Influence of the particle number on the spin dynamics of ultracold atoms

Influence of the particle number on the spin dynamics of ultracold atoms Jannes Heinze,1, ∗ Frank Deuretzbacher,2, † and Daniela Pfannkuche1 1

arXiv:1004.1966v2 [cond-mat.quant-gas] 18 May 2010

2

I. Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, Appelstr. 2, 30167, Hannover, Germany

We study the dependency of the quantum spin dynamics on the particle number in a system of ultracold spin-1 atoms within the single-spatial-mode approximation. We find, for all strengths of the spin-dependent interaction, convergence towards the mean-field dynamics in the thermodynamic limit. The convergence is, however, particularly slow when the spin-changing collisional energy and the quadratic Zeeman energy are equal, i. e. deviations between quantum and mean-field spin dynamics may be extremely large under these conditions. Our estimates show, that quantum corrections to the mean-field dynamics may play a relevant role in experiments with spinor Bose-Einstein condensates. This is especially the case in the regime of few atoms, which may be accessible in optical lattices. Here, spin dynamics is modulated by a beat note at large magnetic fields due to the significant influence of correlated many-body spin states. PACS numbers: 67.85.Fg, 67.85.De, 67.85.Hj, 75.50.Mm

I.

INTRODUCTION

The spin degree of freedom of spinor Bose-Einstein condensates (BECs) alows one to explore magnetism of ultracold quantum fluids [1], which exhibit intriguing phenomena, like the formation of coreless vortices [2] and other spin textures [3–5], and spontaneous symmetry breaking after a quench [6]. Beside the investigation of the ground-state phases [7–10], the main focus of research is on the study of spin dynamics [11–15]. So far most experiments confirmed the mean-field (MF) description, including a nonlinear resonance phenomenon near a critical magnetic field [16–19], which is caused by the interplay of spin-changing collisions and quadratic Zeeman shift. Interestingly, the Hamiltonian of the spin-dependent interatomic interactions appears also in nonlinear quantum optics [9]. As a result, spinor BECs allow one to explore, e. g., four-wave mixing [18, 20] and parametric down conversion [21–23] with matter waves. Moreover, spinchanging collisions are responsible for the dynamical evolution of squeezed collective spin states and entanglement from uncorrelated (product) states [24–28], which provides a way to overcome the standart quantum limit in precision measurements with matter waves. Those beyond-mean-field correlations can be described by means of the Bogoliubov approximation as long as the quantum fluctuations of the spinor field operator are small [21–23, 29]. Correlations limit the validity of the MF approximation. Hence, knowledge of the boundaries of the Gross-Pitaevskii equation (GPE) is of great interest. The validity of the GPE has been proven for weakly interacting spinless bosons in the limit N → ∞ with N a fixed [30, 31], where N is the number of particles and a is the scattering length. We show in this article by means of a numerically exact diagonalization of the effective spin Hamiltonian [9, 10, 29, 32], that the quantum spin dynamics in the single-mode approximation (SMA) converges towards the MF dynamics in the thermodynamic

∗ Electronic † Electronic

address: [email protected] address: [email protected]

limit (TDL). This is, interestingly, the case for all strengths of the spin-dependent interaction (within the SMA). The convergence is, however, particularly slow in a regime, where the spin-changing collisional energy and the quadratic Zeeman energy are equal. We determine the validity time of the initial MF dynamics under these conditions, which grows logarithmically with the number of particles. From this we expect that quantum corrections to the MF dynamics may play an important role in experiments with spinor BECs. Additionally, we discuss a beat-note phenomenon in the spin dynamics of few atoms (N ∼ 10), which may be observed in deep optical lattices. The paper is organized as follows. In Sec. II we outline the method to calculate the quantum spin dynamics. Afterwards, in Sec. III, we discuss few particles. First, in Subsec. III A we apply the formalism to two particles to make the method clear and to discuss similarities with the MF dynamics. Then, we discuss the spin dynamics of three particles in Subsec. III B, which is modulated by a beat note at large magnetic fields. A similar beat-note phenomenon is recovered for few atoms, which is shown in Subsec. III C. In Sec. IV we turn to the comparison of the N -particle quantum dynamics with the MF dynamics. This is done for two typical initial states. First, in Subsec. IV A, we study the initial state, where all atoms are in the m = 0 Zeeman sublevel, and then, in Subsec. IV B, we analyze the initial dynamics of the transversely magnetized state. We finally summarize our results in Sec. V.

II. A.

METHODS

Effective spin Hamiltonian

The two-body interaction between ultracold spin-1 atoms is modeled by a spin-dependent δ potential [7]   Vint. (~r1 − ~r2 ) = δ(~r1 − ~r2 ) ~c0 + ~c2 f~1 · f~2 with the interaction strengths c0 and c2 and the dimensionless spin-1 matrices f~i of atom i = 1, 2. Typically, c0 is one or two

2 orders of magnitude larger than c2 . The atoms are confined by a spin-independent trapping potential Vtrap (~r). Additionally, a homogeneous magnetic field along the z-direction generates the potential
VZ = −~pfz − ~q 4 − fz2 , 2

where p ∝ B and q ∝ B are the coefficients of the linear and quadratic Zeeman energy, respectively. The many-body Hamiltonian is then  X X  ~2 00 H = − ∆i + Vtrap (~ri ) + ~c0 δ(~ri − ~rj ) 2m i i 2. The (2) nonzero matrix elements of T0 are (2)

hF, 0|T0 |F, 0i and

(2)

hF + 2, 0|T0 |F, 0i.

The Wigner-Eckart theorem allows one to calculate the matrix (2) elements of T0 from a special class of matrix elements (2)

hF + q, 0|T0 |F, 0i =

hF, 2; 0, 0|F + q, 0i hF, 2; −F, −q|F + q, −F − qi (2)

Appendix B: Quantum dynamics of the number state at zero K

In the following, we derive the population dynamics of the number state at zero K. In this limiting regime, the Hamiltonian consists only of the interaction Hs , which has the eigenbasis {|F, M i}. The occupation number operator N0 and the initial state |θN i need to be expressed in this basis to calculate the dynamics. The expansion of |θN i is given by |θN i = |0, N, 0i =

N X F =Fmin ,∆F =2

χF |F, 0i.

×hF + q, −F − q|T−q |F, −F i

for q = 0, 2. The brackets hf1 , f2 ; m1 , m2 |f 0 , m0 i are the Clebsch-Gordan coefficients (CGC). Eq. (B5) simplifies the calculation, since the states |F, −F i have a rather simple representation in the occupation number basis |F, −F i = cF a†−

F,F 0

(B2)  with the frequencies ωF = gs F (F + 1) − 2N . The matrix elements hF, 0|N0 |F 0 , 0i are calculated in the following. The occupation number operator N0 can be written in terms of spherical tensor operators r 1 2 (2) N0 = N − T , 3 3 0 is the zeroth component of the one-particle spherwhere (2) ical tensor operator Tq of rank 2. Its five components are T±2 = a†± a∓  1  (2) T±1 = √ a†0 a∓ − a†± a0 2  1  † (2) T0 = √ a+ a+ − 2a†0 a0 + a†− a− . 6 (2)

(B3)

(B4)

(B7)

k=0

After inserting Eqs. (B3), (B4), (B6) and the required CGCs into Eq. (B5), a lengthy calculation leads to the nonzero matrix elements   1 c2F F + 1 (2) √ hF, 0|T0 |F, 0i = 3F − 2N + 6 2 (B8) dF 2F − 1 6 and (2)

hF + 2, 0|T0 |F, 0i = s r 3 cF N − F (F + 1)(F + 2) , 2 cF +2 2 (2F + 1)(2F + 3)

(B9)

where N −F

 N −F 2 2 X 1 k!(k + F )!(N − F − 2k)! 2 = k . (B10) 2 dF k 2N −F −2k k=0

The coefficients χF of the expansion (B1) can be calculated by means of Eq. (B6). Using p
F |F, 0i = 1/ (2F )! F+ |F, −F i

N is proportional to the identity matrix and thus its matrix elements are hF, 0|N |F 0 , 0i = N δF F 0 .


2 i N −F 2 a†+ a†− − a†0 /2 |0, 0, 0i (B6)

N −F  N −F 2 2 X k!(k + F )!(N − F − 2k)! 1 2 = . 2 cF k 2N −F −2k



(2) T0


F h

with the normalization constant

(B1)

Since |θN i has the Fz eigenvalue M = 0, it is a superposition of the states |F, M = 0i. Due to symmetry reasons, the summation runs over F = Fmin , Fmin + 2, . . . , N with Fmin = 0 or 1 if N is even or odd, respectively [33]. The coefficients χF are determined later. By inserting the expansion (B1) into Eq. (8) the evolution becomes X   n0 (t) = χF χF 0 hF, 0|N0 |F 0 , 0i cos (ωF − ωF 0 )t /N

(B5)

(B11)

we obtain p
F χF = hF, 0|0, N, 0i = hF, −F | F− |0, N, 0i/ (2F )! s     N −2F −1 2 1 2F = cF − N! . (B12) 2 F

11 shows, that cF is negligible for F  N . Thus, we assume F  N in the following. The binomials in (B16) are expanded into factorials and approximated using (n − k)! ≈ n!/nk for n  k:

coefficient CF

0.012

 N −F 2  −1  N −ϑ   N −F N N −ϑ 2 ≈ . N −ϑ ϑ−F ϑ 2N 2 2

0.008

0.004

0

0

50

100 total spin F

150

200

FIG. 11: (color online). Exact amplitudes CF [(B13), blue crosses], of 2000 atoms compared to the approximation [(B24), red line].

The evolution is obtained by inserting Eqs. (B8)–(B12) into Eq. (B2). Only terms with |F − F 0 | = 2 lead to non-constant contributions to the evolution: N −2 X

n0 (t)0 =

  CF cos gs (2F + 3)t ,

F =Fmin ,∆F =2

with frequencies gs (2F + 3) = ωF +2 − ωF and amplitudes  1 N −2F CF = (N − F )c2F (N − 1)! 2 s   −1 −1 2F 2F + 4 (F + 1)(F + 2) × . (B13) F F +2 (2F + 1)(2F + 3) The amplitudes (B13) can be approximated for large N . The most involved part is the approximation of cF . The factorials in (B7) can be written as binomial coefficients.   −1 N ϑ k!(k + F )!(N − F − 2k)! = N ! , ϑ−F ϑ 2 where ϑ = 2k + F . We obtain from (B7)  N −F 2   −1 N X 1 N ϑ N! 2 = . ϑ−F c2F 2N −ϑ ϑ−F ϑ 2 2 ϑ=F,∆ϑ=2

(B14) The terms with ϑ ≈ N dominate the sum in (B14). The approximation     1 (k − n/2)2 1 n 1 exp − (B15) ≈√ p 2n k 2 n/4 2π n/4
ϑ holds for large n. Applying this to (ϑ−F )/2 gives  N −F 2  −1  2 N F 2 exp . ϑ−F ϑ 2ϑ 2 ϑ=F (B16) Further, we approximate ϑ = N in the exponential and the square root of (B16). The exponential dependence on F 2 N X 1 N! ≈ 2 cF 2N

r

πϑ 2

We apply (B15) to the right-hand side of (B17) s  1 N −ϑ N − ϑ 2 . ≈ N −ϑ 2 π(N − ϑ) 2 Further, for F  N , one gets to first order    N − F N −ϑ F (N − ϑ) ≈ exp − . N N

(B17)

(B18)

(B19)

We insert the approximations (B17)–(B19) into (B16) and approximate the sum by an integral Z N r ϑ N 1 −F F 2 /(2N ) N ! 1 ≈e e dϑ eF N , c2F 2N 2 F N −ϑ where an additional factor 1/2 accounts for the step size of 2 in the sum. The integral has the value ! r r Z N r ϑ N π F (N − ϑ) eF N = −N eF erf dϑ N −ϑ F N F (B20) with the error function erf(x). Inserting the upper limit N of the integral into (B20) leads √ to erf(0) = 0. With F  N , the lower limit F gives erf( F ) ≈ 1 for F ≥ 5 and thus   √ 1 N! π N 1 F2 √ exp ≈ N . (B21) c2F 2 2 2N F p The maximum of (B21) is at F = N/2. The Gaussian p exp(−F 2 /N ) has a width of N/2, which is an upper bound for the width of cF√ . Now we approximate the other terms in (B13). Since F ≈ N for the dominant contributions, s  r    2F 2F + 4 2F 1 2F ≈ ≈2 , (B22) F F +2 F πF where we used (B15) in the last step. For 1  F  N , the remaining factors become s N −F (F + 1)(F + 2) 1 · ≈ . (B23) N (2F + 1)(2F + 3) 2 Inserting (B21)–(B23) into (B13) leads to   F 1 F2 exp − . CF ≈ N 2N

(B24)

The approximation (B24) and the exact amplitude (B13) are compared for 2000 atoms in Fig. 11.

12

[1] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, A. P. Chikkatur, and W. Ketterle, Nature 396, 345 (1998). [2] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, Nature 443, 312 (2006). [3] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M. StamperKurn, Phys. Rev. Lett. 100, 170403 (2008). [4] M. Vengalattore, J. Guzman, S. R. Leslie, F. Serwane, and D. M. Stamper-Kurn, arXiv:0901.3800. [5] J. Kronj¨ager, C. Becker, P. Soltan-Panahi, K. Bongs, and K. Sengstock, arXiv:0904.2339. [6] A. Lamacraft, Phys. Rev. Lett. 98, 160404 (2007). [7] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998). [8] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998). [9] C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. 81, 5257 (1998). [10] M. Koashi and M. Ueda, Phys. Rev. Lett. 84, 1066 (2000). [11] H. Schmaljohann, M. Erhard, J. Kronj¨ager, M. Kottke, S. van Staa, L. Cacciapuoti, J. J. Arlt, K. Bongs, and K. Sengstock, Phys. Rev. Lett. 92, 040402 (2004). [12] M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, K. M. Fortier, W. Zhang, L. You, and M. S. Chapman, Phys. Rev. Lett. 92, 140403 (2004). [13] T. Kuwamoto, K. Araki,T. Eno, and T. Hirano, Phys. Rev. A 69, 063604 (2004). [14] A. Widera, F. Gerbier, S. F¨olling, T. Gericke, O. Mandel, and I. Bloch, Phys. Rev. Lett. 95, 190405 (2005). [15] A. Widera, F. Gerbier, S. F¨olling, T. Gericke, O. Mandel, and I. Bloch, New. J. Phys. 8, 152 (2006). [16] M.-S. Chang, Q. Qin, W. Zhang, L. You, and M. S. Chapman, Nature Phys. 1, 111 (2005). [17] W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman, and L. You, Phys. Rev. A 72, 013602 (2005). [18] J. Kronj¨ager, C. Becker, P. Navez, K. Bongs, and K. Sengstock, Phys. Rev. Lett. 97, 110404 (2006). [19] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D. Lett, Phys. Rev. Lett. 99, 070403 (2007). [20] E. V. Goldstein and P. Meystre, Phys. Rev. A 59, 3896 (1999). [21] S. R. Leslie, J. Guzman, M. Vengalattore, J. D. Sau, M. L. Cohen, and D. M. Stamper-Kurn, Phys. Rev. A 79, 043631 (2009).

[22] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger, P. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys. Rev. Lett. 103, 195302 (2009). [23] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. Henninger, P. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys. Rev. Lett. 104, 195303 (2010). [24] A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Nature 409, 63 (2001). [25] H. Pu and P. Meystre, Phys. Rev. Lett. 85, 3987 (2000). [26] L.-M. Duan, A. Sørensen, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, 3991 (2000). [27] L.-M. Duan, J. I. Cirac, and P. Zoller, Phys. Rev. A 65, 033619, (2002). ¨ E. M¨ustecaplıo˘glu, M. Zhang, and L. You, Phys. Rev. A 66, [28] O. 033611 (2002). [29] X. Cui, Y. Wang, and F. Zhou, Phys. Rev. A 78, 050701(R) (2008). [30] E. H. Lieb, R. Seiringer, and J. Yngvason, Phys. Rev. A 61, 043602 (2000). [31] L. Erd˝os, B. Schlein, and H.-T. Yau, Phys. Rev. Lett. 98, 040404 (2007). [32] Z. Chen, C. Bao, and Z. Li, J. Phys. Soc. Jpn. 78, 114002 (2009). [33] E. Eisenberg and E. H. Lieb, Phys. Rev. Lett. 89, 220403 (2002). [34] J. Kronj¨ager, C. Becker, M. Brinkmann, R. Walser, P. Navez, K. Bongs, and K. Sengstock, Phys. Rev. A 72, 063619 (2005). [35] J. Kronj¨ager, Ph.D. thesis, University of Hamburg (2007); available online: urn:nbn:de:gbv:18-34281. [36] L. G. Yaffe, Rev. Mod. Phys. 54, 407 (1982). [37] J. Heinze, diploma thesis, University of Hamburg (2009); available online. [38] J. Kronj¨ager, K. Sengstock, and K. Bongs, New. J. Phys. 10, 045028 (2008). [39] C. Weiss and N. Teichmann, Phys. Rev. Lett. 100, 140408 (2008). [40] I. N. Bronstein, K. A. Semendjajew, G. Musiol, and H. M¨uhlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, 2001).