Spin evolution of spin-1 Bose-Einstein condensates
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a y Spin evolution of spin-1 Bose-Einstein condensates
Ma Luo, Zhibing Li, Chengguang Bao ∗ The State Key Laboratory of Optoelectronic Materials and TechnologiesSchool of Physics and EngineeringSun Yat-Sen University, Guangzhou, 510275, P.R. China
An analytical formula is obtained to describe the evolution of the average populations of spincomponents of spin-1 atomic gases. The formula is derived from the exact time-dependent solutionof the Hamiltonian H S = c S without using approximation. Therefore it goes beyond the meanfield theory and provides a general, accurate, and complete description for the whole process of non-dissipative evolution starting from various initial states. The numerical results directly given by theformula coincide qualitatively well with existing experimental data, and also with other theoreticalresults from solving dynamic differential equations. For some special cases of initial state, insteadof undergoing strong oscillation as found previously, the evolution is found to go on very steadily ina very long duration. PACS numbers: 03.75. Fi, 03.65. Fd
The liberation of the freedoms of spin of atoms in opti-cal traps [1, 2, 3, 4, 5] opens a new field, namely spin dy-namics of condensates, which is promising for super-highprecise measurement, quantum computation, and quan-tum information processing. [6, 7, 8] Recently, the evo-lution of spinor condensates has been extensively studiedexperimentally and theoretically. [9, 10, 12, 13, 14] Ini-tially, the condensate was prepared in a Fock-state or acoherent state confined in an optical trap. Then, dueto the spin-dependent interaction, the system begin toevolve where a pair of atoms with spin components 1and -1 can jump to 0 an 0, and vice versa, via scatter-ing. Finally the system will arrive in equilibrium, how-ever the process is not smooth. In 1998, the averagepopulation of each of the spin components µ =1, 0, and-1 was found to depend sensitively on initial states andmay oscillate strongly with time. [12]. This finding wasfurther confirmed by a number of research groups. In2006, in the study of the probability of finding a givennumber of bosons in a given µ state, the ”quantum car-pet” spin-time structure was found. [14] These findingsshow the amazing peculiarity of the spin dynamics. Re-lated theoretic calculations are mostly based on the meanfield theory. Although, in a number of particular cases,theoretical results compares qualitatively well with ex-perimental data, the underlying physics remains to befurther clarified. This paper is a study of the evolutionof the average populations. We shall go beyond the meanfield theory but use strict quantum mechanic many-bodytheory with a full consideration of symmetry. Insteadof solving dynamic differential equations under specifiedinitial condition, we succeed to derive a general analyt-ical formula to describe rigorously the whole process ofevolution (non-dissipative) and is valid for all possibleinitial status. This is reported as follows.It is first assumed that the initial state of N spin-1 ∗ Corresponding author: [email protected] atoms is a Fock-state with populations N , N and N − ,the magnetization M = N − N − . When N and M aregiven, the Fock-state can be simply denoted as | N i . Letthe part of the Hamiltonian responsible for spin evolutionbe H S = c S , where c is a constant, S is the operator oftotal spin. Then, the time evolution readsΨ( t ) = e − iH S t/ ~ | N i = X S e − iS ( S +1) τ | ϑ NS,M ih ϑ NS,M | N i (1)where τ = ct/ ~ , and | ϑ NS,M > is the all-symmetric to-tal spin-state with good quantum numbers S and M .By using the analytical forms of the fractional parent-age coefficients and Clebesh-Gordan coefficients [16, 18],particle 1 can be extracted from the total spin-state as | ϑ NS,M > = X µ χ µ (1)[ A ( N, S, M, µ ) | ϑ N − S +1 , M − µ i + B ( N, S, M, µ ) | ϑ N − S − , M − µ i ] (2)where χ µ (1) is the spin-state of particle 1. The coeffi-cients involved in (1) and (2) are given in the appendix.Inserting (2) into (1), the probability of particle 1 in µ can be obtained, it reads P MN o ,µ ( τ ) = B MN o ,µ + O MN o ,µ ( τ ) (3)where B MN o ,µ = X S P S,Mµ h N | ϑ NS,M ih ϑ NS,M | N i (4) P S,Mµ = ( A ( N, S, M, µ )) + ( B ( N, S, M, µ )) (5) O MN o ,µ ( τ ) = X S O M,SN o ,µ cos(4( S + 3 / τ ) (6) O M,SN o ,µ = 2 A ( N, S, M, µ ) B ( N, S + 2 , M, µ ) ×h N | ϑ NS,M ih ϑ NS +2 ,M | N i (7)The summation covers S = N, N − , · · · · · M ∗ , where M ∗ = M (or M + 1) if N − M is even (or odd).Since the particles are identical, each of them plays thesame role, therefore the average population in µ is just N P MN o ,µ ( τ ) ≡ h a + µ a µ i (this identity has been exactlyproved numerically). In what follows µ = 0 is assumed(the cases with µ = 0 can be thereby understood). Thelabel µ may be neglected from now on if µ = 0.Eq.(3) is an exact consequence of the Hamiltonian H S = c S , no approximation has been introduced, itgives an analytical description of the whole evolution(non-dissipative). There are time dependent and in-dependent terms, it implies an oscillation surroundinga background. It is straight forward from (6) that P MN o ( τ ) = P MN o ( − τ ) = P MN o ( τ + π ), therefore P MN o ( π + τ ) = P MN o ( π − τ ) . It implies that the oscillation is periodicwith the period π and P MN o ( τ ) is symmetric with respectto τ = π . Furthermore, since cos(4( S + 3 / π + τ )) = − cos(4( S + 3 / π − τ )) , O MN o ( τ ) is antisymmetric withrespect to π , we have P MN o ( π + τ ) = 2 B MN o − P MN o ( π − τ ) . Therefore, once P MN o ( τ ) has been known in the domain0 to π/
4, it can be known everywhere. In particu-lar, P MN o (0) = N /N , P MN o ( π ) = B MN o , and P MN o ( π ) =2 B MN o − N /N .In (4) the factor P S,M has an exact analytical form as[18] P S,M = (2 + 1 /N ) S ( S + 1) − − M (2 + 3 /N )(2 S + 3)(2 S −
1) (8)When N is large, P S,M ≈ (1 − ( M/S ) ) . Therefore, B MN o ≈
12 [1 − X S ( MS ) h N | ϑ NS,M ih ϑ NS,M | N i ] ≤
12 (9)In particular, when M → B MN o ≈ . The value 1/2was first obtained numerically by Law, et al [12] , and wassupported by the recent study by Chang, et al [9]. Nowthis value is obtained analytically, and is further foundnot depending on N . When M → N, S must also tendto N , therefore both P S,M and B MN o → O M,SN o in (6) depends on N strongly. There are three representative cases.(i) When N = N − M or 0, O M,SN o is distributed ina narrow domain of S (say, from S a to S b ) as shownin Fig.1a and 1b. In this case, when O M,SN o is roughlyconsidered as a constant in the narrow domain, from (6)we have O MN o ( τ ) ≈ β MN o k max X k =0 cos(4(2 k + S a + 3 / τ ) ≡ β MN o G ( τ )(10)where β MN o is time-independent, k = ( S − S a ) / , k max =( S b − S a ) / G ( τ ) can be exactly rewritten as G ( τ ) = cos(4( S a +3 / k max ) τ ) sin(4( k max +1) τ ) / sin(4 τ )(11) The denominator sin(4 τ ) affects the behavior of G ( τ )strongly. In the neighborhoods of 0, the magnitude of G ( τ ) would be remarkably larger because sin(4 τ ) is small,in particular, G (0) = k max + 1. In the neighborhoods of π/
4, the magnitude of G ( τ ) would also be larger dueto the denominator. However, since G ( π/
4) = 0, therewould be a strong oscillation when τ → π/ N ≈ ( N − M ) / O M,SN o is distributed in abroad domain of S as shown in Fig.1c where O M,SN o and O M,S +2 N o have similar magnitudes but opposite signs. Inthis case, the summation in (6) can be divided into two,similarly we can define ∼ G ( τ ) = k ′ max X k ′ =0 cos(4(4 k ′ + S a + 3 / τ ) − k ′′ max X k ′′ =0 cos(4(4 k ′′ + S a + 7 / τ )= 2 sin(4 τ )sin(8 τ ) · (12)sin(4( S a + 52 + 2 k max ) τ ) sin(8( k max + 1) τ )The feature of ∼ G ( τ ) is greatly different from G ( τ ), inparticular ∼ G (0) = ∼ G ( π/
4) = 0, the denominator sin(8 τ )implies that ∼ G ( τ ) would be large in the neighborhoodof τ ≈ π/
8. This leads to a very different feature ofevolution as shown later.(iii) When N is not close to the above cases, the vari-ation of O M,SN o against S has a band structure as shown inFig.1d, where neighboring O M,SN o and O M,S +2 N o may havethe same or opposite signs.Examples of P MN o ( τ ) calculated from (3) are given inthe follows. Fig.2 shows the evolution in the whole pe-riod 0 to π , where the strong oscillation is concentratedin the neighborhoods of kπ/ kπ/ π/ k is an integer, due to the distinct features of G ( τ )and ∼ G ( τ ). These figures show the symmetry in theperiod. Experimentally, the duration of observation ismuch shorter than π. Evaluate under the Thomas-Fermilimit, when the trap is described by an isotropic har-monic potential with frequency ω/ π , τ = π is associatedwith t period = π ( N/ω ) / X sec, where X = 1 . × (3.86 × ) for Rb ( Na). In what follows τ is onlygiven in a short duration.The cases N = N − M are shown in Fig.3a to 3e. Fig.3a is associated with the experiments by the MIT group(upper panel of Fig.2 of [3]); Fig. 3b and c are the casesthat experiment error emerges which makes M deviatefrom 0 slightly. Fig. 3d and e are associated with theexperiments by GIT group (Fig.1 of [10] ), and Hamburggroup (Fig.5 of [11]), respectively. Where, all P MN o ( τ ) (insolid lines) tend to B MN o = 1/2 or lower (if M is larger)as predicted above.The cases N = 0 are shown in Fig.3f to 3h, respec-tively. Where 3f is associated with the lower panel ofFig.2 of ref. [3] [Stenger98].The cases N = ( N − M ) / M is small the evolution is very steady in avery long period 0 to ∼ π/
8, then a strong oscillationoccurs suddenly in the neighborhood of π/ ∼ G ( τ ). Afterwards, the evolution becomessteady again, and repeatedly.When N is not close to the above cases, two examplesare given in Fig.3k and Fig.3l. The former one is thecase discussed by Law, et al. (shown in Fig.3 of [[12]]).In this case, O M,SN o is nearly chaos (Fig.1d), P MN o ( τ ) oscil-lates with τ with a very high frequency in the beginning,but suddenly disappears, and suddenly recovers, and re-peatedly.In summary, this paper has essentially two findings(1) Going beyond the mean field theory, without thenecessity to solve dynamical equations, a general ana-lytical formula has been derived based on symmetry todescribe the evolution of the average populations P MN o ( τ )initiated from a pure Fock-state. This formula is an ex-act consequence of the Hamiltonian H S = c S with a fullconsideration of symmetry, no approximation is adopted.Therefore the analysis based on this formula can help usto understand better the peculiarity of spin evolution.For examples, one can understand why the oscillationof P MN o ( τ ) becomes very strong in somewhere (in π/ π/ P MN o ( τ ) is symmetric with respect to π/
2, andso on. The results from the formula coincides qualita-tively with existing experimental data or other theoreti-cal results. It is expected that, when accurate experimen-tal data come out, a detailed quantitative comparison canbe made.(2) A special initial state with N = ( N − M ) / M ≈ P MN o ( τ ) is steadyin a very long duration from the begining until τ ≈ π/ Acknowledgments
The support from the NSFC under the grants 10574163and 90306016 are appreciated.
Appendix