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Kinnunen JJ, Baarsma JE, Martikainen JP, Törmä P. The Fulde-Ferrell-Larkin-Ovchinnikov state for ultracold fermions in lattice and harmonic potentials: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2018; 81:046401. [PMID: 29293087 DOI: 10.1088/1361-6633/aaa4ad] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We review the concepts and the present state of theoretical studies of spin-imbalanced superfluidity, in particular the elusive Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, in the context of ultracold quantum gases. The comprehensive presentation of the theoretical basis for the FFLO state that we provide is useful also for research on the interplay between magnetism and superconductivity in other physical systems. We focus on settings that have been predicted to be favourable for the FFLO state, such as optical lattices in various dimensions and spin-orbit coupled systems. These are also the most likely systems for near-future experimental observation of the FFLO state. Theoretical bounds, such as Bloch's and Luttinger's theorems, and experimentally important limitations, such as finite-size effects and trapping potentials, are considered. In addition, we provide a comprehensive review of the various ideas presented for the observation of the FFLO state. We conclude our review with an analysis of the open questions related to the FFLO state, such as its stability, superfluid density, collective modes and extending the FFLO superfluid concept to new types of lattice systems.
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Affiliation(s)
- Jami J Kinnunen
- COMP Center of Excellence, Department of Applied Physics, Aalto University, Fi-00076, Aalto, Finland
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2
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Dutta S, Mueller EJ. Collective Modes of a Soliton Train in a Fermi Superfluid. PHYSICAL REVIEW LETTERS 2017; 118:260402. [PMID: 28707921 DOI: 10.1103/physrevlett.118.260402] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/14/2016] [Indexed: 06/07/2023]
Abstract
We characterize the collective modes of a soliton train in a quasi-one-dimensional Fermi superfluid, using a mean-field formalism. In addition to the expected Goldstone and Higgs modes, we find novel long-lived gapped modes associated with oscillations of the soliton cores. The soliton train has an instability that depends strongly on the interaction strength and the spacing of solitons. It can be stabilized by filling each soliton with an unpaired fermion, thus forming a commensurate Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We find that such a state is always dynamically stable, which paves the way for realizing long-lived FFLO states in experiments via phase imprinting.
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Affiliation(s)
- Shovan Dutta
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA
| | - Erich J Mueller
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA
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3
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Revelle MC, Fry JA, Olsen BA, Hulet RG. 1D to 3D Crossover of a Spin-Imbalanced Fermi Gas. PHYSICAL REVIEW LETTERS 2016; 117:235301. [PMID: 27982646 DOI: 10.1103/physrevlett.117.235301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2016] [Indexed: 06/06/2023]
Abstract
We have characterized the one-dimensional (1D) to three-dimensional (3D) crossover of a two-component spin-imbalanced Fermi gas of ^{6}Li atoms in a 2D optical lattice by varying the lattice tunneling and the interactions. The gas phase separates, and we detect the phase boundaries using in situ imaging of the inhomogeneous density profiles. The locations of the phases are inverted in 1D as compared to 3D, thus providing a clear signature of the crossover. By scaling the tunneling rate t with respect to the pair binding energy ε_{B}, we observe a collapse of the data to a universal crossover point at a scaled tunneling value of t[over ˜]_{c}=0.025(7).
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Affiliation(s)
- Melissa C Revelle
- Department of Physics and Astronomy and Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
| | - Jacob A Fry
- Department of Physics and Astronomy and Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
| | - Ben A Olsen
- Department of Physics and Astronomy and Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
| | - Randall G Hulet
- Department of Physics and Astronomy and Rice Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
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4
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Li X, Liu WV. Physics of higher orbital bands in optical lattices: a review. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2016; 79:116401. [PMID: 27651388 DOI: 10.1088/0034-4885/79/11/116401] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
The orbital degree of freedom plays a fundamental role in understanding the unconventional properties in solid state materials. Experimental progress in quantum atomic gases has demonstrated that high orbitals in optical lattices can be used to construct quantum emulators of exotic models beyond natural crystals, where novel many-body states such as complex Bose-Einstein condensates and topological semimetals emerge. A brief introduction of orbital degrees of freedom in optical lattices is given and a summary of exotic orbital models and resulting many-body phases is provided. Experimental consequences of the novel phases are also discussed.
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Affiliation(s)
- Xiaopeng Li
- Department of Physics, Condensed Matter Theory Center and Joint Quantum Institute, University of Maryland, College Park, MD 20742-4111, USA
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5
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Mitra D, Brown PT, Schauß P, Kondov SS, Bakr WS. Phase Separation and Pair Condensation in a Spin-Imbalanced 2D Fermi Gas. PHYSICAL REVIEW LETTERS 2016; 117:093601. [PMID: 27610853 DOI: 10.1103/physrevlett.117.093601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2016] [Indexed: 06/06/2023]
Abstract
We study a two-component quasi-two-dimensional Fermi gas with imbalanced spin populations. We probe the gas at different interaction strengths and polarizations by measuring the density of each spin component in the trap and the pair momentum distribution after time of flight. For a wide range of experimental parameters, we observe in-trap phase separation characterized by the appearance of a spin-balanced core surrounded by a polarized gas. Our momentum space measurements indicate pair condensation in the imbalanced gas even for large polarizations where phase separation vanishes, pointing to the presence of a polarized pair condensate. Our observation of zero momentum pair condensates in 2D spin-imbalanced gases opens the way to explorations of more exotic superfluid phases that occupy a large part of the phase diagram in lower dimensions.
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Affiliation(s)
- Debayan Mitra
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Peter T Brown
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Peter Schauß
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Stanimir S Kondov
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Waseem S Bakr
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
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6
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Zheng Z, Qu C, Zou X, Zhang C. Fulde-Ferrell Superfluids without Spin Imbalance in Driven Optical Lattices. PHYSICAL REVIEW LETTERS 2016; 116:120403. [PMID: 27058062 DOI: 10.1103/physrevlett.116.120403] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/12/2015] [Indexed: 06/05/2023]
Abstract
Spin-imbalanced ultracold Fermi gases have been widely studied recently as a platform for exploring the long-sought Fulde-Ferrell-Larkin-Ovchinnikov superfluid phases, but so far conclusive evidence has not been found. Here we propose to realize an Fulde-Ferrell (FF) superfluid without spin imbalance in a three-dimensional fermionic cold atom optical lattice, where s- and p-orbital bands of the lattice are coupled by another weak moving optical lattice. Such coupling leads to a spin-independent asymmetric Fermi surface, which, together with the s-wave scattering interaction between two spins, yields an FF type of superfluid pairing. Unlike traditional schemes, our proposal does not rely on the spin imbalance (or an equivalent Zeeman field) to induce the Fermi surface mismatch and provides a completely new route for realizing FF superfluids.
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Affiliation(s)
- Zhen Zheng
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
- Key Laboratory of Quantum Information, and Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China
| | - Chunlei Qu
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Xubo Zou
- Key Laboratory of Quantum Information, and Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China
| | - Chuanwei Zhang
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
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7
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Xu Y, Zhang C. Berezinskii-Kosterlitz-Thouless phase transition in 2D spin-orbit-coupled Fulde-Ferrell superfluids. PHYSICAL REVIEW LETTERS 2015; 114:110401. [PMID: 25839244 DOI: 10.1103/physrevlett.114.110401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2014] [Indexed: 06/04/2023]
Abstract
The experimental observation of traditional Zeeman-field induced Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluids has been hindered by various challenges, in particular, the requirement of low dimensional systems. In 2D, finite temperature phase fluctuations lead to an extremely small Berezinskii-Kosterlitz-Thouless (BKT) transition temperature for FFLO superfluids, raising serious concerns regarding their experimental observability. Recently, it was shown that FFLO superfluids can be realized using a Rashba spin-orbit coupled Fermi gas subject to Zeeman fields, which may also support topological excitations such as Majorana fermions in 2D. Here we address the finite temperature BKT transition issue in this system, which may exhibit gapped, gapless, topological, and gapless topological FF phases. We find a large BKT transition temperature due to large effective superfluid densities, making it possible to observe 2D FF superfluids at finite temperature. In addition, we show that gapless FF superfluids can be stable due to their positive superfluid densities. These findings pave the way for the experimental observation of 2D gapped and gapless FF superfluids and their associated topological excitations at finite temperature.
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Affiliation(s)
- Yong Xu
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Chuanwei Zhang
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
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8
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Heikkinen MOJ, Kim DH, Troyer M, Törmä P. Nonlocal quantum fluctuations and fermionic superfluidity in the imbalanced attractive Hubbard model. PHYSICAL REVIEW LETTERS 2014; 113:185301. [PMID: 25396376 DOI: 10.1103/physrevlett.113.185301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2014] [Indexed: 06/04/2023]
Abstract
We study fermionic superfluidity in strongly anisotropic optical lattices with attractive interactions utilizing the cluster dynamical mean-field theory method, and focusing in particular on the role of nonlocal quantum fluctuations. We show that nonlocal quantum fluctuations impact the BCS superfluid transition dramatically. Moreover, we show that exotic superfluid states with a delicate order parameter structure, such as the Fulde-Ferrell-Larkin-Ovchinnikov phase driven by spin population imbalance, can emerge even in the presence of such strong fluctuations.
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Affiliation(s)
- M O J Heikkinen
- COMP Centre of Excellence and Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland
| | - D-H Kim
- Department of Physics and Photon Science, School of Physics and Chemistry, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea
| | - M Troyer
- Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland
| | - P Törmä
- COMP Centre of Excellence and Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland
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9
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Xu Y, Mao L, Wu B, Zhang C. Dark solitons with Majorana fermions in spin-orbit-coupled Fermi gases. PHYSICAL REVIEW LETTERS 2014; 113:130404. [PMID: 25302874 DOI: 10.1103/physrevlett.113.130404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/14/2014] [Indexed: 06/04/2023]
Abstract
We show that a single dark soliton can exist in a spin-orbit-coupled Fermi gas with a high spin imbalance, where spin-orbit coupling favors uniform superfluids over nonuniform Fulde-Ferrell-Larkin-Ovchinnikov states, leading to dark soliton excitations in highly imbalanced gases. Above a critical spin imbalance, two topological Majorana fermions without interactions can coexist inside a dark soliton, paving a way for manipulating Majorana fermions through controlling solitons. At the topological transition point, the atom density contrast across the soliton suddenly vanishes, suggesting a signature for identifying topological solitons.
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Affiliation(s)
- Yong Xu
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Li Mao
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA and School of Physics and Technology, Wuhan University, Wuhan 430072, China
| | - Biao Wu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China and Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Chuanwei Zhang
- Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA
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10
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Lu H, Baksmaty LO, Bolech CJ, Pu H. Expansion of 1D polarized superfluids: the Fulde-Ferrell-Larkin-Ovchinnikov state reveals itself. PHYSICAL REVIEW LETTERS 2012; 108:225302. [PMID: 23003611 DOI: 10.1103/physrevlett.108.225302] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/17/2012] [Indexed: 06/01/2023]
Abstract
We study the expansion dynamics of a one-dimensional polarized Fermi gas after its sudden release from confinement using both the mean-field Bogoliubov-de Gennes and the numerically unbiased time-evolving block decimation methods. Our results show that experimentally observable spin density modulations, directly related to the presence of a Fulde-Ferrell-Larkin-Ovchinnikov state, develop during the expansion of the cloud. Our work therefore provides a robust theoretical proposal for the detection of this long-sought state.
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Affiliation(s)
- Hong Lu
- Department of Physics and Astronomy, Rice University, 6100 Main Street, MS-61, Houston, Texas 77005, USA
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11
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Kim DH, Kinnunen JJ, Martikainen JP, Törmä P. Exotic superfluid states of lattice fermions in elongated traps. PHYSICAL REVIEW LETTERS 2011; 106:095301. [PMID: 21405631 DOI: 10.1103/physrevlett.106.095301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2010] [Revised: 12/04/2010] [Indexed: 05/30/2023]
Abstract
We present real-space dynamical mean-field theory calculations for attractively interacting fermions in three-dimensional lattices with elongated traps. The critical polarization is found to be 0.8, regardless of the trap elongation. Below the critical polarization, we find unconventional superfluid structures where the polarized superfluid and Fulde-Ferrell-Larkin-Ovchinnikov-type states emerge across the entire core region.
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Affiliation(s)
- D-H Kim
- Department of Applied Physics, Aalto University, Finland
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12
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Spin-imbalance in a one-dimensional Fermi gas. Nature 2010; 467:567-9. [DOI: 10.1038/nature09393] [Citation(s) in RCA: 428] [Impact Index Per Article: 30.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/03/2010] [Accepted: 08/03/2010] [Indexed: 11/08/2022]
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13
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Zapata I, Wunsch B, Zinner NT, Demler E. π phases in balanced fermionic superfluids on spin-dependent optical lattices. PHYSICAL REVIEW LETTERS 2010; 105:095301. [PMID: 20868172 DOI: 10.1103/physrevlett.105.095301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/09/2009] [Revised: 06/18/2010] [Indexed: 05/29/2023]
Abstract
We study a balanced two-component system of ultracold fermions in one dimension with attractive interactions and subject to a spin-dependent optical lattice potential of opposite sign for the two components. We find states with different types of modulated pairing order parameters which are conceptually similar to π phases discussed for superconductor-ferromagnet heterostructures. Increasing the lattice depth induces sharp transitions between states of different parity. While the origin of the order parameter oscillations is similar to the Fulde-Ferrel-Larkin-Ovchinnikov phase for paired states with spin imbalance, the current system is intrinsically stable to phase separation. We discuss experimental requirements for creating and probing these novel phases.
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Affiliation(s)
- I Zapata
- Departamento de Física de Materiales, Universidad Complutense de Madrid, E-28040 Madrid, Spain
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14
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James AJA, Lamacraft A. Non-Fermi-liquid fixed point for an imbalanced gas of fermions in 1+ϵ dimensions. PHYSICAL REVIEW LETTERS 2010; 104:190403. [PMID: 20866949 DOI: 10.1103/physrevlett.104.190403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/27/2009] [Revised: 03/29/2010] [Indexed: 05/29/2023]
Abstract
We consider a gas of two species of fermions with population imbalance. Using the renormalization group in d=1+ϵ spatial dimensions, we show that for spinless fermions and 2>ϵ>0 a fixed point appears at finite attractive coupling where the quasiparticle residue vanishes, and identify this with the transition to Larkin-Ovchinnikov-Fulde-Ferrell order (inhomogeneous superconductivity). When the two species of fermions also carry spin degrees of freedom we find a fixed point indicating a transition to spin density wave order.
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Affiliation(s)
- A J A James
- Department of Physics, University of Virginia, Charlottesville, Virginia 22904-4717, USA
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15
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Loh YL, Trivedi N. Detecting the elusive Larkin-Ovchinnikov modulated superfluid phases for imbalanced Fermi gases in optical lattices. PHYSICAL REVIEW LETTERS 2010; 104:165302. [PMID: 20482061 DOI: 10.1103/physrevlett.104.165302] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2009] [Revised: 02/28/2010] [Indexed: 05/29/2023]
Abstract
A system with unequal populations of up and down fermions may exhibit a Larkin-Ovchinnikov (LO) phase consisting of a periodic arrangement of domain walls where the order parameter changes sign and the excess polarization is localized. We find that the LO phase has a much larger range of stability in a lattice compared to the continuum; in a harmonic trap, the LO phase may involve 80% of the atoms in the trap, and can exist up to an entropy s approximately 0.5k(B) per fermion. We discuss detection of the LO phase (i) in real space by phase-contrast imaging of the periodic excess polarization; (ii) in k space by time-of-flight imaging of the single-particle and pair-momentum distributions; (iii) in energy space from the excess density of states within the gap arising from Andreev bound states in the domain walls.
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Affiliation(s)
- Yen Lee Loh
- Department of Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, Ohio 43210, USA
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Zhao E, Guan XW, Liu WV, Batchelor MT, Oshikawa M. Analytic thermodynamics and thermometry of Gaudin-Yang Fermi gases. PHYSICAL REVIEW LETTERS 2009; 103:140404. [PMID: 19905551 DOI: 10.1103/physrevlett.103.140404] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2009] [Revised: 08/25/2009] [Indexed: 05/28/2023]
Abstract
We study the thermodynamics of a one-dimensional attractive Fermi gas (the Gaudin-Yang model) with spin imbalance. The exact solution has been known from the thermodynamic Bethe ansatz for decades, but it involves an infinite number of coupled nonlinear integral equations whose physics is difficult to extract. Here the solution is analytically reduced to a simple, powerful set of four algebraic equations. The simplified equations become universal and exact in the experimental regime of strong interaction and relatively low temperature. Using the new formulation, we discuss the qualitative features of finite-temperature crossover and make quantitative predictions on the density profiles in traps. We propose a practical two-stage scheme to achieve accurate thermometry for a trapped spin-imbalanced Fermi gas.
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Affiliation(s)
- Erhai Zhao
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
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Edge JM, Cooper NR. Signature of the Fulde-Ferrell-Larkin-Ovchinnikov phase in the collective modes of a trapped ultracold Fermi gas. PHYSICAL REVIEW LETTERS 2009; 103:065301. [PMID: 19792579 DOI: 10.1103/physrevlett.103.065301] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/10/2009] [Indexed: 05/28/2023]
Abstract
We study theoretically the collective modes of a two-component Fermi gas with attractive interactions in a quasi-one-dimensional harmonic trap. We focus on an imbalanced gas in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. Using a mean-field theory, we study the response of the ground state to time-dependent potentials. For potentials with short wavelengths, we find dramatic signatures in the large-scale response of the gas which are characteristic of the FFLO phase. This response provides an effective way to detect the FFLO state in experiments.
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Affiliation(s)
- Jonathan M Edge
- T.C.M. Group, Cavendish Laboratory, Cambridge CB3 0HE, United Kingdom
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18
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Feiguin AE, Heidrich-Meisner F. Pair correlations of a spin-imbalanced Fermi gas on two-leg ladders. PHYSICAL REVIEW LETTERS 2009; 102:076403. [PMID: 19257697 DOI: 10.1103/physrevlett.102.076403] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2008] [Indexed: 05/27/2023]
Abstract
We study the pair correlations of a spin-imbalanced two-leg ladder with attractive interactions, using the density matrix renormalization group method. We identify regions in the phase diagram spanned by the chemical potential and the magnetic field that can harbor Fulde-Ferrell-Larkin-Ovchinnikov- (FFLO-)like physics. Results for the pair structure factor, exhibiting multiple pairing wave vectors, substantiate the presence of FFLO-like correlations. We further discuss phase separation scenarios induced by a harmonic trap, which differ from the case of isolated chains.
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Affiliation(s)
- A E Feiguin
- Microsoft Project Q, University of California, Santa Barbara, California 93106, USA
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