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  • Letter
  • Open Access

Quantum coherent states of interacting Bose-Fermi mixtures in one dimension

J. Clayton Peacock1,2, Aleksandar Ljepoja2, and C. J. Bolech2,3

  • 1Department of Physics, New York University, New York, New York 10003, USA
  • 2Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221, USA
  • 3ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA

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Phys. Rev. Research 4, L022034 – Published 11 May, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L022034

Abstract

We study two-component atomic gas mixtures in one dimension involving both bosons and fermions. When the interspecies interaction is attractive, we report a rich variety of coherent ground-state phases that vary with the intrinsic and relative strength of the interactions. We avoid any artifacts of lattice discretization by developing an implementation of a continuous matrix-product-state Ansatz for mixtures and priorly demonstrate the validity of our approach on the integrable point that exists for mixtures with equal masses and interactions (Lai-Yang model), where we find that the Ansatz correctly and systematically converges towards the exact results.

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References (63)

  1. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269, 198 (1995); C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose-Einstein Condensation in an Atomic Gas with Attractive Interactions, Phys. Rev. Lett. 75, 1687 (1995); 79, 1170(E) (1997); K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein Condensation in a Gas of Sodium Atoms, ibid. 75, 3969 (1995).
  2. B. DeMarco and D. S. Jin, Onset of Fermi degeneracy in a trapped atomic gas, Science 285, 1703 (1999).
  3. A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, Observation of Fermi pressure in a gas of trapped atoms, Science 291, 2570 (2001); F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, Quasipure Bose-Einstein Condensate Immersed in a Fermi Sea, Phys. Rev. Lett. 87, 080403 (2001).
  4. C. Ebner and D. Edwards, The low temperature thermodynamic properties of superfluid solutions of He3 in He4, Phys. Rep. 2, 77 (1971); D. Edwards and M. Pettersen, Lectures on the properties of liquid and solid He3−He4 mixtures at low temperatures, J. Low Temp. Phys. 87, 473523 (1992); S. Aubin, M. H. T. Extavour, S. Myrskog, L. J. LeBlanc, J. Estève, S. Singh, P. Scrutton, D. McKay, R. McKenzie, I. D. Leroux, A. Stummer, and J. H. Thywissen, Trapping fermionic K40 and bosonic Rb87 on a chip, ibid. 140, 377 (2005).
  5. L. Viverit, C. J. Pethick, and H. Smith, Zero-temperature phase diagram of binary boson-fermion mixtures, Phys. Rev. A 61, 053605 (2000).
  6. K. K. Das, Bose-Fermi Mixtures in One Dimension, Phys. Rev. Lett. 90, 170403 (2003).
  7. A. Albus, F. Illuminati, and J. Eisert, Mixtures of bosonic and fermionic atoms in optical lattices, Phys. Rev. A 68, 023606 (2003).
  8. M. A. Cazalilla and A. F. Ho, Instabilities in Binary Mixtures of One-Dimensional Quantum Degenerate Gases, Phys. Rev. Lett. 91, 150403 (2003).
  9. M. A. Cazalilla, A. F. Ho, and T. Giamarchi, Two-Component Fermi Gas on Internal-State-Dependent Optical Lattices, Phys. Rev. Lett. 95, 226402 (2005); N. Oelkers, M. T. Batchelor, M. Bortz, and X.-W. Guan, Bethe ansatz study of one-dimensional Bose and Fermi gases with periodic and hard wall boundary conditions, J. Phys. A: Math. Gen. 39, 1073 (2006); G. Orso, Attractive Fermi Gases with Unequal Spin Populations in Highly Elongated Traps, Phys. Rev. Lett. 98, 070402 (2007); H. Hu, X.-J. Liu, and P. D. Drummond, Phase Diagram of a Strongly Interacting Polarized Fermi Gas in One Dimension, ibid. 98, 070403 (2007); X. W. Guan, M. T. Batchelor, C. Lee, and M. Bortz, Phase transitions and pairing signature in strongly attractive Fermi atomic gases, Phys. Rev. B 76, 085120 (2007); A. E. Feiguin and F. Heidrich-Meisner, Pairing states of a polarized Fermi gas trapped in a one-dimensional optical lattice, ibid. 76, 220508(R) (2007); M. M. Parish, S. K. Baur, E. J. Mueller, and D. A. Huse, Quasi-One-Dimensional Polarized Fermi Superfluids, Phys. Rev. Lett. 99, 250403 (2007); X.-J. Liu, H. Hu, and P. D. Drummond, Finite-temperature phase diagram of a spin-polarized ultracold Fermi gas in a highly elongated harmonic trap, Phys. Rev. A 78, 023601 (2008); M. Casula, D. M. Ceperley, and E. J. Mueller, Quantum Monte Carlo study of one-dimensional trapped fermions with attractive contact interactions, ibid. 78, 033607 (2008); E. Zhao and W. V. Liu, Theory of quasi-one-dimensional imbalanced Fermi gases, ibid. 78, 063605 (2008); A. E. Feiguin and F. Heidrich-Meisner, Pair Correlations of a Spin-Imbalanced Fermi Gas on Two-Leg Ladders, Phys. Rev. Lett. 102, 076403 (2009); J.-S. He, A. Foerster, X. W. Guan, and M. T. Batchelor, Magnetism and quantum phase transitions in spin-1/2 attractive fermions with polarization, New J. Phys. 11, 073009 (2009); P. Kakashvili and C. J. Bolech, Paired states in spin-imbalanced atomic Fermi gases in one dimension, Phys. Rev. A 79, 041603(R) (2009); B. Wang, H.-D. Chen, and S. Das Sarma, Quantum phase diagram of fermion mixtures with population imbalance in one-dimensional optical lattices, ibid. 79, 051604(R) (2009); J. E. Baarsma, K. B. Gubbels, and H. T. C. Stoof, Population and mass imbalance in atomic Fermi gases, ibid. 82, 013624 (2010); G. Orso, E. Burovski, and T. Jolicoeur, Luttinger Liquid of Trimers in Fermi Gases with Unequal Masses, Phys. Rev. Lett. 104, 065301 (2010); L. O. Baksmaty, H. Lu, C. J. Bolech, and H. Pu, Concomitant modulated superfluidity in polarized Fermi gases, Phys. Rev. A 83, 023604 (2011); A Bogoliubov-de Gennes study of trapped spin-imbalanced unitary Fermi gases, New J. Phys. 13, 055014 (2011); K. Sun, J. S. Meyer, D. E. Sheehy, and S. Vishveshwara, Oscillatory pairing of fermions in spin-split traps, Phys. Rev. A 83, 033608 (2011); H. Lu, L. O. Baksmaty, C. J. Bolech, and H. Pu, Expansion of 1D Polarized Superfluids: The Fulde-Ferrell-Larkin-Ovchinnikov State Reveals Itself, Phys. Rev. Lett. 108, 225302 (2012); M. Dalmonte, K. Dieckmann, T. Roscilde, C. Hartl, A. E. Feiguin, U. Schollwöck, and F. Heidrich-Meisner, Dimer, trimer, and Fulde-Ferrell-Larkin-Ovchinnikov liquids in mass- and spin-imbalanced trapped binary mixtures in one dimension, Phys. Rev. A 85, 063608 (2012); C. J. Bolech, F. Heidrich-Meisner, S. Langer, I. P. McCulloch, G. Orso, and M. Rigol, Long-Time Behavior of the Momentum Distribution During the Sudden Expansion of a Spin-Imbalanced Fermi Gas in One Dimension, Phys. Rev. Lett. 109, 110602 (2012); K. B. Gubbels and H. T. C. Stoof, Imbalanced Fermi gases at unitarity, Phys. Rep. 525, 255 (2013); J. Wang, H. Guo, and Q. Chen, Exotic phase separation and phase diagrams of a Fermi-Fermi mixture in a trap at finite temperature, Phys. Rev. A 87, 041601(R) (2013); X.-W. Guan, M. T. Batchelor, and C. Lee, Fermi gases in one dimension: From Bethe ansatz to experiments, Rev. Mod. Phys. 85, 1633 (2013); A. Trenkwalder, C. Kohstall, M. Zaccanti, D. Naik, A. I. Sidorov, F. Schreck, and R. Grimm, Hydrodynamic Expansion of a Strongly Interacting Fermi-Fermi Mixture, Phys. Rev. Lett. 106, 115304 (2011); D. Roscher, J. Braun, and J. E. Drut, Inhomogeneous phases in one-dimensional mass- and spin-imbalanced Fermi gases, Phys. Rev. A 89, 063609 (2014); B. Liu, X. Li, R. G. Hulet, and W. V. Liu, Detecting π-phase superfluids with p-wave symmetry in a quasi-one-dimensional optical lattice, ibid. 94, 031602 (2016); Z. Mei, L. Vidmar, F. Heidrich-Meisner, and C. J. Bolech, Unveiling hidden structure of many-body wave functions of integrable systems via sudden-expansion experiments, ibid. 93, 021607(R) (2016); B. Sundar, J. A. Fry, M. C. Revelle, R. G. Hulet, and K. R. A. Hazzard, Spin-imbalanced ultracold Fermi gases in a two-dimensional array of tubes, ibid. 102, 033311 (2020); F. He, Y.-Z. Jiang, H.-Q. Lin, R. G. Hulet, H. Pu, and X.-W. Guan, Emergence and Disruption of Spin-Charge Separation in One-Dimensional Repulsive Fermions, Phys. Rev. Lett. 125, 190401 (2020).
  10. C. A. Regal, M. Greiner, and D. S. Jin, Observation of Resonance Condensation of Fermionic Atom Pairs, Phys. Rev. Lett. 92, 040403 (2004); M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle, Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance, ibid. 92, 120403 (2004); J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E. Thomas, Evidence for Superfluidity in a Resonantly Interacting Fermi Gas, ibid. 92, 150402 (2004); M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, Collective Excitations of a Degenerate Gas at the BEC-BCS Crossover, ibid. 92, 203201 (2004); T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann, L. Tarruell, S. J. J. M. F. Kokkelmans, and C. Salomon, Experimental Study of the BEC-BCS Crossover Region in Lithium 6, ibid. 93, 050401 (2004); G. B. Partridge, K. E. Strecker, R. I. Kamar, M. W. Jack, and R. G. Hulet, Molecular Probe of Pairing in the BEC-BCS Crossover, ibid. 95, 020404 (2005); M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle, Vortices and superfluidity in a strongly interacting Fermi gas, Nature (London) 435, 1047 (2005); M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Fermionic superfluidity with imbalanced spin populations, Science 311, 492 (2006); G. B. Partridge, W. Li, R. I. Kamar, Y.-A. Liao, and R. G. Hulet, Pairing and phase separation in a polarized Fermi gas, ibid. 311, 503 (2006); G. B. Partridge, W. Li, Y. A. Liao, R. G. Hulet, M. Haque, and H. T. C. Stoof, Deformation of a Trapped Fermi Gas with Unequal Spin Populations, Phys. Rev. Lett. 97, 190407 (2006); M. Jag, M. Zaccanti, M. Cetina, R. S. Lous, F. Schreck, R. Grimm, D. S. Petrov, and J. Levinsen, Observation of a Strong Atom-Dimer Attraction in a Mass-Imbalanced Fermi-Fermi Mixture, ibid. 112, 075302 (2014).
  11. Y.-A. Liao, A. S. C. Rittner, T. Paprotta, W. Li, G. B. Partridge, R. G. Hulet, S. K. Baur, and E. J. Mueller, Spin-imbalance in a one-dimensional Fermi gas, Nature (London) 467, 567 (2010); Y. A. Liao, M. Revelle, T. Paprotta, A. S. C. Rittner, W. Li, G. B. Partridge, and R. G. Hulet, Metastability in Spin-Polarized Fermi Gases, Phys. Rev. Lett. 107, 145305 (2011); G. Pagano, M. Mancini, G. Cappellini, P. Lombardi, F. Schäfer, H. Hu, X.-J. Liu, J. Catani, C. Sias, M. Inguscio, and L. Fallani, A one-dimensional liquid of fermions with tunable spin, Nat. Phys. 10, 198 (2014); B. A. Olsen, M. C. Revelle, J. A. Fry, D. E. Sheehy, and R. G. Hulet, Phase diagram of a strongly interacting spin-imbalanced Fermi gas, Phys. Rev. A 92, 063616 (2015); M. C. Revelle, J. A. Fry, B. A. Olsen, and R. G. Hulet, 1D to 3D Crossover of a Spin-Imbalanced Fermi Gas, Phys. Rev. Lett. 117, 235301 (2016); T. L. Yang, P. Grišins, Y. T. Chang, Z. H. Zhao, C. Y. Shih, T. Giamarchi, and R. G. Hulet, Measurement of the Dynamical Structure Factor of a 1D Interacting Fermi Gas, ibid. 121, 103001 (2018).
  12. C. K. Lai and C. N. Yang, Ground-state energy of a mixture of fermions and bosons in one dimension with a repulsive δ-function interaction, Phys. Rev. A 3, 393 (1971).
  13. M. Olshanii, Atomic Scattering in the Presence of an External Confinement and a Gas of Impenetrable Bosons, Phys. Rev. Lett. 81, 938 (1998).
  14. S. Gautam and S. K. Adhikari, Weak coupling to unitarity crossover in Bose-Fermi mixtures: Mixing-demixing transition and spontaneous symmetry breaking in trapped systems, Phys. Rev. A 100, 023626 (2019).
  15. M. Lewenstein, L. Santos, M. A. Baranov, and H. Fehrmann, Atomic Bose-Fermi Mixtures in an Optical Lattice, Phys. Rev. Lett. 92, 050401 (2004); M. Rizzi and A. Imambekov, Pairing of one-dimensional Bose-Fermi mixtures with unequal masses, Phys. Rev. A 77, 023621 (2008); F. M. Marchetti, T. Jolicoeur, and M. M. Parish, Stability and Pairing in Quasi-One-Dimensional Bose-Fermi Mixtures, Phys. Rev. Lett. 103, 105304 (2009); M. Singh and G. Orso, Enhanced visibility of the Fulde-Ferrell-Larkin-Ovchinnikov state in one-dimensional Bose-Fermi mixtures near the immiscibility point, Phys. Rev. Research 2, 023148 (2020); R. Avella, J. J. Mendoza-Arenas, R. Franco, and J. Silva-Valencia, Mixture of scalar bosons and two-color fermions in one dimension: Superfluid-insulator transitions, Phys. Rev. A 102, 033341 (2020); R. Guerrero-Suarez, J. J. Mendoza-Arenas, R. Franco, and J. Silva-Valencia, Spin-selective insulators in Bose-Fermi mixtures, ibid. 103, 023304 (2021).
  16. M. C. Gutzwiller, Effect of Correlation on the Ferromagnetism of Transition Metals, Phys. Rev. Lett. 10, 159 (1963); W. Krauth, M. Caffarel, and J.-P. Bouchaud, Gutzwiller wave function for a model of strongly interacting bosons, Phys. Rev. B 45, 3137 (1992); K. Sun and C. J. Bolech, Bose-Hubbard model with occupation-parity couplings, ibid. 89, 064506 (2014); R. E. Barfknecht, I. Brouzos, and A. Foerster, Contact and static structure factor for bosonic and fermionic mixtures, Phys. Rev. A 91, 043640 (2015); C. Zhu, L. Chen, H. Hu, X.-J. Liu, and H. Pu, Spin-exchange-induced exotic superfluids in a Bose-Fermi spinor mixture, ibid. 100, 031602(R) (2019).
  17. S. R. White, Density Matrix Formulation for Quantum Renormalization Groups, Phys. Rev. Lett. 69, 2863 (1992); Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993); S. S. Kancharla and C. J. Bolech, Optical response in one-dimensional Mott insulators, ibid. 64, 085119 (2001).
  18. F. Verstraete and J. I. Cirac, Continuous Matrix Product States for Quantum Fields, Phys. Rev. Lett. 104, 190405 (2010).
  19. I. Maruyama and H. Katsura, Continuous matrix product ansatz for the one-dimensional Bose gas with point interaction, J. Phys. Soc. Jpn. 79, 073002 (2010).
  20. T. J. Osborne, J. Eisert, and F. Verstraete, Holographic Quantum States, Phys. Rev. Lett. 105, 260401 (2010).
  21. J. Haegeman, J. I. Cirac, T. J. Osborne, H. Verschelde, and F. Verstraete, Applying the Variational Principle to (1+1)-Dimensional Quantum Field Theories, Phys. Rev. Lett. 105, 251601 (2010).
  22. J. Haegeman, J. I. Cirac, T. J. Osborne, and F. Verstraete, Calculus of continuous matrix product states, Phys. Rev. B 88, 085118 (2013).
  23. D. Draxler, J. Haegeman, T. J. Osborne, V. Stojevic, L. Vanderstraeten, and F. Verstraete, Particles, Holes, and Solitons: A Matrix Product State Approach, Phys. Rev. Lett. 111, 020402 (2013).
  24. R. Hübener, A. Mari, and J. Eisert, Wick's Theorem for Matrix Product States, Phys. Rev. Lett. 110, 040401 (2013).
  25. F. Quijandría, J. J. García-Ripoll, and D. Zueco, Continuous matrix product states for coupled fields: Application to Luttinger liquids and quantum simulators, Phys. Rev. B 90, 235142 (2014).
  26. V. Stojevic, J. Haegeman, I. P. McCulloch, L. Tagliacozzo, and F. Verstraete, Conformal data from finite entanglement scaling, Phys. Rev. B 91, 035120 (2015).
  27. S. S. Chung, K. Sun, and C. J. Bolech, Matrix product ansatz for Fermi fields in one dimension, Phys. Rev. B 91, 121108(R) (2015).
  28. F. Quijandría and D. Zueco, Continuous-matrix-product-state solution for the mixing-demixing transition in one-dimensional quantum fields, Phys. Rev. A 92, 043629 (2015).
  29. Z. Mei and C. J. Bolech, Derivation of matrix product states for the Heisenberg spin chain with open boundary conditions, Phys. Rev. E 95, 032127 (2017).
  30. D. Draxler, J. Haegeman, F. Verstraete, and M. Rizzi, Continuous matrix product states with periodic boundary conditions and an application to atomtronics, Phys. Rev. B 95, 045145 (2017).
  31. M. Ganahl, J. Rincón, and G. Vidal, Continuous Matrix Product States for Quantum Fields: An Energy Minimization Algorithm, Phys. Rev. Lett. 118, 220402 (2017).
  32. S. S. Chung and C. J. Bolech, Multiple phase separation in one-dimensional mixtures of mass- and population-imbalanced attractive Fermi gases, Phys. Rev. A 96, 023609 (2017).
  33. M. Ganahl and G. Vidal, Continuous matrix product states for nonrelativistic quantum fields: A lattice algorithm for inhomogeneous systems, Phys. Rev. B 98, 195105 (2018).
  34. A. Tilloy and J. I. Cirac, Continuous Tensor Network States for Quantum Fields, Phys. Rev. X 9, 021040 (2019).
  35. M. Balanzó-Juandó and G. De las Cuevas, Generalized ansatz for continuous matrix product states, Phys. Rev. A 101, 052312 (2020).
  36. W. Tang, H.-H. Tu, and L. Wang, Continuous Matrix Product Operator Approach to Finite Temperature Quantum States, Phys. Rev. Lett. 125, 170604 (2020).
  37. T. D. Karanikolaou, P. Emonts, and A. Tilloy, Gaussian continuous tensor network states for simple bosonic field theories, Phys. Rev. Research 3, 023059 (2021).
  38. For integrable models solvable with the (non-nested) quantum inverse scattering method, the exact wave functions take the same form but with an additional projector inside the trace that fixes particle number and restores the U(1) invariance [19, 29]. The exact eigenstates can thus be regarded as having a hidden concomitant coherent nature uncovered by cMPS.
  39. This is a particular case of what is known as a quasi Jordan (canonical) form [61]. Notice that our usage of the Jordan form is purely algebraic, to enforce second-degree nilpotency, and we are thus not concerned with the notorious numerical issues presented by defective matrices.
  40. If Γ is singular, we can define Db=Γ+AbΓ using the Moore-Penrose pseudoinverse. This reintroduces the possibility of zero eigenvalues in Rf, but we found that it was not necessary in practice.
  41. We found that the cMPS Ansatz for two-fermion mixtures can be reimplemented more efficiently based on this Bose-Fermi one; the details will be given elsewhere.
  42. The null right eigenvalue of T is given by the identity matrix (with the double index of the direct-product basis interpreted as rows and columns, respectively).
  43. C. N. Yang, Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction, Phys. Rev. Lett. 19, 1312 (1967).
  44. B. Sutherland, Further Results for the Many-Body Problem in One Dimension, Phys. Rev. Lett. 20, 98 (1968).
  45. C. K. Lai, Thermodynamics of a mixture of fermions and bosons in one dimension with a repulsive δfunction potential, J. Math. Phys. (Melville, NY) 15, 954 (1974).
  46. The attractive case (g<0) is also integrable, but the system is expected to be unstable to the formation of solitons, as in the Lieb-Liniger case [62].
  47. A. Imambekov and E. Demler, Exactly solvable case of a one-dimensional Bose–Fermi mixture, Phys. Rev. A 73, 021602(R) (2006); Applications of exact solution for strongly interacting one-dimensional Bose–Fermi mixture: Low-temperature correlation functions, density profiles, and collective modes, Ann. Phys. (Amsterdam) 321, 2390 (2006).
  48. M. T. Batchelor, M. Bortz, X. W. Guan, and N. Oelkers, Exact results for the one-dimensional mixed boson-fermion interacting gas, Phys. Rev. A 72, 061603(R) (2005); X.-W. Guan, M. T. Batchelor, and J.-Y. Lee, Magnetic ordering and quantum statistical effects in strongly repulsive Fermi-Fermi and Bose-Fermi mixtures, ibid. 78, 023621 (2008).
  49. H. Frahm and G. Palacios, Correlation functions of one-dimensional Bose-Fermi mixtures, Phys. Rev. A 72, 061604(R) (2005).
  50. H. Hu, L. Guan, and S. Chen, Strongly interacting Bose–Fermi mixtures in one dimension, New J. Phys. 18, 025009 (2016).
  51. A. Imambekov, C. J. Bolech, M. Lukin, and E. Demler, Breakdown of the local density approximation in interacting systems of cold fermions in strongly anisotropic traps, Phys. Rev. A 74, 053626 (2006).
  52. Also 2m=1 is another popular choice; cf. Ref. [63].
  53. Middle of the range considered in Fig. 1 of Ref. [47].
  54. R. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, NJ, 1972).
  55. L. Tonks, The complete equation of state of one, two and three-dimensional gases of hard elastic spheres, Phys. Rev. 50, 955 (1936); M. Girardeau, Relationship between systems of impenetrable bosons and fermions in one dimension, J. Math. Phys. (Melville, NY) 1, 516 (1960).
  56. N. N. Bogolyubov, Izv. Akad. Nauk. Ser. Fiz. 11, 77 (1947) [On the theory of superfluidity, J. Phys. (USSR) 11, 23 (1947)].
  57. R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963).
  58. L. Mathey, D.-W. Wang, W. Hofstetter, M. D. Lukin, and E. Demler, Luttinger Liquid of Polarons in One-Dimensional Boson-Fermion Mixtures, Phys. Rev. Lett. 93, 120404 (2004); L. Mathey and D.-W. Wang, Phase diagrams of one-dimensional Bose-Fermi mixtures of ultracold atoms, Phys. Rev. A 75, 013612 (2007); L. Mathey, Commensurate mixtures of ultracold atoms in one dimension, Phys. Rev. B 75, 144510 (2007).
  59. K. Sun and C. J. Bolech, Pair tunneling, phase separation, and dimensional crossover in imbalanced fermionic superfluids in a coupled array of tubes, Phys. Rev. A 87, 053622 (2013).
  60. Ohio Supercomputer Center, http://osc.edu/ark:/19495/f5s1ph73.
  61. G. H. Golub and J. H. Wilkinson, Ill-conditioned eigensystems and the computation of the Jordan canonical form, SIAM Rev. 18, 578 (1976).
  62. J. Cuevas, P. G. Kevrekidis, B. A. Malomed, P. Dyke, and R. G. Hulet, Interactions of solitons with a Gaussian barrier: splitting and recombination in quasi-one-dimensional and three-dimensional settings, New J. Phys. 15, 063006 (2013); V. A. Yurovsky, B. A. Malomed, R. G. Hulet, and M. Olshanii, Dissociation of One-Dimensional Matter-Wave Breathers due to Quantum Many-Body Effects, Phys. Rev. Lett. 119, 220401 (2017).
  63. E. H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev. 130, 1605 (1963); E. H. Lieb, Exact analysis of an interacting Bose gas. II. The excitation spectrum, ibid. 130, 1616 (1963).

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Phys. Rev. Research 4, L022034– Published 11 May, 2022
  • Received 4 November 2021
  • Accepted 20 April 2022
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