1
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Houwman JJA, Baillie D, Blakie PB, Natale G, Ferlaino F, Mark MJ. Measurement of the Excitation Spectrum of a Dipolar Gas in the Macrodroplet Regime. PHYSICAL REVIEW LETTERS 2024; 132:103401. [PMID: 38518353 DOI: 10.1103/physrevlett.132.103401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2023] [Revised: 11/27/2023] [Accepted: 01/08/2024] [Indexed: 03/24/2024]
Abstract
The excitation spectrum of a cigar-shaped strongly dipolar quantum gas at the crossover from a Bose-Einstein condensate to a trapped macrodroplet is predicted to exhibit peculiar features-a strong upward shift of low momentum excitation energies together with a strong multiband response for high momenta. By performing Bragg spectroscopy over a wide range of momenta, we observe both key elements and also confirm the predicted stiffening of excitation modes when approaching the macrodroplet regime. Our measurements are in good agreement with numerical calculations taking into account finite size effects.
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Affiliation(s)
- J J A Houwman
- Universität Innsbruck, Institut für Experimentalphysik, 6020 Innsbruck, Austria
| | - D Baillie
- Department of Physics, Centre for Quantum Science, and The Dodd-Walls Centre for Photonic and Quantum Technologies, University of Otago, Dunedin 9016, New Zealand
| | - P B Blakie
- Department of Physics, Centre for Quantum Science, and The Dodd-Walls Centre for Photonic and Quantum Technologies, University of Otago, Dunedin 9016, New Zealand
| | - G Natale
- Universität Innsbruck, Institut für Experimentalphysik, 6020 Innsbruck, Austria
- Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
| | - F Ferlaino
- Universität Innsbruck, Institut für Experimentalphysik, 6020 Innsbruck, Austria
- Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
| | - M J Mark
- Universität Innsbruck, Institut für Experimentalphysik, 6020 Innsbruck, Austria
- Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
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2
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Malomed BA. Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings. ENTROPY (BASEL, SWITZERLAND) 2024; 26:137. [PMID: 38392392 PMCID: PMC10887582 DOI: 10.3390/e26020137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2024] [Revised: 01/26/2024] [Accepted: 01/28/2024] [Indexed: 02/24/2024]
Abstract
This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.
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Affiliation(s)
- Boris A Malomed
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile
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3
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He L, Li H, Yi W, Yu ZQ. Quantum Criticality of Liquid-Gas Transition in a Binary Bose Mixture. PHYSICAL REVIEW LETTERS 2023; 130:193001. [PMID: 37243630 DOI: 10.1103/physrevlett.130.193001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2022] [Revised: 02/19/2023] [Accepted: 04/17/2023] [Indexed: 05/29/2023]
Abstract
Quantum liquid, in the form of a self-bound droplet, is stabilized by a subtle balance between the mean-field contribution and quantum fluctuations. While a liquid-gas transition is expected when such a balance is broken, it remains elusive whether liquid-gas critical points exist in the quantum regime. Here, we study the quantum criticality in a binary Bose mixture undergoing the liquid-gas transition. We show that, beyond a narrow stability window of the self-bound liquid, a liquid-gas coexistence persists, which eventually transits into a homogeneous mixture. Importantly, we identify two distinct critical points where the liquid-gas coexistence terminates. These critical points are characterized by rich critical behaviors in their vicinity, including divergent susceptibility, unique phonon-mode softening, and enhanced density correlations. The liquid-gas transition and the critical points can be readily explored in ultracold atoms confined to a box potential. Our work highlights the thermodynamic approach as a powerful tool in revealing the quantum liquid-gas criticality, and paves the way for further studies of critical phenomena in quantum liquids.
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Affiliation(s)
- Li He
- College of Physics and Electronic Engineering, Shanxi University, Taiyuan 030006, China
| | - Haowei Li
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
| | - Wei Yi
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, Hefei 230026, China
| | - Zeng-Qiang Yu
- Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
- State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China
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4
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Song J, Yan Z, Malomed BA. Formations and dynamics of two-dimensional spinning asymmetric quantum droplets controlled by a PT-symmetric potential. CHAOS (WOODBURY, N.Y.) 2023; 33:033141. [PMID: 37003809 DOI: 10.1063/5.0138420] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2022] [Accepted: 03/03/2023] [Indexed: 06/19/2023]
Abstract
In this paper, vortex solitons are produced for a variety of 2D spinning quantum droplets (QDs) in a PT-symmetric potential, modeled by the amended Gross-Pitaevskii equation with Lee-Huang-Yang corrections. In particular, exact QD states are obtained under certain parameter constraints, providing a guide to finding the respective generic family. In a parameter region of the unbroken PT symmetry, different families of QDs originating from the linear modes are obtained in the form of multipolar and vortex droplets at low and high values of the norm, respectively, and their stability is investigated. In the spinning regime, QDs become asymmetric above a critical rotation frequency, most of them being stable. The effect of the PT-symmetric potential on the spinning and nonspinning QDs is explored by varying the strength of the gain-loss distribution. Generally, spinning QDs trapped in the PT-symmetric potential exhibit asymmetry due to the energy flow affected by the interplay of the gain-loss distribution and rotation. Finally, interactions between spinning or nonspinning QDs are explored, exhibiting elastic collisions under certain conditions.
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Affiliation(s)
- Jin Song
- KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Zhenya Yan
- KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - 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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5
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Kopyciński J, Łebek M, Górecki W, Pawłowski K. Ultrawide Dark Solitons and Droplet-Soliton Coexistence in a Dipolar Bose Gas with Strong Contact Interactions. PHYSICAL REVIEW LETTERS 2023; 130:043401. [PMID: 36763437 DOI: 10.1103/physrevlett.130.043401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/14/2022] [Revised: 11/10/2022] [Accepted: 12/19/2022] [Indexed: 06/18/2023]
Abstract
We look into dark solitons in a quasi-1D dipolar Bose gas and in a quantum droplet. We derive the analytical solitonic solution of a Gross-Pitaevskii-like equation accounting for beyond mean-field effects. The results show there is a certain critical value of the dipolar interactions, for which the width of a motionless soliton diverges. Moreover, there is a peculiar solution of the motionless soliton with a nonzero density minimum. We also present the energy spectrum of these solitons with an additional excitation subbranch appearing. Finally, we perform a series of numerical experiments revealing the coexistence of a dark soliton inside a quantum droplet.
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Affiliation(s)
- Jakub Kopyciński
- Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
| | - Maciej Łebek
- Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
| | - Wojciech Górecki
- Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
| | - Krzysztof Pawłowski
- Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
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6
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Morera I, Ołdziejewski R, Astrakharchik GE, Juliá-Díaz B. Superexchange Liquefaction of Strongly Correlated Lattice Dipolar Bosons. PHYSICAL REVIEW LETTERS 2023; 130:023602. [PMID: 36706388 DOI: 10.1103/physrevlett.130.023602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2022] [Revised: 11/03/2022] [Accepted: 12/07/2022] [Indexed: 06/18/2023]
Abstract
We propose a mechanism for liquid formation in strongly correlated lattice systems. The mechanism is based on an interplay between long-range attraction and superexchange processes. As an example, we study dipolar bosons in one-dimensional optical lattices. We present a perturbative theory and validate it in comparison with full density-matrix renormalization group simulations for the energetic and structural properties of different phases of the system, i.e., self-bound Mott insulator, liquid, and gas. We analyze the nonequilibrium properties and calculate the dynamic structure factor. Its structure differs in compressible and insulating phases. In particular, the low-energy excitations in compressible phases are linear phonons. We extract the speed of sound and analyze its dependence on dipolar interaction and density. We show that it exhibits a nontrivial behavior owing to the breaking of Galilean invariance. We argue that an experimental detection of this previously unknown quantum liquid could provide a fingerprint of the superexchange process and open intriguing possibilities for investigating non-Galilean invariant liquids.
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Affiliation(s)
- Ivan Morera
- Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028 Barcelona, Spain
- Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martí i Franquès 1, E-08028 Barcelona, Spain
| | - Rafał Ołdziejewski
- Max Planck Institute of Quantum Optics, 85748 Garching, Germany
- Munich Center for Quantum Science and Technology, Schellingstrasse 4, 80799 Munich, Germany
| | - Grigori E Astrakharchik
- Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028 Barcelona, Spain
- Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martí i Franquès 1, E-08028 Barcelona, Spain
- Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
| | - Bruno Juliá-Díaz
- Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028 Barcelona, Spain
- Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martí i Franquès 1, E-08028 Barcelona, Spain
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7
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Al Sakkaf L, Al Khawaja U. Reflectionless potentials and resonant scattering of flat-top and thin-top solitons. Phys Rev E 2023; 107:014202. [PMID: 36797884 DOI: 10.1103/physreve.107.014202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2022] [Accepted: 11/16/2022] [Indexed: 01/05/2023]
Abstract
We identify a class of potentials for which the scattering of flat-top solitons and thin-top solitons of the nonlinear Schrödinger equation with dual nonlinearity can be reflectionless. The scattering is characterized by sharp resonances between regimes of full transmission and full quantum reflection. Perturbative expansion in terms of the magnitude of radiation losses leads to the general form of reflectionless potentials. Simulating the scattering of flat-top solitons and thin-top solitons confirms the reflectionless feature of these potentials.
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Affiliation(s)
- L Al Sakkaf
- Department of Physics, United Arab Emirates University, P.O. Box 15551, Al-Ain, United Arab Emirates
| | - U Al Khawaja
- Department of Physics, United Arab Emirates University, P.O. Box 15551, Al-Ain, United Arab Emirates
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8
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Chomaz L, Ferrier-Barbut I, Ferlaino F, Laburthe-Tolra B, Lev BL, Pfau T. Dipolar physics: a review of experiments with magnetic quantum gases. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2022; 86:026401. [PMID: 36583342 DOI: 10.1088/1361-6633/aca814] [Citation(s) in RCA: 20] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2022] [Accepted: 12/02/2022] [Indexed: 06/17/2023]
Abstract
Since the achievement of quantum degeneracy in gases of chromium atoms in 2004, the experimental investigation of ultracold gases made of highly magnetic atoms has blossomed. The field has yielded the observation of many unprecedented phenomena, in particular those in which long-range and anisotropic dipole-dipole interactions (DDIs) play a crucial role. In this review, we aim to present the aspects of the magnetic quantum-gas platform that make it unique for exploring ultracold and quantum physics as well as to give a thorough overview of experimental achievements. Highly magnetic atoms distinguish themselves by the fact that their electronic ground-state configuration possesses a large electronic total angular momentum. This results in a large magnetic moment and a rich electronic transition spectrum. Such transitions are useful for cooling, trapping, and manipulating these atoms. The complex atomic structure and large dipolar moments of these atoms also lead to a dense spectrum of resonances in their two-body scattering behaviour. These resonances can be used to control the interatomic interactions and, in particular, the relative importance of contact over dipolar interactions. These features provide exquisite control knobs for exploring the few- and many-body physics of dipolar quantum gases. The study of dipolar effects in magnetic quantum gases has covered various few-body phenomena that are based on elastic and inelastic anisotropic scattering. Various many-body effects have also been demonstrated. These affect both the shape, stability, dynamics, and excitations of fully polarised repulsive Bose or Fermi gases. Beyond the mean-field instability, strong dipolar interactions competing with slightly weaker contact interactions between magnetic bosons yield new quantum-stabilised states, among which are self-bound droplets, droplet assemblies, and supersolids. Dipolar interactions also deeply affect the physics of atomic gases with an internal degree of freedom as these interactions intrinsically couple spin and atomic motion. Finally, long-range dipolar interactions can stabilise strongly correlated excited states of 1D gases and also impact the physics of lattice-confined systems, both at the spin-polarised level (Hubbard models with off-site interactions) and at the spinful level (XYZ models). In the present manuscript, we aim to provide an extensive overview of the various related experimental achievements up to the present.
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Affiliation(s)
- Lauriane Chomaz
- Institut für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria
- Physikalisches Institut der Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
| | - Igor Ferrier-Barbut
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
- Université Paris-Saclay, Institut d'Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France
| | - Francesca Ferlaino
- Institut für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria
- Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, 6020 Innsbruck, Austria
| | - Bruno Laburthe-Tolra
- Université Sorbonne Paris Nord, Laboratoire de Physique des Lasers, F-93430 Villetaneuse, France
- CNRS, UMR 7538, LPL, F-93430 Villetaneuse, France
| | - Benjamin L Lev
- Departments of Physics and Applied Physics and Ginzton Laboratory, Stanford University, Stanford, CA 94305, United States of America
| | - Tilman Pfau
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
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9
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Pathak MR, Nath A. Droplet to soliton crossover at negative temperature in presence of bi-periodic optical lattices. Sci Rep 2022; 12:18248. [PMID: 36309597 PMCID: PMC9617923 DOI: 10.1038/s41598-022-23026-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2022] [Accepted: 10/24/2022] [Indexed: 11/21/2022] Open
Abstract
It is shown that the phenomenon of negative temperature essentially occurs in Bose-Einstein condensate due to the realization of the upper bound energy state utilizing a combination of expulsive harmonic oscillator and optical lattice potentials. We study the existence of quantum droplets at negative temperature and droplet-to-soliton crossover in the binary Bose-Einstein condensate mixture in the presence of bi-periodic optical lattices and expulsive-BOL confinements. Based on the beyond mean field approximation, we employ the extended Gross-Pitäevskii equation and calculate the exact analytical form of wavefunction solutions for BOL, expulsive-BOL confinements. An interesting transition of quantum droplets from positive to negative temperatures and the droplet-to-soliton crossover by modulating the disorder in BOL potential are illustrated. The affirmation of such crossover is performed by exploring the profile of atomic condensate density which smoothly transits from being a flat top density in optical lattice confinement to a bright soliton for BOL trap. Further, we confirm the crossover by exploring the energy per particle and the variation in the root mean square size of the condensate with respect to the potential depth of the BOL trap. Eventually, all of this aid us to construct a phase diagram in a space between the amplitude of BOL potential depth and particle number which reveals the formation of droplet and soliton phases. In expulsive-BOL confinement, it is seen that the impact of the expulsive trap is insignificant on atomic condensate density in the droplet phase and it becomes prominent in the soliton region. Further, the variation of total energy reveals that the amplitude of the expulsive oscillator strengthens the droplet phase and leads to an increase in the negative temperature of the considered system.
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Affiliation(s)
- Maitri R Pathak
- Indian Institute of Information Technology Vadodara Gujarat India, Gandhinagar, 382 028, India
| | - Ajay Nath
- Indian Institute of Information Technology Vadodara Gujarat India, Gandhinagar, 382 028, India.
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10
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Argüello-Luengo J, González-Tudela A, González-Cuadra D. Tuning Long-Range Fermion-Mediated Interactions in Cold-Atom Quantum Simulators. PHYSICAL REVIEW LETTERS 2022; 129:083401. [PMID: 36053702 DOI: 10.1103/physrevlett.129.083401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/08/2022] [Revised: 06/10/2022] [Accepted: 07/28/2022] [Indexed: 06/15/2023]
Abstract
Engineering long-range interactions in cold-atom quantum simulators can lead to exotic quantum many-body behavior. Fermionic atoms in ultracold atomic mixtures can act as mediators, giving rise to long-range Ruderman-Kittel-Kasuya-Yosida-type interactions characterized by the dimensionality and density of the fermionic gas. Here, we propose several tuning knobs, accessible in current experimental platforms, that allow one to further control the range and shape of the mediated interactions, extending the existing quantum simulation toolbox. In particular, we include an additional optical lattice for the fermionic mediator, as well as anisotropic traps to change its dimensionality in a continuous manner. This allows us to interpolate between power-law and exponential decays, introducing an effective cutoff for the interaction range, as well as to tune the relative interaction strengths at different distances. Finally, we show how our approach allows one to investigate frustrated regimes that were not previously accessible, where symmetry-protected topological phases as well as chiral spin liquids emerge.
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Affiliation(s)
- Javier Argüello-Luengo
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Avinguda Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
| | | | - Daniel González-Cuadra
- Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria
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11
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Luo Z, Liu Y, Li Y, Batle J, Malomed BA. Stability limits for modes held in alternating trapping-expulsive potentials. Phys Rev E 2022; 106:014201. [PMID: 35974589 DOI: 10.1103/physreve.106.014201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2022] [Accepted: 06/09/2022] [Indexed: 06/15/2023]
Abstract
We elaborate a scheme of trapping-expulsion management (TEM), in the form of the quadratic potential periodically switching between confinement and expulsion, as a means of stabilization of two-dimensional dynamical states against the backdrop of the critical collapse driven by the cubic self-attraction with strength g. The TEM scheme may be implemented, as spatially or temporally periodic modulations, in optics or BEC, respectively. The consideration is carried out by dint of numerical simulations and variational approximation (VA). In terms of the VA, the dynamics amounts to a nonlinear Ermakov equation, which, in turn, is tantamount to a linear Mathieu equation. Stability boundaries are found as functions of g and parameters of the periodic modulation of the trapping potential. Below the usual collapse threshold, which is known, in the numerical form, as g<g_{c}^{(num)}≈5.85 (in the standard notation), the stability is limited by the onset of the parametric resonance. This stability limit, including the setup with the self-repulsive sign of the cubic term (g<0), is accurately predicted by the VA. At g>g_{c}^{(num)}, the collapse threshold is found with the help of full numerical simulations. The relative increase of g_{c} above g_{c}^{(num)} is ≈1.5%. It is a meaningful result, even if its size is small, because the collapse threshold is a universal constant which is difficult to change.
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Affiliation(s)
- Zhihuan Luo
- Department of Applied Physics, South China Agricultural University, Guangzhou 510642, China
| | - Yan Liu
- Department of Applied Physics, South China Agricultural University, Guangzhou 510642, China
| | - Yongyao Li
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Josep Batle
- CRISP Centre de Recerca Independent de sa Pobla, C. Albéniz 12, 07420 sa Pobla, Balearic Islands, Spain
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
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12
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Ultradilute Quantum Droplets in the Presence of Higher-Order Quantum Fluctuations. ATOMS 2022. [DOI: 10.3390/atoms10020064] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
We investigate the effects of higher-order quantum fluctuations on the bulk properties of self-bound droplets in three-, two- and one-dimensional binary Bose mixtures using the Hartree–Fock–Bogoliubov theory. We calculate higher-order corrections to the equation of state of the droplet at both zero and finite temperatures. We show that our results for the ground-state energy are in a good agreement with recent quantum Monte Carlo simulations in any dimension. Our study extends to the finite temperature case where it is found that thermal fluctuations may destabilize the droplet state and eventually destroy it. In two dimensions, we reveal that the droplet occurs at temperatures well below the Berezinskii–Kosterlitz–Thouless transition temperature.
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13
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Dynamics of quantum droplets in an external harmonic confinement. Sci Rep 2022; 12:6904. [PMID: 35484174 PMCID: PMC9050709 DOI: 10.1038/s41598-022-10468-6] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/16/2021] [Accepted: 03/31/2022] [Indexed: 11/20/2022] Open
Abstract
Recent theoretical and experimental results show that one-dimensional (1D) weakly interacting atomic Bose-Bose mixtures with repulsive interspecies mean field (MF) interaction are stabilized by attractive quadratic beyond-mean-field (BMF) effects into self-bound quantum droplet (QD) in free space. Here, we construct an exact analytical model to investigate the structure and dynamics of QDs in presence of external harmonic confinement by solving the 1D extended Gross–Pitäevskii equation (eGPE) with temporal variation of MF and BMF interactions. The model provides the analytical form of wavefunction, phase, MF and BMF nonlinearities. The generation of QDs and interesting droplet to soliton transition in presence of regular/expulsive parabolic traps by taking the comparable MF and BMF interactions are illustrated. We derive the phase diagram of the droplet-soliton phase transition between amplitude of MF, BMF interactions and harmonic oscillator frequency. The strength and form of oscillator frequency are identified as key parameter for tuning the compression, fragmentation and transport of droplets. Finally, the stability of the obtained solutions are confirmed from Vakhitov–Kolokolov (VK) criterion and are found stable.
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14
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Hammond A, Lavoine L, Bourdel T. Tunable Three-Body Interactions in Driven Two-Component Bose-Einstein Condensates. PHYSICAL REVIEW LETTERS 2022; 128:083401. [PMID: 35275683 DOI: 10.1103/physrevlett.128.083401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2021] [Revised: 01/21/2022] [Accepted: 02/07/2022] [Indexed: 06/14/2023]
Abstract
We propose and demonstrate the appearance of an effective attractive three-body interaction in coherently driven two-component Bose-Einstein condensates. It originates from the spinor degree of freedom that is affected by a two-body mean-field shift of the driven transition frequency. Importantly, its strength can be controlled with the Rabi-coupling strength and it does not come with additional losses. In the experiment, the three-body interactions are adjusted to play a predominant role in the equation of state of a cigar-shaped trapped condensate. This is confirmed through two striking observations: a downshift of the radial breathing mode frequency and the radial collapses for positive values of the dressed-state scattering length.
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Affiliation(s)
- A Hammond
- Université Paris-Saclay, Institut d'Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France
| | - L Lavoine
- Université Paris-Saclay, Institut d'Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France
| | - T Bourdel
- Université Paris-Saclay, Institut d'Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France
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15
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Sanz J, Frölian A, Chisholm CS, Cabrera CR, Tarruell L. Interaction Control and Bright Solitons in Coherently Coupled Bose-Einstein Condensates. PHYSICAL REVIEW LETTERS 2022; 128:013201. [PMID: 35061464 DOI: 10.1103/physrevlett.128.013201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2019] [Revised: 06/27/2021] [Accepted: 11/15/2021] [Indexed: 06/14/2023]
Abstract
We demonstrate fast control of the interatomic interactions in a Bose-Einstein condensate by coherently coupling two atomic states with intra- and interstate scattering lengths of opposite signs. We measure the elastic and inelastic scattering properties of the system and find good agreement with a theoretical model describing the interactions between dressed states. In the attractive regime, we observe the formation of bright solitons formed by dressed-state atoms. Finally, we study the response of the system to an interaction quench from repulsive to attractive values, and observe how the resulting modulational instability develops into a bright soliton train.
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Affiliation(s)
- J Sanz
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
| | - A Frölian
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
| | - C S Chisholm
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
| | - C R Cabrera
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
| | - L Tarruell
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
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16
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Lavoine L, Hammond A, Recati A, Petrov DS, Bourdel T. Beyond-Mean-Field Effects in Rabi-Coupled Two-Component Bose-Einstein Condensate. PHYSICAL REVIEW LETTERS 2021; 127:203402. [PMID: 34860048 DOI: 10.1103/physrevlett.127.203402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2021] [Revised: 08/09/2021] [Accepted: 10/15/2021] [Indexed: 06/13/2023]
Abstract
We theoretically calculate and experimentally measure the beyond-mean-field (BMF) equation of state in a coherently coupled two-component Bose-Einstein condensate (BEC) in the regime where averaging of the interspecies and intraspecies coupling constants over the hyperfine composition of the single-particle dressed state predicts the exact cancellation of the two-body interaction. We show that with increasing the Rabi-coupling frequency Ω, the BMF energy density crosses over from the nonanalytic Lee-Huang-Yang scaling ∝n^{5/2} to an expansion in integer powers of density, where, in addition to a two-body BMF term ∝n^{2}sqrt[Ω], there emerges a repulsive three-body contribution ∝n^{3}/sqrt[Ω]. We experimentally evidence these two contributions, thanks to their different scaling with Ω, in the expansion of a Rabi-coupled two-component ^{39}K condensate in a waveguide. By studying the expansion with and without Rabi coupling, we reveal an important feature relevant for observing BMF effects and associated phenomena in mixtures with spin-asymmetric losses: Rabi coupling helps preserve the spin composition and thus prevents the system from drifting away from the point of the vanishing mean field.
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Affiliation(s)
- L Lavoine
- Laboratoire Charles Fabry, UMR 8501, Institut d'Optique, CNRS, Université Paris-Saclay, Avenue Augustin Fresnel, 91127 Palaiseau CEDEX, France
| | - A Hammond
- Laboratoire Charles Fabry, UMR 8501, Institut d'Optique, CNRS, Université Paris-Saclay, Avenue Augustin Fresnel, 91127 Palaiseau CEDEX, France
| | - A Recati
- INO-CNR BEC Center and Dipartimento di Fisica, Università degli Studi di Trento, 38123 Povo, Italy and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Trento, Italy
| | - D S Petrov
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| | - T Bourdel
- Laboratoire Charles Fabry, UMR 8501, Institut d'Optique, CNRS, Université Paris-Saclay, Avenue Augustin Fresnel, 91127 Palaiseau CEDEX, France
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17
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Many-body and temperature effects in two-dimensional quantum droplets in Bose-Bose mixtures. Sci Rep 2021; 11:21765. [PMID: 34741072 PMCID: PMC8571397 DOI: 10.1038/s41598-021-01089-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/23/2021] [Accepted: 10/15/2021] [Indexed: 11/25/2022] Open
Abstract
We study the equilibrium properties of self-bound droplets in two-dimensional Bose mixtures employing the time-dependent Hartree–Fock–Bogoliubov theory. This theory allows one to understand both the many-body and temperature effects beyond the Lee–Huang–Yang description. We calculate higher-order corrections to the excitations, the sound velocity, and the energy of the droplet. Our results for the ground-state energy are compared with the diffusion Monte Carlo data and good agreement is found. The behavior of the depletion and anomalous density of the droplet is also discussed. At finite temperature, we show that the droplet emerges at temperatures well below the Berezinskii–Kosterlitz–Thouless transition temperature. The critical temperature strongly depends on the interspecies interactions. Our study is extended to the finite size droplet by numerically solving the generalized finite-temperature Gross-Pitaevskii equation which is obtained self-consistently from our formalism in the framework of the local density approximation.
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18
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Ma Y, Peng C, Cui X. Borromean Droplet in Three-Component Ultracold Bose Gases. PHYSICAL REVIEW LETTERS 2021; 127:043002. [PMID: 34355973 DOI: 10.1103/physrevlett.127.043002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2021] [Revised: 04/21/2021] [Accepted: 06/25/2021] [Indexed: 06/13/2023]
Abstract
We investigate droplet formation in three-component ultracold bosons. In particular, we identify the formation of a Borromean droplet, where only the ternary bosons can form a self-bound droplet while any binary subsystems cannot, as the first example of Borromean binding due to a collective many-body effect. Its formation is facilitated by an additional attractive force induced by the density fluctuation of a third component, which enlarges the mean-field collapse region in comparison to the binary case and renders the formation of a Borromean droplet after incorporating the repulsive force from quantum fluctuations. Outside the Borromean regime, we demonstrate an interesting phenomenon of droplet phase separation due to the competition between ternary and binary droplets. We further show that the transition between different droplets and gas phase can be conveniently tuned by boson numbers and interaction strengths. The study reveals the rich physics of a quantum droplet in three-component boson mixtures and sheds light on the more intriguing many-body bound state formed in multicomponent systems.
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Affiliation(s)
- Yinfeng Ma
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Cheng Peng
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Xiaoling Cui
- Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
- Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China
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19
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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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20
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Bakkali-Hassani B, Maury C, Zou YQ, Le Cerf É, Saint-Jalm R, Castilho PCM, Nascimbene S, Dalibard J, Beugnon J. Realization of a Townes Soliton in a Two-Component Planar Bose Gas. PHYSICAL REVIEW LETTERS 2021; 127:023603. [PMID: 34296923 DOI: 10.1103/physrevlett.127.023603] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2021] [Revised: 05/07/2021] [Accepted: 05/24/2021] [Indexed: 06/13/2023]
Abstract
Most experimental observations of solitons are limited to one-dimensional (1D) situations, where they are naturally stable. For instance, in 1D cold Bose gases, they exist for any attractive interaction strength g and particle number N. By contrast, in two dimensions, solitons appear only for discrete values of gN, the so-called Townes soliton being the most celebrated example. Here, we use a two-component Bose gas to prepare deterministically such a soliton: Starting from a uniform bath of atoms in a given internal state, we imprint the soliton wave function using an optical transfer to another state. We explore various interaction strengths, atom numbers, and sizes and confirm the existence of a solitonic behavior for a specific value of gN and arbitrary sizes, a hallmark of scale invariance.
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Affiliation(s)
- 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
| | - Y-Q Zou
- 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
| | - R Saint-Jalm
- Department of Physics, Ludwig-Maximilians-Universität München, Schellingstrasse 4, D-80799 München, Germany
| | - P C M Castilho
- Instituto de Física de São Carlos, Universidade de São Paulo, CP 369, 13560-970 São Carlos, Brazil
| | - 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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21
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Dong L, Kartashov YV. Rotating Multidimensional Quantum Droplets. PHYSICAL REVIEW LETTERS 2021; 126:244101. [PMID: 34213917 DOI: 10.1103/physrevlett.126.244101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2021] [Revised: 04/05/2021] [Accepted: 05/10/2021] [Indexed: 06/13/2023]
Abstract
We predict a new type of two- and three-dimensional stable quantum droplets persistently rotating in broad external two-dimensional and weakly anharmonic potential. Their evolution is described by the system of the Gross-Pitaevskii equations with Lee-Huang-Yang quantum corrections. Such droplets resemble whispering-gallery modes localized in the polar direction due to nonlinear interactions and, depending on their chemical potential and rotation frequency, they appear in rich variety of shapes, ranging from nearly flat-top or strongly localized rotating wave packets, to crescentlike objects extending nearly over the entire range of polar angles. Above critical rotation frequency quantum droplets transform into vortex droplets (in two dimensions) or vortex tori (in three dimensions), whose topological charge gradually increase with the increase of the modulus of chemical potential, and therefore they belong to the family of nonlinear modes connecting fundamental and vortex quantum droplets. Rotating quantum droplets are exceptionally robust objects, stable practically in the entire range of their existence.
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Affiliation(s)
- Liangwei Dong
- Department of Physics, Shaanxi University of Science & Technology, Xi'an 710021, China
| | - Yaroslav V Kartashov
- Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 108840, Russia
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22
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Skov TG, Skou MG, Jørgensen NB, Arlt JJ. Observation of a Lee-Huang-Yang Fluid. PHYSICAL REVIEW LETTERS 2021; 126:230404. [PMID: 34170163 DOI: 10.1103/physrevlett.126.230404] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2020] [Revised: 04/19/2021] [Accepted: 05/18/2021] [Indexed: 06/13/2023]
Abstract
We observe monopole oscillations in a mixture of Bose-Einstein condensates, where the usually dominant mean-field interactions are canceled. In this case, the system is governed by the next-order Lee-Huang-Yang (LHY) correction to the ground state energy, which describes the effect of quantum fluctuations. Experimentally such a LHY fluid is realized by controlling the atom numbers and interaction strengths in a ^{39}K spin mixture confined in a spherical trap potential. We measure the monopole oscillation frequency as a function of the LHY interaction strength as proposed recently by Jrgensen et al. [Phys. Rev. Lett. 121, 173403 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.173403] and find excellent agreement with simulations of the complete experiment including the excitation procedure and inelastic losses. This confirms that the system and its collective behavior are initially dominated by LHY interactions. Moreover, the monopole oscillation frequency is found to be stable against variations of the involved scattering lengths in a broad region around the ideal values, confirming the stabilizing effect of the LHY interaction. These results pave the way for using the nonlinearity provided by the LHY term in quantum simulation experiments and for investigations beyond the LHY regime.
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Affiliation(s)
- Thomas G Skov
- Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
| | - Magnus G Skou
- Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
| | - Nils B Jørgensen
- Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
| | - Jan J Arlt
- Center for Complex Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
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23
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Naidon P, Petrov DS. Mixed Bubbles in Bose-Bose Mixtures. PHYSICAL REVIEW LETTERS 2021; 126:115301. [PMID: 33798344 DOI: 10.1103/physrevlett.126.115301] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/22/2020] [Revised: 11/17/2020] [Accepted: 01/14/2021] [Indexed: 06/12/2023]
Abstract
Repulsive Bose-Bose mixtures are known to either mix or phase separate into pure components. Here we predict a mixed-bubble regime in which bubbles of the mixed phase coexist with a pure phase of one of the components. This is a beyond-mean-field effect that occurs for unequal masses or unequal intraspecies coupling constants and is due to a competition between the mean-field term, quadratic in densities, and a nonquadratic beyond-mean-field correction. We find parameters of the mixed-bubble regime in all dimensions and discuss implications for current experiments.
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Affiliation(s)
- P Naidon
- Strangeness Nuclear Physics Laboratory, RIKEN Nishina Centre, Wakō 351-0198, Japan
| | - D S Petrov
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
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24
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Self-Evaporation Dynamics of Quantum Droplets in a 41K-87Rb Mixture. APPLIED SCIENCES-BASEL 2021. [DOI: 10.3390/app11020866] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We theoretically investigate the self-evaporation dynamics of quantum droplets in a 41K-87Rb mixture, in free-space. The dynamical formation of the droplet and the effects related to the presence of three-body losses are analyzed by means of numerical simulations. We identify a regime of parameters allowing for the observation of the droplet self-evaporation in a feasible experimental setup.
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25
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Morera I, Astrakharchik GE, Polls A, Juliá-Díaz B. Universal Dimerized Quantum Droplets in a One-Dimensional Lattice. PHYSICAL REVIEW LETTERS 2021; 126:023001. [PMID: 33512190 DOI: 10.1103/physrevlett.126.023001] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/12/2020] [Accepted: 12/14/2020] [Indexed: 06/12/2023]
Abstract
The ground-state properties of two-component bosonic mixtures in a one-dimensional optical lattice are studied both from few- and many-body perspectives. We rely directly on a microscopic Hamiltonian with attractive intercomponent and repulsive intracomponent interactions to demonstrate the formation of a quantum liquid. We reveal that its formation and stability can be interpreted in terms of finite-range interactions between dimers. We derive an effective model of composite bosons (dimers) which correctly captures both the few- and many-body properties and validate it against exact results obtained by the density matrix renormalization group method for the full Hamiltonian. The threshold for the formation of the liquid coincides with the appearance of a bound state in the dimer-dimer problem and possesses a universality in terms of the two-body parameters of the dimer-dimer interaction, namely, scattering length and effective range. For sufficiently strong effective dimer-dimer repulsion we observe fermionization of the dimers which form an effective Tonks-Girardeau state and identify conditions for the formation of a solitonic solution. Our predictions are relevant to experiments with dipolar atoms and two-component mixtures.
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Affiliation(s)
- Ivan Morera
- Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028 Barcelona, Spain
- Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martí i Franquès 1, Barcelona 08028, Spain
| | - Grigori E Astrakharchik
- Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
| | - Artur Polls
- Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028 Barcelona, Spain
- Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martí i Franquès 1, Barcelona 08028, Spain
| | - Bruno Juliá-Díaz
- Departament de Física Quàntica i Astrofísica, Facultat de Física, Universitat de Barcelona, E-08028 Barcelona, Spain
- Institut de Ciències del Cosmos, Universitat de Barcelona, ICCUB, Martí i Franquès 1, Barcelona 08028, Spain
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26
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Böttcher F, Schmidt JN, Hertkorn J, Ng KSH, Graham SD, Guo M, Langen T, Pfau T. New states of matter with fine-tuned interactions: quantum droplets and dipolar supersolids. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2021; 84:012403. [PMID: 33176284 DOI: 10.1088/1361-6633/abc9ab] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
Quantum fluctuations can stabilize Bose-Einstein condensates (BEC) against the mean-field collapse. Stabilization of the condensate has been observed in quantum degenerate Bose-Bose mixtures and dipolar BECs. The fine-tuning of the interatomic interactions can lead to the emergence of two new states of matter: liquid-like self-bound quantum droplets and supersolid crystals formed from these droplets. We review the properties of these exotic states of matter and summarize the experimental progress made using dipolar quantum gases and Bose-Bose mixtures. We conclude with an outline of important open questions that could be addressed in the future.
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Affiliation(s)
- Fabian Böttcher
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Jan-Niklas Schmidt
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Jens Hertkorn
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Kevin S H Ng
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Sean D Graham
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Mingyang Guo
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Tim Langen
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
| | - Tilman Pfau
- Physikalisches Institut and Center for Integrated Quantum Science and Technology, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
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27
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Hu H, Liu XJ. Consistent Theory of Self-Bound Quantum Droplets with Bosonic Pairing. PHYSICAL REVIEW LETTERS 2020; 125:195302. [PMID: 33216582 DOI: 10.1103/physrevlett.125.195302] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2020] [Accepted: 10/06/2020] [Indexed: 06/11/2023]
Abstract
We revisit the Bogoliubov theory of quantum droplets proposed by Petrov [Phys. Rev. Lett. 115, 155302 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.155302] for an ultracold Bose-Bose mixture, where the mean-field collapse is stabilized by the Lee-Huang-Yang quantum fluctuations. We show that a loophole in Petrov's theory, i.e., the ignorance of the softening complex Bogoliubov spectrum, can be naturally removed by the introduction of bosonic pairing. The pairing leads to weaker mean-field attractions, and also a stronger Lee-Huang-Yang term in the case of unequal intraspecies interactions. As a result, the equilibrium density for the formation of self-bound droplets significantly decreases in the deep droplet regime, in agreement with a recent observation from diffusion Monte Carlo simulations. Our construction of a consistent Bogoliubov theory paves the way to understand the puzzling low critical number of small quantum droplets observed in the experiment [C. Cabrera et al., Science 359, 301 (2018)SCIEAS0036-807510.1126/science.aao5686].
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Affiliation(s)
- Hui Hu
- Centre for Quantum Technology Theory, Swinburne University of Technology, Melbourne, Victoria 3122, Australia
| | - Xia-Ji Liu
- Centre for Quantum Technology Theory, Swinburne University of Technology, Melbourne, Victoria 3122, Australia
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28
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He L, Gao P, Yu ZQ. Normal-Superfluid Phase Separation in Spin-Half Bosons at Finite Temperature. PHYSICAL REVIEW LETTERS 2020; 125:055301. [PMID: 32794831 DOI: 10.1103/physrevlett.125.055301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2019] [Accepted: 07/10/2020] [Indexed: 06/11/2023]
Abstract
For pseudospin-half bosons with interspin attraction and intraspin repulsion, the normal phase and Bose condensed phase can coexist at finite temperature. The homogeneous system is unstable against the spinodal decomposition within a medium density interval, and, consequently, a normal-superfluid phase separation takes place. The isothermal equation of state shows a characteristic plateau in the P-V (pressure-volume) diagram, which is reminiscent of a classical gas-liquid transition, although, unlike the latter, the coexistence lines never terminate at a critical point as temperature increases. In a harmonic trap, the phase separation can be revealed by the density profile of the atomic cloud, which exhibits a sudden jump across the phase boundary.
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Affiliation(s)
- Li He
- College of Physics and Electronic Engineering, Shanxi University, Taiyuan 030006, China
| | - Peipei Gao
- Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
- State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China
| | - Zeng-Qiang Yu
- Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
- State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China
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29
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Marchukov OV, Malomed BA, Dunjko V, Ruhl J, Olshanii M, Hulet RG, Yurovsky VA. Quantum Fluctuations of the Center of Mass and Relative Parameters of Nonlinear Schrödinger Breathers. PHYSICAL REVIEW LETTERS 2020; 125:050405. [PMID: 32794875 DOI: 10.1103/physrevlett.125.050405] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2019] [Revised: 03/06/2020] [Accepted: 07/02/2020] [Indexed: 06/11/2023]
Abstract
We study quantum fluctuations of macroscopic parameters of a nonlinear Schrödinger breather-a nonlinear superposition of two solitons, which can be created by the application of a fourfold quench of the scattering length to the fundamental soliton in a self-attractive quasi-one-dimensional Bose gas. The fluctuations are analyzed in the framework of the Bogoliubov approach in the limit of a large number of atoms N, using two models of the vacuum state: white noise and correlated noise. The latter model, closer to the ab initio setting by construction, leads to a reasonable agreement, within 20% accuracy, with fluctuations of the relative velocity of constituent solitons obtained from the exact Bethe-ansatz results [Phys. Rev. Lett. 119, 220401 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.220401] in the opposite low-N limit (for N≤23). We thus confirm, for macroscopic N, the breather dissociation time to be within the limits of current cold-atom experiments. Fluctuations of soliton masses, phases, and positions are also evaluated and may have experimental implications.
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Affiliation(s)
- Oleksandr V Marchukov
- Institute for Applied Physics, Technical University of Darmstadt, 64289 Darmstadt, Germany
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, 6997801 Tel Aviv, Israel
| | - Boris 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
| | - Vanja Dunjko
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - Joanna Ruhl
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - Maxim Olshanii
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - Randall G Hulet
- Department of Physics and Astronomy, Rice University, Houston,Texas 77005, USA
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30
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Static and Dynamic Properties of a Few Spin 1/2 Interacting Fermions Trapped in a Harmonic Potential. MATHEMATICS 2020. [DOI: 10.3390/math8071196] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We provide a detailed study of the properties of a few interacting spin 1 / 2 fermions trapped in a one-dimensional harmonic oscillator potential. The interaction is assumed to be well represented by a contact delta potential. Numerical results obtained by means of direct diagonalization techniques are combined with analytical expressions for both the non-interacting and strongly interacting regime. The N = 2 case is used to benchmark our numerical techniques with the known exact solution of the problem. After a detailed description of the numerical methods, in a tutorial-like manner, we present the static properties of the system for N = 2 , 3 , 4 and 5 particles, e.g., low-energy spectrum, one-body density matrix, ground-state densities. Then, we consider dynamical properties of the system exploring first the excitation of the breathing mode, using the dynamical structure function and corresponding sum-rules, and then a sudden quench of the interaction strength.
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31
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Mestrom PMA, Colussi VE, Secker T, Groeneveld GP, Kokkelmans SJJMF. van der Waals Universality near a Quantum Tricritical Point. PHYSICAL REVIEW LETTERS 2020; 124:143401. [PMID: 32338969 DOI: 10.1103/physrevlett.124.143401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2019] [Accepted: 03/10/2020] [Indexed: 06/11/2023]
Abstract
We study the three-body scattering hypervolume D of atoms whose scattering length a is on the order of or smaller than the typical range r_{vdW} of the van der Waals attraction. We find that the real part of D behaves universally in this weakly interacting regime (|a|/r_{vdW}≲1) in the absence of trimer resonances. This universality originates from hard-spherelike collisions that dominate elastic three-body scattering. We use this result to make quantitative predictions for the thermodynamics and elementary excitations of an atomic Bose-Einstein condensate in the vicinity of a quantum tricritical point, including quantum droplets stabilized by effective three-body interactions.
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Affiliation(s)
- P M A Mestrom
- Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, Netherlands
| | - V E Colussi
- Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, Netherlands
| | - T Secker
- Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, Netherlands
| | - G P Groeneveld
- Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, Netherlands
| | - S J J M F Kokkelmans
- Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, Netherlands
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32
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A Dual-Species Bose-Einstein Condensate with Attractive Interspecies Interactions. CONDENSED MATTER 2020. [DOI: 10.3390/condmat5010021] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
We report on the production of a 41 K- 87 Rb dual-species Bose–Einstein condensate with tunable interspecies interaction and we study the mixture in the attractive regime; i.e., for negative values of the interspecies scattering length a 12 . The binary condensate is prepared in the ground state and confined in a pure optical trap. We exploit Feshbach resonances for tuning the value of a 12 . After compensating the gravitational sag between the two species with a magnetic field gradient, we drive the mixture into the attractive regime. We let the system evolve both in free space and in an optical waveguide. In both geometries, for strong attractive interactions, we observe the formation of self-bound states, recognizable as quantum droplets. Our findings prove that robust, long-lived droplet states can be realized in attractive two-species mixtures, despite the two atomic components possibly experiencing different potentials.
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33
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Abstract
This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center; the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross–Pitaevskii equation (GPE), which combines the attractive potential ∼ r − 2 and the quartic self-repulsive nonlinearity, induced by the Lee–Huang–Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity ∼ r − 4 / 3 at r → 0 . Modes with embedded angular momentum exist too, but they are unstable. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity ∼ r − 2 / ( 4 − D ) . Such states may be considered the results of screening a “bare” delta-functional attractive potential by the respective nonlinearities.
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34
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Ołdziejewski R, Górecki W, Pawłowski K, Rzążewski K. Strongly Correlated Quantum Droplets in Quasi-1D Dipolar Bose Gas. PHYSICAL REVIEW LETTERS 2020; 124:090401. [PMID: 32202868 DOI: 10.1103/physrevlett.124.090401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2019] [Accepted: 01/27/2020] [Indexed: 06/10/2023]
Abstract
We exploit a few- to many-body approach to study strongly interacting dipolar bosons in the quasi-one-dimensional system. The dipoles attract each other while the short range interactions are repulsive. Solving numerically the multiatom Schrödinger equation, we discover that such systems can exhibit not only the well-known bright soliton solutions but also novel quantum droplets for a strongly coupled case. For larger systems, basing on microscopic properties of the found few-body solution, we propose a new equation for a density amplitude of atoms. It accounts for fermionization for strongly repelling bosons by incorporating the Lieb-Liniger energy in a local density approximation and approaches the standard Gross-Pitaevskii equation (GPE) in the weakly interacting limit. Not only does such a framework provide an alternative mechanism of the droplet stability, but it also introduces means to further analyze this previously unexplored quantum phase. In the limiting strong repulsion case, yet another simple multiatom model is proposed. We stress that the celebrated Lee-Huang-Yang term in the GPE is not applicable in this case.
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Affiliation(s)
- Rafał Ołdziejewski
- Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
| | - Wojciech Górecki
- Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
| | - Krzysztof Pawłowski
- Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
| | - Kazimierz Rzążewski
- Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
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35
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Abstract
We demonstrate that the commonly known concept which treats solitons as nonsingular solutions produced by the interplay of nonlinear self-attraction and linear dispersion may be extended to include modes with a relatively weak singularity at the central point, which keeps their integral norm convergent. Such states are generated by self-repulsion, which should be strong enough, represented by septimal, quintic, and usual cubic terms in the framework of the one-, two-, and three-dimensional (1D, 2D, and 3D) nonlinear Schrödinger equations (NLSEs), respectively. Although such solutions seem counterintuitive, we demonstrate that they admit a straightforward interpretation as a result of screening of an additionally introduced attractive δ-functional potential by the defocusing nonlinearity. The strength ("bare charge") of the attractive potential is infinite in 1D, finite in 2D, and vanishingly small in 3D. Analytical asymptotics of the singular solitons at small and large distances are found, entire shapes of the solitons being produced in a numerical form. Complete stability of the singular modes is accurately predicted by the anti-Vakhitov-Kolokolov criterion (under the assumption that it applies to the model), as verified by means of numerical methods. In 2D, the NLSE with a quintic self-focusing term admits singular-soliton solutions with intrinsic vorticity too, but they are fully unstable. We also mention that dissipative singular solitons can be produced by the model with a complex coefficient in front of the nonlinear term.
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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
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
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36
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Roccuzzo SM, Gallemí A, Recati A, Stringari S. Rotating a Supersolid Dipolar Gas. PHYSICAL REVIEW LETTERS 2020; 124:045702. [PMID: 32058751 DOI: 10.1103/physrevlett.124.045702] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2019] [Indexed: 06/10/2023]
Abstract
Distinctive features of supersolids show up in their rotational properties. We calculate the moment of inertia of a harmonically trapped dipolar Bose-Einstein condensed gas as a function of the tunable scattering length parameter, providing the transition from the (fully) superfluid to the supersolid phase and eventually to an incoherent crystal of self-bound droplets. The transition from the superfluid to the supersolid phase is characterized by a jump in the moment of inertia, revealing its first order nature. In the case of elongated trapping in the plane of rotation, we show that the moment of inertia determines the value of the frequency of the scissors mode, which is significantly affected by the reduction of superfluidity in the supersolid phase. The case of an in-plane isotropic trapping is instead well suited to study the formation of quantized vortices, which are shown to be characterized, in the supersolid phase, by a sizeable deformed core, caused by the presence of the surrounding density peaks.
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Affiliation(s)
- S M Roccuzzo
- INO-CNR BEC Center and Dipartimento di Fisica, Università degli Studi di Trento, 38123 Povo, Italy and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Trento, Italy
| | - A Gallemí
- INO-CNR BEC Center and Dipartimento di Fisica, Università degli Studi di Trento, 38123 Povo, Italy and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Trento, Italy
| | - A Recati
- INO-CNR BEC Center and Dipartimento di Fisica, Università degli Studi di Trento, 38123 Povo, Italy and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Trento, Italy
| | - S Stringari
- INO-CNR BEC Center and Dipartimento di Fisica, Università degli Studi di Trento, 38123 Povo, Italy and Trento Institute for Fundamental Physics and Applications, INFN, 38123 Trento, Italy
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37
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Modulational Instability, Inter-Component Asymmetry, and Formation of Quantum Droplets in One-Dimensional Binary Bose Gases. Symmetry (Basel) 2020. [DOI: 10.3390/sym12010174] [Citation(s) in RCA: 29] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
Quantum droplets are ultradilute liquid states that emerge from the competitive interplay of two Hamiltonian terms, the mean-field energy and beyond-mean-field correction, in a weakly interacting binary Bose gas. We relate the formation of droplets in symmetric and asymmetric two-component one-dimensional boson systems to the modulational instability of a spatially uniform state driven by the beyond-mean-field term. Asymmetry between the components may be caused by their unequal populations or unequal intra-component interaction strengths. Stability of both symmetric and asymmetric droplets is investigated. Robustness of the symmetric solutions against symmetry-breaking perturbations is confirmed.
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38
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Sowiński T, Ángel García-March M. One-dimensional mixtures of several ultracold atoms: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2019; 82:104401. [PMID: 31404916 DOI: 10.1088/1361-6633/ab3a80] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Recent theoretical and experimental progress on studying one-dimensional systems of bosonic, fermionic, and Bose-Fermi mixtures of a few ultracold atoms confined in traps is reviewed in the broad context of mesoscopic quantum physics. We pay special attention to limiting cases of very strong or very weak interactions and transitions between them. For bosonic mixtures, we describe the developments in systems of three and four atoms as well as different extensions to larger numbers of particles. We also briefly review progress in the case of spinor Bose gases of a few atoms. For fermionic mixtures, we discuss a special role of spin and present a detailed discussion of the two- and three-atom cases. We discuss the advantages and disadvantages of different computation methods applied to systems with intermediate interactions. In the case of very strong repulsion, close to the infinite limit, we discuss approaches based on effective spin chain descriptions. We also report on recent studies on higher-spin mixtures and inter-component attractive forces. For both statistics, we pay particular attention to impurity problems and mass imbalance cases. Finally, we describe the recent advances on trapped Bose-Fermi mixtures, which allow for a theoretical combination of previous concepts, well illustrating the importance of quantum statistics and inter-particle interactions. Lastly, we report on fundamental questions related to the subject which we believe will inspire further theoretical developments and experimental verification.
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Affiliation(s)
- Tomasz Sowiński
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw, Poland
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39
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Zhang X, Xu X, Zheng Y, Chen Z, Liu B, Huang C, Malomed BA, Li Y. Semidiscrete Quantum Droplets and Vortices. PHYSICAL REVIEW LETTERS 2019; 123:133901. [PMID: 31697515 DOI: 10.1103/physrevlett.123.133901] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2019] [Revised: 08/01/2019] [Indexed: 06/10/2023]
Abstract
We consider a binary bosonic condensate with weak mean-field (MF) residual repulsion, loaded in an array of nearly one-dimensional traps coupled by transverse hopping. With the MF force balanced by the effectively one-dimensional attraction, induced in each trap by the Lee-Hung-Yang correction (produced by quantum fluctuations around the MF state), stable on-site- and intersite-centered semidiscrete quantum droplets (QDs) emerge in the array, as fundamental ones and self-trapped vortices, with winding numbers, at least, up to five, in both tightly bound and quasicontinuum forms. The application of a relatively strong trapping potential leads to squeezing transitions, which increase the number of sites in fundamental QDs and eventually replace vortex modes by fundamental or dipole ones. The results provide the first realization of stable semidiscrete vortex QDs, including ones with multiple vorticity.
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Affiliation(s)
- Xiliang Zhang
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Xiaoxi Xu
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Yiyin Zheng
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Zhaopin Chen
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and the Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
| | - Bin Liu
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Chunqing Huang
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Boris A Malomed
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and the Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
| | - Yongyao Li
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and the Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
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40
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Qin YH, Zhao LC, Ling L. Nondegenerate bound-state solitons in multicomponent Bose-Einstein condensates. Phys Rev E 2019; 100:022212. [PMID: 31574652 DOI: 10.1103/physreve.100.022212] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/15/2019] [Indexed: 06/10/2023]
Abstract
We investigate nondegenerate bound-state solitons systematically in multicomponent Bose-Einstein condensates, through developing the Darboux transformation method to derive exact soliton solutions analytically. In particular, we show that bright solitons with nodes correspond to the excited bound states in effective quantum wells, in sharp contrast to the bright solitons and dark solitons reported before (which usually correspond to ground state and free state, respectively). We further demonstrate that bound-state solitons with nodes are induced by incoherent superposition of solitons in different components. Moreover, we reveal that the interactions between these bound-state solitons are usually inelastic, caused by the incoherent interactions between solitons in different components and the coherent interactions between solitons in the same component. Additionally, the detailed spectral stability analysis demonstrates the stability of nondegenerate bound-state solitons. The bound-state solitons can be used to study many different physical problems, such as beating dynamics, spin-orbit coupling effects, quantum fluctuations, and even quantum entanglement states.
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Affiliation(s)
- Yan-Hong Qin
- School of Physics, Northwest University, Xi'an 710127, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
| | - Li-Chen Zhao
- School of Physics, Northwest University, Xi'an 710127, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
| | - Liming Ling
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
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41
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Kartashov YV, Malomed BA, Torner L. Metastability of Quantum Droplet Clusters. PHYSICAL REVIEW LETTERS 2019; 122:193902. [PMID: 31144921 DOI: 10.1103/physrevlett.122.193902] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2019] [Revised: 03/25/2019] [Indexed: 06/09/2023]
Abstract
We show that metastable ring-shaped clusters can be constructed from two-dimensional quantum droplets in systems described by the Gross-Pitaevskii equations augmented with Lee-Huang-Yang quantum corrections. The clusters exhibit dynamical behavior ranging from contraction to rotation with simultaneous periodic pulsations, or expansion, depending on the initial radius of the necklace pattern and phase shift between adjacent quantum droplets. We show that, using an energy-minimization analysis, one can predict equilibrium values of the cluster radius that correspond to rotation without radial pulsations. In such a regime, the clusters evolve as metastable states, withstanding abrupt variations in the underlying scattering lengths and keeping their azimuthal symmetry in the course of evolution, even in the presence of considerable perturbations.
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Affiliation(s)
- Yaroslav V Kartashov
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
- Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow 108840, Russia
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Centre for Light-Matter Interaction, Tel Aviv University, P. O. Box 39040 Tel Aviv, Israel
| | - Lluis Torner
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
- Universitat Politecnica de Catalunya, 08034 Barcelona, Spain
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42
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Abstract
Solitons and breathers are nonlinear modes that exist in a wide range of physical systems. They are fundamental solutions of a number of nonlinear wave evolution equations, including the unidirectional nonlinear Schrödinger equation (NLSE). We report the observation of slanted solitons and breathers propagating at an angle with respect to the direction of propagation of the wave field. As the coherence is diagonal, the scale in the crest direction becomes finite; consequently, beam dynamics form. Spatiotemporal measurements of the water surface elevation are obtained by stereo-reconstructing the positions of the floating markers placed on a regular lattice and recorded with two synchronized high-speed cameras. Experimental results, based on the predictions obtained from the (2D + 1) hyperbolic NLSE equation, are in excellent agreement with the theory. Our study proves the existence of such unique and coherent wave packets and has serious implications for practical applications in optical sciences and physical oceanography. Moreover, unstable wave fields in this geometry may explain the formation of directional large-amplitude rogue waves with a finite crest length within a wide range of nonlinear dispersive media, such as Bose-Einstein condensates, solids, plasma, hydrodynamics, and optics.
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43
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Parisi L, Astrakharchik GE, Giorgini S. Liquid State of One-Dimensional Bose Mixtures: A Quantum Monte Carlo Study. PHYSICAL REVIEW LETTERS 2019; 122:105302. [PMID: 30932674 DOI: 10.1103/physrevlett.122.105302] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/19/2018] [Indexed: 06/09/2023]
Abstract
By using exact quantum Monte Carlo methods we calculate the ground-state properties of the liquid phase in one-dimensional Bose mixtures with contact interactions. We find that the liquid state can be formed if the ratio of coupling strengths between interspecies attractive and intraspecies repulsive interactions exceeds a critical value. As a function of this ratio we determine the density where the energy per particle has a minimum and the one where the compressibility diverges, thereby identifying the equilibrium density and the spinodal point in the phase diagram of the homogeneous liquid. Furthermore, in the stable liquid state, we calculate the chemical potential, the speed of sound, as well as structural and coherence properties, such as the pair correlation function, the static structure factor, and the one-body density matrix, thus providing a detailed description of the bulk region in self-bound droplets.
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Affiliation(s)
- L Parisi
- Dipartimento di Fisica, Università di Trento and CNR-INO BEC Center, I-38050 Povo, Trento, Italy
| | - G E Astrakharchik
- Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, Spain
| | - S Giorgini
- Dipartimento di Fisica, Università di Trento and CNR-INO BEC Center, I-38050 Povo, Trento, Italy
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44
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Ferioli G, Semeghini G, Masi L, Giusti G, Modugno G, Inguscio M, Gallemí A, Recati A, Fattori M. Collisions of Self-Bound Quantum Droplets. PHYSICAL REVIEW LETTERS 2019; 122:090401. [PMID: 30932536 DOI: 10.1103/physrevlett.122.090401] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2018] [Indexed: 06/09/2023]
Abstract
We report on the study of binary collisions between quantum droplets formed by an attractive mixture of ultracold atoms. We distinguish two main outcomes of the collision, i.e., merging and separation, depending on the velocity of the colliding pair. The critical velocity v_{c} that discriminates between the two cases displays a different dependence on the atom number N for small and large droplets. By comparing our experimental results with numerical simulations, we show that the nonmonotonic behavior of v_{c}(N) is due to the crossover from a compressible to an incompressible regime, where the collisional dynamics is governed by different energy scales, i.e., the droplet binding energy and the surface tension. These results also provide the first evidence of the liquidlike nature of quantum droplets in the large N limit, where their behavior closely resembles that of classical liquid droplets.
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Affiliation(s)
- Giovanni Ferioli
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - Giulia Semeghini
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - Leonardo Masi
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - Giovanni Giusti
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
| | - Giovanni Modugno
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - Massimo Inguscio
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - Albert Gallemí
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
- Trento Institute for Fundamental Physics and Applications, INFN, 38123, Trento, Italy
| | - Alessio Recati
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
- Trento Institute for Fundamental Physics and Applications, INFN, 38123, Trento, Italy
| | - Marco Fattori
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
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45
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Jørgensen NB, Bruun GM, Arlt JJ. Dilute Fluid Governed by Quantum Fluctuations. PHYSICAL REVIEW LETTERS 2018; 121:173403. [PMID: 30411960 DOI: 10.1103/physrevlett.121.173403] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/10/2018] [Indexed: 06/08/2023]
Abstract
Understanding the effects of interactions in complex quantum systems beyond the mean-field paradigm constitutes a fundamental problem in physics. We show how the atom numbers and interactions in a Bose-Bose mixture can be tuned to cancel mean-field interactions completely. The resulting system is entirely governed by quantum fluctuations-specifically the Lee-Huang-Yang correlations. We derive an effective one-component Gross-Pitaevskii equation for this system, which is shown to be very accurate by comparison with a full two-component description. This allows us to show how the Lee-Huang-Yang correlation energy can be accurately measured using two powerful probes of atomic gases: collective excitations and radio-frequency spectroscopy. Importantly, the behavior of the system is robust against deviations from the atom number and interaction criteria for canceling the mean-field interactions. This shows that it is feasible to realize a setting where quantum fluctuations are not masked by mean-field forces, allowing investigations of the Lee-Huang-Yang correction at unprecedented precision.
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Affiliation(s)
- Nils B Jørgensen
- Institut for Fysik og Astronomi, Aarhus Universitet, 8000 Aarhus C, Denmark
| | - Georg M Bruun
- Institut for Fysik og Astronomi, Aarhus Universitet, 8000 Aarhus C, Denmark
| | - Jan J Arlt
- Institut for Fysik og Astronomi, Aarhus Universitet, 8000 Aarhus C, Denmark
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46
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Wilson KE, Westerberg N, Valiente M, Duncan CW, Wright EM, Öhberg P, Faccio D. Observation of Photon Droplets and Their Dynamics. PHYSICAL REVIEW LETTERS 2018; 121:133903. [PMID: 30312099 DOI: 10.1103/physrevlett.121.133903] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2018] [Indexed: 06/08/2023]
Abstract
We present experimental evidence of photon droplets in an attractive (focusing) nonlocal nonlinear medium. Photon droplets are self-bound, finite-sized states of light that are robust to size and shape perturbations due to a balance of competing attractive and repulsive forces. It has recently been shown theoretically, via a multipole expansion of the nonlocal nonlinearity, that the self-bound state arises due to competition between the s-wave and d-wave nonlinear terms, together with diffraction. The theoretical photon droplet framework encompasses both a solitonlike stationary ground state and the nonsolitonlike dynamics that ensue when the system is displaced from equilibrium, i.e., driven into an excited state. We present numerics and experiments supporting the existence of these photon droplet states and measurements of the dynamical evolution of the photon droplet orbital angular momentum.
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Affiliation(s)
- Kali E Wilson
- Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
| | - Niclas Westerberg
- Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
| | - Manuel Valiente
- Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
| | - Callum W Duncan
- Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
| | - Ewan M Wright
- Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
- College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA
| | - Patrik Öhberg
- Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
| | - Daniele Faccio
- Institute of Photonics and Quantum Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom
- School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom
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47
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Schmidt F, Mayer D, Bouton Q, Adam D, Lausch T, Spethmann N, Widera A. Quantum Spin Dynamics of Individual Neutral Impurities Coupled to a Bose-Einstein Condensate. PHYSICAL REVIEW LETTERS 2018; 121:130403. [PMID: 30312071 DOI: 10.1103/physrevlett.121.130403] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2018] [Indexed: 06/08/2023]
Abstract
We report on spin dynamics of individual, localized neutral impurities immersed in a Bose-Einstein condensate. Single cesium atoms are transported into a cloud of rubidium atoms and thermalize with the bath, and the ensuing spin exchange between localized impurities with quasispin F_{i}=3 and bath atoms with F_{b}=1 is resolved. Comparing our data to numerical simulations of spin dynamics, we find that, for gas densities in the Bose-Einstein condensate regime, the dynamics is dominated by the condensed fraction of the cloud. We spatially resolve the density overlap of impurities and gas by the spin population of impurities. Finally, we trace the coherence of impurities prepared in a coherent superposition of internal states when coupled to a gas of different densities. For our choice of states, we show that, despite high bath densities and, thus, fast thermalization rates, the impurity coherence is not affected by the bath, realizing a regime of sympathetic cooling while maintaining internal state coherence. Our work paves the way toward the nondestructive probing of quantum many-body systems via localized impurities.
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Affiliation(s)
- Felix Schmidt
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Germany
| | - Daniel Mayer
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Germany
| | - Quentin Bouton
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Germany
| | - Daniel Adam
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Germany
| | - Tobias Lausch
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Germany
| | - Nicolas Spethmann
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Germany
| | - Artur Widera
- Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, Germany
- Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, 67663 Kaiserslautern, Germany
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48
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Semeghini G, Ferioli G, Masi L, Mazzinghi C, Wolswijk L, Minardi F, Modugno M, Modugno G, Inguscio M, Fattori M. Self-Bound Quantum Droplets of Atomic Mixtures in Free Space. PHYSICAL REVIEW LETTERS 2018; 120:235301. [PMID: 29932719 DOI: 10.1103/physrevlett.120.235301] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/25/2017] [Indexed: 06/08/2023]
Abstract
Self-bound quantum droplets are a newly discovered phase in the context of ultracold atoms. In this Letter, we report their experimental realization following the original proposal by Petrov [Phys. Rev. Lett. 115, 155302 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.155302], using an attractive bosonic mixture. In this system, spherical droplets form due to the balance of competing attractive and repulsive forces, provided by the mean-field energy close to the collapse threshold and the first-order correction due to quantum fluctuations. Thanks to an optical levitating potential with negligible residual confinement, we observe self-bound droplets in free space, and we characterize the conditions for their formation as well as their size and composition. This work sets the stage for future studies on quantum droplets, from the measurement of their peculiar excitation spectrum to the exploration of their superfluid nature.
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Affiliation(s)
- G Semeghini
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - G Ferioli
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - L Masi
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - C Mazzinghi
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
| | - L Wolswijk
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
| | - F Minardi
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
- Dipartimento di Fisica e Astronomia, Università di Bologna, 40127 Bologna, Italy
| | - M Modugno
- Departamento de Física Teórica e Historia de la Ciencia, Universidad del Pais Vasco UPV/EHU, 48080 Bilbao, Spain
- IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain
| | - G Modugno
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - M Inguscio
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
| | - M Fattori
- LENS and Dipartimento di Fisica e Astronomia, Università di Firenze, 50019 Sesto Fiorentino, Italy
- CNR Istituto Nazionale Ottica, 50019 Sesto Fiorentino, Italy
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49
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Suppression of Quantum-Mechanical Collapse in Bosonic Gases with Intrinsic Repulsion: A Brief Review. CONDENSED MATTER 2018. [DOI: 10.3390/condmat3020015] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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50
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Ferrier-Barbut I, Wenzel M, Böttcher F, Langen T, Isoard M, Stringari S, Pfau T. Scissors Mode of Dipolar Quantum Droplets of Dysprosium Atoms. PHYSICAL REVIEW LETTERS 2018; 120:160402. [PMID: 29756918 DOI: 10.1103/physrevlett.120.160402] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2017] [Indexed: 06/08/2023]
Abstract
We report on the observation of the scissors mode of a single dipolar quantum droplet. The existence of this mode is due to the breaking of the rotational symmetry by the dipole-dipole interaction, which is fixed along an external homogeneous magnetic field. By modulating the orientation of this magnetic field, we introduce a new spectroscopic technique for studying dipolar quantum droplets. This provides a precise probe for interactions in the system, allowing us to extract a background scattering length for ^{164}Dy of 69(4)a_{0}. Our results establish an analogy between quantum droplets and atomic nuclei, where the existence of the scissors mode is also only due to internal interactions. They further open the possibility to explore physics beyond the available theoretical models for strongly dipolar quantum gases.
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Affiliation(s)
- Igor Ferrier-Barbut
- 5. Physikalisches Institut and Center for Integrated Quantum Science and Technology IQST, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
| | - Matthias Wenzel
- 5. Physikalisches Institut and Center for Integrated Quantum Science and Technology IQST, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
| | - Fabian Böttcher
- 5. Physikalisches Institut and Center for Integrated Quantum Science and Technology IQST, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
| | - Tim Langen
- 5. Physikalisches Institut and Center for Integrated Quantum Science and Technology IQST, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
| | - Mathieu Isoard
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
| | - Sandro Stringari
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy
| | - Tilman Pfau
- 5. Physikalisches Institut and Center for Integrated Quantum Science and Technology IQST, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
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