week ending PRL 107, 195306 (2011) PHYSICAL REVIEW LETTERS 4 NOVEMBER 2011

Quantum Phase Transition in an Antiferromagnetic Spinor Bose-Einstein Condensate

E. M. Bookjans, A. Vinit, and C. Raman* School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA (Received 15 July 2011; published 4 November 2011) We have experimentally observed the dynamics of an antiferromagnetic sodium Bose-Einstein condensate quenched through a quantum phase transition. Using an off-resonant microwave field coupling the F 1 and F 2 atomic hyperfine levels, we rapidly switched the quadratic energy shift q from ¼ ¼ positive to negative values. At q 0, the system undergoes a transition from a polar to antiferromagnetic ¼ phase. We measured the dynamical evolution of the population in the F 1, m 0 state in the vicinity ¼ F ¼ of this transition point and observed a mixed state of all 3 hyperfine components for q<0. We also observed the coarsening dynamics of the instability for q<0, as it nucleated small domains that grew to the axial size of the cloud.

DOI: 10.1103/PhysRevLett.107.195306 PACS numbers: 67.85.Fg, 05.30.Rt, 67.85.De, 67.85.Hj

A quantum phase transition describes a many-body sys- levels shift independently, spin conservation leads to the tem whose ground state can be tuned through a point of cancellation of the linear energy shifts, such that only the nonanalyticity [1]. Quantum gases afford the possibility to quadratic energy shift q is important for spinor BECs. realize such phase transitions in the laboratory, as well as to Therefore, q plays the role of an external parameter, and explore the dynamical evolution of the state of the system the combination of c2 and q realizes a rich phase diagram of by directly controlling the tuning parameters. In particular, possibilities [6]. Various quantum phases and dynamics spinor Bose-Einstein condensates possess a vector order have been observed for both c2 > 0 and c2 < 0 [7–13]. parameter with additional degrees of freedom relevant to For an antiferromagnetic spinor BEC constrained to this problem. By changing external fields dynamically, one have zero net magnetization, the ground state solution is can observe and quantify a host of nonequilibrium phe- a nematic order parameter É. It varies smoothly with q for nomena, including spin domain formation and aggrega- all values except q 0, which divides the phase diagram tion, topological defect creation, and possible dynamic into two regions. For¼ q>0, the ground state is a polar scaling laws [2]. condensate consisting of a single component—the mF 0 In this work we examine a first-order phase transi- 2 ¼ spin projection that minimizes F^ z . For q<0, the ground tion associated with the quadratic energy shift, which state maximizes the same quantityh i through a superposition is an essential parameter in spinor physics. For a spin-1 of two components mF 1, a so-called antiferromag- Bose-Einstein condensate (BEC), the spin-dependent netic phase [2]. The symmetry¼Æ properties of the ground Hamiltonian can be written in a mean-field representa- state therefore change discontinuously, defining a first- tion as order phase boundary. H c n F^ 2 q F^ 2 ; Exactly at the phase transition point, the many-body sp ¼ 2 h i þ h zi ground state is a condensate of boson pairs forming a spin ^ where F and F^ z are the vector spin-1 operator and its singlet state and possessing super-Poissonian spin fluctua- z projection, respectively, n is the particle density, c2 is tions. Near this boundary, the mean-field wave function É the spin-dependent interaction coefficient [3], and q is the undergoes collapse and revival dynamics triggered by quan- 1 energy difference 2 E 1 E 1 E0, where Ei is the tum fluctuations [14–16]. Controllably accessing q 0 ð þ þ À ÞÀ 2 ¼ energy of the atomic level for the spin m i component would restore the full S symmetry of the nematic order F ¼ of F 1. The spin-dependent interaction coefficient c2 parameter, which has unusual topological defects such as arises¼ from spin-changing collisions that can convert half-quantum vortices [17–19]. However, experimentally two mF 0 atoms into an mF 1 pair and vice versa, this requires low magnetic fields where spinor condensates a process¼ constrained by the conservation¼Æ of angular mo- are susceptible to ambient magnetic field noise that can mentum. It determines the nature of the ground state— mask interaction-related phenomena. In this Letter, we antiferromagnetic for c2 > 0 or ferromagnetic for c2 < 0. investigated the dynamical instability of an F 1, mF 0 The quadratic energy shift q is usually determined by an antiferromagnetic sodium BEC that is rapidly¼ quenched¼ external magnetic field B through the second-order Zeeman across the boundary from q>0 to q<0. The quadratic effect and is B2. However, it can also be tuned by using a energy shift q was tuned by an additional microwave dress- microwave dressing/ field [4,5], a feature we exploit in this ing field that allowed us to access the q<0 regime. work to uncover a previously unexplored phase transition in Microwave dressing of ferromagnetic 87Rb has been an antiferromagnetic 23Na spinor gas. While in general the used to tune spin mixing dynamics into the resonant regime

0031-9007=11=107(19)=195306(5) 195306-1 Ó 2011 American Physical Society week ending PRL 107, 195306 (2011) PHYSICAL REVIEW LETTERS 4 NOVEMBER 2011 in optical lattices [4] and for the study of spontaneous separating the 3 components spatially [7]. After a total time magnetization [5]. Unlike ferromagnetic systems, in anti- of flight of 25 ms, we optically pumped the atoms into the ferromagnetic condensates one generally expects unmag- F 2 state and imaged them on the F 2 F 3 ¼ ¼ ! 0 ¼ netized domains to form as a result of mF 1 pair cycling transition ensuring an equal imaging sensitivity production. Pair formation dynamics have¼Æ also been to each spin component. Examples of the temporal evolu- studied in antiferromagnetic F 2 87Rb spinors where tion for q h 3:2 Hz and q h 17:4 Hz are ¼ ¼ ÂÀ ¼ ÂÀ the signs of both c2 and q are reversed with respect to shown in Figs. 1(b) and 1(c), respectively. They indicate the F 1 manifold [12,20,21], although the spinor life- that a pure m 0 condensate was unstable, evolving into a ¼ time is¼ limited by hyperfine state changing collisions. The superposition of all 3 components that preserved the over- F 1 spin system, by contrast, is intrinsically stable and all zero net magnetization. In order to quantify the q amenable¼ to studies of long time scale dynamics in the dependence of the instability, we fitted data such as shown transition region—in the present work we observed evolu- in Fig. 1 to a sigmoid function and determined both the tion times ranging from 30 ms to over 2 s. crossover time T1=2 and the final saturation value f0;min, 6 We prepared BECs with up to 3 10 sodium atoms where f0 T1=2 1=2 1 f0;min . We defined the mea-  14 3 ð Þ¼ ð þ Þ with a peak atom density n 5:4 10 cm in an sured instability rate to be À q 1=T1=2. 0 À ð Þ optical dipole trap created by a¼ single focused far-detuned At zero microwave power, the quadratic energy shift was 1064 nm laser beam. The measured axial trapping fre- q = c n 0:02. Under these conditions, the gas was B ð 2 0Þ¼ quency was 8 Hz, with inferred radial frequencies of about relatively stable against spin relaxation, i.e., the creation 600 Hz, which correspond to Thomas-Fermi radii of 4 and of mF 1 pairs. This stability is a characteristic of ¼Æ 270 "m, respectively. The condensate is initially prepared quantum antiferromagnetism [2]. For a homogeneous in a magnetic trap from which it was transferred into the optical trap as described in earlier work [22]. In order to a) create a pure mF 0 condensate, we adiabatically swept the frequency of¼ an rf magnetic field across the m F ¼ 1 mF 0 transition at a bias magnetic field of 13 G [À23].! The¼ bias was then ramped to a final value of B 97 mG in 30 ms, which was defined as the starting point¼ (t 0) of our experiment. At this¼ magnetic field, the quadratic Zeeman shift is b) q h 2:5 Hz [7], far smaller than the spin-dependent B ¼  interaction energy c2n0 h 130 Hz [24]. The radial Thomas-Fermi radius of our¼ cloud, 4 "m, is only a factor of 3 larger than the spin-healing length $s =p2Mc2n 1:3 "m, which is the width of a typical spin¼ @ domain wall ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi [25]. Therefore, our experiment is performed mostly in a quasi-1D geometry, in which only axial spin structures are c) expected. At t 0 we instantly turned on an oscillating micro- wave field¼ within tens of microseconds, far shorter than any dynamical time scale relevant to the problem. We used microwave powers between 0 and 7.5 W, which allowed us to tune q from its initial value of qB to a final value q q q 18:5 Hz, where q is the quadratic en- ¼ B þ M ¼À M ergy shift due to the ac Stark shift caused by the microwave FIG. 1. Quenching dynamics. For different final values of q, field [4]. This field created an instability in the initial the m 0 fraction (circles) is plotted versus time after the F ¼ mF 0 spin state. By holding the condensate at a fixed quench. The solid lines are fits of the mF 0 fraction to a value¼ of q and varying the hold time after the quench was sigmoid function. In (a) no microwave dressing¼ field is applied. initiated, we could observe the temporal evolution of the These data show the relaxation in the absence of a quench (q h 2:5 Hz). The equilibration to a pure m 1 fractional population in mF 0. This corresponded to a ¼  F ¼Æ measurement of the squared amplitude¼ of the z component cloud occurs after 2.1 s. In (b) the gas is quenched to q h 3:2 Hz, showing that the population decay is faster. In¼ of the nematic order parameter. The relative populations in (c)ÂÀ the gas is quenched to q h 17:4 Hz and rapidly the m 1; 0; 1 states were determined by Stern- ¼ ÂÀ F ¼À þ reaches a quasiequilibrium state containing all 3 spin compo- Gerlach time-of-flight images. After an expansion of nents in roughly equal proportions, i.e., an order parameter 3–3.5 ms, we pulsed on a Stern-Gerlach field gradient for delocalized over the S2 sphere. Each data point in the figure a duration of 2–4.5 ms perpendicular to the axial direction corresponded to a separate run of the experiment.

195306-2 week ending PRL 107, 195306 (2011) PHYSICAL REVIEW LETTERS 4 NOVEMBER 2011 system and 0 magnon excitations¼ (spin waves) by an energy gap Á k q k q 2c2n , where 2 2 ¼ ð þ Þð þ þ Þ  k [26]. The k 0 mode is the most unstable mode, k  @2M ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and, for q c2n, the energy gap p2qc2n h 25 Hz for our parameters.  ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffi As q 0, quantum fluctuations destabilize the pure ! mF 0 state: The fraction of atoms in the mF 1 states should¼ reach of the order of 1 within a time ¼Æ1:5s[15].  FIG. 2. Final mF 0 fraction versus instability rate À. The This time scale is consistent with the slow rate of relaxa- ¼ 1 open circles indicate data for which the final mF 0 distribution tion ( 0:5sÀ ) that we observed in our experiment [see consisted purely of thermal atoms. ¼ Fig. 1(a) ]. However, we cannot definitively rule out other mechanisms, including thermal fluctuations (our BEC had a thermal fraction of 40%) and imperfect transfer to the segregation. However, it may provide useful insights into the short time scale behavior following the quench for mF 0 state. For hold times longer than 2 s, the cloud ¼ larger q , where a mixture of all 3 components was ob- separated into two nonoverlapping mF 1 spin domains j j along the long axis of our trap, an indication¼Æ that small served. The system is described by a single, macroscopic residual magnetic field gradients might have been present. quantum rotor with angular momentum L, moment of 2 For c n0 to q<0 causes V to change sign, resulting in dependence of the mF 0 fraction, f0 t , followed a back- ¼ ð Þ rapid dispersion of a wave packet initially highly localized ward S-shaped curve that saturated at a value f0;min de- in angle near  0. The resulting wave packet dynamics pending on q. The data sets could be roughly divided into 2 ¼ regions: For positive [Fig. 1(a)] and slightly negative q are mostly classical in character and consist of a rapid dispersion in  followed by sparse, periodic revivals at [Fig. 1(b)], we observed a slow decay to a value f0;min of time t [28]. For our parameters t 350 s, consider- nearly zero. For these data, an examination of the spatial rev rev  distribution of the three hyperfine components showed that ably longer than our observation time [16]. For short times, the 1 states had separated from one another along the as in Fig. 1(c), we observed only the rapid dispersion Æ phase, and we interpret the measured value of 0:3 for axial direction. This is most likely due to the residual  magnetic field gradients mentioned earlier. each spin component to be the result of wave packet For more negative q [Fig. 1(c)], however, the behavior dispersion that tends to equalize the spin populations. The measured instability rate À has been plotted versus was dramatically different—f0 approached a final value of the final quadratic energy q q q in Fig. 3 for q f0;min 0:3 within a time as little as 30 ms and remained ¼ B þ M roughly constant over 200 ms [27]. Thus, for the range of q ranging from 2:5 to 18:5 Hz. The data show a steep rise in the pair formationþ À rate by a factor of nearly 100 as q explored in this work, the instability was observed to create c2n0 a mixed state of all 3 components which appeared to be varied from 0:02 to 0:15, indicating that we had 1 þ À metastable on a time scale much longer than ÀÀ . When crossed a phase boundary in the dynamical evolution of accounting for residual thermal atoms, the estimated con- the system. Figure 3 also shows the predicted maximal densate fraction m 0 is slightly less than 0.3. instability rate for c2n0 quantum rotor model that is valid in the single mode inhomogeneous density profile in determining the modes approximation [16]. Such a model cannot describe spin that were populated [12,29]. No clear indication of reso- domain formation and therefore is inapplicable to the data nances were visible in our data. Moreover, this theory does for small q discussed above, for which we observed spin not account for possible spin exchange processes between j j

195306-3 week ending PRL 107, 195306 (2011) PHYSICAL REVIEW LETTERS 4 NOVEMBER 2011

a) b) c)d) e)

axial f) 250

200 m]

FIG. 3. Quenching through the quantum phase transition. The µ 150 formation rate of mF 1 atom pairs is plotted versus the quadratic energy shift¼Æq (circles) determined by fitting 100 Width [ the temporal evolution of the mF 0 fraction to a sigmoid 50 function. The error bars give the statistical¼ uncertainty in the fit. The instability rate dramatically increases below the transi- 0 tion point at q 0. Also shown (dotted line) is the predicted 0 20 40 60 80 100 120 Hold time [ms] instability rate from¼ Bogoliubov theory for a uniform gas. The inset shows the same data plotted on a semilog scale. FIG. 4. Spatial dynamics of the instability. Absorption images of the condensate taken at a time of flight (TOF) of 25 ms for a quench to q h 17:4 Hz for different hold times: (a) 15, the condensate and the residual thermal cloud, which could ¼ ÂÀ play a role in our observations [30]. (b) 20, (c) 25, (d) 30, and (e) 150 ms. From top to bottom, the In order to gain an understanding of the spatial dynamics, images show the mF 1, mF 0, and mF 1 spin state distribution. The width¼À of the images¼ is 1 mm. In¼þ (f) the width of one can think of the inhomogeneous mF 0 condensate as ¼ the mF 1 component after a TOF 25 ms, determined by a locally varying gain medium for the pair formation insta- the 1=e¼Àwidth of a Gaussian fit, is plotted¼ as a function of hold bility. For very short times after the quench, depletion of the time (triangles). The fit to a sigmoid function (line) is included to gain can be neglected, and one may write the growth rate guide the eye. The appearance of small mF 1 domains, of the m 1 atom pair number forq<0 by using a local which grow outward with hold time, contradicts¼Æ a homogenous F ¼Æ density approximation as À q q 2c n r~ =h, Bogoliubov theory, which predicts a uniform instability rate local ¼ j j½ þ 2 ð ފ across the cloud. where n r~ is the spatial density profilep offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the mF 0 cloud with theð Þ maximum gain occurring at the cloud¼ center. In this regime, the inhomogeneous gain acts as a nonlinear mentioned earlier, which arises through the inhomogene- spatial mode coupler that converts energy from long to ous density profile. In addition, we note that even in the short wavelengths, i.e., exhibits a tendency to nucleate homogeneous theory, À is nonzero over a finite range of small-sized domains. We observed this to occur in our wave vectors 0 k

195306-4 week ending PRL 107, 195306 (2011) PHYSICAL REVIEW LETTERS 4 NOVEMBER 2011 which could not be quantified clearly due to the presence of [12] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. undamped mF 0 density fluctuations in the initial state Henninger, P. Hyllus, W. Ertmer, L. Santos, and J. J. caused by nonadiabaticity¼ in the initial transfer to the Arlt, Phys. Rev. Lett. 103, 195302 (2009). optical trap. [13] J. Kronjager, C. Becker, P. Soltan-Panahi, K. Bongs, and In conclusion, we have tuned an antiferromagnetic con- K. Sengstock, Phys. Rev. Lett. 105, 090402 (2010). [14] C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett. 81, densate through a phase boundary and quantified in detail 5257 (1998). the rate of instability in its vicinity. Future work will [15] X. Cui, Y. Wang, and F. Zhou, Phys. Rev. A 78, 050701 explore the phase coherence between the dynamically (2008). created mF 1 spin components in relation to topologi- [16] R. Barnett, J. D. Sau, and S. Das Sarma, Phys. Rev. A 82, ¼Æ cal defect formation. 031602 (2010). We thank Carlos Sa de Melo for valuable discussions [17] F. Zhou, Phys. Rev. Lett. 87, 080401 (2001). and Jason Gilbertson, Sean Dixon, and Diego Remolina for [18] K. C. Wright, L. S. Leslie, A. Hansen, and N. P. Bigelow, technical assistance. This work was supported by the U.S. Phys. Rev. Lett. 102, 030405 (2009). Department of Energy. [19] J. Ruostekoski and J. R. Anglin, Phys. Rev. Lett. 91, 190402 (2003). [20] H. Schmaljohann, M. Erhard, J. Kronjager, M. Kottke, S. vanStaa, L. Cacciapuoti, J. J. Arlt, K. Bongs, and K. Sengstock, Phys. Rev. Lett. 92, 040402 (2004). *[email protected] [21] C. Klempt, O. Topic, G. Gebreyesus, M. Scherer, T. [1] S. Sachdev, Quantum Phase Transitions (Cambridge Henninger, P. Hyllus, W. Ertmer, L. Santos, and J. J. University Press, Cambridge, England, 1999). Arlt, Phys. Rev. Lett. 104, 195303 (2010). [2] M. Ueda and Y. Kawaguchi, arXiv:1001.2072. [22] D. S. Naik and C. Raman, Phys. Rev. A 71, 033617 (2005). [3] In terms of atomic parameters, c 4% 2 a a , where [23] M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, 2 ¼ 3M@ ð 2 À 0Þ M is the atomic mass and a2;0 are the triplet and singlet C. G. Townsend, and W. Ketterle, Phys. Rev. Lett. 78, 582 scattering lengths, respectively. (1997). [4] F. Gerbier, A. Widera, S. Fo¨lling, O. Mandel, and I. Bloch, [24] The peak atom density n0 was determined from the Phys. Rev. A 73, 041602 (2006). time-of-flight expansion velocity. We used the value c2 52 3 ¼ [5] S. R. Leslie, J. Guzman, M. Vengalattore, J. D. Sau, M. L. 1:6 10À Jm [10]. Cohen, and D. M. Stamper-Kurn, Phys. Rev. A 79, 043631 [25] D. M. Stamper-Kurn, Ph.D. thesis, MIT, 1999. (2009). [26] H. Saito and M. Ueda, Phys. Rev. A 72, 023610 (2005). [6] W. Zhang, S. Yi, and L. You, New J. Phys. 5, 77 (2003). [27] We could not observe the gas for longer times due to [7] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H.-J. Miesner, losses caused by residual excitation to the F 2 manifold A. P. Chikkatur, and W. Ketterle, Nature (London) 396, at the microwave power and detuning used.¼ 345 (1998). [28] H. Stapelfeldt and T. Seideman, Rev. Mod. Phys. 75, 543 [8] M. S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer, (2003). K. M. Fortier, W. Zhang, L. You, and M. S. Chapman, [29] M. Scherer, B. Lucke, G. Gebreyesus, O. Topic, F. Phys. Rev. Lett. 92, 140403 (2004). Deuretzbacher, W. Ertmer, L. Santos, J. J. Arlt, and C. [9] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, Klempt, Phys. Rev. Lett. 105, 135302 (2010). and D. M. Stamper-Kurn, Nature (London) 443, 312 [30] J. M. McGuirk, D. M. Harber, H. J. Lewandowski, and (2006). E. A. Cornell, Phys. Rev. Lett. 91, 150402 (2003). [10] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D. [31] The domain size could be slightly larger in situ by 20% Lett, Phys. Rev. Lett. 99, 070403 (2007). due to a weak focusing of the m 1 cloud in the time of [11] Y. Liu, S. Jung, S. E. Maxwell, L. D. Turner, E. Tiesinga, flight caused by the Stern-Gerlach¼À field. and P. D. Lett, Phys. Rev. Lett. 102, 125301 (2009). [32] A. Bray, Adv. Phys. 43, 357 (1994).

195306-5