Tunneling Dynamics of Dipolar Bosonic System with Periodically Modulated s-wave Scattering

Transcription

1 Commun. Theor. Phys. 61 (2014) Vol. 61, No. 5, May 1, 2014 Tunneling Dynamics of Dipolar Bosonic System with Periodically Modulated s-wave Scattering YU Zi-Fa ( Ù) and XUE Ju-Kui ( û) Key Laboratory of Atomic & Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou , China (Received September 25, 2013; revised manuscript received January 10, 2014) Abstract Within the mean-field three-site Bose Hubbard model, the tunneling dynamics of dipolar bosonic gas with a periodically modulation of s-wave scattering is investigated. The system experiences complex and rich coherent tunneling (CT)-coherent destruction of tunneling (CDT) transitions resulting from the correlated effect among the next-neighbor dipole-dipole interaction, the on-site interaction and the modulated s-wave scattering. In particular, The region of the modulated s-wave scattering for generating CT (CDT) is the widest (narrowest) when the on-site interaction and the next-neighbor dipole-dipole interaction have some correlated values, which are closely related to the tunneling energy and the interaction energy of the system. The correlated values for appearing CDT can be theoretically gained from the tunneling energy and the interaction energy of the system. PACS numbers: Lm, Kk, Be Key words: ultra-cold dipolar gas, triple-well, coherent destruction of tunneling 1 Introduction Breakthroughs in the experimental realization of the dipolar condensate of 52 Cr, [ Er, [3 and 164 Dy [4 5 atoms whose dipole moment are 6µ B, 7µ B, and 10µ B, respectively, in which µ B is Bohr s magneton, and the recent astounding progress in experiments with the KRb polar molecular gas [6 offer a fascinating playground to design and achieve exceedingly rich and complex quantum phenomena. [7 8 The major features of the dipoledipole interaction (DDI) coupling in inter-site dipolar condensates [9 13 are its long range character and anisotropy, so the DDI introduces many unique and novel phase diagrams, such as the supersolid phase and the insulating checkerboard phase. [14 18 In particular, a triplewell system supports a simple model for exhibiting the effect of DDI. The quantum tunneling and the transportation in a triple-well system have been attracting great interests and were investigated extensively. [19 28 The phenomenon of CDT [29 34 by a driving field has been a highlight of quantum dynamics control during the past decade, which is surprising and is discovered first by Grossman et al. [29 30 It has been observed in different physical system experimentally. [35 36 Modulation traditionally focuses on an external periodic potential, in which the BEC has Mott-insulator phase, [37 38 but a new kind of modulation was already demonstrated in experiment [39 which is the modulated interatomic interactions (or scattering length) of a BEC periodically in space with laser beams [40 by optical Feshbach resonance. [41 43 The modulated interatomic interaction has also been studied in the different system previously. Interestingly, this modulation makes it possible to precisely control the number of bosons allowed to tunnel for a two-mode BEC, [44 which can also lead to pair superfluid phases, defect-free Mott states, and holon and doublon superfluids in optical lattices. [45 However, a full understanding of periodically modulated interactions bosonic system with both short-range interactions and next-neighbor DDI is still missing. In this paper, the tunneling dynamics of dipolar bsonic gas in a periodically modulated s-wave scattering triplewell is investigated. The correlated effect among the nextneighbor dipolar-dipolar interaction (DDI), the on-site interaction and the modulated s-wave scattering results in the complex transitions from CT to CDT. More interestingly, when the on-site interaction and the DDI have some correlated values, the region of the modulated s-wave scattering for generating CT (CDT) is the widest (narrowest). According to the tunneling energy and the interaction energy, we theoretically demonstrate the correlated values of the strongest CT and the weakest CDT between the next-neighbor DDI and the on-site interaction. 2 Mean-Field Model We consider N dipolar bosons loaded in a triple well (see Fig. 1), with magnetic dipole moment orientated along a given direction by an external magnetic field. Due to the on-site interaction and the inter-site next-neighbor dipole-dipole interaction (DDI) between bosons, a dilute gas of bosons can be described by a Bose Hubbard (HB) Supported by the National Natural Science Foundation of China under Grant Nos and , by the Natural Science Foundation of Gansu province under Grant No. 2011GS04358, and by Creation of Science and Technology of Northwest Normal University, China under Grant Nos. NWNU-KJCXGC-03-48, NWNU-LKQN-10-27, NWNU-LKQN-12-12, and NWNU-LKQN xuejk@nwnu.edu.cn c 2014 Chinese Physical Society and IOP Publishing Ltd

2 566 Communications in Theoretical Physics Vol. 61 model: [37,44 46 Ĥ q = v[â 2 (â 1 + â 3 ) + H.c. + U 2N + U 1 N 3 ˆn i (ˆn i 1) i=1 (ˆn 1ˆn 2 + ˆn 3ˆn χ ˆn 1ˆn 3 ), (1) where â i (â i) for i = 1, 2, 3 are the creation (annihilation) operators in the i-th localized state, ˆn i = â i â i denotes the number operator at site i. The energy parameters in this equation are the tunneling constant v, the effective on-site interaction U, and the inter-site DDI U 1 (U 1 = Uij dd (i j)). 4 χ 8 refers to the next-neighbor DDI depending on the trap geometry, [19 20 here, we set χ = 8. [20 The on-site parameter U is given by two contributions: one arises from the s-wave scattering U s = U s (a) and the second one is due to the on-site dipole-dipole interaction Uii dd, where a is the s-wave scattering length, i.e., U = U s (a) + Uii dd. As demonstrated experimentally,[9 10 while the s-wave scattering U s (a), i.e., a, can be modulated by the Feshbach resonance technique, the dipoledipole interactions Uii dd and Uij dd can be contorted by changing the well width and the orientation of dipoles. Here, in order to manipulate the quantum tunneling of dipolar system, we apply a periodic modulation of the interatomic on-site s-wave interaction, i.e., a a(t). Then, U can be assumed to be [41 45 U = U 0 + Acosωt, (2) where, the time dependent term A cos(ωt) results from the periodic modulation of the s-wave scattering length a, A is the modulated strength, and ω is the modulated frequency. In our study, we focus on the case with U 0 > 0 and U 1 > 0 (repulsive inter-site DDI). When any thermal effects can be neglected at zero temperature and the number of bosons N is sufficiently large, it is suitable to make the following coherent substitutes b i = â i / N, (3) where b i is probability amplitude of atoms in the i-th well and the total probability 3 i=1 b i 2 = 1 is conserved. So, we can acquire a mean field Hamiltonian H mf = Ĥq N = v[b 2 (b 1 + b 3 ) + b 2 (b 1 + b 3 ) + U 3 b i 4 2 i=1 vb 2, (7) where we have used the natural unit = 1. The parameters v, A, ω, U 0, and U 1 have the same units of energy and we can treat them as dimensionless parameters because the ratios between these parameters are essential. 3 Tunneling Dynamics As is well known, CT and CDT are the two dynamic factors of periodically modulated system. In order to present the detailed tunneling dynamics of dipolar bosonic system with periodically modulated s-wave scattering, we numerically solve the fully nonlinear Eqs. (5) (7) with the particles initially localized in the first well. Here, we focus on high-frequency modulations, i.e., ω max{u 0, U 1, v}, as discussed in the following section. The variation of the number of atoms b i 2 in the i-th well with normalized time ωt are plotted in Figs. 2 4 for different A/ω, U 0 /ω, and U 1 /ω. Note that, if the number of atoms b 1 2 in the first well oscillates between zero and one, then it demonstrates no suppression (i.e., CT occurs), however, if b 1 2 1, then the particles are completely suppressed in the first potential well, namely, CDT occurs. Fig. 1 Schematic diagram of triple-well potential structure. + U 1 ( b 1 2 b b 3 2 b χ b 1 2 b 3 2). (4) The system can be described by a three-mode Gross Pitaevskii equation iḃ1 = [U 0 + Acos(ωt) b 1 2 b 1 + U 1 ( b χ b 3 2) b 1 vb 2, (5) iḃ2 = [U 0 + Acos(ωt) b 2 2 b 2 + U 1 ( b b 3 2 )b 2 vb 1 vb 3, (6) iḃ3 = [U 0 + Acos(ωt) b 3 2 b 3 + U 1 ( b χ b 1 2) b 3 Fig. 2 The number of atoms b i 2 at the i-th well with U 1/ω = 0 and v/ω = From the first panel to the sixth panel: A/ω = 0,2.00, 2.25, 2.50, 3.00, 4.00, respectively. From the first column to the third column: U 0/ω = 0, 0.04, 0.16, respectively. Figure 2 depicts the tunneling dynamics in the case of inter-site DDI U 1 /ω = 0. As A/ω increases (from

3 No. 5 Communications in Theoretical Physics top to bottom for a fixed column), when U0 /ω is small (e.g., U0 /ω = 0, 0.04), the system undergoes CT-CDTCT transitions, but when U0 /ω is sufficiently large (e.g., U0 /ω = 0.16), the system only experiences CT-CDT transitions. For a fixed A/ω (from left to right for a fixed panel), when A/ω is small, the system is in CT region, however, when A/ω is large, the system can undergo CTCDT-CT transitions (e.g., A/ω = 2.25, 3.00) or CT-CDT transitions (e.g., A/ω = 4.00). We also find that the system is in CDT region as A/ω = 2.5 which satisfies J0 (A/ω) 0, where J0 is the 0-th order Bessel function. In short, the modulated s-wave scattering can result in the system converting between CT and CDT, and U0 /ω can change the range of modulated s-wave scattering for appearing CDT. 567 first well h b1 2 i for different U0 /ω or U1 /ω is more clearly demonstrated in Fig. 5, in which if h b1 2 i 1/3 or 1/2, the system is in CT region (particles tunnel among three wells or between the first well and the third well), and if h b1 2 i 1.0, the system is in CDT region (particles are all localized in the first well). One can find that, the CT and CDT regions in U1 /ω(u0 /ω) A/ω plane are separated by the vertical lines of (A/ω)0 = 2.5, 5.6,..., which is determined by J0 [(A/ω)0 0. On the other hand, when A/ω (A/ω)0, the system is in CDT region wherever U0 /ω or U1 /ω is. Out of these regions, the system can either be in CT or CDT region depending on U1 /ω or U0 /ω. In these regions, for a fixed A/ω, the system can undergo CT-CDT or CDT-CT-CDT or CT-CDT-CT-CDT transitions as U1 /ω or U0 /ω increases. The CT regions shrink as A/ω increases. For fixed U1 /ω and U0 /ω, when U0 /ω (U1 /ω) is small, the system can undergo CT-CDT-CT- transitions as A/ω increases, however, when U0 /ω (U1 /ω) is large, the system can only experience CT-CDT transitions as A/ω increases. Fig. 3 The number of atoms bi 2 at the i-th well with U0 /ω = 0 and v/ω = From the first panel to the sixth panel: A/ω = 0, 2.00, 2.25, 2.50, 3.00, 4.00, respectively. From the first column to the third column: U1 /ω = 0, 0.10, 0.16, respectively. In Fig. 3, we show the detailed tunneling dynamics with only considering the next-neighbor DDI, which is similar to the results of U1 /ω = 0 presented in Fig. 2. When we consider the correlation between DDI and onsite interaction, the tunneling dynamics have some different characters as shown in Fig. 4, where we set U0 /ω = 0.08 and vary U1 /ω. When U1 /ω U0 /ω (the first column of Fig. 4) or U1 /ω U0 /ω (the third column of Fig. 4), the system can only undergo CT-CDT transitions as A/ω increases. When U1 /ω U0 /ω, the system experiences CT-CDT-CT transitions, and CDT occurs at A/ω = 2.5, which satisfies J0 (A/ω) 0. That is, the correlation of on-site interaction and DDI can result in the system undergoing complex transitions between CT and CDT. The CT and CDT diagram in U1 /ω(u0 /ω) A/ω plane obtained by the long-time averaging of particles in the Fig. 4 The number of atoms bi 2 at the i-th well with U0 /ω = 0.08 and v/ω = From the first panel to the sixth panel: A/ω = 0, 2.00, 2.25, 2.50, 3.00, 4.00, respectively. From the first column to the third column: U1 /ω = 0, 0.08, 0.16, respectively. In order to show clearly how the atomic interaction influences the dynamics of the tunneling, the CT and CDT diagram in U1 /ω U0 /ω plane obtained by the long-time averaging of particles in the first well h b1 2 i for different A/ω is plotted in Fig. 6. More interestingly, one can find that the CT can occur in the regions adjacent to U0 = U1 and U0 = U1 /χ, out of these regions, the CDT is revealed, and the width of these CT regions depends on A/ω, in particular, when A/ω = 2.50 (Fig. 6(d)), only CDT can be discovered, which is also shown in Fig. 5. So the correlation among the DDI, the on-site interaction and the

4 568 Communications in Theoretical Physics Vol. 61 modulated s-wave scattering has significantly influence on the tunneling dynamics. Fig. 5 The CT and CDT diagram in U 1/ω(U 0/ω) A/ω plane obtained by the long-time averaging of particles in the first well b 1 2 for different U 0/ω or U 1/ω and v/ω = The first panel: U 0/ω = 0,0.04, 0.08, respectively. The second panel: U 1/ω = 0,0.04, 0.08, respectively. The total normalized time used for averaging is in dimensionless units. Fig. 6 The CT and CDT diagram in U 1/ω U 0/ω plane obtained by the long-time averaging of particles in the first well b 1 2 for different A/ω with v/ω = 0.05, and A/ω = 0, 2.00, 2.25, 2.50, 3.00, 4.00, respectively. The total normalized time used for averaging is in dimensionless units. 4 Tunneling Energy and Interaction Energy To have a deep insight into the tunneling dynamics obtained in Figs. 2 6, in this section, we focus on the tunneling energy and the interaction energy of the system because they are two basic concepts and tools in understanding tunneling dynamics. It is difficult for us to obtain exactly the tunneling energy and the interaction energy except for the zero-modulation case. However, in the high-frequency region (ω max{v, U 0, U 1 }), the driving field varies so rapidly that the wave function is not able to respond this variation appreciably and thus can be approximately described by the slowly varying function of time. So we take advantage of the transformation [47 b i = exp [ iasin(ωt) c i 2 c i. (8) ω Using the formula J 0 ω ( c i 2 c j 2 ) = 1 T T/2 T/2 e i[asin(ωt)( ci 2 c j 2 ) dt, (9) where J 0 (x) is the 0-th order Bessel function, averages out the high-frequency terms, [48 49 we acquire a nonmodulated nonlinear model, ic 1 = [U 0 c U 1 ( c χ c 3 2) c 1 vj 0 ω ( c 1 2 c 2 2 ) c 2, (10) ic 2 = vj 0 ω ( c 2 2 c 1 2 ) + [U 0 c U 1 ( c c 3 2 )c 2 vj 0 ω ( c 2 2 c 3 2 ) c 3, ( ic 3 = [U 0 c U 1 c χ c 1 2) c 3 (11) vj 0 ω ( c 3 2 c 2 2 ) c 2. (12) From the non-modulated nonlinear model Eqs. (10) (12), we can obtain the tunneling energy E tun and the interaction energy E int of this system: E tun = vj 0 ω ( c 1 2 c 2 2 ) (c 2c 1 + c 2 c 1) + vj 0 ω ( c 3 2 c 2 2 ) (c 2c 3 + c 2 c 3), (13) E int = U (U 1 U 0 )( c c 3 2 ) c 2 2 ( U1 ) + χ U 0 c 1 2 c 3 2. (14) We find that, in order to generate the CDT, the tunneling energy E tun should have the minimum values, namely, J 0 ω ( c 1 2 c 2 2 ) = 0, according to the tunneling energy Eq. (13). At the moment, the population difference between the first well and the second well c 1 2 c 2 2 is close to 1, so the values of the modulated s-wave scattering A/ω for appearing CDT lag behind the null points of the 0-th order Bessel function. The null points of the 0-th order Bessel function are 2.4, 5.5,... The numerical results demonstrate c 1

5 No. 5 Communications in Theoretical Physics 569 that the CDT occurs at the modulated s-wave scattering (A/ω) 0 = 2.5, 5.6,..., as shown in Figs. 5 and 6(d), which are in good agreement with the results obtained by J 0 (A/ω) 0. With the help of the interaction energy of the system given by Eq. (14), the enhancement of the CT near U 0 = U 1 or U 0 U 1 /χ shown in Fig. 6 can be understood. Clearly, when U 0 U 1 or U 0 U 1 /χ, the interaction energy E int has the minimum values, where the crossover from CDT to CT is obtained. The CT occurs in the nearest-neighbor sites for U 0 = U 1, and it exists in the next-nearest-neighbor sites for U 0 = U 1 /χ. This is also in good agree with the numerical results shown in Fig. 6. In a word, the complex transitions between CT and CDT are related to the tunneling energy and the interaction energy of the system. They result from the correlation among the DDI, the on-site interaction and the modulated s-wave scattering. 5 Conclusion In conclusion, we have studied the tunneling dynamics of three weakly coupled dipolar gas subjecting to a periodical modulation of s-wave scattering under the mean-field Bose Hubbard model. The system experiences rich transitions from CT to CDT because of the correlated effect among the next-neighbor DDI, the on-site interaction and the modulated s-wave scattering. In particular, when the on-site interaction and the next-neighbor DDI correlate, the region of the modulated s-wave scattering for appearing CT (CDT) is the widest (narrowest) as U 0 = U 1 or U 0 = U 1 /χ. These phenomena are closely related to the tunneling energy and the interaction energy of the system. References [1 A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, Phys. Rev. Lett. 94 (2005) [2 B. Pasquiou, G. Bismut, E. Maréchal, P. Pedri, L. Vernac, O. Gorceix, and B. Laburthe-Tolra, Phys. Rev. Lett. 106 (2011) [3 K. Aikawa, A. Frisch, M. Mark, S. Baier, A. Rietzler, R. Grimm, and F. Ferlaino, Phys. Rev. Lett. 108 (2012) [4 M.W. Lu, N.Q. Burdick, S.H. Youn, and B.L. Lev, Phys. Rev. Lett. 107 (2011) [5 M.W. Lu, N.Q. Burdick, and B.L. Lev, Phys. Rev. Lett. 108 (2012) [6 K.K. Ni, S. Ospelkaus, M.H.G. de Miranda, A. Peér, B. Neyenhuis, J.J. Zirbe, S. Kotochigova, P.S. Julienne, D.S. Jin, and J. Ye, Science 322 (2008) 231. [7 M.A. Baranov, Phys. Rep. 464 (2008) 71. [8 M. Klawunn and L. Santos, Phys. Rev. A 80 (2009) [9 T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, Nature (London) 448 (2007) 672. [10 M. Fattori, G. Roati, B. Deissler, C. D Errico, M. Zaccanti, M. Jona-Lasinio, L. Santos, M. Inguscio, and G. Modugno, Phys. Rev. Lett. 101 (2008) [11 Z.W. Xie, Z.X. Cao, E.I. Kats, and W.M. Liu, Phys. Rev. A 71 (2005) [12 A.X. Zhang and J.K. Xue, Phys. Rev. A 82 (2010) [13 K. Góral, L. Santos, and M. Lewenstein, Phys. Rev. Lett. 88 (2002) [14 C. Menotti, C. Trefzger, and M. Lewenstein, Phys. Rev. Lett. 98 (2007) ; Phys. Rev. A 78 (2008) [15 I. Danshita and A.R. Sá de Melo Carlos, Phys. Rev. Lett. 103 (2009) [16 S. Yi, T. Li, and C.P. Sun, Phys. Rev. Lett. 98 (2007) [17 O.P. Sah and J. Manta, Phys. Plasmas 16 (2009) [18 D. Yamamoto, I. Danshita, and C.A.R. Sá de Melo, Phys. Rev. A 85 (2012) [19 D. Peter, K. Pawlowski, T. Pfau, and K. Pzazewski, J. Phys. B: At. Mol. Opt. Phys. 45 (2012) [20 T. Lahaye, T. Pfau, and L. Santos, Phys. Rev. Lett. 104 (2010) [21 A. Benseny, S. Fernández-Vidal, J. Bagudà, R. Corbalán, A. Picón, L. Roso, G. Birkl, and J. Mompart, Phys. Rev. A 82 (2010) [22 W. Lu and S. Wang, Chem. Phys. 368 (2010) 93. [23 J.A. Stickney, D.Z. Anderson, and A.A. Zozulya, Phys. Rev. A 75 (2007) [24 A.X. Zhang and J.K. Xue, J. Phys. B 45 (2012) [25 G.B. Lu, W.H. Hai, and Q.T. Xie, Phys. Rev. A 83 (2011) [26 M.L. Zou, G.B. Lu, W.H. Hai, and R.L. Zou, J. Phys. B: At. Mol. Opt. Phys. 46 (2013) [27 Y. Wu and X. Yang, Phys. Rev. Lett. 98 (2007) [28 B. Liu, L.B. Fu, S.P. Yang, and J. Liu, Phys. Rev. A 75 (2007) [29 F. Grossmann, T. Dittrich, P. Jung, and P. Hänggi, Phys. Rev. Lett. 67 (1991) 516; Z. Phys. B: Condens. Matter 84 (1991) 315. [30 F. Grossmann and P. Hänggi, Europhys. Lett. 18 (1992) 571. [31 R. Bavli and H. Metiu, Phys. Rev. Lett. 69 (1992) [32 X.B. Luo, Q.T. Xie, and B. Wu, Phys. Rev. A 77 (2008) [33 Y.Y. Zhang, J.P. Hu, B.A. Bernevig, X.R. Wang, X.C. Xie, and W.M. Liu, Phys. Rev. Lett. 102 (2009) [34 J.B. Gong, L. Morales-Molina, and P. Hänggi, Phys. Rev. Lett. 103 (2009) [35 G. Della Valle, M. Ornigotti, E. Cianci, and V. Foglietti, Phys. Rev. Lett. 98 (2007) [36 E. Kierig, U. Schnorrberger, A. Schietinger, J. Tomkovic, and M.K. Oberthaler, Phys. Rev. Lett. 100 (2008) [37 D. Jaksch, C. Bruder, J.I. Cirac, C.W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81 (1998) 3108.

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