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Dicke-Type Phase Transition in a Spin-Orbit-Coupled Bose&Ndash

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Dicke-Type Phase Transition in a Spin-Orbit-Coupled Bose&Ndash ARTICLE Received 16 Jan 2014 | Accepted 30 Apr 2014 | Published 4 Jun 2014 DOI: 10.1038/ncomms5023 Dicke-type phase transition in a spin-orbit-coupled Bose–Einstein condensate Chris Hamner1, Chunlei Qu2, Yongping Zhang1,3, JiaJia Chang1, Ming Gong2,4, Chuanwei Zhang1,2 & Peter Engels1 Spin-orbit-coupled Bose–Einstein condensates (BECs) provide a powerful tool to investigate interesting gauge field-related phenomena. Here we study the ground state properties of such a system and show that it can be mapped to the well-known Dicke model in quantum optics, which describes the interactions between an ensemble of atoms and an optical field. A central prediction of the Dicke model is a quantum phase transition between a superradiant phase and a normal phase. We detect this transition in a spin-orbit-coupled BEC by measuring various physical quantities across the phase transition. These quantities include the spin polarization, the relative occupation of the nearly degenerate single-particle states, the quantity analogous to the photon field occupation and the period of a collective oscillation (quadrupole mode). The applicability of the Dicke model to spin-orbit-coupled BECs may lead to interesting applications in quantum optics and quantum information science. 1 Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164, USA. 2 Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080, USA. 3 Quantum System Unit, Okinawa Institute of Science and Technology, Okinawa 904-0495, Japan. 4 Department of Physics and Center of Coherence, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China. Correspondence and requests for materials should be addressed to C.Z. (email: [email protected]) or to P.E. (email: [email protected]). NATURE COMMUNICATIONS | 5:4023 | DOI: 10.1038/ncomms5023 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5023 ltracold atomic gases afford unique opportunities to Results simulate quantum optical and condensed matter phe- Theoretical description of the SO-coupled BEC. The Raman Unomena, many of which are difficult to observe in their dressing scheme is based on coupling two atomic hyperfine states 1,2 ‘ original contexts . Over the past decade, much theoretical and in such a way that a momentum transfer of 2 kR in the x experimental progress in implementing quantum simulations direction is accompanied with the change of the hyperfine states, ‘ with atomic gases has been achieved, exploiting the flexibility and where kR is the photon recoil momentum. The dynamics in the tunability of these systems. The recent generation of spin-orbit y and z directions are decoupled, which allows us to consider a (SO) coupling in Bose–Einstein condensates (BECs)3–6 and Fermi 1D system in our following discussions (see Methods). The two gases7–9 has brought the simulation of a large class of gauge field- coupled hyperfine states are regarded as the two orientations of a related physics into reach, such as the spin Hall effect10–13. With pseudo-spin 1/2 system. The Raman dressed BEC is governed by such achievements, SO-coupled ultracold atomic gases have the 1D Gross–Pitaevskii (G-P) equation with the Hamiltonian emerged as excellent platforms to simulate topological insulators, HSO ¼ Hs þ HI. Here Hs is the single-particle Hamiltonian and in topological superconductors/superfluids and so on, which have the basis of the uncoupled states can be written as important applications for the design of next-generation spin- ! ‘ 2 2 d O based atomtronic devices and for topological quantum 2m ðkx þ kRÞ þ 2 2 H ¼ 2 þ V : ð1Þ computation14,15. s O ‘ 2 d t 2 2m ðkx À kRÞ À 2 Recently, the ground state properties of a BEC with one- dimensional (1D) or two-dimensional (2D) SO coupling have O is the Raman coupling strength and d is the detuning of the been analysed theoretically. These investigations have predicted a Raman drive from the level splitting. The recoil energy is defined 16–23 2 2 2 2 plane wave or stripe phase for different parameter regimes , as ER ¼ ‘ kR/2m. kx is the quasi-momentum and Vt ¼ moxx /2 is agreeing with the experimental observations3. In the plane wave the external harmonic trap. The many-body interactions between phase of such a SO-coupled BEC, the atomic spins collectively atoms are described by ! interact with the motional degrees of freedom in the external X X trapping field, providing a possible analogy to the well-known H ¼ diag g j c j 2; g j c j 2 ; ð2Þ quantum Dicke model. The Dicke model24, proposed nearly 60 I "s s #s s s¼";# s¼";# years ago, describes the interaction between an ensemble of two- 25 level atoms and an optical field . For atom–photon interaction where gab are the effective 1D interaction parameters (see strengths greater than a threshold value, the ensemble of atoms Methods)30,31. The presence of the interatomic interactions is favours to interact with the optical field collectively as a large spin crucial to observe the Dicke phase transition. For 87Rb atoms, the and the system shows an interesting superradiant phase with a differences between the spin-dependent nonlinear coefficients are macroscopic occupation of photons and non-vanishing spin very small and contribute only small modifications to the polarization26–28. Even though this model has been solved and is collective behaviour (see Methods). well understood theoretically, the experimental observation was The band structure of the non-interacting system with Oo4ER achieved only recently by coupling a BEC to an optical cavity29. and d ¼ 0 has two degenerate local minima at quasi-momenta 2 1/2 In this work, we experimentally investigate the ground state ±q, where q ¼ kR(1 À (O/4ER) ) . The spin polarization of properties of the plane wave phase of a SO-coupled BEC and these two states is finite and opposite to each other. An ensemble show that an insightful analogy to the quantum optical Dicke of non-interacting atoms occupies both states equally and thus model can be constructed. The SO coupling in a BEC is realized has zero average spin polarization and quasi-momentum. When with a Raman dressing scheme. The system exhibits coupling the nonlinear interactions are taken into account, a superposition between momentum states and the collective atomic spin, which state with components located at both degenerate minima is analogous to the coupling between the photon field and the generally has an increased energy and is thus not the many- atomic spin in the Dicke model. This analogy is depicted in Fig. 1. body ground state32. The ground state of the BEC is obtained By changing the Raman coupling strength, the system can be when the atoms occupy one of the degenerate single-particle driven across a quantum phase transition from a spin-polarized ground states (L or R). This is depicted in the inset of Fig. 2a. phase, marked by a non-zero quasi-momentum, to a spin- The mean field energy associated with a spin-flip in an balanced phase with zero quasi-momentum, akin to the transition interacting, harmonically trapped BEC is determined by the from superradiant to normal phases in Dicke model. Measure- coupling between the atomic spin and the many-body ground ments of various physical quantities in these two phases are state harmonic mode. This situation is similar to that of many presented. two-level atoms interacting with a single photon field in an ab Figure 1 | Analogy between standard Dicke model and SO-coupled BEC. (a) Standard Dicke model describing the interaction of an ensemble of two-level atoms in an optical cavity. The optical mode in the cavity couples two atomic spin states. (b) SO-coupled BEC in an external trap. Two spins states are coupled by two counter-propagating Raman lasers. 2 NATURE COMMUNICATIONS | 5:4023 | DOI: 10.1038/ncomms5023 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5023 ARTICLE a expressed in terms of the harmonic trap mode (see Methods). ‘ ‘ 1/2 w 1.00 Setting px ¼ kx ¼ i(mox /2) (a À a), the N-particle Hamilto- nian can be written as rffiffiffiffiffiffiffiffiffiffiffi 0.75 ‘ ‘ y 2 ox y O HDicke¼N oxa a þ ikR a À a Jz þ ‘ Jx m ð3Þ 0.50 4G d 4G þ 3 þ J þ 3 J2 þ const: LorR N‘ ‘ z N2‘ 2 z |Spin polarization| 0.25 where the uniform approximation has been adopted to treat the nonlinear interaction term, G3¼nðg"" À g##Þ=4, n is the local 0.00 density. PJx,z are theP collective spin operators defined as ‘ i ‘ i y 0 1234567 Jx¼ =2 sx; Jz¼ =2 sz: a a is the occupation number of Raman coupling (E ) the harmonic trap mode. The differences between the interaction R energies contribute an effective detuning term and a nonlinear 2 b Ω Ω Ω Ω Ω term in the large spin operator, Jz . However, these terms are small =2.3ER =2.8ER =3.3ER =3.64ER =4.5ER 1 for the experimental states chosen and thus are ignored in the d |1,–1〉 following analysis. For ¼ 0, the Hamiltonian of the first line in equation (3) is equivalent to the Dicke model24. A quantum phase 2hk R transition between the normal phase and a superradiant phase |1,0〉 can be driven by changing the Raman coupling strength O. Number density (a.u.) The critical point for the phase transition can be derived using 0 34–36 the standard mean field approximation yielding Oc ¼ 4ER Figure 2 | Spin polarization and corresponding quasi-momentum from 2 (note that the Jz term yields a small correction À 4G3/N to Oc, time-of-flight image.
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