Coherent seeding of the dynamics of a spinor Bose-Einstein condensate: from quantum to classical behavior
CCoherent seeding of the dynamics of a spinor Bose-Einstein condensate:from quantum to classical behavior
Bertrand Evrard, An Qu, Jean Dalibard and Fabrice Gerbier
Laboratoire Kastler Brossel, Coll`ege de France, CNRS,ENS-PSL Research University, Sorbonne Universit´e,11 Place Marcelin Berthelot, 75005 Paris, France (Dated: February 26, 2021)We present experiments revealing the competing effect of quantum fluctuations and of a coherentseed in the dynamics of a spin-1 Bose-Einstein condensate, and discuss the relevance of a mean-field description of our system. We first explore a near-equilibrium situation, where the mean-fieldequations can be linearized around a fixed point corresponding to all atoms in the same Zeemanstate m = 0. Preparing the system at this classical fixed point, we observe a reversible dynamicstriggered by quantum fluctuations, which cannot be understood within a classical framework. Wedemonstrate that the classical description becomes accurate provided a coherent seed of a fewatoms only is present in the other Zeeman states m = ±
1. In a second regime characterized by astrong non-linearity of the mean-field equations, we observe a collapse dynamics driven by quantumfluctuations. This behavior cannot be accounted for by a classical description and persists for alarge range of initial states. We show that all our experimental results can be explained with asemi-classical description (truncated Wigner approximation), using stochastic classical variables tomodel the quantum noise.
I. INTRODUCTION
The mean-field approximation is an essential tool ofmany-body physics. In this approach, the interactionof a single body with the rest of the system is treatedin an averaged way, neglecting fluctuations around themean and erasing any spatial correlations. The originalmany-body problem is then reduced to a much simplerone-body problem, a tremendous simplification enablinga basic analysis of the problem at hand. The accuracyof the averaging improves with the number of particlesin direct interaction. Consequently, the mean-field treat-ment is well suited for highly connected systems, whileimportant deviations are common for systems with shortrange interactions in reduced dimensions.When applied to bosonic quantum systems, a mean-field approach often entails another important approx-imation where intrinsic quantum fluctuations (and thecorrelations they induce) are neglected. Since quantumfluctuations are reflected in the non-commutativity of ob-servables, field operators in the second-quantization for-malism are replaced by commuting c -numbers. A pos-sible improvement consists in replacing the field opera-tors by classical stochastic fields [1–5], with a statisticsproperly chosen to be as close as possible to the originalquantum problem. Such a semi-classical approach al-lows to account quantitatively for quantum fluctuations,while keeping the inherent simplicity of the mean-fieldequations.In this Letter, we study the role of quantum fluctua-tions and the emergence of mean-field behavior in a quan-tum spinor Bose-Einstein condensate [6]. The atoms arecondensed in the same spatial mode and interact all-to-all. The mean-field approach is thus well appropriate tostudy the dynamics in the spin sector, and has indeedbeen successfully used to describe several situations, ei- ther at [7, 8] or out-of [9–14] equilibrium. More recently,several experiments addressed the dynamics of a conden-sate prepared in an unstable configuration, achieving ahigh sensitivity to both classical and quantum fluctua-tions [15–31].Here, our goal is twofold. First, we reveal the effect ofquantum fluctuations in two different dynamical regimes,corresponding to persistent oscillations or relaxation toa stationary state [31]. Second, we address the relevanceof a classical field description by comparing our experi-mental results systematically with three theoretical ap-proaches. In the fully classical picture (C), we derivemean-field equations of motion and solve them for well-defined initial conditions, possibly including a coherentseed. In the semi-classical picture (SC), we keep the samemean-field equations of motion but for fluctuating initialconditions, with a probability distribution designed tomodel the quantum noise of the initial state. Finally,we perform a fully quantum treatment (Q), consistingin a numerical resolution of the many-body Schr¨odingerequation. II. SPINOR BOSE-EINSTEIN CONDENSATES
We work with Bose-Einstein condensates of N spin-1sodium atoms in a tight optical trap. Due to the strongconfinement, all atoms share the same spatial wave func-tion ψ ( r ) [32], such that the spin is the only relevantdegree of freedom. In this regime, the Hamiltonian de-scribing the spin-spin interaction is (up to an additiveconstant) [6, 32–34]ˆ H int = U s N N (cid:88) i,j =1 ˆ s i · ˆ s j = U s N ˆ S . (1) a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b Here ˆ s i denotes the spin of atom i , ˆ S = (cid:80) i ˆ s i the to-tal spin, and U s the spin-spin interaction energy. In thesingle-mode limit, the spin-spin interaction is given by U s = (4 π (cid:126) a s N/M ) (cid:82) d r | ψ ( r ) | , where a s is a spin-dependent scattering length, M is the mass of a sodiumatom, and the spin-independent spatial mode ψ is thelowest energy solution of the time-independent Gross-Pitaevskii equation [35]. Note that technical fluctuationsof the atom number N translate into fluctuations of U s (other factors, such as fluctuations of the trap geometry,can also contribute to the latter). As will be discussed inmore detail in Section IV, these technical fluctuations addto the intrinsic relaxation due to quantum fluctuationsand thereby play a significant role in the interpretationof the experiments.We use a magnetic field B aligned along the z axisto shift the energies of the individual Zeeman states | m (cid:105) , the eigenstates of ˆ s z with eigenvalues m = 0 , ± B , the Zeeman Hamiltonian isˆ H Z = (cid:80) Ni =1 p ˆ s zi + q ˆ s zi , where p ∝ B and q ∝ B are thelinear and quadratic Zeeman shifts, respectively. Notic-ing that [ ˆ S z , ˆ H int ] = 0 , the first term in ˆ H Z is a constantof motion that can be removed by a unitary transforma-tion. The total Hamiltonian thus reads [6]ˆ H = ˆ H int + ˆ H Z = U s N ˆ S + q (cid:16) ˆ N +1 + ˆ N − (cid:17) , (2)where ˆ N m is the number of atoms in | m (cid:105) .Under a mean-field approximation, the annihilationoperators ˆ a m are replaced by the c -numbers √ N ζ m = √ N m exp( iφ m ). By convention we set φ = 0, and wefocus on the situation S z = 0. We define the mean num-ber of (+1 , −
1) pairs N p = ( N +1 + N − ) /
2, and take itsnormalized value n p = N p /N and the conjugate phase θ = φ +1 + φ − as dynamical variables. In terms of thesevariables, the mean-field equations of motion are [9] (cid:126) ˙ n p = − U s n p (1 − n p ) sin θ , (3) (cid:126) ˙ θ = − q + 2 U s (4 n p − θ ) . (4)At t = 0, the BEC is prepared in a generalized coherentspin state | ψ ini (cid:105) = ( (cid:80) m ζ ini ,m | m (cid:105) ) ⊗ N , with ζ ini = √ n seed e i θ ini+ η ini2 √ − n seed √ n seed e i θ ini − η ini2 , (5)where n seed = N seed /N and N seed is the number of atomsin the m = ± η = φ +1 − φ − evolves as η ( t ) = η ini − pt/ (cid:126) and does not play anyimportant role in the following. We focus on the behaviorof N p ( t ) as a function of time.We notice that the state with all atoms in m = 0 ( i.e. N seed = 0 and hence n p = 0) is stationary accordingto Eq. (3,4). However, this state is not an eigenstateof ˆ H int and thus not a stationary state of the quantumequation of motion. In the absence of a seed, we iden-tified in Ref. [31] two different regimes for the ensuingnon-classical dynamics: • For U s /N (cid:28) q , the dynamics is reversible: Thenumber of pairs N p ( t ) oscillates with a small am-plitude. • For q (cid:28) U s /N , the dynamics is strongly dampedand N p ( t ) relaxes to a stationary value.Here, we revisit these experiments to investigate the ef-fect of a coherent seeding of the m = ± III. REVERSIBLE DYNAMICS a. Theoretical predictions
We focus first on the sit-uation where U s /N (cid:28) q (cid:28) U s and n seed (cid:28)
1. In thiscase, the reduced number of pairs n p remains small atall times. Linearizing the mean-field Eqs. (3,4), we ob-tain [36] N (C)p ( t ) ≈ U s q sin ( ωt ) cos (cid:18) θ ini (cid:19) N seed , (6)where ω ≈ √ qU s . Note that the oscillation frequency ω is independent on the initial conditions θ ini and N seed . InSec. IV, we investigate a regime, where the frequency ofthe classical solution increases with N seed , with dramaticconsequences on the semi-classical dynamics.To improve the prediction (6) and account for quan-tum fluctuations, we use a semi-classical approach, thetruncated Wigner approximation [2–5, 30]. The proba-bility amplitudes ζ ini ,m are treated as complex randomvariables which sample the initial Wigner distribution ofthe initial state at t = 0. The amplitudes are then prop-agated according to the mean-field equations of motion.Averaging the mean-field predictions over the fluctua-tions of ζ ini , we find [30, 36] N (SC)p ( t ) ≈ U s q sin ( ωt ) (cid:20) (cid:18) θ ini (cid:19) N seed + 1 (cid:21) . (7)In analogy with quantum optics, the term ∝ N seed inEqs. (6,7) describes “stimulated emission” from the mode m = 0 to the modes m = ±
1, while the additional term inEq. (7) can be interpreted as “spontaneous emission”. Wehave verified numerically that the SC results are in goodagreement with a fully quantum treatment. Moreover,comparing equations (6) and (7), we notice that unlessthe initial phase is chosen such that θ ini ≈ π , a largeseed N seed (cid:29) N . In fact,seeding with a few atoms N seed ≈ − θ = 0is sufficient to reach a 90 % agreement between the twoapproaches. b. Experimental sequence We prepare a BEC in thestate m = 0 using evaporative cooling in a crossed lasertrap with a large magnetic field B = 1 G ( q (cid:29) U s ). Afterevaporation, the BEC contains N ≈ m = 0, with N p ≈
100 residual thermal atomsin m = ±
1. We then turn on a strong magnetic fieldgradient to pull the m = ± N p (cid:28) n seed to bein a given m = ± q (cid:29) U s and θ ini can be tuned keeping n p = n seed (see SupplementaryMaterial [36] for more details). In this way, we are ableto prepare any coherent spin state given by Eq. (5), upto the phase η ini which is irrelevant for the experimentsdescribed here. The main imperfection in the prepara-tion originates from the fluctuations of the total atomnumber δN ≈ . N , which induce ≈
10% relative fluc-tuations on N seed . The magnetic field is then quenched tothe desired value, and we let the system evolve for a time t before measuring the population of each Zeeman stateusing a combination of Stern-Gerlach separation and flu-orescence imaging with a detection sensitivity around 1 . c. Experimental results In Fig. 1, we show the timeevolution of N p ( t ) for various initial states. In Fig. 1(a),we do not seed the dynamics. We observe an oscilla-tion of N p ( t ), not captured by the classical descriptionof Eq. (6), but in good agreement with the semi-classicalpredictions (7) or with the numerical resolution of theSchr¨odinger equation. In Fig. 1(b), we prepare a seedwith N seed ≈ . ± .
03 (inferred from a calibration ofthe rf power) and θ i ≈
0. Compared to (a), the amplitudeof the oscillations is doubled, in good agreement with (7).In Fig. 1(c), we set N seed ≈ . ± . θ ini ≈
0. Theamplitude of the oscillations is further increased, and nowalso well reproduced by the fully classical treatment (6).In all cases (a,b,c), the condition N p ( t ) (cid:28) N remainsfulfilled at all times. The validity of Eqs. (6,7) and theindependence of the oscillation frequency on N seed (ascan be seen from Fig. 1) follow.We investigate the role of the initial phase θ ini inFig. 2. In Fig. 2 (a), we plot the variation of N p ( T / T = π/ω the period of oscillations, against N seed for three values of θ ini . For N seed (cid:28)
1, we observe a sat-uration of N p ( T /
2) at a value independent of θ ini , con-sistent with the SC prediction (7). For such small seeds,the dynamics is triggered by quantum fluctuations. Forlarger seeds, unless the anti-phase-matching condition θ ini ≈ π is fulfilled (red curves), stimulated emission be-comes dominant and the fully classical description is ac-curate. We observe a linear increase of N p ( T /
2) until thesmall-depletion approximation used to derive Eqs. (6,7)becomes inconsistent. For our data, this occurs for thepoint N seed ≈
100 , θ ini ≈
0. In this case, an exact resolu-tion of the mean-field equations (3,4) provides accurateresults. In Fig. 2 (b), we set N seed ≈ . θ ini . We measure oscillations of N p ( T /
2) in goodagreement with Eqs. (6,7). a N p b N p c t [ms] N p FIG. 1. Evolution of the number of (+1 , −
1) pairs N p (circles)for q/h ≈ . ± .
03 Hz, N ≈ ±
190 atoms and variousseed sizes: N seed ≈
0; 0 .
25; 1 . θ ini ≈
0. The solid lines are numeri-cal solutions of the Schr¨odinger equation with the many-bodyHamiltonian in Eq.(2) using U s /h = 9 . IV. RELAXATION DYNAMICS a. Theoretical prediction
We now investigate the re-laxation dynamics in a very small magnetic field, suchthat q (cid:28) U s /N . In this regime, the quadratic Zeemanshift q is negligible and we set it to zero for the cal-culation. However, the assumption n p (cid:28) q = 0, the mean-field equationsof motion can be solved directly. Taking for simplicity θ ini = 0, we find [36] n (C)p ( t ) = 14 − − n seed t ) , (8)with an oscillation frequencyΩ = 4 U s (cid:126) (cid:112) n seed (1 − n seed ) . (9)The non-linear dependence of Ω with n seed reflects thenon-linearity of the mean-field equations, and has dra-matic consequences when one takes into account quan-tum fluctuations. The seeds spontaneously created fromthe vacuum of pairs induce random shifts of the oscil-lation frequency around its mean-field value. Averagingover many realizations therefore results in an intrinsicdephasing of the oscillations predicted in Eq. (8). Moreprecisely, for the generalized coherent spin state prepared − − N seed a N p ( T / ) θ i [rad] N p ( T / ) b FIG. 2. (a) Number of pairs produced after half a pe-riod of evolution versus N seed for q/h ≈ . ± .
03 Hz and N ≈ ± θ ini ≈
0; 2 .
2; and 3 . N seed is inferredfrom the calibration of the rf power. The solid lines are thesemi-classical predictions given by Eq. (7) with U s /h ≈
12 Hz,assuming N p (cid:28) N . For large N seed , this approximationbreaks down, but a numerical solution of the non-linear classi-cal mean-field Eqs. (3,4) with fixed initial conditions, becomesrelevant. This fully classical treatment is shown as dashedlines. (b) Scan of the initial phase θ ini after half a period ofevolution for N seed ≈ . in our experiment, the initial number of atoms in the m = ± N +1 , ini + N − , ini = Σ follows a binomialdistribution of mean 2 N seed (quantum partition noise).We use the random variable Σ as an initial conditionto solve the mean-field equations (3,4), i.e. substituting n seed in Eq. (8) with Σ / (2 N ). After averaging over thepartition noise, we obtain for N seed (cid:29) n (SC)p ( t ) ≈ − − n seed t )e − ( γ c t ) , (10)with a collapse rate γ c = 2 U s √ N (cid:126) | − n seed | . (11)The analytic formula (10) agrees very well with the nu-merical solution of the many-body Schr¨odinger equationfor N seed (cid:38)
1. The case N seed (cid:28) n p to 1 /
4, butwith a different asymptotic behavior, n p − / ∝ /t .In a related work [ ? ], it was shown that Poissonianfluctuations of the atom number in each mode of a two-component BEC caused a Gaussian decay of the two-timecorrelation function. For the spin-1 and two-componentcases, a similar mechanism is at work. The combination . . a n p . . b n p . . c n p . . d n p . . e t [ms] n p FIG. 3. Evolution of the fraction of (+1 , −
1) pairs n p = N p /N in a negligible magnetic field, for N ≈ ±
12 atomsand various seedings: N seed = 0; 0 .
54; 2 .
1; 4 .
9; 12 .
8; from (a)to (e). The initial phase is always set to θ ini ≈
0. The solidlines are numerical solutions of the Schr¨odinger equation for U s /h = 24 . of non-linearities due to interactions and of quantum par-tition noise leads to dephasing and relaxation.In an actual experiment, the relaxation of N p is alsoenhanced by purely classical noise sources of technicalorigin. In our case, we identify shot-to-shot fluctuationsof U s (see Section II) as a significant additional mecha-nism contributing to the blurring of the oscillations. Toaccount for this phenomenon, we average Eq. (10) over aGaussian distribution of U s with variance δU s . The re-sulting n p ( t ) has the same functional form as in Eq. (10)with the replacement γ c → Γ = (cid:113) γ + γ , (12)with a technical blurring rate γ t = 4 δU s (cid:126) (cid:112) n seed (1 − n seed ) . (13)For small enough seeds n seed (cid:28) /
4, the total dephasing a N seed ω / ( π ) [ H z ] b N seed γ [ s − ] FIG. 4. Frequency (a) and relaxation rate (b) of the spin-mixing dynamics in a negligible magnetic field. The circles areobtained from a fit to the data of Fig. 3, with the error barsindicating the 95% confidence interval. In (a), the red dashedline corresponds to the frequency ω predicted by the mean-field treatment. In (b), the dash-dotted blue line correspondsto the rate γ c of the collapse driven by quantum fluctuations,the red dashed line is the damping rate γ t due to technicalfluctuations, and the solid purple line corresponds to the totaldamping rate Γ = [ γ + γ ] / . We use the value δU s /U s =0 . ± .
04, obtained from a fit to the data. rate can be writtenΓ ≈ γ c (cid:115) (cid:18) δU s U s (cid:19) N seed . (14)This indicates a crossover from quantum to classical de-phasing for seed sizes N ∗ ≈ U s / (2 δU s ) . b. Experimental considerations In order to achievethe “zero field” regime
N q (cid:28) U s experimentally, the bestoption is to reduce the atom number. Indeed, the densityand therefore U s , cannot be arbitrarily increased due toundesired inelastic processes. Reducing the applied mag-netic field further is not feasible due to ambiant strayfields and environment-induced fluctuations (at the sub-mG level in our experiment). Therefore, we lower N by more than one order of magnitude with respect tothe previous sections and prepare mesoscopic BECs of N ≈ ±
12 atoms. We also slightly tighten the trap inorder to achieve U s /h ≈ . c. Experimental results We show in figure 3 the re-laxation dynamics of n p for various seed sizes n seed . Weobserve an acceleration of the initial dynamics for in-creasing n seed and the emergence of rapidly damped os-cillations. Eventually, n p relaxes to the stationary value ≈ / U s takenas a fit parameter are overall in good agreement with thedata, although they slightly underestimate the dampingrate for the largest seed N seed = 12 . N seed in the range we have explored experi-mentally. This observation is explained by the SC the-ory including technical fluctuations. Indeed, the slowdecrease of γ c with N seed is compensated by the increaseof γ t . Using δU s /U s ≈ .
13 as determined in Fig. (4),we find a “quantum-classical crossover” for seed sizesaround N ∗ ≈
15, close to the largest value we exploredexperimentally. For small seeds N seed (cid:46)
5, our measure-ments are consistent with a collapse driven primarily byquantum fluctuations. On the contrary, for the largest N seed ≈ .
8, classical technical dephasing is the domi-nant damping mechanism.
V. CONCLUSION
We investigated the dynamics of a spin-1 BEC pre-pared with a majority of atoms in the Zeeman state m = 0 and possibly small coherent seeds in the m = ± N seed (cid:38)
2) areused to seed the dynamics.We also studied the dynamics in a negligible magneticfield. In this second regime, the combination of non-linear mean-field equations and quantum noise leads tothe relaxation of the spin populations. When the size ofthe seed increases, the intrinsic damping rate γ c decreasesand the mean-field picture becomes more and more rel-evant. However, it eventually fails for sufficiently longtimes. Experimentally, technical noise sources provideadditional dephasing mechanisms of purely classical ori-gin that can be completely described in the mean-fieldapproach. In our experiment, we identify the fluctua-tions of the total atom number as the leading blurringmechanism when the seed size exceeds a dozen atoms.All the experiments presented in this Letter are wellcaptured by a semi-classical theory, where quantum fluc-tuations are modeled using stochastic classical variables.An interesting direction for future work would be to testexperimentally the validity of such a semi-classical de-scription in other contexts, in particular in a chaoticregime [31, 38, 39]. [1] C. Gardiner and P. Zoller, Quantum noise (Springer Sci-ence, 2004). [2] A. Polkovnikov, Annals of Physics , 1790 (2010). [3] M. J. Steel, M. K. Olsen, L. I. Plimak, P. D. Drummond,S. M. Tan, M. J. Collett, D. F. Walls, and R. Graham,Phys. Rev. A , 4824 (1998).[4] A. Sinatra, C. Lobo, and Y. Castin, Journal of Physics B:Atomic, Molecular and Optical Physics , 3599 (2002).[5] R. Mathew and E. Tiesinga, Phys. Rev. A , 013604(2017).[6] Y. Kawaguchi and M. Ueda, Physics Reports , 253(2012).[7] W. Zhang, S. Yi, and L. You, New Journal of Physics ,77 (2003).[8] D. Jacob, L. Shao, V. Corre, T. Zibold, L. De Sarlo,E. Mimoun, J. Dalibard, and F. Gerbier, Phys. Rev. A , 061601 (2012).[9] W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman,and L. You, Phys. Rev. A , 013602 (2005).[10] M.-S. Chang, Q. Qin, W. Zhang, L. You, and M. S. Chap-man, Nature Physics , 111 (2005).[11] J. Kronj¨ager, C. Becker, M. Brinkmann, R. Walser,P. Navez, K. Bongs, and K. Sengstock, Phys. Rev. A , 063619 (2005).[12] J. Kronj¨ager, C. Becker, P. Navez, K. Bongs, and K. Sen-gstock, Phys. Rev. Lett. , 110404 (2006).[13] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D.Lett, Phys. Rev. Lett. , 070403 (2007).[14] Y. Liu, E. Gomez, S. E. Maxwell, L. D. Turner,E. Tiesinga, and P. D. Lett, Phys. Rev. Lett. , 225301(2009).[15] C. Klempt, G. Gebreyesus, M. Scherer, T. Henninger,P. Hyllus, W. Ertmer, L. Santos, and J. J. Arlt, Phys.Rev. Lett. , 195303 (2010).[16] E. M. Bookjans, C. D. Hamley, and M. S. Chapman,Phys. Rev. Lett. , 210406 (2011).[17] B. L¨ucke, M. Scherer, J. Kruse, L. Pezze, F. Deuret-zbacher, P. Hyllus, J. Peise, W. Ertmer, J. Arlt, L. San-tos, et al., Science , 773 (2011).[18] C. D. Hamley, C. Gerving, T. Hoang, E. Bookjans, andM. S. Chapman, Nat. Phys. , 305 (2012).[19] B. L¨ucke, J. Peise, G. Vitagliano, J. Arlt, L. Santos,G. T´oth, and C. Klempt, Phys. Rev. Lett. , 155304(2014).[20] D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J.Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler,Phys. Rev. Lett. , 013001 (2016).[21] D. Linnemann, J. Schulz, W. Muessel, P. Kunkel, M. Pr¨ufer, A. Fr¨olian, H. Strobel, and M. Oberthaler,Quantum Science and Technology , 044009 (2017).[22] P. Kunkel, M. Pr¨ufer, H. Strobel, D. Linnemann,A. Fr¨olian, T. Gasenzer, M. G¨arttner, and M. K.Oberthaler, Science , 413 (2018).[23] M. Fadel, T. Zibold, B. D´ecamps, and P. Treutlein, Sci-ence , 409 (2018).[24] K. Lange, J. Peise, B. L¨ucke, I. Kruse, G. Vitagliano,I. Apellaniz, M. Kleinmann, G. T´oth, and C. Klempt,Science , 416 (2018).[25] T. Tian, H.-X. Yang, L.-Y. Qiu, H.-Y. Liang, Y.-B. Yang,Y. Xu, and L.-M. Duan, Phys. Rev. Lett. , 043001(2020).[26] H.-X. Yang, T. Tian, Y.-B. Yang, L.-Y. Qiu, H.-Y. Liang,A.-J. Chu, C. B. Da˘g, Y. Xu, Y. Liu, and L.-M. Duan,Phys. Rev. A , 013622 (2019).[27] A. Qu, B. Evrard, J. Dalibard, and F. Gerbier, Phys.Rev. Lett. , 033401 (2020).[28] G. I. Mias, N. R. Cooper, and S. Girvin, Phys. Rev. A , 023616 (2008).[29] X. Cui, Y. Wang, and F. Zhou, Phys. Rev. A , 050701(2008).[30] J. P. Wrubel, A. Schwettmann, D. P. Fahey, Z. Glass-man, H. Pechkis, P. Griffin, R. Barnett, E. Tiesinga, andP. Lett, Phys. Rev. A , 023620 (2018).[31] B. Evrard, A. Qu, J. Dalibard, and F. Gerbier, Phys.Rev. Lett. , 063401 (2021).[32] S. Yi, ¨O. M¨ustecaplıo˘glu, C.-P. Sun, and L. You, Phys.Rev. A , 011601 (2002).[33] T. Ohmi and K. Machida, Journal of the Physical Societyof Japan , 1822 (1998).[34] T.-L. Ho, Phys. Rev. Lett. , 742 (1998).[35] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Reviews of Modern Physics , 463 (1999).[36] for more details see Supplemental Material, wich includesthe reference [40].[37] C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. , 5257 (1998).[38] M. Rautenberg and M. G¨arttner, Phys. Rev. A ,053604 (2020).[39] J. Tomkoviˇc, W. Muessel, H. Strobel, S. L¨ock,P. Schlagheck, R. Ketzmerick, and M. K. Oberthaler,Phys. Rev. A , 011602 (2017).[40] S. Uchino, M. Kobayashi, and M. Ueda, Phys. Rev. A , 063632 (2010). Supplemental Material:Coherent seeding of the dynamics of a spinor Bose-Einstein condensate:from quantum to classical behavior
I. INITIAL STATE PREPARATIONA. Oscillating regime
We prepare the spinor BEC at t = 0 in a generalizedcoherent spin state | ψ ini (cid:105) = ( (cid:80) m ζ ini ,m | m (cid:105) ) ⊗ N , ζ ini = √ n seed e i θ ini+ η ini2 √ − n seed √ n seed e i θ ini − η ini2 . (S1)We prepare this state starting from | m = 0 (cid:105) using acombination of magnetic field ramps and resonant radio-frequency (rf) pulses. In details, we first pulse a rffield resonant with the Zeeman splitting to populate the m = ± n seed = sin (Ω rf t ) / rf is the rf Rabi frequency and t the pulse duration. At this stage, we have prepared acoherent spin state of the form (S1) with θ ini ≈ π .To change θ ini , we let the system evolve in a field B = 0 . q/h ≈
70 Hz) for a time t < h/ (2 q ), be-fore quenching the magnetic field down to 28 ± q/h ≈ .
22 Hz) in t = 4 ms to achieve the desiredregime U s /N (cid:28) q (cid:28) U s . Interactions are negligible( U s /h ≈
10 Hz hence U s t , /h (cid:28) θ = − qt / (cid:126) while themagnetic field is held constant, and ∆ θ = − (cid:82) q ( t ) dt/ (cid:126) during the quench. This results in an initial phase θ ini = π − qt / (cid:126) + ∆ θ that is fully tunable from 0 to2 π by varying t . B. Relaxing regime
We prepare mesoscopic BECs of N ≈
124 atoms inthe same initial spin state as before. We lower the mag-netic field down to B = 4 . ± . q/h ≈ t = 20 ms. The ramp time corresponds to the timeneeded for the damping of eddy currents in the vacuumchamber. Because of the small atom number, the effectsof the spin dependent interactions are negligible over theramp ( U s /h ≈ U s t /h (cid:28)
1) and the evo-lution of the state is essentially another phase shift of θ ,which can be compensated for by varying t . For theseexperiments, we always choose t such that θ ini ≈ U s /h ≈ →
24 Hz). By performing nu-merical simulations of the sequence with the many-bodySchr¨odinger equation, we have checked that the ramp canbe considered instantaneous to a good approximation.
II. CLASSICAL AND SEMI-CLASSICALDYNAMICS
We detail here the calculations of the dynamics of N p ( t ) given in the main text. We use a classical (C)approach based on the mean-field approximation and asemi-classical (SC) approach inspired by the truncatedWigner approximation (TWA). In both frameworks, theannihilation operators ˆ a m are replaced by c -numbers α m = √ N ζ m , with N the number of condensed atomsand ζ a spin-1 wavefunction (normalized to unity) pa-rameterized as ζ = √ n p e i θ + η (cid:112) − n p √ n p e i θ − η . (S2)Here n p = ( N +1 + N − ) / (2 N ) denotes the average num-ber of m = ± N p = N n p ), and we have restricted ourselves to the sit-uation N +1 = N − . We also have chosen ζ real withoutloss of generality.The mean field equations of motion for a spin-1 con-densate in the single-mode regime take the form [1, 2] (cid:126) ˙ n p = − U s n p (1 − n p ) sin θ (S3) (cid:126) ˙ θ = − q + 2 U s (4 n p − θ ) . (S4)The mean-field energy per atom is given by E s = 2 U s n p (1 − n p )(1 + cos θ ) + 2 qn p . (S5)The energy E s is a constant of motion, a fact that we willused repeatedly in the following. A. Dynamics in the oscillating regime
In this section we derive the evolution of N p ( t ) forthe oscillating regime q (cid:29) U s /N . We assume N seed (cid:28) N , i.e. the situation where quantum fluctuations mayplay a significant role. For N seed ∼ N , a fully classicaltreatment is accurate. a. Classical solution : Assuming n p (cid:28)
1, we lin-earize Eqs. (S3) and (S5), (cid:126) ˙ n p ≈ − U s n p sin θ (S6) E s ≈ (cid:16) U s (1 + cos θ ) + 2 q (cid:17) n p . (S7)We use the second equation to express cos θ as a functionof n p and of the constants q, U s , E s . Substituting in thefirst equation, we obtain a differential equation on n p only, ˙ n = − ω [ n p − α ] + A , where (cid:126) ω = (cid:112) q ( q + 2 U s ) , α = E s ( q + U s )2( (cid:126) ω ) , (S8)and where A is constant. Differentiating one more time,we find that either n p is constant or it obeys the harmonicequation ¨ n p + 4 ω ( n p − α ) = 0. The evolution is thus aharmonic motion at frequency 2 ω , n p ( t ) ≈ n seed + 2( α − n seed ) sin ( ωt ) , (S9)with the initial conditions n p (0) = n seed and θ (0) = θ ini .If we also assume (as in the experiments we performed)that q (cid:28) U s , we have E s ≈ U s n seed cos ( θ ini / (cid:29) q ,and α ≈ E s / (4 q ) (cid:29)
1. Eq. (S9) then reduces to n p ( t ) ≈ n seed + 2 U s n seed q cos ( θ ini /
2) sin ( ωt ) , i.e. to Eq. (6) in the main text. b. Semi-classical picture : We now consider the ef-fect of quantum fluctuations within the TWA [3–7]. Inthis method, the c -numbers α m used instead of the an-nihilation operators ˆ a m in the mean-field approximationare treated as complex random variables. At t = 0, thesevariables sample the Wigner distribution of the initialstate | ψ i (cid:105) . Their mean values are given by¯ α ini = N √ n seed e i θ ini+ η ini2 √ − n seed √ n seed e i θ ini − η ini2 . (S10)In the limit N seed (cid:28) N , the calculation can be simplifiedby neglecting the depletion of the mode m = 0. For the m = ± | ψ ini (cid:105) ≈ √ N ! (cid:89) m = ± e ¯ α m, ini ˆ a † m − ¯ α ∗ m, ini ˆ a m ˆ a † N | vac (cid:105) . (S11)For t >
0, the equations of evolution (S3,S4) remainvalid in the TWA. The solution for initial conditions α ± , ini is thus given by Eq. (S9) with the substitution4 N seed cos ( θ ini / → | α +1 , ini + α ∗− , ini | .To average over the initial distribution of α ± , ini ,we recall that the Wigner distribution average (cid:104)O ( α m , α ∗ m ) (cid:105) Wig of an operator O is equal to the expec-tation value (cid:104)O sym (ˆ a m , ˆ a † m ) (cid:105) of the corresponding sym-metrically ordered operator O sym [3]. We obtain (cid:104) α +1 , ini α ∗− , ini (cid:105) Wig = (cid:104) ˆ a +1 ˆ a †− (cid:105) = ¯ α +1 , ini ¯ α ∗− , ini , (S12) (cid:104)| α m, ini | (cid:105) Wig = 12 (cid:104) ˆ a † m ˆ a m + ˆ a m ˆ a † m (cid:105) = | ¯ α m, ini | + 12 . (S13)This leads to (cid:104) N p ( t ) (cid:105) ≈ U s q sin ( ωt ) (cid:0) | ¯ α +1 , ini + | ¯ α ∗− , ini | + 1 (cid:1) , which gives Eq. (7) in the main text.As a final remark, we note that the Bogoliubov methodis also well suited to study the regime that we investi-gated here, and leads to the same result [8–10]. B. Relaxation dynamics
We now discuss the regime q (cid:28) U s /N , in which weobserve a relaxation of the number of pairs N p to a sta-tionary value. In this regime, the quantum fluctuationsplay an important role even for N seed (cid:29)
1. We will thusconsider that N seed (cid:29) N − N seed (cid:29)
1. For sim-plicity, we will focus on the situation θ ini = 0, for whichthe effect of the seed is maximal. The case with no seedhas been treated using an exact diagonalization of theHamiltonian [10] or the TWA [6]. a. Classical solution In order to simplify the calcu-lation, we neglect completely the quadratic Zeeman shift.In this regime q (cid:28) U s /N , the Zeeman term indeed playsno significant role even for the fully quantum model. In-troducing the auxiliary variable x = 4 n p −
1, the equa-tions of motion and the energy become (cid:126) ˙ x = − U s (1 − x ) sin θ , (S14) (cid:126) ˙ θ = 2 U s x (1 + cos θ ) , (S15) E s = U s − x )(1 + cos θ ) = cst . (S16)We combine the first and last equations to obtain˙ x = − E s (cid:126) sin θ θ . (S17)Differentiating this equation, we eliminate the phase θ and obtain a simple harmonic equation, ¨ x = − Ω x , withan oscillation frequency (cid:126) Ω = √ U s E s . For the initialconditions n p (0) = n seed and θ (0) = 0, we have (cid:126) Ω =2 U s (cid:112) − x and x ( t ) = x cos(Ω t ) with x = 4 n seed − b. Quantum partition noise: The initial state | ψ ini (cid:105) = 1 √ N ! (cid:34) (cid:88) m =0 , ± ζ m ˆ a † m (cid:35) N | vac (cid:105) , is characterized by fluctuations of the number of ± | ζ +1 | = | ζ − | = √ N seed and θ i = 0. We introduce the sumΣ = N +1 + N − , its relative value s = Σ /N and thedifference ∆ = N +1 − N − . The components of ζ arerelated to the average ¯Σ of Σ by | ζ ± | = ¯Σ2 , | ζ | = N − ¯Σ . (S18)The joint distribution of Σ and ∆ in the initial coherentspin state is P (Σ , ∆) = N ! (cid:0) Σ+∆2 (cid:1) ! (cid:0) Σ − ∆2 (cid:1) !( N − Σ)! (cid:16) ¯ s (cid:17) Σ (1 − ¯ s ) N − Σ . (S19)We deduce from Eq. (S19) the distribution of Σ, P (Σ) = N !Σ!( N − Σ)! ¯ s Σ (1 − ¯ s ) N − Σ . (S20)with Σ ∈ [0 , N ]. The normalization follows from thebinomial formula.For large N and Σ away from the extreme values 0 , N ,the binomial distribution is well approximated by a con-tinuous Gaussian distribution P (Σ) ≈ N √ πσ e − ( s − ¯ s )22 σ = p ( s ) ds. (S21)with a step size ds = 1 /N and a standard deviation σ = (cid:114) ¯ s (1 − ¯ s ) N = (cid:114) n seed (1 − n seed ) N . (S22)One can check the normalization of both distributions, N (cid:88) Σ=0 P (Σ) → (cid:90) f ( s ) ds ≈ (cid:90) + ∞−∞ √ π e − u du = 1 . To extend the lower boundary to −∞ , we require ¯ s/σ = √ N × (cid:112) ¯ s/ (1 − ¯ s ) (cid:29)
1, or N ¯ s = 2 N seed (cid:29) c. Semi-classical picture of the dynamics: Similarlyto what we have done in Sec. II A, we average the meanfield solution (S3,S4) with 2 n seed → s over the probabilitydistribution p ( s ) in Eq. (S21). This amounts to computethe integral I = 12 (cid:90) s cos[Ω( s ) t ] p ( s ) ds. (S23)We use the fact that p ( s ) is sharply peaked around ¯ s ,with a width ∼ /N much narrower than the scale ofvariation of the rest of the integrand s cos[Ω( s ) t ]. Asa result, we extend the integral boundaries to ±∞ , set s ≈ ¯ s and expand the frequency Ω( s ) to first order,Ω( s ) ≈ ¯Ω + ¯Ω (cid:48) ( s − ¯ s ) + O ( (cid:15) ) , (S24)where ¯Ω = Ω(¯ s ) and ¯Ω (cid:48) = Ω (cid:48) (¯ s ) = (2 U s / (cid:126) ) × (1 − s ) / (cid:112) ¯ s (1 − ¯ s ). With straightforward manipulations, we cast I in theform of the Fourier transform of a Gaussian function,which is readily calculated. We find I = 12 ¯ s cos[ ¯Ω t ] e − ( γ c t ) , (S25)with a damping rate γ c = | ¯Ω (cid:48) σ | = 2 U s √ N (cid:126) | − s | . (S26)Using ¯ s = 2 n seed , this gives Eq. (11) in the main text. d. Classical fluctuations of Ω : In addition to the in-trinsic dephasing originating from quantum fluctuations,any technical fluctuations of Ω will also contribute tothe observed relaxation. We consider here the dominantsource of classical blurring in our experiment, namelyfluctuations of the interaction strength U s mainly due toshot-to-shot atom number fluctuations.We model these fluctuations by considering a fluctu-ating interaction strength U (cid:48) s = U s + δU s x , with U s theaverage value, δU s the standard deviation of the noise,and x a centered Gaussian random variable of varianceunity. This leads to a fluctuating oscillation frequencyΩ( x ) = ¯Ω(1 + x · δU s /U s ). We neglect the fluctua-tions of γ c , which is legitimate for N seed (cid:29) γ c (cid:28) ¯Ω. Averaging over the Gaussian probability distri-bution p ( x ), we find that I = (cid:68) cos[Ω( x ) t ]e − ( γ c t ) (cid:69) x = cos[ ¯Ω t ] e − ( γ t t ) − ( γ c t ) , (S27)with a classical (technical) damping rate given by γ t = ¯Ω δU s U s . (S28)From Eqs. (S27,S28) we obtain Eqs. (12,13) given in themain text. [1] W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman,and L. You, Phys. Rev. A , 013602 (2005).[2] Y. Kawaguchi and M. Ueda, Physics Reports , 253(2012).[3] A. Polkovnikov, Annals of Physics , 1790 (2010).[4] M. J. Steel, M. K. Olsen, L. I. Plimak, P. D. Drummond,S. M. Tan, M. J. Collett, D. F. Walls, and R. Graham,Phys. Rev. A , 4824 (1998).[5] A. Sinatra, C. Lobo, and Y. Castin, Journal of Physics B:Atomic, Molecular and Optical Physics , 3599 (2002).[6] R. Mathew and E. Tiesinga, Phys. Rev. A , 013604 (2017).[7] J. P. Wrubel, A. Schwettmann, D. P. Fahey, Z. Glass-man, H. Pechkis, P. Griffin, R. Barnett, E. Tiesinga, andP. Lett, Phys. Rev. A , 023620 (2018).[8] G. I. Mias, N. R. Cooper, and S. Girvin, Phys. Rev. A , 023616 (2008).[9] S. Uchino, M. Kobayashi, and M. Ueda, Phys. Rev. A81