Low-Lying Energy Levels of a One-Dimensional Weakly Interacting Bose Gas under Zero Boundary Conditions


  • M. D. Tomchenko Bogolyubov Institute for Theoretical Physics, Nat. Acad. of Sci. of Ukraine




interacting bosons, Bogoliubov method, zero boundary conditions


We diagonalize the second-quantized Hamiltonian of a one-dimensional Bose gas with a non-point repulsive interatomic potential and zero boundary conditions. At a weak coupling, the solutions for the ground-state energy E0 and the dispersion law E(k) coincide with the Bogoliubov solutions for a periodic system. In this case, the single-particle density matrix F1(x, x′) at T = 0 is close to the solution for a periodic system and, at T > 0, is significantly different from it. We also obtain that the wave function ⟨w(x, t)⟩ of the effective condensate is close to a constant √︀N0/L inside the system and vanishes on the boundaries (here, N0 is the number of atoms in the effective condensate, and L is the size of the system). We find the criterion of applicability of the method, according to which the method works for a finite system at very low temperature and with a weak coupling (a weak interaction or a large concentration).


N.N. Bogoliubov, On the theory of superfluidity. J. Phys. USSR 11, 23 (1947).

R. Feynman. Atomic theory of the two-fluid model of liquid helium. Phys. Rev. 94, 262 (1954). https://doi.org/10.1103/PhysRev.94.262

N.N. Bogoliubov, D.N. Zubarev. The wave function of the lowest state of a system of interacting Bose particles. Sov. Phys. JETP 1, 83 (1955).

K. Brueckner. Theory of Nuclear Structure (Methuen, 1959).

E.H. Lieb, W. Liniger. Exact analysis of an interacting Bose gas. I. The general solution and the ground state. Phys. Rev. 130, 1605 (1963). https://doi.org/10.1103/PhysRev.130.1605

M.D. Tomchenko. Microstructure of He II in the presence of boundaries. Ukr. J. Phys. 59, 123 (2014). https://doi.org/10.15407/ujpe59.02.0123

M. Gaudin. Boundary energy of a Bose gas in one dimension. Phys. Rev. A 4, 386 (1971). https://doi.org/10.1103/PhysRevA.4.386

M. Tomchenko. Point bosons in a one-dimensional box: the ground state, excitations and thermodynamics. J. Phys. A: Math. Theor. 48, 365003 (2015). https://doi.org/10.1088/1751-8113/48/36/365003

M. Tomchenko. Quasimomentum of an elementary excitation for a system of point bosons with zero boundary conditions. arXiv:1705.10565 [cond-mat.quant-gas].

M.A. Cazalilla. Low-energy properties of a one-dimensional system of interacting bosons with boundaries.EPL 59, 793 (2002). https://doi.org/10.1209/epl/i2002-00112-5

M.A. Cazalilla. Bosonizing one-dimensional cold atomic gases. J. Phys. B: At. Mol. Opt. Phys. 37, S1 (2004). https://doi.org/10.1088/0953-4075/37/7/051

M.D. Girardeau, R. Arnowitt. Theory of many-boson systems: Pair theory. Phys. Rev. 113, 755 (1959). https://doi.org/10.1103/PhysRev.113.755

N.N. Bogoliubov. Quasi-Averages in Problems of Statistical Mechanics (Dubna report D-781, 1961) (in Russian).

N.N. Bogoliubov, Lectures on Quantum Statistics, vol. 2: Quasi-Averages (Gordon and Breach, 1970) [ISBN: 0-677-20570-8].

A.L. Fetter. Nonuniform states of an imperfect Bose gas. Ann. Phys. 70, 67 (1972). https://doi.org/10.1016/0003-4916(72)90330-2

C.W. Gardiner. Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross–Pitaevskii equation for a highly condensed Bose gas. Phys. Rev. A 56, 1414 (1997). https://doi.org/10.1103/PhysRevA.56.1414

M.D. Girardeau. Comment on "Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross–Pitaevskii equation for a highly condensed Bose gas". Phys. Rev. A 58, 775 (1998). https://doi.org/10.1103/PhysRevA.58.775

A.G. Leggett. Bose-Einstein condensation in the alkali gases: Some fundamental concepts. Rev. Mod. Phys. 73, 307 (2001). https://doi.org/10.1103/RevModPhys.73.307

V.A. Zagrebnov, J.-B. Bru. The Bogoliubov model of weakly imperfect Bose gas. Phys. Rep. 350, 291 (2001). https://doi.org/10.1016/S0370-1573(00)00132-0

V.A. Zagrebnov. The Bogoliubov theory of weakly imperfect Bose gas and its modern development in: N.N. Bogoliubov, Collection of scientific works in 12 volumes, ed. by A.D. Sukhanov (Nauka, 2007), v. 8. (in Russian)[ISBN: 978-5020339422, 978-5-02-035723-5].

A. Rovenchak. Weakly-interacting bosons in a trap within approximate second quantization approach. J. Low Temp. Phys. 148, 411 (2007). https://doi.org/10.1007/s10909-007-9406-x

A. Rovenchak. Effective Hamiltonian and excitation spectrum of harmonically trapped bosons. Low Temp. Phys. 42, 36 (2016). https://doi.org/10.1063/1.4939154

V.B. Bobrov, A.G. Zagorodny, S.A. Trigger. Coulomb interaction potential and Bose–Einstein condensate. Low Temp. Phys. 41, 901 (2015). https://doi.org/10.1063/1.4936669

J. Sato, E. Kaminishi, T. Deguchi. Finite-size scaling behavior of Bose–Einstein condensation in the 1D Bose gas. arXiv:1303.2775 [cond-mat.quant-gas].

J. Grond, A.I. Streltsov, A.U.J. Lode, K. Sakmann, L.S. Cederbaum, O.E. Alon. Excitation spectra of many-body systems by linear response: General theory and applications to trapped condensates. Phys. Rev. A 88, 023606 (2013). https://doi.org/10.1103/PhysRevA.88.023606

J.W. Kane, L.P. Kadanoff. Long-range order in superfluid helium. Phys. Rev. 155, 80 (1967). https://doi.org/10.1103/PhysRev.155.80

P.C. Hohenberg. Existence of long-range order in one and two dimensions. Phys. Rev. 158, 383 (1967). https://doi.org/10.1103/PhysRev.158.383

U.R. Fischer. Existence of long-range order for trapping interacting bosons. Phys. Rev. Lett. 89, 280402 (2002). https://doi.org/10.1103/PhysRevLett.89.280402

A.I. Bugrij, V.M. Loktev. On the theory of Bose–Einstein condensation of quasiparticles: On the possibility of condensation of ferromagnons at high temperatures. Low Temp. Phys. 33, 37 (2007). https://doi.org/10.1063/1.2409633

D.A. Kirzhnits. Superconductivity and elementary particles. Sov. Phys. Usp. 21, 470 (1978). https://doi.org/10.1070/PU1978v021n05ABEH005556

A. Griffin. BEC and the new world of coherent matter waves, in Theoretical Physics at the End of the Twentieth Century, ed. by Y. Saint-Aubin and L. Vinet (Springer, 2002). [ISBN: 0387953116, 978-0387953113]. https://doi.org/10.1007/978-1-4757-3671-7_4

O. Penrose, L. Onsager. Bose–Einstein condensation and liquid helium. Phys. Rev. 104, 576 (1956). https://doi.org/10.1103/PhysRev.104.576

M. Tomchenko. On a fragmented condensate in a uniform Bose system. arXiv:1808.08203 [cond-mat.quant-gas].

N.N. Bogoliubov. Lectures on Quantum Statistics, vol. 1: Quantum Statistics (Gordon and Breach, 1967) [ISBN: 0677200307, 9780677200309].

E.P. Gross. Unified theory of interacting bosons. Phys. Rev. 106, 161 (1957). https://doi.org/10.1103/PhysRev.106.161

E.P. Gross. Structure of a quantized vortex in boson systems. Nuovo Cimento 20, 454 (1961). https://doi.org/10.1007/BF02731494

L.P. Pitaevskii. Vortex lines in an imperfect Bose gas. Sov. Phys. JETP 13, 451 (1961).

M. Tomchenko. Expansions of the interatomic potential under various boundary conditions and the transition to the thermodynamic limit. arXiv:1403.8014 [cond-mat.other].

S.N. Bose. Plancks gesetz und lichtquantenhypothese. Z. Phys. 26, 178 (1924). https://doi.org/10.1007/BF01327326

C.J. Pethick, H. Smith. Bose–Einstein Condensation in Dilute Gases (Cambridge Univ. Press, 2008), Chap. 15. https://doi.org/10.1017/CBO9780511802850

W. Ketterle, N.J. van Druten. Bose–Einstein condensation of a finite number of particles trapped in one or three dimensions. Phys. Rev. A 54, 656 (1996). https://doi.org/10.1103/PhysRevA.54.656

M. Tomchenko. Bose–Einstein condensation in a one-dimensional system of interacting bosons. J. Low Temp. Phys. 182, 170 (2016). https://doi.org/10.1007/s10909-015-1435-2

M. Schwartz. Off-diagonal long-range behavior of interacting Bose systems. Phys. Rev. B 15, 1399 (1977). https://doi.org/10.1103/PhysRevB.15.1399

F.D.M. Haldane. Effective harmonic-fluid approach to low-energy properties of one-dimensional quantum fluids. Phys. Rev. Lett. 47, 1840 (1981). https://doi.org/10.1103/PhysRevLett.47.1840

D.S. Petrov, D.M. Gangardt, G.V. Shlyapnikov. Low-dimensional trapped gases. J. Phys. IV France 116, 3 (2004). https://doi.org/10.1051/jp4:2004116001

V.N. Popov. On the theory of the superfluidity of two- and one-dimensional bose systems. Theor. Math. Phys. 11, 565 (1972). https://doi.org/10.1007/BF01028373

A. Berkovich, G. Murthy. Time-dependent multipoint correlation functions of the nonlinear Schr?odinger model. Phys. Lett. A 142, 121 (1989). https://doi.org/10.1016/0375-9601(89)90172-2

C. Mora, Y. Castin. Extension of Bogoliubov theory to quasicondensates. Phys. Rev. A 67, 053615 (2003). https://doi.org/10.1103/PhysRevA.67.053615

V. Dunjko, M. Olshanii. A Hermite–Pad?e perspective on Gell-Mann–Low renormalization group: an application to the correlation function of Lieb–Liniger gas. arXiv:0910.0565 [cond-mat.quant-gas].

I. Bouchoule, N.J. van Druten, C.I. Westbrook. Atom chips and one-dimensional Bose gases. arXiv:0901.3303 [physics.atom-ph].

J.-S. Caux, P. Calabrese, N.A. Slavnov. One-particle dynamical correlations in the one-dimensional Bose gas. J. Stat. Mech. P01008 (2007). https://doi.org/10.1088/1742-5468/2007/01/P01008

M.T. Batchelor, X.W. Guan, N. Oelkers, C. Lee. The 1D interacting Bose gas in a hard wall box. J. Phys. A: Math. Gen. 38, 7787 (2005). https://doi.org/10.1088/0305-4470/38/36/001

T. Giamarchi. Quantum Physics in One Dimension (Clarendon Press, 2003) [ISBN: 0-19-852500-1]. https://doi.org/10.1093/acprof:oso/9780198525004.001.0001

J.M. Vogels, K. Xu, C. Raman, J.R. Abo-Shaeer, W. Ketterle. Experimental observation of the Bogoliubov transformation for a Bose–Einstein condensed gas. Phys. Rev. Lett. 88, 060402 (2002). https://doi.org/10.1103/PhysRevLett.88.060402




How to Cite

Tomchenko, M. D. (2019). Low-Lying Energy Levels of a One-Dimensional Weakly Interacting Bose Gas under Zero Boundary Conditions. Ukrainian Journal of Physics, 64(3), 250. https://doi.org/10.15407/ujpe64.3.250



Structure of materials