Bose-Hubbard model with attractive interactions and inhomogeneous lattice potential Mukesh Khanore and Bishwajyoti Dey Citation: AIP Conference Proceedings 1591, 139 (2014); doi: 10.1063/1.4872521 View online: http://dx.doi.org/10.1063/1.4872521 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1591?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Long cycles in the infinite-range-hopping Bose–Hubbard model J. Math. Phys. 50, 073301 (2009); 10.1063/1.3158836 Accurate determination of the superfluidinsulator transition in the onedimensional BoseHubbard model AIP Conf. Proc. 1076, 292 (2008); 10.1063/1.3046265 Effect of Quantum Correction in the BoseHubbard Model AIP Conf. Proc. 850, 53 (2006); 10.1063/1.2354603 The GHS interaction model for strong attractive potentials Phys. Fluids 7, 1173 (1995); 10.1063/1.868774 Nematic ordering in the lattice model with attractive isotropic interaction J. Chem. Phys. 67, 948 (1977); 10.1063/1.434920 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 196.1.114.238 On: Fri, 02 May 2014 07:22:14 Bose-Hubbard model with attractive interactions and inhomogeneous lattice potential Mukesh Khanore and Bishwajyoti Dey* Department of Physics, University of Pune, Pune-411007, India. *Email: bdey@physics.unipune.ac.in Abstract. We consider the Bose-Hubbard model of attractive Bose gas in an optical lattice potential and in an additional inhomogeneous double well magnetic trap potential. We calculate the energy spectrum and the on-site number fluctuation of the system using numerical exact diagonalization procedure. Our results shows very clearly the role played by the degenerate double well minimum of the trap potential on the nature of instability of the attractive Bose-Hubbard model and the phase-transition picture of the system. Keywords: Attractive Bose-Hubbard model, Double Well trap, exact diagonalization PACS: 03.75.Lm, 67.85.Hj INTRODUCTION Bose-Einstein condensates (BEC) are ideals systems for studying many-body physics in the weak as well as strong interaction regime of the condensed matter systems. Since the condensate is free from impurities, it is useful to study various phenomena like Bloch oscillations, quantum phase transitions etc. in the system. BECs have been experimentally produced in a wide range of trapping potentials and loaded into various optical lattices [1, 2]. The nature of interaction between the neutral condensate atoms, whether attractive or repulsive, can be experimentally controlled by Feshbach resonance mechanism. The loading of the condensates on the optical lattice allows the condensate atoms to move in a periodic potential. For deep optical lattice, the atoms can be trapped at the lattice sites. The trapped Bose gas with attractive interaction loaded in an optical lattice can be described by the attractive Bose-Hubbard model hamiltonian given by H { J ¦b i  i, j ! † bj  ª ¦ «¬H i i 1 º nˆi  U nˆi (nˆi  1)» 2 ¼ where J is the hopping matrix to the neighboring sites, U is the strength of the onsite interactions due to Swave scattering, i ranges from 1 to M where M is lattice sites, H i is the additional inhomogeneous double well confining magnetic trap potential, Hi ª§ M  1 · 2 § M  ' ·º «¨ i  ¸» ¸ ¨ 2 ¹ © 2 ¹¼» ¬«© 2 Ȝ is the measure of the curvature and 0 ”ǻ is the offset of the lattice from the center of the confining potential. The nature of the potential is shown in Fig. 1. The behavior of the ground state of the Bose gas with attractive interactions and trapped in a symmetric double well potential shows that the systems experience second-order phase transition [3]. In the absence of trapping potential, for J >> | U | the condensate shows large phase coherence and for U > 0 the systems undergoes Mott-insulator transition at a critical value of U / J when the on-site number fluctuation įni ĺ 0 G ni {  nˆi 2 !   nˆi ! 2 and all phase coherence vanishes [4]. For U < 0, the attractive interaction leads to an instability of the phase-coherent condensate which however is complicated by the presence of the double-well confining trap. For the case of symmetric wells, the Solid State Physics AIP Conf. Proc. 1591, 139-141 (2014); doi: 10.1063/1.4872521 © 2014 AIP Publishing LLC 978-0-7354-1225-5/$30.00 139 to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject 196.1.114.238 On: Fri, 02 May 2014 07:22:14 system is unable to decide which well to choose and this leads to quantum superposition of the two possibilities. This superposition state is associated with large number fluctuations leading to įQi ĺQi where ni Ł< n̂i >. In this paper we examine the instability of the phase coherent condensate in presence of inhomogeneous double-well lattice potential. In order to treat the strong interaction regime we have calculated the energy spectrum of the Hamiltonian (Eq.(1)) by numerical exact diagonalization method. In the Fock state basis there are (N + M -1)!=[N!(M -1)!] states, where N is the number of atoms and M is the lattice sites. In these states the Hamiltonian reduces to a sparse matrix and the exact eigenvalues and eigenstates are calculated by diagonalizing the Hamiltonian matrix using LAPACK subroutine (DSYEV). The onsite number fluctuation įni and site occupation number ni are calculated in the ground state of the system. FIGURE 1. Plot of the double well potential for ǻ = 0.9. RESULTS Fig. 2 shows the results of the exact calculations for the single well (harmonic) inhomogeneous lattice potential. For U/J = -0.5, the number fluctuations įn3 increases to a large value and then decreases to zero when the atoms accumulates into a single site. For the same value of U / J the site occupation n3 also shows a transition to higher value. The inset shows the energy difference between the first excited state and the ground state. The degeneracy of these two states is also lifted at U/J = -0.5. Since the potential H i has degenerate minimum near sites i = 2 and i = 5 (Fig. 1), we plot number fluctuations įni for sites i = 2 and i = 5 in Fig. 3. Both the curves shows a peak for U / J = -0.85 and then it decreases. For U/ J < -0.85 the system is unable to choose between the sites and accordingly both įni and ni (Fig. 4) shows an oscillatory behavior between i = 2 and i = 5 sites. This form quantum superposition between the states with large number fluctuations įQi ĺQi . From Fig. 3 and Fig. 4 we can see that indeed įQi Ł ni for U / J = -0.85. This leads to a phase coherence condensate. On the other hand for U / J < -0.85, įQi ĺ 0 and ni reaches saturation value N. This means all phase coherence vanishes leading to Mott-insulator transition. FIGURE 2. Plot of n3 and įn3 for Ȝ = 0.02 J and ǻ = 0.9 for M=6 , N=10 as a function of U / J. The inset .shows energy difference between the first excited state and the ground state. FIGURE 3. Plot shows įn2 , įn5 as a function of U / J. The inset plot shows energy difference between the first excited state and the ground state as a function U / J for Ȝ = 0.02 J and ǻ = 0.9 for M=6 , N=10 140 to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject 196.1.114.238 On: Fri, 02 May 2014 07:22:14 FIGURE 4. Plot shows n2 , n5 as a function of U / J. for Ȝ = 0.02 J and ǻ = 0.9 for M=6 , N=10 CONCLUSION In conclusion, we have studied the ground state properties of Bose-Hubbard model of attractive Bose gas in an optical lattice and double-well lattice potential. The eigenvalues and eigenstates of the system are obtained by numerical exact diagonalization method. The site occupation and the number fluctuations are calculated in the ground state of the system. The attractive interactions and doublewell nature of the trap potential shows interesting behaviour of the Bose gas, such as, phase coherent condensate, Mott-insulator transition and oscillatory behaviour of the number fluctuation and site occupation number which are very different from the repulsive interaction case. ACKNOWLEDGMENTS B. Dey thanks DST for financial assistance through a research project. Mukesh Khanore thanks Aniruddha Kibey and Bhooshan Gadre for many helpful discussions. REFERENCES 1. 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