Two-body physics in quasi-low-dimensional atomic gases under spin–orbit coupling

Abstract

One of the most dynamic directions in ultracold atomic gas research is the study of low-dimensional physics in quasi-low-dimensional geometries, where atoms are confined in strongly anisotropic traps. Recently, interest has significantly intensified with the realization of synthetic spin–orbit coupling (SOC). As a first step toward understanding the SOC effect in quasi-low-dimensional systems, the solution of two-body problems in different trapping geometries and different types of SOC has attracted great attention in the past few years. In this review, we discuss both the scattering-state and the bound-state solutions of two-body problems in quasi-one and quasi-two dimensions. We show that the degrees of freedom in tightly confined dimensions, in particular with the presence of SOC, may significantly affect system properties. Specifically, in a quasi-one-dimensional atomic gas, a one-dimensional SOC can shift the positions of confinement-induced resonances whereas, in quasitwo-dimensional gases, a Rashba-type SOC tends to increase the two-body binding energy, such that more excited states in the tightly confined direction are occupied and the system is driven further away from a purely two-dimensional gas. The effects of the excited states can be incorporated by adopting an effective low-dimensional Hamiltonian having the form of a two-channel model. With the bare parameters fixed by two-body solutions, this effective Hamiltonian leads to qualitatively different many-body properties compared to a purely low-dimensional model.

Keywords

artificial gauge field / synthetic spin–orbit coupling / quasi-low dimensional sysem

References

[1] Y. J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, J. V. Porto, and I. B. Spielman, Bose–Einstein condensate in a uniform light-induced vector potential, Phys. Rev. Lett. 102(13), 130401 (2009)

[2] Y. J. Lin, K. Jiménez-García, and I. B. Spielman, Spin–orbit-coupled Bose–Einstein condensates, Nature 471(7336), 83 (2011)

[3] J. Y. Zhang, S. C. Ji, Z. Chen, L. Zhang, Z. D. Du, B. Yan, G. S. Pan, B. Zhao, Y. J. Deng, H. Zhai, S. Chen, and J. W. Pan, Collective dipole oscillations of a spin–orbit coupled Bose–Einstein condensate, Phys. Rev. Lett. 109(11), 115301 (2012)

[4] C. Qu, C. Hamner, M. Gong, C. Zhang, and P. Engels, Non-equilibrium spin dynamics and Zitterbewegung in quenched spin–orbit coupled Bose–Einstein condensates, Phys. Rev. A 88, 021604(R) (2013)

[5] S. C. Ji, J. Y. Zhang, L. Zhang, Z. D. Du, W. Zheng, Y. J. Deng, H. Zhai, S. Chen, and J. W. Pan, Experimental determination of the finite-temperature phase diagram of a spin–orbit coupled Bose gas, Nat. Phys. 10(4), 314 (2014)

[6] P. Wang, Z. Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, Spin-orbit coupled degenerate Fermi gases, Phys. Rev. Lett. 109(9), 095301 (2012)

[7] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Spin-injection spectroscopy of a spin–orbit coupled Fermi gas, Phys. Rev. Lett. 109(9), 095302 (2012)

[8] Z. Fu, L. Huang, Z. Meng, P. Wang, L. Zhang, S. Zhang, H. Zhai, P. Zhang, and J. Zhang, Production of Feshbach molecules induced by spin–orbit coupling in Fermi gases, Nat. Phys. 10(2), 110 (2014)

[9] P. J. Wang and J. Zhang, Spin–orbit coupling in Bose–Einstein condensate and degenerate Fermi gases, Front. Phys. 9(5), 612 (2014)

[10] H. Zhai, Spin–orbit coupled quantum gases, Int. J. Mod. Phys. B 26(01), 1230001 (2012)

[11] V. Galitski and I. B. Spielman, Spin–orbit coupling in quantum gases, Nature 494(7435), 49 (2013)

[12] X. Zhou, Y. Li, Z. Cai, and C. Wu, Unconventional states of bosons with the synthetic spin–orbit coupling, J. Phys. B 46(13), 134001 (2013)

[13] N. Goldman, G. Juzeliūnas, P. Öhberg, and I. B. Spielman, Light-induced gauge fields for ultracold atoms, Rep. Prog. Phys. 77(12), 126401 (2014)

[14] H. Zhai, Degenerate quantum gases with spin–orbit coupling: A review, Rep. Prog. Phys. 78(2), 026001 (2015)

[15] J. Zhang, H. Hu, X. J. Liu, and H. Pu, Fermi gases with synthetic spin–orbit coupling, Annu. Rev. Cold At. Mol. 2, 81 (2014)

[16] Y. Xu and C. Zhang, Topological Fulde–Ferrell superfluids of a spin–orbit coupled Fermi gas, Int. J. Mod. Phys. B 29(01), 1530001 (2015)

[17] W. Yi, W. Zhang, and X. Cui, Pairing superfluidity in spin–orbit coupled ultracold Fermi gases, Sci. China- Phys. Mech. Astron. 58(1), 014201 (2015)

[18] J. P. Vyasanakere and V. B. Shenoy, Bound states of two spin-1/2 fermions in a synthetic non-Abelian gauge field, Phys. Rev. B 83(9), 094515 (2011)

[19] J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, BCS–BEC crossover induced by a synthetic non-Abelian gauge field, Phys. Rev. B 84(1), 014512 (2011)

[20] X. Cui, Mixed-partial-wave scattering with spin–orbit coupling and validity of pseudopotentials, Phys. Rev. A 85(2), 022705 (2012)

[21] Z. F. Xu and L. You, Dynamical generation of arbitrary spin–orbit couplings for neutral atoms, Phys. Rev. A 85(4), 043605 (2012)

[22] F. Wu, R. Zhang, T. S. Deng, W. Zhang, W. Yi, and G. C. Guo, BCS–BEC crossover and quantum phase transition in an ultracold Fermi gas under spin–orbit coupling, Phys. Rev. A 89(6), 063610 (2014)

[23] Z. Y. Shi, X. Cui, and H. Zhai, Universal trimers induced by spin–orbit coupling in ultracold Fermi gases, Phys. Rev. Lett. 112(1), 013201 (2014)

[24] X. Cui, and W. Yi, Universal Borromean binding in spin–orbit-coupled ultracold fermi gases, Phys. Rev. X 4(3), 031026 (2014)

[25] M. Sato, Y. Takahashi, and S. Fujimoto, Non-Abelian topological order in s-wave superfluids of ultracold fermionic atoms, Phys. Rev. Lett. 103(2), 020401 (2009)

[26] J. P. Kestner and L. M. Duan, Conditions of low dimensionality for strongly interacting atoms under a transverse trap, Phys. Rev. A 74(5), 053606 (2006)

[27] J. P. Kestner and L. M. Duan, Effective low-dimensional Hamiltonian for strongly interacting atoms in a transverse trap, Phys. Rev. A 76(6), 063610 (2007)

[28] Z. Q. Yu and H. Zhai, Spin–orbit coupled Fermi gases across a Feshbach resonance, Phys. Rev. Lett. 107(19), 195305 (2011)

[29] R. Zhang, F. Wu, J. R. Tang, G. C. Guo, W. Yi, and W. Zhang, Significance of dressed molecules in a quasi-two-dimensional Fermi gas with spin–orbit coupling, Phys. Rev. A 87(3), 033629 (2013)

[30] T. Stöferle, H. Moritz, K. Günter, M. Köhl, and T. Esslinger, Molecules of fermionic atoms in an optical lattice, Phys. Rev. Lett. 96(3), 030401 (2006)

[31] B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, Tonks–Girardeau gas of ultracold atoms in an optical lattice, Nature 429(6989), 227 (2004)

[32] T. Kinoshita, T. Wenger, and D. S. Weiss, Observation of a one-dimensional Tonks–Girardeau gas, Science 305(5687), 1125 (2004)

[33] M. Köhl, H. Moritz, T. Stöferle, K. Günter, and T. Esslinger, Fermionic atoms in a three dimensional optical lattice: Observing Fermi surfaces, dynamics, and interactions, Phys. Rev. Lett. 94(8), 080403 (2005)

[34] Y. Castin, Simple theoretical tools for low dimension Bose gases, J. Phys. IV 116, 89 (2004)

[35] D. E. Sheehy and L. Radzihovsky, Quantum decoupling transition in a one-dimensional Feshbach-resonant superfluid, Phys. Rev. Lett. 95(13), 130401 (2005)

[36] E. Orignac and R. Citro, Phase transitions in the boson–fermion resonance model in one dimension, Phys. Rev. A 73(6), 063611 (2006)

[37] M. Olshanii, Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons, Phys. Rev. Lett. 81(5), 938 (1998)

[38] T. Bergeman, M. G. Moore, and M. Olshanii, Atom–atom scattering under cylindrical harmonic confinement: Numerical and analytic studies of the confinement induced resonance, Phys. Rev. Lett. 91(16), 163201 (2003)

[39] T. Busch, B. G. Englert, K. Rzażewski, and M. Wilkens, Two cold atoms in a harmonic trap, Found. Phys. 28(4), 549 (1998)

[40] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov, Bose–Einstein condensation in quasi-2D trapped gases, Phys. Rev. Lett. 84(12), 2551 (2000)

[41] D. S. Petrov and G. V. Shlyapnikov, Interatomic collisions in a tightly confined Bose gas, Phys. Rev. A 64(1), 012706 (2001)

[42] P. O. Fedichev, M. J. Bijlsma, and P. Zoller, Extended molecules and geometric scattering resonances in optical lattices, Phys. Rev. Lett. 92(8), 080401 (2004)

[43] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser, Resonance superfluidity in a quantum degenerate Fermi gas, Phys. Rev. Lett. 87(12), 120406 (2001)

[44] M. H. Szymańska, K. Góral, T. Köhler, and K. Burnett, Conventional character of the BCS–BEC crossover in ultracold gases of K⁴⁰, Phys. Rev. A 72(1), 013610 (2005)

[45] Q. Chen, J. Stajic, S. Tan, and K. Levin, BCS–BEC crossover: From high temperature superconductors to ultracold superfluids, Phys. Rep. 412(1), 1 (2005)

[46] Q. Chen and J. Wang, Pseudogap phenomena in ultracold atomic Fermi gases, Front. Phys. 9(5), 570 (2014)

[47] L. M. Duan, Effective Hamiltonian for fermions in an optical lattice across a Feshbach resonance, Phys. Rev. Lett. 95(24), 243202 (2005)

[48] R. B. Diener and T. L. Ho, Fermions in optical lattices swept across Feshbach resonances, Phys. Rev. Lett. 96(1), 010402 (2006)

[49] C. A. Regal, M. Greiner, and D. S. Jin, Observation of resonance condensation of fermionic atom pairs, Phys. Rev. Lett. 92(4), 040403 (2004)

[50] K. Huang, Statistical Mechanics, 2nd Ed., New York: Wiley, 1987

[51] L. Tonks, The complete equation of state of one, two and three-dimensional gases of hard elastic spheres, Phys. Rev. 50(10), 955 (1936)

[52] H. Moritz, T. Stöferle, K. Günter, M. Köhl, and T. Esslinger, Confinement induced molecules in a 1D Fermi gas, Phys. Rev. Lett. 94(21), 210401 (2005)

[53] J. K. Chin, D. E. Miller, Y. Liu, C. Stan, W. Setiawan, C. Sanner, K. Xu, and W. Ketterle, Evidence for superfluidity of ultracold fermions in an optical lattice, Nature 443(7114), 961 (2006)

[54] Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, and J. Dalibard, Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas, Nature 441(7097), 1118 (2006)

[55] W. Zhang, G. D. Lin, and L. M. Duan, BCS–BEC crossover of a quasi-two-dimensional Fermi gas: The significance of dressed molecules, Phys. Rev. A 77(6), 063613 (2008)

[56] P. Dyke, E. D. Kuhnle, S. Whitlock, H. Hu, M. Mark, S. Hoinka, M. Lingham, P. Hannaford, and C. J. Vale, Crossover from 2D to 3D in a weakly interacting Fermi gas, Phys. Rev. Lett. 106(10), 105304 (2011)

[57] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, 2nd Ed., New York: Dover, 1975

[58] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 3rd Ed., Oxford: Butterworth–Heinemann, 1999

[59] Y. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, Influence on inelastic processes of the phase transition in a weakly collisional two-dimensional Bose gas, Sov. Phys. JETP 66, 480 (1987)

[60] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, 2nd Ed., New York: McGraw-Hill, 1995

[61] W. Zhang and P. Zhang, Confinement-induced resonances in quasi-one-dimensional traps with transverse anisotropy, Phys. Rev. A 83(5), 053615 (2011)

[62] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, 2nd Ed., New York: Gordon and Breach, 1986

[63] Z. Idziaszek and T. Calarco, Analytical solutions for the dynamics of two trapped interacting ultracold atoms, Phys. Rev. A 74(2), 022712 (2006)

[64] X. J. Liu and H. Hu, Topological superfluid in one-dimensional spin–orbit-coupled atomic Fermi gases, Phys. Rev. A 85(3), 033622 (2012)

[65] J. R. Taylor, Scattering Theory, 2nd Ed., New York: Wiley, 1972

[66] P. Zhang, L. Zhang, and W. Zhang, Interatomic collisions in two-dimensional and quasi-two-dimensional confinements with spin–orbit coupling, Phys. Rev. A 86(4), 042707 (2012)

[67] Y. C. Zhang, S. W. Song, and W. M. Liu, The confinement induced resonance in spin–orbit coupled cold atoms with Raman coupling, Sci. Rep. 4, 4992 (2014)

[68] R. Zhang and W. Zhang, Effective Hamiltonians for quasi-one-dimensional Fermi gases with spin–orbit coupling, Phys. Rev. A 88(5), 053605 (2013)

[69] L. Dell'Anna, G. Mazzarella, and L. Salasnich, Condensate fraction of a resonant Fermi gas with spin–orbit coupling in three and two dimensions, Phys. Rev. A 84(3), 033633 (2011)

[70] A. J. Leggett, Modern Trends in the Theory of Condensed Matter, 2nd Ed., Berlin: Springer-Verlag, 1980

[71] P. Nozières and S. Schmitt-Rink, Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity, J. Low Temp. Phys. 59(3), 195 (1985)

[72] C. A. R. S. de Melo, M. Randeria, and J. R. Engelbrecht, Crossover from BCS to Bose superconductivity– transition temparetaure and time-dependent Ginzburg–Landau theory, Phys. Rev. Lett. 71, 3202 (1993)

[73] J. Zhou, W. Zhang, and W. Yi, Topological superfluid in a trapped two-dimensional polarized Fermi gas with spin–orbit coupling, Phys. Rev. A 84(6), 063603 (2011)

[74] W. Yi and L. M. Duan, BCS–BEC crossover and quantum phase transition for Li6 and K40 atoms across the Feshbach resonance, Phys. Rev. A 73(6), 063607 (2006)

[75] P. Fulde and R. A. Ferrell, Superconductivity in a strong spin-exchange field, Phys. Rev. 135(3A), A550 (1964)

[76] A. I. Larkin and Y. N. Ovchinnikov, Inhomogeneous state of superconductors, Sov. Phys. JETP 20, 762 (1965)

[77] W. V. Liu and F. Wilczek, Interior gap superfluidity, Phys. Rev. Lett. 90(4), 047002 (2003)

[78] C. Zhang, S. Tewari, R. M. Lutchyn, and S. Das Sarma, px+ipy superfluid from s-wave interactions of fermionic cold atoms, Phys. Rev. Lett. 101(16), 160401 (2008)

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