Abstract
Spinorbitcoupled Bose–Einstein condensates (BECs) provide a powerful tool to investigate interesting gauge fieldrelated phenomena. Here we study the ground state properties of such a system and show that it can be mapped to the wellknown 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 spinorbitcoupled BEC by measuring various physical quantities across the phase transition. These quantities include the spin polarization, the relative occupation of the nearly degenerate singleparticle 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 spinorbitcoupled BECs may lead to interesting applications in quantum optics and quantum information science.
Introduction
Ultracold atomic gases afford unique opportunities to simulate quantum optical and condensed matter phenomena, many of which are difficult to observe in their original contexts^{1,2}. Over the past decade, much theoretical and experimental progress in implementing quantum simulations with atomic gases has been achieved, exploiting the flexibility and tunability of these systems. The recent generation of spinorbit (SO) coupling in Bose–Einstein condensates (BECs)^{3,4,5,6} and Fermi gases^{7,8,9} has brought the simulation of a large class of gauge fieldrelated physics into reach, such as the spin Hall effect^{10,11,12,13}. With such achievements, SOcoupled ultracold atomic gases have emerged as excellent platforms to simulate topological insulators, topological superconductors/superfluids and so on, which have important applications for the design of nextgeneration spinbased atomtronic devices and for topological quantum computation^{14,15}.
Recently, the ground state properties of a BEC with onedimensional (1D) or twodimensional (2D) SO coupling have been analysed theoretically. These investigations have predicted a plane wave or stripe phase for different parameter regimes^{16,17,18,19,20,21,22,23}, agreeing with the experimental observations^{3}. In the plane wave phase of such a SOcoupled BEC, the atomic spins collectively interact with the motional degrees of freedom in the external trapping field, providing a possible analogy to the wellknown quantum Dicke model. The Dicke model^{24}, proposed nearly 60 years ago, describes the interaction between an ensemble of twolevel atoms and an optical field^{25}. For atom–photon interaction strengths greater than a threshold value, the ensemble of atoms favours to interact with the optical field collectively as a large spin and the system shows an interesting superradiant phase with a macroscopic occupation of photons and nonvanishing spin polarization^{26,27,28}. Even though this model has been solved and is well understood theoretically, the experimental observation was achieved only recently by coupling a BEC to an optical cavity^{29}.
In this work, we experimentally investigate the ground state properties of the plane wave phase of a SOcoupled BEC and show that an insightful analogy to the quantum optical Dicke model can be constructed. The SO coupling in a BEC is realized with a Raman dressing scheme. The system exhibits coupling between momentum states and the collective atomic spin, which is analogous to the coupling between the photon field and the atomic spin in the Dicke model. This analogy is depicted in Fig. 1. By changing the Raman coupling strength, the system can be driven across a quantum phase transition from a spinpolarized phase, marked by a nonzero quasimomentum, to a spinbalanced phase with zero quasimomentum, akin to the transition from superradiant to normal phases in Dicke model. Measurements of various physical quantities in these two phases are presented.
Results
Theoretical description of the SOcoupled BEC
The Raman dressing scheme is based on coupling two atomic hyperfine states in such a way that a momentum transfer of 2ℏk_{R} in the x direction is accompanied with the change of the hyperfine states, where ℏk_{R} is the photon recoil momentum. The dynamics in the y and z directions are decoupled, which allows us to consider a 1D system in our following discussions (see Methods). The two coupled hyperfine states are regarded as the two orientations of a pseudospin 1/2 system. The Raman dressed BEC is governed by the 1D Gross–Pitaevskii (GP) equation with the Hamiltonian H_{SO}=H_{s}+H_{I}. Here H_{s} is the singleparticle Hamiltonian and in the basis of the uncoupled states can be written as
Ω is the Raman coupling strength and δ is the detuning of the Raman drive from the level splitting. The recoil energy is defined as E_{R}=ℏ^{2}k_{R}^{2}/2m. k_{x} is the quasimomentum and V_{t}=mω_{x}^{2}x^{2}/2 is the external harmonic trap. The manybody interactions between atoms are described by
where g_{αβ} are the effective 1D interaction parameters (see Methods)^{30,31}. The presence of the interatomic interactions is crucial to observe the Dicke phase transition. For ^{87}Rb atoms, the differences between the spindependent nonlinear coefficients are very small and contribute only small modifications to the collective behaviour (see Methods).
The band structure of the noninteracting system with Ω<4E_{R} and δ=0 has two degenerate local minima at quasimomenta ±q, where q=k_{R}(1−(Ω/4E_{R})^{2})^{1/2}. The spin polarization of these two states is finite and opposite to each other. An ensemble of noninteracting atoms occupies both states equally and thus has zero average spin polarization and quasimomentum. When the nonlinear interactions are taken into account, a superposition state with components located at both degenerate minima generally has an increased energy and is thus not the manybody ground state^{32}. The ground state of the BEC is obtained when the atoms occupy one of the degenerate singleparticle ground states (L or R). This is depicted in the inset of Fig. 2a.
The mean field energy associated with a spinflip in an interacting, harmonically trapped BEC is determined by the coupling between the atomic spin and the manybody ground state harmonic mode. This situation is similar to that of many twolevel atoms interacting with a single photon field in an optical cavity^{24}. As shown in the following, the interactioninduced spontaneous symmetry breaking of the ground state allows the mapping of the SOcoupled BEC to the wellknown Dicke model, leading to the prediction of an intriguing quantum phase transition when the Raman coupling strength is varied^{33}.
Mapping to the Dicke model
The realization of the Dicke transition requires two conditions: a single mode that interacts with all atoms and the thermodynamic (large atom number) limit. While the latter is naturally fulfilled in our SOcoupled BECs with N>10^{4} atoms, the first condition is driven by the manybody interaction between atoms. Note that the mean field interaction energy contains two types of terms (see Methods): a term proportional to the overall interaction strength (∝g_{↑↑}+g_{↓↓}+2g_{↑↓}) that prefers a uniform density (the plane wave phase) and a term proportional to the difference of the interaction strengths in different spin states (∝g_{↑↑}−g_{↑↓}) that prefers spatial modulation of the density. In the experiment, the first term dominates for a reasonably large Ω and the plane wave phase is preferred (that is, possesses the lowest energy)^{22}. In this phase, all atoms collectively interact with a single plane wave mode, similar to atoms confined in a cavity interacting only with the cavity mode.
To see the connection between the SOcoupled BEC and the Dicke model, the interacting manybody ground state is first expressed in terms of the harmonic trap mode (see Methods). Setting p_{x}=ℏk_{x}=i(mω_{x}ℏ/2)^{1/2}(a^{†}−a), the Nparticle Hamiltonian can be written as
where the uniform approximation has been adopted to treat the nonlinear interaction term, , n is the local density. J_{x,z} are the collective spin operators defined as is the occupation number of the harmonic trap mode. The differences between the interaction energies contribute an effective detuning term and a nonlinear term in the large spin operator, . However, these terms are small for the experimental states chosen and thus are ignored in the following analysis. For δ=0, the Hamiltonian of the first line in equation (3) is equivalent to the Dicke model^{24}. A quantum phase transition between the normal phase and a superradiant phase can be driven by changing the Raman coupling strength Ω.
The critical point for the phase transition can be derived using the standard mean field approximation^{34,35,36} yielding Ω_{c}=4E_{R} (note that the term yields a small correction −4G_{3}/N to Ω_{c}, which is neglected here). When Ω<Ω_{c}, the Dicke model predicts that the dependence of the order parameter on the Raman coupling strength is . For Ω>Ω_{c}, one obtains 〈J_{z}〉/jℏ=0 and 〈σ_{x}〉=〈J_{x}〉/jℏ=1 (j=N/2). This scaling is confirmed in our numerical simulation of the GP equation. The ground state of the BEC is obtained through an imaginary time evolution. The spin polarization can be calculated as . The absolute value of 〈σ_{z}〉 is taken since ±q are spontaneously chosen. The scaling of 〈J_{z}〉/(jℏ) is shown in Fig. 2 and clearly is consistent with the experimental data described in the following.
Experimental procedure
To experimentally probe this system, we adiabatically Raman dress a BEC of ^{87}Rb atoms in the and hyperfine states (see Methods). A magnetic bias field is applied that generates a sufficiently large quadratic Zeeman splitting such that the F=1,m_{F}=1〉 state can be neglected. The system can thus be treated as an effective twostate system. To analyse the bare state composition, all lasers are switched off and the atoms are imaged after timeofflight in a Stern–Gerlach field. This separates the bare states along the vertical axis of the images. The absolute value of the spin polarization, given by (N_{↑}–N_{↓})/(N_{↑}+N_{↓}), and the quasimomentum are directly measured for condensed atoms at ±q. The experimentally measured absolute value of the spin polarization for various Raman coupling strengths Ω spanning the quantum phase transition is shown in Fig. 2a.
An experimental investigation of the collective choice of the atoms to occupy a single minimum in the dispersion relation is presented in Fig. 3. For this data, the BEC is adiabatically prepared in a Raman dressed state with Ω=3E_{R} and a detuning δ of either ±5.4E_{R}. The detuning is then linearly swept to a final value in 50 ms. The left/right relative occupation of the BEC, defined by (N_{+q}–N_{−q})/(N_{+q}+N_{−q}), provides a measure for the relative occupation of the dressed states at the quasimomentum of the two dispersion minima. The data presented in Fig. 3 exhibits a hysteretic effect dependent on the sign of the initial detuning ±δ. The width of the hysteresis depends on the chosen sweep rate of the detuning (see Methods). While these sweeps are slow enough for the BEC to follow a chosen minimum of the band structure, the transition between the two possible minima is nonadiabatic. Sweeps causing a reversal of the energetic order of the two minima eventually generate heating of the BEC, shown in Fig. 3c. Only values of δ where the BEC is not heated are plotted in Fig. 3a,b. Recently, a similar energy dispersion exhibiting a double well structure was also observed for a BEC in a shaken optical lattice, where the analogous spontaneous occupation of a single band mimimum was also found^{37}. However, such a system lacks the linear coupling between momentum and spin presented in our system (Equation 1), and thus cannot be mapped to the Dicke model.
From the effective Dicke model, we also obtain the quantity analogous to the photon field excitation, as well as the ground state energy of the system. The photon number N_{p} is related to the quasimomentum q of the SOcoupled BEC: N_{p}=q^{2} after a constant coefficient is scaled to unity (see Methods). As seen from the experimental measurements and numerical simulations in Fig. 4a, the superradiant phase has a macroscopic photon excitation, while the photon field vanishes in the normal phase. The ground state energy E_{s} of a single particle, as obtained from GP simulations, is plotted in Fig. 4b and is in agreement with the predictions of the Dicke model. The secondorder derivative of E_{s} is discontinuous at the critical point Ω=4E_{R}, reinforcing the nature of the quantum phase transition. The red circles in Fig. 4b show the densitydependent mean field interaction energy in the numerical simulations, contributing ~0.2E_{R} to the total energy.
Collective excitations
It is well known that various physical quantities may change dramatically across a quantum critical point^{38}. As a particular example, we investigate the quadrupole collective excitation of a SOcoupled BEC. In BECs without SO coupling, the quadrupole excitation frequency only depends on the trapping geometry and on the ratio of the kinetic energy to the trapping energy^{39}. Measurements and GP simulations of the quadrupolemode frequencies for SOcoupled BECs are shown in Fig. 5 as a function of the Raman coupling strength. For this data, the quadrupole oscillation period is scaled by the period measured for an offresonant case . This removes a dependence on the trapping geometry that changes with the Raman coupling strength in the experiment. A peak in the oscillation period around the phase transition is observed. In the GP simulations, a quadrupole oscillation amplitude of ~0.2ℏk_{R} was used, whereas the oscillation amplitude in the experimental results varied from 0.15ℏk_{R} to 0.55ℏk_{R}. In SOcoupled systems, the hydrodynamicmode frequencies depend strongly on the oscillation amplitude^{4}. This may contribute to the variation of the experimental data. The numerical results reveal that the oscillations of the BEC are undamped in both spinbalanced and polarized phases, but show strong damping in the transition region. The behaviour of the quadrupole mode provides an additional signature of the quantum phase transition. Similar experimental behaviour has been observed for the collective dipole motion^{4}.
Finite detuning
While the analysis above has focused on the case of δ=0, a finite detuning leads to the realization of the generalized Dicke Hamiltonian^{40}. For finite detuning, the sharp quantum phase transition vanishes and the BEC is always in a superradiant (spinpolarized) phase, as shown in Fig. 6. The inclusion of this parameter demonstrates the completeness in the mapping of the SOcoupled BEC to a Dicketype system.
Discussion
We have demonstrated that varying the Raman coupling strength can drive a harmonically trapped SOcoupled BEC across a quantum phase transition from a spinbalanced to a spinpolarized ground state. This is a realization of the longsought phase transition from a normal to a superradiant phase in the quantum Dicke model. The corresponding spin polarization and photon field occupation are calculated for the Dicke model and agree with the experimental measurements and the numerical simulations. The ground state energy is found to be consistent with our numerical simulations as well. Using SOcoupled BECs to study the Dicke model physics may have important applications in quantum information and quantum optics including spin squeezing, quantum entanglement and so on.
Methods
Experimental setup and parameters
For the experiments presented in this article, the BECs are held in a crossed optical dipole trap with frequencies {ω_{x}, ω_{y}, ω_{z},} within the range of 2π × {12–34,134,178} Hz, respectively. Depending on the Raman coupling strength and crossed dipole trap potential, BECs of 2 × 10^{4} to 10 × 10^{4} atoms are loaded into the dressed state. Small atom numbers are favourable in this trapping geometry to reduce collisions between atoms differing by 2ℏk_{R} during timeofflight. The Raman beams, intersecting under an angle of π/2, are individually oriented at an angle of π/4 from the long axis of the BEC. They are operated between the D1 and D2 lines of ^{87}Rb in the range of 782.5−790 nm. A 10G magnetic bias field is applied along the long axis of the BEC and produces a quadratic Zeeman shift of ~7.4E_{R}. The BEC is imaged after performing Stern–Gerlach separation during a 11.5ms timeofflight expansion. The images directly reveal the spin and momentum distributions.
For each value of Ω, δ=0 is experimentally identified by finding the Raman laser detuning for which the spin polarization is minimized (normal phase) or for which the BEC switches between the two degenerate states (superradiant phase). This method compensates for the presence of the third atomic hyperfine state (1, +1〉) in the F=1 manifold. The magnetic bias field was actively stabilized using a technique similar to the one described in ref. 41.
Experimental preparation methods
To experimentally measure the absolute value of the spin polarization 〈σ_{z}〉 and the photon field occupation (shown in Figs 2 and 4), we start with a BEC in the bare state and adiabatically turn on the Raman dressing at δ=5.4E_{R}(−5.4E_{R}). The initial state and the dressed state have similar spin compositions. This is followed by a linear sweep of the detuning to a final value in 50–100 ms. An example of results obtained with this loading method is shown in Fig. 7 for the normal phase of the Dicke model with Ω=6E_{R}. This loading method produces BECs with only minimal excitations when the final detuning value is chosen near δ=0. As seen in Fig. 3, this method can cause a hysteresislike feature at small δ.
An alternative loading method, as opposed to ramps of δ, is investigated by fixing the laser detuning while Ω is increased. The resulting L/R relative occupation measurement is shown in Fig. 8. Here the BEC is loaded to the desired parameters in ~60 ms by only increasing the Raman beam intensities. This is followed by a 150ms wait time, allowing for excitations to be dissipated. While this wait time is long, there may still be some residual small amplitude collective excitations, such as dipole oscillations. When this technique is used in Fig. 8, the BEC occupies only one of the two minima for all but small detunings. In the range where the BEC fails to load to a single point in the dispersion relation, the quasimomentum (shown in Fig. 8b, strongly deviates from the singleparticle expectation and the BEC acquires excitations. While this loading method does not result in the occupation of the ground state at δ=0, the BEC does choose a single minimum for detunings much smaller than the chemical potential μ, which is on the order of 0.55E_{R} for the data shown. Similar results are obtained by adding sufficient wait time following the linear sweeps of δ performed for Fig. 3. A related effect has been observed in ref. 42.
The quadrupole frequency measurements begin with a BEC in the F, m_{F}〉=1, −1〉 state in the crossed dipole trap. The two Raman beam intensities are linearly increased, one after the other, over 30–150 ms. This loading method, performed with the Raman beams at 782.5 nm, generates large quadrupole and small dipole excitations. To simplify the analysis, only the first few oscillations of the quadrupole mode are fit to an undamped oscillation. In the numerical simulations, the quadrupole mode is excited by a sudden jump of the trap frequency along the SO coupling direction. Evolution time is added to observe the temporal oscillation of the condensate width that reveals the quadrupolemode frequency.
Dimension reduction from 3D to 1D
The dynamics of SOcoupled BECs and their mapping to the Dicke model are studied on the basis of the 1D GP equation, although the experimental system is three dimensional (3D) with strong confinements along two directions. The SO coupling is along the elongated direction of the trap, therefore the transverse degrees of freedom do not couple with the internal spin states. Assuming the harmonic ground states along the transverse directions, the effective 1D nonlinear interaction coefficients, g_{1D}, can be approximately obtained from 3D through g_{1D}=g_{3D}/l_{y}l_{z}, where are the harmonic characteristic lengths along the transverse directions and is the 3D nonlinear interaction coefficient. For the study of the quadrupole mode, we simulate the 2D GP equation in the xy plane to compare with the experimental observations, where the dimension reduction gives g_{2D}=g_{3D}/l_{z}.
Mapping to the Dicketype Hamiltonian
In the mean field approximation, the singleparticle Hamiltonian is given by
As discussed in the main text, the large nonlinear interactions enable the atoms to collectively occupy the same manybody ground state, forming a single mode. For an ensemble of N interacting bosonic atoms, we define the collective spin operators and . Substituting these collective operators into the Nparticle Hamiltonian and using the harmonic mode operator, , we obtain
which corresponds to the generalized Dicke model.
The interaction term in the mean field approximation is
which can also be formally expressed using collective operators. We define the variables , and (ref. 22), where n is the average density of the BEC. For the parameters used in our experiment, we have , thus G_{3}=2G_{2}≈0 and .
Because of the normalization condition of the BEC wave function, it is easy to see that and (J_{z}=Nℏ/2 when all atoms are in the spinup state). With these notations, the Hamiltonian can be mapped to
Note that is proportional to N, thus all the terms in the above Hamiltonian scale as N.
Dicke phase transition
The critical point for the phase transition of the Dicke model can be derived using the mean field coherent state^{36}, where the mean field ansatz of the ground state wave function is given by
Here the spatial coherent state α〉 is defined by aα〉=αa〉, and the spin coherent state θ〉 is defined as
with j=N/2 for spin 1/2 atoms and θ ∈[0,2π].
In the absence of the detuning term , the ground state energy of the Dicke Hamiltonian is
where u and v are the real and imaginary parts of α, that is, α=u+iv. Minimizing E(θ,α) with respect to u and v leads to
Thus the ground state energy becomes
Further minimization of E(θ,α) with respect to θ leads to
Defining Ω_{c}=4E_{R}(1−G_{3}/NE_{R}), we obtain two different regions:

1
Ω>Ω_{c}: there is only one solution that minimizes the mean field energy: cosθ=0 and sinθ=−1. In this case, the spin polarization is zero and the corresponding phase is the spinbalanced normal phase.

2
Ω<Ω_{c}: the energy is minimized for and there are two possible values for , corresponding to a BEC occupying the left or right band minimum.
Ground state properties in the Dicke model
In the following, we neglect the constant mean field energy terms and the small G_{3} terms, therefore Ω_{c}≈4E_{R}. In the region Ω>Ω_{c}, the system is in the normal phase and v=u=0. The mean photon number is
The spin polarizations become
The ground state energy per particle is
We denote the first two terms as the singleparticle energy E_{s}.
In the region Ω<Ω_{c}, the system is in the superradiant phase with
and the mean photon number
where is the quasimomentum of the BEC. Note that the average photon number depends on the trapping frequency and in the main text we rescale it to . The spin polarization in the superradiant phase is
Finally, the ground state energy per particle is
We denote the first term as the singleparticle energy E_{s}. The energy expressions for Ω<4E_{R} and Ω>4E_{R} are consistent, they are continuous at Ω=4E_{R}.
Additional information
How to cite this article: Hamner, C. et al. Dicketype phase transition in a spinorbit coupled BoseEinstein condensate. Nat. Commun. 5:4023 doi: 10.1038/ncomms5023 (2014).
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Acknowledgements
C.H., J.C. and P.E. acknowledge funding from NSF and ARO. C.Q., Y.Z., M.G. and C.Z. are supported by ARO (W911NF1210334), DARPAYFA (N660011014025), AFOSR (FA95501310045) and NSFPHY (1249293). M.G. is supported by Hong Kong RGC/GRF Projects (Nos. 401011 and 2130352) and the Chinese University of Hong Kong (CUHK) Focused Investments Scheme.
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C.H., Y.Z., C.Z. and P.E. conceived the experiment and theoretical modelling; C.H., J.C. and P.E. performed the experiments; C.Q., Y.Z., M.G. performed the theoretical calculations; C.Z. and P.E. supervised the project.
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Hamner, C., Qu, C., Zhang, Y. et al. Dicketype phase transition in a spinorbitcoupled Bose–Einstein condensate. Nat Commun 5, 4023 (2014). https://doi.org/10.1038/ncomms5023
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DOI: https://doi.org/10.1038/ncomms5023
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