symbiotic solitons are predicted by the VA quite accurately and are stable. Close to a boundary of the existence region of the solitons (which may feature several connected branches), there are broad solitons which are not well approximated by the VA and are unstable.
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Affiliation(s)
- Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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37
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Cormack SC, Schumayer D, Hutchinson DAW. Rotons in interacting ultracold bose gases. PHYSICAL REVIEW LETTERS 2011; 107:140401. [PMID: 22107177 DOI: 10.1103/physrevlett.107.140401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2011] [Revised: 07/11/2011] [Indexed: 05/31/2023]
Abstract
In three dimensions, noninteracting bosons undergo Bose-Einstein condensation at a critical temperature, T(c), which is slightly shifted by ΔT(c), if the particles interact. We calculate the excitation spectrum of interacting Bose systems, (4)He and (87)Rb, and show that a roton minimum emerges in the spectrum above a threshold value of the gas parameter. We provide a general theoretical argument for why the roton minimum and the maximal upward critical temperature shift are related. We also suggest two experimental avenues to observe rotons in condensates. These results, based upon a path-integral Monte Carlo approach, provide a microscopic explanation of the shift in the critical temperature and also show that a roton minimum does emerge in the excitation spectrum of particles with a structureless, short-range, two-body interaction.
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Affiliation(s)
- Samuel C Cormack
- Jack Dodd Centre for Quantum Technology, Department of Physics, University of Otago, Dunedin, New Zealand
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38
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Navon N, Piatecki S, Günter K, Rem B, Nguyen TC, Chevy F, Krauth W, Salomon C. Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas. PHYSICAL REVIEW LETTERS 2011; 107:135301. [PMID: 22026867 DOI: 10.1103/physrevlett.107.135301] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2011] [Revised: 06/21/2011] [Indexed: 05/31/2023]
Abstract
We measure the zero-temperature equation of state of a homogeneous Bose gas of (7)Li atoms by analyzing the in situ density distributions of trapped samples. For increasing repulsive interactions our data show a clear departure from mean-field theory and provide a quantitative test of the many-body corrections first predicted in 1957 by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957).]. We further probe the dynamic response of the Bose gas to a varying interaction strength and compare it to simple theoretical models. We deduce a lower bound for the value of the universal constant ξ > 0.44(8) that would characterize the universal Bose gas at the unitary limit.
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Affiliation(s)
- Nir Navon
- Laboratoire Kastler Brossel, CNRS, UPMC, École Normale Supérieure, Paris, France.
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Gross N, Shotan Z, Kokkelmans S, Khaykovich L. Nuclear-spin-independent short-range three-body physics in ultracold atoms. PHYSICAL REVIEW LETTERS 2010; 105:103203. [PMID: 20867519 DOI: 10.1103/physrevlett.105.103203] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2010] [Revised: 05/31/2010] [Indexed: 05/29/2023]
Abstract
We investigate three-body recombination loss across a Feshbach resonance in a gas of ultracold 7Li atoms prepared in the absolute ground state and perform a comparison with previously reported results of a different nuclear-spin state [N. Gross, Phys. Rev. Lett. 103, 163202 (2009)]. We extend the previously reported universality in three-body recombination loss across a Feshbach resonance to the absolute ground state. We show that the positions and widths of recombination minima and Efimov resonances are identical for both states which indicates that the short-range physics is nuclear-spin independent.
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Affiliation(s)
- Noam Gross
- Department of Physics, Bar-Ilan University, Ramat-Gan, 52900 Israel
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Altin PA, Robins NP, Döring D, Debs JE, Poldy R, Figl C, Close JD. 85Rb tunable-interaction Bose-Einstein condensate machine. THE REVIEW OF SCIENTIFIC INSTRUMENTS 2010; 81:063103. [PMID: 20590221 DOI: 10.1063/1.3430538] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
We describe our experimental setup for creating stable Bose-Einstein condensates (BECs) of (85)Rb with tunable interparticle interactions. We use sympathetic cooling with (87)Rb in two stages, initially in a tight Ioffe-Pritchard magnetic trap and subsequently in a weak, large-volume, crossed optical dipole trap, using the 155 G Feshbach resonance to manipulate the elastic and inelastic scattering properties of the (85)Rb atoms. Typical (85)Rb condensates contain 4 x 10(4) atoms with a scattering length of a=+200a(0). Many aspects of the design presented here could be adapted to other dual-species BEC machines, including those involving degenerate Fermi-Bose mixtures. Our minimalist apparatus is well suited to experiments on dual-species and spinor Rb condensates, and has several simplifications over the (85)Rb BEC machine at JILA, which we discuss at the end of this article.
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Affiliation(s)
- P A Altin
- Australian Centre for Quantum Atom Optics, Australian National University, Australian Capital Territory 0200, Australia.
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Lahaye T, Pfau T, Santos L. Mesoscopic ensembles of polar bosons in triple-well potentials. PHYSICAL REVIEW LETTERS 2010; 104:170404. [PMID: 20482095 DOI: 10.1103/physrevlett.104.170404] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/27/2009] [Revised: 02/25/2010] [Indexed: 05/29/2023]
Abstract
Mesoscopic dipolar Bose gases in triple-well potentials offer a minimal system for the analysis of the nonlocal character of the dipolar interaction. We show that this nonlocal character may be clearly revealed by a variety of possible ground-state phases. In addition, an appropriate control of short-range and dipolar interactions may lead to novel scenarios for the dynamics of polar bosons in lattices, including the dynamical creation of mesoscopic quantum superpositions, which may be employed in the design of Heisenberg-limited atom interferometers.
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Affiliation(s)
- T Lahaye
- Université de Toulouse, UPS, Laboratoire Collisions Agrégats Réactivité, IRSAMC; F-31062 Toulouse, France
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42
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Pollack SE, Dries D, Hulet RG. Universality in Three- and Four-Body Bound States of Ultracold Atoms. Science 2009; 326:1683-5. [DOI: 10.1126/science.1182840] [Citation(s) in RCA: 270] [Impact Index Per Article: 18.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Affiliation(s)
- Scott E. Pollack
- Department of Physics and Astronomy and Rice Quantum Institute, Rice University, Houston, TX 77005, USA
| | - Daniel Dries
- Department of Physics and Astronomy and Rice Quantum Institute, Rice University, Houston, TX 77005, USA
| | - Randall G. Hulet
- Department of Physics and Astronomy and Rice Quantum Institute, Rice University, Houston, TX 77005, USA
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43
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Song JL, Zhou F. Ground state properties of cold bosonic atoms at large scattering lengths. PHYSICAL REVIEW LETTERS 2009; 103:025302. [PMID: 19659218 DOI: 10.1103/physrevlett.103.025302] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2009] [Indexed: 05/28/2023]
Abstract
In this Letter, we study bosonic atoms at large scattering lengths using a variational method where the condensate amplitude is a variational parameter. We further examine momentum distribution functions, chemical potentials, the speed of sound, and spatial density profiles of cold bosonic atoms in a trap in this limit. The latter two properties turn out to bear similarities to those of Fermi gases. The estimates obtained here are applicable near Feshbach resonances, particularly when the fraction of atoms forming three-body structures is small and can be tested in future cold atom experiments.
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Affiliation(s)
- Jun Liang Song
- Department of Physics and Astronomy, The University of British Columbia, Vancouver, B. C., Canada V6T1Z1
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