Abstract
Orbital physics plays a significant role for a vast number of important phenomena in complex condensedmatter systems, including hightemperature superconductivity and unconventional magnetism. In contrast, phenomena in superfluids—in particular in ultracold quantum gases—are typically well described by the lowest orbital and a real order parameter^{1}. Here, we report on the observation of a multiorbital superfluid phase with a complex order parameter in binary spin mixtures. In this unconventional superfluid, the local phase angle of the complex order parameter is continuously twisted between neighbouring lattice sites. The nature of this twisted superfluid quantum phase is an interactioninduced admixture of the porbital contributions favoured by the graphenelike band structure of the hexagonal optical lattice used in the experiment. We observe a secondorder quantum phase transition between the normal superfluid and the twisted superfluid phase, which is accompanied by a symmetry breaking in momentum space. The experimental results are consistent with calculated phase diagrams and reveal fundamentally new aspects of orbital superfluidity in quantum gas mixtures. Our studies might bridge the gap between conventional superfluidity and complex phenomena of orbital physics.
Main
The topological properties of graphene and its remarkable band structure have recently opened a new field in physics^{2}. The linear dispersion relation at the Dirac points proves to be a fascinating key aspect of this material, as it gives rise to phenomena such as quasirelativistic particles^{3} and an anomalous quantum Hall effect^{4,5}. The possibility to realize hexagonal optical lattices^{6} enables the emulation of graphenelike physics with ultracold atoms^{7,8,9}. In particular, loading bosonic particles in such lattices renders completely new possibilities such as studying nextnearestneighbour processes and tunnelling blockades in multicomponent systems^{6}. In general, optical lattices have proven to be a versatile tool to simulate Hubbardlike systems and actively drive and monitor quantum phase transitions^{10,11,12,13,14}. The important role of higher orbitals has recently been demonstrated for quantum gas mixtures in the strongly correlated regime^{15,16,17,18}. However, for weakly interacting systems such as superfluids, higher orbitals are generally expected to have only marginal effects. So far, orbital superfluidity has been observed only in excited states with limited lifetimes^{19,20,21}.
Here, we demonstrate the realization of a bosonic superfluid ground state where higherorbital physics plays a crucial role. In conventional superfluids (Fig. 1a), the local phase angle ϕ(r) of the order parameter
is constant (represented by arrows), where n(r) denotes the particle density. Therefore, the order parameter φ(r) can be chosen as real. In contrast, the observed twisted superfluid (TSF) ground state reveals a nontrivial complex superfluid order parameter, where the phase factor e^{iϕ(r)} is continuously twisted in the complex plane (Fig. 1b). As we shall demonstrate, this unconventional behaviour arises from a strong interactioninduced coupling of s and p orbitals at zero quasimomentum. This is fundamentally different to previous studies where all atoms are excited to the metastable p orbital and condense at finite quasimomentum^{19,20,21,22,23,24,25}. Most strikingly, even a quantum phase transition between the normal superfluid (NSF) and the TSF phase is directly observed in our experiment. It is driven by the competition between intra and interspecies interactions in binary mixtures coupling s and p bands. More precisely, in shallow lattices the intraspecies interaction W_{s p} dominates, whereas the interspecies interaction V _{s p} dominates for intermediate lattice depths, which is further elaborated below. For the experimental realization, we use a 1:1 mixture of ultracold ^{87}Rb atoms in two spin states F,m_{F}〉of the hyperfine manifold F=1,2 with Zeeman states m_{F} (see Methods). The repulsively interacting atoms are confined in a spindependent hexagonal optical lattice^{6}.
As a central aspect, the formation of the twisted superfluid phase originates from both the spin dependence and the specific topology of the hexagonal lattice. The topology leads to a graphenelike band structure with the particular feature that s and p bands are separated only by the order of the tunnelling energy. In addition, the spin dependence induces an individual sublattice structure for different F,m_{F}〉 states. This leads to an alternating density modulation for spin states (see Fig. 1c). The mutual interaction between different spin states induces a redistribution of both species, leading to an admixture of the p orbital. The combination of both topology and state dependence causes a strong coupling of s and p bands for the case of spin mixtures.
In the following, we first explain how the TSF phase can be identified experimentally and subsequently discuss theoretical phase diagrams as well as experimental results in detail. The NSF phase possesses the expected sixfold rotational symmetry in momentum space. This is observed in experiments through timeofflight (TOF) imaging for all singlecomponent spin states and is exemplarily shown for 1,−1〉 in Fig. 2a. In stark contrast, the twisted superfluid phase is accompanied by a symmetry breaking in momentum space, which appears as an alternating pattern in the firstorder momentum peaks (Fig. 2b). This reduced threefold rotational symmetry reflects the occurrence of a twisted complex phase factor e^{iϕ(r)}, which we observe only for mixtures of two spin states. We observe no significant decay of the alternating pattern for holding times up to 100 ms. This is comparable to the lifetime of the normal superfluid, reflecting the groundstate nature of the twisted superfluid phase. Figure 2b shows the results for a 1:1 mixture of 1,−1〉 and 1,+1〉 atoms, where the spin states are separated in the experiment by a Stern–Gerlach field. The occurrence of the twisted superfluid phase is clearly visible for very small lattice depths. Here, both components show a complementary momentum distribution reflecting the opposite phase twist as indicated in Fig. 1b. For increasing lattice depths, the transition to the normal superfluid phase is observed by the restoration of the sixfold rotational symmetry (Fig. 2b). Finally, the overall interference contrast vanishes as the system approaches the Mott insulator phase, where the atoms localize on individual lattice sites.
Remarkably, the clear signature of the TSF phase persists even for pband admixtures as small as a few per cent, which is a typical value in our experiment. This relies on the fact that the firstorder momentumpeaks of the p orbital are much stronger than those of the s orbital, leading to a strong amplification of the pband contributions, which can be clearly identified in the TOF images (see Fig. 2c). The observed alternating pattern for the twisted superfluid is caused by the quantum interference of s and pband contributions at zero quasimomentum (see equation (4)), which enables an extremely sensitive probing of the local phase properties through TOF. Note that the interference requires that s and porbital contributions occupy the same quasimomentum state (here q=0).
The connection between the hybridization of s and p orbitals and the transition to the TSF phase can be explained as follows. In general, a superposition between s orbital s〉 and p orbital p〉 can be written as
where the coefficients n_{s} and n_{p}denote the fraction of atoms in the s and the p orbital, respectively, and N the total number of particles. The global phase angle θ between the two orbitals is crucial for the formation of the twisted superfluid. It takes the value which minimizes the energy of the system and can lead to two different physical situations: For θ=0 (or π), the system is in the normal superfluid phase, where no interference takes place and the alternating momentum pattern vanishes (Fig. 2c). In contrast, causes a destructive interference of the firstorder peaks, thereby revealing the twist of the local phases ϕ(r). Thus, the global phase angle θ takes the important role of an order parameter describing the phase transition between NSF and TSF, where the TSF phase is defined by a nonzero value of θ.
We explore this phase transition theoretically using a multiband meanfield approach (see Methods), which leads to the phase diagrams presented in Fig. 3. The phase diagrams show the results for different binary spin mixtures. In agreement with the experimental results, the twisted superfluid emerges only in shallow lattices. It is important to mention that, in our hexagonal optical lattice configuration, different spin states preferably occupy different sublattices and therefore the interplay of intra and interspecies interaction strongly depends on the spin mixture considered. In particular, the phase diagrams in Fig. 3b demonstrate that the TSF phase area is drastically reduced for spin mixtures predominantly occupying the same sublattice (mixture C) in comparison with spin mixtures where each component occupies a different sublattice (mixtures A and B). In addition, the occurrence of the TSF phase depends on the admixture of the pband orbital, where n_{p} vanishes at zero lattice depth and is expected to reach a few per cent for intermediate lattice depths under experimental conditions. This also explains the absence of the TSF phase for singlecomponent samples (Fig. 2a) where the population of the p orbital is negligible.
In the following, we experimentally investigate the phase diagrams above. As an experimental indicator characterizing the NSF and TSF phase, we define a triangular interference contrast , which is illustrated in Fig. 4. It serves as a measure of the order parameter θof the NSF–TSF transition, where corresponds to the TSF phase. For spin mixture A, the TSF phase is clearly resolved for V _{0}<4 E_{rec} (Fig. 4a), where E_{rec} is the recoil energy. As expected for symmetry reasons, both components exhibit the same triangular contrast . In the limit of zero lattice depth, the admixture n_{p} is negligible, which leads to a vanishing . As n_{p}increases with the lattice depth and vanishes at the phase boundary, a maximum can be observed in the triangular interference contrast. In accordance with the phase diagrams presented in Fig. 3, mixture B exhibits a similar behaviour to mixture A, where in both cases different sublattices are occupied by the constituents (Fig. 4b). The substantial difference of the predicted TSF phase areas for mixtures B and C is also clearly revealed in our experiment.
To gain further insight into the underlying processes of the NSF–TSF transition, we turn back to its theoretical description. The observed quantum phase transition is entirely driven by the competition between intra and interspecies interactions and can thus occur at zero temperatures. We apply meanfield theory, where we restrict our analysis to an effective twomode Hamiltonian^{25} (see Methods). For the superfluid order parameter, we consider s and pband contributions described by equation (2), where the order parameter of the transition θ is the relative phase angle between the two orbitals. In the calculation, higher bands and nonzeroquasimomentum states can be ignored to first order.
For simplicity, we discuss here a mixture of 1,0〉 with 1,−1〉 atoms (see Fig. 3a). In this case, the symmetry of s and pband wavefunctions for 1,0〉leads to only two competing θdependent terms in the energy functional, namely, the interspecies interaction and the intraspecies interaction w_{s p}(θ)=2N^{2}W_{s p} n_{s}n_{p}cos(2θ). Here, N is the number of particles in each component; V _{s p} and W_{s p} are integrals of s and porbital wavefunctions with V _{s p}<0 and W_{s p}>0(see Methods). Physically, v_{s p} describes a collision of two particles with different spin states, where one particle is promoted from the s to the p band, whereas the intraspecies interaction w_{s p} promotes a pair of particles of the same species from the s to the p orbital. For sufficiently small pband admixtures n_{p}, the interspecies interaction v_{s p} dominates and therefore θ = 0 minimizes the energy, which corresponds to a normal superfluid. However, for a critical value of n_{p} the energy functional v_{s p}(θ)+w_{s p}(θ) no longer exhibits a minimum at θ = 0. As a central result, this defines the phase boundary n_{p}^{crit} of the NSF–TSF transition, which is given for the considered case by
The phase transition is of second order as the second derivative of the energy functional is discontinuous at this boundary. In the applied theory, the Hamiltonian is invariant under the transformation θ → −θ causing a twofold degenerate ground state in the twisted phase. This corresponds to the two possible orientations of the triangular pattern with opposite signs of (see Fig. 2b,c). However, in the experiment always the same one of the two ground states is observed. Further studies are necessary to investigate this nonspontaneous symmetry breaking.
Our study of a new type of complex superfluid phase paves the way for further investigations of the interplay between orbital physics and strong correlations. In particular, a possible competition between the twisted superfluid and the strongly correlated Mott insulator phase can be realized by increasing the interactions, for example by means of Feshbach resonances. Moreover, further insight into the roles of intra and interspecies interactions can be gained using binary mixtures consisting of two different atomic states, where the two interactions differ considerably from each other. In addition, dynamically driven phase transitions may be observable in our systems by preparing a dynamical superposition of s and p orbitals for one of the two spin components through microwave coupling.
Methods
Creation of spindependent hexagonal lattices.
The spindependent hexagonal lattice is realized by intersection of three coplanar laser beams under an angle of 120°. The laser beams are derived from a Ti:sapphire laser operated at a wavelength λ_{L}=830 nm (red detuned), where each beam is linearly polarized within the plane of intersection. Orthogonal to the plane, we apply a retroreflected onedimensional lattice at V _{1D}=8.8 E_{rec} operated at the same wavelength (for details see ref. 6). The recoil energy E_{rec} is defined as E_{rec}=ℏ^{2}k_{L}^{2}/2mwith the wavevector of the laser k_{L}=2π/λ_{L} and the mass m of an ^{87}Rb atom.
Preparation and detection schemes for spin mixtures.
We start with a Bose–Einstein condensate of typically several 10^{5}atoms in the stretched state 1,−1〉, which is confined in a nearly isotropic crossed dipole trap with a trap frequency ω≈2π×90 Hz. For these experimental parameters we expect a maximum filling factor of approximately four to six particles per lattice unit cell, which decreases towards the edge of the system owing to the inhomogeneous trapping potential. The preparation of the different pure and mixed spin states is carried out with the aid of radiofrequency and/or microwave sweeps. After the state preparation we apply a homogeneous magnetic field of 1.1 G to suppress spin dynamics^{26} and ramp up the optical lattice within 55 ms using an exponential ramp. Within the ramping time the coherence between different spin states is lost. To separate different spin components during 27 ms TOF, a Stern–Gerlach gradient field is applied before absorption imaging. The density distribution after TOF ρ_{TOF} reflects the momentum distribution in the lattice and can be calculated using the Fourier transform of Bloch wavefunctions
with . Whereas for θ=0 or π the third term vanishes, it causes an interference effect for other phase angles θ (for example for F,m_{F}=0〉 as shown in Fig. 2c the function is real and is imaginary). The simulated TOF images in Fig. 2c show the time evolution of the ensemble taking the finite trap size into account.
To verify independently the emerging interference pattern for different values of θ, we carry out a microwave excitation of the spin state 2,−2〉 to 1,−1〉. In this way we create a superposition of s and p orbitals in the 1,−1〉state, which evolves in time t as , where the oscillation frequency of the triangular interference contrast matches the energy difference of s and p bands ΔE_{s p}. The observed features show the same pattern as shown in Fig. 2 when replacing θ by the timedependent expression θ → −ΔE_{s p}t/ℏ.
Theoretical model.
To first order, N particles of the spin species σ experience the interaction with the noninteracting density Mϕ_{σ′}(r)^{2} of the other species σ′ with M atoms. Thus, we can write the effective Hamiltonian for the spin state σ as
where is the operator for kinetic and potential energy and is the bosonic field operator. The interaction strength between two spin states σ=F,m_{F}〉 and σ′=F′,m_{F′}〉 is labelled by g_{σ σ′}=4πℏ^{2}a_{σ σ′}/m with an swave scattering length a_{σ σ′}≈100a_{0} (a_{0} is the Bohr radius). For shallow lattices, we assume that only s and pband Bloch functions φ_{s,p} with quasimomentum q=0 contribute. For a large total number of particles and weak interactions, we apply meanfield theory and expand the field operators according to equation (2)
where φ_{s,p} are real functions. The energy functional can be divided into a θindependent and a θdependent part, where the latter is given by H_{θ}(n_{p},θ)=v_{s p}+w_{s p}+x_{s p} with
These terms depend on the interspecies integral And the intraspecies integrals , and , where the latter two vanish for symmetric spin states F,m_{F}=0〉 owing to parity. Without loss of generality, we choose the arbitrary sign of φ_{p} such that v_{s p} exhibits a minimum for θ = 0 corresponding to V _{s p}<0. The phase boundary of the phase transition between normal and twisted superfluid phases is defined by
When approaching the Mott insulator transition, higherquasimomentum states become occupied and the twomode description presented here is no longer fully valid. This could explain quantitative deviations between theory and experiment.
Change history
15 November 2011
In the version of this Letter originally published online, the squareroot signs in the formula in Fig. 3a were displayed incorrectly. This has been corrected in all versions of the Letter.
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Acknowledgements
The work has been funded by Deutsche Forschungsgemeinschaft grants FOR 801 and GRK 1355 as well as by the Landesexzellenzinitiative Hamburg, which is supported by the Joachim Herz Stiftung.
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The experimental work and data analysis were done by P.SP., J.S., DS.L., P.W. and K.S. DS.L. and P.SP. carried out the theoretical calculations. P.SP. and DS.L. wrote the manuscript with substantial contributions by all authors.
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SoltanPanahi, P., Lühmann, DS., Struck, J. et al. Quantum phase transition to unconventional multiorbital superfluidity in optical lattices. Nature Phys 8, 71–75 (2012). https://doi.org/10.1038/nphys2128
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